Chapter 1
An Overview of Corporate Finance and
The Financial Environment
ANSWERS TO ENDOFCHAPTER QUESTIONS
11 a. A proprietorship, or sole proprietorship, is a business owned by one
individual. A partnership exists when two or more persons associate to
conduct a business. In contrast, a corporation is a legal entity created by
a state. The corporation is separate and distinct from its owners and
managers.
b. In a limited partnership, limited partners' liabilities, investment returns and control
are limited, while general partners have unlimited liability and control. A limited
liability partnership (LLP), sometimes called a limited liability company (LLC),
combines the limited liability advantage of a corporation with the tax advantages of a
partnership. A professional corporation (PC), known in some states as a
professional association (PA), has most of the benefits of incorporation but the
participants are not relieved of professional (malpractice) liability.
c. Stockholder wealth maximization is the appropriate goal for management decisions.
The risk and timing associated with expected earnings per share and cash flows are
considered in order to maximize the price of the firm's common stock.
d. A money market is a financial market for debt securities with maturities of less than
one year (shortterm). The New York money market is the world's largest. Capital
markets are the financial markets for longterm debt and corporate stocks. The New
York Stock Exchange is an example of a capital market. Primary markets are the
markets in which newly issued securities are sold for the first time. Secondary
markets are where securities are resold after initial issue in the primary market. The
New York Stock Exchange is a secondary market.
Answers and Solutions: 1 1 e. In private markets, transactions are worked out directly between two parties and
structured in any manner that appeals to them. Bank loans and private placements
of debt with insurance companies are examples of private market transactions. In
public markets, standardized contracts are traded on organized exchanges.
Securities that are issued in public markets, such as common stock and corporate
bonds, are ultimately held by a large number of individuals. Private market
securities are more tailormade but less liquid, whereas public market securities are
more liquid but subject to greater standardization. Derivatives are claims whose
value depends on what happens to the value of some other asset. Futures and
options are two important types of derivatives, and their values depend on what
happens to the prices of other assets, say IBM stock, Japanese yen, or pork bellies.
Therefore, the value of a derivative security is derived from the value of an
underlying real asset.
f. An investment banker is a middleman between businesses and savers.
Investment banking houses assist in the design of corporate securities and
then sell them to savers (investors) in the primary markets. Financial
service corporations offer a wide range of financial services such as
brokerage operations, insurance, and commercial banking. A financial
intermediary buys securities with funds that it obtains by issuing its own
securities. An example is a common stock mutual fund that buys
common stocks with funds obtained by issuing shares in the mutual fund.
g. A mutual fund is a corporation that sells shares in the fund and uses the
proceeds to buy stocks, longterm bonds, or shortterm debt instruments.
The resulting dividends, interest, and capital gains are distributed to the
fund's shareholders after the deduction of operating expenses. Different
funds are designed to meet different objectives. Money market funds are
mutual funds which invest in shortterm debt instruments and offer their
shareholders check writing privileges; thus, they are essentially
interestbearing checking accounts.
h. Physical location exchanges, such as the New York Stock Exchange, facilitate
communication between buyers and sellers of securities. Each physical location
exchange is a physical entity at a particular location and is governed by an elected
board of governors. A computer/telephone network, such as Nasdaq, consists of all
the facilities that provide for security transactions not conducted at a physical
location exchange. These facilities are, basically, the communications network that
links the buyers and sellers.
i. An open outcry auction is a method of matching buyers and sellers. In an auction,
the buyers and sellers are facetoface, with each stating the prices and which they
will buy or sell. In a dealer market, a dealer holds an inventory of the security and
makes a market by offering to buy or sell. Others who wish to buy or sell can see
Answers and Solutions: 1 2 the offers made by the dealers, and can contact the dealer of their choice to arrange a
transaction. In an ECN, orders from potential buyers and sellers are automatically
matched, and the transaction is automatically completed.
j. Production opportunities are the returns available within an economy from
investment in productive assets. The higher the production opportunities, the more
producers would be willing to pay for required capital. Consumption time
preferences refer to the preferred pattern of consumption. Consumer's time
preferences for consumption establish how much consumption they are willing to
defer, and hence save, at different levels of interest.
k. The real riskfree rate is that interest rate which equalizes the aggregate supply of,
and demand for, riskless securities in an economy with zero inflation. The real
riskfree rate could also be called the pure rate of interest since it is the rate of interest
that would exist on very shortterm, defaultfree U.S. Treasury securities if the
expected rate of inflation were zero. It has been estimated that this rate of interest,
denoted by r*, has fluctuated in recent years in the United States in the range of 2 to 4
percent. The nominal riskfree rate of interest, denoted by rRF, is the real riskfree
rate plus a premium for expected inflation. The shortterm nominal riskfree rate is
usually approximated by the U.S. Treasury bill rate, while the longterm nominal
riskfree rate is approximated by the rate on U.S. Treasury bonds. Note that while
Tbonds are free of default and liquidity risks, they are subject to risks due to changes
in the general level of interest rates.
l. The inflation premium is the premium added to the real riskfree rate of interest to
compensate for the expected loss of purchasing power. The inflation premium is the
average rate of inflation expected over the life of the security. Default risk is the
risk that a borrower will not pay the interest and/or principal on a loan as they
become due. Thus, a default risk premium (DRP) is added to the real riskfree rate
to compensate investors for bearing default risk. Liquidity refers to a firm's cash
and marketable securities position, and to its ability to meet maturing obligations. A
liquid asset is any asset that can be quickly sold and converted to cash at its "fair"
value. Active markets provide liquidity. A liquidity premium is added to the real
riskfree rate of interest, in addition to other premiums, if a security is not liquid.
m. Interest rate risk arises from the fact that bond prices decline when interest rates rise.
Under these circumstances, selling a bond prior to maturity will result in a capital loss,
and the longer the term to maturity, the larger the loss. Thus, a maturity risk
premium must be added to the real riskfree rate of interest to compensate for interest
rate risk. Reinvestment rate risk occurs when a shortterm debt security must be
"rolled over." If interest rates have fallen, the reinvestment of principal will be at a
lower rate, with correspondingly lower interest payments and ending value. Note
that longterm debt securities also have some reinvestment rate risk because their
interest payments have to be reinvested at prevailing rates.
Answers and Solutions: 1 3 n. The term structure of interest rates is the relationship between yield to maturity and
term to maturity for bonds of a single risk class. The yield curve is the curve that
results when yield to maturity is plotted on the Yaxis with term to maturity on the
Xaxis.
o. When the yield curve slopes upward, it is said to be "normal," because it is like this
most of the time. Conversely, a downwardsloping yield curve is termed
"abnormal" or "inverted."
p. The expectations theory states that the slope of the yield curve depends on
expectations about future inflation rates and interest rates. Thus, if the annual rate
of inflation and future interest rates are expected to increase, the yield curve will be
upward sloping, whereas the curve will be downward sloping if the annual rates are
expected to decrease.
r. A foreign trade deficit occurs when businesses and individuals in the U. S. import
more goods from foreign countries than are exported. Trade deficits must be
financed, and the main source of financing is debt. Therefore, as the trade deficit
increases, the debt financing increases, driving up interest rates. U. S. interest rates
must be competitive with foreign interest rates; if the Federal Reserve attempts to set
interest rates lower than foreign rates, foreigners will sell U.S. bonds, decreasing
bond prices, resulting in higher U. S. rates. Thus, if the trade deficit is large relative
to the size of the overall economy, it may hinder the Fed's ability to combat a
recession by lowering interest rates.
12 Sole proprietorship, partnership, and corporation are the three principal forms of business
organization. The advantages of the first two include the ease and low cost of formation.
The advantages of the corporation include limited liability, indefinite life, ease of
ownership transfer, and access to capital markets.
The disadvantages of a sole proprietorship are (1) difficulty in obtaining large sums
of capital; (2) unlimited personal liability for business debts; and (3) limited life. The
disadvantages of a partnership are (1) unlimited liability, (2) limited life, (3) difficulty of
transferring ownership, and (4) difficulty of raising large amounts of capital. The
disadvantages of a corporation are (1) double taxation of earnings and (2) requirements to
file state and federal reports for registration, which are expensive, complex and
timeconsuming.
13 The three primary determinants of a firm's cash flows are: (1) sales revenues; (2)
operating expenses, such as raw materials costs and labor costs; and (3) the necessary
investments in operating capital, such as buildings, equipment, and inventory.
14 Financial intermediaries are business organizations that receive funds in one form and
repackage them for the use of those who need funds. Through financial intermediation,
Answers and Solutions: 1 4 resources are allocated more effectively, and the real output of the economy is thereby
increased.
15 Shortterm rates are more volatile because (1) the Fed operates mainly in the shortterm
sector, hence Federal Reserve intervention has its major effect here, and (2) longterm
rates reflect the average expected inflation rate over the next 20 to 30 years, and this
average does not change as radically as yeartoyear expectations.
16 a. If transfers between the two markets were costly, interest rates would be
different in the two areas. Area Y, with the relatively young population,
would have less in savings accumulation and stronger loan demand. Area
O, with the relatively old population, would have more savings accumula
tion and weaker loan demand as the members of the older population have
already purchased their houses, and are less consumption oriented. Thus,
supply/demand equilibrium would be at a higher rate of interest in Area Y.
b. Yes. Nationwide branching, and so forth, would reduce the cost of financial
transfers between the areas. Thus, funds would flow from Area O with excess
relative supply to Area Y with excess relative demand. This flow would increase the
interest rate in Area O and decrease the interest rate in Y until the rates were roughly
equal, the difference being the transfer cost.
17 a. The immediate effect on the yield curve would be to lower interest rates in
the shortterm end of the market, since the Fed deals primarily in that
market segment. However, people would expect higher future inflation,
which would raise longterm rates. The result would be a much steeper
yield curve.
b. If the policy is maintained, the expanded money supply will result in increased rates
of inflation and increased inflationary expectations. This will cause investors to
increase the inflation premium on all debt securities, and the entire yield curve would
rise; that is, all rates would be higher.
Answers and Solutions: 1 5 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
11 r* = 3%; I1 = 2%; I2 = 4%; I3 = 4%; MRP = 0; rT2 = ?; rT3 = ?
r = r* + IP + DRP + LP + MRP.
Since these are Treasury securities, DRP = LP = 0.
rT2 = r* + IP2
IP2 = (2% + 4%)/2 = 3%
rT2 = 3% + 3% = 6%.
rT3 = r* + IP3
IP3 = (2% + 4% + 4%)/3 = 3.33%
rT3 = 3% + 3.33% = 6.33%.
12 rT10 = 6%; rC10 = 8%; LP = 0.5%; DRP = ?
r = r* + IP + DRP + LP + MRP.
rT10 = 6% = r* + IP + MRP; DRP = LP = 0.
rC10 = 8% = r* + IP + DRP + 0.5% + MRP.
Because both bonds are 10year bonds the inflation premium and maturity risk premium
on both bonds are equal. The only difference between them is the liquidity and default
risk premiums.
rC10 = 8% = r* + IP + MRP + 0.5% + DRP. But we know from above that r* + IP +
MRP = 6%; therefore,
rC10 = 8% = 6% + 0.5% + DRP
1.5% = DRP.
13 r* = 3%; IP = 3%; rT2 = 6.2%; MRP2 = ?
rT2 = k* + IP + MRP = 6.2%
rT2 = 3% + 3% + MRP = 6.2%
MRP = 0.2%.
Answers and Solutions: 1 6 14 r = r* + IP + MRP + DRP + LP.
r* = 0.03.
IP = [0.03 + 0.04 + (5)(0.035)]/7 = 0.035.
MRP = 0.0005(6) = 0.003.
DRP = 0.
LP = 0.
r = 0.03 + 0.035 + 0.003 = 0.068 = 6.8%.
15 First, note that we will use the equation rt = 3% + IPt + MRPt. We have the data needed
to find the IPs:
8% + 5% + 4% + 4% + 4% 25%
IP5 = = = 5%.
5 5
8% + 5%
IP2 = = 6.5%.
2
Now we can substitute into the equation:
r2 = 3% + 6.5% + MRP2 = 10%. r5 = 3% + 5% + MRP5 = 10%.
Now we can solve for the MRPs, and find the difference:
MRP5 = 10%  8% = 2%. MRP2 = 10%  9.5% = 0.5%.
Difference = (2%  0.5%) = 1.5%.
Answers and Solutions: 1 7 16 Basic relevant equations:
rt = r* + IPt + DRPt + MRPt + LPt.
But here IP is the only premium, so rt = r* + IPt.
IPt = Avg. inflation = (I1 + I2 + ...)/N.
We know that I1 = IP1 = 3% and r* = 2%. Therefore,
r1 = 2% + 3% = 5%. r3 = r1 + 2% = 5% + 2% = 7%. But,
r3 = r* + IP3 = 2% + IP3 = 7%, so
IP3 = 7%  2% = 5%.
We also know that It = Constant after t = 1.
We can set up this table:
r* I Avg. I = IPt r = r* + IPt
1 2 3 3%/1 = 3% 5%
2 2 I (3% + I)/2 = IP2
3 2 I (3% + I + I)/3 = IP3 r3 = 7%, so IP3 = 7%  2% = 5%.
Avg. I = IP3 = (3% + 2I)/3 = 5%
2I = 12%
I = 6%.
Answers and Solutions: 1 8 17 a. Real
Years to RiskFree
Maturity Rate (r*) IP** MRP rT = r* + IP + MRP
1 2% 7.00% 0.2% 9.20%
2 2 6.00 0.4
8.40
3 2 5.00 0.6
7.60
4 2 4.50 0.8
7.30
5 2 4.20 1.0
7.20
10 2 3.60 1.0
6.60
20 2 3.30 1.0 6.30
**The computation of the inflation premium is as follows:
Expected Average
Year Inflation Expected Inflation
1 7% 7.00%
2 5 6.00
3 3 5.00
4 3 4.50
5 3 4.20
10 3 3.60
20 3 3.30
For example, the calculation for 3 years is as follows:
7% + 5% + 3%
= 5.00%.
3
Thus, the yield curve would be as follows:
Answers and Solutions: 1 9 Interest Rate
(%)
11.0
10.5
10.0
9.5
9.0
8.5
LILCO
8.0
7.5
Exxon
7.0
6.5
Tbonds
0 2 4 6 8 10 12 14 16 18 20
Years to Maturity
b. The interest rate on the Exxon bonds has the same components as the Treasury
securities, except that the Exxon bonds have default risk, so a default risk premium
must be included. Therefore,
rExxon = r* + IP + MRP + DRP.
For a strong company such as Exxon, the default risk premium is virtually zero for
shortterm bonds. However, as time to maturity increases, the probability of default,
although still small, is sufficient to warrant a default premium. Thus, the yield risk
curve for the Exxon bonds will rise above the yield curve for the Treasury securities.
In the graph, the default risk premium was assumed to be 1.0 percentage point on the
20year Exxon bonds. The return should equal 6.3% + 1% = 7.3%.
c. LILCO bonds would have significantly more default risk than either Treasury
securities or Exxon bonds, and the risk of default would increase over time due to
possible financial deterioration. In this example, the default risk premium was
assumed to be 1.0 percentage point on the 1year LILCO bonds and 2.0 percentage
points on the 20year bonds. The 20year return should equal 6.3% + 2% = 8.3%.
Answers and Solutions: 1 10 SOLUTION TO SPREADSHEET PROBLEM
18 The detailed solution for the spreadsheet problem is available both on the instructor's
resource CDROM (in the file Solution for FM11 Ch 01 P08 Build a Model.xls) and on
the instructor's side of the textbook's web site, http://brigham.swcollege.com.
Answers and Solutions: 1 11 MINI CASE
Assume that you recently graduated with a degree in finance and have just reported to work as an investment advisor at the brokerage firm of Balik and Kiefer Inc. One of the firm's clients is Michelle Dellatorre, a professional tennis player who has just come to the United States from Chile. Dellatorre is a highly ranked tennis player who would like to start a company to produce and market apparel that she designs. She also expects to invest substantial amounts of money through Balik and Kiefer. Dellatorre is also very bright, and, therefore, she would like to understand, in general terms, what will happen to her money. Your boss has developed the following set of questions which you must ask and answer to explain the U.S. financial system to Dellatorre.
a. Why is corporate finance important to all managers?
Answer: Corporate finance provides the skills managers need to: (1) identify and select the corporate
strategies and individual projects that add value to their firm; and (2) forecast the funding
requirements of their company, and devise strategies for acquiring those funds.
b. Describe the organizational forms a company might have as it evolves from a startup to a major
corporation. List the advantages and disadvantages of each form.
Answer: The three main forms of business organization are (1) sole proprietorships, (2) partnerships, and
(3) corporations. In addition, several hybrid forms are gaining popularity. These hybrid forms
are the limited partnership, the limited liability partnership, the professional corporation, and the s
corporation.
The proprietorship has three important advantages: (1) it is easily and inexpensively
formed, (2) it is subject to few government regulations, and (3) the business pays no
corporate income taxes. The proprietorship also has three important limitations: (1)
it is difficult for a proprietorship to obtain large sums of capital; (2) the proprietor has
unlimited personal liability for the business's debts, and (3) the life of a business
organized as a proprietorship is limited to the life of the individual who created it.
The major advantage of a partnership is its low cost and ease of formation. The
disadvantages are similar to those associated with proprietorships: (1) unlimited liability, (2)
limited life of the organization, (3) difficulty of transferring ownership, and (4) difficulty of
raising large amounts of capital. The tax treatment of a partnership is similar to that for
proprietorships, which is often an advantage.
The corporate form of business has three major advantages: (1) unlimited life, (2) easy
transferability of ownership interest, and (3) limited liability. While the corporate form offers
significant advantages over proprietorships and partnerships, it does have two primary
disadvantages: (1) corporate earnings may be subject to double taxation and (2) setting up a
corporation and filing the many required state and federal reports is more complex and
timeconsuming than for a proprietorship or a partnership.
Answers and Solutions: 2  12 In a limited partnership, the limited partners are liable only for the amount of their investment
in the partnership; however, the limited partners typically have no control. The limited liability
partnership form of organization combines the limited liability advantage of a corporation with
the tax advantages of a partnership. Professional corporations provide most of the benefits of
incorporation but do not relieve the participants of professional liability. S corporations are
similar in many ways to limited liability partnerships, but LLPS frequently offer more flexibility
and benefits to their owners.
c. How do corporations "go public" and continue to grow? What are agency problems?
Answer: A company goes public when it sells stock to the public in an initial public as the firm grows, it
might issue additional stock or debt. An agency problem occurs when the managers of the firm act
in their own self interests and not in the interests of the shareholders.
d. What should be the primary objective of managers?
Answer: The corporation's primary goal is stockholder wealth maximization, which translates to
maximizing the price of the firm's common stock.
d. 1. Do firms have any responsibilities to society at large?
Answer: Firms have an ethical responsibility to provide a safe working environment, to avoid polluting the
air or water, and to produce safe products. However, the most significant costincreasing actions
will have to be put on a mandatory rather than a voluntary basis to ensure that the burden falls
uniformly on all businesses.
d. 2. Is stock price maximization good or bad for society?
Answer: The same actions that maximize stock prices also benefit society. Stock price maximization
requires efficient, lowcost operations that produce highquality goods and services at the lowest
possible cost. Stock price maximization requires the development of products and services that
consumers want and need, so the profit motive leads to new technology, to new products, and to
new jobs. Also, stock price maximization necessitates efficient and courteous service, adequate
stocks of merchandise, and welllocated business establishmentsfactors that are all necessary to
make sales, which are necessary for profits.
d. 3. Should firms behave ethically?
Answer: Yes. Results of a recent study indicate that the executives of most major firms in the United
States believe that firms do try to maintain high ethical standards in all of their business dealings.
Furthermore, most executives believe that there is a positive correlation between ethics and
longrun profitability. Conflicts often arise between profits and ethics. Companies must deal
with these conflicts on a regular basis, and a failure to handle the situation properly can lead to
Answers and Solutions: 2  13 huge product liability suits and even to bankruptcy. There is no room for unethical behavior in
the business world.
e. What three aspects of cash flows affect the value of any investment?
Answer: (1) amount of expected cash flows; (2) timing of the cash flow stream; and (3) riskiness of the
cash flows.
f. What are free cash flows? What are the three determinants of free cash flows?
Answer: free cash flows are the cash flows available for distribution to all investors (stockholders and
creditors) after paying expenses (including taxes) and making the necessary investments to
support growth. Three factors determine cash flows: (1) current level and growth rates of sales; (2)
operating expenses; and (3) capital expenses.
g. What is the weighted average cost of capital? What affects it?
Answer: The weighted average cost of capital (WACC) is the average rate of return required by all of the
company's investors (stockholders and creditors). It is affected by the firm's capital structure,
interest rates, the firm's risk, and the market's overall attitude toward risk. h. How do free cash flows and the weighted average cost of capital interact to determine a
firm's value?
Answer: A firm's value is the sum of all future expected free cash flows, converted into today's dollars.
FCF1 FCF2 FCF
Value = + + ... .
(1 + WACC)1 (1 + WACC) 2 (1 + WACC)
i. What are financial assets? Describe some financial instruments.
Answer: Financial assets are pieces of paper with contractual obligations. Some shortterm (i.e., they
mature in less than a year) are instruments with low default risk are u.s. treasury bills, banker's
acceptances, commercial paper, negotiable CDs, and eurodollar deposits. Commercial loans
(which have maturities up to seven years) have rates that are usually tied to the prime rate (i.e., the
rate that U.S. banks charge to their best customers) or LIBOR (the London Interbank Offered Rate,
which is the rate that banks in the U.K. charge one another. U.S. treasury notes and bonds have
maturities from two to thirty years; they are free of default risk. Mortgages have maturities up to
thirty years. Municipal bonds have maturities of up to thirty years; their interest is exempt from
most taxes. Corporate bonds have maturities up to forty years. Municipal and corporate
bonds are subject to default risk. Some preferred stocks have no maturity date, some do have a
specific maturity date. Common stock has no maturity date, and is riskier than preferred stock.
j. Who are the providers (savers) and users (borrowers) of capital? How is capital
transferred between savers and borrowers?
Answers and Solutions: 2  14 Answer: Households are net savers. Nonfinancial corporations are net borrowers. Governments are net
borrowers, although the U.S. government is a net saver when it runs a surplus. Nonfinancial
corporations (i.e., financial intermediaries) are slightly net borrowers, but they are almost
breakeven. Capital is transferred through: (1) direct transfer (e.g., corporation issues commercial
paper to insurance company); (2) an investment banking house (e.g., IPO, seasoned equity
offering, or debt placement); (3) a financial intermediary (e.g., individual deposits money in bank,
bank makes commercial loan to a company).
k. List some financial intermediaries.
Answer: Commercial banks, savings & loans, mutual savings banks, and credit unions, life insurance
companies, mutual funds, and pension funds are financial intermediaries.
l. What are some different types of markets?
Answer: A market is a method of exchanging one asset (usually cash) for another asset. Some types of
markets are: physical assets vs. financial assets; spot versus future markets; money versus capital
markets; primary versus secondary markets.
m. How are secondary markets organized?
Answer: They are categorized by "location" (physical location exchanges or computer/telephone networks)
and by the way that orders from buyers and sellers are matched (open outcry auctions, dealers (i.e.,
market makers), and electronic communications networks (ECNS).
m. 1. List some physical location markets and some computer/telephone networks.
Answer: Physical location exchanges include the NYSE, AMEX, CBOT, and Tokyo stock exchange.
Computer/telephone networks include Nasdaq, government bond markets, and foreign exchange
markets.
m. 2. Explain the differences between open outcry auctions, dealer markets, and electronic
communications networks (ECNS).
Answer: The NYSE and AMEX are the two largest auction markets for stocks (NYSE is a modified
auction, with a "specialist"). Participants have a seat on the exchange, meet facetoface, and
place orders for themselves or for their clients; e.g., CBOT. Some orders are market orders,
which are executed at the current market price, some are limit orders, which specify that the trade
should occur only at a certain price within a certain time period (or the trade does not occur at all).
In dealer markets, "dealers" keep an inventory of the stock (or other financial asset) and place bid
and ask "advertisements," which are prices at which they are willing to buy and sell. A
computerized quotation system keeps track of bid and ask prices, but does not automatically
match buyers and sellers. Some examples of dealer markets are the Nasdaq national market,
Answers and Solutions: 2  15 the Nasdaq small cap market, the London SEAQ, and the German Neuer market. ECNS are
computerized systems that match orders from buyers and sellers and automatically execute the
trades. Some examples are Instinet (US, stocks), Eurex (SwissGerman, futures contracts), sets
(London, stocks). In the old days, securities were kept in a safe behind the counter, and passed
"over the counter" when they were sold. Now the OTC market is the equivalent of a computer
bulletin board, which allows potential buyers and sellers to post an offer. However, the OTC has
no dealers and very poor liquidity.
n. What do we call the price that a borrower must pay for debt capital? What is the price of
equity capital? What are the four most fundamental factors that affect the cost of money,
or the general level of interest rates, in the economy?
Answer: The interest rate is the price paid for borrowed capital, while the return on equity capital comes
in the form of dividends plus capital gains. The return that investors require on capital depends
on (1) production opportunities, (2) time preferences for consumption, (3) risk, and (4) inflation.
Production opportunities refer to the returns that are available from investment in productive
assets: the more productive a producer firm believes its assets will be, the more it will be willing
to pay for the capital necessary to acquire those assets.
Time preference for consumption refers to consumers' preferences for current consumption
versus savings for future consumption: consumers with low preferences for current consumption
will be willing to lend at a lower rate than consumers with a high preference for current
consumption.
Inflation refers to the tendency of prices to rise, and the higher the expected rate of inflation,
the larger the required rate of return.
Risk, in a money and capital market context, refers to the chance that a loan will not be repaid
as promisedthe higher the perceived default risk, the higher the required rate of return.
Risk is also linked to the maturity and liquidity of a security. The longer the
maturity and the less liquid (marketable) the security, the higher the required rate of
return, other things constant.
The preceding discussion related to the general level of money costs, but the level of interest
rates will also be influenced by such things as fed policy, fiscal and foreign trade deficits, and the
level of economic activity. Also, individual securities will have higher yields than the riskfree
rate because of the addition of various premiums as discussed below.
Answers and Solutions: 2  16 o. What is the real riskfree rate of interest (r*) and the nominal riskfree rate (rRF)? How
are these two rates measured?
Answer: Keep these equations in mind as we discuss interest rates. We will define the terms as we go
along:
r = r* + IP + DRP + LP + MRP.
rRF = r* + IP.
The real riskfree rate, r*, is the rate that would exist on defaultfree securities in the absence of
inflation.
The nominal riskfree rate, rrf, is equal to the real riskfree rate plus an inflation premium
which is equal to the average rate of inflation expected over the life of the security.
There is no truly riskless security, but the closest thing is a shortterm U. S. Treasury bill
(tbill), which is free of most risks. The real riskfree rate, r*, is estimated by subtracting the
expected rate of inflation from the rate on shortterm treasury securities. It is generally assumed
that r* is in the range of 1 to 4 percentage points. The tbond rate is used as a proxy for the
longterm riskfree rate. However, we know that all longterm bonds contain interest rate risk,
so the tbond rate is not really riskless. It is, however, free of default risk.
p. Define the terms inflation premium (IP), default risk premium (DRP), liquidity premium
(LP), and maturity risk premium (MRP). Which of these premiums is included when
determining the interest rate on (1) shortterm U.S. treasury securities, (2) longterm U.S.
treasury securities, (3) shortterm corporate securities, and (4) longterm corporate
securities? Explain how the premiums would vary over time and among the different
securities listed above.
Answer: The inflation premium (IP) is a premium added to the real riskfree rate of interest to compensate
for expected inflation.
The default risk premium (DRP) is a premium based on the probability that the issuer will
default on the loan, and it is measured by the difference between the interest rate on a U.S.
treasury bond and a corporate bond of equal maturity and marketability.
A liquid asset is one that can be sold at a predictable price on short notice; a liquidity
premium is added to the rate of interest on securities that are not liquid.
The maturity risk premium (MRP) is a premium which reflects interest rate risk; longerterm
securities have more interest rate risk (the risk of capital loss due to rising interest rates) than do
shorterterm securities, and the MRP is added to reflect this risk.
1. Shortterm treasury securities include only an inflation premium.
2. Longterm treasury securities contain an inflation premium plus a maturity risk premium.
Note that the inflation premium added to longterm securities will differ from that for
shortterm securities unless the rate of inflation is expected to remain constant.
3. The rate on shortterm corporate securities is equal to the real riskfree rate plus premiums for
inflation, default risk, and liquidity. The size of the default and liquidity premiums will
vary depending on the financial strength of the issuing corporation and its degree of liquidity,
with larger corporations generally having greater liquidity because of more active trading.
Answers and Solutions: 2  17 4. The rate for longterm corporate securities also includes a premium for maturity risk. Thus,
longterm corporate securities generally carry the highest yields of these four types of
securities.
q. What is the term structure of interest rates? What is a yield curve?
Answer: The term structure of interest rates is the relationship between interest rates, or
yields, and maturities of securities. When this relationship is graphed, the resulting
curve is called a yield curve.
r. Suppose most investors expect the inflation rate to be 5 percent next year, 6
percent the following year, and 8 percent thereafter. The real riskfree rate is 3
percent. The maturity risk premium is zero for securities that mature in 1 year
or less, 0.1 percent for 2year securities, and then the MRP increases by 0.1
percent per year thereafter for 20 years, after which it is stable. What is the
interest rate on 1year, 10year, and 20year treasury securities? Draw a yield
curve with these data. What factors can explain why this constructed yield
curve is upward sloping?
Answer: Step 1: find the average expected inflation rate over years 1 to 20:
Yr 1: IP = 5.0%.
Yr 10: IP = (5 + 6 + 8 + 8 + 8 + ... + 8)/10 = 7.5%.
Yr 20: IP = (5 + 6 + 8 + 8 + ... + 8)/20 = 7.75%.
Step 2: find the maturity premium in each year:
Yr 1: MRP = 0.0%.
Yr 10: MRP = 0.1 × 9 = 0.9%.
Yr 20: MRP = 0.1 × 19 = 1.9%.
Step 3: sum the IPS and MRPS, and add r* = 3%:
Yr 1: rRF = 3% + 5.0% + 0.0% = 8.0%.
Yr 10: rRF = 3% + 7.5% + 0.9% = 11.4%.
Yr 20: rRF = 3% + 7.75% + 1.9% = 12.65%.
The shape of the yield curve depends primarily on two factors:
(1) expectations about future inflation and (2) the relative riskiness of securities with different
maturities.
Answers and Solutions: 2  18 Interest
rate (%)
13
12
11
10
9
8
0 1 5 10 15 20
Years to maturity
The constructed yield curve is upward sloping. This is due to increasing expected inflation
and an increasing maturity risk premium.
s. At any given time, how would the yield curve facing an AAArated company compare with
the yield curve for U. S. Treasury securities? At any given time, how would the yield curve
facing a BBrated company compare with the yield curve for U. S. Treasury securities?
Draw a graph to illustrate your answer.
Answer: The yield curve normally slopes upward, indicating that shortterm interest rates are lower than
longterm interest rates. Yield curves can be drawn for government securities or for the
securities of any corporation, but corporate yield curves will always lie above government yield
curves, and the riskier the corporation, the higher its yield curve. The spread between a
corporate yield curve and the treasury curve widens as the corporate bond rating decreases.
Answers and Solutions: 2  19 4  20
Hypothetical Treasury and
Corporate Yield Curves
Interest
Rate (%)
15
BBRated
10
AAARated
Treasury
6.0%
5 5.9% yield curve
5.2%
Years to
0
maturity
0 1 5 10 15 20
Copyright ? 2002 by Harcourt, Inc. All rights reserved.
t. What is the pure expectations theory? What does the pure expectations theory
imply about the term structure of interest rates?
Answer: The pure expectations theory assumes that investors establish bond prices and interest rates
strictly on the basis of expectations for interest rates. This means that they are indifferent with
respect to maturity in the sense that they do not view longterm bonds as being riskier than
shortterm bonds. If this were true, then the maturity risk premium would be zero, and longterm
interest rates would simply be a weighted average of current and expected future shortterm
interest rates. If the pure expectations theory is correct, you can use the yield curve to "back
out" expected future interest rates.
Answers and Solutions: 2  20 u. Finally, Dellatorre is also interested in investing in countries other than the
United States. Describe the various types of risks that arise when investing
overseas.
Answer: First, Dellatorre should consider country risk, which refers to the risk that arises from investing or
doing business in a particular country. This risk depends on the country's economic, political,
and social environment. Country risk also includes the risk that property will be expropriated
without adequate compensation, as well as new host country stipulations about local production,
sourcing or hiring practices, and damage or destruction of facilities due to internal strife.
Second, Dellatorre should consider exchange rate risk. Dellatorre needs to keep in mind
when investing overseas that more often than not the security will be denominated in a currency
other than the dollar, which means that the value of the investment will depend on what happens
to exchange rates. Two factors can lead to exchange rate fluctuations. Changes in relative inflation
will lead to changes in exchange rates. Also, an increase in country risk will also cause the
country's currency to fall. Consequently, inflation risk, country risk, and exchange rate risk are
all interrelated.
Answers and Solutions: 2  21 Chapter 2
Time Value of Money ANSWERS TO ENDOFCHAPTER QUESTIONS
21 a. PV (present value) is the value today of a future payment, or stream of payments, discounted at
the appropriate rate of interest. PV is also the beginning amount that will grow to some future
value. The parameter i is the periodic interest rate that an account pays. The parameter INT is
the dollars of interest earned each period. FVn (future value) is the ending amount in an account,
where n is the number of periods the money is left in the account. PVAn is the value today of a
future stream of equal payments (an annuity) and FVAn is the ending value of a stream of equal
payments, where n is the number of payments of the annuity. PMT is equal to the dollar amount
of an equal, or constant cash flow (an annuity). In the EAR equation, m is used to denote the
number of compounding periods per year, while iNom is the nominal, or quoted, interest rate.
b. FVIFi,n is the future value interest factor for a lump sum left in an account for n periods paying i
percent interest per period. PVIFi,n is the present value interest factor for a lump sum received n
periods in the future discounted at i percent per period. FVIFAi,n is the future value interest
factor for an ordinary annuity of n periodic payments paying i percent interest per period.
PVIFAi,n is the present value interest factor for an ordinary annuity of n periodic payments
discounted at i percent interest per period. All the above factors represent the appropriate PV or
FVn when the lump sum or ordinary annuity payment is $1. Note that the above factors can also
be defined using formulas.
c. The opportunity cost rate (i) of an investment is the rate of return available on the best alternative
investment of similar risk.
d. An annuity is a series of payments of a fixed amount for a specified number of periods. A single
sum, or lump sum payment, as opposed to an annuity, consists of one payment occurring now or
at some future time. A cash flow can be an inflow (a receipt) or an outflow (a deposit, a cost, or an
amount paid). We distinguish between the terms cash flow and PMT. We use the term cash
flow for uneven streams, while we use the term PMT for annuities, or constant payment amounts.
An uneven cash flow stream is a series of cash flows in which the amount varies from one period
to the next. The PV (or FVn) of an uneven payment stream is merely the sum of the present
values (or future values) of each individual payment.
e. An ordinary annuity has payments occurring at the end of each period. A deferred annuity is just
another name for an ordinary annuity. An annuity due has payments occurring at the beginning
of each period. Most financial calculators will accommodate either type of annuity. The
payment period must be equal to the compounding period.
f. A perpetuity is a series of payments of a fixed amount that last indefinitely. In other words, a
perpetuity is an annuity where n equals infinity. Consol is another term for perpetuity. Consols
were originally bonds issued by England in 1815 to consolidate past debt.
g. An outflow is a deposit, a cost, or an amount paid, while an inflow is a receipt. A time line is an
important tool used in time value of money analysis; it is a graphical representation which is used
to show the timing of cash flows. The terminal value is the future value of an uneven cash flow
stream.
Answers and Solutions: 2  22 h. Compounding is the process of finding the future value of a single payment or series of payments.
Discounting is the process of finding the present value of a single payment or series of payments;
it is the reverse of compounding.
i. Annual compounding means that interest is paid once a year. In semiannual, quarterly, monthly,
and daily compounding, interest is paid 2, 4, 12, and 365 times per year respectively. When
compounding occurs more frequently than once a year, you earn interest on interest more often,
thus increasing the future value. The more frequent the compounding, the higher the future
value.
j. The effective annual rate is the rate that, under annual compounding, would have produced the
same future value at the end of 1 year as was produced by more frequent compounding, say
quarterly. The nominal (quoted) interest rate, iNom, is the rate of interest stated in a contract. If
the compounding occurs annually, the effective annual rate and the nominal rate are the same. If
compounding occurs more frequently, the effective annual rate is greater than the nominal rate.
The nominal annual interest rate is also called the annual percentage rate, or APR. The periodic
rate, iPER, is the rate charged by a lender or paid by a borrower each period. It can be a rate per
year, per 6month period, per quarter, per month, per day, or per any other time interval (usually
one year or less).
k. An amortization schedule is a table that breaks down the periodic fixed payment of an installment
loan into its principal and interest components. The principal component of each payment
reduces the remaining principal balance. The interest component is the interest payment on the
beginningofperiod principal balance. An amortized loan is one that is repaid in equal periodic
amounts (or "killed off" over time).
22 The opportunity cost rate is the rate of interest one could earn on an alternative investment with a risk
equal to the risk of the investment in question. This is the value of i in the TVM equations, and it is
shown on the top of a time line, between the first and second tick marks. It is not a single ratethe
opportunity cost rate varies depending on the riskiness and maturity of an investment, and it also
varies from year to year depending on inflationary expectations.
23 True. The second series is an uneven payment stream, but it contains an annuity of $400 for 8 years.
The series could also be thought of as a $100 annuity for 10 years plus an additional payment of $100
in Year 2, plus additional payments of $300 in Years 3 through 10.
24 True, because of compounding effectsgrowth on growth. The following example demonstrates the
point. The annual growth rate is i in the following equation:
$1(1 + i)10 = $2.
The term (1 + i)10 is the FVIF for i percent, 10 years. We can find i in one of two ways:
1. Using a financial calculator input N = 10, PV = 1, PMT = 0, FV = 2, and I = ?. Solving for I
you obtain 7.18%.
2. Using a financial calculator, input N = 10, I = 10, PV = 1, PMT = 0, and FV = ?.
Solving for FV you obtain $2.59. This formulation recognizes the "interest on
interest" phenomenon.
25 For the same stated rate, daily compounding is best. You would earn more "interest on interest."
Answers and Solutions: 2  23 Answers and Solutions: 2  24 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
21 a. 0 6% 1
  $500(1.06) = $530.00.
500 FV = ?
b. 0 6% 1 2
   $500(1.06)2 = $561.80.
500 FV = ?
c. 0 6% 1
  $500(1/1.06) = $471.70.
PV = ? 500
d. 0 6% 1 2
   $500(1/1.06)2 = $445.00.
PV = ? 500
6% 22 a. 0 1 2 3 4 5 6 7 500 (FVIF6%,10 ) =
$ 8 9 10
          
$500 (1.7908) = $895.40.
500 FV =?
b. 012% 1 2 3 4 5 6 7 8
$500 (FVIF12%,10 ) =9 10
          
$500 (3.1058) = $1,552.90.
500 FV =?
c. 6%
0 1 2 3 4 5 6 $500 (FVIF6%,10 ) = 9
7 8 10
          
$500 (0.5584) = $279.20.
PV = ? 500
d. 12%
0 1 2 3 4 5 6 7 8 9 10
          
PV = ? 1,552.90
$1,552.90(PVIF12%,10) = $1,552.90(PVIF6%,10) =
$1,552.90(0.3220) = $500.03; i = 6%: $1,552.90(0.5584) = $867.14.
The present value is the value today of a sum of money to be received in the future. For
example, the value today of $1,552.90 to be received 10 years in the future is
about $500 at an interest rate of 12 percent, but it is approximately $867 if the
interest rate is 6 percent. Therefore, if you had $500 today and invested it at 12
percent, you would end up with $1,552.90 in 10 years. The present value
depends on the interest rate because the interest rate determines the amount of
interest you forgo by not having the money today.
7%
Answers and Solutions: 2  25 23 a. ?
  $400 = $200(FVIF7%,n)
200 400 2 = FVIF7%,n
n 10 years.
With a financial calculator, enter I = 7, PV = 200, PMT = 0, and FV = 400. Then press the N
key to find N = 10.24. Override I with the other values to find N = 7.27, 4.19, and 1.00.
b. 10% ?
  2 = FVIF10%,n
200 400 n 7 years.
c. 18% ?
  2 = FVIF18%,n
200 400 n 4 years.
d. 100% ?
  2 = FVIF100%,n
200 400 n = 1 year.
24 The general formula is FVAn = PMT(FVIFAi,n).
a. 10%
0 1 2 3 4 5 6 7 8 9 10
          
400 400 400 400 400 400 400 400 400 400
FV = ?
FVA10 = ($400)15.9374 = $6,374.96.
With a financial calculator, enter N = 10, I = 10, PV = 0, and PMT = 400. Then press the FV
key to find FV = $6,374.97.
b. 0 5% 1 2 3 4 5
      ($200)5.5256 = $1,105.12.
200 200 200 200 200
FV = ?
With a financial calculator, enter N = 5, I = 5, PV = 0, and PMT =
200. Then press the FV key to find FV = $1,105.13.
0%
c. 0 1 2 3 4 5
      ($400)5 = $2,000.00.
400 400 400 400 400
FV = ?
With a financial calculator, enter N = 5, I = 0, PV = 0, and PMT =
400. Then press the FV key to find FV = $2,000.
Answers and Solutions: 2  26 d. To solve Part d using a financial calculator, repeat the procedures discussed in Parts a, b, and c,
but first switch the calculator to "BEG" mode. Make sure you switch the calculator back to
"END" mode after working the problem.
(1) 0 10% 1 2 3 4 5 6 7 8 9 10
          
400 400 400 400 400 400 400 400 400 400 FV =?
FVAn(Annuity due) = PMT(FVIFAi,n)(1 + i). Therefore,
FVA10 = $400(15.9374)(1.10) = $7,012.46.
(2) 0 5% 1 2 3 4 5
     
200 200 200 200 200 FV = ?
FVA5 = $200(5.5256)(1.05) = $1,160.38.
(3) 0 0% 1 2 3 4 5
     
400 400 400 400 400 FV = ?
FVA5 = $400(5)(1.00) = $2,000.00.
25 The general formula is PVAn = PMT(PVIFAi,n).
a. 0
10% 1 2 3 4 5 6 7 8 9 10
          
PV = ? 400 400 400 400 400 400 400 400 400 400
PV = $400 (6.1446) = $2,457.83.
Answers and Solutions: 2  27 With a financial calculator, simply enter the known values and then press the key for the
unknowns. Except for rounding errors, the answers are as given below.
b. 0
5% 1 2 3 4 5
      $200(4.3295) = $865.90.
PV = ? 200 200 200 200 200
c. 0%
0 1 2 3 4 5
      $400(5) = $2,000.00.
PV = ? 400 400 400 400 400
10%
d. (1) 0 1 2 3 4 5 6 7 8 9 10
          
400 400 400 400 400 400 400 400 400 400 PV =?
PVAn (Annuity due) = PMT(PVIFAi,n)(1 + i). Therefore,
$400(6.1446)(1.10) = $2,703.62.
(2) 0 5% 1 2 3 4 5
     
200 200 200 200 200 200
PVAn (Annuity due) = $200(4.3295)(1.05) = $909.20.
(3) 0 0% 1 2 3 4 5
     
400 400 400 400 400
PV = ?
PVAn (Annuity due)= $400(5)(1.00) = $2,000.00.
26 a. Cash Stream A Cash Stream B 8%
0 1 2 3 4 5 0 1 2 3 8%
4 5
           
PV = ? 100 400 400 400 300 PV = ? 300 400 400 400 100
With a financial calculator, simply enter the cash flows (be sure to enter CF0 = 0), enter I = 8, and
press the NPV key to find NPV = PV = $1,251.25 for the first problem. Override I = 8 with I = 0 to
find the next PV for Cash Stream A. Repeat for Cash Stream B to get NPV = PV = $1,300.32.
b. PVA = $100 + $400 + $400 + $400 + $300 = $1,600.
PVB = $300 + $400 + $400 + $400 + $100 = $1,600
Answers and Solutions: 2  28 27 These problems can all be solved using a financial calculator by entering the known values shown on
the time lines and then pressing the I button.
a. 0 1

i=? 
+700 749
7 percent: $700 = $749(PVIFi,1); PVIFi,1 = 0.9346.
b. 0 1
i=?
  7 percent.
700 +749
c. 0 10
i=?
 
+85,000 201,229
$201,229/$85,000 = 2.3674 = FVIFi,10; i = 9%.
i=?
d. 0 1 2 3 4 5
     
+9,000 2,684.80 2,684.80 2,684.80 2,684.80 2,684.80
$9,000/$2,684.80 = 3.3522 = PVIFAi,5; i = 15%.
Answers and Solutions: 2  29 28 a. 0 12% 1 2 3 4 5
     
500 FV = ?
With a financial calculator, enter N = 5, I = 12, PV = 500, and PMT = 0, and then press FV to
obtain FV = $881.17. With a regular calculator, proceed as follows:
Fvn = PV(1 + i)n = $500(1.12)5 = $500(1.7623) = $881.15.
b. 0
6%
1 2 3 4 5 6 7 8 9 10
          
500 FV =?
Enter the time line values into a financial calculator to obtain FV = $895.42, or
mn
i
PVn= PV 1 +
m
2 5)
(
0.12
= $500 1 + = $500(1.06)10
2
= $500(FVIF6%, 10) = $500(1.7908) = $895.40.
c. 0 4 8 12 16 20
 3%     
500 FV = ?
Enter the time line values into a financial calculator to obtain FV = $903.06, or
4 (5)
0.12
FVn = $500 1 + = $500(1.03)20 = $500(1.8061) = $903.05.
4
d. 0 1% 12 24 36 48 60
     
500 ?
Enter the time line values into a financial calculator to obtain FV = $908.35, or
12(5)
0.12
FVn = $500 1 + = $500(1.01)60 = $500(1.8167) = $908.35.
12
Answers and Solutions: 2  30 29 a. 0 2 4 6 8 10
 6%     
PV = ? 500
Enter the time line values into a financial calculator to obtain PV = $279.20, or
mn 2 (5)
1
PV = FVn 1 = $500
i 0.12
1 + 1+
m 2
10
1
= $500 = $500(PVIF6%, 10) = $500(0.5584) = $279.20.
1.06
b. 0 4 8 12 16 20
 3%     
PV = ? 500
Enter the time line values into a financial calculator to obtain PV = $276.84, or
4 (5)
20
1 1
PV = $500 = $500 = $500(0.5537) = $276.85.
0.12 1.03
1 +
4
c. 0 1 2 12
 1%   ··· 
PV = ? 500
Enter the time line values into a financial calculator to obtain PV = $443.72, or
12(1)
1
PV = $500
0.12
1+
12
12
1
= $500 = $500(1.01)12 = $500(0.8874) = $443.70.
1.01
Answers and Solutions: 2  31 210 a. 0 1 2 3 9 10 6%
    ···  
400 400 400 400 400
FV = ?
Enter N = 5 × 2 = 10, I = 12/2 = 6, PV = 0, PMT = 400, and then press FV to get FV = $5,272.32.
b. Now the number of periods is calculated as N = 5 x 4 = 20, I = 12/4 = 3, PV = 0, and PMT =
200. The calculator solution is $5,374.07.
Note that the solution assumes that the nominal interest rate is compounded at the annuity period.
c. The annuity in Part b earns more because some of the money is on deposit for a longer period of
time and thus earns more interest. Also, because compounding is more frequent, more interest is
earned on interest.
211 a. Universal Bank: Effective rate = 7%.
Regional Bank:
4
0.06
Effective rate = 1 +  1.0 = (1.015)4 1.0
4
= 1.0614 1.0 = 0.0614 = 6.14%.
With a financial calculator, you can use the interest rate conversion feature to obtain the same
answer. You would choose the Universal Bank.
b. If funds must be left on deposit until the end of the compounding period (1 year for Universal and
1 quarter for Regional), and you think there is a high probability that you will make a withdrawal
during the year, the Regional account might be preferable. For example, if the withdrawal is
made after 10 months, you would earn nothing on the Universal account but (1.015)3  1.0 =
4.57% on the Regional account.
Ten or more years ago, most banks and S&Ls were set up as described above, but now
virtually all are computerized and pay interest from the day of deposit to the day of withdrawal,
provided at least $1 is in the account at the end of the period.
Answers and Solutions: 2  32 212 a. With a financial calculator, enter N = 5, I = 10, PV = 25000, and FV = 0, and then press the PMT
key to get PMT = $6,594.94. Then go through the amortization procedure as described in your
calculator manual to get the entries for the amortization table.
Repayment Remaining
Year Payment Interest of Principal Balance
1 $ 6,594.94 $2,500.00 $ 4,094.94 $20,905.06
2 6,594.94 2,090.51 4,504.43 16,400.63
3 6,594.94 1,640.06 4,954.88 11,445.75
4 6,594.94 1,144.58 5,450.36 5,995.39
5 6,594.93* 599.54 5,995.39 0
$32,974.69 $7,974.69 $25,000.00
*The last payment must be smaller to force the ending balance to zero.
b. Here the loan size is doubled, so the payments also double in size to $13,189.87.
c. The annual payment on a $50,000, 10year loan at 10 percent interest would be $8,137.27.
Because the payments are spread out over a longer time period, more interest must be paid on the
loan, which raises the amount of each payment. The total interest paid on the 10year loan is
$31,372.70 versus interest of $15,949.37 on the 5year loan.
213 a. 1997 ? 1998 1999 2000 2001 2002
     
6 12 (in millions)
With a calculator, enter N = 5, PV = 6, PMT = 0, FV = 12, and then solve for I = 14.87%.
b. The calculation described in the quotation fails to take account of the compounding effect. It can
be demonstrated to be incorrect as follows:
$6,000,000(1.20)5 = $6,000,000(2.4883) = $14,929,800,
which is greater than $12 million. Thus, the annual growth rate is less than 20 percent; in
fact, it is about 15 percent, as shown in Part a.
Answers and Solutions: 2  33 214 0 1 2 3 4 5 6 7 8 9 10= ?
i
          
4 8 (in millions)
$4,000,000/$8,000,000 = 0.50, which is slightly less than the PVIFi,n for 7 percent in 10 years. Thus,
the expected rate of return is just over 7 percent. With a calculator, enter N = 10, PV = 4, PMT =
0, FV = 8, and then solve for I = 7.18%.
215 0 1 2 3
i=?
4 30
     ··· 
85,000 8,273.59 8,273.59 8,273.59 8,273.59 8,273.59
$85,000/$8,273.59 = 10.2737 = PVIFAi,n for a 30year annuity.
With a calculator, enter N = 30, PV = 85000, PMT = 8273.59, FV = 0, and then solve for I = 9%.
216 a. 0 1 2 3
7% 4
    
PV = ? 10,000 10,000 10,000 10,000
With a calculator, enter N = 4, I = 7, PMT = 10000, and FV = 0. Then press PV to get PV =
$33,872.11.
b. (1) At this point, we have a 3year, 7% annuity whose value is $26,243.16. You can also
think of the problem as follows:
$33,872(1.07) $10,000 = $26,243.04.
(2) Zero after the last withdrawal.
217 0 1 2 ?
 9%   ··· 
12,000 1,500 1,500 1,500
PVA n = PMT(PVIFA i,n ).
$12,000 = $1,500(PVIFA 9%,n )
PVIFA 9%,n = 8.000.
With a calculator, enter I = 9, PV = 12000, PMT = 1500, and FV = 0. Press N to get N = 14.77 15
years. Therefore, it will take approximately 15 years to pay back the loan.
Answers and Solutions: 2  34 218 0 1 2 3 4 5 6 12%
      
1,250 1,250 1,250 1,250 1,250 ?
FV = 10,000
With a financial calculator, get a "ballpark" estimate of the years by entering I = 12, PV = 0, PMT =
1250, and FV = 10000, and then pressing the N key to find N = 5.94 years. This answer assumes
that a payment of $1,250 will be made 94/100th of the way through Year 5.
Now find the FV of $1,250 for 5 years at 12%; it is $7,941.06. Compound this value for 1 year at
12% to obtain the value in the account after 6 years and before the last payment is made; it is
$7,941.06(1.12) = $8,893.99. Thus, you will have to make a payment of $10,000  $8,893.99 =
$1,106.01 at Year 6, so the answer is: it will take 6 years, and $1,106.01 is the amount of the last
payment.
219 PV = $100/0.07 = $1,428.57. PV = $100/0.14 = $714.29.
When the interest rate is doubled, the PV of the perpetuity is halved.
220 0
8.24%
1 2 3 4
    
PV = ? 50 50 50 1,050
Discount rate: Effective rate on bank deposit:
EAR = (1 + 0.08/4)4  1 = 8.24%.
Find PV of above stream at 8.24%:
PV = $893.26 using the cash flow register.
Also get PV = $893.26 using the TVM register, inputting N = 4, I = 8.24, PMT = 50, and FV = 1000.
Answers and Solutions: 2  35 221 This can be done with a calculator by specifying an interest rate of 5% per period for 20 periods with
1 payment per period, or 10% interest, 20 periods, 2 payments per year. Either way, we get the
payment each 6 months:
N = 10 × 2 = 20.
I = 10%/2 = 5.
PV = 10000.
FV = 0.
Solve for PMT = $802.43. Set up amortization table:
Pmt of
Period Beg Bal Payment Interest Principal End Bal
1 $10,000.00 $802.43 $500.00 $302.43 $9,697.57
2 9,697.57 802.43 484.88
$984.88
You can also work the problem with a calculator having an amortization function. Find the interest
in each 6month period, sum them, and you have the answer. Even simpler, with some calculators
such as the HP17B, just input 2 for periods and press INT to get the interest during the first year,
$984.88. The HP10B does the same thing.
222 First, find PMT by using a financial calculator: N = 5, I/YR = 15, PV = 1000000, and FV = 0.
Solve for PMT = $298,315.55. Then set up the amortization table:
Beginning Ending
Year Balance Payment Interest Principal Balance
1 $1,000,000.00 $298,315.55 $150,000.00 $148,315.55 $851,684.45
2 851,684.45 298,315.55 127,752.67 170,562.88 681,121.57
Fraction that is principal = $170,562.88/$298,315.55 = 0.5718 = 57.18%.
Answers and Solutions: 2  36 223 a. Begin with a time line:
6mos. 0 1 2 3 4 5 6 8 10 12 14 16 18 20
Years 0 1 2 3 4 5
6% 6 7 8 9 10
                    
100 100 100 100 100 FVA
Since the first payment is made today, we have a 5period annuity due. The applicable interest rate
is I = 12/2 = 6 per period, N = 5, PV = 0, and PMT = 100. Setting the calculator on "BEG," we
find FVA (Annuity due) = $597.53. That will be the value at the 5th 6month period, which is t =
2.5. Now we must compound out to t = 10, or for 7.5 years at an EAR of 12.36%, or 15
semiannual periods at 6%.
$597.53 20  5 = 15 periods @ 6% $1,432.02,
or $597.53 10  2.5 = 7.5 years @ 12.36% $1,432.02.
b. 1 10 years
0 3% 1 2 3 4 5 40 quarters
      ··· 
PMT PMT PMT PMT PMT FV = 1,432.02
The time line depicting the problem is shown above. Because the payments only occur for 5
periods throughout the 40 quarters, this problem cannot be immediately solved as an annuity
problem. The problem can be solved in two steps:
(1) Discount the $1,432.02 back to the end of Quarter 5 to obtain the PV of that future amount
at Quarter 5.
(2) Then solve for PMT using the value solved in Step 1 as the FV of the fiveperiod annuity
due.
Step 1: Input the following into your calculator: N = 35, I = 3, PMT = 0, FV = 1432.02,
and solve for PV at Quarter 5. PV = $508.92.
Step 2: The PV found in Step 1 is now the FV for the calculations in this step. Change your
calculator to the BEGIN mode. Input the following into your calculator: N = 5, I
= 3, PV = 0, FV = 508.92, and solve for PMT = $93.07.
Answers and Solutions: 2  37 224 Here we want to have the same effective annual rate on the credit extended as on the bank loan that
will be used to finance the credit extension.
First, we must find the EAR = EFF% on the bank loan. Enter NOM% = 15, N = P/YR = 12, and
press EFF% to get EAR = 16.08%.
Now recognize that giving 3 months of credit is equivalent to quarterly compoundinginterest is
earned at the end of the quarter, so it is available to earn interest during the next quarter. Therefore,
enter P/YR = 4, EFF% = EAR = 16.08%, and press NOM% to find the nominal rate of 15.19 percent.
Therefore, if a 15.19 percent nominal rate is charged and credit is given for 3 months, the cost of
the bank loan will be covered.
Alternative solution: We need to find the effective annual rate (EAR) the bank is charging first.
Then, we can use this EAR to calculate the nominal rate that should be quoted to the customers.
Bank EAR: EAR = (1 + iNom/m)m  1 = (1 + 0.15/12)12  1 = 16.08%.
Nominal rate that should be quoted to customers:
16.08% = (1 + iNom/4)4  1
1.1608 = (1 + iNom/4)4
1.0380 = 1 + iNom/4
iNom = 0.0380(4) = 15.19%.
Answers and Solutions: 2  38 225 Information given:
1. Will save for 10 years, then receive payments for 25 years.
2. Wants payments of $40,000 per year in today's dollars for first payment only. Real income will
decline. Inflation will be 5 percent. Therefore, to find the inflated fixed payments, we have this
time line:
0 5 10
 5%          
40,000 FV = ?
Enter N = 10, I = 5, PV = 40000, PMT = 0, and press FV to get FV = $65,155.79.
3. He now has $100,000 in an account which pays 8 percent, annual compounding. We need to
find the FV of the $100,000 after 10 years. Enter N = 10, I = 8, PV = 100000, PMT = 0, and
press FV to get FV = $215,892.50.
4. He wants to withdraw, or have payments of, $65,155.79 per year for 25 years, with the first
payment made at the beginning of the first retirement year. So, we have a 25year annuity due
with PMT = 65,155.79, at an interest rate of 8 percent. (The interest rate is 8 percent annually,
so no adjustment is required.) Set the calculator to "BEG" mode, then enter N = 25, I = 8, PMT
= 65155.79, FV = 0, and press PV to get PV = $751,165.35. This amount must be on hand to
make the 25 payments.
5. Since the original $100,000, which grows to $215,892.50, will be available, we must save enough
to accumulate $751,165.35  $215,892.50 = $535,272.85.
6. The $535,272.85 is the FV of a 10year ordinary annuity. The payments will be deposited in the
bank and earn 8 percent interest. Therefore, set the calculator to "END" mode and enter N = 10,
I = 8, PV = 0, FV = 535272.85, and press PMT to find PMT = $36,949.61.
Answers and Solutions: 2  39 SOLUTION TO SPREADSHEET PROBLEM
226 The detailed solution for the spreadsheet problem is available both on the instructor's resource
CDROM (in the file Solution for Ch 02 P26 Build a Model.xls) and on the instructor's side of the
textbook's web site, brigham.swcollege.com.
Answers and Solutions: 2  40 MINI CASE
Assume that you are nearing graduation and that you have applied for a job with a local bank. As part of the bank's evaluation process, you have been asked to take an examination which covers several financial analysis techniques. The first section of the test addresses discounted cash flow analysis. See how you would do by answering the following questions.
a. Draw time lines for (a) a $100 lump sum cash flow at the end of year 2, (b) an ordinary
annuity of $100 per year for 3 years, and (c) an uneven cash flow stream of $50, $100, $75,
and $50 at the end of years 0 through 3.
Answer: (Begin by discussing basic discounted cash flow concepts, terminology, and solution methods.)
A time line is a graphical representation which is used to show the timing of cash flows. The
tick marks represent end of periods (often years), so time 0 is today; time 1 is the end of the first
year, or 1 year from today; and so on.
0 1 2 year
 i%  
lump sum
100 cash flow
0 1 2 3
 i%    annuity
100 100 100
0 1 2 3
 i%   
uneven cash flow stream
50 100 75 50
A lump sum is a single flow; for example, a $100 inflow in year 2, as shown in the top time line.
An annuity is a series of equal cash flows occurring over equal intervals, as illustrated in the
middle time line. An uneven cash flow stream is an irregular series of cash flows which do not
constitute an annuity, as in the lower time line. 50 represents a cash outflow rather than a
receipt or inflow.
Answers and Solutions: 3 41 b. 1. What is the future value of an initial $100 after 3 years if it is invested in an account paying
10 percent annual interest?
Answer: Show dollars corresponding to question mark, calculated as follows:
0 1 2
3 10%
   
100 FV
=?
After 1 year:
FV1 = PV + i1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $110.00.
Similarly:
FV2 = FV1 + i2 = FV1 + FV1(i) = FV1(1 + i)
= $110(1.10) = $121.00 = PV(1 + i)(1 + i) = PV(1 + i)2.
FV3 = FV2 + i3 = FV2 + FV2(i) = FV2(1 + i)
= $121(1.10)=$133.10=PV(1 + i)2(1 + i)=PV(1 + i)3.
In general, we see that:
FVn = PV(1 + i)n,
SO FV3 = $100(1.10)3 = $100(1.3310) = $133.10.
Note that this equation has 4 variables: FVn, PV, i, and n. Here we know all except FVn, so we
solve for FVn. We will, however, often solve for one of the other three variables. By far, the
easiest way to work all time value problems is with a financial calculator. Just plug in any 3 of
the four values and find the 4th.
Finding future values (moving to the right along the time line) is called compounding. Note that
there are 3 ways of finding FV3: using a regular calculator, financial calculator, or spreadsheets.
For simple problems, we show only the regular calculator and financial calculator methods.
(1) regular calculator:
1. $100(1.10)(1.10)(1.10) = $133.10.
2. $100(1.10)3 = $133.10.
(2) financial calculator:
This is especially efficient for more complex problems, including exam problems. Input
the following values: N = 3, I = 10, PV = 100, pmt = 0, and solve for FV = $133.10.
Answers and Solutions: 3 42 b. 2. What is the present value of $100 to be received in 3 years if the appropriate interest rate is
10 percent?
Answer: Finding present values, or discounting (moving to the left along the time line), is the reverse of
compounding, and the basic present value equation is the reciprocal of the compounding equation:
0 1 2
10%
3
   
PV = ? 100
FVn = PV(1 + i)n transforms to:
n
FVn 1
PV = = FVn = FVn(1 + i)n
(1 + i) n 1+ i
thus:
3
1
PV = $100 = $100(PVIFi,n) = (0.7513) = $75.13.
1.10
The same methods used for finding future values are also used to find present values.
Using a financial calculator input N = 3, I = 10, pmt = 0, FV = 100, and then solve for PV =
$75.13.
c. We sometimes need to find how long it will take a sum of money (or anything else) to grow
to some specified amount. For example, if a company's sales are growing at a rate of 20
percent per year, how long will it take sales to double?
Answer: We have this situation in time line format:
0 1 2 3 3.8 4
20%
     
12 2
Say we want to find out how long it will take us to double our money at an interest rate of
20%. We can use any numbers, say $1 and $2, with this equation:
FVn = $2 = $1(1 + i)n = $1(1.20)n.
(1.2)n = $2/$1 = 2
n LN(1.2) = LN(2)
n = LN(2)/LN(1.2)
n = 0.693/0.182 = 3.8.
Answers and Solutions: 3 43 Alternatively, we could use a financial calculator. We would plug I = 20, PV = 1, PMT = 0, and FV = 2 into our calculator, and then press the N button to find the number of years it would take 1
(or any other beginning amount) to
FV double when growth occurs at a 20% rate. The answer is 3.8 years, but some calculators will round this value 2
up to the next highest whole number. The graph also shows what is happening.
1
3.8
0 1 2 3 4
Year
Answers and Solutions: 3 44 d. If you want an investment to double in three years, what interest rate must it earn?
Answer: 0 1 2 3
   
1 2
1(1 + i) 1(1 + i)2 1(1 + i)3
FV = $1(1 + i)3 = $2.
$1(1 + i)3 = $2.
(1 + i)3 = $2/$1 = 2.
1+i = (2)1/3
1+i = 1.2599
i = 25.99%.
Use a financial calculator to solve: enter N = 3, PV = 1, PMT = 0, FV = 2, then press the I
button to find I = 25.99%.
Calculators can find interest rates quite easily, even when periods and/or interest rates are not
even numbers, and when uneven cash flow streams are involved. (With uneven cash flows, we
must use the "CFLO" function, and the interest rate is called the IRR, or "internal rate of return;"
we will use this feature in capital budgeting.)
e. What is the difference between an ordinary annuity and an annuity due? What type of
annuity is shown below? How would you change it to the other type of annuity?
0 1 2 3
   
100 100 100
Answer: This is an ordinary annuityit has its payments at the end of each period; that is, the first payment
is made 1 period from today. Conversely, an annuity due has its first payment today. In other
words, an ordinary annuity has endofperiod payments, while an annuity due has
beginningofperiod payments.
The annuity shown above is an ordinary annuity. To convert it to an annuity due, shift each
payment to the left, so you end up with a payment under the 0 but none under the 3.
Answers and Solutions: 3 45 f. 1. What is the future value of a 3year ordinary annuity of $100 if the appropriate interest rate
is 10 percent?
Answer: 0 1 2 3
 10%   
100 100 100
110
121
$331
Go through the following discussion. One approach would be to treat each annuity flow
as a lump sum. Here we have
FVAn = $100(1) + $100(1.10) + $100(1.10)2
= $100[1 + (1.10) + (1.10)2] = $100(3.3100) = $331.00.
Using a financial calculator, N = 3, I = 10, PV = 0, PMT = 100. This gives FV = $331.00.
f. 2. What is the present value of the annuity?
Answer: 0 10% 1 2 3
   
100 100 100
90.91
82.64
75.13
$248.68
The present value of the annuity is $248.68. Using a financial calculator, input N = 3, I =
10, PMT = 100, FV = 0, and press the PV button.
Spreadsheets are useful for time lines with multiple cash flows.
The following spreadsheet shows this problem:
A B C D
1 0 1 2 3
2 100 100 100
3 248.69
The excel formula in cell A3 is = NPV(10%,B2:D2). This gives a result of 248.69. Note
that the interest rate can be either 10% or 0.10, not just 10. Also, note that the
range does not include any cash flow at time zero.
Excel also has special functions for annuities. For ordinary annuities, the excel formula is =
PV(interest rate, number of periods, payment). In this problem, = PV(10%,3,100), gives a
result of 248.96. For the future value, it would be = FV(10%,3,100), with a result of 331.
Answers and Solutions: 3 46 f. 3. What would the future and present values be if the annuity were an annuity due?
Answer: If the annuity were an annuity due, each payment would be shifted to the left, so each payment is
compounded over an additional period or discounted back over one less period.
To find the future value of an annuity due use the following formula:
FVAn(Annuity Due) = FVAn(1 + i).
In our situation, the future value of the annuity due is $364.10:
FVA3(Annuity Due) = $331.00(1.10)1 = $364.10.
This same result could be obtained by using the time line: $133.10 + $121.00 + $110.00 =
$364.10.
The best way to work annuity due problems is to switch your calculator to "beg" or beginning
or "due" mode, and go through the normal process. Note that it's critical to remember to change
back to "end" mode after working an annuity due problem with your calculator.
This formula could be used to find the present value of an annuity due:
PVAn(Annuity Due) = PVAn(1 + i) = PMT(PVIFAi,n)(1 + i).
In our situation, the present value of the annuity due is $273.56:
PVA3(Annuity Due) = $248.69(1.10)1 = $273.56.
The Excel function is = PV(10%,3,100,0,1). The fourth term, 0, tells Excel there are no
additional cash flows. The fifth term, 1, tells Excel it is an annuity due. The result is $273.56.
A similar modification gives the future value: = FV(10%,3,100,0,1), with a result of
364.10.
Answers and Solutions: 3 47 g. What is the present value of the following uneven cash flow stream? The appropriate
interest rate is 10 percent, compounded annually.
0 1 2 3 4 years
    
0 100 300 300 50
Answer: Here we have an uneven cash flow stream. The most straightforward approach is to find the PVs
of each cash flow and then sum them as shown below:
0 10% 1 2 3 4 years
    
100 300 300 50
90.91
247.93
225.39
(34.15)
530.08
Note (1) that the $50 year 4 outflow remains an outflow even when discounted. There are
numerous ways of finding the present value of an uneven cash flow stream. But by far the
easiest way to deal with uneven cash flow streams is with a financial calculator or a spreadsheet.
Calculators have a function which on the HP 17B is called "CFLO," for "cash flow." other
calculators could use other designations such as cf0 and CFi, but they explain how to use them in
the manual. You would input the cash flows, so they are in the calculator's memory, then input
the interest rate, I, and then press the NPV or PV button to find the present value.
Spreadsheets are especially useful for uneven cash flows. The following spreadsheet shows
this problem:
A B C D E
1 0 1 2 3 4
2 100 300 300 50
3 530.09
The Excel formula in cell A3 is = NPV(10%,B2:E2), with a result of 530.09.
h. 1. Define (a) the stated, or quoted, or nominal rate, (iNom), and (b) the periodic rate (iPer).
ANSWER: The quoted, or nominal, rate is merely the quoted percentage rate of return. The periodic rate is
the rate charged by a lender or paid by a borrower each period (periodic rate = inom/m).
Answers and Solutions: 3 48 h. 2. Will the future value be larger or smaller if we compound an initial amount more often
than annually, for example, every 6 months, or semiannually, holding the stated interest rate
constant? Why?
Answer: Accounts that pay interest more frequently than once a year, for example, semiannually, quarterly,
or daily, have future values that are higher because interest is earned on interest more often.
Virtually all banks now pay interest daily on passbook and money fund accounts, so they use daily
compounding.
h. 3. What is the future value of $100 after 5 years under 12 percent annual compounding?
Semiannual compounding? Quarterly compounding? Monthly compounding? Daily
compounding
Answer: Under annual compounding, the $100 is compounded over 5 annual periods at a 12.0 percent
periodic rate:
iNom = 12%.
mn 1*5
i 0.12
FVn = PV1 + Nom
= $100 1 + = $100(1.12)5 = $176.23.
m 1
Under semiannual compounding, the $100 is compounded over 10 semiannual periods at a 6.0
percent periodic rate:
iNom = 12%.
mn 2*5
i 0.12
FVn = PV1 + Nom
= $100 1 + = $100(1.06)10 = $179.08.
m 2
quarterly: FVn = $100(1.03)20 = $180.61.
monthly: FVn = $100(1.01)60 = $181.67.
daily: FVn = $100(1+ 0.12/365)365*5 = $182.19.
Answers and Solutions: 3 49 h. 4. What is the effective annual rate (EAR)? What is the ear for a nominal rate of 12 percent,
compounded semiannually? Compounded quarterly? Compounded monthly?
Compounded daily?
Answer: The effective annual rate is the annual rate that causes the PV to grow to the same FV as under
multiperiod compounding. For 12 percent semiannual compounding, the ear is 12.36 percent:
m
1 + i Nom
EAR = Effective Annual Rate =  1.0.
m
IF iNom = 12% and interest is compounded semiannually, then:
2
0.12 2
EAR = 1 +  1.0 = (1.06) 1.0 = 1.1236 1.0 = 0.1236 = 12,36%.
2
For quarterly compounding, the effective annual rate is:
(1.03)4  1.0 = 12.55%.
For monthly compounding, the effective annual rate is:
(1.01)12  1.0 = 12.55%.
For daily compounding, the effective annual rate is:
(1 + 0.12/365)365  1.0 = 12.75%.
i. Will the effective annual rate ever be equal to the nominal (quoted) rate?
Answer: If annual compounding is used, then the nominal rate will be equal to the effective annual rate.
If more frequent compounding is used, the effective annual rate will be above the nominal rate.
Answers and Solutions: 3 50 j. 1. Construct an amortization schedule for a $1,000, 10 percent annual rate loan with 3 equal
installments.
2. What is the annual interest expense for the borrower, and the annual interest income for the
lender, during year 2?
Answer: To begin, note that the face amount of the loan, $1,000, is the present value of a 3year annuity at
a 10 percent rate:
0 10% 1 2
3
   
1,000 PMT PMT PMT
1 2 3
1 1 1
PVA3 = PMT + PMT + PMT
1 + i 1 + i 1 + i
$1,000 = PMT(1 + i)1 + PMT(1 + i)2 + PMT(1 + i)3
= PMT(1.10)1 + PMT(1.10)2 + PMT(1.10)3.
We have an equation with only one unknown, so we can solve it to find PMT. The easy way is
with a financial calculator. Input n = 3, i = 10, PV = 1,000, FV = 0, and then press the PMT
button to get PMT = 402.1148036, rounded to $402.11.
Now make the following points regarding the amortization schedule:
· The $402.11 annual payment includes both interest and principal. Interest in the
first year is calculated as follows:
1st year interest = i × beginning balance = 0.1 × $1,000 = $100.
· The repayment of principal is the difference between the $402.11 annual payment
and the interest payment:
1st year principal repayment = $402.11  $100 = $302.11.
· The loan balance at the end of the first year is:
1st year ending balance = beginning balance principal repayment
= $1,000  $302.11 = $697.89.
· We would continue these steps in the following years.
· Notice that the interest each year declines because the beginning loan balance is
declining. Since the payment is constant, but the interest component is declining,
the principal repayment portion is increasing each year.
Answers and Solutions: 3 51 · The interest component is an expense which is deductible to a business or a
homeowner, and it is taxable income to the lender. If you buy a house, you will
get a schedule constructed like ours, but longer, with 30 × 12 = 360 monthly
payments if you get a 30year, fixed rate mortgage.
· The payment may have to be increased by a few cents in the final year to take
care of rounding errors and make the final payment produce a zero ending
balance.
· The lender received a 10% rate of interest on the average amount of money that
was invested each year, and the $1,000 loan was paid off. This is what
amortization schedules are designed to do.
· Most financial calculators have amortization functions built in.
k. Suppose on January 1 you deposit $100 in an account that pays a nominal, or quoted,
interest rate of 11.33463 percent, with interest added (compounded) daily. How much will
you have in your account on October 1, or after 9 months?
Answer: The daily periodic interest rate is rPer = 11.3346%/365 = 0.031054%. There are 273 days between
January 1 and October 1. Calculate FV as follows:
FV273 = $100(1.00031054)273
= $108.85.
Using a financial calculator, input n = 273, i = 0.031054, PV = 100, and PMT = 0.
Pressing FV gives $108.85.
An alternative approach would be to first determine the effective annual rate of interest, with
daily compounding, using the formula:
365
0.1133463
EAR = 1 +  1 = 0.12 = 12.0%.
365
(Some calculators, e.g., the hp 10b and 17b, have this equation built in under the ICNV [interest
conversion] function.)
Thus, if you left your money on deposit for an entire year, you would earn $12 of
interest, and you would end up with $112. The question, though, is this: how
much will be in your account on October 1, 2002?
Here you will be leaving the money on deposit for 9/12 = 3/4 = 0.75 of a year.
0 12% 0.75 1
  
100 FV = ? 112
You would use the regular setup, but with a fractional exponent:
Answers and Solutions: 3 52 FV0.75 = $100(1.12)0.75 = $100(1.088713) = $108.87.
This is slightly different from our earlier answer, because n is actually 273/365 = 0.7479 rather than 0.75.
Fractional time periods
Thus far all of our examples have dealt with full years. Now we are going to look at the situation when we are dealing with fractional years, such as 9 months, or 10 years. In these situations, proceed as follows:
· As always, start by drawing a time line so you can visualize the situation.
· Then think about the interest ratethe nominal rate, the compounding periods per
year, and the effective annual rate. If you have been given a nominal rate, you
may have to convert to the ear, using this formula:
m
i
EAR = 1 + Nom  1 .
m
· If you have the effective annual rateeither because it was given to you or after
you calculated it with the formulathen you can find the PV of a lump sum by
applying this equation:
t
1
PV = FVt .
1 + EAR
· Here t can be a fraction of a year, such as 0.75, if you need to find the PV of
$1,000 due in 9 months, or 450/365 = 1.2328767 if the payment is due in 450
days.
· If you have an annuity with payments different from once a year, say every month,
you can always work it out as a series of lump sums. That procedure always
works. We can also use annuity formulas and calculator functions, but you have
to be careful.
Answers and Solutions: 3 53 l. 1. What is the value at the end of year 3 of the following cash flow stream if the quoted interest
rate is 10 percent, compounded semiannually?
0 1 2 3 YEARS
      
100 100 100
Answer: 0 1 2
5% 3
      
100 100 100
110.25 = 100(1.05)2
121.55 = 100(1.05)4
331.80
Here we have a different situation. The payments occur annually, but compounding occurs each
6 months. Thus, we cannot use normal annuity valuation techniques. There are two
approaches that can be applied: (1) treat the cash flows as lump sums, as was done above, or (2)
treat the cash flows as an ordinary annuity, but use the effective annual rate:
m 2
0.10
EAR = 1 + i Nom  1 = 1 +  1 = 10.25%.
m 2
Now we have this 3period annuity:
FVA3 = $100(1.1025)2 + $100(1.1025)1 + $100 = $331.80.
You can plug in n = 3, I = 10.25, PV = 0, and PMT = 100, and then press the FV button to find
FV = $331.80.
l. 2. What is the PV of the same stream?
Answer: 0 1 2 3 5%
      
100 100 100
90.70
82.27 PV = 100(1.05)4
74.62
247.59
PV = $100(2.4759) = $247.59 AT 10.25%.
To use a financial calculator, input N = 3, I = 10.25, PMT = 100, FV = 0, and then press the PV
key to find PV = $247.59.
Answers and Solutions: 3 54 l. 3. Is the stream an annuity?
Answer: The payment stream is an annuity in the sense of constant amounts at regular intervals, but the
intervals do not correspond with the compounding periods. This kind of situation occurs often.
In this situation the interest is compounded semiannually, so with a quoted rate of 10%, the ear
will be 10.25%. Here we could find the effective rate and then treat it as an annuity. Enter N =
3, I = 10.25, PMT = 100, and FV = 0. Now press PV to get $247.59.
l. 4. An important rule is that you should never show a nominal rate on a time line or use it in
calculations unless what condition holds? (Hint: think of annual compounding, when iNom
= EAR = iPer.) What would be wrong with your answer to questions l(1) and l(2) if you used
the nominal rate (10%) rather than the periodic rate (iNom /2 = 10%/2 = 5%)?
Answer: iNom can only be used in the calculations when annual compounding occurs. If the nominal rate of
10% was used to discount the payment stream the present value would be overstated by $272.32 
$247.59 = $24.73.
Answers and Solutions: 3 55 m. Suppose someone offered to sell you a note calling for the payment of $1,000 15 months from
today. They offer to sell it to you for $850. You have $850 in a bank time deposit which
pays a 6.76649 percent nominal rate with daily compounding, which is a 7 percent effective
annual interest rate, and you plan to leave the money in the bank unless you buy the note.
The note is not riskyyou are sure it will be paid on schedule. Should you buy the note?
Check the decision in three ways: (1) by comparing your future value if you buy the note
versus leaving your money in the bank, (2) by comparing the PV of the note with your
current bank account, and (3) by comparing the ear on the note versus that of the bank
account.
Answer: You can solve this problem in three ways(1) by compounding the $850 now in the bank for 15
months and comparing that FV with the $1,000 the note will pay, (2) by finding the PV of the note
and then comparing it with the $850 cost, and (3) finding the effective annual rate of return on the
note and comparing that rate with the 7% you are now earning, which is your opportunity cost of
capital. All three procedures lead to the same conclusion. Here is the time line:
0 1 1.25
7%
  
850 1,000
(1) FV = $850(1.07)1.25 = $925.01 = amount in bank after 15 months versus $1,000 if you buy
the note. (Again, you can find this value with a financial calculator. Note that certain
calculators like the hp 12c perform a straightline interpolation for values in a fractional
time period analysis rather than an effective interest rate interpolation. The value that the
hp 12c calculates is $925.42.) This procedure indicates that you should buy the note.
Alternatively, 15 months = (1.25 years)(365 days per year) = 456.25 456 days.
FV456 = $850(1.00018538)456
= $924.97.
The slight difference is due to using n = 456 rather than n = 456.25.
(2) PV = $1,000/(1.07)1.25 = $918.90. Since the present value of the note is greater than the
$850 cost, it is a good deal. You should buy it.
Alternatively, PV = $1000/(1.00018538)456 = $918.95.
(3) FVn = PV(1 + i)n, SO $1,000 = $850(1 + i)1.25 = $1,000. Since we have an equation with
one unknown, we can solve it for i. You will get a value of i = 13.88%. The easy way is
to plug values into your calculator. Since this return is greater than your 7% opportunity
cost, you should buy the note. This action will raise the rate of return on your asset
portfolio.
Alternatively, we could solve the following equation:
$1,000 = $850(1 + i)456 for a daily i = 0.00035646,
With a result of EAR = EFF% = (1.00035646)365  1 = 13.89%.
Answers and Solutions: 3 56 Chapter 3
Financial Statements, Cash Flow, and Taxes
ANSWERS TO ENDOFCHAPTER QUESTIONS
31 a. The annual report is a report issued annually by a corporation to its stockholders. It contains
basic financial statements, as well as management's opinion of the past year's operations and the
firm's future prospects. A firm's balance sheet is a statement of the firm's financial position at a
specific point in time. It specifically lists the firm's assets on the lefthand side of the balance
sheet, while the righthand side shows its liabilities and equity, or the claims against these assets.
An income statement is a statement summarizing the firm's revenues and expenses over an
accounting period. Net sales are shown at the top of each statement, after which various costs,
including income taxes, are subtracted to obtain the net income available to common stockholders.
The bottom of the statement reports earnings and dividends per share.
b. Common Stockholders' Equity (Net Worth) is the capital supplied by common
stockholderscapital stock, paidin capital, retained earnings, and, occasionally, certain reserves.
Paidin capital is the difference between the stock's par value and what stockholders paid when
they bought newly issued shares. Retained earnings is the portion of the firm's earnings that
have been saved rather than paid out as dividends.
c. The statement of retained earnings shows how much of the firm's earnings were retained in the
business rather than paid out in dividends. Note that retained earnings represents a claim against
assets, not assets per se. Firms retain earnings primarily to expand the business, not to
accumulate cash in a bank account. The statement of cash flows reports the impact of a firm's
operating, investing, and financing activities on cash flows over an accounting period.
d. Depreciation is a noncash charge against tangible assets, such as buildings or machines. It is
taken for the purpose of showing an asset's estimated dollar cost of the capital equipment used up
in the production process. Amortization is a noncash charge against intangible assets, such as
goodwill. EBITDA is earnings before interest, taxes, depreciation, and amortization.
e. Operating current assets are the current assets used to support operations, such as cash, accounts
receivable, and inventory. It does not include shortterm investments. Operating current
liabilities are the current liabilities that are a natural consequence of the firm's operations, such as
accounts payable and accruals. It does not include notes payable or any other shortterm debt
that charges interest. Net operating working capital is operating current assets minus operating
current liabilities. Total net operating capital is sum of net operating working capital and
operating longterm assets, such as net plant and equipment. Operating capital also is equal to
the net amount of capital raised from investors. This is the amount of interestbearing debt plus
preferred stock plus common equity minus shortterm investments.
f. Accounting profit is a firm's net income as reported on its income statement. Net cash flow, as
opposed to accounting net income, is the sum of net income plus noncash adjustments. NOPAT,
net operating profit after taxes, is the amount of profit a company would generate if it had no debt
Answers and Solutions: 3 57 and no financial assets. Free cash flow is the cash flow actually available for distribution to
investors after the company has made all investments in fixed assets and working capital
necessary to sustain ongoing operations.
g. Market value added is the difference between the market value of the firm (i.e., the sum of the
market value of common equity, the market value of debt, and the market value of preferred stock)
and the book value of the firm's common equity, debt, and preferred stock. If the book values of
debt and preferred stock are equal to their market values, then MVA is also equal to the difference
between the market value of equity and the amount of equity capital that investors supplied.
Economic value added represents the residual income that remains after the cost of all capital,
including equity capital, has been deducted.
h. A progressive tax means the higher one's income, the larger the percentage paid in taxes.
Taxable income is defined as gross income less a set of exemptions and deductions which are
spelled out in the instructions to the tax forms individuals must file. Marginal tax rate is defined
as the tax rate on the last unit of income. Average tax rate is calculated by taking the total
amount of tax paid divided by taxable income.
i. Capital gain (loss) is the profit (loss) from the sale of a capital asset for more (less) than its
purchase price. Ordinary corporate operating losses can be carried backward for 2 years or
forward for 20 years to offset taxable income in a given year.
j. Improper accumulation is the retention of earnings by a business for the purpose of enabling
stockholders to avoid personal income taxes on dividends. An S corporation is a small
corporation which, under Subchapter S of the Internal Revenue Code, elects to be taxed as a
proprietorship or a partnership yet retains limited liability and other benefits of the corporate form
of organization.
32 The four financial statements contained in most annual reports are the balance sheet, income statement,
statement of retained earnings, and statement of cash flows.
33 No, because the $20 million of retained earnings would probably not be held as cash. The retained
earnings figure represents the reinvestment of earnings by the firm. Consequently, the $20 million
would be an investment in all of the firm's assets.
35 Operating capital is the amount of interest bearing debt, preferred stock, and common equity used to
acquire the company's net operating assets. Without this capital a firm cannot exist, as there is no
source of funds with which to finance operations.
36 NOPAT is the amount of net income a company would generate if it had no debt
and held no financial assets. NOPAT is a better measure of the performance of
a company's operations because debt lowers income. In order to get a true
reflection of a company's operating performance, one would want to take out
debt to get a clearer picture of the situation.
37 Free cash flow is the cash flow actually available for distribution to investors after
the company has made all the investments in fixed assets and working capital
necessary to sustain ongoing operations. It is the most important measure of
cash flows because it shows the exact amount available to all investors.
Answers and Solutions: 3 58 38 If the business were organized as a partnership or a proprietorship, its income could be taken out by
the owners without being subject to double taxation. Also, if you expected to have losses for a few
years while the company was getting started, if you were not incorporated, and if you had outside
income, the business losses could be used to offset your other income and reduce your total tax bill.
These factors would lead you to not incorporate the business. An alternative would be to organize as
an S Corporation, if requirements are met.
Answers and Solutions: 3 59 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
31 Corporate yield = 9%; T = 35.5%
AT yield = 9%(1  T)
= 9%(0.645) = 5.76%.
32 Corporate bond yields 8%. Municipal bond yields 6%.
Equivalent pretax yield Yield on muni
=
on taxable bond (1  T )
6%
8% =
(1  T )
0.08  0.08T = 0.06
 0.08T = 0.02
T = 25%.
32 Income $365,000
Less Interest deduction (50,000)
Plus: Dividends receiveda 4,500
Taxable income $319,500
a
For a corporation, 70% of dividends received are excluded from taxes; therefore, taxable dividends
are calculated as $15,000(1  0.70) = $4,500.
Tax = $22,250 + ($319,500  $100,000)(0.39) = $22,250 + $85,605 = $107,855.
Aftertax income:
Taxable income $319,500
Taxes (107,855)
Plus Nontaxable dividends receivedb 10,500
Net income $222,145
b
Nontaxable dividends are calculated as $15,000 x 0.7 = $10,500.
The company's marginal tax rate is 39 percent. The company's average tax rate is
$107,855/$319,500 = 33.76%.
34 a. Tax = $3,400,000 + ($10,500,000  $10,000,000)(0.35) = $3,575,000.
b. Tax = $1,000,000(0.35) = $350,000.
Answers and Solutions: 3 60 c. Tax = ($1,000,000)0.30(0.35) = $105,000.
35 AT yield on FLA bond = 5%.
AT yield on AT&T bond = 7.5%  Taxes = 7.5%  7.5%(0.35) = 4.875%.
Check: Invest $10,000 @ 7.5% = $750 interest.
Pay 35% tax, so AT income = $750(1  T) = $750(0.65) = $487.50.
AT rate of return = $487.50/$10,000 = 4.875%.
AT yield on AT&T preferred stock:
AT yield = 6%  Taxes = 6%  0.3(6%)(0.35) = 6%  0.63% = 5.37%.
Therefore, invest in AT&T preferred stock. We could make this a harder problem by asking for the
tax rate that would cause the company to prefer the Florida bond or the AT&T bond.
36 EBIT = $750,000; DEP = $200,000; 100% Equity; T = 40%
NI = ?; NCF = ?; OCF = ?
First, determine net income by setting up an income statement:
EBIT $750,000
Interest 0
EBT $750,000
Taxes (40%) 300,000
NI $450,000
NCF = NI + DEP = $450,000 + $200,000 = $650,000.
Answers and Solutions: 3 61 37 a. Income Statement
Sales revenues $12,000,000
Costs except
depreciation 9,000,000
Depreciation 1,500,000
EBT $ 1,500,000
Taxes (40%) 600,000
Net income $ 900,000
Add back depreciation 1,500,000
Net cash flow $ 2,400,000
b. If depreciation doubled, taxable income would fall to zero and taxes would be zero. Thus, net
income would decrease to zero, but net cash flow would rise to $3,000,000. Menendez would
save $600,000 in taxes, thus increasing its cash flow:
CF = T(Depreciation) = 0.4($1,500,000) = $600,000.
c. If depreciation were halved, taxable income would rise to $2,250,000 and taxes to $900,000.
Therefore, net income would rise to $1,350,000, but net cash flow would fall to $2,100,000.
d. You should prefer to have higher depreciation charges and higher cash flows. Net cash flows are
the funds that are available to the owners to withdraw from the firm and, therefore, cash flows
should be more important to them than net income.
Answers and Solutions: 3 62 38 a. NOPAT = EBIT(1  Tax rate)
= $150,000,000(0.6)
= $90,000,000.
b. NOWC03 = Operating CA operating CL
= $360,000,000  ($90,000,000 + $60,000,000)
= $210,000,000.
NOWC04 = $372,000,000  $180,000,000 = $192,000,000.
Net plant Net operating
c. Operating capital03 = +
and equipment working capital
= $250,000,000 + $210,000,000
= $460,000,000.
Operating capital04 = $300,000,000 + $192,000,000
= $492,000,000.
d. FCF = NOPAT  Net investment in operating capital
= $90,000,000  ($492,000,000  $460,000,000)
= $58,000,000.
e. The large increase in dividends for 2004 can most likely be attributed to a large increase in free
cash flow from 2003 to 2004, since FCF represents the amount of cash available to be paid out to
stockholders after the company has made all investments in fixed assets and working capital
necessary to sustain the business.
39 Prior Years 2002 2003
Profit earned $150,000 $150,000
Carryback credit 150,000 150,000
Adjusted profit $ 0 $ 0
Tax previously
paid (40%) 60,000 60,000
Tax refund: Taxes
Answers and Solutions: 3 63 previously paid $ 60,000 $ 60,000
Total check from U.S. Treasury = $60,000 + $60,000 = $120,000.
Future Years 2005 2006 2007 2008 2009
Estimated
profit $150,000 $150,000 $150,000 $150,000 $150,000
Carryforward
credit 150,000 150,000 50,000 0 0
Adjusted
profit $ 0 $ 0 $100,000 $150,000 $150,000
Tax (at 40%) 0 $ 0 $ 40,000 $ 60,000 $ 60,000
Answers and Solutions: 3 64 SOLUTION TO SPREADSHEET PROBLEM 313 The detailed solution for the spreadsheet problem is available both on the
instructor's resource CDROM (in the file Solution for FM11 Ch 03 P13 Build a
Model.xls) and on the instructor's side of the book's web site,
http://brigham.swcollege.com.
Answers and Solutions: 3 65 MINI CASE
Donna Jamison, a recent graduate of the University of Tennessee with four years of banking experience, was recently brought in as assistant to the chairman of the board of Computron Industries, a manufacturer of electronic calculators.
The company doubled its plant capacity, opened new sales offices outside its home territory, and launched an expensive advertising campaign. Computron's results were not satisfactory, to put it mildly. Its board of directors, which consisted of its president and vicepresident plus its major stockholders (who were all local business people), was most upset when directors learned how the expansion was going. Suppliers were being paid late and were unhappy, and the bank was complaining about the deteriorating situation and threatening to cut off credit. As a result, Al Watkins, Computron's president, was informed that changes would have to be made, and quickly, or he would be fired. Also, at the board's insistence Donna Jamison was brought in and given the job of assistant to Fred Campo, a retired banker who was Computron's chairman and largest stockholder. Campo agreed to give up a few of his golfing days and to help nurse the company back to health, with Jamison's help.
Jamison began by gathering financial statements and other data. Assume that you are Jamison's assistant, and you must help her answer the following questions for Campo.
Balance Sheets
Assets 2003 2004
Cash $ 9,000 $ 7,282
Shortterm investments. 48,600 20,000
Accounts receivable 351,200 632,160
Inventories 715,200 1,287,360
total current assets $ 1,124,000 $ 1,946,802
Gross fixed assets 491,000 1,202,950
Less: accumulated depreciation 146,200 263,160
net fixed assets $ 344,800 $ 939,790
Total assets $ 1,468,800 $ 2,886,592
Liabilities and equity 2003 2004
Accounts payable $ 145,600 $ 324,000
Notes payable 200,000 720,000
Accruals 136,000 284,960
total current liabilities $ 481,600 $ 1,328,960
Longterm debt 323,432 1,000,000
Common stock (100,000 shares) 460,000 460,000
Retained earnings 203,768 97,632
total equity $ 663,768 $ 557,632
Total liabilities and equity $ 1,468,800 $ 2,886,592
Answers and Solutions: 4  66 Income Statements
2003 2004 Sales $ 3,432,000 $ 5,834,400 Cost of goods sold 2,864,000 4,980,000 Other expenses 340,000 720,000 Depreciation 18,900 116,960
total operating costs $ 3,222,900 $ 5,816,960
EBIT $ 209,100 $ 17,440 Interest expense 62,500 176,000
EBT $ 146,600 $ (158,560) Taxes (40%) 58,640 (63,424) Net income $ 87,960 $ (95,136)
Other data 2002 2003 Stock price $ 8.50 $ 6.00 Shares outstanding 100,000 100,000 EPS $ 0.880 $ (0.951) DPS $ 0.220 $ 0.110
Statement of retained earnings, 2004
Balance of retained earnings, 12/31/2003 $ 203,768
add: net income, 2004 $ (95,136)
less: dividend paid, 2004 $ (11,000)
Balance of retained earnings, 12/31/2004 $ 97,632
Answers and Solutions: 4  67 Statement of Cash Flows
Operating activities
Net income $ (95,136)
Adjustments:
noncash adjustments:
depreciation 116,960
changes in working capital:
change in accounts receivable (280,960)
change in inventories (572,160)
change in accounts payable 178,400
change in accruals 148,960
Net cash provided by operating activities $ (503,936)
Longterm investing activities
Cash used to acquire fixed assets $ (711,950)
Financing activities
change in short term investments $ 28,600
change in notes payable $ 520,000
change in longterm debt $ 676,568
change in common stock $ 
payment of cash dividends $ (11,000)
Net cash provided by financing activities $ 1,214,168
Summary
Net change in cash $ (1,718)
Cash at beginning of year 9,000
Cash at end of year $ 7,282
a. What effect did the expansion have on sales and net income? What effect did the
expansion have on the asset side of the balance sheet? What effect did it have on liabilities
and equity?
Answer: Sales increased by over by over $2.4 million, but net income fell by over $190,000. Assets
almost doubled. Debt and funds provided by suppliers increased, but retained earnings fell due
to the year's loss.
b. What do you conclude from the statement of cash flows?
Answer: Net CF from operations = $503,936, because of negative net income and increases in working
capital. The firm spent $711,950 on FA. The firm borrowed heavily and sold some shortterm
investments to meet its cash requirements. Even after borrowing, the cash account fell by
$1,718.
Answers and Solutions: 4  68 c. What is free cash flow? Why is it important? What are the five uses of FCF?
Answer: FCF is the amount of cash available from operations for distribution to all investors (including
stockholders and debtholders) after making the necessary investments to support operations. A
company's value depends upon the amount of FCF it can generate.
1. Pay interest on debt.
2. Pay back principal on debt.
3. Pay dividends.
4. Buy back stock.
5. Buy nonoperating assets (e.g., marketable securities, investments in other companies, etc.)
d. What are operating current assets? What are operating current liabilities? How much
net operating working capital and total net operating capital does Computron have?
Answer: Operating current assets are the CA needed to support operations. OP CA include: cash,
inventory, receivables. OP CA exclude: shortterm investments, because these are not a part of
operations. Operating current liabilities are the CL resulting as a normal part of operations. OP
CL include: accounts payable and accruals. OP CA exclude: notes payable, because this is a
source of financing, not a part of operations.
NOWC = operating CA operating CL
NOWC04 = ($7,282 + $632,160 + $1,287,360)  ($324,000 + $284,960)
= $1,317,842.
NOWC03 = $793,800.
Total operating working capital = NOWC + net fixed assets.
Operating capital in 2004 = $1,317,842 + $939,790
= $2,257,632.
Operating capital in 2003 = $1,138,600. e. What are Computron's net operating profit after taxes (NOPAT) and free cash flow (FCF)?
ANSWER: NOPAT = EBIT(1  TAX RATE)
NOPAT04 = $17,440(1  0.4)
= $10,464.
NOPAT03 = $125,460.
FCF = NOPAT  NET INVESTMENT IN CAPITAL
= $10,464  ($2,257,632  $1,138,600)
= $10,464  $1,119,032
= $1,108,568.
Answers and Solutions: 4  69 f. Calculate Computron's return on invested capital. Computron has a 10% cost of capital
(WACC). Do you think Computron's growth added value?
ANSWER: ROIC = NOPAT / TOTAL NET OPERATING CAPITAL.
ROIC04 = $10,464 / $2,257,632
= 0.5%.
ROIC03 = 11.0%.
The ROIC of 0.5% is less than the WACC of 10%. Investors did not get the return they require.
Note: high growth usually causes negative FCF (due to investment in capital), but that's ok if
ROIC > WACC. For example, home depot has high growth, negative FCF, but a high ROIC.
g. Jamison also has asked you to estimate Computron's EVA. She estimates that the aftertax cost of
capital was 10 percent in both years.
ANSWER: EVA = NOPAT (WACC)(CAPITAL).
EVA04 = $10,464  (0.1)($2,257,632)
= $10,464  $225,763
= $215,299.
EVA03 = $125,460  (0.10)($1,138,600)
= $125,460  $113,860
= $11,600.
h. What happened to Computron's market value added (MVA)?
Answer: MVA = market value of the firm  book value of the firm.
Market value = (# shares of stock)(price per share) + value of debt.
Book value = total common equity + value of debt.
If the market value of debt is close to the book value of debt, then MVA is market value of equity
minus book value of equity. Assume market value of debt equals book value of debt.
Market value of equity 2003 = (100,000)($6.00) = $600,000.
Book value of equity 2003 = $557,632.
MVA03 = $600,000  $557,632 = $42,368.
MVA02 = $850,000  $663,768 = $186,232.
i. Assume that a corporation has $100,000 of taxable income from operations plus
$5,000 of interest income and $10,000 of dividend income. What is the company's
tax liability?
Answers and Solutions: 4  70 Answer: Calculation of the company's tax liability:
Taxable operating income $100,000
Taxable interest income 5,000
Taxable dividend income (0.3 × $10,000) 3,000
Total taxable income $108,000
Tax = $22,250 + ($108,000  $100,000)0.39 = $25,370.
taxable dividend income = dividends  exclusion
= $10,000  0.7($10,000)
= $3,000.
j. Assume that you are in the 27 percent marginal tax bracket and that you have
$5,000 to invest. You have narrowed your investment choices down to California
bonds with a yield of 7 percent or equally risky Exxon bonds with a yield of 10
percent. Which one should you choose and why? At what marginal tax rate would
you be indifferent to the choice between California and Exxon bonds?
Answer: Aftertax return income at t = 27%:
Exxon = 0.10($5,000)  (0.10)($5,000)(0.27) = $365.
California = 0.07($5,000)  $0 = $350.
Alternatively, calculate aftertax yields:
AT yieldExxon = 10.0%(1  t) = 10%(1  0.27) = 7.3%.
AT yieldCalif. = 7.0%.
At what marginal tax rate would you be indifferent?
7.0% = 10.0%(1  t). Solve for t.
7.0% = 10.0%  10.0%(t)
10.0%(t) = 3%
t = 30%.
Answers and Solutions: 4  71 Chapter 4
Risk and Return: The Basics
ANSWERS TO ENDOFCHAPTER QUESTIONS
41 a. Standalone risk is only a part of total risk and pertains to the risk an investor takes by holding
only one asset. Risk is the chance that some unfavorable event will occur. For instance, the
risk of an asset is essentially the chance that the asset's cash flows will be unfavorable or less than
expected. A probability distribution is a listing, chart or graph of all possible outcomes, such as
expected rates of return, with a probability assigned to each outcome. When in graph form, the
tighter the probability distribution, the less uncertain the outcome.
b. The expected rate of return ( r ) is the expected value of a probability distribution of expected
returns.
c. A continuous probability distribution contains an infinite number of outcomes and is graphed
from  and +.
d. The standard deviation () is a statistical measure of the variability of a set of observations. The
variance (2) of the probability distribution is the sum of the squared deviations about the
expected value adjusted for deviation. The coefficient of variation (CV) is equal to the standard
deviation divided by the expected return; it is a standardized risk measure which allows
comparisons between investments having different expected returns and standard deviations.
e. A risk averse investor dislikes risk and requires a higher rate of return as an inducement to buy
riskier securities. A realized return is the actual return an investor receives on their investment.
It can be quite different than their expected return.
f. A risk premium is the difference between the rate of return on a riskfree asset and the expected
return on Stock i which has higher risk. The market risk premium is the difference between the
expected return on the market and the riskfree rate.
g. CAPM is a model based upon the proposition that any stock's required rate of return is equal to
the risk free rate of return plus a risk premium reflecting only the risk remaining after
diversification.
h. The expected return on a portfolio. r p, is simply the weightedaverage expected return of the
individual stocks in the portfolio, with the weights being the fraction of total portfolio value
invested in each stock. The market portfolio is a portfolio consisting of all stocks.
i. Correlation is the tendency of two variables to move together. A correlation coefficient () of
+1.0 means that the two variables move up and down in perfect synchronization, while a
coefficient of 1.0 means the variables always move in opposite directions. A correlation
coefficient of zero suggests that the two variables are not related to one another; that is, they are
independent.
Answers and Solutions: 4  72 j. Market risk is that part of a security's total risk that cannot be eliminated by diversification. It is
measured by the beta coefficient. Diversifiable risk is also known as company specific risk, that
part of a security's total risk associated with random events not affecting the market as a whole.
This risk can be eliminated by proper diversification. The relevant risk of a stock is its
contribution to the riskiness of a welldiversified portfolio.
k. The beta coefficient is a measure of a stock's market risk, or the extent to which the returns on a
given stock move with the stock market. The average stock's beta would move on average with
the market so it would have a beta of 1.0.
l. The security market line (SML) represents in a graphical form, the relationship between the risk
of an asset as measured by its beta and the required rates of return for individual securities. The
SML equation is essentially the CAPM, ri = rRF + bi(rM  rRF).
m. The slope of the SML equation is (rM  rRF), the market risk premium. The slope of the SML
reflects the degree of risk aversion in the economy. The greater the average investors aversion to
risk, then the steeper the slope, the higher the risk premium for all stocks, and the higher the
required return.
42 a. The probability distribution for complete certainty is a vertical line.
b. The probability distribution for total uncertainty is the X axis from  to +.
43 Security A is less risky if held in a diversified portfolio because of its lower beta and negative
correlation with other stocks. In a singleasset portfolio, Security A would be more risky because A
> B and CVA > CVB.
44 a. No, it is not riskless. The portfolio would be free of default risk and liquidity risk, but inflation
could erode the portfolio's purchasing power. If the actual inflation rate is greater than that
expected, interest rates in general will rise to incorporate a larger inflation premium (IP) and the
value of the portfolio would decline.
b. No, you would be subject to reinvestment rate risk. You might expect to "roll over" the Treasury
bills at a constant (or even increasing) rate of interest, but if interest rates fall, your investment
income will decrease.
c. A U.S. governmentbacked bond that provided interest with constant purchasing power (that is, an
indexed bond) would be close to riskless.
45 The risk premium on a high beta stock would increase more.
RPj = Risk Premium for Stock j = (rM  rRF)bj.
If risk aversion increases, the slope of the SML will increase, and so will the market risk premium (rM
rRF). The product (rM rRF)bj is the risk premium of the jth stock. If bj is low (say, 0.5), then the
product will be small; RPj will increase by only half the increase in RPM. However, if bj is large (say,
2.0), then its risk premium will rise by twice the increase in RPM.
46 According to the Security Market Line (SML) equation, an increase in beta will increase a company's
expected return by an amount equal to the market risk premium times the change in beta. For
example, assume that the riskfree rate is 6 percent, and the market risk premium is 5 percent. If the
Answers and Solutions: 4  73 company's beta doubles from 0.8 to 1.6 its expected return increases from 10 percent to 14 percent.
Therefore, in general, a company's expected return will not double when its beta doubles.
47 Yes, if the portfolio's beta is equal to zero. In practice, however, it may be impossible to find
individual stocks that have a nonpositive beta. In this case it would also be impossible to have a stock
portfolio with a zero beta. Even if such a portfolio could be constructed, investors would probably
be better off just purchasing Treasury bills, or other zero beta investments.
Answers and Solutions: 4  74 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
41 r = (0.1)(50%) + (0.2)(5%) + (0.4)(16%) + (0.2)(25%) + (0.1)(60%)
= 11.40%.
2 = (50%  11.40%)2(0.1) + (5%  11.40%)2(0.2) + (16%  11.40%)2(0.4)
+ (25%  11.40%)2(0.2) + (60%  11.40%)2(0.1)
2
= 712.44; = 26.69%.
26.69%
CV = = 2.34.
11.40%
42 Investment Beta
$35,000 0.8
40,000 1.4
Total $75,000
($35,000/$75,000)(0.8) + ($40,000/$75,000)(1.4) = 1.12.
43 rRF = 5%; RPM = 6%; rM = ?
rM = 5% + (6%)1 = 11%.
rs when b = 1.2 = ?
rs = 5% + 6%(1.2) = 12.2%.
44 rRF = 6%; rM = 13%; b = 0.7; rs = ?
rs = rRF + (rM  rRF)b
= 6% + (13%  6%)0.7
= 10.9%.
Answers and Solutions: 4  75 45 a. r m= (0.3)(15%) + (0.4)(9%) + (0.3)(18%) = 13.5%.
r j= (0.3)(20%) + (0.4)(5%) + (0.3)(12%) = 11.6%.
b. M = [(0.3)(15%  13.5%)2 + (0.4)(9%  13.5%)2 + (0.3)(18% 13.5%)2]1/2
= 14.85% = 3.85%.
J = [(0.3)(20%  11.6%)2 + (0.4)(5%  11.6%)2 + (0.3)(12%  11.6%)2]1/2
= 38.64% = 6.22%.
3.85%
c. CVM = = 0.29.
13.5%
6.22%
CVJ = = 0.54.
11.6%
46 a. rA = rRF + (rM  rRF)bA
12% = 5% + (10%  5%)bA
12% = 5% + 5%(bA)
7% = 5%(bA)
1.4 = bA.
b. rA = 5% + 5%(bA)
rA = 5% + 5%(2)
rA = 15%.
47 a. ri = rRF + (rM  rRF)bi = 9% + (14%  9%)1.3 = 15.5%.
b. 1. rRF increases to 10%:
rM increases by 1 percentage point, from 14% to 15%.
ri = rRF + (rM  rRF)bi = 10% + (15%  10%)1.3 = 16.5%.
2. rRF decreases to 8%:
rM decreases by 1%, from 14% to 13%.
ri = rRF + (rM  rRF)bi = 8% + (13%  8%)1.3 = 14.5%.
c. 1. rM increases to 16%:
ri = rRF + (rM  rRF)bi = 9% + (16%  9%)1.3 = 18.1%.
2. rM decreases to 13%:
ri = rRF + (rM  rRF)bi = 9% + (13%  9%)1.3 = 14.2%.
Answers and Solutions: 4  76 $142,500 $7,500 48 Old portfolio beta = (b) + (1.00)
$150,000 $150,000
1.12 = 0.95b + 0.05
1.07 = 0.95b
1.13 = b.
New portfolio beta = 0.95(1.13) + 0.05(1.75) = 1.16.
Alternative Solutions:
1. Old portfolio beta = 1.12 = (0.05)b1 + (0.05)b2 +...+ (0.05)b20
1.12 = (bi)(0.05)
bi = 1.12/0.05 = 22.4.
New portfolio beta = (22.4  1.0 + 1.75)(0.05) = 1.1575 = 1.16.
2. bi excluding the stock with the beta equal to 1.0 is 22.4  1.0 = 21.4, so the beta of the portfolio
excluding this stock is b = 21.4/19 = 1.1263. The beta of the new portfolio is:
1.1263(0.95) + 1.75(0.05) = 1.1575 = 1.16.
$400,000 $600,000 49 Portfolio beta = (1.50) + (0.50)
$4,000,000 $4,000,000
$1,000,000 $2,000,000
+ (1.25) + (0.75)
$4,000,000 $4,000,000
= 0.1)(1.5) + (0.15)(0.50) + (0.25)(1.25) + (0.5)(0.75)
= 0.15  0.075 + 0.3125 + 0.375 = 0.7625.
rp = rRF + (rM  rRF)(bp) = 6% + (14%  6%)(0.7625) = 12.1%.
Answers and Solutions: 4  77 Alternative solution: First compute the return for each stock using the CAPM equation [rRF + (rM 
rRF)b], and then compute the weighted average of these returns.
rRF = 6% and rM  rRF = 8%.
Stock Investment Beta r = rRF + (rM  rRF)b Weight
A $ 400,000 1.50 18% 0.10
B 600,000 (0.50) 2 0.15
C 1,000,000 1.25 16 0.25
D 2,000,000 0.75 12 0.50
Total $4,000,000 1.00
rp = 18%(0.10) + 2%(0.15) + 16%(0.25) + 12%(0.50) = 12.1%.
410 First, calculate the beta of what remains after selling the stock:
bp = 1.1 = ($100,000/$2,000,000)0.9 + ($1,900,000/$2,000,000)bR
1.1 = 0.045 + (0.95)bR
bR = 1.1105.
bN = (0.95)1.1105 + (0.05)1.4 = 1.125.
411 We know that bR = 1.50, bS = 0.75, rM = 13%, rRF = 7%.
ri = rRF + (rM  rRF)bi = 7% + (13%  7%)bi.
rR = 7% + 6%(1.50) = 16.0%
rS = 7% + 6%(0.75) = 11.5
4.5%
412 The answers to a, b, c, and d are given below:
rA rB Portfolio
2000 (18.00%) (14.50%) (16.25%)
2001 33.00 21.80
27.40
2002 15.00 30.50
22.75
2003 (0.50) (7.60)
(4.05)
Answers and Solutions: 4  78 2004 27.00 26.30
26.65
Mean 11.30 11.30
11.30
Std Dev 20.79 20.78
20.13
CV 1.84 1.84
1.78
e. A riskaverse investor would choose the portfolio over either Stock A or Stock B alone, since the
portfolio offers the same expected return but with less risk. This result occurs because returns on
A and B are not perfectly positively correlated (AB = 0.88).
413 a. bX = 1.3471; bY = 0.6508.
b. rX = 6% + (5%)1.3471 = 12.7355%.
rY = 6% + (5%)0.6508 = 9.2540%.
c. bp = 0.8(1.3471) + 0.2(0.6508) = 1.2078.
rp = 6% + (5%)1.2078 = 12.04%.
Alternatively,
rp = 0.8(12.7355%) + 0.2(9.254%) = 12.04%.
d. Stock X is undervalued, because its expected return exceeds its required rate of return.
Answers and Solutions: 4  79 SOLUTION TO SPREADSHEET PROBLEM
414 The detailed solution for the spreadsheet problem is available both on the instructor's resource
CDROM (in the file Solution for FM11 Ch 04 P14 Build a Model.xls) and on the instructor's side of
the textbook's web site, brigham.swcollege.com.
Answers and Solutions: 4  80 MINI CASE
Assume that you recently graduated with a major in finance, and you just landed a job as a financial planner with Barney Smith Inc., a large financial services corporation. Your first assignment is to invest $100,000 for a client. Because the funds are to be invested in a business at the end of one year, you have been instructed to plan for a oneyear holding period. Further, your boss has restricted you to the following investment alternatives, shown with their probabilities and associated outcomes. (Disregard for now the items at the bottom of the data; you will fill in the blanks later.)
Returns On Alternative Investments
Estimated Rate Of Return
State of the T Alta Repo Am.
Market 2stock
economy prob. Bills Inds Men Foam portfolio portfolio
Recession 0.1 8.0% 22.0% 28.0% 10.0%* 13.0% 3.0%
Below avg 0.2 8.0 2.0 14.7 10.0 1.0
Average 0.4 8.0 20.0 0.0 7.0 15.0 10.0
Above avg 0.2 8.0 35.0 10.0 45.0 29.0
Boom 0.1 8.0 50.0 20.0 30.0 43.0 15.0
rhat ( r ) 1.7% 13.8% 15.0%
Std dev () 0.0 13.4 18.8 15.3
Coef of var (cv) 7.9 1.4 1.0
beta (b) 0.86 0.68
*Note that the estimated returns of American Foam do not always move in the same direction as the overall economy. For example, when the economy is below average, consumers purchase fewer mattresses than they would if the economy was stronger. However, if the economy is in a flatout recession, a large number of consumers who were planning to purchase a more expensive inner spring mattress may purchase, instead, a cheaper foam mattress. Under these circumstances, we would expect American Foam's stock price to be higher if there is a recession than if the economy was just below average.
Barney Smith's economic forecasting staff has developed probability estimates for the state of the economy, and its security analysts have developed a sophisticated computer program which was used to estimate the rate of return on each alternative under each state of the economy. Alta Industries is an electronics firm; Repo Men collects pastdue debts; and American Foam manufactures mattresses and other foam products. Barney Smith also maintains an "index fund" which owns a marketweighted fraction of all publicly traded stocks; you can invest in that fund, and thus obtain average stock market results. Given the situation as described, answer the following questions.
Answers and Solutions: 5  81 a. What are investment returns? What is the return on an investment that costs
$1,000 and is sold after one year for $1,100?
Answer: Investment return measures the financial results of an investment. They may be expressed in
either dollar terms or percentage terms. The dollar return is $1,100  $1,000 = $100. The percentage return is $100/$1,000 = 0.10 =
10%.
b. 1. Why is the tbill's return independent of the state of the economy? Do tbills promise a
completely riskfree return?
Answer: The 8 percent tbill return does not depend on the state of the economy because the treasury must
(and will) redeem the bills at par regardless of the state of the economy.
The tbills are riskfree in the default risk sense because the 8 percent return will be realized
in all possible economic states. However, remember that this return is composed of the real
riskfree rate, say 3 percent, plus an inflation premium, say 5 percent. Since there is uncertainty
about inflation, it is unlikely that the realized real rate of return would equal the expected 3
percent. For example, if inflation averaged 6 percent over the year, then the realized real return
would only be 8%  6% = 2%, not the expected 3%. Thus, in terms of purchasing power, tbills
are not riskless.
Also, if you invested in a portfolio of Tbills, and rates then declined, your nominal income
would fall; that is, tbills are exposed to reinvestment rate risk. So, we conclude that there are no
truly riskfree securities in the United States. If the treasury sold inflationindexed, taxexempt
bonds, they would be truly riskless, but all actual securities are exposed to some type of risk.
b. 2. Why are Alta Ind.'s returns expected to move with the economy whereas Repo
Men's are expected to move counter to the economy?
Answer: Alta Industries' returns move with, hence are positively correlated with, the economy, because the
firm's sales, and hence profits, will generally experience the same type of ups and downs as the
economy. If the economy is booming, so will Alta. On the other hand, Repo Men is considered
by many investors to be a hedge against both bad times and high inflation, so if the stock market
crashes, investors in this stock should do relatively well. Stocks such as Repo Men are thus
negatively correlated with (move counter to) the economy. (note: in actuality, it is almost
impossible to find stocks that are expected to move counter to the economy. Even Repo Men
shares have positive (but low) correlation with the market.)
Answers and Solutions: 5  82 c. Calculate the expected rate of return on each alternative and fill in the blanks on the row for
r in the table above.
Answer: The expected rate of return, r , is expressed as follows:
n
r= P r .
i =1
i i
Here Pi is the probability of occurrence of the ith state, r i is the estimated rate of return for
that state, and n is the number of states. Here is the calculation for Alta Inds.:
r Alta Inds = 0.1(22.0%) + 0.2(2.0%) + 0.4(20.0%) + 0.2(35.0%) + 0.1(50.0%)
= 17.4%.
We use the same formula to calculate r's for the other alternatives:
r Tbills = 8.0%.
r Repo Men = 1.7%.
r Am Foam = 13.8%.
r M = 15.0%.
d. You should recognize that basing a decision solely on expected returns is only appropriate
for riskneutral individuals. Since your client, like virtually everyone, is risk averse, the
Answers and Solutions: 5  83 riskiness of each alternative is an important aspect of the decision. One possible measure
of risk is the standard deviation of returns.
1. Calculate this value for each alternative, and fill in the blank on the row for in the table
above.
Answer: The standard deviation is calculated as follows:
n
= (r  r )
i =1
i
2
i
Pi .
Alta = [(22.0  17.4)2(0.1) + (2.0  17.4)2(0.2) + (20.0  17.4)2(0.4)
+ (35.0  17.4)2(0.2) + (50.0  17.4)2(0.1)]0.5
= 401.4 = 20.0%.
Here are the standard deviations for the other alternatives:
Tbills = 0.0%.
Repo = 13.4%.
Am Foam = 18.8%.
M = 15.3%.
d. 2. What type of risk is measured by the standard deviation?
Answer: The standard deviation is a measure of a security's (or a portfolio's) standalone risk. The larger
the standard deviation, the higher the probability that actual realized returns will fall far below the
expected return, and that losses rather than profits will be incurred.
d. 3. Draw a graph which shows roughly the shape of the probability distributions for Alta Inds,
Am Foam, and Tbills.
Answer:
Answers and Solutions: 5  84 Probability of
Occurrence
TBills
ALTA INDS
AM FOAM
60 45 30 15 0 15 30 45 60
Rate of Return (%)
Based on these data, Alta Inds is the most risky investment, tbills the least risky.
e. Suppose you suddenly remembered that the coefficient of variation (CV) is generally
regarded as being a better measure of standalone risk than the standard deviation when the
alternatives being considered have widely differing expected returns. Calculate the missing
CVs, and fill in the blanks on the row for CV in the table above. Does the CV produce the
same risk rankings as the standard deviation?
Answer: The coefficient of variation (CV) is a standardized measure of dispersion about the expected value;
it shows the amount of risk per unit of return.
CV =
.
r
CVTbills = 0.0%/8.0% = 0.0.
CVAlta Inds = 20.0%/17.4% = 1.1.
Answers and Solutions: 5  85 CVRepo Men = 13.4%/1.7% = 7.9.
CVAm Foam = 18.8%/13.8% = 1.4.
CVM = 15.3%/15.0% = 1.0.
When we measure risk per unit of return, Repo Men, with its low expected return, becomes
the most risky stock. The CV is a better measure of an asset's standalone risk than because
CV considers both the expected value and the dispersion of a distributiona security with a low
expected return and a low standard deviation could have a higher chance of a loss than one with a
high but a high r .
f. Suppose you created a 2stock portfolio by investing $50,000 in Alta Inds and
$50,000 in Repo Men.
1. Calculate the expected return ( r p), the standard deviation (p), and the coefficient of
variation (cvp) for this portfolio and fill in the appropriate blanks in the table above.
Answer: To find the expected rate of return on the twostock portfolio, we first calculate the rate of return
on the portfolio in each state of the economy. Since we have half of our money in each stock,
the portfolio's return will be a weighted average in each type of economy. For a recession, we
have: rp = 0.5(22%) + 0.5(28%) = 3%. We would do similar calculations for the other states
of the economy, and get these results:
State Portfolio
Recession 3.0%
Below Average 6.4
Average 10.0
Above Average 12.5
Boom 15.0
Now we can multiply probabilities times outcomes in each state to get the expected return on
this twostock portfolio, 9.6%.
Alternatively, we could apply this formula,
R = wi x ri = 0.5(17.4%) + 0.5(1.7%) = 9.6%,
Answers and Solutions: 5  86 Which finds r as the weighted average of the expected returns of the individual securities in the
portfolio.
It is tempting to find the standard deviation of the portfolio as the weighted average of the
standard deviations of the individual securities, as follows:
p wi(i) + wj(j) = 0.5(20%) + 0.5(13.4%) = 16.7%.
However, this is not correctit is necessary to use a different formula, the one for that we used
earlier, applied to the twostock portfolio's returns.
The portfolio's depends jointly on (1) each security's and (2) the correlation between the
securities' returns. The best way to approach the problem is to estimate the portfolio's risk and
return in each state of the economy, and then to estimate p with the formula. Given the
distribution of returns for the portfolio, we can calculate the portfolio's and CV as shown below:
p = [(3.0  9.6)2(0.1) + (6.4  9.6)2(0.2) + (10.0  9.6)2(0.4)
+ (12.5  9.6)2(0.2) + (15.0  9.6)2(0.1)]0.5
= 3.3%.
CVp = 3.3%/9.6% = 0.3.
f. 2. How does the riskiness of this 2stock portfolio compare with the riskiness of the individual
stocks if they were held in isolation?
Answer: Using either or CV as our standalone risk measure, the standalone risk of the portfolio is
significantly less than the standalone risk of the individual stocks. This is because the two
stocks are negatively correlatedwhen Alta Inds is doing poorly, Repo Men is doing well, and
vice versa. Combining the two stocks diversifies away some of the risk inherent in each stock if
it were held in isolation, i.e., in a 1stock portfolio.
g. Suppose an investor starts with a portfolio consisting of one randomly selected
stock. What would happen (1) to the riskiness and (2) to the expected return of
the portfolio as more and more randomly selected stocks were added to the
portfolio? What is the implication for investors? Draw a graph of the two
portfolios to illustrate your answer.
Answer:
Answers and Solutions: 5  87 Density
Portfolio of stocks
with rp = 16%
One
Stock
%
0 16 Return
The standard deviation gets smaller as more stocks are combined in the portfolio, while rp (the
portfolio's return) remains constant. Thus, by adding stocks to your portfolio, which initially started
as a 1stock portfolio, risk has been reduced.
In the real world, stocks are positively correlated with one anotherif the economy does well,
so do stocks in general, and vice versa. Correlation coefficients between stocks generally range
from +0.5 to +0.7. A single stock selected at random would on average have a standard
deviation of about 35 percent. As additional stocks are added to the portfolio, the portfolio's
standard deviation decreases because the added stocks are not perfectly positively correlated.
However, as more and more stocks are added, each new stock has less of a riskreducing impact,
and eventually adding additional stocks has virtually no effect on the portfolio's risk as measured
by . In fact, stabilizes at about 20.4 percent when 40 or more randomly selected stocks are
added. Thus, by combining stocks into welldiversified portfolios, investors can eliminate
almost onehalf the riskiness of holding individual stocks. (Note: it is not completely costless
to diversify, so even the largest institutional investors hold less than all stocks. Even index funds
generally hold a smaller portfolio which is highly correlated with an index such as the S&P 500
rather than hold all the stocks in the index.)
The implication is clear: investors should hold welldiversified portfolios of stocks rather
than individual stocks. (In fact, individuals can hold diversified portfolios through mutual fund
investments.) By doing so, they can eliminate about half of the riskiness inherent in individual
stocks.
h. 1. Should portfolio effects impact the way investors think about the riskiness of individual
stocks?
Answer: Portfolio diversification does affect investors' views of risk. A stock's standalone risk as
measured by its or CV, may be important to an undiversified investor, but it is not relevant to a
welldiversified investor. A rational, riskaverse investor is more interested in the impact that the
stock has on the riskiness of his or her portfolio than on the stock's standalone risk. Standalone
Answers and Solutions: 5  88 risk is composed of diversifiable risk, which can be eliminated by holding the stock in a
welldiversified portfolio, and the risk that remains is called market risk because it is present even
when the entire market portfolio is held.
h. 2. If you decided to hold a 1stock portfolio, and consequently were exposed to more risk than
diversified investors, could you expect to be compensated for all of your risk; that is, could
you earn a risk premium on that part of your risk that you could have eliminated by
diversifying?
Answer: If you hold a onestock portfolio, you will be exposed to a high degree of risk, but you won't be
compensated for it. If the return were high enough to compensate you for your high risk, it
would be a bargain for more rational, diversified investors. They would start buying it, and these
buy orders would drive the price up and the return down. Thus, you simply could not find stocks
in the market with returns high enough to compensate you for the stock's diversifiable risk.
i. How is market risk measured for individual securities? How are beta coefficients
calculated?
Answer: Market risk, which is relevant for stocks held in welldiversified portfolios, is defined
as the contribution of a security to the overall riskiness of the portfolio. It is
measured by a stock's beta coefficient, which measures the stock's volatility relative
to the market. Run a regression with returns on the stock in question plotted on the y axis and returns on the
market portfolio plotted on the x axis. The slope of the regression line, which measures relative volatility, is defined as the stock's
beta coefficient, or b.
j. Suppose you have the following historical returns for the stock market and for
another company, P.Q. Unlimited. Explain how to calculate beta, and use the
historical stock returns to calculate the beta for PQU. Interpret your results.
Answers and Solutions: 5  89 YEAR MARKET PQU
1 25.7% 40.0%
2 8.0% 15.0%
3 11.0% 15.0%
4 15.0% 35.0%
5 32.5% 10.0%
6 13.7% 30.0%
7 40.0% 42.0%
8 10.0% 10.0%
9 10.8% 25.0%
10 13.1% 25.0%
Answer: Betas are calculated as the slope of the "characteristic" line, which is the regression
line showing the relationship between a given stock and the general stock market.
PQU
40%
20%
0% rM
40% 20% 0% 20% 40%
20%
r PQU = 0.83r M + 0.03
40% 2
R = 0.36
Show the graph with the regression results. Point out that the beta is the slope
coeeficient, which is 0.83. State that an average stock, by definition, moves with the
market. Beta coefficients measure the relative volatility of a given stock relative to
the stock market. The average stock's beta is 1.0. Most stocks have betas in the
range of 0.5 to 1.5. Theoretically, betas can be negative, but in the real world they
are generally positive.
In practice, 4 or 5 years of monthly data, with 60 observations, would generally be used.
Some analysts use 52 weeks of weekly data. Point out that the r2 of 0.36 is slightly higher than
the typical value of about 0.29. A portfolio would have an r2 greater than 0.9.
Answers and Solutions: 5  90 k. The expected rates of return and the beta coefficients of the alternatives as
supplied by barney smith's computer program are as follows:
Security Return ( r ) Risk (Beta)
Alta Inds 17.4% 1.29
Market 15.0 1.00
Am. Foam 13.8 0.68
TBills 8.0 0.00
Repo Men 1.7 (0.86)
(1) Do the expected returns appear to be related to each alternative's market risk?
(2) Is it possible to choose among the alternatives on the basis of the
information developed thus far?
Answer: The expected returns are related to each alternative's market riskthat is, the higher the alternative's
rate of return the higher its beta. Also, note that tbills have 0 risk. We do not yet have enough information to choose among the various alternatives. We need to
know the required rates of return on these alternatives and compare them with their
expected returns.
l. 1. Write out the security market line (SML) equation, use it to calculate the required rate of
return on each alternative, and then graph the relationship between the expected and
required rates of return.
Answer: Here is the SML equation:
ri = rrf + (rm  rrf)bi.
If we use the tbill yield as a proxy for the riskfree rate, then rRF = 8%. Further, our
estimate of rm = rm is 15%. Thus, the required rates of return for the alternatives are as follows:
Alta Inds: 8% + (15%  8%)1.29 = 17.03% 17.0%.
Market: 8% + (15%  8%)1.00 = 15.0%.
Am Foam : 8% +(15%  8%)0.68 = 12.76% 12.8%.
TBills: 8% + (15%  8%)1.29 = 17.03% 17.0%.
Repo Men: 8% + (15%  8%)0.86 = 1.98% 2%.
l. 2. How do the expected rates of return compare with the required rates of return?
Answer: We have the following relationships:
Answers and Solutions: 5  91 Expected Required
Return Return
SECURITY (r ) (r) CONDITION
Alta Inds 17.4% 17.0% Undervalued: r > R
Market 15.0 15.0 Fairly Valued (Market Equilibrium)
Am Foam 13.8 12.8 Undervalued: r> R
TBills 8.0 8.0 Fairly Valued
Repo Men 1.7 2.0 Overvalued: R > r
SML: r i = r + RP M b i
RF
Required and Expected Rates of
= 8% + 7%(b i )
25%
20% Alta Inds.
15% Am. Foam Market
Return
10%
5% TBills
0% Repo Men
5%
10%
3 2 1 0 1 2 3
Beta
(Note: the plot looks somewhat unusual in that the x axis extends to the left of zero. We have a
negative beta stock, hence a required return that is less than the riskfree rate.) The tbills and
market portfolio plot on the SML, Alta Inds. And Am. Foam plot above it, and Repo Men plots
below it. Thus, the tbills and the market portfolio promise a fair return, Alta Inds and Am.
Foam are good deals because they have expected returns above their required returns, and Repo
Men has an expected return below its required return.
l. 3. Does the fact that Repo Men has an expected return which is less than the tbill
rate make any sense?
Answer: Repo Men is an interesting stock. Its negative beta indicates negative market riskincluding it
in a portfolio of "normal" stocks will lower the portfolio's risk. Therefore, its required rate of
return is below the riskfree rate. Basically, this means that Repo Men is a valuable security to
rational, welldiversified investors. To see why, consider this question: would any rational
investor ever make an investment which has a negative expected return? The answer is
"yes"just think of the purchase of a life or fire insurance policy. The fire insurance policy has a
negative expected return because of commissions and insurance company profits, but businesses
buy fire insurance because they pay off at a time when normal operations are in bad shape. Life
insurance is similarit has a high return when work income ceases. A negative beta stock is
conceptually similar to an insurance policy.
Answers and Solutions: 5  92 l. 4. What would be the market risk and the required return of a 5050 portfolio of
Alta Inds and Repo Men? Of Alta Inds and Am. Foam?
Answer: Note that the beta of a portfolio is simply the weighted average of the betas of the stocks in the
portfolio. Thus, the beta of a portfolio with 50 percent Alta Inds and 50 percent Repo Men is:
n
bp = w b .
i =1
i i
bp = 0.5(bAlta) + 0.5(bRepo) = 0.5(1.29) + 0.5(0.86)
= 0.215,
rp = rRF + (rM  rRF)bp = 8.0% + (15.0%  8.0%)(0.215)
= 8.0% + 7%(0.215) = 9.51% 9.5%.
For a portfolio consisting of 50% Alta Inds plus 50% Am. Foam, the required return
would be 14.9%:
bp = 0.5(1.29) + 0.5(0.68) = 0.985.
rp = 8.0% + 7%(0.985) = 14.9%.
Answers and Solutions: 5  93 m. 1. Suppose investors raised their inflation expectations by 3 percentage points over current
estimates as reflected in the 8 percent tbill rate. What effect would higher inflation have
on the SML and on the returns required on high and lowrisk securities?
Answer:
Required and Expected
Rates of Return (%)
40
35
Increased Risk Aversion
30
Increased Inflation
25
20
15 Original Situation
10
5
Beta
0.00 0.50 1.00 1.50 2.00
Here we have plotted the SML for betas ranging from 0 to 2.0. The base case SML is based on
r RF = 8% and r M = 15%. If inflation expectations increase by 3 percentage points, with no
change in risk aversion, then the entire SML is shifted upward (parallel to the base case SML) by
3 percentage points. Now, r RF = 11%, r M = 18%, and all securities' required returns rise by
3 percentage points. Note that the market risk premium, rm  r RF , remains at 7 percentage
points.
m. 2. Suppose instead that investors' risk aversion increased enough to cause the
market risk premium to increase by 3 percentage points. (inflation remains
constant.) What effect would this have on the SML and on returns of high and
lowrisk securities?
Answer: When investors' risk aversion increases, the SML is rotated upward about the yintercept ( r RF ).
r RF remains at 8 percent, but now r M increases to 18 percent, so the market risk premium
increases to 10 percent. The required rate of return will rise sharply on highrisk (highbeta)
stocks, but not much on lowbeta securities.
Answers and Solutions: 5  94 Optional question (cover if time is available)
Financial managers are more concerned with investment decisions relating to real assets such as plant and equipment than with investments in financial assets such as securities. How does the analysis that we have gone through relate to real asset investment decisions, especially corporate capital budgeting decisions?
Answer: There is a great deal of similarity between your financial asset decisions and a firm's capital
budgeting decisions. Here is the linkage:
1. A company may be thought of as a portfolio of assets. If the company diversifies its assets,
and especially if it invests in some projects that tend to do well when others are doing badly,
it can lower the variability of its returns.
2. Companies obtain their investment funds from investors, who buy the firm's
stocks and bonds. When investors buy these securities, they require a risk
premium which is based on the company's risk as they (investors) see it. Further,
since investors in general hold welldiversified portfolios of stocks and bonds, the
risk that is relevant to them is the security's market risk, not its standalone risk.
Thus, investors view the risk of the firm from a market risk perspective.
3. Therefore, when a manager makes a decision to build a new plant, the riskiness of
the investment in the plant that is relevant to the firm's investors (its owners) is its
market risk, not its standalone risk. Accordingly, managers need to know how
physical asset investment decisions affect their firm's beta coefficient. A
particular asset may look quite risky when viewed in isolation, but if its returns
are negatively correlated with returns on most other stocks, the asset may really
have low risk. We will discuss all this in more detail in our capital budgeting
discussions.
Answers and Solutions: 5  95 Chapter 5
Risk and Return: Portfolio Theory and Asset Pricing
Models ANSWERS TO ENDOFCHAPTER QUESTIONS
51 a. A portfolio is made up of a group of individual assets held in combination. An asset that would
be relatively risky if held in isolation may have little, or even no risk if held in a welldiversified
portfolio.
The feasible, or attainable, set represents all portfolios that can be constructed from a given set of
stocks. This set is only efficient for part of its combinations.
An efficient portfolio is that portfolio which provides the highest expected return for any degree
of risk. Alternatively, the efficient portfolio is that which provides the lowest degree of risk for
any expected return.
The efficient frontier is the set of efficient portfolios out of the full set of potential portfolios. On
a graph, the efficient frontier constitutes the boundary line of the set of potential portfolios.
b. An indifference curve is the risk/return tradeoff function for a particular investor and reflects that
investor's attitude toward risk. The indifference curve specifies an investor's required rate of return
for a given level of risk. The greater the slope of the indifference curve, the greater is the
investor's risk aversion.
The optimal portfolio for an investor is the point at which the efficient set of portfoliosthe
efficient frontieris just tangent to the investor's indifference curve. This point marks the
highest level of satisfaction an investor can attain given the set of potential portfolios.
c. The Capital Asset Pricing Model (CAPM) is a general equilibrium market model developed to
analyze the relationship between risk and required rates of return on assets when they are held in
welldiversified portfolios. The SML is part of the CAPM.
The Capital Market Line (CML) specifies the efficient set of portfolios an investor can attain by
combining a riskfree asset and the risky market portfolio M. The CML states that the expected
return on any efficient portfolio is equal to the riskless rate plus a risk premium, and thus
describes a linear relationship between expected return and risk.
d. The characteristic line for a particular stock is obtained by regressing the historical returns on that
stock against the historical returns on the general stock market. The slope of the characteristic
line is the stock's beta, which measures the amount by which the stock's expected return increases
for a given increase in the expected return on the market.
The beta coefficient (b) is a measure of a stock's market risk. It measures the stock's volatility
relative to an average stock, which has a beta of 1.0.
e. Arbitrage Pricing Theory (APT) is an approach to measuring the equilibrium risk/return
relationship for a given stock as a function of multiple factors, rather than the single factor (the
market return) used by the CAPM. The APT is based on complex mathematical and statistical
Answers and Solutions: 5  96 theory, but can account for several factors (such as GNP and the level of inflation) in determining
the required return for a particular stock.
The FamaFrench 3factor model has one factor for the excess market return (the market return
minus the risk free rate), a second factor for size (defined as the return on a portfolio of small
firms minus the return on a portfolio of big firms), and a third factor for the booktomarket effect
(defined as the return on a portfolio of firms with a high booktomarket ratio minus the return on
a portfolio of firms with a low booktomarket ratio).
Most people don't behave rationally in all aspects of their personal lives, and behavioral finance
assume that investors have the same types of psychological behaviors in their financial lives as in
their personal lives.
52 Security A is less risky if held in a diversified portfolio because of its lower beta and negative
correlation with other stocks. In a singleasset portfolio, Security A would be more risky because A
> B and CVA > CVB.
Answers and Solutions: 5  97 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
51 a. A plot of the approximate regression line is shown in the following figure:
rX (%)
30
25
20
15
10
5
rY
0
30 20 10 0 10 20 30 40 50
5
10
15
20
Using Excel, the regression equation estimates are: Beta = 0.56; Intercept = 0.037; R2 = 0.96.
b. The arithmetic average return for Stock X is calculated as follows:
(14.0 + 23.0 + ... + 18.2)
r Avg = = 10.6%.
7
The arithmetic average rate of return on the market portfolio, determined similarly, is 12.1%.
For Stock X, the estimated standard deviation is 13.1 percent:
(14.0  10.6) 2 + (23.0  10.6) 2 + ... + (18.2  10.6) 2
X = = 13.1%.
7 1
The standard deviation of returns for the market portfolio is similarly determined to be 22.6
percent. The results are summarized below:
Answers and Solutions: 5  98 Stock X Market Portfolio
Average return, r Avg 10.6% 12.1%
Standard deviation, 13.1 22.6
Several points should be noted: (1) M over this particular period is higher than the historic
average M of about 15 percent, indicating that the stock market was relatively volatile during this
period; (2) Stock X, with X = 13.1%, has much less total risk than an average stock, with Avg =
22.6%; and (3) this example demonstrates that it is possible for a very lowrisk single stock to
have less risk than a portfolio of average stocks, since X < M.
c. Since Stock X is in equilibrium and plots on the Security Market Line (SML), and given the
further assumption that r X = r X and r M = r M and this assumption often does not holdthen
this equation must hold:
r X = rRF + (r  rRF )b X .
This equation can be solved for the riskfree rate, rRF, which is the only unknown:
10.6 = rRF + (12.1  rRF ) 0.56
10.6 = rRF + 6.8  0.56 rRF
0.44 rRF = 10.6  6.8
rRF = 3.8 / 0.44 = 8.6%.
Answers and Solutions: 5  99 d. The SML is plotted below. Data on the riskfree security (bRF = 0,
rRF = 8.6%) and Security X (bX = 0.56, r X = 10.6%) provide the two points through which the
SML can be drawn. rM provides a third point.
r(%)
k(%)
20
rX X = 10.6%
k = 10.6%
kM = 12.1%
rRF = 8.6%10
kRF = 8.6
1.0 2.0 Beta
e. In theory, you would be indifferent between the two stocks. Since they have the same beta, their
relevant risks are identical, and in equilibrium they should provide the same returns. The two
stocks would be represented by a single point on the SML. Stock Y, with the higher standard
deviation, has more diversifiable risk, but this risk will be eliminated in a welldiversified
portfolio, so the market will compensate the investor only for bearing market or relevant risk. In
practice, it is possible that Stock Y would have a slightly higher required return, but this premium
for diversifiable risk would be small.
52
Answers and Solutions: 5  100 a. The regression graph is shown above. Using a speadsheet, we find b = 0.62.
rk y(%)
S (%)
45
30
15
30 15 15 30 45 k M (%)
rM(%)
15
b. Because b = 0.62, Stock Y is about 62 percent as volatile as the market; thus, its relative risk is
about 62 percent of that of an average firm.
c. 1. Total risk ( 2 ) would be greater because the second term of the firm's risk equation,
Y
2 = b 2 2 + eY , would be greater.
Y Y M
2
2. CAPM assumes that companyspecific risk will be eliminated in a portfolio, so the
risk premium under the CAPM would not be affected.
d. 1. The stock's variance would not change, but the risk of the stock to an investor holding a
diversified portfolio would be greatly reduced.
2. It would now have a negative correlation with rM.
3. Because of a relative scarcity of such stocks and the beneficial net effect on portfolios that
include it, its "risk premium" is likely to be very low or even negative. Theoretically, it
should be negative.
iM i 53 a. ri = rRF + (rM  rRF )b i = rRF + (rM  rRF ) .
M
Answers and Solutions: 5  101
r M  rRF r r
b. CML: r p = rRF + p . SML: ri = rRF + M RF
riM i .
M M
With some arranging, the similarities between the CML and SML are obvious. When in this
form, both have the same market price of risk, or slope,(rM  rRF)/M.
The measure of risk in the CML is p. Since the CML applies only to efficient portfolios, p
not only represents the portfolio's total risk, but also its market risk. However, the SML
applies to all portfolios and individual securities. Thus, the appropriate risk measure is not i,
the total risk, but the market risk, which in this form of the SML is riMi, and is less than for all
assets except those which are perfectly positively correlated with the market, and hence have riM =
+1.0.
54 a. Using the CAPM:
ri = rRF + (rM  rRF)bi = 7% + (1.1)(6.5%) = 14.15%
b. Using the 3factor model:
ri = rRF + (rM rrf)bi + (rSMB)ci + (rHML)di
= 7% + (1.1)(6.5%) + (5%)(0.7) + (4%)(0.3) = 16.45%
Answers and Solutions: 5  102 MINI CASE
To begin, briefly review the Chapter 4 Mini Case. Then, extend your knowledge of risk and return by answering the following questions.
a. Suppose asset A has an expected return of 10 percent and a standard deviation of 20 percent.
Asset B has an expected return of 16 percent and a standard deviation of 40 percent. If the
correlation between A and B is 0.4, what are the expected return and standard deviation for
a portfolio comprised of 30 percent asset A and 70 percent asset B?
Answer:
rP = w A rA + (1  w A ) rB
^ ^ ^
= 0.3(0.1) + 0.7(0.16)
= 0.142 = 14.2%.
p = WA 2 + (1  WA ) 2 2 + 2 WA (1  WA ) AB A B
2
A B
= 0.3 2 (0.2 2 ) + 0.7 2 (0.4 2 ) + 2(0.3)(0.7)(0.4)(0.2)(0.4)
= 0.309
b. Plot the attainable portfolios for a correlation of 0.4. Now plot the attainable portfolios for
correlations of +1.0 and 1.0.
Answer:
AB = +0.4: Attainable Set of
Risk/Return Combinations
20%
Expected return
15%
10%
5%
0%
0% 10% 20% 30% 40%
Risk, p
Answers and Solutions: 6  103 AB = +1.0: Attainable Set of
Risk/Return Combinations
20%
Expected return
15%
10%
5%
0%
0% 10% 20% 30% 40%
Risk, p
AB = 1.0: Attainable Set of Risk/Return
Combinations
20% Expected return
15%
10%
5%
0%
0% 10% 20% 30% 40%
Risk, p
Answers and Solutions: 6  104 c. Suppose a riskfree asset has an expected return of 5 percent. By definition, its standard
deviation is zero, and its correlation with any other asset is also zero. Using only asset A and the
riskfree asset, plot the attainable portfolios.
Answer:
Attainable Set of Risk/Return
Combinations with RiskFree Asset
15%
Expected return
10%
5%
0%
0% 5% 10% 15% 20%
Risk, p
Answers and Solutions: 6  105 d. Construct a reasonable, but hypothetical, graph which shows risk, as measured by
portfolio standard deviation, on the x axis and expected rate of return on the y axis.
Now add an illustrative feasible (or attainable) set of portfolios, and show what
portion of the feasible set is efficient. What makes a particular portfolio efficient?
Don't worry about specific values when constructing the graphmerely illustrate
how things look with "reasonable" data.
Answer:
Expected Portfolio
Return
Expected Portfolio
^
Return, kp
^
rP
B
Efficient Set (A,B)
C
A
D
Feasible, or
Attainable, Set
E
risk, Pp
Risk,
The figure above shows the feasible set of portfolios. The points B, C, D, and E represent single
securities (or portfolios containing only one security). All the other points in the shaded area,
including its boundaries, represent portfolios of two or more securities. The shaded area is
called the feasible, or attainable, set.
The boundary AB defines the efficient set of portfolios, which is also called the efficient
frontier. Portfolios to the left of the efficient set are not possible because they lie outside the
attainable set. Portfolios to the right of the boundary line (interior portfolios) are inefficient
because some other portfolio would provide either a higher return with the same degree of risk or
a lower level of risk for the same rate of return.
e. Now add a set of indifference curves to the graph created for part B. What do these
curves represent? What is the optimal portfolio for this investor? Finally, add a
second set of indifference curves which leads to the selection of a different optimal
portfolio. Why do the two investors choose different portfolios?
Answers and Solutions: 6  106 Expected Portfolio
Return,
Expected Portfolio
^^
Return, kp
r p
B
I B2
C
I B1
Optimal
Portfolio
I A3 Investor B
I A2
A
D
I A1
Optimal
Portfolio
Investor A
E
risk, P
Risk, p
Answer:
The figure above shows the indifference curves for two hypothetical investors, A and B. To
determine the optimal portfolio for a particular investor, we must know the investor's attitude
towards risk as reflected in his or her risk/return tradeoff function, or indifference curve. Curves
Ia1, Ia2, and Ia3 represent the indifference curves for individual A, with the higher curve (Ia3)
denoting a greater level of satisfaction (or utility). Thus, Ia3 is better than Ia2 for any level of risk.
The optimal portfolio is found at the tangency point between the efficient set of portfolios and
one of the investor's indifference curves. This tangency point marks the highest level of
satisfaction the investor can attain. The arrows point toward the optimal portfolios for both
investors A and B.
The investors choose different optimal portfolios because their risk aversion is different.
Investor A chooses the portfolio with the lower expected return, but the riskiness of that portfolio
is also lower than investor's B optimal portfolio, because investor a is more risk averse.
f. What is the capital asset pricing model (CAPM)? What are the assumptions that underlie
the model?
Answer: The Capital Asset Pricing Model (CAPM) is an equilibrium model which specifies the
relationship between risk and required rates of return on assets when they are held in
welldiversified portfolios. The CAPM requires an extensive set of assumptions:
· All investors are singleperiod expected utility of terminal wealth maximizers, who
choose among alternative portfolios on the basis of each portfolio's expected
return and standard deviation.
Answers and Solutions: 6  107 · All investors can borrow or lend an unlimited amount at a given riskfree rate of
interest.
· Investors have homogeneous expectations (that is, investors have identical
estimates of the expected values, variances, and covariances of returns among all
assets).
· All assets are perfectly divisible and perfectly marketable at the going price, and
there are no transactions costs.
· There are no taxes.
· All investors are price takers (that is, all investors assume that their own buying and
selling activity will not affect stock prices).
· The quantities of all assets are given and fixed.
g. Now add the riskfree asset. What impact does this have on the efficient frontier?
Answer: The riskfree asset by definition has zero risk, and hence = 0%, so it is plotted on the vertical
axis. Now, given the possibility of investing in the riskfree asset, investors can create new
portfolios that combine the riskfree asset with a portfolio of risky assets. This enables them to
achieve any combination of risk and return that lies along any straight line connecting rRF with
any portfolio in the feasible set of risky portfolios. However, the straight line connecting rRF
with m, the point of tangency between the line and the portfolio's efficient set curve, is the one
that all investors would choose. Since all portfolios on the line rRFmz are preferred to the other
risky portfolio opportunities on the efficient frontier AB, the points on the line rRFmz now
represent the best attainable combinations of risk and return. Any combination under the rRFmz
line offers less return for the same amount of risk, or offers more risk for the same amount of
return. Thus, everybody wants to hold portfolios which are located on the rRFmz line.
h. Write out the equation for the capital market line (CML) and draw it on the graph.
Interpret the CML. Now add a set of indifference curves, and illustrate how an investor's
optimal portfolio is some combination of the risky portfolio and the riskfree asset. What is
the composition of the risky portfolio?
Answers and Solutions: 6  108 Expected Portfolio
Return, Portfolio
Expected
^
Return, k p^ Z
rp
B
M
A
krRF
RF
risk, Pp
Risk,
Answer: The line rRFmz in the figure above is called the capital market line (CML). It has an intercept of
rRF and a slope of ( r M  rRF ) / M . Therefore the equation for the capital market line may be
expressed as follows:
^
r M  rRF
CML: r p = rRF + p .
M
The CML tells us that the expected rate of return on any efficient portfolio (that is, any portfolio
on the CML) is equal to the riskfree rate plus a risk premium, and the risk premium is equal to
( r M  rRF ) / M multiplied by the portfolio's standard deviation, p . Thus, the CML specifies a
linear relationship between expected return and risk, with the slope of the CML being equal to the
expected return on the market portfolio of risky stocks, r M , minus the riskfree rate, rRF, which is
called the market risk premium, all divided by the standard deviation of returns on the market
portfolio, m.
Answers and Solutions: 6  109 Expected Rate
of Return,
Expected Rate
^
^
r
of Return,pk p
I I CML
3 2 I1
rRFRF
k
Optimal
Portfolio
Risk, p
The figure above shows a set of indifference curves (i1, i2, and i3), with i1 touching the CML.
This point of tangency defines the optimal portfolio for this investor, and he or she will buy a
combination of the market portfolio and the riskfree asset.
The risky portfolio, m, must contain every asset in exact proportion to that asset's fraction of
the total market value of all assets; that is, if security g is x percent of the total market value of all
securities, x percent of the market portfolio must consist of security g.
i. What is a characteristic line? How is this line used to estimate a stock's beta coefficient?
Write out and explain the formula that relates total risk, market risk, and diversifiable risk.
Answer: Betas are calculated as the slope of the characteristic line, which is the regression line formed by
plotting returns on a given stock on the y axis against returns on the general stock market on the x
Answers and Solutions: 6  110 axis. In practice, 5 years of monthly data, with 60 observations, would be used, and a computer
would be used to obtain a least squares regression line.
The relationship between stock J's total risk, market risk, and diversifiable risk can be
expressed as follows:
TOTAL RISK = VARIANCE = MARKET RISK + DIVERSIFIABLE RISK
2 =
J b 22
J M + 2
eJ
Here J is the variance or total risk of stock j, 2 is the variance of the market, bj is stock J's
2
M
beta coefficient, and 2 is the variance of stock J's regression error term. If stock J is held in
eJ
isolation, then the investor must bear its total risk. However, when stock J is held as part of a
welldiversified portfolio, the regression error term, 2 is driven to zero; hence, only the market
eJ
risk remains.
j. What are two potential tests that can be conducted to verify the CAPM? What are the
results of such tests? What is roll's critique of CAPM tests?
Answer: Since the CAPM was developed on the basis of a set of unrealistic assumptions, empirical tests
should be used to verify the CAPM. The first test looks for stability in historical betas. If betas
have been stable in the past for a particular stock, then its historical beta would probably be a
good proxy for its exante, or expected beta. Empirical work concludes that the betas of individual
securities are not good estimators of their future risk, but that betas of portfolios of ten or more
randomly selected stocks are reasonably stable, hence that past portfolio betas are good estimators
of future portfolio volatility.
The second type of test is based on the slope of the SML. As we have seen, the CAPM states
that a linear relationship exists between a security's required rate of return and its beta. Further,
when the SML is graphed, the vertical axis intercept should be rRF, and the required rate of return
for a stock (or portfolio) with beta = 1.0 should be rm, the required rate of return on the market.
Various researchers have attempted to test the validity of the CAPM model by calculating betas
and realized rates of return, plotting these values in graphs, and then observing whether or not (1)
the intercept is equal to rRF, (2) the regression line is linear, and (3) the SML passes through the
point b = 1.0, rm. Evidence shows a moreorless linear relationship between realized returns and
market risk, but the slope is less than predicted. Tests that attempt to assess the relative
importance of market and companyspecific risk do not yield definitive results, so the irrelevance
of diversifiable risk specified in the CAPM model can be questioned.
Roll questioned whether it is even conceptually possible to test the CAPM. Roll showed
that the linear relationship which prior researchers had observed in graphs resulted from the
mathematical properties of the models being tested, hence that a finding of linearity proved
nothing about the validity of the CAPM. Roll's work did not disprove the CAPM theory, but he
did show that it is virtually impossible to prove that investors behave in accordance with the
theory.
In general, evidence seems to support the CAPM model when it is applied to portfolios, but
the evidence is less convincing when the CAPM is applied to individual stocks.
Nevertheless, the CAPM provides a rational way to think about risk and return as long as one
recognizes the limitations of the CAPM when using it in practice.
k. Briefly explain the difference between the CAPM and the arbitrage pricing theory
Answers and Solutions: 6  111 (APT).
Answer: The CAPM is a singlefactor model, while the Arbitrage Pricing Theory (APT) can include any
number of risk factors. It is likely that the required return is dependent on many fundamental
factors such as the GNP growth, expected inflation, and changes in tax laws, and that different
groups of stocks are affected differently by these factors. Thus, the apt seems to have a stronger
theoretical footing than does the CAPM. However, the apt faces several major hurdles in
implementation, the most severe being that the apt does not identify the relevant factorsa
complex mathematical procedure called factor analysis must be used to identify the factors. To
date, it appears that only three or four factors are required in the apt, but much more research is
required before the apt is fully understood and presents a true challenge to the CAPM.
l. Suppose you are given the following information. The beta of company, bi, is 0.9, the risk
free rate, rRF, is 6.8%, and the expected market premium, rMrRF, is 6.3%. Because your
company is larger than average and more successful than average (i.e., it has a lower
booktomarket ratio), you think the FamaFrench 3factor model might be more
appropriate than the CAPM. You estimate the additional coefficients from the
FamaFrench 3factor model: the coefficient for the size effect, Ci, is 0.5, and the coefficient
for the booktomarket effect, di, is 0.3. If the expected value of the size factor is 4% and
the expected value of the booktomarket factor is 5%, what is the required return using the
FamaFrench 3factor model? (assume that Ai = 0.0.) What is the required return using
CAPM?
Answer: The FamaFrench model:
ri = rRF + (rm  rRF)bi + (rsmb)ci + (rhmb)dj
ri = 6.8% + (6.3%)(0.9) + (4%)(0.5) + (5%)(0.3)
= 8.97%
The CAPM:
ri = rrf + (rm  rrf)bi
ri = 6.8% + (6.3%)(0.9)
= 12.47%
Answers and Solutions: 6  112 Chapter 6
Bonds and Their Valuation
ANSWERS TO ENDOFCHAPTER QUESTIONS
61 a. A bond is a promissory note issued by a business or a governmental unit. Treasury bonds,
sometimes referred to as government bonds, are issued by the Federal government and are not
exposed to default risk. Corporate bonds are issued by corporations and are exposed to default
risk. Different corporate bonds have different levels of default risk, depending on the issuing
company's characteristics and on the terms of the specific bond. Municipal bonds are issued by
state and local governments. The interest earned on most municipal bonds is exempt from
federal taxes, and also from state taxes if the holder is a resident of the issuing state. Foreign
bonds are issued by foreign governments or foreign corporations. These bonds are not only
exposed to default risk, but are also exposed to an additional risk if the bonds are denominated in
a currency other than that of the investor's home currency.
b. The par value is the nominal or face value of a stock or bond. The par value of a bond generally
represents the amount of money that the firm borrows and promises to repay at some future date.
The par value of a bond is often $1,000, but can be $5,000 or more. The maturity date is the date
when the bond's par value is repaid to the bondholder. Maturity dates generally range from 10 to
40 years from the time of issue. A call provision may be written into a bond contract, giving the
issuer the right to redeem the bonds under specific conditions prior to the normal maturity date.
A bond's coupon, or coupon payment, is the dollar amount of interest paid to each bondholder on
the interest payment dates. The coupon is so named because bonds used to have dated coupons
attached to them which investors could tear off and redeem on the interest payment dates. The
coupon interest rate is the stated rate of interest on a bond.
c. In some cases, a bond's coupon payment may vary over time. These bonds are called floating
rate bonds. Floating rate debt is popular with investors because the market value of the debt is
stabilized. It is advantageous to corporations because firms can issue longterm debt without
committing themselves to paying a historically high interest rate for the entire life of the loan.
Zero coupon bonds pay no coupons at all, but are offered at a substantial discount below their par
values and hence provide capital appreciation rather than interest income. In general, any bond
originally offered at a price significantly below its par value is called an original issue discount
bond (OID).
d. Most bonds contain a call provision, which gives the issuing corporation the right to call the
bonds for redemption. The call provision generally states that if the bonds are called, the
company must pay the bondholders an amount greater than the par value, a call premium.
Redeemable bonds give investors the right to sell the bonds back to the corporation at a price that
is usually close to the par value. If interest rates rise, investors can redeem the bonds and
reinvest at the higher rates. A sinking fund provision facilitates the orderly retirement of a bond
issue. This can be achieved in one of two ways: The company can call in for redemption (at par
value) a certain percentage of bonds each year. The company may buy the required amount of
bonds on the open market.
e. Convertible bonds are securities that are convertible into shares of common stock, at a fixed price,
at the option of the bondholder. Bonds issued with warrants are similar to convertibles.
Warrants are options which permit the holder to buy stock for a stated price, thereby providing a
Answers and Solutions: 6  113 capital gain if the stock price rises. Income bonds pay interest only if the interest is earned.
These securities cannot bankrupt a company, but from an investor's standpoint they are riskier
than "regular" bonds. The interest rate of an indexed, or purchasing power, bond is based on an
inflation index such as the consumer price index (CPI), so the interest paid rises automatically
when the inflation rate rises, thus protecting the bondholders against inflation.
f. Bond prices and interest rates are inversely related; that is, they tend to move in the opposite
direction from one another. A fixedrate bond will sell at par when its coupon interest rate is
equal to the going rate of interest, rd. When the going rate of interest is above the coupon rate, a
fixedrate bond will sell at a "discount" below its par value. If current interest rates are below the
coupon rate, a fixedrate bond will sell at a "premium" above its par value.
g. The current yield on a bond is the annual coupon payment divided by the current market price.
YTM, or yield to maturity, is the rate of interest earned on a bond if it is held to maturity. Yield
to call (YTC) is the rate of interest earned on a bond if it is called. If current interest rates are
well below an outstanding callable bond's coupon rate, the YTC may be a more relevant estimate
of expected return than the YTM, since the bond is likely to be called.
h. The shorter the maturity of the bond, the greater the risk of a decrease in interest rates. The risk
of a decline in income due to a drop in interest rates is called reinvestment rate risk. Interest
rates fluctuate over time, and people or firms who invest in bonds are exposed to risk from
changing interest rates, or interest rate risk. The longer the maturity of the bond, the greater the
exposure to interest rate risk. Interest rate risk relates to the value of the bonds in a portfolio,
while reinvestment rate risk relates to the income the portfolio produces. No fixedrate bond can
be considered totally riskless. Bond portfolio managers try to balance these two risks, but some
risk always exists in any bond. Another important risk associated with bonds is default risk. If
the issuer defaults, investors receive less than the promised return on the bond. Default risk is
influenced by both the financial strength of the issuer and the terms of the bond contract,
especially whether collateral has been pledged to secure the bond. The greater the default risk,
the higher the bond's yield to maturity.
i. Corporations can influence the default risk of their bonds by changing the type of bonds they
issue. Under a mortgage bond, the corporation pledges certain assets as security for the bond.
All such bonds are written subject to an indenture, which is a legal document that spells out in
detail the rights of both the bondholders and the corporation. A debenture is an unsecured
bond, and as such, it provides no lien against specific property as security for the obligation.
Debenture holders are, therefore, general creditors whose claims are protected by property not
otherwise pledged. Subordinated debentures have claims on assets, in the event of bankruptcy,
only after senior debt as named in the subordinated debt's indenture has been paid off.
Subordinated debentures may be subordinated to designated notes payable or to all other debt.
j. A development bond is a taxexempt bond sold by state and local governments whose proceeds
are made available to corporations for specific uses deemed (by Congress) to be in the public
interest. Municipalities can insure their bonds, in which an insurance company guarantees to pay
the coupon and principal payments should the issuer default. This reduces the risk to investors
who are willing to accept a lower coupon rate for an insured bond issue visavis an uninsured
issue. Bond issues are normally assigned quality ratings by major rating agencies, such as
Moody's Investors Service and Standard & Poor's Corporation. These ratings reflect the
probability that a bond will go into default. Aaa (Moody's) and AAA (S&P) are the highest
ratings. Rating assignments are based on qualitative and quantitative factors including the firm's
debt/assets ratio, current ratio, and coverage ratios. Because a bond's rating is an indicator of its
Answers and Solutions: 6  114 default risk, the rating has a direct, measurable influence on the bond's interest rate and the firm's
cost of debt capital. Junk bonds are highrisk, highyield bonds issued to finance leveraged
buyouts, mergers, or troubled companies. Most bonds are purchased by institutional investors
rather than individuals, and many institutions are restricted to investment grade bonds, securities
with ratings of Baa/BBB or above.
62 False. Shortterm bond prices are less sensitive than longterm bond prices to interest rate changes
because funds invested in shortterm bonds can be reinvested at the new interest rate sooner than
funds tied up in longterm bonds.
63 The price of the bond will fall and its YTM will rise if interest rates rise. If the bond still has a long
term to maturity, its YTM will reflect longterm rates. Of course, the bond's price will be less
affected by a change in interest rates if it has been outstanding a long time and matures shortly.
While this is true, it should be noted that the YTM will increase only for buyers who purchase the
bond after the change in interest rates and not for buyers who purchased previous to the change. If
the bond is purchased and held to maturity, the bondholder's YTM will not change, regardless of what
happens to interest rates.
64 If interest rates decline significantly, the values of callable bonds will not rise by as much as those of
bonds without the call provision. It is likely that the bonds would be called by the issuer before
maturity, so that the issuer can take advantage of the new, lower rates.
65 From the corporation's viewpoint, one important factor in establishing a sinking fund is that its own
bonds generally have a higher yield than do government bonds; hence, the company saves more
interest by retiring its own bonds than it could earn by buying government bonds. This factor causes
firms to favor the second procedure. Investors also would prefer the annual retirement procedure if
they thought that interest rates were more likely to rise than to fall, but they would prefer the
government bond purchases program if they thought rates were likely to fall. In addition,
bondholders recognize that, under the government bond purchase scheme, each bondholder would be
entitled to a given amount of cash from the liquidation of the sinking fund if the firm should go into
default, whereas under the annual retirement plan, some of the holders would receive a cash benefit
while others would benefit only indirectly from the fact that there would be fewer bonds outstanding.
On balance, investors seem to have little reason for choosing one method over the other, while
the annual retirement method is clearly more beneficial to the firm. The consequence has been a
pronounced trend toward annual retirement and away from the accumulation scheme.
Answers and Solutions: 6  115 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
61 With your financial calculator, enter the following:
N = 10; I = YTM = 9%; PMT = 0.08 × 1,000 = 80; FV = 1000; PV = VB = ?
PV = $935.82.
Alternatively,
VB = $80(PVIFA9%,10) + $1,000(PVIF9%,10)
= $80((1 1/1.0910)/0.09) + $1,000(1/1.0910)
= $80(6.4177) + $1,000(0.4224)
= $513.42 + $422.40 = $935.82.
62 With your financial calculator, enter the following:
N = 12; PV = 850; PMT = 0.10 × 1,000 = 100; FV = 1000; I = YTM = ?
YTM = 12.48%.
63 With your financial calculator, enter the following to find YTM:
N = 10 × 2 = 20; PV = 1100; PMT = 0.08/2 × 1,000 = 40; FV = 1000; I = YTM = ?
YTM = 3.31% × 2 = 6.62%.
With your financial calculator, enter the following to find YTC:
N = 5 × 2 = 10; PV = 1100; PMT = 0.08/2 × 1,000 = 40; FV = 1050; I = YTC = ?
YTC = 3.24% × 2 = 6.49%.
64 With your financial calculator, enter the following to find the current value of the bonds, so you can
then calculate their current yield:
N = 7; I = YTM = 8; PMT = 0.09 × 1,000 = 90; FV = 1000; PV = VB = ?
PV = $1,052.06. Current yield = $90/$1,052.06 = 8.55%.
Alternatively,
VB = $90(PVIFA8%,7) + $1,000(PVIF8%,7)
= $90((1 1/1.087)/0.08) + $1,000(1/1.087)
= $90(5.2064) + $1,000(0.5835)
= $468.58 + $583.50 = $1,052.08.
Current yield = $90/$1,052.08 = 8.55%.
Answers and Solutions: 6  116 65 The problem asks you to find the price of a bond, given the following facts:
N = 16; I = 8.5/2 = 4.25; PMT = 45; FV = 1000.
With a financial calculator, solve for PV = $1,028.60
66 a. VB = PMT(PVIFAi,n) + FV(PVIFi,n)
= PMT((1 1/(1+in))/i) + FV(1/(1+i)n)
1. 5%: Bond L: VB = $100(10.3797) + $1,000(0.4810) = $1,518.97.
Bond S: VB = ($100 + $1,000)(0.9524) = $1,047.64.
2. 8%: Bond L: VB = $100(8.5595) + $1,000(0.3152) = $1,171.15.
Bond S: VB = ($100 + $1,000)(0.9259) = $1,018.49.
3. 12%: Bond L: VB = $100(6.8109) + $1,000(0.1827) = $863.79.
Bond S: VB = ($100 + $1,000)(0.8929) = $982.19.
Calculator solutions:
1. 5%: Bond L: Input N = 15, I = 5, PMT = 100, FV = 1000, PV = ?, PV = $1,518.98.
Bond S: Change N = 1, PV = ? PV = $1,047.62.
2. 8%: Bond L: From Bond S inputs, change N = 15 and I = 8, PV = ?, PV = $1,171.19.
Bond S: Change N = 1, PV = ? PV = $1,018.52.
3. 12%: Bond L: From Bond S inputs, change N = 15 and I = 12, PV = ? PV = $863.78.
Bond S: Change N = 1, PV = ? PV = $982.14.
b. Think about a bond that matures in one month. Its present value is influenced primarily by the
maturity value, which will be received in only one month. Even if interest rates double, the price
of the bond will still be close to $1,000. A oneyear bond's value would fluctuate more than the
onemonth bond's value because of the difference in the timing of receipts. However, its value
would still be fairly close to $1,000 even if interest rates doubled. A longterm bond paying
semiannual coupons, on the other hand, will be dominated by distant receipts, receipts which are
multiplied by 1/(1 + rd/2)t, and if rd increases, these multipliers will decrease significantly.
Another way to view this problem is from an opportunity point of view. A onemonth bond can
be reinvested at the new rate very quickly, and hence the opportunity to invest at this new rate is
not lost; however, the longterm bond locks in subnormal returns for a long period of time.
Answers and Solutions: 6  117 N
INT M 67 a. VB = (1 + r
t =1 d)
t
+
(1 + r d ) N
= PMT((1 1/(1+rdn))/rd) + FV(1/(1+rd)n).
M = $1,000. INT = 0.09($1,000) = $90.
1. $829= $90((1 1/(1+rd4))/rd) + $1,000(1/(1+rd)4).
The YTM can be found by trialanderror. If the YTM was 9 percent, the bond value
would be its maturity value. Since the bond sells at a discount, the YTM must
be greater than 9 percent. Let's try 10 percent.
At 10%, VB = $285.29 + $683.00
= $968.29.
$968.29 > $829.00; therefore, the bond's YTM is greater than 10 percent.
Try 15 percent.
At 15%, VB = $256.95 + $571.80
= $828.75.
Therefore, the bond's YTM is approximately 15 percent.
2. $1,104 = $90((1 1/(1+rd4))/rd) + $1,000(1/(1+rd)4).
The bond is selling at a premium; therefore, the YTM must be below 9 percent. Try 6
percent.
At 6%, VB = $311.86 + $792.10
= $1,103.96.
Therefore, when the bond is selling for $1,104, its YTM is approximately 6 percent.
Calculator solution:
1. Input N = 4, PV = 829, PMT = 90, FV = 1000, I = ? I = 14.99%.
2. Change PV = 1104, I = ? I = 6.00%.
b. Yes. At a price of $829, the yield to maturity, 15 percent, is greater than your required rate of
return of 12 percent. If your required rate of return were 12 percent, you should be willing to
buy the bond at any price below $908.88.
Answers and Solutions: 6  118 68 $1,000 = $140((1 1/(1+rd6))/rd) + $1,090(1/(1+rd)6).
Try 18 percent:
PV18% = $140(3.4976) + $1,090(0.3704) = $489.66 + $403.74 = $893.40.
18 percent is too high.
Try 15 percent:
PV15% = $140(3.7845) + $1,090(0.4323) = $529.83 + $471.21 = $1,001.04.
15 percent is slightly low.
The rate of return is approximately 15.03 percent, found with a calculator using the following inputs.
N = 6; PV = 1000; PMT = 140; FV = 1090; I = ? Solve for I = 15.03%.
69 a. Using a financial calculator, input the following:
N = 20, PV = 1100, PMT = 60, FV = 1000, and solve for I = 5.1849%.
However, this is a periodic rate. The nominal annual rate = 5.1849%(2) = 10.3699% 10.37%.
b. The current yield = $120/$1,100 = 10.91%.
c. YTM = Current Yield + Capital Gains (Loss) Yield
10.37% = 10.91% + Capital Loss Yield
0.54% = Capital Loss Yield.
d. Using a financial calculator, input the following:
N = 8, PV = 1100, PMT = 60, FV = 1060, and solve for I = 5.0748%.
However, this is a periodic rate. The nominal annual rate = 5.0748%(2) = 10.1495% 10.15%.
Answers and Solutions: 6  119 610 The problem asks you to solve for the YTM, given the following facts:
N = 5, PMT = 80, and FV = 1000. In order to solve for I we need PV.
However, you are also given that the current yield is equal to 8.21%. Given this information, we can
find PV.
Current yield = Annual interest/Current price
0.0821 = $80/PV
PV = $80/0.0821 = $974.42.
Now, solve for the YTM with a financial calculator:
N = 5, PV = 974.42, PMT = 80, and FV = 1000. Solve for I = YTM = 8.65%.
611 The problem asks you to solve for the current yield, given the following facts: N = 14, I = 10.5883/2
= 5.2942, PV = 1020, and FV = 1000. In order to solve for the current yield we need to find PMT.
With a financial calculator, we find PMT = $55.00. However, because the bond is a semiannual
coupon bond this amount needs to be multiplied by 2 to obtain the annual interest payment:
$55.00(2) = $110.00. Finally, find the current yield as follows:
Current yield = Annual interest/Current Price = $110/$1,020 = 10.78%.
612 The bond is selling at a large premium, which means that its coupon rate is much higher than the
going rate of interest. Therefore, the bond is likely to be calledit is more likely to be called than to
remain outstanding until it matures. Thus, it will probably provide a return equal to the YTC rather
than the YTM. So, there is no point in calculating the YTMjust calculate the YTC. Enter these
values:
N = 10, PV = 1353.54, PMT = 70, FV = 1050, and then solve for I.
The periodic rate is 3.24 percent, so the nominal YTC is 2 x 3.24% = 6.47%. This would be close to
the going rate, and it is about what the firm would have to pay on new bonds.
Answers and Solutions: 6  120 613 a. The bonds now have an 8year, or a 16semiannual period, maturity, and their value is calculated
as follows:
16
$50 $1,000
VB = (1.03 )
t =1
t
+
(1.03 )16
= $50(12.5611) + $1,000(0.6232)
= $628.06 + $623.20 = $1,251.26.
Calculator solution: Input N = 16, I = 3, PMT = 50, FV = 1000,
PV = ? PV = $1,251.22.
b. VB = $50(10.1059) + $1,000(0.3936) = $505.30 + $393.60 = $898.90.
Calculator solution: Change inputs from Part a to I = 6, PV = ?
PV = $898.94.
c. The price of the bond will decline toward $1,000, hitting $1,000 (plus accrued interest) at the
maturity date 8 years (16 sixmonth periods) hence.
614
Price at 8% Price at 7% Pctge. change
10year, 10% annual coupon $1,134.20 $1,210.71 6.75%
10year zero 463.19 508.35 9.75
5year zero 680.58 712.99 4.76
30year zero 99.38 131.37 32.19
$100 perpetuity 1,250.00 1,428.57 14.29
Answers and Solutions: 6  121 615 a.
t Price of Bond C Price of Bond Z
0 $1,012.79 $ 693.04
1 1,010.02 759.57
2 1,006.98 832.49
3 1,003.65 912.41
4 1,000.00 1,000.00
b.
Bond Value Time Path
($)
1,100
Bond C
1,000 · · · ·
900 ·
Bond Z
·
800
·
700
Years
0 1 2 3 4
Answers and Solutions: 6  122 SOLUTION TO SPREADSHEET PROBLEM
616 The detailed solution for the problem is available both on the instructor's resource CDROM (in the
file Solution for FM11 Ch 06 P16 Build a Model.xls) and on the instructor's side of the book's web
site, brigham.swcollege.com.
Answers and Solutions: 6  123 Answers and Solutions: 7  124 MINI CASE
Sam Strother and Shawna Tibbs are vicepresidents of Mutual of Seattle Insurance Company and codirectors of the company's pension fund management division. A major new client, the Northwestern Municipal Alliance, has requested that Mutual of Seattle present an investment seminar to the mayors of the represented cities, and Strother and Tibbs, who will make the actual presentation, have asked you to help them by answering the following questions. Because the Boeing Company operates in one of the league's cities, you are to work Boeing into the presentation.
a. What are the key features of a bond?
Answer:
1. Par or face value. We generally assume a $1,000 par value, but par can be anything, and
often $5,000 or more is used. With registered bonds, which is what are issued today, if you
bought $50,000 worth, that amount would appear on the certificate.
2. Coupon rate. The dollar coupon is the "rent" on the money borrowed, which is generally the
par value of the bond. The coupon rate is the annual interest payment divided by the par
value, and it is generally set at the value of r on the day the bond is issued.
3. Maturity. This is the number of years until the bond matures and the issuer must repay the
loan (return the par value).
4. Issue date. This is the date the bonds were issued.
5. Default risk is inherent in all bonds except treasury bondswill the issuer have the cash to
make the promised payments? Bonds are rated from AAA to D, and the lower the rating the
riskier the bond, the higher its default risk premium, and, consequently, the higher its required
rate of return, r.
Answers and Solutions: 7  125 b. What are call provisions and sinking fund provisions? Do these provisions make bonds
more or less risky?
Answer: A call provision is a provision in a bond contract that gives the issuing corporation the right to
redeem the bonds under specified terms prior to the normal maturity date. The call provision
generally states that the company must pay the bondholders an amount greater than the par value
if they are called. The additional sum, which is called a call premium, is typically set equal to
one year's interest if the bonds are called during the first year, and the premium declines at a
constant rate of INT/n each year thereafter.
A sinking fund provision is a provision in a bond contract that requires the issuer to retire a
portion of the bond issue each year. A sinking fund provision facilitates the orderly retirement of
the bond issue.
The call privilege is valuable to the firm but potentially detrimental to the investor, especially
if the bonds were issued in a period when interest rates were cyclically high. Therefore, bonds
with a call provision are riskier than those without a call provision. Accordingly, the interest rate
on a new issue of callable bonds will exceed that on a new issue of noncallable bonds.
Although sinking funds are designed to protect bondholders by ensuring that an issue is
retired in an orderly fashion, it must be recognized that sinking funds will at times work to the
detriment of bondholders. On balance, however, bonds that provide for a sinking fund are
regarded as being safer than those without such a provision, so at the time they are issued sinking
fund bonds have lower coupon rates than otherwise similar bonds without sinking funds.
Answers and Solutions: 7  126 c. How is the value of any asset whose value is based on expected future cash flows
determined?
Answer: 0 1 2 3 n
    ··· 
CF1 CF2 CF3 CFn
PV CF1
PV CF2
The value of an asset is merely the present value of its expected future cash flows:
n
VALUE = PV =
CF1
1
+
CF2
(1 + r ) (1 + r ) 2
+
CF3
(1 + r ) 3
+ ... +
CFn
(1 + r ) n
= CFt
t = 1 (1 + r )
t
.
If the cash flows have widely varying risk, or if the yield curve is not horizontal, which signifies
that interest rates are expected to change over the life of the cash flows, it would be logical for
each period's cash flow to have a different discount rate. However, it is very difficult to make
such adjustments; hence it is common practice to use a single discount rate for all cash flows.
The discount rate is the opportunity cost of capital; that is, it is the rate of return that could be
obtained on alternative investments of similar risk. Thus, the discount rate depends primarily on
factors discussed back in chapter 1:
ri = r* + IP + LP + MRP + DRP.
Answers and Solutions: 7  127 d. How is the value of a bond determined? What is the value of a 10year, $1,000 par
value bond with a 10 percent annual coupon if its required rate of return is 10
percent?
Answer: A bond has a specific cash flow pattern consisting of a stream of constant interest payments plus
the return of par at maturity. The annual coupon payment is the cash flow: pmt = (coupon rate)
× (par value) = 0.1($1,000) = $100.
For a 10year, 10 percent annual coupon bond, the bond's value is found as follows:
0 1 2 3 9
10%
10
    ···  
100 100 100 100 100
90.91 + 1,000
82.64
.
.
.
38.55
385.54
1,000.00
Expressed as an equation, we have:
$100 $100 $1,000
VB = 1
+ ... + 10
+
(1 + r ) (1 + r ) (1 + r )10
= $90.91 + . . . + $38.55 + $385.54 = $1,000.
or:
VB = $100(PVIFA10%,10) + $1,000(PVIF10%,10)
= $100 ((11/(1+.1)10)/0.10)+ $1,000 (1/(1+0.10)10).
The bond consists of a 10year, 10% annuity of $100 per year plus a $1,000 lump sum payment at
t = 10:
PV Annuity = $ 614.46
PV Maturity Value = 385.54
Value Of Bond = $1,000.00
The mathematics of bond valuation is programmed into financial calculators which do the
operation in one step, so the easy way to solve bond valuation problems is with a financial
calculator. Input n = 10, rd = i = 10, PMT = 100, and FV = 1000, and then press PV to find the
bond's value, $1,000. Then change n from 10 to 1 and press PV to get the value of the 1year
bond, which is also $1,000.
Answers and Solutions: 7  128 e. 1. What would be the value of the bond described in part d if, just after it had been issued, the
expected inflation rate rose by 3 percentage points, causing investors to require a 13 percent
return? Would we now have a discount or a premium bond?
Answer: with a financial calculator, just change the value of r = i from 10% to 13%, and press the PV
button to determine the value of the bond:
10year = $837.21.
Using the formulas, we would have, at r = 13 percent,
VB(10YR) = $100(PVIFA13%,10) + $1,000(PVIF13%,10)
= $100 ((1 1/(1+0.13)10)/0.13) + $1,000 (1/(1+0.13)10)
= $542.62 + $294.59 = $837.21.
In a situation like this, where the required rate of return, r, rises above the coupon rate, the bonds'
values fall below par, so they sell at a discount.
e. 2. What would happen to the bonds' value if inflation fell, and rd declined to 7 percent?
Would we now have a premium or a discount bond?
Answer: In the second situation, where r falls to 7 percent, the price of the bond rises above par. Just
change r from 13% to 7%. We see that the 10year bond's value rises to $1,210.71.
With tables, we have:
VB(10YR) = $100(PVIFA7%,10) + $1,000(PVIF7%,10)
= $100 ((1 1/(1+0.07)10)/0.07) + $1,000 (1/(1+0.07)10)
= $702.36 + $508.35 = $1,210.71.
Thus, when the required rate of return falls below the coupon rate, the bonds' value rises above
par, or to a premium. Further, the longer the maturity, the greater the price effect of any given
interest rate change.
Answers and Solutions: 7  129 e. 3. What would happen to the value of the 10year bond over time if the required rate of
return remained at 13 percent, or if it remained at
7 percent? (Hint: with a financial calculator, enter PMT, I, FV, and N, and then
change (override) n to see what happens to the PV as the bond approaches maturity.)
Answer: Assuming that interest rates remain at the new levels (either 7% or 13%), we could find the bond's
value as time passes, and as the maturity date approaches. If we then plotted the data, we would
find the situation shown below:
Bond Value ($)
rd = 7%.
rd = 10%. M
837
rd = 13%.
775
30 25 20 15 10 5 0
Years remaining to Maturity
At maturity, the value of any bond must equal its par value (plus accrued interest).
Therefore, if interest rates, hence the required rate of return, remain constant over time, then a
bond's value must move toward its par value as the maturity date approaches, so the value of a
premium bond decreases to $1,000, and the value of a discount bond increases to $1,000 (barring
default).
Answers and Solutions: 7  130 f. 1. What is the yield to maturity on a 10year, 9 percent annual coupon, $1,000 par value bond
that sells for $887.00? That sells for $1,134.20? What does the fact that a bond sells at a
discount or at a premium tell you about the relationship between rd and the bond's coupon
rate?
Answer: The yield to maturity (YTM) is that discount rate which equates the present value of a bond's cash
flows to its price. In other words, it is the promised rate of return on the bond. (Note that the
expected rate of return is less than the YTM if some probability of default exists.) On a time line,
we have the following situation when the bond sells for $887:
0 1 9
10
  ···  
90 90 90
PV1 1,000
.
. r=?
PV1
PVM
SUM = PV = 887
We want to find r in this equation:
INT INT M
V B = PV = 1
+ ... + N
+ .
(1 + r ) (1 + r ) (1 + r ) N
We know n = 10, PV = 887, pmt = 90, and FV = 1000, so we have an equation with one unknown,
r. We can solve for r by entering the known data into a financial calculator and then pressing the
I = r button. The YTM is found to be 10.91%.
Alternatively, we could use present value interest factors:
$887 = $90(PVIFAr,10) + $1,000(PVIFr,10)
= $90 ((1 1/(1+r)10)/r) + $1,000 (1/(1+r)10)
.
We would substitute for various interest rates, in a trialanderror manner, until we found the rate
that produces the equality. This is tiresome, and the procedure will not give an exact answer
unless the YTM is a whole number. Consequently, in the real world everyone uses financial
calculators.
We can tell from the bond's price, even before we begin the calculations, that the YTM must be
above the 9% coupon rate. We know this because the bond is selling at a discount, and discount
bonds always have r > coupon rate.
If the bond were priced at $1,134.20, then it would be selling at a premium. In that case, it
must have a YTM that is below the 9 percent coupon rate, because all premium bonds must have
coupons which exceed the going interest rate. Going through the same procedures as
beforeplugging the appropriate values into a financial calculator and then pressing the r = I
button, we find that at a price of $1,134.20, r = YTM = 7.08%.
f. 2. What are the total return, the current yield, and the capital gains yield for the discount
bond? (Assume the bond is held to maturity and the company does not default on
Answers and Solutions: 7  131 the bond.)
Answer: The current yield is defined as follows:
Annual coupon interest payment
Current Yield = .
Current price of the bond
The capital gains yield is defined as follows:
Expected Change in bond' s price
Capital gains yield = .
Beginning  of  year price
The total expected return is the sum of the current yield and the expected capital gains yield:
Expected Expected
Expected capital
= + .
Total Return current yield gains yield
The term yield to maturity, or YTM, is often used in discussing bonds. It is simply the expected
total return (assuming no default risk), so r = expected total return = expected YTM.
Recall also that securities have required returns, r, which depend on a number of factors:
Required return = r = r* + IP + LP + MRP + DRP.
We know that (1) security markets are normally in equilibrium, and (2) that for equilibrium to
exist, the expected return, r = YTM, as seen by the marginal investor, must be equal to the
required return, r. If that equality does not hold, then buying and selling will occur until it does
hold, and equilibrium is established. Therefore, for the marginal investor:
r = YTM = r.
For our 9% coupon, 10year bond selling at a price of $887 with a YTM of 10.91%, the current
yield is:
$90
Current yield = = 0.1015 = 10.15%.
$887
Knowing the current yield and the total return, we can find the capital gains yield:
YTM = current yield + capital gains yield
And
Capital gains yield = YTM  current yield = 10.91%  10.15% = 0.76%.
The capital gains yield calculation can be checked by asking this question: "What is the
expected value of the bond 1 year from now, assuming that interest rates remain at current
levels?" This is the same as asking, "What is the value of a 9year, 9 percent annual coupon
Answers and Solutions: 7  132 bond if its YTM (its required rate of return) is 10.91 percent?" The answer, using the bond
valuation function of a calculator, is $893.87. With this data, we can now calculate the bond's
capital gains yield as follows:
Capital Gains Yield = ( V B  V B )/ V B
1 0 0
= ($893.87  $887)/$887 = 0.0077 = 0.77%,
This agrees with our earlier calculation (except for rounding). When the bond is selling for
$1,134.20 and providing a total return of r = YTM = 7.08%, we have this situation:
Current Yield = $90/$1,134.20 = 7.94%
and
Capital Gains Yield = 7.08%  7.94% = 0.86%.
The bond provides a current yield that exceeds the total return, but a purchaser would incur a
small capital loss each year, and this loss would exactly offset the excess current yield and force
the total return to equal the required rate.
g. What is interest rate (or price) risk? Which bond has more interest rate risk, an annual
payment 1year bond or a 10year bond? Why?
Answer: Interest rate risk, which is often just called price risk, is the risk that a bond will lose value as the
result of an increase in interest rates. Earlier, we developed the following values for a 10 percent,
annual coupon bond:
Maturity
r 1Year Change
10Year Change 38.6%
5% $1,048 $1,386
4.8% 1,000 25.1%
10 1,000
15 956 4.4% 749
A 5 percentage point increase in r causes the value of the 1year bond to decline by only 4.8
percent, but the 10year bond declines in value by more than 38 percent. Thus, the 10year bond
has more interest rate price risk.
Answers and Solutions: 7  133 Bond Value Interest Rate Price Risk for 10 Percent Coupon
($) Bonds with Different Maturities
1,800
1,400
1,000
1Year
5Year
10Year
20Year
30Year
600
5 6 7 8 9 10 11 12 13 14 15
Interest Rate (%)
The graph above shows the relationship between bond values and interest rates for a 10
percent, annual coupon bond with different maturities. The longer the maturity, the greater the
change in value for a given change in interest rates, rd.
h. What is reinvestment rate risk? Which has more reinvestment rate risk, a 1year
bond or a 10year bond?
Answer: Investment rate risk is defined as the risk that cash flows (interest plus principal repayments) will
have to be reinvested in the future at rates lower than today's rate. To illustrate, suppose you just
won the lottery and now have $500,000. You plan to invest the money and then live on the
income from your investments. Suppose you buy a 1year bond with a YTM of 10 percent.
Your income will be $50,000 during the first year. Then, after 1 year, you will receive your
$500,000 when the bond matures, and you will then have to reinvest this amount. If rates have
fallen to 3 percent, then your income will fall from $50,000 to $15,000. On the other hand, had
you bought 30year bonds that yielded 10%, your income would have remained constant at
$50,000 per year. Clearly, buying bonds that have short maturities carries reinvestment rate risk.
Note that long maturity bonds also have reinvestment rate risk, but the risk applies only to the
coupon payments, and not to the principal amount. Since the coupon payments are significantly
less than the principal amount, the reinvestment rate risk on a longterm bond is significantly less
than on a shortterm bond.
Answers and Solutions: 7  134 i. How does the equation for valuing a bond change if semiannual payments are made? Find
the value of a 10year, semiannual payment, 10 percent coupon bond if nominal rd = 13%.
Answer: In reality, virtually all bonds issued in the U.S. have semiannual coupons and are valued using the
setup shown below:
1 2 N YEARS
0 1 2 3 4 2N1 2N SA PERIODS
     ···  
INT/2 INT/2 INT/2 INT/2 INT/2 INT/2
M
PV1
.
.
.
PVN
PVM
VBOND = sum of PVs
We would use this equation to find the bond's value:
2N
INT/ 2 M
VB = (1 + r
t =1 d / 2) t
+
(1 + r d / 2 )2 N
.
The payment stream consists of an annuity of 2n payments plus a lump sum equal to the maturity
value.
To find the value of the 10year, semiannual payment bond, semiannual interest = annual
coupon/2 = $100/2 = $50 and n = 2 (years to maturity) = 2(10) = 20. To find the value of the
bond with a financial calculator, enter n = 20, rd/2 = I = 5, pmt = 50, FV = 1000, and then press
PV to determine the value of the bond. Its value is $1,000.
You could then change r = I to see what happens to the bond's value as r changes, and plot the
valuesthe graph would look like the one we developed earlier.
For example, if r rose to 13%, we would input I= 6.5 rather than 5%, and find the 10year
bond's value to be $834.72. If r fell to 7%, then input I = 3.5 and press PV to find the bond's new
value, $1,213.19.
We would find the values with a financial calculator, but they could also be found with
formulas. Thus:
V10YEAR= $50(PVIFA5%,20) + $1,000(PVIF5%,20)
= $50 ((1 1/(1+0.05)20)/0.065) + $1,000 (1/(1+0.05)20)
= $50(12.4622) + $1,000(0.37689) = $623.11 + $376.89 = $1,000.00.
At a 13 percent required return:
V10YEAR = $50(PVIFA6.5%,20) + $1,000(PVIF6.5%,20)
= $50 ((1 1/(1+0.065)20)/0.065) + $1,000 (1/(1+0.065)20)
= $834.72.
Answers and Solutions: 7  135 At a 7 percent required return:
V10YEAR = $50(PVIFA3.5%,20) + $1,000(PVIF3.5%,20)
= $50 ((1 1/(1+0.035)20)/0.035) + $1,000 (1/(1+0.035)20)
= $1,213.19.
j. Suppose you could buy, for $1,000, either a 10 percent, 10year, annual payment
bond or a 10 percent, 10year, semiannual payment bond. They are equally risky.
Which would you prefer? If $1,000 is the proper price for the semiannual bond,
what is the equilibrium price for the annual payment bond?
Answer: The semiannual payment bond would be better. Its EAR would be:
m 2
r 0.10
EAR = 1 + Nom 1= 1+  1 = 10.25%.
m 2
An EAR of 10.25% is clearly better than one of 10.0%, which is what the annual payment bond
offers. You, and everyone else, would prefer it.
If the going rate of interest on semiannual bonds is rNom = 10%, with an EAR of 10.25%, then
it would not be appropriate to find the value of the annual payment bond using a 10% EAR. If
the annual payment bond were traded in the market, its value would be found using 10.25%,
because investors would insist on getting the same EAR on the two bonds, because their risk is
the same. Therefore, you could find the value of the annual payment bond, using 10.25%, with
your calculator. It would be $984.80 versus $1,000 for the semiannual payment bond.
Note that, if the annual payment bond were selling for $984.80 in the market, its EAR would
be 10.25%. This value can be found by entering n = 10, PV = 984.80, pmt = 100, and FV =
1000 into a financial calculator and then pressing the r = I button to find the answer, 10.25%.
With this rate, and the $984.80 price, the annual and semiannual payment bonds would be in
equilibriuminvestors would get the same rate of return on either bond, so there would not be a
tendency to sell one and buy the other (as there would be if they were both priced at $1,000.)
Answers and Solutions: 7  136 k. Suppose a 10year, 10 percent, semiannual coupon bond with a par value of $1,000 is
currently selling for $1,135.90, producing a nominal yield to maturity of 8 percent.
However, the bond can be called after 5 years for a price of $1,050.
k. 1. What is the bond's nominal yield to call (YTC)?
Answer: If the bond were called, bondholders would receive $1,050 at the end of year 5. Thus, the time
line would look like this:
0 1 2 3 4 5
     
50 50 50 50 50 50 50 50 50 50
1,050
PV1
.
.
PV4
PV5C
PV5CP
1,135.90 = sum of PVs
The easiest way to find the YTC on this bond is to input values into your calculator: n = 10; PV
= 1135.90; pmt = 50; and FV = 1050, which is the par value plus a call premium of $50; and then
press the r = I button to find I = 3.765%. However, this is the 6month rate, so we would find the
nominal rate on the bond as follows:
rNom = 2(3.765%) = 7.5301% 7.5%.
This 7.5% is the rate brokers would quote if you asked about buying the bond.
You could also calculate the EAR on the bond:
EAR = (1.03765)2  1 = 7.672%.
Usually, people in the bond business just talk about nominal rates, which is OK so long as all the
bonds being compared are on a semiannual payment basis. When you start making comparisons
among investments with different payment patterns, though, it is important to convert to EARs.
Answers and Solutions: 7  137 k. 2. If you bought this bond, do you think you would be more likely to earn the YTM or
the YTC? Why?
Answer: Since the coupon rate is 10% versus YTC = rd = 7.53%, it would pay the company to call the bond,
get rid of the obligation to pay $100 per year in interest, and sell replacement bonds whose
interest would be only $75.30 per year. Therefore, if interest rates remain at the current level
until the call date, the bond will surely be called, so investors should expect to earn 7.53%. In
general, investors should expect to earn the YTC on premium bonds, but to earn the YTM on par
and discount bonds. (Bond brokers publish lists of the bonds they have for sale; they quote
YTM or YTC depending on whether the bond sells at a premium or a discount.)
l. Boeing's bonds were issued with a yield to maturity of 7.5 percent. Does the yield
to maturity represent the promised or expected return on the bond?
Answer: The yield to maturity is the rate of return earned on a bond if it is held to maturity. It can be
viewed as the bond's promised rate of return, which is the return that investors will receive if all
the promised payments are made. The yield to maturity equals the expected rate of return only if
(1) the probability of default is zero and (2) the bond cannot be called. For bonds where there is
some default risk, or where the bond may be called, there is some probability that the promised
payments to maturity will not be received, in which case, the promised yield to maturity will
differ from the expected return.
m. Boeing's bonds were rated AA by S&P. Would you consider these bonds
investment grade or junk bonds?
Answer: The Boeing bonds would be investment grade bonds. TripleA doubleA, singleA, and tripleB
bonds are considered investment grade. DoubleB and lowerrated bonds are considered
speculative, or junk bonds, because they have a significant probability of going into default. Many
financial institutions are prohibited from buying junk bonds.
Answers and Solutions: 7  138 n. What factors determine a company's bond rating?
Answer: Bond ratings are based on both qualitative and quantitative factors, some of which are listed
below.
1. Financial performancedetermined by ratios such as the debt, TIE, FCC, and current ratios.
2. Provisions in the bond contract:
A. Secured vs. Unsecured debt
B. Senior vs. Subordinated debt
C. Guarantee provisions
D. Sinking fund provisions
E. Debt maturity
3. Other factors:
A. Earnings stability
B. Regulatory environment
C. Potential product liability
D. Accounting policy
Answers and Solutions: 7  139 o. If this firm were to default on the bonds, would the company be immediately liquidated?
Would the bondholders be assured of receiving all of their promised payments?
Answer: When a business becomes insolvent, it does not have enough cash to meet scheduled interest and
principal payments. A decision must then be made whether to dissolve the firm through
liquidation or to permit it to reorganize and thus stay alive.
The decision to force a firm to liquidate or to permit it to reorganize depends on whether the
value of the reorganized firm is likely to be greater than the value of the firm's assets if they were
sold off piecemeal. In a reorganization, a committee of unsecured creditors is appointed by the
court to negotiate with management on the terms of a potential reorganization. The
reorganization plan may call for a restructuring of the firm's debt, in which case the interest rate
may be reduced, the term to maturity lengthened, or some of the debt may be exchanged for
equity. The point of the restructuring is to reduce the financial charges to a level that the firm's
cash flows can support.
If the firm is deemed to be too far gone to be saved, it will be liquidated and the priority of
claims would be as follows:
1. Secured creditors.
2. Trustee's costs.
3. Expenses incurred after bankruptcy was filed.
4. Wages due workers, up to a limit of $2,000 per worker.
5. Claims for unpaid contributions to employee benefit plans.
6. Unsecured claims for customer deposits up to $900 per customer.
7. Federal, state, and local taxes.
8. Unfunded pension plan liabilities.
9. General unsecured creditors.
10. Preferred stockholders, up to the par value of their stock.
11. Common stockholders, if anything is left.
If the firm's assets are worth more "alive" than "dead," the company would be reorganized.
Its bondholders, however, would expect to take a "hit." Thus, they would not expect to receive
all their promised payments. If the firm is deemed to be too far gone to be saved, it would be
liquidated.
Answers and Solutions: 7  140 Chapter 7
Stocks and Their Valuation
ANSWERS TO ENDOFCHAPTER QUESTIONS
71 a. A proxy is a document giving one person the authority to act for another, typically the power to
vote shares of common stock. If earnings are poor and stockholders are dissatisfied, an outside
group may solicit the proxies in an effort to overthrow management and take control of the
business, known as a proxy fight. A takeover is an action whereby a person or group succeeds in
ousting a firm's management and taking control of the company. The preemptive right gives the
current shareholders the right to purchase any new shares issued in proportion to their current
holdings. The preemptive right may or may not be required by state law. When granted, the
preemptive right enables current owners to maintain their proportionate share of ownership and
control of the business. It also prevents the sale of shares at low prices to new stockholders
which would dilute the value of the previously issued shares. Classified stock is sometimes
created by a firm to meet special needs and circumstances. Generally, when special
classifications of stock are used, one type is designated "Class A", another as "Class B", and so on.
Class A might be entitled to receive dividends before dividends can be paid on Class B stock.
Class B might have the exclusive right to vote. Founders' shares are stock owned by the firm's
founders that have sole voting rights but restricted dividends for a specified number of years.
b. Some companies are so small that their common stocks are not actively traded; they are owned by
only a few people, usually the companies' managers. Such firms are said to be closely held
corporations. In contrast, the stocks of most larger companies are owned by a large number of
investors, most of whom are not active in management. Such companies are said to be publicly
owned corporations.
c. The secondary market deals with trading in previously issued, or outstanding, shares of
established, publicly owned companies. The company receives no new money when sales are
made in the secondary market. The primary market handles additional shares sold by established,
publicly owned companies. Companies can raise additional capital by selling in this market.
Going public is the act of selling stock to the public at large by a closely held corporation or its
principal stockholders, and this market is often termed the initial public offering (IPO) market.
^
d. Intrinsic value ( P0 ) is the present value of the expected future cash flows. The market price (P0)
is the price at which an asset can be sold.
e. The required rate of return on common stock, denoted by rs, is the minimum acceptable rate of
return considering both its riskiness and the returns available on other investments. The
expected rate of return, denoted by rs , is the rate of return expected on a stock given its current
price and expected future cash flows. If the stock is in equilibrium, the required rate of return
will equal the expected rate of return. The realized (actual) rate of return, denoted by r s , is the
rate of return that was actually realized at the end of some holding period. Although expected
Answers and Solutions: 7  141 and required rates of return must always be positive, realized rates of return over some periods
may be negative.
f. The capital gains yield results from changing prices and is calculated as (P1  P0)/P0, where P0 is
the beginningofperiod price and P1 is the endofperiod price. For a constant growth stock, the
capital gains yield is g, the constant growth rate. The dividend yield on a stock can be defined as
either the endofperiod dividend divided by the beginningofperiod price, or the ratio of the
current dividend to the current price. Valuation formulas use the former definition. The
expected total return, or expected rate of return, is the expected capital gains yield plus the
expected dividend yield on a stock. The expected total return on a bond is the yield to maturity.
g. Normal, or constant, growth occurs when a firm's earnings and dividends grow at some constant
rate forever. One category of nonconstant growth stock is a "supernormal" growth stock which
has one or more years of growth above that of the economy as a whole, but at some point the
growth rate will fall to the "normal" rate. This occurs, generally, as part of a firm's normal life
cycle. A zero growth stock has constant earnings and dividends; thus, the expected dividend
payment is fixed, just as a bond's coupon payment. Since the company is presumed to continue
operations indefinitely, the dividend stream is a perpetuity. A perpetuity is a security on which
the principal never has to be repaid.
h. Equilibrium is the condition under which the expected return on a security is just equal to its
required return, r = r, and the price is stable. The Efficient Markets Hypothesis (EMH) states (1)
that stocks are always in equilibrium and (2) that it is impossible for an investor to consistently
"beat the market." In essence, the theory holds that the price of a stock will adjust almost
immediately in response to any new developments. In other words, the EMH assumes that all
important information regarding a stock is reflected in the price of that stock. Financial theorists
generally define three forms of market efficiency: weakform, semistrongform, and strongform.
Weakform efficiency assumes that all information contained in past price movements is fully
reflected in current market prices. Thus, information about recent trends in a stock's price is of
no use in selecting a stock. Semistrongform efficiency states that current market prices reflect
all publicly available information. Therefore, the only way to gain abnormal returns on a stock
is to possess inside information about the company's stock. Strongform efficiency assumes that
all information pertaining to a stock, whether public or inside information, is reflected in current
market prices. Thus, no investors would be able to earn abnormal returns in the stock market.
i. Preferred stock is a hybridit is similar to bonds in some respects and to common stock in other
respects. Preferred dividends are similar to interest payments on bonds in that they are fixed in
amount and generally must be paid before common stock dividends can be paid. If the preferred
dividend is not earned, the directors can omit it without throwing the company into bankruptcy.
So, although preferred stock has a fixed payment like bonds, a failure to make this payment will
not lead to bankruptcy. Most preferred stocks entitle their owners to regular fixed dividend
payments.
72 True. The value of a share of stock is the PV of its expected future dividends. If the two investors
expect the same future dividend stream, and they agree on the stock's riskiness, then they should reach
similar conclusions as to the stock's value.
73 A perpetual bond is similar to a nogrowth stock and to a share of preferred stock in the following
ways:
Answers and Solutions: 7  142 1. All three derive their values from a series of cash inflowscoupon payments from the perpetual
bond, and dividends from both types of stock.
2. All three are assumed to have indefinite lives with no maturity value (M) for the perpetual bond
and no capital gains yield for the stocks.
SOLUTIONS TO ENDOFCHAPTER PROBLEMS
71 D0 = $1.50; g13 = 5%; gn = 10%; D1 through D5 = ?
D1 = D0(1 + g1) = $1.50(1.05) = $1.5750.
D2 = D0(1 + g1)(1 + g2) = $1.50(1.05)2 = $1.6538.
D3 = D0(1 + g1)(1 + g2)(1 + g3) = $1.50(1.05)3 = $1.7364.
D4 = D0(1 + g1)(1 + g2)(1 + g3)(1 + gn) = $1.50(1.05)3(1.10) = $1.9101.
D5 = D0(1 + g1)(1 + g2)(1 + g3)(1 + gn)2 = $1.50(1.05)3(1.10)2 = $2.1011.
72 D1 = $0.50; g = 7%; rs = 15%; P0 = ?
^
^ D1 $0.50
P0 = = = $6.25.
rs  g 0.15  0.07
73 P0 = $20; D0 = $1.00; g = 10%; P1 = ?;
^ r s= ?
^
P1 = P0(1 + g) = $20(1.10) = $22.
D1 $1.00(1.10)
rs = +g= + 0.10
P0 $20
$1.10
= + 0.10 = 15.50%. r s = 15.50%.
$20
74 Dps = $5.00; Vps = $60; rps = ?
D ps $5.00
rps = = = 8.33%.
v ps $60.00
Answers and Solutions: 7  143 75 0 1 2 3
   
D0 = 2.00 D1 D2 D3
^
P2
Step 1: Calculate the required rate of return on the stock:
rs = rRF + (rM  rRF)b = 7.5% + (4%)1.2 = 12.3%.
Step 2: Calculate the expected dividends:
D0 = $2.00
D1 = $2.00(1.20) = $2.40
D2 = $2.00(1.20)2 = $2.88
D3 = $2.88(1.07) = $3.08
Step 3: Calculate the PV of the expected dividends:
PVDiv = $2.40/(1.123) + $2.88/(1.123)2 = $2.14 + $2.28 = $4.42.
^
Step 4: Calculate P2 :
^
P2 = D3/(rs  g) = $3.08/(0.123  0.07) = $58.11.
^
Step 5: Calculate the PV of P2 :
PV = $58.11/(1.123)2 = $46.08.
Step 6: Sum the PVs to obtain the stock's price:
^
P0 = $4.42 + $46.08 = $50.50.
Alternatively, using a financial calculator, input the following:
CF0 = 0, CF1 = 2.40, and CF2 = 60.99 (2.88 + 58.11) and then enter I = 12.3 to solve for NPV =
$50.50.
76 The problem asks you to determine the constant growth rate, given the following facts: P0 = $80, D1
= $4, and rs = 14%. Use the constant growth rate formula to calculate g:
Answers and Solutions: 7  144 D1
rs= +g
P0
$4
0.14 = +g
$80
g = 0.09 = 9%.
77 ^
The problem asks you to determine the value of P3 , given the following facts: D1 = $2, b = 0.9, rRF
= 5.6%, RPM = 6%, and P0 = $25. Proceed as follows:
Step 1: Calculate the required rate of return:
rs = rRF + (rM  rRF)b = 5.6% + (6%)0.9 = 11%.
Step 2: Use the constant growth rate formula to calculate g:
D1
rs = +g
P0
$2
0.11 = +g
$25
g = 0.03 = 3%.
^
Step 3: Calculate P3 :
^
P3 = P0(1 + g)3 = $25(1.03)3 = $27.3182 $27.32.
^
Alternatively, you could calculate D4 and then use the constant growth rate formula to solve for P3 :
D4 = D1(1 + g)3 = $2.00(1.03)3 = $2.1855.
^
P3 = $2.1855/(0.11  0.03) = $27.3188 $27.32.
78 Vps = Dps/rps; therefore, rps = Dps/Vps.
a. rps = $8/$60 = 13.3%.
b. rps = $8/$80 = 10%.
c. rps = $8/$100 = 8%.
Answers and Solutions: 7  145 d. rps = $8/$140 = 5.7%.
^ D1 D 0 (1 + g ) $5[1 + ( 0.05)] $5(0.95) $4.75 79 P0 = = = = = = $23.75.
rs  g rs  g 0.15  ( 0.05)] 0.15 + 0.05 0.20
710 a. ri = rRF + (rM  rRF)bi.
rC = 9% + (13%  9%)0.4 = 10.6%. rD = 9% + (13%  9%)0.5 = 7%.
Note that rD is below the riskfree rate. But since this stock is like an insurance policy because it
"pays off" when something bad happens (the market falls), the low return is not unreasonable.
b. In this situation, the expected rate of return is as follows:
r c= D1/P0 + g = $1.50/$25 + 4% = 10%.
However, the required rate of return is 10.6 percent. Investors will seek to sell the stock,
dropping its price to the following:
^ $1.50
PC = = $22.73.
0.106  0.04
$1.50
At this point, r c= + 4% = 10.6%, and the stock will be in equilibrium.
$22.73
711 D0 = $1, rS = 7% + 6% = 13%, g1 = 50%, g2 = 25%, gn = 6%.
0 rs = 13% 1 2 3 4
    
g1 = 50% g2 = 25% g1 = 6%
1.50 1.875 1.9875
1.327 + 28.393 = 1.9875/(0.13  0.06)
= 30.268
23.704
$25.03
Answers and Solutions: 7  146 712 Calculate the dividend stream and place them on a time line. Also, calculate the price of the stock at
the end of the supernormal growth period, and include it, along with the dividend to be paid at t = 5, as
CF5. Then, enter the cash flows as shown on the time line into the cash flow register, enter the
required rate of return as I = 15, and then find the value of the stock using the NPV calculation. Be
sure to enter CF0 = 0, or else your answer will be incorrect.
D0 = 0; D1 = 0, D2 = 0, D3 = 1.00
D4 = 1.00(1.5) = 1.5; D5 = 1.00(1.5)2 = 2.25; D6 = 1.00(1.5)2(1.08)
= $2.43.
^
P0 = ?
0 rs = 15% 1 2
g = 50% 3 g = 8% 4 5 6
      
1.00 1.50 2.25 2.43
0.66 34.71
0.86 36.96 0.15  0.08
18.38
^
$19.89 = P0
^
P5 = D6/(rs  g) = 2.43/(0.15  0.08) = 34.71. This is the price of the stock at the end of Year 5.
CF0 = 0; CF12 = 0; CF3 = 1.0; CF4 = 1.5; CF5 = 36.96; I = 15%.
With these cash flows in the CFLO register, press NPV to get the value of the stock today: NPV =
$19.89.
D ps $10 713 a. Vps = = = $125.
rps 0.08
$10
b. Vps = = $83.33.
0.12
Answers and Solutions: 7  147 714 0 1 2 3 4 g = 5%
    
D0 = 2.00 D1 D2 D3 D4
^
P3
a. D1 = $2(1.05) = $2.10. D2 = $2(1.05)2 = $2.21. D3 = $2(1.05)3 = $2.32.
b. PV = $2.10(0.8929) + $2.21(0.7972) + $2.32(0.7118) = $5.29.
Calculator solution: Input 0, 2.10, 2.21, and 2.32 into the cash flow register, input I = 12, PV = ?
PV = $5.29.
c. $34.73(0.7118) = $24.72.
Calculator solution: Input 0, 0, 0, and 34.73 into the cash flow register, I = 12, PV = ? PV =
$24.72.
d. $24.72 + $5.29 = $30.01 = Maximum price you should pay for the stock.
D (1 + g ) D1 $2.10
e. P0 = 0
^ = = = $30.00.
rs  g rs  g 0.12  0.05
f. The value of the stock is not dependent upon the holding period. The value calculated in Parts a
through d is the value for a 3year holding period. It is equal to the value calculated in Part e
except for a small rounding error. Any other holding period would produce the same value of
^ ^
P0 ; that is, P0 = $30.00.
715 a. g = $1.1449/$1.07  1.0 = 7%.
Calculator solution: Input N = 1, PV = 1.07, PMT = 0, FV = 1.1449,
I = ? I = 7.00%.
b. $1.07/$21.40 = 5%.
c. r s= D1/P0 + g = $1.07/$21.40 + 7% = 5% + 7% = 12%.
Answers and Solutions: 7  148 ^ $2(1  0.05) $1.90 716 a. 1. P0 = = $9.50.
0.15 + 0.05 0.20
2. ^
P0 = $2/0.15 = $13.33.
^ $2(1.05) $2.10
3. P0 = = = $21.00.
0.15  0.05 0.10
^ $2(1.10) $2.20
4. P0 = = = $44.00.
0.15  0.10 0.05
b. 1. ^
P0 = $2.30/0 = Undefined.
2. ^
P0 = $2.40/(0.05) = $48, which is nonsense.
These results show that the formula does not make sense if the required rate of return is equal to
or less than the expected growth rate.
c. No.
Answers and Solutions: 7  149 717 a. End of Year: 0 1 2 3
r = 12% 4 5 6
      
g = 15% g = 5%
D0 = 1.75 D1 D2 D3 D4 D5 D6
Dt = D0(1 + g)t
D1 = $1.75(1.15)1 = $2.01.
D2 = $1.75(1.15)2 = $1.75(1.3225) = $2.31.
D3 = $1.75(1.15)3 = $1.75(1.5209) = $2.66.
D4 = $1.75(1.15)4 = $1.75(1.7490) = $3.06.
D5 = $1.75(1.15)5 = $1.75(2.0114) = $3.52.
b. Step 1
5
D
PV of dividends = (1 + rt ) t .
t =1 s
PV D1 = $2.01(PVIF12%,1) = $2.01(0.8929) = $1.79
PV D2 = $2.31(PVIF12%,2) = $2.31(0.7972) = $1.84
PV D3 = $2.66(PVIF12%,3) = $2.66(0.7118) = $1.89
PV D4 = $3.06(PVIF12%,4) = $3.06(0.6355) = $1.94
PV D5 = $3.52(PVIF12%,5) = $3.52(0.5674) = $2.00
PV of dividends = $9.46
Step 2
D6 D5 (1 + g n ) $3.52(1.05) $3.70
^
P5 = = = = = $52.80.
rs  g n rs  g n 0.12  0.05 0.07
This is the price of the stock 5 years from now. The PV of this price, discounted back 5 years, is
as follows:
^
PV of P5 = $52.80(PVIF12%,5) = $52.80(0.5674) = $29.96.
Step 3
The price of the stock today is as follows:
^
P0 ^
= PV dividends Years 1 through 5 + PV of P5
= $9.46 + $29.96 = $39.42.
This problem could also be solved by substituting the proper values into the following equation:
5
5
D 0 (1 + g s ) t D6 1
^
P0 = (1 + rs ) t
+r g 1+ r .
t =1 s n s
Answers and Solutions: 7  150 Calculator solution: Input 0, 2.01, 2.31, 2.66, 3.06, 56.32 (3.52 + 52.80) into the cash flow
register, input I = 12, PV = ? PV = $39.43.
c. First Year
D1/P0 = $2.01/$39.42 = 5.10%
Capital gains yield = 6.90%
Expected total return = 12.00%
Sixth Year
D6/P5 = $3.70/$52.80 = 7.00%
Capital gains yield = 5.00
Expected total return = 12.00%
*We know that r is 12 percent, and the dividend yield is 5.10 percent; therefore, the capital gains
yield must be 6.90 percent.
The main points to note here are as follows:
1. The total yield is always 12 percent (except for rounding errors).
2. The capital gains yield starts relatively high, then declines as the supernormal growth period
approaches its end. The dividend yield rises.
3. After t=5, the stock will grow at a 5 percent rate. The dividend yield will equal 7 percent,
the capital gains yield will equal 5 percent, and the total return will be 12 percent.
Answers and Solutions: 7  151 718 a. Part 1. Graphical representation of the problem:
Supernormal Normal
growth growth
0 1 2 3
    
D0 D1 ^
(D2 + P2 ) D3 D
PVD1
PVD2
PV P2
^
P0
D1 = D0(1 + gs) = $1.6(1.20) = $1.92.
D2 = D0(1 + gs)2 = $1.60(1.20)2 = $2.304.
^ D3 D 2 (1 + g n ) $2.304 (1.06)
P2 = = = = $61.06.
rs  g n rs  g n 0.10  0.06
P0 = PV(D1) + PV(D2) + PV( P2 )
^ ^
D1 D2 ^
P2
= + +
(1 + rs ) (1 + rs ) 2 (1 + rs ) 2
= $1.92(0.9091) + $2.304(0.8264) + $61.06(0.8264) = $54.11.
Calculator solution: Input 0, 1.92, 63.364(2.304 + 61.06) into the cash flow register, input I = 10,
PV = ? PV = $54.11.
Part 2.
Expected dividend yield: D1/P0 = $1.92/$54.11 = 3.55%.
^ ^
Capital gains yield: First, find P1 which equals the sum of the present values of D2 and P2 ,
discounted for one year.
^ ^ $2.304 + $61.06
P1 = D2(PVIF10%, 1) + P2 (PVIF10%, 1) = = $57.60.
(1.10)1
Calculator solution: Input 0, 63.364(2.304 + 61.06) into the cash flow register, input I = 10, PV
= ? PV = $57.60.
Second, find the capital gains yield:
^
P1  P0 $57.60  $54.11
= = 6.45%.
P0 $54.11
Answers and Solutions: 7  152 Dividend yield = 3.55%
Capital gains yield = 6.45
10.00% = rs.
b. Due to the longer period of supernormal growth, the value of the stock will be higher for each
year. Although the total return will remain the same, rs = 10%, the distribution between dividend
yield and capital gains yield will differ: The dividend yield will start off lower and the capital
gains yield will start off higher for the 5year supernormal growth condition, relative to the 2year
supernormal growth state. The dividend yield will increase and the capital gains yield will
decline over the 5year period until dividend yield = 4% and capital gains yield = 6%.
c. Throughout the supernormal growth period, the total yield will be 10 percent, but the dividend
yield is relatively low during the early years of the supernormal growth period and the capital
gains yield is relatively high. As we near the end of the supernormal growth period, the capital
gains yield declines and the dividend yield rises. After the supernormal growth period has
ended, the capital gains yield will equal gn = 6%. The total yield must equal rs = 10%, so the
dividend yield must equal 10%  6% = 4%.
d. Some investors need cash dividends (retired people) while others would prefer growth. Also,
investors must pay taxes each year on the dividends received during the year, while taxes on
capital gains can be delayed until the gain is actually realized.
719 a. rs = rRF + (rM  rRF)b = 11% + (14%  11%)1.5 = 15.5%.
^
P0 = D1/(rs  g) = $2.25/(0.155  0.05) = $21.43.
b. rs = 9% + (12%  9%)1.5 = 13.5%. ^
P0 = $2.25/(0.135  0.05) = $26.47.
c. rs = 9% + (11%  9%)1.5 = 12.0%. ^
P0 = $2.25/(0.12  0.05) = $32.14.
d. New data given: rRF = 9%; rM = 11%; g = 6%, b = 1.3.
rs = rRF + (rM  rRF)b = 9% + (11%  9%)1.3 = 11.6%.
^
P0 = D1/(rs  g) = $2.27/(0.116  0.06) = $40.54.
Answers and Solutions: 7  153 SOLUTION TO SPREADSHEET PROBLEM
720 The detailed solution for the problem is available both on the instructor's resource CDROM (in the
file Solution for FM11 Ch 07 P20 Build a Model.xls) and on the instructor's side of the web site,
brigham.swcollege.com.
Answers and Solutions: 7  154 MINI CASE
Sam Strother and Shawna Tibbs are senior vice presidents of the Mutual of Seattle. They are codirectors of the company's pension fund management division, with Strother having responsibility for fixed income securities (primarily bonds) and Tibbs being responsible for equity investments. A major new client, the Northwestern Municipal League, has requested that Mutual of Seattle present an investment seminar to the mayors of the represented cities, and Strother and Tibbs, who will make the actual presentation, have asked you to help them.
To illustrate the common stock valuation process, Strother and Tibbs have asked you to analyze the Temp Force Company, an employment agency that supplies word processor operators and computer programmers to businesses with temporarily heavy workloads. You are to answer the following questions.
a. Describe briefly the legal rights and privileges of common stockholders.
Answer: The common stockholders are the owners of a corporation, and as such, they have certain rights
and privileges as described below.
1. Ownership implies control. Thus, a firm's common stockholders have the right to
elect its firm's directors, who in turn elect the officers who manage the business.
2. Common stockholders often have the right, called the preemptive right, to purchase
any additional shares sold by the firm. In some states, the preemptive right is
automatically included in every corporate charter; in others, it is necessary to insert it
specifically into the charter.
b. 1. Write out a formula that can be used to value any stock, regardless of its dividend
pattern.
Answer: The value of any stock is the present value of its expected dividend stream:
^ D1 D2 D3 D
P0 = + + + + .
(1 + rs ) t
(1 + rs ) (1 + rs ) 3
(1 + rs )
However, some stocks have dividend growth patterns which allow them to be valued using
shortcut formulas.
Answers and Solutions: 8  155 b. 2. What is a constant growth stock? How are constant growth stocks valued?
Answer: A constant growth stock is one whose dividends are expected to grow at a constant rate forever.
"Constant growth" means that the best estimate of the future growth rate is some constant number,
not that we really expect growth to be the same each and every year. Many companies have
dividends which are expected to grow steadily into the foreseeable future, and such companies are
valued as constant growth stocks.
For a constant growth stock:
D1 = D0(1 + g), D2 = D1(1 + g) = D0(1 + g)2, and so on.
With this regular dividend pattern, the general stock valuation model can be simplified to
the following very important equation:
^ D1 D 0 (1 + g )
P0 = = .
rs  g rs  g
This is the wellknown "Gordon," or "constantgrowth" model for valuing stocks. Here D1, is
the next expected dividend, which is assumed to be paid 1 year from now, rs is the required rate of
return on the stock, and g is the constant growth rate.
b. 3. What happens if a company has a constant g which exceeds its rs? Will many stocks have
expected g > rs in the short run (i.e., for the next few years)? In the long run (i.e., forever)?
Answer: The model is derived mathematically, and the derivation requires that rs > g. If g is greater than rs,
the model gives a negative stock price, which is nonsensical. The model simply cannot be used
unless (1) rs > g, (2) g is expected to be constant, and (3) g can reasonably be expected to continue
indefinitely.
Stocks may have periods of supernormal growth, where gs > rs; however, this growth rate
cannot be sustained indefinitely. In the longrun, g < rs.
c. Assume that temp force has a beta coefficient of 1.2, that the riskfree rate (the yield
on Tbonds) is 7 percent, and that the market risk premium is 5 percent. What is the
required rate of return on the firm's stock?
Answer: Here we use the SML to calculate temp force's required rate of return:
rs = rRF + (rM rRF)bTemp Force = 7% + (12%  7%)(1.2)
= 7% + (5%)(1.2) = 7% + 6% = 13%.
Answers and Solutions: 8  156 d. Assume that Temp Force is a constant growth company whose last dividend (D0, which was
paid yesterday) was $2.00, and whose dividend is expected to grow indefinitely at a 6 percent
rate.
d. 1. What is the firm's expected dividend stream over the next 3 years?
Answer: Temp Force is a constant growth stock, and its dividend is expected to grow at a constant rate of 6
percent per year. Expressed as a time line, we have the following setup. Just enter 2 in your
calculator; then keep multiplying by 1 + g = 1.06 to get D1, D2, and D3:
0 rs = 13% 1 2
3 4
    
g = 6%
D0 = 2.00 2.12 2.247 2.382
1.88
1.76
1.65
.
.
.
d. 2. What is the firm's current stock price?
Answer: We could extend the time line on out forever, find the value of Temp Force's dividends for every
year on out into the future, and then the PV of each dividend, discounted at r = 13%. For
example, the PV of D1 is $1.76106; the PV of D2 is $1.75973; and so forth. Note that the
dividend payments increase with time, but as long as rs > g, the present values decrease with time.
If we extended the graph on out forever and then summed the PVs of the dividends, we would
have the value of the stock. However, since the stock is growing at a constant rate, its value can
be estimated using the constant growth model:
^ D1 $2.12 $2.12
P0 = = = = $30.29.
rs  g 0.13  0.06 0.07
d. 3. What is the stock's expected value one
year from now?
Answer: After one year, D1 will have been paid, so the expected dividend stream will then be D2, D3, D4,
and so on. Thus, the expected value one year from now is $32.10:
^ D2 $2.247 $2.247
P1 = = = = $32.10.
( rs  g) (0.13  0.06) 0.07
Answers and Solutions: 8  157 d. 4. What are the expected dividend yield, the capital gains yield, and the total return
during the first year?
Answer: The expected dividend yield in any year n is
Dn
Dividend Yield = ,
^
P n 1
While the expected capital gains yield is
^ ^
( Pn  Pn 1 ) Dn
Capital Gains Yield = =r .
^
Pn 1 Pn 1
Thus, the dividend yield in the first year is 10 percent, while the capital gains yield is 6
percent:
Total return = 13.0%
Dividend yield = $2.12/$30.29 = 7.0%
Capital gains yield = 6.0%
e. Now assume that the stock is currently selling at $30.29. What is the expected
rate of return on the stock?
Answer: The constant growth model can be rearranged to this form:
D1
r s= +g.
P0
Here the current price of the stock is known, and we solve for the expected return. For
Temp Force:
r s= $2.12/$30.29 + 0.060 = 0.070 + 0.060 = 13%.
Answers and Solutions: 8  158 f. What would the stock price be if its dividends were expected to have zero growth?
Answer: If Temp Force's dividends were not expected to grow at all, then its dividend
stream would be a perpetuity. Perpetuities are valued as shown below:
0 rs = 13% 1 2 3
   
g = 0%
2.00 2.00 2.00
1.77
1.57
1.39
.
.
.
P0 = 15.38
P0 = PMT/r = $2.00/0.13 = $15.38.
Note that if a preferred stock is a perpetuity, it may be valued with this formula.
Answers and Solutions: 8  159 g. Now assume that Temp Force is expected to experience supernormal growth of 30
percent for the next 3 years, then to return to its longrun constant growth rate of 6
percent. What is the stock's value under these conditions? What is its expected
dividend yield and capital gains yield be in year 1? In year 4?
Answer: Temp Force is no longer a constant growth stock, so the constant growth model is not applicable.
Note, however, that the stock is expected to become a constant growth stock in 3 years. Thus, it
has a nonconstant growth period followed by constant growth. The easiest way to value such
nonconstant growth stocks is to set the situation up on a time line as shown below:
0r = 13% 1 2
s
3 4
    
g = 30% g = 30% g = 30% g = 6%
2.600 3.380 4.394
4.658
2.301
2.647
3.045
^ 4.658
46.116 P3 = $66.54 =
54.109 0.13  0.06
Simply enter $2 and multiply by (1.30) to get D1 = $2.60; multiply that result by 1.3 to get D2 =
$3.38, and so forth. Then recognize that after year 3, Temp Force becomes a constant growth
^ ^
stock, and at that point P3 can be found using the constant growth model. P3 is the present
value as of t = 3 of the dividends in year 4 and beyond.
^
With the cash flows for D1, D2, D3, and P3 shown on the time line, we discount each value
back to year 0, and the sum of these four PVs is the value of the stock today, P0 = $54.109.
The dividend yield in year 1 is 4.80 percent, and the capital gains yield is 8.2
percent:
$2.600
Dividend yield = = 0.0480 = 4.8%.
$54.109
Capital gains yield = 13.00%  4.8% = 8.2%.
During the nonconstant growth period, the dividend yields and capital gains yields are not
constant, and the capital gains yield does not equal g. However, after year 3, the stock becomes
a constant growth stock, with g = capital gains yield = 6.0% and dividend yield = 13.0%  6.0% =
7.0%.
Answers and Solutions: 8  160 h. Is the stock price based more on longterm or shortterm expectations? Answer this
by finding the percentage of Temp Force current stock price based on dividends
expected more than three years in the future.
$46.116 Answer: = 85.2%.
$54.109
Stock price is based more on longterm expectations, as is evident by the fact that over 85
percent of temp force stock price is determined by dividends expected more than
three years from now.
i. Suppose Temp Force is expected to experience zero growth during the first 3 years
and then to resume its steadystate growth of 6 percent in the fourth year. What is
the stock's value now? What is its expected dividend yield and its capital gains
yield in year 1? In year 4?
Answer: Now we have this situation:
0
rs = 13% 1 2
3 4
    
g = 0% g = 0% g = 0% g = 6%
2.00 2.00 2.00 2.00
2.12
1.77
1.57
1.39
20.99 ^ 2.12
P3 = 30.29 =
^
25.72 = P0 0.07
During year 1:
$2.00
Dividend Yield = = 0.0778 = 7.78%.
$25.72
Capital Gains Yield = 13.00%  7.78% = 5.22%.
Again, in year 4 temp force becomes a constant growth stock; hence g = capital gains yield =
6.0% and dividend yield = 7.0%.
Answers and Solutions: 8  161 j. Finally, assume that Temp Force's earnings and dividends are expected to decline by
a constant 6 percent per year, that is, g = 6%. Why would anyone be willing to buy
such a stock, and at what price should it sell? What would be the dividend yield and
capital gains yield in each year?
Answer: The company is earning something and paying some dividends, so it clearly has a value greater
than zero. That value can be found with the constant growth formula, but where g is negative:
D1 D 0 (1 + g ) $2.00(0.94) $1.88
P0 = = = = = $9.89.
rS  g rS  g 0.13  ( 0.06) 0.19
Since it is a constant growth stock:
g = Capital Gains Yield = 6.0%,
hence:
Dividend Yield = 13.0%  (6.0%) = 19.0%.
As a check:
$1.88
Dividend Yield = = 0.190 = 19.0%.
$9.89
The dividend and capital gains yields are constant over time, but a high (19.0 percent)
dividend yield is needed to offset the negative capital gains yield.
k. What is market mutliple analysis?
Answer: Analysts often use the P/E multiple (the price per share divided by the earnings per share) or the
P/CF multiple (price per share divided by cash flow per share, which is the earnings per share plus
the dividends per share) to value stocks. For example, estimate the average P/E ratio of
comparable firms. This is the P/E multiple. Multiply this average P/E ratio by the expected
earnings of the company to estimate its stock price. The entity value (V) is the market value of
equity (# shares of stock multiplied by the price per share) plus the value of debt. Pick a measure,
such as EBITDA, sales, customers, eyeballs, etc. Calculate the average entity ratio for a sample of
comparable firms. For example, V/EBITDA, V/customers. Then find the entity value of the
firm in question. For example, multiply the firm's sales by the V/sales multiple, or multiply the
firm's # of customers by the V/customers ratio. The result is the total value of the firm. Subtract
the firm's debt to get the total value of equity. Divide by the number of shares to get the price per
share. There are problems with market multiple analysis. (1) It is often hard to find comparable
firms. (2) The average ratio for the sample of comparable firms often has a wide range. For
example, the average P/E ratio might be 20, but the range could be from 10 to 50. How do you
know whether your firm should be compared to the low, average, or high performers?
l. Why do stock prices change? Suppose the expected D1 is $2, the growth rate is 5
Answers and Solutions: 8  162 percent, and rs is 10 percent. Using the constant growth model, what is the impact
on stock price if g is 4 percent or 6 percent? If rs is 9 percent or 11 percent?
Answer: Using the constant growth model, the price of a stock is P0 = D1 / (rs g). If estimates of g
change, then the price will change. If estimates of the required return on stock change, then the
stock price will change. Notice that rs = rRF + (rpm)bi, so rs will change if there are changes in
inflation expectations, risk aversion, or company risk. The following table shows the stock price
for various levels of g and rs.
g g g
rs 4% 5% 6%
9% 40.00 50.00 66.67
10% 33.33 40.00 50.00
11% 28.57 33.33 40.00
m. What does market equilibrium mean?
Answer: Equilibrium means stable, no tendency to change. Market equilibrium means that prices are
stableat its current price, there is no general tendency for people to want to buy or to sell a
security that is in equilibrium. Also, when equilibrium exists, the expected rate of return will be
equal to the required rate of return:
r = D1/P0 + g = r = rRF + (rM  rRF)b.
n. If equilibrium does not exist, how will it be established?
Answer: Securities will be bought and sold until the equilibrium price is established.
o. What is the efficient markets hypothesis, what are its three forms, and what are its
implications?
Answer: The EMH in general is the hypothesis that securities are normally in equilibrium, and are "priced
fairly," making it impossible to "beat the market."
Weakform efficiency says that investors cannot profit from looking at past movements in
stock pricesthe fact that stocks went down for the last few days is no reason to think that they
will go up (or down) in the future. This form has been proven pretty well by empirical tests,
even though people still employ "technical analysis."
Semistrongform efficiency says that all publicly available information is reflected in stock
prices, hence that it won't do much good to pore over annual reports trying to find undervalued
stocks. This one is (I think) largely true, but superior analysts can still obtain and process new
information fast enough to gain a small advantage.
Strongform efficiency says that all information, even inside information, is embedded in
stock prices. This form does not holdinsiders know more, and could take advantage of that
information to make abnormal profits in the markets. Trading on the basis of insider information
is illegal.
Answers and Solutions: 8  163 p. Temp Force recently issued preferred stock. It pays an annual dividend of $5,
and the issue price was $50 per share. What is the expected return to an
investor on this preferred stock?
D ps Answer: r ps =
Vps
$5
=
$50
= 10%.
Answers and Solutions: 8  164 Chapter 8
Financial Options and Their Valuation
ANSWERS TO ENDOFCHAPTER QUESTIONS
81 a. An option is a contract which gives its holder the right to buy or sell an asset at some
predetermined price within a specified period of time. A call option allows the holder to buy the
asset, while a put option allows the holder to sell the asset.
b. A simple measure of an option's value is its exercise value. The exercise value is
equal to the current price of the stock (underlying the option) less the striking
price of the option. The strike price is the price stated in the option contract at
which the security can be bought (or sold). For example, if the underlying
stock sells for $50 and the striking price is $20, the exercise value of the option
would be $30.
c. The BlackScholes Option Pricing Model is widely used by option traders to value options. It is
derived from the concept of a riskless hedge. By buying shares of a stock and simultaneously
selling call options on that stock, the investor will create a riskfree investment position. This
riskless return must equal the riskfree rate or an arbitrage opportunity would exist. People
would take advantage of this opportunity until the equilibrium level estimated by the
BlackScholes model was reached.
82 The market value of an option is typically higher than its exercise value due to the speculative nature
of the investment. Options allow investors to gain a high degree of personal leverage when buying
securities. The option allows the investor to limit his or her loss but amplify his or her return. The
exact amount this protection is worth is the premium over the exercise value.
83 (1) An increase in stock price causes an increase in the value of a call option. (2) An increase in
exercise price causes a decrease in the value of a call option. (3) An increase in the time to expiration
causes an increase in the value of a call option. (4) An increase in the riskfree rate causes an increase
in the value of a call option. (1) An increase in the variance of stock return causes an increase in the
value of a call option.
Answers and Solutions: 8  165 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
81 P = $15; X = $15; t = 0.5; rRF = 0.06; 2 = 0.12; d1 = 0.24495;
d2 = 0.0000; N(d1) = 0.59675; N(d2) = 0.500000; V = ?
Using the BlackScholes Option Pricing Model, you calculate the option's value as:
r t
V = P[N(d1)]  Xe RF [N(d2)]
= $15(0.59675)  $15e(0.10)(0.5)(0.50000)
= $8.95128  $15(0.9512)(0.50000)
= $1.6729 $1.67.
82 Option's exercise price = $15; Exercise value = $22; Premium value = $5;
V = ? P0 = ?
Premium = Market price of option  Exercise value
$5 = V  $22
V = $27.
Exercise value = P0  Exercise price
$22 = P0  $15
P0 = $37.
ln (P/X) + [rRF + ( 2 / 2)]t ln ($30 /$35) + [0.05 + (0.25/ 2)](0 .333333) 83 d1 = = =  0.3319.
t 0.5 0.33333)
d2 = d1 s (t)0.5 = 0.3319 0.5(0.33333)0.5 = 0.6206.
N(d1) = 0.3700 (from Excel NORMSDIST function).
N(d2) = 0.2674 (from Excel NORMSDIST function).
r t
V = P[N(d1)]  Xe RF [N(d2)]
= $30(0.3700)  $35e(0.05)(0.33333)(0.2674)
= $11.1000  $9.2043
= $1.8957 $1.90.
Answers and Solutions: 8  166 84 The stock's range of payoffs in one year is $26  $16 = $10. At expiration, the option will be worth
$26  $21 = $5 if the stock price is $26, and zero if the stock price $16. The range of payoffs for the
stock option is $5 0 = $5.
Equalize the range to find the number of shares of stock: Option range / Stock range = $5/$10 = 0.5.
With 0.5 shares, the stock's payoff will be either $13 or $8. The portfolio's payoff will be $13  $5 =
$8, or $8 0 = $8.
The present value of $8 at the daily compounded riskfree rate is: PV = $8 / (1+ (0.05/365))365 =
$7.610.
The option price is the current value of the stock in the portfolio minus the PV of the payoff:
V = 0.5($20)  $7.610 = $2.39.
85 The stock's range of payoffs in six months is $18  $13 = $5. At expiration, the option will be worth
$18  $14 = $4 if the stock price is $18, and zero if the stock price $13. The range of payoffs for the
stock option is $4 0 = $5.
Equalize the range to find the number of shares of stock: Option range / Stock range = $4/$5 = 0.8.
With 0.8 shares, the stock's payoff will be either 0.8($18) = $14.40 or 0.8($13) = $10.40. The
portfolio's payoff will be $14.4  $4 = $10.40, or $10.40 0 = $10.40.
The present value of $10.40 at the daily compounded riskfree rate is: PV = $10.40 / (1+
(0.06/365))365/2 = $10.093.
The option price is the current value of the stock in the portfolio minus the PV of the payoff:
V = 0.8($15)  $10.093 = $1.907 .$1.91.
86 Put = V P + X exp(rRF t)
= $6.56  $33 + $32 e0.06(1)
= $6.56  $33 + $30.136 = $3.696 $3.70.
Answers and Solutions: 8  167 SOLUTION TO SPREADSHEET PROBLEMS
87 The detailed solution for the problem is available both on the instructor's resource CDROM (in the
file Solution for FM 11 Ch 08 P07 Build a Model.xls) and on the instructor's side of the textbook's
web site, http://brigham.swcollege.com.
Answers and Solutions: 8  168 MINI CASE
Assume that you have just been hired as a financial analyst by Triple Trice Inc., a midsized California company that specializes in creating exotic clothing. Since no one at Triple Trice is familiar with the basics of financial options, you have been asked to prepare a brief report that the firm's executives could use to gain at least a cursory understanding of the topic.
To begin, you gathered some outside materials the subject and used these materials to draft a list of pertinent questions that need to be answered. In fact, one possible approach to the paper is to use a questionandanswer format. Now that the questions have been drafted, you have to develop the answers.
a. What is a financial option? What is the single most important characteristic of an
option?
Answer: A financial option is a contract which gives its holder the right to buy (or sell) an asset at a
predetermined price within a specified period of time. An option's most important characteristic
is that it does not obligate its owner to take any action; it merely gives the owner the right to buy
or sell an asset.
b. Options have a unique set of terminology. Define the following terms: (1) call
option; (2) put option; (3) exercise price; (4) striking, or strike, price; (5) option price;
(6) expiration date; (7) exercise value; (8) covered option; (9) naked option; (10)
inthemoney call; (11) outofthemoney call; and (12) LEAPS.
Answer: 1. A call option is an option to buy a specified number of shares of a security within some
future period.
2. A put option is an option to sell a specified number of shares of a security within some
future period.
3. Exercise price is another name for strike price, the price stated in the option contract at
which the security can be bought (or sold).
4. The strike price is the price stated in the option contract at which the security can be bought
(or sold).
5. The option price is the market price of the option contract.
6. The expiration date is the date the option matures.
7. The exercise value is the value of a call option if it were exercised today, and it is equal to
the current stock price minus the strike price. Note: the exercise value is zero if the
stock price is less than the strike price.
8. A covered option is a call option written against stock held in an investor's portfolio.
9. A naked option is an option sold without the stock to back it up.
Answers and Solutions: 9 169 10. An inthemoney call is a call option whose exercise price is less than the current price of
the underlying stock.
11. An outofthemoney call is a call option whose exercise price exceeds the current stock
price.
12. LEAPS stands for longterm equity anticipation securities. They are similar to
conventional options except they are longterm options with maturities of up to 2½ years.
c. Consider Triple Trice's call option with a $25 strike price. The following table
contains historical values for this option at different stock prices:
Stock Price Call Option Price
$25 $ 3.00
30 7.50
35 12.00
40 16.50
45 21.00
50 25.50
1. Create a table which shows (a) stock price, (b) strike price, (c) exercise value, (d)
option price, and (e) the premium of option price over exercise value.
Answer: Price Of Strike Exercise Value Market Price Premium
Stock Price Of Option Of Option
(D)  (C) =
(A) (B) (A)  (B) = (C) (D) (E)
$25.00 $25.00 $ 0.00 $ 3.00 $3.00
30.00 25.00 5.00 7.50 2.50
35.00 25.00 10.00 12.00 2.00
40.00 25.00 15.00 16.50 1.50
45.00 25.00 20.00 21.00 1.00
50.00 25.00 25.00 25.50 0.50
c. 2. What happens to the premium of option price over exercise value as the stock price
rises? Why?
Answer: As the table shows, the premium of the option price over the exercise value declines as the stock
price increases. This is due to the declining degree of leverage provided by options as the
underlying stock prices increase, and to the greater loss potential of options at higher option
prices.
Answers and Solutions: 9 170 d. In 1973, Fischer Black and Myron Scholes developed the BlackScholes Option
Pricing Model (OPM).
1. What assumptions underlie the OPM?
Answer: The assumptions which underlie the OPM are as follows:
· The stock underlying the call option provides no dividends during the life of the
option.
· No transactions costs are involved with the sale or purchase of either the stock or
the option.
· The shortterm, riskfree interest rate is known and is constant during the life of
the option.
· Security buyers may borrow any fraction of the purchase price at the shortterm,
riskfree rate.
· Shortterm selling is permitted without penalty, and sellers receive immediately
the full cash proceeds at today's price for securities sold short.
· The call option can be exercised only on its expiration date.
· Security trading takes place in continuous time, and stock prices move randomly
in continuous time.
Answers and Solutions: 9 171 d. 2. Write out the three equations that constitute the model.
Answer: The OPM consists of the following three equations:
V = P[N(d1)  Xe  rRF t [N(d2)].
ln( P/X ) + [ rRF + ( 2 /2)]t
d1 = .
t
d2 = d1  t .
Here,
V = current value of a call option with time t until expiration.
P = current price of the underlying stock.
N(di) = probability that a deviation less than di will occur in a standard normal distribution.
Thus, N(d1) and N(d2) represent areas under a standard normal distribution function.
X = exercise, or strike, price of the option.
e 2.7183.
rRF = riskfree interest rate.
t = time until the option expires (the option period).
ln(P/X) = natural logarithm of P/X.
2 = variance of the rate of return on the stock.
Answers and Solutions: 9 172 d. 3. What is the value of the following call option according to the OPM?
Stock Price = $27.00.
Exercise Price = $25.00
Time To Expiration = 6 Months.
RiskFree Rate = 6.0%.
Stock Return Variance = 0.11.
Answer: the input variables are:
P = $27.00; X = $25.00; rRF = 6.0%; t = 6 months = 0.5 years; and 2 = 0.11.
Now, we proceed to use the OPM:
V = $27[N(d1)]  $25e(0.06)(0.5)[N(d2)].
ln($27/$25) + [(0.06 + 0.11/2)](0.5)
d1 =
(0.3317)(0.7071)
0.0770 + 0.0575
= = 0.5736.
0.2345
d2 = d1  (0.3317)(0.7071) = d1  0.2345
= 0.5736  0.2345 = 0.3391.
N(d1) = N(0.5736) = 0.5000 + 0.2168 = 0.7168.
N(d2) = N(0.3391) = 0.5000 + 0.1327 = 0.6327.
Therefore,
V = $27(0.7168)  $25e0.03(0.6327) = $19.3536  $25(0.97045)(0.6327)
= $19.3536  $15.3500 = $4.0036 $4.00.
Thus, under the OPM, the value of the call option is about $4.00.
Answers and Solutions: 9 173 e. What impact does each of the following call option parameters have on the value of a
call option?
1. Current Stock Price
2. Exercise Price
3. Option's Term To Maturity
4. RiskFree Rate
5. Variability Of The Stock Price
Answer: 1. The value of a call option increases (decreases) as the current stock price increases
(decreases).
2. As the exercise price of the option increases (decreases), the value of the option
decreases (increases).
3. As the expiration date of the option is lengthened, the value of the option increases.
This is because the value of the option depends on the chance of a stock price
increase, and the longer the option period, the higher the stock price can climb.
4. As the riskfree rate increases, the value of the option tends to increase as well.
Since increases in the riskfree rate tend to decrease the present value of the option's
exercise price, they also tend to increase the current value of the option.
5. The greater the variance in the underlying stock price, the greater the possibility that
the stock's price will exceed the exercise price of the option; thus, the more valuable
the option will be.
f. What is putcall parity?
Answer: Putcall parity specifies the relationship between puts, calls, and the underlying stock price that
must hold to prevent arbitrage:
Put + Stock = Call + PV Of Exercise Price
Answers and Solutions: 9 174 Chapter 9
The Cost of Capital
ANSWERS TO ENDOFCHAPTER QUESTIONS
91 a. The weighted average cost of capital, WACC, is the weighted average of the aftertax component
costs of capitaldebt, preferred stock, and common equity. Each weighting factor is the
proportion of that type of capital in the optimal, or target, capital structure. The aftertax cost of
debt, rd(1  T), is the relevant cost to the firm of new debt financing. Since interest is deductible
from taxable income, the aftertax cost of debt to the firm is less than the beforetax cost. Thus,
rd(1  T) is the appropriate component cost of debt (in the weighted average cost of capital).
b. The cost of preferred stock, rps, is the cost to the firm of issuing new preferred stock. For
perpetual preferred, it is the preferred dividend, Dps, divided by the net issuing price, Pn. Note
that no tax adjustments are made when calculating the component cost of preferred stock because,
unlike interest payments on debt, dividend payments on preferred stock are not tax deductible.
The cost of new common equity, re, is the cost to the firm of equity obtained by selling new
common stock. It is, essentially, the cost of retained earnings adjusted for flotation costs.
Flotation costs are the costs that the firm incurs when it issues new securities. The amount
actually available to the firm for capital investment from the sale of new securities is the sales
price of the securities less flotation costs. Note that flotation costs consist of (1) direct expenses
such as printing costs and brokerage commissions, (2) any price reduction due to increasing the
supply of stock, and (3) any drop in price due to informational asymmetries.
c. The target capital structure is the relative amount of debt, preferred stock, and common equity that
the firm desires. The WACC should be based on these target weights.
d. There are considerable costs when a company issues a new security, including fees to an
investment banker and legal fees. These costs are called flotation costs. The cost of new
common equity is higher than that of common equity raised internally by reinvesting earnings.
Project's financed with external equity must earn a higher rate of return, since they project must
cover the flotation costs.
92 The WACC is an average cost because it is a weighted average of the firm's component costs of
capital. However, each component cost is a marginal cost; that is, the cost of new capital. Thus, the
WACC is the weighted average marginal cost of capital.
93 Probable Effect on
rd(1  T) rs WACC
a. The corporate tax rate is lowered. + 0 +
b. The Federal Reserve tightens credit. + + +
c. The firm uses more debt; that is, it
increases its debt/assets ratio. + + 0
Answers and Solutions: 9 175 d. The firm doubles the amount of capital
it raises during the year. 0 or + 0 or + 0 or +
e. The firm expands into a risky
new area. + + +
f. Investors become more risk averse. + + +
94 Standalone risk views a project's risk in isolation, hence without regard to portfolio effects;
withinfirm risk, also called corporate risk, views project risk within the context of the firm's portfolio
of assets; and market risk (beta) recognizes that the firm's stockholders hold diversified portfolios of
stocks. In theory, market risk should be most relevant because of its direct effect on stock prices.
95 If a company's composite WACC estimate were 10 percent, its managers might use 10 percent to
evaluate averagerisk projects, 12 percent for those with highrisk, and 8 percent for lowrisk projects.
Unfortunately, given the data, there is no completely satisfactory way to specify exactly how much
higher or lower we should go in setting riskadjusted costs of capital.
Answers and Solutions: 9 176 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
91 40% Debt; 60% Equity; rd = 9%; T = 40%; WACC = 9.96%; rs = ?
WACC = (wd)(rd)(1  T) + (wce)(rs)
9.96% = (0.4)(9%)(1  0.4) + (0.6)rs
9.96% = 2.16% + 0.6rs
7.8% = 0.6rs
rs = 13%.
92 Vps = $50; Dps = $3.80; F = 5%; rps = ?
D ps
rps =
Vps (1  F)
$3.80
=
$50(1  0.05)
$3.80
= = 8%.
$47.50
93 P0 = $30; D1 = $3.00; g = 5%; rs = ?
D1
rs = + g = + 0.05 = 15%.
P0
94 a. rd(1  T) = 13%(1  0) = 13.00%.
b. rd(1  T) = 13%(0.80) = 10.40%.
c. rd(1  T) = 13%(0.65) = 8.45%.
95 rd(1  T) = 0.12(0.65) = 7.80%.
$100(0.11) $11 $11 96 rps = = = = 11.94%.
$97.00(1  0.05) $97.00(0.95) $92.15
97 Enter these values: N = 60, PV = 515.16, PMT = 30, and FV = 1000, to get I = 6% = periodic rate.
The nominal rate is 6%(2) = 12%, and the aftertax component cost of debt is 12%(0.6) = 7.2%.
Answers and Solutions: 9 177 D1 $2.14 98 a. rs = +g= + 7% = 9.3% + 7% = 16.3%.
P0 $23
b. rs = rRF + (rM  rRF)b
= 9% + (13%  9%)1.6 = 9% + (4%)1.6 = 9% + 6.4% = 15.4%.
c. rs = Bond rate + Risk premium = 12% + 4% = 16%.
d. The bondyieldplusriskpremium approach and the CAPM method both resulted in lower cost of
equity values than the DCF method. The firm's cost of equity should be estimated to be about
15.9 percent, which is the average of the three methods.
99 a. $6.50 = $4.42(1+g)5
(1+g)5 = 6.50/4.42 = 1.471
(1+g) = 1.471(1/5) = 1.080
g = 8%.
Alternatively, with a financial calculator, input N = 5, PV = 4.42, PMT = 0, FV = 6.50, and then
solve for I = 8.02% 8%.
b. D1 = D0(1 + g) = $2.60(1.08) = $2.81.
c. rs = D1/P0 + g = $2.81/$36.00 + 8% = 15.81%.
D1 910 a. rs = +g
P0
$3.60
0.09 = +g
$60.00
0.09 = 0.06 + g
g = 3%.
b. Current EPS $5.400
Less: Dividends per share 3.600
Retained earnings per share $1.800
Rate of return × 0.090
Increase in EPS $0.162
Current EPS 5.400
Next year's EPS $5.562
Alternatively, EPS1 = EPS0(1 + g) = $5.40(1.03) = $5.562.
Answers and Solutions: 9 178 911 a. Common equity needed:
0.5($30,000,000) = $15,000,000.
b. Cost using rs:
AfterTax
Percent × Cost = Product
Debt 0.50 4.8%* 2.4%
Common equity 0.50 12.0 6.0
WACC =
8.4%
*8%(1  T) = 8%(0.6) = 4.8%.
c. rs and the WACC will increase due to the flotation costs of new equity.
912 The book and market value of the current liabilities are both $10,000,000.
The bonds have a value of
V = $60(PVIFA10%,20) + $1,000(PVIF10%,20)
= $60([1/0.10][1/(0.1*(1+0.10)20)]) + $1,000((1+0.10)20)
= $60(8.5136) + $1,000(0.1486)
= $510.82 + $148.60 = $659.42.
Alternatively, using a financial calculator, input N = 20, I = 10, PMT = 60, and FV = 1000 to arrive at
a PV = $659.46.
The total market value of the longterm debt is 30,000($659.46) = $19,783,800.
There are 1 million shares of stock outstanding, and the stock sells for $60 per share. Therefore,
the market value of the equity is $60,000,000.
The market value capital structure is thus:
Shortterm debt $10,000,000 11.14%
Longterm debt 19,783,800 22.03
Common equity 60,000,000 66.83
$89,783,800 100.00%
Answers and Solutions: 9 179 913 Several steps are involved in the solution of this problem. Our solution follows:
Step 1.
Establish a set of market value capital structure weights. In this case, A/P and accruals, and also
shortterm debt, may be disregarded because the firm does not use these as a source of permanent
financing.
Debt:
The longterm debt has a market value found as follows:
40
$40 $1,000
V0 = t
+ = $699,
t =1 (1.06) (1.06) 40
or 0.699($30,000,000) = $20,970,000 in total.
Preferred Stock:
The preferred has a value of
$2
Pps = = $72.73.
0.11 / 4
There are $5,000,000/$100 = 50,000 shares of preferred outstanding, so the total market value of the
preferred is
50,000($72.73) = $3,636,500.
Common Stock:
The market value of the common stock is
4,000,000($20) = $80,000,000.
Therefore, here is the firm's market value capital structure, which we assume to be optimal:
Longterm debt $ 20,970,000 20.05%
Preferred stock 3,636,500 3.48
Common equity 80,000,000 76.47
$104,606,500 100.00%
We would round these weights to 20 percent debt, 4 percent preferred, and 76 percent common equity.
Step 2.
Establish cost rates for the various capital structure components.
Debt cost:
rd(1  T) = 12%(0.6) = 7.2%.
Answers and Solutions: 9 180 Preferred cost:
Annual dividend on new preferred = 11%($100) = $11. Therefore,
rps = $11/$100(1  0.05) = $11/$95 = 11.6%.
Common equity cost:
There are three basic ways of estimating rs: CAPM, DCF, and risk premium over own bonds. None of the methods is very exact.
CAPM:
We would use rRF = Tbond rate = 10%. For RPM, we would use 4.5% to 5.5%. For beta, we would use a beta in the 1.3 to 1.7 range. Combining these values, we obtain this range of values for rs:
Highest: rs = 10% + (5.5)(1.7) = 19.35%.
Lowest: rs = 10% + (4.5)(1.3) = 15.85%.
Midpoint: rs = 10% + (5.0)(1.5) = 17.50%.
DCE:
The company seems to be in a rapid, nonconstant growth situation, but we do not have the inputs necessary to develop a nonconstant rs. Therefore, we will use the constant growth model but temper our growth rate; that is, think of it as a longterm average g that may well be higher in the immediate than in the more distant future.
Data exist that would permit us to calculate historic growth rates, but problems would clearly arise, because the growth rate has been variable and also because gEPS gDPS. For the problem at hand, we would simply disregard historic growth rates, except for a discussion about calculating them as an exercise.
We could use as a growth estimator this method:
g = b(r) = 0.5(24%) = 12%.
It would not be appropriate to base g on the 30% ROE, because investors do not expect that rate.
Finally, we could use the analysts' forecasted g range, 10 to 15 percent. The dividend yield is D1/P0. Assuming g = 12%,
D1 $1(1.12)
= = 5.6%.
P0 $20
One could look at a range of yields, based on P in the range of $17 to $23, but because we believe in efficient markets, we would use P0 = $20. Thus, the DCF model suggests a rs in the range of 15.6 to 20.6 percent:
Highest: rs = 5.6% + 15% = 20.6%.
Lowest: rs = 5.6% + 10% = 15.6%.
Midpoint: rs = 5.6% + 12.5% = 18.1%.
Answers and Solutions: 9 181 Generalized risk premium.
Highest: rs = 12% + 6% = 18%.
Lowest: rs = 12% + 4% = 16%.
Midpoint: rs = 12% + 5% = 17%.
Based on the three midpoint estimates, we have rs in this range:
CAPM 17.5%
DCF 18.1%
Risk Premium 17.0%
Step 3.
Calculate the WACC:
WACC = (D/V)(rdAT) + (P/V)(rps) + (S/V)(rs or re)
= 0.20(rdAT) + 0.04(rps) + 0.76(rs or re).
It would be appropriate to calculate a range of WACCs based on the ranges of component costs, but to
save time, we shall assume rdAT = 7.2%, rps = 11.6%, and rs = 17.5%. With these cost rates, here is
the WACC calculation:
WACC = 0.2(7.2%) + 0.04(11.6%) + 0.76(17.5%) = 15.2%.
914 P0 = $30; D1 = $3.00; g = 5%; F = 10%; rs = ?
rs = [D1/(1F) P0] + g = [3/(10.10)(30)] + 0.05 = 16.1%.
915 Enter these values: N = 20, PV =1000(10.02)=980, PMT = 90(1.4)=54, and
FV = 1000, to get I = 5.57%, which is the aftertax component cost of debt.
Answers and Solutions: 9 182 SPREADSHEET PROBLEM
916 The detailed solution for the problem is available both on the instructor's resource CDROM (in the
file Solution for FM11 Ch 09 P16 Build a Model.xls) and on the instructor's side of the web site,
http://brigham.swcollege.com.
Answers and Solutions: 9 183 MINI CASE
During the last few years, Harry Davis Industries has been too constrained by the high cost of capital to make many capital investments. Recently, though, capital costs have been declining, and the company has decided to look seriously at a major expansion program that had been proposed by the marketing department. Assume that you are an assistant to Leigh Jones, the financial vicepresident. Your first task is to estimate Harry Davis' cost of capital. Jones has provided you with the following data, which she believes may be relevant to your task:
1. The firm's tax rate is 40 percent.
2. The current price of Harry Davis' 12 percent coupon, semiannual payment,
noncallable bonds with 15 years remaining to maturity is $1,153.72. Harry
Davis does not use shortterm interestbearing debt on a permanent basis. New
bonds would be privately placed with no flotation cost.
3. The current price of the firm's 10 percent, $100 par value, quarterly dividend, perpetual preferred
stock is $113.10. Harry Davis would incur flotation costs of $2.00 per share on a new issue.
4. Harry Davis' common stock is currently selling at $50 per share. Its last dividend (d0) was $4.19,
and dividends are expected to grow at a constant rate of 5 percent in the foreseeable future. Harry
Davis' beta is 1.2; the yield on tbonds is 7 percent; and the market risk premium is estimated to be
6 percent. For the bondyieldplusriskpremium approach, the firm uses a 4 percentage point risk
premium.
5. Harry Davis' target capital structure is 30 percent longterm debt, 10 percent preferred stock, and
60 percent common equity.
To structure the task somewhat, Jones has asked you to answer the following questions.
Answers and Solutions: 10  184 a. 1. What sources of capital should be included when you estimate Harry Davis' weighted
average cost of capital (WACC)?
Answer: The WACC is used primarily for making longterm capital investment decisions,
i.e., for capital budgeting. Thus, the WACC should include the types of capital
used to pay for longterm assets, and this is typically longterm debt, preferred
stock (if used), and common stock. Shortterm sources of capital consist of (1)
spontaneous, noninterestbearing liabilities such as accounts payable and
accruals and (2) shortterm interestbearing debt, such as notes payable. If the
firm uses shortterm interestbearing debt to acquire fixed assets rather than
just to finance working capital needs, then the WACC should include a
shortterm debt component. Noninterestbearing debt is generally not included
in the cost of capital estimate because these funds are netted out when
determining investment needs, that is, net rather than gross working capital is
included in capital expenditures.
a. 2. Should the component costs be figured on a beforetax or an aftertax basis?
Answer: Stockholders are concerned primarily with those corporate cash flows that are available for their
use, namely, those cash flows available to pay dividends or for reinvestment. Since dividends
are paid from and reinvestment is made with aftertax dollars, all cash flow and rate of return
calculations should be done on an aftertax basis.
a. 3. Should the costs be historical (embedded) costs or new (marginal) costs?
Answer: In financial management, the cost of capital is used primarily to make decisions which involve
raising new capital. Thus, the relevant component costs are today's marginal costs rather than
historical costs.
Answers and Solutions: 10  185 b. What is the market interest rate on Harry Davis' debt and its component cost of debt?
Answer: Harry Davis' 12 percent bond with 15 years to maturity is currently selling for $1,153.72. Thus,
its yield to maturity is 10 percent:
0 1 2 3
29 30
    ···  
1,153.72 60 60 60 60
60
1,000
Enter n = 30, PV = 1153.72, pmt = 60, and FV = 1000, and then press the i button to find rd/2 = i
= 5.0%. Since this is a semiannual rate, multiply by 2 to find the annual rate, rd = 10%, the
pretax cost of debt.
Since interest is tax deductible, Uncle Sam, in effect, pays part of the cost, and Harry Davis'
relevant component cost of debt is the aftertax cost:
rd(1  T) = 10.0%(1  0.40) = 10.0%(0.60) = 6.0%.
Optional Question
Should flotation costs be included in the estimate?
Answer: The actual component cost of new debt will be somewhat higher than 6 percent because the firm
will incur flotation costs in selling the new issue. However, flotation costs are typically small on
public debt issues, and, more important, most debt is placed directly with banks, insurance
companies, and the like, and in this case flotation costs are almost nonexistent.
Optional Question
Should you use the nominal cost of debt or the effective annual cost?
Answer: Our 10 percent pretax estimate is the nominal cost of debt. Since the firm's debt has semiannual
coupons, its effective annual rate is 10.25 percent:
(1.05)2  1.0 = 1.1025  1.0 = 0.1025 = 10.25%.
However, nominal rates are generally used. The reason is that the cost of capital is used in
capital budgeting, and capital budgeting cash flows are generally assumed to occur at yearend.
Therefore, using nominal rates makes the treatment of the capital budgeting discount rate and cash
flows consistent.
Answers and Solutions: 10  186 c. 1. What is the firm's cost of preferred stock?
Answer: Since the preferred issue is perpetual, its cost is estimated as follows:
D ps 0.1($100) $10
rps = = = = 0.090 = 9.0%.
Pn $133.10  $2.00 $111.10
Note (1) that flotation costs for preferred are significant, so they are included here, (2) that since
preferred dividends are not deductible to the issuer, there is no need for a tax adjustment, and (3)
that we could have estimated the effective annual cost of the preferred, but as in the case of debt,
the nominal cost is generally used.
c. 2. Harry Davis' preferred stock is riskier to investors than its debt, yet the preferred's
yield to investors is lower than the yield to maturity on the debt. Does this suggest
that you have made a mistake? (Hint: think about taxes.)
Answer: Corporate investors own most preferred stock, because 70 percent of preferred dividends received
by corporations are nontaxable. Therefore, preferred often has a lower beforetax yield than the
beforetax yield on debt issued by the same company. Note, though, that the aftertax yield to a
corporate investor, and the aftertax cost to the issuer, are higher on preferred stock than on debt.
d. 1. What are the two primary ways companies raise common equity?
Answer: A firm can raise common equity in two ways: (1) by retaining earnings and (2) by issuing new
common stock.
d. 2. Why is there a cost associated with reinvested earnings?
Answer: Management may either pay out earnings in the form of dividends or else retain earnings for
reinvestment in the business. If part of the earnings is retained, an opportunity cost is incurred:
stockholders could have received those earnings as dividends and then invested that money in
stocks, bonds, real estate, and so on.
d. 3. Harry Davis doesn't plan to issue new shares of common stock. Using the CAPM
approach, what is Harry Davis' estimated cost of equity?
Answer: rs = 0.07 + (0.06)1.2 = 14.2%.
e. 1. What is the estimated cost of equity using the discounted cash flow (DCF) approach?
D1 D 0 (1 + g ) $4.19 (1.05) Answer: rs = = +g = + 0.05 = 13.8%.
P0 P0 $50
Answers and Solutions: 10  187 e. 2. Suppose the firm has historically earned 15 percent on equity (ROE) and retained 35
percent of earnings, and investors expect this situation to continue in the future.
How could you use this information to estimate the future dividend growth rate, and
what growth rate would you get? Is this consistent with the 5 percent growth rate
given earlier?
Answer: Another method for estimating the growth rate is to use the retention growth model:
g = (1  Payout Ratio)ROE
In this case g = (0.35)0.15 = 5.25%. This is consistent with the 5% rate given earlier.
e. 3. Could the DCF method be applied if the growth rate was not constant? How?
Answer: yes, you could use the DCF using nonconstant growth. You would find the PV of the dividends
during the nonconstant growth period and add this value to the PV of the series of inflows when
growth is assumed to become constant.
f. What is the cost of equity based on the bondyieldplusriskpremium method?
Answer: rs = company's own bond yield + risk premium.
First find the YTM of the bond:
Enter n = 30, PV = 1153.72, pmt = 60, and FV = 1000, and then press the i button to find r/2 = i =
5%. Since this is a semiannual rate, multiply by 2 to find the annual rate, r = 10%.
The assumed risk premium is 4%, thus
rs = 0.10 + 0.04 = 14%.
g. What is your final estimate for the cost of equity, rs?
Answer: The final estimate for the cost of equity would simply be the average of the values found using the
above three methods.
CAPM 14.2%
DCF 13.8
BOND YIELD + R.P. 14.0
AVERAGE 14.0%
h. What is Harry Davis' weighted average cost of capital (WACC)?
Answer: WACC= wdrd(1  T) + wpsrps + wce(rs)
Answers and Solutions: 10  188 = 0.3(0.10)(0.6) + 0.1(0.09) + 0.6(0.14)
= 0.111 = 11.1%.
i. What factors influence Harry Davis' composite WACC?
Answer: There are factors that the firm cannot control and those that they can control that influence
WACC.
Factors The Firm Cannot Control:
}
Level Of Interest Rates
Tax Rates Market Conditions
Factors The Firm Can Control:
Capital Structure Policy
Dividend Policy
Investment Policy
j. Should the company use the composite WACC as the hurdle rate for each of its projects?
Answer: No. The composite WACC reflects the risk of an average project undertaken by the firm.
Therefore, the WACC only represents the "hurdle rate" for a typical project with average risk.
Different projects have different risks. The project's WACC should be adjusted to reflect the
project's risk.
k. What procedures are used to determine the riskadjusted cost of capital for a particular
division? What approaches are used to measure a division's beta?
Answer: The following procedures can be used to determine a division's riskadjusted cost of capital:
(1) Subjective adjustments to the firm's composite WACC.
(2) Attempt to estimate what the cost of capital would be if the division were a standalone
firm. This requires estimating the division's beta.
The following approaches can be used to measure a division's beta:
(1) Pure play approach. Find several publicly traded companies exclusively in the project's
business. Then, use the average of their betas as a proxy for the project's beta. (It's hard
to find such companies.)
(2) Accounting beta approach. Run a regression between the project's ROA and the S&P
index ROA. Accounting betas are correlated (0.5  0.6) with market betas. However,
you normally can't get data on new project ROAs before the capital budgeting decision has
been made.
Answers and Solutions: 10  189 l. Harry Davis is interested in establishing a new division, which will focus primarily on
developing new internetbased projects. In trying to determine the cost of capital for
this new division, you discover that standalone firms involved in similar projects
have on average the following characteristics:
· Their capital structure is 10 percent debt and 90 percent common equity.
· Their cost of debt is typically 12 percent.
· The beta is 1.7.
given this information, what would your estimate be for the division's cost of
capital?
Answer:
rs DIV. = rRF + (rM  rRF)bDIV.
= 7% + (6%)1.7 = 17.2%.
WACCDIV. = Wdrd(1  T) + Wcrs
= 0.1(12%)(0.6) + 0.9(17.2%)
= 16.2%.
The division's WACC = 16.2% vs. The corporate WACC = 11.1%. The division's
market risk is greater than the firm's average projects. Typical projects within this
division would be accepted if their returns are above 16.2 percent.
m. What are three types of project risk? How is each type of risk used?
Answer: The three types of project risk are:
StandAlone Risk
Corporate Risk
Market Risk
Market risk is theoretically best in most situations. However, creditors, customers,
suppliers, and employees are more affected by corporate risk. Therefore, corporate
risk is also relevant. Standalone risk is the easiest type of risk to measure.
Taking on a project with a high degree of either standalone or corporate risk will not
necessarily affect the firm's market risk. However, if the project has highly uncertain returns, and
if those returns are highly correlated with returns on the firm's other assets and with most other
assets in the economy, the project will have a high degree of all types of risk.
n. Explain in words why new common stock that is raised externally has a higher
percentage cost than equity that is raised internally by reivesting earnings.
Answer: The company is raising money in order to make an investment. The money has a cost, and this
cost is based primarily on the investors' required rate of return, considering risk and alternative
investment opportunities. So, the new investment must provide a return at least equal to the
investors' opportunity cost.
If the company raises capital by selling stock, the company doesn't get all of the money that
investors put up. For example, if investors put up $100,000, and if they expect a 15 percent return
Answers and Solutions: 10  190 on that $100,000, then $15,000 of profits must be generated. But if flotation costs are 20 percent
($20,000), then the company will receive only $80,000 of the $100,000 investors put up. That
$80,000 must then produce a $15,000 profit, or a 15/80 = 18.75% rate of return versus a 15
percent return on equity raised as retained earnings.
o. 1. Harry Davis estimates that if it issues new common stock, the flotation cost will be 15
percent. Harry Davis incorporates the flotation costs into the DCF approach. What is the
estimated cost of newly issued common stock, taking into account the flotation cost?
Answer:
D0 (1 + g) + g
re =
P0 (1  F)
$4.19(1.05)
= + 5.0%
$50(1  0.15)
$4.40
= + 5.0% = 15.4%.
$42.50
o. 2. Suppose Harry Davis issues 30year debt with a par value of $1,000 and a coupon rate of
10%, paid annually. If flotation costs are 2 percent, what is the aftertax cost of debt for
the new bond?
Answer: Using a financial calculator, n = 30, PV = (10.02)(1000) = 980, pmt = (10.40)(100) = 60, FV =
1000. The resulting i is 6.15%, which is the aftertax cost of debt.
p. What four common mistakes in estimating the WACC should Harry Davis avoid?
Answer: 1. Don't use the coupon rate on a firm's existing debt as the pretax cost of debt. Use the
current cost of debt.
2. When estimating the risk premium for the CAPM approach, don't subtract the current
longterm tbond rate from the historical average return on stocks.
For example, the historical average return on stocks has been about 12.7%. If inflation
has driven the current riskfree rate up to 10%, it would be wrong to conclude that the current
market risk premium is 12.7%  10% = 2.7%. In all likelihood, inflation would also have
driven up the expected return on the market. Therefore, the historical return on the market
would not be a good estimate of the current expected return on the market.
3. Don't use book weights to estimte the weights for the capital structure. Use the target
capital structure to determine the weights for the WACC. If you don't have the target
weights, then use market value rather than book value to obtain the weights. Use the book
value of debt only as a last resort.
4. Always remember that capital components are sources of funding that come from investors.
If it's not a source of funding from an investor, then it's not a capital component.
Answers and Solutions: 10  191 Chapter 10
The Basics of Capital Budgeting
Evaluating Cash Flows
ANSWERS TO ENDOFCHAPTER QUESTIONS
101 a. Capital budgeting is the whole process of analyzing projects and deciding whether they should be
included in the capital budget. This process is of fundamental importance to the success or
failure of the firm as the fixed asset investment decisions chart the course of a company for many
years into the future. The payback, or payback period, is the number of years it takes a firm to
recover its project investment. Payback may be calculated with either raw cash flows (regular
payback) or discounted cash flows (discounted payback). In either case, payback does not
capture a project's entire cash flow stream and is thus not the preferred evaluation method. Note,
however, that the payback does measure a project's liquidity, and hence many firms use it as a risk
measure.
b. Mutually exclusive projects cannot be performed at the same time. We can choose either Project
1 or Project 2, or we can reject both, but we cannot accept both projects. Independent projects
can be accepted or rejected individually.
c. The net present value (NPV) and internal rate of return (IRR) techniques are discounted cash flow
(DCF) evaluation techniques. These are called DCF methods because they explicitly recognize
the time value of money. NPV is the present value of the project's expected future cash flows
(both inflows and outflows), discounted at the appropriate cost of capital. NPV is a direct
measure of the value of the project to shareholders. The internal rate of return (IRR) is the
discount rate that equates the present value of the expected future cash inflows and outflows.
IRR measures the rate of return on a project, but it assumes that all cash flows can be reinvested at
the IRR rate.
d. The modified internal rate of return (MIRR) assumes that cash flows from all projects are
reinvested at the cost of capital as opposed to the project's own IRR. This makes the modified
internal rate of return a better indicator of a project's true profitability. The profitability index is
found by dividing the project's PV of future cash flows by its initial cost. A profitability index
greater than 1 is equivalent to a positive NPV project.
e. An NPV profile is the plot of a project's NPV versus its cost of capital. The crossover rate is the
cost of capital at which the NPV profiles for two projects intersect.
f. Capital projects with nonnormal cash flows have a large cash outflow either sometime during or
at the end of their lives. A common problem encountered when evaluating projects with
nonnormal cash flows is multiple IRRs. A project has normal cash flows if one or more cash
outflows (costs) are followed by a series of cash inflows.
g. The hurdle rate is the project cost of capital, or discount rate. It is the rate used in
discounting future cash flows in the NPV method, and it is the rate that is
compared to the IRR. The mathematics of the NPV method imply that project
cash flows are reinvested at the cost of capital while the IRR method assumes
Answers and Solutions: 10  192 reinvestment at the IRR. Since project cash flows can be replaced by new
external capital which costs r, the proper reinvestment rate assumption is the
cost of capital, and thus the best capital budget decision rule is NPV. The
postaudit is the final aspect of the capital budgeting process. The postaudit is a
feedback process in which the actual results are compared with those predicted
in the original capital budgeting analysis. The postaudit has several purposes,
the most important being to improve forecasts and improve operations.
h. A replacement chain is a method of comparing mutually exclusive projects that have unequal lives.
Each project is replicated such that they will both terminate in a common year. If projects with
lives of 3 years and 5 years are being evaluated, the 3year project would be replicated 5 times
and the 5year project replicated 3 times; thus, both projects would terminate in 15 years. Not all
projects maximize their NPV if operated over their engineering lives and therefore it may be best
to terminate a project prior to its potential life. The economic life is the number of years a
project should be operated to maximize its NPV, and is often less than the maximum potential life.
Capital rationing occurs when management places a constraint on the size of the firm's capital
budget during a particular period.
102 Project classification schemes can be used to indicate how much analysis is required to evaluate a
given project, the level of the executive who must approve the project, and the cost of capital that
should be used to calculate the project's NPV. Thus, classification schemes can increase the
efficiency of the capital budgeting process.
103 The NPV is obtained by discounting future cash flows, and the discounting process actually
compounds the interest rate over time. Thus, an increase in the discount rate has a much greater
impact on a cash flow in Year 5 than on a cash flow in Year 1.
104 This question is related to Question 103 and the same rationale applies. With regard to the second
part of the question, the answer is no; the IRR rankings are constant and independent of the firm's cost
of capital.
105 The NPV and IRR methods both involve compound interest, and the mathematics of discounting
requires an assumption about reinvestment rates. The NPV method assumes reinvestment at the cost
of capital, while the IRR method assumes reinvestment at the IRR. MIRR is a modified version of
IRR which assumes reinvestment at the cost of capital.
106 Generally, the failure to employ common life analysis in such situations will bias the NPV against the
shorter project because it "gets no credit" for profits beyond its initial life, even though it could
possibly be "renewed" and thus provide additional NPV.
Answers and Solutions: 10  193 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
101 a. $52,125/$12,000 = 4.3438, so the payback is about 4 years.
b. Project K's discounted payback period is calculated as follows:
Annual Discounted @12%
Period Cash Flows Cash Flows Cumulative
0 ($52,125) ($52,125.00) ($52,125.00)
1 12,000 10,714.80 (41,410.20)
2 12,000 9,566.40 (31,843.80)
3 12,000 8,541.60 (23,302.20)
4 12,000 7,626.00 (15,676.20)
5 12,000 6,808.80 (8,867.40)
6 12,000 6,079.20 (2,788.20)
7 12,000 5,427.60 2,639.40
8 12,000 4,846.80 7,486.20
$2,788.20
The discounted payback period is 6 + years, or 6.51 years.
$5,427.60
Alternatively, since the annual cash flows are the same, one can divide $12,000 by 1.12 (the discount
rate = 12%) to arrive at CF1 and then continue to divide by 1.12 seven more times to obtain the
discounted cash flows (Column 3 values). The remainder of the analysis would be the same.
c. NPV = $52,125 + $12,000[(1/i)(1/(i*(1+i)n)]
= $52,125 + $12,000[(1/0.12)(1/(0.12*(1+0.12)8)]
= $52,125 + $12,000(4.9676) = $7,486.20.
Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 12, and
then solve for NPV = $7,486.68.
d. Financial calculator: Input the appropriate cash flows into the cash flow
register and then solve for IRR = 16%.
Answers and Solutions: 10  194 e. MIRR: PV Costs = $52,125.
FV Inflows:
PV FV
0 1 2 3 4 5 6
12% 7 8
        
12,000 12,000 12,000 12,000 12,000 12,000 12,000 12,000
13,440
15,053
16,859
18,882
21,148
23,686
26,528 52,125 MIRR = 13.89% 147,596
Financial calculator: Obtain the FVA by inputting N = 8, I = 12, PV = 0, PMT = 12000, and then solve for FV = $147,596. The MIRR can be obtained by inputting N = 8, PV = 52125, PMT = 0, FV = 147596, and then solving for I = 13.89%.
Answers and Solutions: 10  195 102 Project A:
Using a financial calculator, enter the following:
CF0 = 15000000
CF1 = 5000000
CF2 = 10000000
CF3 = 20000000
I = 10; NPV = $12,836,213.
Change I = 10 to I = 5; NPV = $16,108,952.
Change I = 5 to I = 15; NPV = $10,059,587.
Project B:
Using a financial calculator, enter the following:
CF0 = 15000000
CF1 = 20000000
CF2 = 10000000
CF3 = 6000000
I = 10; NPV = $15,954,170.
Change I = 10 to I = 5; NPV = $18,300,939.
Change I = 5 to I = 15; NPV = $13,897,838.
103 Truck:
NPV = $17,100 + $5,100(PVIFA14%,5)
= $17,100 + $5,100(3.4331) = $17,100 + $17,509
= $409. (Accept)
Answers and Solutions: 10  196 Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 14, and
then solve for NPV = $409.
Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for
IRR = 14.99% 15%.
MIRR: PV Costs = $17,100.
FV Inflows:
PV FV
14%
0 1 2 3 4 5
     
5,100 5,100 5,100 5,100 5,100
5,814
6,628
7,556
8,614
17,100 MIRR = 14.54% (Accept) 33,712
Financial calculator: Obtain the FVA by inputting N = 5, I = 14, PV = 0, PMT = 5100, and then
solve for FV = $33,712. The MIRR can be obtained by inputting N = 5, PV = 17100, PMT = 0, FV
= 33712, and then solving for I = 14.54%.
Pulley:
NPV = $22,430 + $7,500(3.4331) = $22,430 + $25,748
= $3,318. (Accept)
Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 14, and
then solve for NPV = $3,318.
Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for
IRR = 20%.
MIRR: PV Costs = $22,430.
Answers and Solutions: 10  197 FV Inflows:
PV FV
0 1 2 3
14% 4 5
     
7,500 7,500 7,500 7,500 7,500
8,550
9,747
11,112
12,667
22,430 MIRR = 17.19% (Accept) 49,576
Financial calculator: Obtain the FVA by inputting N = 5, I = 14, PV = 0, PMT = 7500, and then
solve for FV = $49,576. The MIRR can be obtained by inputting N = 5, PV = 22430, PMT = 0, FV
= 49576, and then solving for I = 17.19%.
104 Electricpowered:
NPVE = $22,000 + $6,290 [(1/i)(1/(i*(1+i)n)]
= $22,000 + $6,290 [(1/0.12)(1/(0.12*(1+0.12)6)]
= $22,000 + $6,290(4.1114) = $22,000 + $25,861 = $3,861.
Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 12, and
then solve for NPV = $3,861.
Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for
IRR = 18%.
Gaspowered:
NPVG = $17,500 + $5,000 [(1/i)(1/(i*(1+i)n)]
= $17,500 + $5,000 [(1/0.12)(1/(0.12*(1+0.12)6)]
= $17,500 + $5,000(4.1114) = $17,500 + $20,557 = $3,057.
Financial calculator: Input the appropriate cash flows into the cash flow register, input I = 12, and
then solve for NPV = $3,057.
Financial calculator: Input the appropriate cash flows into the cash flow register and then solve for
IRR = 17.97% 18%.
The firm should purchase the electricpowered forklift because it has a higher NPV than the
gaspowered forklift. The company gets a high rate of return (18% > r = 12%) on a larger
investment.
Answers and Solutions: 10  198 105 Financial calculator solution, NPV:
Project S
Inputs 5 12 3000 0
N I PV PMT FV
Output = 10,814.33
NPVS = $10,814.33  $10,000 = $814.33.
Project L
Inputs 5 12 7400 0
N I PV PMT FV
Output = 26,675.34
NPVL = $26,675.34  $25,000 = $1,675.34.
Financial calculator solution, IRR:
Input CF0 = 10000, CF1 = 3000, Nj = 5, IRRS = ? IRRS = 15.24%.
Input CF0 = 25000, CF1 = 7400, Nj = 5, IRRL = ? IRRL = 14.67%.
Financial calculator solution, MIRR:
Project S
Inputs 5 12 0 3000
N I PV PMT FV
Output = 19,058.54
PV costsS = $10,000.
FV inflowsS = $19,058.54.
Inputs 5 10000 0 19058.54
N I PV PMT FV
Output = 13.77
MIRRS = 13.77%.
Project L
Inputs 5 12 0 7400
N I PV PMT FV
Answers and Solutions: 10  199 Output = 47,011.07
PV costsL = $25,000.
FV inflowsL = $47,011.07.
Inputs 5 25000 0 47011.07
N I PV PMT FV
Output = 13.46
MIRRL = 13.46%.
$10,814.33 $26,675.34 PIS = = 1.081. PIL = = 1.067.
$10,000 $25,000
Thus, NPVL > NPVS, IRRS > IRRL, MIRRS > MIRRL, and PIS > PIL. The scale difference between Projects S and L result in the IRR, MIRR, and PI favoring S over L. However, NPV favors Project L, and hence L should be chosen.
Answers and Solutions: 10  200 106 Project X: 0 1 2 3
12% 4
    
1,000 100 300 400
700.00
448.00
376.32
140.49
1,664.81
1,000 13.59% = MIRRX`
$1,000 = $1,664.81/(1 + MIRRX)4.
Project Y: 0 1 2 3
12%
4
    
1,000 1,000 100 50
50.00
56.00
125.44
1,404.93
1,636.37
1,000 13.10% = MIRRY
$1,000 = $1,636.37/(1 + MIRRY)4.
Thus, since MIRRX > MIRRY, Project X should be chosen.
Alternative step: You could calculate NPVs, see that Project X has the higher NPV, and just
calculate MIRRX.
NPVX = $58.02 and NPVY = $39.94.
107 a. Purchase price $ 900,000
Installation 165,000
Initial outlay $1,065,000
CF0 = 1065000; CF15 = 350000; I = 14; NPV = ?
NPV = $136,578; IRR = 19.22%.
b. Ignoring environmental concerns, the project should be undertaken because its
NPV is positive and its IRR is greater than the firm's cost of capital.
c. Environmental effects could be added by estimating penalties or any other cash outflows that
might be imposed on the firm to help return the land to its previous state (if possible). These
outflows could be so large as to cause the project to have a negative NPVin which case the
project should not be undertaken.
Answers and Solutions: 10  201 108 a.
NPV
($)
1,000
900
800
700
600
500
Project A
400
300
200 Project B
100 Cost of
Capital (%)
5 10 15 20 25 30
100
200
300
r NPVA NPVB
0.0% $890 $399
10.0 283 179
12.0 200 146
18.1 0 62
20.0 (49) 41
24.0 (138) 0
30.0 (238) (51)
b. IRRA = 18.1%; IRRB = 24.0%.
c. At r = 10%, Project A has the greater NPV, specifically $283.34 as compared to
Project B's NPV of $178.60. Thus, Project A would be selected. At r = 17%,
Project B has an NPV of $75.95 which is higher than Project A's NPV of $31.05.
Thus, choose Project B if r = 17%.
Answers and Solutions: 10  202 d. Here is the MIRR for Project A when r = 10%:
PV costs = $300 + $387/(1.10)1 + $193/(1.10)2
+ $100/(1.10)3 + $180/(1.10)7 = $978.82.
TV inflows = $600(1.10)3 + $600(1.10)2 + $850(1.10)1 = $2,459.60.
Now, MIRR is that discount rate which forces the TV of $2,459.60 in 7 years to equal $978.82:
$952.00 = $2,547.60(1+MIRR)7.
MIRRA = 14.07%.
Similarly, MIRRB = 15.89%.
At r = 17%,
MIRRA = 17.57%.
MIRRB = 19.91%.
e. To find the crossover rate, construct a Project which is the difference in the two projects' cash
flows:
Project =
Year CFA  CFB
0 $105
1 (521)
2 (327)
3 (234)
4 466
5 466
6 716
7 (180)
IRR = Crossover rate = 14.53%.
Projects A and B are mutually exclusive, thus, only one of the projects can be chosen. As long
as the cost of capital is greater than the crossover rate, both the NPV and IRR
methods will lead to the same project selection. However, if the cost of capital is
less than the crossover rate the two methods lead to different project selectionsa
conflict exists. When a conflict exists the NPV method must be used.
Because of the sign changes and the size of the cash flows, Project has multiple IRRs.
Thus, a calculator's IRR function will not work. One could use the trial and error method of
entering different discount rates until NPV = $0. However, an HP can be "tricked" into giving
the roots. After you have keyed Project Delta's cash flows into the g register of an HP10B, you
will see an "ErrorSoln" message. Now enter 10 STO IRR/YR and the 14.53% IRR is
found. Then enter 100 STO IRR/YR to obtain IRR = 456.22%. Similarly, Excel or Lotus
123 can also be used.
109 a. Incremental Cash
Year Plan B Plan A Flow (B  A)
0 ($10,000,000) ($10,000,000) $ 0
1 1,750,000 12,000,000 (10,250,000)
Answers and Solutions: 10  203 220 1,750,000 0 1,750,000
If the firm goes with Plan B, it will forgo $10,250,000 in Year 1, but will receive $1,750,000 per
year in Years 220.
b. If the firm could invest the incremental $10,250,000 at a return of 16.07%, it would receive cash
flows of $1,750,000. If we set up an amortization schedule, we would find that payments of
$1,750,000 per year for 19 years would amortize a loan of $10,250,000 at 16.0665%.
Financial calculator solution:
Inputs 19 10250000 1750000 0
N I PV PMT FV
Output = 16.0665
c. Yes, assuming (1) equal risk among projects, and (2) that the cost of capital is a constant and does
not vary with the amount of capital raised.
d. See graph. If the cost of capital is less than 16.07%, then Plan B should be accepted; if r >
16.07%, then Plan A is preferred.
NPV ( Mi l l i ons of Dol l ar s)
25
B
20
15
10 Cr ossover Rat e = 16. 07%
A
I RRB = 16. 7%
5
I RRA = 20%
Cost of
Capi t al ( %)
5 10 15 20 25
Answers and Solutions: 10  204 1010 a. Financial calculator solution:
Plan A
Inputs 20 10 8000000 0
N I PV PMT FV
Output = 68,108,510
NPVA = $68,108,510  $50,000,000 = $18,108,510.
Plan B
Inputs 20 10 3400000 0
N I PV PMT FV
Output = 28,946,117
NPVB = $28,946,117  $15,000,000 = $13,946,117.
Plan A
Inputs 20 50000000 8000000 0
N I PV PMT FV
Output = 15.03
IRRA = 15.03%.
Plan B
Inputs 20 15000000 3400000 0
N I PV PMT FV
Output = 22.26
IRRB = 22.26%.
Answers and Solutions: 10  205 b. If the company takes Plan A rather than B, its cash flows will be (in millions of dollars):
Cash Flows Cash Flows Project
Year from A from B Cash
Flows
0 ($50) ($15.0)
($35.0)
1 8 3.4
4.6
2 8 3.4
4.6
. . .
.
. . .
.
. . . .
20 8 3.4 4.6
So, Project has a "cost" of $35,000,000 and "inflows" of $4,600,000 per year for 20 years.
Inputs 20 10 4600000 0
N I PV PMT FV
Output = 39,162,393
NPV = $39,162,393  $35,000,000 = $4,162,393.
Inputs 2 35000000 4600000 0
N I PV PMT FV
Output = 11.71
IRR = 11.71%.
Since IRR > r, and since we should accept . This means accept the larger project (Project A).
In addition, when dealing with mutually exclusive projects, we use the NPV method for choosing
the best project.
Answers and Solutions: 10  206 c.
N P V ( Mi l l i o n s o f D o l l a r s )
125
A
100 Cr oss over Rat e = 11. 7%
75 B
50 I RRA = 1 5. 03%
I RRB = 2 2 . 2 6 %
25
5 10 15 20 25 30
C o s t o f C a p i t a l ( %)
25
I RR = 11 . 7%
50
d. The NPV method implicitly assumes that the opportunity exists to reinvest the cash flows
generated by a project at the cost of capital, while use of the IRR method implies the opportunity
to reinvest at the IRR. If the firm's cost of capital is constant at 10 percent, all projects with an
NPV > 0 will be accepted by the firm. As cash flows come in from these projects, the firm will
either pay them out to investors, or use them as a substitute for outside capital which costs 10
percent. Thus, since these cash flows are expected to save the firm 10 percent, this is their
opportunity cost reinvestment rate.
The IRR method assumes reinvestment at the internal rate of return itself, which is an
incorrect assumption, given a constant expected future cost of capital, and ready access to capital
markets.
1011 a. The project's expected cash flows are as follows (in millions of dollars):
Time Net Cash Flow
0 ($ 4.4)
1 27.7
2 (25.0)
Answers and Solutions: 10  207 We can construct the following NPV profile:
NP V ( Mi l l i o n s o f Do l l a r s )
Ma x i mu m
3 NP V a t 8 0 . 5 %
2
1
Di s c o u n t
Ra t e ( %)
10 20 80.5 420
1
I RR 1 = 9 . 2 % I RR2 = 4 20%
2
3
NPV appr oac hes  $4. 0 as
t he cost of capi t al
4 appr oaches
4. 4
Discount Rate NPV
0% ($1,700,000)
9 (29,156)
10 120,661
50 2,955,556
100 3,200,000
200 2,055,556
300 962,500
400 140,000
410 70,204
420 2,367
430 (63,581)
The table above was constructed using a financial calculator with the following inputs: CF0 =
4400000, CF1 = 27700000, CF2 = 25000000, and I = discount rate to solve for the NPV.
b. If r = 8%, reject the project since NPV < 0. But if r = 14%, accept the project because NPV > 0.
c. Other possible projects with multiple rates of return could be nuclear power plants where disposal
of radioactive wastes is required at the end of the project's life, or leveraged leases where the
borrowed funds are repaid at the end of the lease life. (See Chapter 20 for more information on
leases.)
d. Here is the MIRR for the project when r = 8%:
PV costs = $4,400,000 + $25,000,000/(1.08)2 = $25,833,470.51.
Answers and Solutions: 10  208 TV inflows = $27,700,000(1.08)1 = $29,916,000.00.
Now, MIRR is that discount rate which forces the PV of the TV of $29,916,000 over 2 years to
equal $25,833,470.51:
$25,833,470.51 = $29,916,000(PVIFr,2).
Inputs 2 25833470.51 0 29916000
N I PV PMT FV
Output = 7.61
MIRR = 7.61%.
At r = 14%,
Inputs 2 23636688.21 0 31578000
N I PV PMT FV
Output = 15.58
MIRR = 15.58%.
PV costs = $4,400,000 + $25,000,000/(1.14)2 = $23,636,688.21.
TV inflows = $27,700,000(1.14)1 = $31,578,000.
Now, MIRR is that discount rate which forces the PV of the TV of $31,578,000 over 2 years to
equal $23,636,688.21:
$23,636,688.21 = $31,578,000(PVIFr,2).
Yes. The MIRR method leads to the same conclusion as the NPV method. Reject the project if
r = 8%, which is greater than the corresponding MIRR of 7.61%, and accept the
project if r = 14%, which is less than the corresponding MIRR of 15.58%.
1012 a. The IRRs of the two alternatives are undefined. To calculate an IRR, the cash flow stream must
include both cash inflows and outflows.
b. The PV of costs for the conveyor system is ($911,067), while the PV of costs for the forklift
system is ($838,834). Thus, the forklift system is expected to be ($838,834)  ($911,067) =
$72,233 less costly than the conveyor system, and hence the forklift trucks should be used.
Financial calculator solution:
Input: CF0 = 500000, CF1 = 120000, Nj = 4, CF2 = 20000, I = 8, NPVC = ? NPVC =
911,067.
Answers and Solutions: 10  209 Input: CF0 = 200000, CF1 = 160000, N1 = 5, I = 8, NPVF = ? NPVF = 838,834.
1013 a. Payback A (cash flows in thousands):
Annual
Period Cash Flows Cumulative
0 ($25,000) ($25,000)
1 5,000 (20,000)
2 10,000 (10,000)
3 15,000 5,000
4 20,000 25,000
PaybackA = 2 + $10,000/$15,000 = 2.67 years.
Payback B (cash flows in thousands):
Annual
Period Cash Flows Cumulative
0 ($25,000) $25,000)
1 20,000 (5,000)
2 10,000 5,000
3 8,000 13,000
4 6,000 19,000
PaybackB = 1 + $5,000/$10,000 = 1.50 years.
Answers and Solutions: 10  210 b. Discounted payback A (cash flows in thousands):
Annual Discounted @10%
Period Cash Flows Cash Flows Cumulative
0 ($25,000) ($25,000.00) ($25,000.00)
1 5,000 4,545.45 ( 20,454.55)
2 10,000 8,264.46 ( 12,190.09)
3 15,000 11,269.72 ( 920.37)
4 20,000 13,660.27 12,739.90
Discounted PaybackA = 3 + $920.37/$13,660.27 = 3.07 years.
Discounted payback B (cash flows in thousands):
Annual Discounted @10%
Period Cash Flows Cash Flows Cumulative
0 ($25,000) ($25,000.00) ($25,000.00)
1 20,000 18,181.82 ( 6,818.18)
2 10,000 8,264.46 1,446.28
3 8,000 6,010.52 7,456.80
4 6,000 4,098.08 11,554.88
Discounted PaybackB = 1 + $6,818.18/$8,264.46 = 1.825 years.
c. NPVA = $12,739,908; IRRA = 27.27%.
NPVB = $11,554,880; IRRB = 36.15%.
Both projects have positive NPVs, so both projects should be undertaken.
d. At a discount rate of 5%, NPVA = $18,243,813.
At a discount rate of 5%, NPVB = $14,964,829.
At a discount rate of 5%, Project A has the higher NPV; consequently, it should be accepted.
e. At a discount rate of 15%, NPVA = $8,207,071.
At a discount rate of 15%, NPVB = $8,643,390.
At a discount rate of 15%, Project B has the higher NPV; consequently, it should be accepted.
f. Project =
Year CFA  CFB
0 $ 0
1 (15)
2 0
3 7
4 14
IRR = Crossover rate = 13.5254% 13.53%.
g. Use 3 steps to calculate MIRRA @ r = 10%:
Answers and Solutions: 10  211 Step 1: Calculate the NPV of the uneven cash flow stream, so its FV can then be calculated.
With a financial calculator, enter the cash flow stream into the cash flow registers,
then enter I = 10, and solve for NPV = $37,739,908.
Step 2: Calculate the FV of the cash flow stream as follows:
Enter N = 4, I = 10, PV = 37739908, and PMT = 0 to solve for FV = $55,255,000.
Step 3: Calculate MIRRA as follows:
Enter N = 4, PV = 25000000, PMT = 0, and FV = 55255000 to solve for I = 21.93%.
Use 3 steps to calculate MIRRB @ r = 10%:
Step 1: Calculate the NPV of the uneven cash flow stream, so its FV can then be calculated.
With a financial calculator, enter the cash flow stream into the cash flow registers,
then enter I = 10, and solve for NPV = $36,554,880.
Step 2: Calculate the FV of the cash flow stream as follows:
Enter N = 4, I = 10, PV = 36554880, and PMT = 0 to solve for FV = $53,520,000.
Step 3: Calculate MIRRB as follows:
Enter N = 4, PV = 25000000, PMT = 0, and FV = 53520000 to solve for I = 20.96%.
According to the MIRR approach, if the 2 projects were mutually exclusive, Project A would be chosen because it has the higher MIRR. This is consistent with the NPV approach.
Answers and Solutions: 10  212 1014 Plane A: Expected life = 5 years; Cost = $100 million; NCF = $30 million; COC = 12%.
Plane B: Expected life = 10 years; Cost = $132 million; NCF = $25 million; COC = 12%.
0 1 2 3 4 5 6 7 8 9 10
A:           
100 30 30 30 30 30 30 30 30 30 30
100
70
Enter these values into the cash flow register: CF0 = 100; CF14 = 30; CF5 = 70; CF610 = 30.
Then enter I = 12, and press the NPV key to get NPVA = 12.764 $12.76 million.
0 1 2 3 4 5 6 7 8 9 10
B:           
132 25 25 25 25 25 25 25 25 25 25
Enter these cash flows into the cash flow register, along with the interest rate, and press the NPV
key to get NPVB = 9.256 $9.26 million.
Project A is the better project and will increase the company's value by $12.76 million.
1015 0 1 2 3 4 5 6 7 8
A:         
10 4 4 4 4 4 4 4 4
10
6
Machine A's simple NPV is calculated as follows: Enter CF0 = 10 and CF14 = 4. Then enter I
= 10, and press the NPV key to get NPVA = $2.679 million. However, this does not consider the
fact that the project can be repeated again. Enter these values into the cash flow register: CF0
= 10; CF13 = 4; CF4 = 6; CF58 = 4. Then enter I = 10, and press the NPV key to get Extended
NPVA = $4.5096 $4.51 million.
0 1 2 3 4 5 6 7 8
B:         
15 3.5 3.5 3.5 3.5 3.5 3.5 3.5 3.5
Enter these cash flows into the cash flow register, along with the interest rate, and press the NPV
key to get NPVB = $3.672 $3.67 million.
Machine A is the better project and will increase the company's value by $4.51 million.
Answers and Solutions: 10  213 1016 a. Using a financial calculator, input the following: CF0 = 190000, CF1 = 87000, Nj = 3, and I =
14 to solve for NPV1903 = $11,981.99 $11,982 (for 3 years).
Adjusted NPV1903 = $11,982 + $11,982/(1.14)3 = $20,070.
Using a financial calculator, input the following: CF0 = 360000, CF1 = 98300, Nj = 6, and I =
14 to solve for NPV3606 = $22,256.02 $22,256 (for 6 years).
Both new machines have positive NPVs, hence the old machine should be replaced. Further,
since its adjusted NPV is greater, choose Model 3606.
1017 a. NPV of termination after Year t:
NPV0 = $22,500 + $22,500 = 0.
Using a financial calculator, input the following: CF0 = 22500, CF1 = 23750, and I = 10 to
solve for NPV1 = $909.09 $909.
Using a financial calculator, input the following: CF0 = 22500, CF1 = 6250, CF2 = 20250, and I
= 10 to solve for NPV2 = $82.64 $83.
Using a financial calculator, input the following: CF0 = 22500, CF1 = 6250, Nj = 2, CF3 =
17250, and I = 10 to solve for NPV3 = $1,307.29 $1,307.
Using a financial calculator, input the following: CF0 = 22500, CF1 = 6250, Nj = 3, CF4 =
11250, and I = 10 to solve for NPV4 = $726.73 $727.
Using a financial calculator, input the following: CF0 = 22500, CF1 = 6250, Nj = 5, and I = 10
to solve for NPV5 = $1,192.42 $1,192.
The firm should operate the truck for 3 years, NPV3 = $1,307.
b. No. Salvage possibilities could only raise NPV and IRR. The value of the firm is maximized
by terminating the project after Year 3.
Answers and Solutions: 10  214 SOLUTION TO SPREADSHEET PROBLEM
1018 The detailed solution for the problem is available both on the instructor's resource CDROM (in the
file Solution for FM11 Ch 10 P18 Build a Model.xls) and on the instructor's side of the web site,
brigham.swcollege.com.
Answers and Solutions: 10  215 MINI CASE
You have just graduated from the MBA program of a large university, and one of your favorite courses was "Today's Entrepreneurs." In fact, you enjoyed it so much you have decided you want to "be your own boss." While you were in the master's program, your grandfather died and left you $300,000 to do with as you please. You are not an inventor and you do not have a trade skill that you can market; however, you have decided that you would like to purchase at least one established franchise in the fast foods area, maybe two (if profitable). The problem is that you have never been one to stay with any project for too long, so you figure that your time frame is three years. After three years you will sell off your investment and go on to something else.
You have narrowed your selection down to two choices; (1) Franchise L: Lisa's Soups, Salads, & Stuff and (2) Franchise S: Sam's Wonderful Fried Chicken. The net cash flows shown below include the price you would receive for selling the franchise in year 3 and the forecast of how each franchise will do over the threeyear period. Franchise L's cash flows will start off slowly but will increase rather quickly as people become more health conscious, while Franchise S's cash flows will start off high but will trail off as other chicken competitors enter the marketplace and as people become more health conscious and avoid fried foods. Franchise L serves breakfast and lunch, while franchise S serves only dinner, so it is possible for you to invest in both franchises. You see these franchises as perfect complements to one another: you could attract both the lunch and dinner crowds and the health conscious and not so health conscious crowds with the franchises directly competing against one another.
Here are the projects' net cash flows (in thousands of dollars):
Expected Net Cash Flow
Year Franchise L Franchise S
0 ($100) ($100)
1 10 70
2 60 50
3 80 20
Depreciation, salvage values, net working capital requirements, and tax effects are all included in these cash flows.
You also have made subjective risk assessments of each franchise, and concluded that both franchises have risk characteristics that require a return of 10 percent. You must now determine whether one or both of the projects should be accepted.
Answers and Solutions: 11  216 a. What is capital budgeting?
Answer: Capital budgeting is the process of analyzing additions to fixed assets. Capital budgeting is
important because, more than anything else, fixed asset investment decisions chart a company's
course for the future. Conceptually, the capital budgeting process is identical to the decision
process used by individuals making investment decisions. These steps are involved:
1. Estimate the cash flowsinterest and maturity value or dividends in the case of bonds and
stocks, operating cash flows in the case of capital projects.
2. Assess the riskiness of the cash flows.
3. Determine the appropriate discount rate, based on the riskiness of the cash flows and the
general level of interest rates. This is called the project cost of capital in capital budgeting.
4. Evaluate the cash flows.
b. What is the difference between independent and mutually exclusive projects?
Answer: Projects are independent if the cash flows of one are not affected by the acceptance of the other.
Conversely, two projects are mutually exclusive if acceptance of one impacts adversely the cash
flows of the other; that is, at most one of two or more such projects may be accepted. Put
another way, when projects are mutually exclusive it means that they do the same job. For
example, a forklift truck versus a conveyor system to move materials, or a bridge versus a ferry
boat.
Projects with normal cash flows have outflows, or costs, in the first year (or years) followed
by a series of inflows. Projects with nonnormal cash flows have one or more outflows after the
inflow stream has begun. Here are some examples:
Inflow (+) Or Outflow () In Year
0 1 2 3 4 5
Normal  + + + + +
  + + + +
   + + +
Nonnormal  + + + + 
 + +  + 
+ + +   
Answers and Solutions: 11  217 c. 1. What is the payback period? Find the paybacks for franchises L and S.
Answer: The payback period is the expected number of years required to recover a project's cost. We
calculate the payback by developing the cumulative cash flows as shown below for project l (in
thousands of dollars):
Expected NCF
Year Annual Cumulative
0 ($100) ($100)
1 10 (90)
2 60 Payback is between t = 2
(30)
3 80 50 and t = 3
0 1 2 3
   
100 10 60 80
90 30 +50
Franchise L's $100 investment has not been recovered at the end of year 2, but it has been more
than recovered by the end of year 3. Thus, the recovery period is between 2 and 3 years. If we
assume that the cash flows occur evenly over the year, then the investment is recovered $30/$80 =
0.375 0.4 into year 3. Therefore, paybackL = 2.4 years. Similarly, paybackS = 1.6 years.
c. 2. What is the rationale for the payback method? According to the payback criterion, which
franchise or franchises should be accepted if the firm's maximum acceptable payback is 2
years, and if franchises L and S are independent? If they are mutually exclusive?
Answer: Payback represents a type of "breakeven" analysis: the payback period tells us when the project
will break even in a cash flow sense. With a required payback of 2 years, franchise S is
acceptable, but franchise L is not. Whether the two projects are independent or mutually
exclusive makes no difference in this case.
Answers and Solutions: 11  218 c. 3. What is the difference between the regular and discounted payback periods?
Answer: Discounted payback is similar to payback except that discounted rather than raw cash flows are
used.
Setup for franchise L's discounted payback, assuming a 10% cost of capital:
Expected Net Cash Flows
Year Raw Discounted Cumulative
0 ($100) ($100.00) ($100.00)
1 10 9.09 (90.91)
2 60 49.59 (41.32)
3 80 60.11 18.79
Discounted PaybackL = 2 + ($41.32/$60.11) = 2.69 = 2.7 years.
Versus 2.4 years for the regular payback.
c. 4. What is the main disadvantage of discounted payback? Is the payback method of any real
usefulness in capital budgeting decisions?
Answer: Regular payback has two critical deficiencies: (1) it ignores the time value of money, and (2) it
ignores the cash flows that occur after the payback period. Discounted payback does consider
the time value of money, but it still fails to consider cash flows after the payback period; hence it
has a basic flaw. In spite of its deficiency, many firms today still calculate the discounted
payback and give some weight to it when making capital budgeting decisions. However,
payback is not generally used as the primary decision tool. Rather, it is used as a rough measure
of a project's liquidity and riskiness.
Answers and Solutions: 11  219 d. 1. Define the term net present value (NPV). What is each franchise's NPV?
Answer: The net present value (NPV) is simply the sum of the present values of a project's cash flows:
n
CF
NPV = (1 + rt) t .
t =0
Franchise L'S NPV is $18.79:
0 1 2 3
 10%  
(100.00) 10 60 80
9.09
49.59
60.11
18.79 = NPVL
NPVs are easy to determine using a calculator with an NPV function. Enter the cash flows
sequentially, with outflows entered as negatives; enter the cost of capital; and then press the NPV
button to obtain the project's NPV, $18.78 (note the penny rounding difference). The NPV of
franchise S is NPVS = $19.98.
d. 2. What is the rationale behind the NPV method? According to NPV, which franchise or
franchises should be accepted if they are independent? Mutually exclusive?
Answer: The rationale behind the NPV method is straightforward: if a project has NPV = $0, then the
project generates exactly enough cash flows (1) to recover the cost of the investment and (2) to
enable investors to earn their required rates of return (the opportunity cost of capital). If NPV = $0,
then in a financial (but not an accounting) sense, the project breaks even. If the NPV is positive,
then more than enough cash flow is generated, and conversely if NPV is negative.
Consider franchise L's cash inflows, which total $150. They are sufficient (1) to return the
$100 initial investment, (2) to provide investors with their 10 percent aggregate opportunity cost
of capital, and (3) to still have $18.79 left over on a present value basis. This $18.79 excess PV
belongs to the shareholdersthe debtholders' claims are fixed, so the shareholders' wealth will be
increased by $18.79 if franchise L is accepted. Similarly, Axis's shareholders gain $19.98 in
value if franchise S is accepted.
If franchises L and S are independent, then both should be accepted, because they both add to
shareholders' wealth, hence to the stock price. If the franchises are mutually exclusive, then
franchise S should be chosen over L, because s adds more to the value of the firm.
d. 3. Would the NPVs change if the cost of capital changed?
Answer: The NPV of a project is dependent on the cost of capital used. Thus, if the cost of capital changed,
the NPV of each project would change. NPV declines as r increases, and NPV rises as r falls.
Answers and Solutions: 11  220 e. 1. Define the term Internal Rate Of Return (IRR). What is each franchise's IRR?
Answer: The internal rate of return (IRR) is that discount rate which forces the NPV of a project to equal
zero:
0 1 2 3
 IRR  
CF0 CF1 CF2 CF3
PVCF1
PVCF2
PVCF3
0 = SUM OF PVs = NPV.
Expressed as an equation, we have:
n
CF
IRR: (1 + IRR ) t
t = $0 = NPV.
t =0
Note that the IRR equation is the same as the NPV equation, except that to find the IRR the
equation is solved for the particular discount rate, IRR, which forces the project's
NPV to equal zero (the IRR) rather than using the cost of capital (r) in the
denominator and finding NPV. Thus, the two approaches differ in only one respect:
in the NPV method, a discount rate is specified (the project's cost of capital) and the
equation is solved for NPV, while in the IRR method, the NPV is specified to equal
zero and the discount rate (IRR) which forces this equality is found.
Franchise L's IRR is 18.1 percent:
0 1 2
18.1% 3
   
100.00 10 60 80
8.47
43.02
48.57
$ 0.06 $0 if IRRL = 18.1% is used as the discount rate.
therefore, IRRL 18.1%.
A financial calculator is extremely helpful when calculating IRRs. The cash flows are entered
sequentially, and then the IRR button is pressed. For franchise S, IRRS 23.6%. Note that
with many calculators, you can enter the cash flows into the cash flow register, also enter r = i,
and then calculate both NPV and IRR by pressing the appropriate buttons.
e. 2. How is the IRR on a project related to the YTM on a bond?
Answers and Solutions: 11  221 Answer: The IRR is to a capital project what the YTM is to a bond. It is the expected rate of return on the
project, just as the YTM is the promised rate of return on a bond.
e. 3. What is the logic behind the IRR method? According to IRR, which franchises should be
accepted if they are independent? Mutually exclusive?
Answer: IRR measures a project's profitability in the rate of return sense: if a project's IRR equals its cost
of capital, then its cash flows are just sufficient to provide investors with their required rates of
return. An IRR greater than r implies an economic profit, which accrues to the firm's
shareholders, while an IRR less than r indicates an economic loss, or a project that will not earn
enough to cover its cost of capital.
Projects' IRRs are compared to their costs of capital, or hurdle rates. Since franchises L and
S both have a hurdle rate of 10 percent, and since both have IRRs greater than that hurdle rate,
both should be accepted if they are independent. However, if they are mutually exclusive,
franchise S would be selected, because it has the higher IRR.
e. 4. Would the franchises' IRRs change if the cost of capital changed?
Answer: IRRs are independent of the cost of capital. Therefore, neither IRRS nor IRRL would change if r
changed. However, the acceptability of the franchises could changeL would be rejected if r
were above 18.1%, and S would also be rejected if r were above 23.6%.
f. 1. Draw NPV profiles for franchises L and S. At what discount rate do the profiles cross?
Answer: the NPV profiles are plotted in the figure below.
Note the following points:
1. The yintercept is the project's NPV when r = 0%. This is $50 for L and $40 for S.
2. The xintercept is the project's IRR. This is 18.1 percent for l and 23.6 percent for S.
3. NPV profiles are curves rather than straight lines. To see this, note that these profiles
approach cost = $100 as r approaches infinity.
4. From the figure below, it appears that the crossover point is between 8 and 9 percent. The
precise value is approximately 8.7 percent. One can calculate the crossover rate by (1)
going back to the data on the problem, finding the cash flow differences for each year, (2)
entering those differences into the cash flow register, and (3) pressing the IRR button to get
the crossover rate, 8.68% 8.7%.
Answers and Solutions: 11  222 60
50
40
Crossover
Point = 8.7%
30
20
S
10
IRRS = 23.6%
L
0 Discount Rate (%)
0 5 10 15 20 23.6
10
IRRL = 18.1%
r NPVL NPVS
0% $50 $40
5 33
29
10 19
20
15 7
12
20 (4)
5
f. 2. Look at your NPV profile graph without referring to the actual NPVs and IRRs. Which
franchise or franchises should be accepted if they are independent? Mutually exclusive?
Explain. Are your answers correct at any cost of capital less than 23.6 percent?
Answer: The NPV profiles show that the IRR and NPV criteria lead to the same accept/reject decision for
any independent project. Consider franchise L. It intersects the xaxis at its IRR, 18.1 percent.
According to the IRR rule, L is acceptable if r is less than 18.1 percent. Also, at any r less than
18.1 percent, L's NPV profile will be above the x axis, so its NPV will be greater than $0. Thus,
for any independent project, NPV and IRR lead to the same accept/reject decision.
Now assume that L and S are mutually exclusive. In this case, a conflict might arise. First,
note that IRRS = 23.6% > 18.1% = therefore, regardless of the size of r, project S would be
ranked higher by the IRR criterion. However, the NPV profiles show that NPVL > NPVS if r is
less than 8.7 percent. Therefore, for any r below the 8.7% crossover rate, say r = 7 percent, the
NPV rule says choose L, but the IRR rule says choose S. Thus, if r is less than the crossover rate,
a ranking conflict occurs.
Answers and Solutions: 11  223 g. 1. What is the underlying cause of ranking conflicts between NPV and IRR?
Answer: For normal projects' NPV profiles to cross, one project must have both a higher vertical axis
intercept and a steeper slope than the other. A project's vertical axis intercept typically depends
on (1) the size of the project and (2) the size and timing pattern of the cash flowslarge projects,
and ones with large distant cash flows, would generally be expected to have relatively high
vertical axis intercepts. The slope of the NPV profile depends entirely on the timing pattern of
the cash flowslongterm projects have steeper NPV profiles than shortterm ones. Thus, we
conclude that NPV profiles can cross in two situations: (1) when mutually exclusive projects
differ in scale (or size) and (2) when the projects' cash flows differ in terms of the timing pattern
of their cash flows (as for franchises L and S).
g. 2. What is the "reinvestment rate assumption", and how does it affect the NPV versus IRR
conflict?
Answer: The underlying cause of ranking conflicts is the reinvestment rate assumption. All DCF methods
implicitly assume that cash flows can be reinvested at some rate, regardless of what is actually
done with the cash flows. Discounting is the reverse of compounding. Since compounding
assumes reinvestment, so does discounting. NPV and IRR are both found by discounting, so
they both implicitly assume some discount rate. Inherent in the NPV calculation is the
assumption that cash flows can be reinvested at the project's cost of capital, while the IRR
calculation assumes reinvestment at the IRR rate.
Answers and Solutions: 11  224 g. 3. Which method is the best? Why?
Answer: Whether NPV or IRR gives better rankings depends on which has the better reinvestment rate
assumption. Normally, the NPV's assumption is better. The reason is as follows: a project's
cash inflows are generally used as substitutes for outside capital, that is, projects' cash flows
replace outside capital and, hence, save the firm the cost of outside capital. Therefore, in an
opportunity cost sense, a project's cash flows are reinvested at the cost of capital. To see this
graphically, think of the following situation: assume the firm's cost of capital is a constant 10%
within the relevant range of financing considered, and it has projects available as shown in the
graph below:
Percent
I RRA = 25%
25
I RRB = 20%
20
IRRC = 15%
15
IRRD = 12%
MCC
10
IRRE = 8%
5 I RRF = 5%
Dollars Raised and Invested
What projects will be accepted, by either NPV or IRR? Projects A, B, C, and D.
If the same situation exists year after year, at what rate of return will cash flows from earlier
years' investments be reinvested? Capital budgeting decisions are made in this sequence: (1)
the company would say, "we can take on A, B, C, and D and finance them with 10% money, so
let's do it." (2) then, it would get cash flows from earlier years' projects. What would it do with
those cash flows? It would use them in lieu of raising money that costs 10%, so it would save
10%. Therefore, 10% is the opportunity cost of the cash flows. In effect, cash flows are
reinvested at the 10% cost of capital.
Note, however, that NPV and IRR always give the same accept/reject decisions for
independent projects, so IRR can be used just as well as NPV when independent projects are
being evaluated. The NPV versus IRR conflict arises only if mutually exclusive projects are
involved.
Answers and Solutions: 11  225 h. 1. Define the term Modified IRR (MIRR). Find the MIRRs for franchises L and S.
Answer: MIRR is that discount rate which equates the present value of the terminal value of the inflows,
compounded at the cost of capital, to the present value of the costs. Here is the setup for
calculating franchise L's modified IRR:
0 1 2 3 r = 10%
   
PV Of Costs = (100.00) 10 60 80.00
66.00
12.10
TV OF INFLOWS
= 158.10
MIRR = ?
PV Of TV = 100.00
$158.10
= $100 = .
(1 + MIRR ) 3
n
n
COF
CIFt (1 + r ) n  t
TV
PV costs =
n
= (1 + r )t t = t =1 .
(1 + MIRR ) t =0 (1 + MIRR ) n
After you calculate the TV, enter n = 3, PV = 100, pmt = 0, fv = 158.1, and then press i to get the
answer, MIRRL = 16.5%. We could calculate MIRRS similarly: = 16.9%. Thus, franchise S is
ranked higher than L. This result is consistent with the NPV decision.
h. 2. What are the MIRR's advantages and disadvantages visavis the regular IRR? What are
the MIRR's advantages and disadvantages visavis the NPV?
Answer: MIRR is a better rate of return measure than IRR for two reasons: (1) it correctly assumes
reinvestment at the project's cost of capital rather than at its IRR. (2) MIRR avoids the problem
of multiple IRRsthere can be only one MIRR for a given project.
MIRR does not always lead to the same decision as NPV when mutually exclusive projects
are being considered. In particular, small projects often have a higher MIRR, but a lower NPV,
than larger projects. Thus, MIRR is not a perfect substitute for NPV, and NPV remains the
single best decision rule. However, MIRR is superior to the regular IRR, and if a rate of return
measure is needed, MIRR should be used.
Business executives agree. As noted in the text, business executives prefer to compare
projects' rates of return to comparing their NPVs. This is an empirical fact. As a result,
financial managers are substituting MIRR for IRR in their discussions with other corporate
executives. This fact was brought out in the October 1989 FMA meetings, where executives
from Du Pont, Hershey, and Ameritech, among others, all reported a switch from IRR to MIRR.
Answers and Solutions: 11  226 i. As a separate project (project P), you are considering sponsoring a pavilion at the upcoming
world's fair. The pavilion would cost $800,000, and it is expected to result in $5 million of
incremental cash inflows during its 1 year of operation. However, it would then take
another year, and $5 million of costs, to demolish the site and return it to its original
condition. Thus, project P's expected net cash flows look like this (in millions of dollars):
Year Net Cash Flows
0 ($0.8)
1 5.0
2 (5.0)
The project is estimated to be of average risk, so its cost of capital is 10 percent.
i. 1. What are normal and nonnormal cash flows?
Answer: Normal cash flows begin with a negative cash flow (or a series of negative cash flows), switch to
positive cash flows, and then remain positive. They have only one change in sign. (Note:
normal cash flows can also start with positive cash flows, switch to negative cash flows, and then
remain negative.) Nonnormal cash flows have more than one sign change. For example, they
may start with negative cash flows, switch to positive, and then switch back to negative.
i. 2. What is project P's NPV? What is its IRR? Its MIRR?
Answer: Here is the time line for the cash flows, and the NPV:
0 1 2 10%
  
800,000 5,000,000 5,000,000
NPVP = $386,776.86.
We can find the NPV by entering the cash flows into the cash flow register, entering i = 10, and
then pressing the NPV button. However, calculating the IRR presents a problem. With the
cash flows in the register, press the IRR button. An hp10b financial calculator will give the
message "errorsoln." This means that project P has multiple IRRs. An HP17B will ask for a
guess. If you guess 10%, the calculator will produce IRR = 25%. If you guess a high number,
such as 200%, it will produce the second IRR, 400%1. The MIRR of project P = 5.6%, and is
found by computing the discount rate that equates the terminal value ($5.5 million) to the present
value of cost ($4.93 million).
1
Looking at the figure below, if you guess an IRR to the left of the peak NPV rate, the lower IRR will appear. If you guess IRR > peak NPV rate, the higher IRR will appear.
Answers and Solutions: 11  227 i. 3. Draw project P's NPV profile. Does project P have normal or nonnormal cash flows?
Should this project be accepted?
Answer: You could put the cash flows in your calculator and then enter a series of i values, get an NPV for
each, and then plot the points to construct the NPV profile. We used a spreadsheet program to
automate the process and then to draw the profile. Note that the profile crosses the xaxis twice,
at 25% and at 400%, signifying two IRRs. Which IRR is correct? In one sense, they both
areboth cause the project's NPV to equal zero. However, in another sense, both are
wrongneither has any economic or financial significance.
Project P has nonnormal cash flows; that is, it has more than one change of signs in the cash
flows. Without this nonnormal cash flow pattern, we would not have the multiple IRRs.
Since project P's NPV is negative, the project should be rejected, even though both IRRs
(25% and 400%) are greater than the project's 10 percent cost of capital. The MIRR of 5.6%
also supports the decision that the project should be rejected.
NPV
500
375
250
125
Cost of Capital, r (%)
100 200 300 400 500 600
125
250
375
Answers and Solutions: 11  228 j. In an unrelated analysis, you have the opportunity to choose between the following two
mutually exclusive projects:
Expected Net Cash Flows
Year Project S Project L
0 ($100,000) ($100,000)
1 60,000 33,500
2 60,000 33,500
3  33,500
4  33,500
The projects provide a necessary service, so whichever one is selected is expected to be
repeated into the foreseeable future. Both projects have a 10 percent cost of capital.
j. 1. What is each project's initial npv without replication?
Answer: The NPVs, found with a financial calculator, are calculated as follows:
Input the following: CF0 = 100000, CF1 = 60000, NJ = 2, AND I = 10 to solve for NPVS =
$4,132.23 $4,132.
Input the following: CF0 = 100000, CF1 = 33500, NJ = 4, AND I = 10 to solve for NPVL =
$6,190.49 $6,190.
However, if we make our decision based on the raw NPVs, we would be biasing the decision
against the shorter project. Since the projects are expected to be replicated, if we initially choose
project S, it would be repeated after 2 years. However, the raw NPVs do not reflect the
replication cash flows.
Answers and Solutions: 11  229 j. 2. Now apply the replacement chain approach to determine the projects' extended NPVs.
Which project should be chosen?
Answer: The simple replacement chain approach assumes that the projects will be replicated out to a
common life. Since project S has a 2year life and L has a 4year life, the shortest common life
is 4 years. Project L's common life NPV is its raw NPV:
Common Life NPVL = $6,190.
However, project S would be replicated in year 2, and if we assume that the replicated project's
cash flows are identical to the first set of cash flows, then the replicated NPV is also $4,132, but it
"comes in" in year 2. We can put project S's cash flow situation on a time line:
0 1 2
10%
3 4
    
4,132 4,312
3,415
7,547
Here we see that S's common life NPV is NPVS = $7,547.
Thus, when compared over a 4year common life, project s has the higher NPV, hence it
should be chosen. Project s would have the higher NPV over any common life.
j. 3. Now assume that the cost to replicate project S in 2 years will increase to $105,000 because
of inflationary pressures. How should the analysis be handled now, and which project
should be chosen?
Answer: If the cost of project S is expected to increase, the replication project is not identical to the
original, and the EAA approach cannot be used. In this situation, we would put the cash flows
on a time line as follows:
0 1 2 3
4 r = 10%
    
100,000 60,000 60,000 60,000
60,000
105,000
 45,000
Common Life NPVS = $3,415.
With this change, the common life NPV of project s is less than that for project
L, and hence project L should be chosen.
k. You are also considering another project which has a physical life of 3 years; that is, the
machinery will be totally worn out after 3 years. However, if the project were terminated
prior to the end of 3 years, the machinery would have a positive salvage value. Here are
the project's estimated cash flows:
Answers and Solutions: 11  230 Initial Investment EndOfYear
And OperatingNet Salvage
Year Cash Flows Value_
0 ($5,000) $5,000
1 2,100 3,100
2 2,000 2,000
3 1,750 0
Using the 10 percent cost of capital, what is the project's NPV if it is operated
for the full 3 years? Would the NPV change if the company planned to
terminate the project at the end of year 2? At the end of year 1? What is the
project's optimal (economic) life? Answer: Here are the time lines for the 3 alternative lives:
No termination:
0 1 2
3 10%
   
5,000 2,100 2,000 1,750
0
1,750
NPV = $123.
Terminate after 2 years:
0 1 2
 10%  
5,000 2,100 2,000
2,000
4,000
NPV = $215.
Terminate after 1 year:
0 1
 10% 
5,000 2,100
3,100
5,200
NPV = $273.
We see (1) that the project is acceptable only if operated for 2 years, and (2) that a project's
Answers and Solutions: 11  231 engineering life does not always equal its economic life.
l. After examining all the potential projects, you discover that there are many more projects
this year with positive NPVs than in a normal year. What two problems might this extra
large capital budget cause?
You only have a limited amount of capital to commit to projects. If you have to raise external
capital to fund some of these other positive NPV projects, then you may be faced with
an increasing cost of capital. This is called an increasing marginal cost of capital
schedule, and it also happens to companies when they exhaust their internal sources
of funds and have to go to external capital markets for their finding. This increased
cost of capital may cause you to reject projects that you might otherwise accept
because with your increased cost of capital, some projects may be negative NPV
when they would otherwise be positive NPV in a normal year.
Another effect of this large capital budget is that you may choose to ration capitali.e. not fund
all of the projects. This is called capital rationing, and companies and investors do this when for
whatever reason they put a cap on the funds they are willing to invest in new projects.
Answers and Solutions: 11  232 Chapter 11
Cash Flow Estimation
and Risk Analysis
ANSWERS TO ENDOFCHAPTER QUESTIONS
111 a. Cash flow, which is the relevant financial variable, represents the actual flow of cash.
Accounting income, on the other hand, reports accounting data as defined by Generally Accepted
Accounting Principles (GAAP).
b. Incremental cash flows are those cash flows that arise solely from the asset that is being evaluated.
For example, assume an existing machine generates revenues of $1,000 per year and expenses of
$600 per year. A machine being considered as a replacement would generate revenues of $1,000
per year and expenses of $400 per year. On an incremental basis, the new machine would not
increase revenues at all, but would decrease expenses by $200 per year. Thus, the annual
incremental cash flow is a beforetax savings of $200. A sunk cost is one that has already
occurred and is not affected by the capital project decision. Sunk costs are not relevant to capital
budgeting decisions. Within the context of this chapter, an opportunity cost is a cash flow that a
firm must forgo to accept a project. For example, if the project requires the use of a building that
could otherwise be sold, the market value of the building is an opportunity cost of the project.
c. Net operating working capital changes are the increases in current operating assets resulting from
accepting a project less the resulting increases in current operating liabilities, or accruals and
accounts payable. A net operating working capital change must be financed just as a firm must
finance its increases in fixed assets. Salvage value is the market value of an asset after its useful
life. Salvage values and their tax effects must be included in project cash flow estimation.
d. The real rate of return (rr), or, for that matter the real cost of capital, contains no adjustment for
expected inflation. If net cash flows from a project do not include inflation adjustments, then the
cash flows should be discounted at the real cost of capital. In a similar manner, the IRR resulting
from real net cash flows should be compared with the real cost of capital. Conversely, the
nominal rate of return (rn) does include an inflation adjustment (premium). Thus if nominal rates
of return are used in the capital budgeting process, the net cash flows must also be nominal.
e. Sensitivity analysis indicates exactly how much NPV will change in response to a given change in
an input variable, other things held constant. Sensitivity analysis is sometimes called "what if"
analysis because it answers this type of question. Scenario analysis is a shorter version of
simulation analysis that uses only a few outcomes. Often the outcomes considered are optimistic,
pessimistic and most likely. Monte Carlo simulation analysis is a risk analysis technique in
which a computer is used to simulate probable future events and thus to estimate the profitability
and risk of a project.
f. A riskadjusted discount rate incorporates the riskiness of the project's cash flows. The cost of
capital to the firm reflects the average risk of the firm's existing projects. Thus, new projects
that are riskier than existing projects should have a higher riskadjusted discount rate.
Answers and Solutions: 11  233 Conversely, projects with less risk should have a lower riskadjusted discount rate. This
adjustment process also applies to a firm's divisions. Risk differences are difficult to quantify,
thus risk adjustments are often subjective in nature. A project's cost of capital is its riskadjusted
discount rate for that project.
112 Only cash can be spent or reinvested, and since accounting profits do not represent cash, they are of
less fundamental importance than cash flows for investment analysis. Recall that in the stock
valuation chapters we focused on dividends and free cash flows, which represent cash flows, rather
than on earnings per share, which represent accounting profits.
113 Since the cost of capital includes a premium for expected inflation, failure to adjust cash flows means
that the denominator, but not the numerator, rises with inflation, and this lowers the calculated NPV.
114 Capital budgeting analysis should only include those cash flows which will be affected by the decision.
Sunk costs are unrecoverable and cannot be changed, so they have no bearing on the capital budgeting
decision. Opportunity costs represent the cash flows the firm gives up by investing in this project
rather than its next best alternative, and externalities are the cash flows (both positive and negative) to
other projects that result from the firm taking on this project. These cash flows occur only because
the firm took on the capital budgeting project; therefore, they must be included in the analysis.
115 When a firm takes on a new capital budgeting project, it typically must increase its investment in
receivables and inventories, over and above the increase in payables and accruals, thus increasing its
net operating working capital. Since this increase must be financed, it is included as an outflow in
Year 0 of the analysis. At the end of the project's life, inventories are depleted and receivables are
collected. Thus, there is a decrease in NOWC, which is treated as an inflow.
116 Simulation analysis involves working with continuous probability distributions, and the output of a
simulation analysis is a distribution of net present values or rates of return. Scenario analysis
involves picking several points on the various probability distributions and determining cash flows or
rates of return for these points. Sensitivity analysis involves determining the extent to which cash
flows change, given a change in one particular input variable. Simulation analysis is expensive.
Therefore, it would more than likely be employed in the decision for the $200 million investment in a
satellite system than in the decision for the $12,000 truck.
Answers and Solutions: 11  234 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
111 Equipment $ 9,000,000
NWC Investment 3,000,000
Initial investment outlay $12,000,000
112 Operating Cash Flows: t = 1
Sales revenues $10,000,000
Operating costs 7,000,000
Depreciation 2,000,000
Operating income before taxes $ 1,000,000
Taxes (40%) 400,000
Operating income after taxes $ 600,000
Add back depreciation 2,000,000
Operating cash flow $ 2,600,000
113 Equipment's original cost $20,000,000
Depreciation (80%) 16,000,000
Book value $ 4,000,000
Gain on sale = $5,000,000  $4,000,000 = $1,000,000.
Tax on gain = $1,000,000(0.4) = $400,000.
Answers and Solutions: 11  235 AT net salvage value = $5,000,000  $400,000 = $4,600,000.
Answers and Solutions: 11  236 114 a. The net cost is $126,000:
Price ($108,000)
Modification (12,500)
Increase in NWC (5,500)
Cash outlay for new machine ($126,000)
b. The operating cash flows follow:
Year 1 Year 2 Year 3
1. Aftertax savings $28,600 $28,600 $28,600
2. Depreciation tax savings 13,918 18,979 6,326
Net cash flow $42,518 $47,579 $34,926
Notes:
1. The aftertax cost savings is $44,000(1  T) = $44,000(0.65)
= $28,600.
2. The depreciation expense in each year is the depreciable basis, $120,500, times the MACRS
allowance percentages of 0.33, 0.45, and 0.15 for Years 1, 2, and 3, respectively.
Depreciation expense in Years 1, 2, and 3 is $39,765, $54,225, and $18,075. The
depreciation tax savings is calculated as the tax rate (35%) times the depreciation expense in
each year.
c. The terminal year cash flow is $50,702:
Salvage value $65,000
Tax on SV* (19,798)
Return of NWC 5,500
$50,702
BV in Year 4 = $120,500(0.07) = $8,435.
*Tax on SV = ($65,000  $8,435)(0.35) = $19,798.
d. The project has an NPV of $10,841; thus, it should be accepted.
Year Net Cash Flow PV @ 12%
0 ($126,000) ($126,000)
1 42,518 37,963
2 47,579 37,930
3 85,628 60,948
NPV = $ 10,841
Alternatively, place the cash flows on a time line:
0 1 2 3 12%
   
126,000 42,518 47,579 34,926
Answers and Solutions: 11  237 50,702
85,628
With a financial calculator, input the appropriate cash flows into the cash flow register, input I =
12, and then solve for NPV = $10,841.
115 a. The net cost is $89,000:
Price ($70,000)
Modification (15,000)
Change in NWC (4,000)
($89,000)
b. The operating cash flows follow:
Year 1 Year 2 Year 3
Aftertax savings $15,000 $15,000 $15,000
Depreciation shield 11,220 15,300 5,100
Net cash flow $26,220 $30,300 $20,100
Notes:
1. The aftertax cost savings is $25,000(1 T) = $25,000(0.6)
= $15,000.
2. The depreciation expense in each year is the depreciable basis, $85,000, times the MACRS
allowance percentage of 0.33, 0.45, and 0.15 for Years 1, 2 and 3, respectively. Depreciation
expense in Years 1, 2, and 3 is $28,050, $38,250, and $12,750. The depreciation shield is
calculated as the tax rate (40%) times the depreciation expense in each year.
c. The additional endofproject cash flow is $24,380:
Salvage value $30,000
Tax on SV* (9,620)
Return of NWC 4,000
$24,380
*Tax on SV = ($30,000  $5,950)(0.4) = $9,620.
Note that the remaining BV in Year 4 = $85,000(0.07) = $5,950.
d. The project has an NPV of $6,705. Thus, it should not be accepted.
Year Net Cash Flow PV @ 10%
0 ($89,000) ($89,000)
1 26,220 23,836
2 30,300 25,041
3 44,480 33,418
NPV = ($ 6,705)
Answers and Solutions: 11  238 Alternatively, with a financial calculator, input the following: CF0 = 89000, CF1 = 26220, CF2
= 30300, CF3 = 44480, and I = 10 to solve for NPV = $6,703.83.
116 a. Sales = 1,000($138) $138,000
Cost = 1,000($105) 105,000
Net before tax $ 33,000
Taxes (34%) 11,220
Net after tax $ 21,780
Not considering inflation, NPV is $4,800. This value is calculated as
$21,780
$150,000 + = $4,800.
0.15
Considering inflation, the real cost of capital is calculated as follows:
(1 + rr)(1 + i) = 1.15
(1 + rr)(1.06) = 1.15
rr = 0.0849.
Thus, the NPV considering inflation is calculated as
$21,780
$150,000 + = $106,537.
0.0849
After adjusting for expected inflation, we see that the project has a positive NPV and should be
accepted. This demonstrates the bias that inflation can induce into the capital budgeting process:
Inflation is already reflected in the denominator (the cost of capital), so it must also be reflected in
the numerator.
b. If part of the costs were fixed, and hence did not rise with inflation, then sales revenues would rise
faster than total costs. However, when the plant wears out and must be replaced, inflation will
cause the replacement cost to jump, necessitating a sharp output price increase to cover the now
higher depreciation charges.
117 E(NPV) = 0.05($70) + 0.20($25) + 0.50($12) + 0.20($20) + 0.05($30)
= $3.5 + $5.0 + $6.0 + $4.0 + $1.5
= $3.0 million.
NPV = [0.05($70  $3)2 + 0.20($25  $3)2 + 0.50($12  $3)2
+ 0.20($20  $3)2 + 0.05($30  $3)2]0.5
= $23.622 million.
$23.622
CV = = 7.874.
$3.0
Answers and Solutions: 11  239 118 a. Expected annual cash flows:
Project A: Probable
Probability × Cash Flow = Cash Flow
0.2 $6,000 $1,200
0.6 6,750 4,050
0.2 7,500 1,500
Expected annual cash flow = $6,750
Project B: Probable
Probability × Cash Flow = Cash Flow
0.2 $ 0 $ 0
0.6 6,750 4,050
0.2 18,000 3,600
Expected annual cash flow = $7,650
Coefficient of variation:
Standard deviation NPV
CV = =
Expected value Expected NPV
Project A:
A = ($750 ) 2 (0.2) + ($0 ) 2 (0.6) + ($750 ) 2 (0.2) = $474.34.
Project B:
B = ($7,650 ) 2 (0.2) + ($900 ) 2 (0.6) + ($10,350 ) 2 (0.2)
= $5,797.84.
CVA = $474.34/$6,750 = 0.0703.
CVB = $5,797.84/$7,650 = 0.7579.
b. Project B is the riskier project because it has the greater variability in its probable cash flows,
whether measured by the standard deviation or the coefficient of variation. Hence, Project B is
evaluated at the 12 percent cost of capital, while Project A requires only a 10 percent cost of
capital.
Project A: With a financial calculator, input the appropriate cash flows into the cash flow register,
input I = 10, and then solve for NPV = $10,036.25.
Project B: With a financial calculator, input the appropriate cash flows into the cash flow register,
input I = 12, and then solve for NPV = $11,624.01.
Project B has the higher NPV; therefore, the firm should accept Project B.
c. The portfolio effects from Project B would tend to make it less risky than otherwise. This would
tend to reinforce the decision to accept Project B. Again, if Project B were negatively correlated
with the GDP (Project B is profitable when the economy is down), then it is less risky and Project
B's acceptance is reinforced.
Answers and Solutions: 11  240 119 a. First, note that with symmetric probability distributions, the middle value of each distribution is
the expected value. Therefore,
Expected Values
Sales (units) 200
Sales price $13,500
Sales in dollars $2,700,000
Costs (200 x $6,000) 1,200,000
Earnings before taxes $1,500,000
Taxes (40%) 600,000
Net income $ 900,000 =Cash flow under the assumption
used in the problem.
8
$900,000
0= t
 $4,000,000.
t =1 (1 + IRR )
Using a financial calculator, input the following: CF0 = 4000000, CF1 = 900000, and Nj = 8, to
solve for IRR = 15.29%.
Expected IRR = 15.29% 15.3%.
Assuming complete independence between the distributions, and normality, it would be possible
to derive IRR statistically. Alternatively, we could employ simulation to develop a distribution of
IRRs, hence IRR. There is no easy way to get IRR.
b. NPV = $900,000(PVIFA15%,8)  $4,000,000.
Using a financial calculator, input the following: CF0 = 4000000, CF1 = 900000, Nj = 8, and I =
15 to solve for NPV = $38,589.36. Again, there is no easy way to estimate NPV.
c. (1) a. Calculate developmental costs. The 44 random number value, coming between 30
and 70, indicates that the costs for this run should be taken to be $4 million.
b. Calculate the project life. The 17, being less than 20, indicates that a 3year life
should be used.
(2) a. Estimate unit sales. The 16 indicates sales of 100 units.
b. Estimate the sales price. The 58 indicates a sales price of $13,500.
c. Estimate the cost per unit. The 1 indicates a cost of $5,000.
d. Now estimate the aftertax cash flow for Year 1. It is
[100($13,500)  100($5,000)](1  0.4) = $510,000 = CF1.
(3) Repeat the process for Year 2. Sales will be 200 with a random number of 79; the price
will be $13,500 with a random number of 83; and the cost will be $7,000 with a random
Answers and Solutions: 11  241 number of 86:
[200($13,500)  200($7,000)](0.6) = $780,000 = CF2.
(4) Repeat the process for Year 3. Sales will be 100 units with a random number of 19; the
price will be $13,500 with a random number of 62; and the cost will be $5,000 with a
random number of 6:
[100($13,500)  100($5,000)](0.6) = $510,000 = CF3.
$510,000 $780,000 $510,000 (5) a. 0= + +  $4,000,000
(1 + IRR )1 (1 + IRR ) 2 (1 + IRR ) 3
IRR = 31.55%.
Alternatively, with a financial calculator, input the following: CF0 = 4000000, CF1 =
510000, CF2 = 780000, CF3 = 510000, and solve for IRR = 31.55%.
$510,000 $780,000 $510,000
b. NPV = + +  $4,000,000.
1 2
(1.15) (1.15) (1.15) 3
With a financial calculator, input the following: CF0 = 4000000, CF1 = 510000, CF2
= 780000, CF3 = 510000, and I = 15 to solve for NPV = $2,631,396.40.
The results of this run are very bad because the project's life is so short. Had
the life turned out (by chance) to be 13 years, the longest possible life, the IRR would
have been about 25%, and the NPV would have been about $1 million.
Answers and Solutions: 11  242 (6) & (7) The computer would store NPVs and IRRs for the different trials, then display them
as frequency distributions:
Probability
of occurrence
X
XX
XXXX
XXXXXXXX
XXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXX
0 E(NPV)
NPV
Probability
of occurrence
X
XX
XXXX
XXXXXXXX
XXXXXXXXXXXXXXX
XXXXXXXXXXXXXXXXXXX
0 E(NPV)
NPV
The distribution would be reasonably symmetrical because all the input data were
from symmetrical distributions. One often finds, however, that the input and output
distributions are badly skewed. The frequency values would also be used to
calculate NPV and IRR; these values would be printed out and available for
analysis.
Answers and Solutions: 11  243 1110 a. The resulting decision tree is:
NPV
t=0 t=1 t=2 t=3 P NPV Product
$3,000,000 0.24 $881,718 $211,612
($1,000,000) P = 0.5
P = 0.80 1,500,000 0.24 (185,952) (44,628)
($500,000) P = 0.5
P = 0.60
100,000 0.12 (376,709) (45,205)
($10,000) P = 0.20
0 0.40
(10,000) (4,000)
P = 0.40 1.00 Exp. NPV = $117,779
The NPV of the top path is:
$3,000,000 $1,000,000 $500,000
3
 2
  $10,000 = $881,718.
(1.12) (1.12) (1.12)1
Using a financial calculator, input the following: CF0 = 10000,
CF1 = 500000, CF2 = 1000000, CF3 = 3000000, and I = 12 to solve for NPV = $881,718.29
$881,718. The other NPVs were determined in the same manner. If the project is of average risk, it
should be accepted because the expected NPV of the total project is positive.
b. 2NPV = 0.24($881,718  $117,779)2 + 0.24($185,952  $117,779)2
+ 0.12($376,709  $117,779)2 + 0.4($10,000  $117,779)2
= 198,078,470,853.
NPV = $445,060.
$445,060
CVNPV = = 3.78.
$117,779
Since the CV is 3.78 for this project, while the firm's average project has a CV of 1.0 to
2.0, this project is of high risk.
Answers and Solutions: 11  244 SOLUTION TO SPREADSHEET PROBLEM
1111 The detailed solution for the problem is available both on the instructor's resource CDROM (in the
file Solution for FM11 Ch 11 P11 Build a Model.xls) and on the instructor's side of the web site,
http://brigham.swcollege.com.
Answers and Solutions: 11  245 MINI CASE
Shrieves Casting Company is considering adding a new line to its product mix, and the capital budgeting analysis is being conducted by Sidney Johnson, a recently graduated MBA. The production line would be set up in unused space in Shrieves' main plant. The machinery's invoice price would be approximately $200,000; another $10,000 in shipping charges would be required; and it would cost an additional $30,000 to install the equipment. The machinery has an economic life of 4 years, and Shrieves has obtained a special tax ruling which places the equipment in the MACRS 3year class. The machinery is expected to have a salvage value of $25,000 after 4 years of use.
THE NEW LINE WOULD GENERATE INCREMENTAL SALES OF 1,250 UNITS PER YEAR FOR FOUR YEARS AT AN INCREMENTAL COST OF $100 PER UNIT IN THE FIRST YEAR, EXCLUDING DEPRECIATION. EACH UNIT CAN BE SOLD FOR $200 IN THE FIRST YEAR. THE SALES PRICE AND COST ARE EXPECTED TO INCREASE BY 3% PER YEAR DUE TO INFLATION. FURTHER, TO HANDLE THE NEW LINE, THE FIRM'S NET OPERATING WORKING CAPITAL WOULD HAVE TO INCREASE BY AN AMOUNT EQUAL TO 12% OF SALES REVENUES. THE FIRM'S TAX RATE IS 40 PERCENT, AND ITS OVERALL WEIGHTED AVERAGE COST OF CAPITAL IS 10 PERCENT.
a. Define "incremental cash flow."
Answer: This is the firm's cash flow with the project minus the firm's cash flow without the project.
a. 1. Should you subtract interest expense or dividends when calculating project cash
flow?
Answer: The cash flow statement should not include interest expense or dividends. The return required
by the investors furnishing the capital is already accounted for when we apply the 10 percent cost
of capital discount rate, hence including financing flows would be "double counting." Put
another way, if we deducted capital costs in the table, and thus reduced the bottom line cash flows,
and then discounted those CFS by the cost of capital, we would, in effect, be subtracting capital
costs twice.
a. 2. Suppose the firm had spent $100,000 last year to rehabilitate the production line site.
Should this cost be included in the analysis? Explain.
Answer: The $100,000 cost to rehabilitate the production line site was incurred last year, and presumably
also expensed for tax purposes. Since, it is a sunk cost, it should not be included in the analysis.
a. 3. Now assume that the plant space could be leased out to another firm at $25,000 a year.
Should this be included in the analysis? If so, how?
Answer: If the plant space could be leased out to another firm, then if Shrieves accepts this project, it
would forgo the opportunity to receive $25,000 in annual cash flows. This represents an
Answers and Solutions: 12  246 opportunity cost to the project, and it should be included in the analysis. Note that the
opportunity cost cash flow must be net of taxes, so it would be a $25,000(1  t) = $25,000(0.6) =
$15,000 annual outflow.
a. 4. Finally, assume that the new product line is expected to decrease sales of the firm's other
lines by $50,000 per year. Should this be considered in the analysis? If so, how?
Answer: If a project affects the cash flows of another project, this is an "externality" which must be
considered in the analysis. If the firm's sales would be reduced by $50,000, then the net cash
flow loss would be a cost to the project. Note that this annual loss would not be the full $50,000,
because Shrieves would save on cash operating costs if its sales dropped. Note also that
externalities can be positive as well as negative.
b. Disregard the assumptions in part a. What is Shrieves' depreciable basis?
Answer: Get the depreciation rates from table 112 in the book. Note that because of the halfyear
convention, a 3year project is depreciated over 4 calendar years:
YEAR RATE × BASIS = DEPRECIATION
1 0.33 $240 $ 79
2 0.45 240 108
3 0.15 240 36
4 0.07 240 17
$240
c. Calculate the annual sales revenues and costs (other than depreciation). Why is it
important to include inflation when estimating cash flows?
Answer: With an inflation rate of 3%, the annual revenues and costs are:
Year 1 Year 2 Year 3 Year 4
Units 1250 1250 1250 1250
Unit Price $200.00 $206.00 $212.18 $218.55
Unit Cost $100.00 $103.00 $106.09 $109.27
Sales $250,000 $257,500 $265,225 $273,188
Costs $125,000 $128,750 $132,613 $136,588
The cost of capital is a nominal cost; i.e., it includes a premium for inflation. In other words, it
is larger than the real cost of capital. Similarly, nominal cash flows (those that are inflated) are
larger than real cash flows. If you discount the low, real cash flows with the high, nominal rate,
then the resulting NPV is too low. Therefore, you should always discount nominal cash flows
with a nominal rate, and real cash flows with a real rate. In theory, you could do either way and
get the correct answer. However, there is no accurate way to convert a nominal cost of capital to
a real cost. Therefore, you should inflate cash flows and then discount at the nominal rate.
Answers and Solutions: 12  247 c. Calculate the annual sales revenues and costs (other than depreciation). Why is it
important to include inflation when estimating cash flows?
Answer: With an inflation rate of 3%, the annual revenues and costs are:
Here are the annual operating cash flows (in thousands of dollars):
1 2 3 4
Net Revenues $125 $125 $125 $125
Depreciation 79 108 36 17
BeforeTax Income $ 46 $ 17 $ 89 $108
Taxes (40%) 18 7 36 43
Net Income $ 28 $ 10 $ 53 $ 65
Plus Depreciation 79 108 36 17
Net Operating CF $107 $118 $ 89 $ 82
d. Construct annual incremental operating cash flow statements.
Answer:
Year 1 Year 2 Year 3 Year 4
Sales $250,000 $257,500 $265,225 $273,188
Costs $125,000 $128,750 $132,613 $136,588
Depreciation $79,200 $108,000 $36,000 $16,800
Op. EBIT $45,800 $20,750 $96,612 $119,800
Taxes (40%) $18,320 $8,300 $38,645 $47,920
NOPAT $27,480 $12,450 $57,967 $71,880
Depreciation $79,200 $108,000 $36,000 $16,800
Net Operating CF $106,680 $120,450 $93,967 $88,680
e. Estimate the required net operating working capital for each year, and the cash flow due to
investments in net operating working capital.
Answer: The project requires a level of net operating working capital in the amount equal to 12% of the
next year's sales. Any increase in NOWC is a negative cash flow, and any decrease is a positive
cash flow.
Year 0 Year 1 Year 2 Year 3 Year 4
Sales $250,000 $257,500 $265,225 $273,188
NOWC (% of sales) $30,000 $30,900 $31,827 $32,783 $0
CF due to NOWC) ($30,000) ($900) ($927) ($956) $32,783
f. Calculate the aftertax salvage cash flow.
Answers and Solutions: 12  248 Answer: When the project is terminated at the end of year 4, the equipment can be sold for $25,000. But,
since it has been depreciated to a $0 book value, taxes must be paid on the full salvage value.
For this project, the aftertax salvage cash flow is:
Salvage Value $25,000
Tax On Salvage Value (10,000)
Net AfterTax Salvage Cash Flow $15,000
g. Calculate the net cash flows for each year? Based on these cash flows, what are the
project's NPV, IRR, MIRR, and payback? Do these indicators suggest that the project
should be undertaken?
Answer: The net cash flows are:
Year 0 Year 1 Year 2 Year 3 Year 4
Initial Outlay ($240,000)
Operating Cash Flows $106,680 $120,450 $93,967 $88,680
CF Due To NOWC ($30,000) ($900) ($927) ($956) $32,783
Salvage Cash Flows $15,000
Net Cash Flows ($270,000) $105,780 $119,523 $93,011 $136,463
NPV = $88,030
IRR = 23.9%
MIRR = 18.0%
Payback = 2.5
h. What does the term "risk" mean in the context of capital budgeting, to what extent can risk
be quantified, and when risk is quantified, is the quantification based primarily on statistical
analysis of historical data or on subjective, judgmental estimates?
Answer: Risk throughout finance relates to uncertainty about future events, and in capital budgeting, this
means the future profitability of a project. For certain types of projects, it is possible to look back
at historical data and to statistically analyze the riskiness of the investment. This is often true
when the investment involves an expansion decision; for example, if Sears were opening a new
store, if Citibank were opening a new branch, or if GM were expanding its Chevrolet plant, then
past experience could be a useful guide to future risk. Similarly, a company that is considering
going into a new business might be able to look at historical data on existing firms in that industry
to get an idea about the riskiness of its proposed investment. However, there are times when it is
impossible to obtain historical data regarding proposed investments; for example, if GM were
considering the development of an electric auto, not much relevant historical data for assessing
the riskiness of the project would be available. Rather, GM would have to rely primarily on the
judgment of its executives, and they, in turn would have to rely on their experience in developing,
manufacturing, and marketing new products. We will try to quantify risk analysis, but you must
Answers and Solutions: 12  249 recognize at the outset that some of the data used in the analysis will necessarily be based on
subjective judgments rather than on hard statistical observations.
i. 1. What are the three types of risk that are relevant in capital
budgeting?
2. How is each of these risk types measured, and how do they relate to one another?
Answer: Here are the three types of project risk:
· Standalone risk is the project's total risk if it were operated independently.
Standalone risk ignores both the firm's diversification among projects and investors'
diversification among firms. Standalone risk is measured either by the project's
standard deviation of NPV (NPV) or its coefficient of variation of NPV (CVNPV).
Note that other profitability measures, such as IRR and MIRR, can also be used to
obtain standalone risk estimates.
· Withinfirm risk is the total riskiness of the project giving consideration to the firm's
other projects, that is, to diversification within the firm. It is the contribution of the
project to the firm's total risk, and it is a function of (a) the project's standard
deviation of NPV and (b) the correlation of the projects' returns with those of the rest
of the firm. Withinfirm risk is often called corporate risk, and it is measured by the
project's corporate beta, which is the slope of the regression line formed by plotting
returns on the project versus returns on the firm.
· Market risk is the riskiness of the project to a welldiversified investor, hence it
considers the diversification inherent in stockholders' portfolios. It is measured by
the project's market beta, which is the slope of the regression line formed by plotting
returns on the project versus returns on the market.
i. 3. How is each type of risk used in the capital budgeting process?
Answer: Because management's primary goal is shareholder wealth maximization, the most relevant risk
for capital projects is market risk. However, creditors, customers, suppliers, and employees are
all affected by a firm's total risk. Since these parties influence the firm's profitability, a project's
withinfirm risk should not be completely ignored.
Unfortunately, by far the easiest type of risk to measure is a project's standalone risk.
Thus, firms often focus on this type of risk when making capital budgeting
decisions. However, this focus does not necessarily lead to poor decisions,
because most projects that a firm undertakes are in its core business. In this
situation, a project's standalone risk is likely to be highly correlated with its
withinfirm risk, which in turn is likely to be highly correlated with its market
risk.
Answers and Solutions: 12  250 j. 1. What is sensitivity analysis?
Answer: Sensitivity analysis measures the effect of changes in a particular variable, say revenues, on a
project's NPV. To perform a sensitivity analysis, all variables are fixed at their expected values
except one. This one variable is then changed, often by specified percentages, and the resulting
effect on NPV is noted. (One could allow more than one variable to change, but this then
merges sensitivity analysis into scenario analysis.)
j. 2. Perform a sensitivity analysis on the unit sales, salvage value, and cost of capital for the
project. Assume that each of these variables can vary from its base case, or expected, value
by plus and minus 10, 20, and 30 percent. Include a sensitivity diagram, and discuss the
results.
Answer: The sensitivity data are given here in tabular form (in thousands of dollars):
NPV Deviation From Base Case
Deviation
From Units
Base Case WACC Sold Salvage
30% $113,288 $16,668 $84,956
15% $100,310 $52,348 $86,493
0% $88,030 $88,030 $88,030
15% $76,398 $123,711 $89,567
30% $65,371 $159,392 $91,103
Range 47,916 176,060 6,147
We generated these data with a spreadsheet model in the file ch 11 mini case.xls.
Answers and Solutions: 12  251 WACC
Sensitivity Analysis Units Sold
Salvage
$180,000
$160,000
$140,000
$120,000
$100,000 NPV
$80,000
$60,000
$40,000
$20,000
$0
40% 20% 0% 20% 40%
Deviation from BaseCase Value
A. The sensitivity lines intersect at 0% change and the base case NPV, $81,573. Since all other
variables are set at their base case, or expected values, the zero change situation is the base
case.
B. The plots for unit sales and salvage value are upward sloping, indicating that higher variable
values lead to higher NPVs. Conversely, the plot for cost of capital is downward sloping,
because a higher cost of capital leads to a lower NPV.
C. The plot of unit sales is much steeper than that for salvage value. This indicates that NPV is
more sensitive to changes in unit sales than to changes in salvage value.
D. Steeper sensitivity lines indicate greater risk. Thus, in comparing two projects, the one with
the steeper lines is considered to be riskier.
Answers and Solutions: 12  252 j. 3. What is the primary weakness of sensitivity analysis? What is its primary usefulness?
Answer: The two primary disadvantages of sensitivity analysis are (1) that it does not reflect the effects of
diversification and (2) that it does not incorporate any information about the possible magnitudes
of the forecast errors. Thus, a sensitivity analysis might indicate that a project's NPV is highly
sensitive to the sales forecast, hence that the project is quite risky, but if the project's sales, hence
its revenues, are fixed by a longterm contract, then sales variations may actually contribute little
to the project's risk. It also ignores any relationships between variables, such as unit sales and
sales price. Therefore, in many situations, sensitivity analysis is not a particularly good indicator of
risk. However, sensitivity analysis does identify those variables which
potentially have the greatest impact on profitability, and this helps management
focus its attention on those variables that are probably most important.
k. Assume that Sidney Johnson is confident of her estimates of all the variables that affect the
project's cash flows except unit sales and sales price: if product acceptance is poor, unit
sales would be only 900 units a year and the unit price would only be $160; a strong
consumer response would produce sales of 1,600 units and a unit price of $240. Sidney
believes that there is a 25 percent chance of poor acceptance, a 25 percent chance of
excellent acceptance, and a 50 percent chance of average acceptance (the base case).
k. 1. What is scenario analysis?
Answer: Scenario analysis examines several possible situations, usually worst case, most likely case, and
best case. It provides a range of possible outcomes.
k. 2. What is the worstcase NPV? The bestcase NPV?
k. 3. Use the worst, most likely, and bestcase NPVs and probabilities of occurrence to find the
project's expected NPV, standard deviation, and coefficient of variation.
Answer: We used a spreadsheet model to develop the scenarios (in thousands of dollars), which are
summarized below:
Scenario Probability Unit Sales Unit Price NPV
Best Case 25% 1600 $240 $278,965
Base Case 50% 1250 $200 $88,030
Worst Case 25% 900 $160 ($48,514)
Expected NPV = $101,628
Standard Deviation = $116,577
Coefficient Of Variation =
Std Dev / Expected NPV = 1.15
Answers and Solutions: 12  253 l. Are there problems with scenario analysis? Define simulation analysis, and discuss its
principal advantages and disadvantages.
Answer: Scenario analysis examines several possible scenarios, usually worst case, most likely case, and
best case. Thus, it usually considers only 3 possible outcomes. Obviously the world is much
more complex, and most projects have an almost infinite number of possible outcomes.
Simulation analysis is a type of scenario analysis which uses a relatively powerful financial
planning software such as interactive financial planning system (IFPs) or @risk (a spreadsheet
addin). Simple simulations can also be conducted with other spreadsheet addins, such as
Simtools. Here the uncertain cash flow variables (such as unit sales) are entered as continuous
probability distribution parameters rather than as point values. Then, the computer uses a
random number generator to select values for the uncertain variables on the basis of their
designated distributions. Once all of the variable values have been selected, they are combined,
and an NPV is calculated. The process is repeated many times, say 1,000, with new values
selected from the distributions for each run. The end result is a probability distribution of NPV
based on a sample of 1,000 values. The software can graph the distribution as well as print out
summary statistics such as expected NPV and NPV. Simulation provides the decision maker
with a better idea of the profitability of a project than does scenario analysis because it
incorporates many more possible outcomes.
Although simulation analysis is technically refined, its usefulness is limited because
managers are often unable to accurately specify the variables' probability
distributions. Further, the correlations among the uncertain variables must be
specified, along with the correlations over time. If managers are unable to do
this with much confidence, then the results of simulation analyses are of limited
value.
Recognize also that neither sensitivity, scenario, nor simulation analysis provides a decision
rulethey may indicate that a project is relatively risky, but they do not indicate
whether the project's expected return is sufficient to compensate for its risk. Finally, remember that sensitivity, scenario, and simulation analyses all focus on
standalone risk, which is not the most relevant risk in capital budgeting
analysis.
m. 1. Assume that Shrieves' average project has a coefficient of variation in the range of 0.2  0.4.
Would the new line be classified as high risk, average risk, or low risk? What type of risk
is being measured here?
Answer: The project has a CV of 0.57, which is above the average range of 0.20.4, so it falls into the high
risk category. The CV measures a project's standalone riskit is merely a measure of the
variability of returns (as measured by NPV) about the expected return.
m. 2. Shrieves typically adds or subtracts 3 percentage points to the overall cost of capital to
adjust for risk. Should the new furniture line be accepted?
Answers and Solutions: 12  254 Answer: Since the project is judged to have aboveaverage risk, its differential riskadjusted, or project,
cost of capital would be 13 percent. At this discount rate, its NPV would be $60,541, so it would
still be acceptable. If it were a low risk project, its cost of capital would be 7 percent, its NPV
would be $104,975, and it would be an even more profitable project on a riskadjusted basis.
m. 3. Are there any subjective risk factors that should be considered before the final decision is
made?
Answer: A numerical analysis such as this one may not capture all of the risk factors inherent in the project.
If the project has a potential for bringing on harmful lawsuits, then it might be riskier than first
assessed. Also, if the project's assets can be redeployed within the firm or can be easily sold,
then, as a result of "abandonment possibilities," the project may be less risky than the analysis
indicates.
Answers and Solutions: 12  255 Chapter 12
Real Options
ANSWERS TO ENDOFCHAPTER QUESTIONS
121 a. Real options occur when managers can influence the size and risk of a project's cash flows by
taking different actions during the project's life. They are referred to as real options because
they deal with real as opposed to financial assets. They are also called managerial options because
they give opportunities to managers to respond to changing market conditions. Sometimes they
are called strategic options because they often deal with strategic issues. Finally, they are also
called embedded options because they are a part of another project.
b. Investment timing options give companies the option to delay a project rather than implement it
immediately. This option to wait allows a company to reduce the uncertainty of market
conditions before it decides to implement the project. Capacity options allow a company to
change the capacity of their output in response to changing market conditions. This includes the
option to contract or expand production. Growth options allow a company to expand if market
demand is higher than expected. This includes the opportunity to expand into different
geographic markets and the opportunity to introduce complementary or secondgeneration
products. It also includes the option to abandon a project if market conditions deteriorate too
much.
c. Decision trees are a form of scenario analysis in which different actions are taken in different
scenarios.
122 Postponing the project means that cash flows come later rather than sooner; however, waiting may
allow you to take advantage of changing conditions. It might make sense, however, to proceed today if
there are important advantages to being the first competitor to enter a market.
123 Timing options make it less likely that a project will be accepted today. Often, if a firm can delay a
decision, it can increase the expected NPV of a project.
124 Having the option to abandon a project makes it more likely that the project will be accepted today.
SOLUTIONS TO ENDOFCHAPTER PROBLEMS
Answers and Solutions: 12  256 121 a. 0 1 2 20
· · ·
20 3 3 3
NPV = $1.074 million.
b. Wait 1 year:
PV @
0 1 2 3 21 Yr. 1
Tax imposed  r= 13%    ··· 
50% Prob. 0 20 2.2 2.2 2.2 15.45
Tax not imposed     ··· 
50% Prob. 0 20 3.8 3.8 3.8 26.69
Tax imposed: NPV @ Yr. 1 = (20 + 15.45)/(1.13) = 4.027
Tax not imposed: NPV @ Yr 1 = (20 + 26.69)/ (1.13) = 5.920
Expected NPV = .5(4.027) + .5(5.920) = 0.947
Note though, that if the tax is imposed, the NPV of the project is negative and therefore would not
be undertaken. The value of this option of waiting one year is evaluated as 0.5($0) + (0.5)($ 5.920)
= $2.96 million.
Since the NPV of waiting one year is greater than going ahead and proceeding with the project today,
it makes sense to wait.
Answers and Solutions: 12  257 122 a. 0 1 2 3 4
10%
8 4 4 4 4
NPV = $4.6795 million.
b. Wait 2 years:
PV @
0 1 2 3 4 5 6 Yr. 2
 r = 10%      
10% Prob. 0 0 9 2.2 2.2 2.2 2.2 $6.974
      
90% Prob. 0 0 9 4.2 4.2 4.2 4.2 $13.313
Low CF scenario: NPV = (9 + 6.974)/(1.1)2 = $1.674
High CF scenario: NPV = (9 + 13.313)/(1.1)2 = $3.564
Expected NPV = .1(1.674) + .9(3.564) = 3.040
If the cash flows are only $2.2 million, the NPV of the project is negative and, thus,
would not be undertaken. The value of the option of waiting two years is
evaluated as 0.10($0) + 0.90($3.564) = $3.208 million.
Since the NPV of waiting two years is less than going ahead and proceeding with
the project today, it makes sense to drill today.
Answers and Solutions: 12  258 123 a. 0 1 2 20
13%
· · ·
300 40 40 40
NPV = $19.0099 million. Don't purchase.
b. Wait 1 year:
NPV @
0 1 2 3 4 21 Yr. 0
 r = 13%     ··· 
50% Prob. 0 300 30 30 30 30 $78.9889
     ··· 
50% Prob. 0 300 50 50 50 50 45.3430
If the cash flows are only $30 million per year, the NPV of the project is negative. However, we've
not considered the fact that the company could then be sold for $280 million. The decision tree
would then look like this:
NPV @
0 r = 13% 1 2 3 4 21 Yr. 0
     ··· 
50% Prob. 0 300 30 30 + 280 0 0 $27.1468
     ··· 
50% Prob. 0 300 50 50 50 50 45.3430
The expected NPV of waiting 1 year is 0.5($27.1468) + 0.5($45.3430) = $9.0981 million.
Given the option to sell, it makes sense to wait 1 year before deciding whether to make the
acquisition.
Answers and Solutions: 12  259 124 a. 0 1 14 12%
15
  ···  
6,200,000 600,000 600,000 600,000
Using a financial calculator, input the following data: CF0 = 6,200,000;
CF115 = 600,000; I = 12; and then solve for NPV = $2,113,481.31.
b. 0 1
12% 14 15
  ···  
6,200,000 1,200,000 1,200,000 1,200,000
Using a financial calculator, input the following data: CF0 = 6,200,000;
CF115 = 1,200,000; I = 12; and then solve for NPV = $1,973,037.39.
c. If they proceed with the project today, the project's expected NPV = (0.5 × $2,113,481.31) + (0.5 ×
$1,973,037.39) = $70,221.96. So, Hart Enterprises would not do it.
d. Since the project's NPV with the tax is negative, if the tax were imposed the firm would abandon
the project. Thus, the decision tree looks like this:
NPV @
0 1 2 15 Yr. 0
50% Prob.  r= 12%   ··· 
Taxes 6,200,000 6,000,000 0 0 $ 842,857.14
No Taxes    ··· 
50% Prob. 6,200,000 1,200,000 1,200,000 1,200,000 1,973,037.39
Expected NPV $ 565,090.13
Yes, the existence of the abandonment option changes the expected NPV of the project from
negative to positive. Given this option the firm would take on the project because its expected
NPV is $565,090.13.
e. NPV @
0 1 Yr. 0
50% Prob.  r = 12% 
Taxes NPV = ? 1,500,000 $ 0.00
+300,000 = NPV @ t = 1 }wouldn't do
No Taxes  
50% Prob. NPV = ? 1,500,000 2,232,142.86
+4,000,000 = NPV @ t = 1Expected NPV $1,116,071.43
If the firm pays $1,116,071.43 for the option to purchase the land, then the NPV of the project is
exactly equal to zero. So the firm would not pay any more than this for the option.
Answers and Solutions: 12  260 125 P = PV of all expected future cash flows if project is delayed. From Problem 153 we know that PV
@ Year 1 of Tax Imposed scenario is $15.45 and PV @ Year 1 of Tax Not Imposed Scenario is $26.69.
So the PV is:
P = [0.5(15.45)+ 0.5(26.690] / 1.13 = $18.646.
X = $20.
t = 1.
rRF = 0.08.
2 = 0.0687.
d1 = ln[18.646/20] + [0.08 + .5(.0687)](1) = 0.1688
(.0687)0.5 (1)0.5
d2 = 0.1688  (.0687)0.5 (1)0.5 = 0.0933
From Excel function NORMSDIST, or approximated from Table 13E1 in Extension to Chapter 13:
N(d1) = 0.5670
N(d2) = 0.4628
Using the BlackScholes Option Pricing Model, you calculate the option's value as:
r t
V = P[N(d1)]  Xe RF [N(d2)]
= $18.646(0.5670)  $20e(0.08)(1)(0.4628)
= $10.572  $8.544
= $2.028 million.
Answers and Solutions: 12  261 126 P = PV of all expected future cash flows if project is delayed. From Problem 134 we know that PV
@ Year 2 of Low CF Scenario is $6.974 and PV @ Year 2 of High CF Scenario is $13.313. So the
PV is:
P = [0.1(6.974)+ 0.9(13.313] / 1.102 = $10.479.
X = $9.
t = 2.
rRF = 0.06.
2 = 0.0111.
d1 = ln[10.479/9] + [0.06 + .5(.0111)](2) = 1.9010
(.0111)0.5 (2)0.5
d2 = 1.9010  (.0111)0.5 (2)0.5 = 1.7520
From Excel function NORMSDIST, or approximated from Table 12E1 in Extension to Chapter 12:
N(d1) = 0.9713
N(d2) = 0.9601
Using the BlackScholes Option Pricing Model, you calculate the option's value as:
r t
V = P[N(d1)]  Xe RF [N(d2)]
= $10.479(0.9713)  $9e(0.06)(2)(0.9601)
= $10.178  $7.664
= $2.514 million.
Answers and Solutions: 12  262 SOLUTION TO SPREADSHEET PROBLEMS
127 The detailed solution for the problem is available both on the instructor's resource CDROM (in the
file Solution for FM11 Ch 12 P7 Build a Model.xls) and on the instructor's side of the textbook's web
site, http://brigham.swcollege.com.
Answers and Solutions: 12  263 MINI CASE
Assume that you have just been hired as a financial analyst by Tropical Sweets Inc., a midsized California company that specializes in creating exotic candies from tropical fruits such as mangoes, papayas, and dates. The firm's CEO, George Yamaguchi, recently returned from an industry corporate executive conference in San Francisco, and one of the sessions he attended was on real options. Since no one at Tropical Sweets is familiar with the basics of real options, Yamaguchi has asked you to prepare a brief report that the firm's executives could use to gain at least a cursory understanding of the topics.
To begin, you gathered some outside materials the subject and used these materials to draft a list of pertinent questions that need to be answered. In fact, one possible approach to the paper is to use a questionandanswer format. Now that the questions have been drafted, you have to develop the answers.
a. What are some types of real options?
Answer: 1. Investment timing options
2. Growth options
a. Expansion of existing product line
b. New products
c. New geographic markets
3. Abandonment options
a. Contraction
b. Temporary suspension
c. Complete abandonment
4. Flexibility options.
b. What are five possible procedures for analyzing a real option?
Answer: 1. DCF analysis of expected cash flows, ignoring option.
2. Qualitatively assess the value of the real option.
3. Decision tree analysis.
4. Use a model for a corresponding financial option, if possible.
5. Use financial engineering techniques if a corresponding financial option is not available.
Answers and Solutions: 13  264 c. Tropical Sweets is considering a project that will cost $70 million and will generate
expected cash flows of $30 per year for three years. The cost of capital for this type
of project is 10 percent and the riskfree rate is 6 percent. After discussions with the
marketing department, you learn that there is a 30 percent chance of high demand,
with future cash flows of $45 million per year. There is a 40 percent chance of
average demand, with cash flows of $30 million per year. If demand is low (a 30
percent chance), cash flows will be only $15 million per year. What is the expected
NPV?
Answer: Initial Cost = $70 Million
Expected Cash Flows = $30 Million Per Year For Three Years
Cost Of Capital = 10%
PV Of Expected CFs = $74.61 Million
Expected NPV = $74.61  $70
= $4.61 Million
Alternatively, one could calculate the NPV of each scenario:
Demand Probability Annual Cash Flow
High 30% $45
Average 40% $30
Low 30% $15
Find NPV of each scenario:
PV High: N=3 I=10 PV=? PMT=45 FV=0
PV= 111.91
NPV High = $111.91  $70 = $41.91 Million.
PV Average: N=3 I=10 PV=? PMT=30 FV=0
PV= 74.61
NPV Average = $74.61  $70 = $4.71 Million.
PV Low: N=3 I=10 PV=? PMT=15 FV=0
PV= 37.30
NPV Low = $37.30  $70 = $32.70 Million.
Find Expected NPV:
E(NPV)=.3($41.91)+.4($4.61)+.3($32.70)
E(PV)= $4.61.
d. Now suppose this project has an investment timing option, since it can be delayed for
a year. The cost will still be $70 million at the end of the year, and the cash flows
for the scenarios will still last three years. However, Tropical Sweets will know the
level of demand, and will implement the project only if it adds value to the company.
Answers and Solutions: 13  265 Perform a qualitative assessment of the investment timing option's value.
Answer: If we immediately proceed with the project, its expected NPV is $4.61 million. However,
the project is very risky. If demand is high, NPV will be $41.91 million. If demand is
average, NPV will be $4.61 million. If demand is low, NPV will be $32.70 million. However,
if we wait one year, we will find out additional information regarding demand. If demand is
low, we won't implement project. If we wait, the upfront cost and cash flows will stay the
same, except they will be shifted ahead by a year.
The value of any real option increases if the underlying project is very risky or if there is a
long time before you must exercise the option.
This project is risky and has one year before we must decide, so the option to wait is probably
valuable.
e. Use decision tree analysis to calculate the NPV of the project with the investment
timing option.
Answer: The project will be implemented only if demand is average or high.
Here is the time line:
0 1 2 3 4
High $0 $70 $45 $45 $45
Average $0 $70 $30 $30 $30
Low $0 $0 $0 $0 $0
To find the NPVC, discount the cost at the riskfree rate of 6 percent since it is
known for certain, and discount the other risky cash flows at the 10 percent cost
of capital.
High: NPV = $70/1.06 + $45/1.102 + $45/1.103 +$45/1.104 = $35.70
Average: NPV = $70/1.06 + $30/1.102 + $30/1.103 +$30/1.104 = $1.79
Low: NPV = $0.
Expected NPV = 0.3($35.70) + 0.4($1.79) + 0.3($0) = $11.42.
Since this is much greater than the NPV of immediate implementation (which is $4.61 million) we
should wait. In other words, implementing immediately gives an expected NPV of $4.61 million,
but implementing immediately means we give up the option to wait, which is worth $11.42
million.
Answers and Solutions: 13  266 f. Use a financial option pricing model to estimate the value of the investment timing
option.
Answer: The option to wait resembles a financial call option we get to "buy" the project for $70 million
in one year if value of project in one year is greater than $70 million. This is like a call option
with an exercise price of $70 million and an expiration date of one year.
X = Exercise Price = Cost Of Implement Project = $70 Million.
RRF = RiskFree Rate = 6%.
T = Time To Maturity = 1 year.
P = Current Price Of Stock = Current Value Of The Project's Future Cash Flows.
2 = Variance Of Stock Return = Variance Of Project's Rate Of Return.
We explain how to calculate P and 2 below.
Just as the price of a stock is the present value of all the stock's future cash flows, the "price" of
the real option is the present value of all the project's cash flows that occur beyond the exercise
date. Notice that the exercise cost of an option does not affect the stock price. Similarly, the
cost to implement the real option does not affect the current value of the underlying asset (which
is the PV of the project's cash flows). It will be helpful in later steps if we break the calculation
into two parts. First, we find the value of all cash flows beyond the exercise date discounted
back to the exercise date. Then we find the expected present value of those values.
Step 1: Find the value of all cash flows beyond the exercise date discounted back to the exercise
date. Here is the time line. The exercise date is year 1, so we discount all future cash flows
back to year 1.
0 1 2 3 4
High $45 $45 $45
Average $30 $30 $30
Low $15 $15 $15
High: PV1 = $45/1.10 + $45/1.102 + $45/1.103 = $111.91
Average: PV1 = $30/1.10 + $30/1.102 + $30/1.103 = $74.61
Low: PV1 = $15/1.10 + $15/1.102 + $15/1.103 = $37.30
The current expected present value, P, is:
P = 0.3[$111.91/1.1] + 0.4[$74.61/1.1] + 0.3[$37.30/1.1] = $67.82.
For a stock option, 2 is the variance of the stock return, not the variance of the stock price.
Therefore, for a real option we need the variance of the project's rate of return. There are three
ways to estimate this variance. First, we can use subjective judgment. Second, we can
calculate the project's return in each scenario and then calculate the return's variance. This is the
direct approach. Third, we know the projects value at each scenario at the expiration date, and
we know the current value of the project. Thus, we can find a variance of project return that
gives the range of project values that can occur at expiration. This is the indirect approach.
Following is an explanation of each approach.
Subjective estimate:
The typical stock has 2 of about 12%. Most projects will be somewhat riskier than the firm,
since the risk of the firm reflects the diversification that comes from having many projects.
Subjectively scale the variance of the company's stock return up or down to reflect the risk of the
Answers and Solutions: 13  267 project. The company in our example has a stock with a variance of 10%, so we might expect the project to have a variance in the range of 12% to 19%.
Direct approach: From our previous analysis, we know the current value of the project and the value for each scenario at the time the option expires (year 1). Here is the time line:
Current Value Value At Expiration
Year 0 Year 1 High $67.82 $111.91 Average $67.82 $74.61 Low $67.82 $37.30 The annual rate of return is: High: Return = ($111.91/$67.82) 1 = 65%. High: Average = ($74.61/$67.82) 1 = 10%. High: Return = ($37.30/$67.82) 1 = 45%.
Expected Return = 0.3(0.65) + 0.4(0.10) + 0.3(0.45)
= 10%.
2 = 0.3(0.650.10)2 + 0.4(0.100.10)2 + 0.3(0.450.10)2
= 0.182 = 18.2%.
The direct approach gives an estimate of 18.2% for the variance of the project's return.
Answers and Solutions: 13  268 The indirect approach: Given a current stock price and an anticipated range of possible stock prices at some point in the future, we can use our knowledge of the distribution of stock returns (which is lognormal) to relate the variance of the stock's rate of return to the range of possible outcomes for stock price. To use this formula, we need the coefficient of variation of stock price at the time the option expires. To calculate the coefficient of variation, we need the expected stock price and the standard deviation of the stock price (both of these are measured at the time the option expires). For the real option, we need the expected value of the project's cash flows at the date the real option expires, and the standard deviation of the project's value at the date the real option expires. We previously calculated the value of the project at the time the option expires, and we can use this to calculate the expected value and the standard deviation.
Value At Expiration
Year 1 High $111.91 Average $74.61 Low $37.30
Expected Value =.3($111.91)+.4($74.61)+.3($37.3)
= $74.61. value = [.3($111.91$74.61)2 + .4($74.61$74.61)2
+ .3($37.30$74.61)2]1/2
= $28.90.
Coefficient Of Variation = CV = Expected Value / value CV = $74.61 / $28.90 = 0.39.
Here is a formula for the variance of a stock's return, if you know the coefficient of variation of the expected stock price at some point in the future. The CV should be for the entire project, including all scenarios: 2 = LN[CV2 + 1]/T = LN[0.392 + 1]/1 = 14.2%.
Now, we proceed to use the OPM:
V = $67.83[N(d1)]  $70e(0.06)(1)[N(d2)].
Answers and Solutions: 13  269 ln($67.83/$70) + [( 0.06 + 0.142/2 )](15)
d1 =
0.5 0.5
(.142 ) (1)
= 0.2641.
d2 = d1  (0.142)0.5(1)0.5 = 0.2641  0.3768
= 0.1127.
N(d1) = N(0.2641) = 0.6041.
N(d2) = N(0.1127) = 0.4551.
therefore,
V = $67.83(0.6041)  $70e0.06(0.4551)
= $10.98.
g. Now suppose the cost of the project is $75 million and the project cannot be
delayed. But if Tropical Sweets implements the project, then Tropical Sweets will
have a growth option. It will have the opportunity to replicate the original project at
the end of its life. What is the total expected NPV of the two projects if both are
implemented?
Answer: Suppose the cost of the project is $75 million instead of $70 million, and there is no option to
wait.
NPV = PV of future cash flows  cost
= $74.61  $75 = $0.39 million.
The project now looks like a loser. Using NPV analysis:
NPV = NPV Of Original Project + NPV Of Replication Project
= $0.39 + $0.39/(1+0.10)3
= $0.39 + $0.30 = $0.69.
Still looks like a loser, but you will only implement project 2 if demand is high. We might have
chosen to discount the cost of the replication project at the riskfree rate, and this would have
made the NPV even lower.
Answers and Solutions: 13  270 h. Tropical Sweets will replicate the original project only if demand is high. Using
decision tree analysis, estimate the value of the project with the growth option.
Answer: The future cash flows of the optimal decisions are shown below. The cash flow in year 3 for the
high demand scenario is the cash flow from the original project and the cost of the replication
project.
0 1 2 3 4 5 6
High $75 $45 $45 $45 $70 $45 $45 $45
Average $75 $30 $30 $30 $0 $0 $0
Low $75 $15 $15 $15 $0 $0 $0
To find the NPV, we discount the risky cash flows at the 10 percent cost of capital, and the
nonrisky cost to replicate (i.e., the $75 million) at the riskfree rate.
NPV high = $75 + $45/1.10 + $45/1.102 + $45/1.103 + $45/1.104
+ $45/1.105 + $45/1.106  $75/1.063
= $58.02
NPV average = $75 + $30/1.10 + $30/1.102 + $30/1.103 = $0.39
NPV average = $75 + $15/1.10 + $15/1.102 + $15/1.103 = $37.70
Expected NPV = 0.3($58.02) + 0.4($0.39) + 0.3($37.70) = $5.94.
Thus, the option to replicate adds enough value that the project now has a positive NPV.
i. Use a financial option model to estimate the value of the growth option.
Answer: X = Exercise Price = Cost Of Implement Project = $75 million.
RRF = RiskFree Rate = 6%.
T = Time To Maturity = 3 years.
P = Current Price Of Stock = Current Value Of The Project's Future Cash Flows.
2 = Variance Of Stock Return = Variance Of Project's Rate Of Return.
We explain how to calculate P and 2 below.
Step 1: Find the value of all cash flows beyond the exercise date discounted back to the exercise
date. Here is the time line. The exercise date is year 1, so we discount all future cash flows
back to year 3.
0 1 2 3 4 5 6
High $45 $45 $45
Average $30 $30 $30
Low $15 $15 $15
High: PV3 = $45/1.10 + $45/1.102 + $45/1.103 = $111.91
Average: PV3 = $30/1.10 + $30/1.102 + $30/1.103 = $74.61
Low: PV3 = $15/1.10 + $15/1.102 + $15/1.103 = $37.30
Answers and Solutions: 13  271 The current expected present value, P, is: P = 0.3[$111.91/1.13] + 0.4[$74.61/1.13] + 0.3[$37.30/1.13] = $56.05.
Direct approach for estimating 2:
From our previous analysis, we know the current value of the project and the value for each scenario at the time the option expires (year 3). Here is the time line:
Current Value Value At Expiration
Year 0 Year 3 High $56.02 $111.91 Average $56.02 $74.61 Low $56.02 $37.30
The annual rate of return is: High: Return = ($111.91/$56.02)(1/3) 1 = 25.9%. High: Average = ($74.61/$56.02)(1/3) 1 = 10%. High: Return = ($37.30/$56.02)(1/3) 1 = 12.7%.
Expected Return = 0.3(0.259) + 0.4(0.10) + 0.3(0.127)
= 8.0%.
2 = 0.3(0..2590.08)2 + 0.4(0.100.08)2 + 0.3(0.1270.08)2
= 0.182 = 2.3%.
This is lower than the variance found for the previous option because the dispersion of cash flows for the replication project is the same as for the original, even though the replication occurs much later. Therefore, the rate of return for the replication is less volatile. We do sensitivity analysis later.
The indirect approach: First, find the coefficient of variation for the value of the project at the time the option expires (year 3).
We previously calculated the value of the project at the time the option expires, and we can use this to calculate the expected value and the standard deviation.
Value At Expiration
Year 3 High $111.91 Average $74.61 Low $37.30
Answers and Solutions: 13  272 Expected Value =.3($111.91)+.4($74.61)+.3($37.3)
= $74.61.
value = [.3($111.91$74.61)2 + .4($74.61$74.61)2
+ .3($37.30$74.61)2]1/2
= $28.90.
Coefficient Of Variation = CV = Expected Value / value
CV = $74.61 / $28.90 = 0.39.
To find the variance of the project's rate or return, we use the formula below:
2 = LN[CV2 + 1]/T = LN[0.392 + 1]/3 = 4.7%.
Now, we proceed to use the OPM:
V = $56.06[N(d1)]  $75e(0.06)(3)[N(d2)].
ln($56.06/$75) + [( 0.06 + 0.047/2)]( 3)
d1 =
0.5 0.5
( 0.047 ) ( 3)
= 0.1085.
d2 = d1  (0.047)0.5(3)0.5 = .1085  0.3755
= 0.4840.
N(d1) = N(0.1080) = 0.4568.
N(d2) = N(0.4835) = 0.3142.
Therefore,
V = $56.06(0.4568)  $75e(0.06)(3)(0.3142)
= $5.92.
Total Value = NPV Of Project 1 + Value Of Growth Option
=$0.39 + $5.92
= $5.5 million
j. What happens to the value of the growth option if the variance of the project's return
is 14.2 percent? What if it is 50 percent? How might this explain the high
valuations of many dot.com companies?
Answer: If risk, defined by 2, goes up, then value of growth option goes up (see the file ch 12 mini
case.xls for calculations):
2 = 4.7%, option value = $5.92
2 = 14.2%, option value = $12.10
2 = 50%, option value = $24.09
If the future profitability of dot.com companies is very volatile (i.e., there is the potential for very
high profits), then a company with a real option on those profits might have a very high value for
its growth option.
Answers and Solutions: 13  273 Chapter 13
Analysis of Financial Statements
ANSWERS TO ENDOFCHAPTER QUESTIONS
131 a. A liquidity ratio is a ratio that shows the relationship of a firm's cash and other current assets to its
current liabilities. The current ratio is found by dividing current assets by current liabilities. It
indicates the extent to which current liabilities are covered by those assets expected to be
converted to cash in the near future. The quick, or acid test, ratio is found by taking current
assets less inventories and then dividing by current liabilities.
b. Asset management ratios are a set of ratios that measure how effectively a firm is managing its
assets. The inventory turnover ratio is sales divided by inventories. Days sales outstanding is
used to appraise accounts receivable and indicates the length of time the firm must wait after
making a sale before receiving cash. It is found by dividing receivables by average sales per
day. The fixed assets turnover ratio measures how effectively the firm uses its plant and
equipment. It is the ratio of sales to net fixed assets. Total assets turnover ratio measures the
turnover of all the firm's assets; it is calculated by dividing sales by total assets.
c. Financial leverage ratios measure the use of debt financing. The debt ratio is the ratio of total
debt to total assets, it measures the percentage of funds provided by creditors. The
timesinterestearned ratio is determined by dividing earnings before interest and taxes by the
interest charges. This ratio measures the extent to which operating income can decline before
the firm is unable to meet its annual interest costs. The EBITDA coverage ratio is similar to the
timesinterestearned ratio, but it recognizes that many firms lease assets and also must make
sinking fund payments. It is found by adding EBITDA and lease payments then dividing this
total by interest charges, lease payments, and sinking fund payments over one minus the tax rate.
d. Profitability ratios are a group of ratios, which show the combined effects of liquidity, asset
management, and debt on operations. The profit margin on sales, calculated by dividing net
income by sales, gives the profit per dollar of sales. Basic earning power is calculated by
dividing EBIT by total assets. This ratio shows the raw earning power of the firm's assets,
before the influence of taxes and leverage. Return on total assets is the ratio of net income to
total assets. Return on common equity is found by dividing net income into common equity.
e. Market value ratios relate the firm's stock price to its earnings and book value per share. The
price/earnings ratio is calculated by dividing price per share by earnings per sharethis shows
how much investors are willing to pay per dollar of reported profits. The price/cash flow is
calculated by dividing price per share by cash flow per share. This shows how much investors
are willing to pay per dollar of cash flow. Markettobook ratio is simply the market price per
share divided by the book value per share. Book value per share is common equity divided by
the number of shares outstanding.
f. Trend analysis is an analysis of a firm's financial ratios over time. It is used to estimate the
likelihood of improvement or deterioration in its financial situation. Comparative ratio analysis
Answers and Solutions: 13  274 is when a firm compares its ratios to other leading companies in the same industry. This technique
is also known as benchmarking.
g. The Du Pont chart is a chart designed to show the relationships among return on investment, asset
turnover, the profit margin, and leverage. The Du Pont equation is a formula, which shows that the
rate of return on assets can be found as the product of the profit margin times the total assets
turnover.
h. Window dressing is a technique employed by firms to make their financial statements look better
than they really are. Seasonal factors can distort ratio analysis. At certain times of the year a
firm may have excessive inventories in preparation of a "season" of high demand. Therefore an
inventory turnover ratio taken at this time as opposed to after the season will be radically
distorted.
132 The emphasis of the various types of analysts is by no means uniform nor should it be. Management
is interested in all types of ratios for two reasons. First, the ratios point out weaknesses that should
be strengthened; second, management recognizes that the other parties are interested in all the ratios
and that financial appearances must be kept up if the firm is to be regarded highly by creditors and
equity investors. Equity investors are interested primarily in profitability, but they examine the other
ratios to get information on the riskiness of equity commitments. Longterm creditors are more
interested in the debt ratio, TIE, and fixedcharge coverage ratios, as well as the profitability ratios.
Shortterm creditors emphasize liquidity and look most carefully at the liquidity ratios.
133 Given that sales have not changed, a decrease in the total assets turnover means that the company's
assets have increased. Also, the fact that the fixed assets turnover ratio remained constant implies
that the company increased its current assets. Since the company's current ratio increased, and yet,
its quick ratio is unchanged means that the company has increased its inventories.
134 Differences in the amounts of assets necessary to generate a dollar of sales cause asset turnover ratios
to vary among industries. For example, a steel company needs a greater number of dollars in assets
to produce a dollar in sales than does a grocery store chain. Also, profit margins and turnover ratios
may vary due to differences in the amount of expenses incurred to produce sales. For example, one
would expect a grocery store chain to spend more per dollar of sales than does a steel company.
Often, a large turnover will be associated with a low profit margin, and vice versa.
135 a. Cash, receivables, and inventories, as well as current liabilities, vary over the year for firms with
seasonal sales patterns. Therefore, those ratios that examine balance sheet figures will vary
unless averages (monthly ones are best) are used.
b. Common equity is determined at a point in time, say December 31, 2004. Profits are earned over
time, say during 2004. If a firm is growing rapidly, yearend equity will be much larger than
beginningofyear equity, so the calculated rate of return on equity will be different depending on
whether endofyear, beginningofyear, or average common equity is used as the denominator.
Average common equity is conceptually the best figure to use. In public utility rate cases, people
are reported to have deliberately used endofyear or beginningofyear equity to make returns on
equity appear excessive or inadequate. Similar problems can arise when a firm is being
evaluated.
136 Firms within the same industry may employ different accounting techniques, which make it difficult
to compare financial ratios. More fundamentally, comparisons may be misleading if firms in the
Answers and Solutions: 13  275 same industry differ in their other investments. For example, comparing Pepsico and CocaCola may be misleading because apart from their soft drink business, Pepsi also owns other businesses such as FritoLay, Pizza Hut, Taco Bell, and KFC.
Answers and Solutions: 13  276 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
CA CA  I 131 CA = $3,000,000; = 1.5; = 1.0;
CL CL
CL = ?; I = ?
CA
= 1.5
CL
$3,000,000
= 1.5
CL
1.5 CL = $3,000,000
CL = $2,000,000.
CA  I
= 1.0
CL
$3,000,000  I
= 1.0
$2,000,000
$3,000,000  I = $2,000,000
I = $1,000,000.
132 DSO = 40 days; ADS = $20,000; AR = ?
AR
DSO =
S
365
AR
40 =
$20,000
AR = $800,000.
133 A/E = 2.4; D/A = ?
Answers and Solutions: 13  277
D 1
= 1 
A A
E
D 1
= 1 
A 2.4
D
= 0.5833 = 58.33%.
A 134 ROA = 10%; PM = 2%; ROE = 15%; S/TA = ?; A/E = ?
ROA = NI/A; PM = NI/S; ROE = NI/E
ROA = PM × S/TA
NI/A = NI/S × S/TA
10% = 2% × S/TA
S/TA = 5.
ROE = PM × S/TA × TA/E
NI/E = NI/S × S/TA × TA/E
15% = 2% × 5 × TA/E
15% = 10% × TA/E
TA/E = 1.5.
135 We are given ROA = 3% and Sales/Total assets = 1.5×.
From Du Pont equation: ROA = Profit margin × Total assets turnover
3% = Profit margin (1.5)
Answers and Solutions: 13  278 Profit margin = 3%/1.5 = 2%.
We can also calculate the company's debt ratio in a similar manner, given the facts of the problem.
We are given ROA(NI/A) and ROE(NI/E); if we use the reciprocal of ROE we have the following
equation:
E NI E D E
= _ and =1 , so
A A NI A A
E 1
= 3% _
A 0.05
E
= 60% .
A
D
= 1  0.60 = 0.40 = 40% .
A
Alternatively,
ROE = ROA × EM
5% = 3% × EM
EM = 5%/3% = 5/3 = TA/E.
Take reciprocal:
E/TA = 3/5 = 60%;
therefore,
D/A = 1  0.60 = 0.40 = 40%.
Thus, the firm's profit margin = 2% and its debt ratio = 40%.
$1,312,500 136 Present current ratio = = 2.5.
$525,000
$1,312,500 + NP
Minimum current ratio = = 2.0.
$525,000 + NP
$1,312,500 + NP = $1,050,000 + 2NP
NP = $262,500.
Answers and Solutions: 13  279 Shortterm debt can increase by a maximum of $262,500 without violating a 2 to 1 current ratio,
assuming that the entire increase in notes payable is used to increase current assets. Since we
assumed that the additional funds would be used to increase inventory, the inventory account will
increase to $637,500, and current assets will total $1,575,000.
Quick ratio = ($1,575,000  $637,500)/$787,500 = $937,500/$787,500 = 1.19×.
Current assets $810,000 137 1. = 3.0× = 3.0×
Current liabilities Current liabilities
Current liabilities = $270,000.
Current assets  Inventories $810,000  Inventories
2. = 1.4× = 1.4×
Current liabilities $270,000
Inventories = $432,000.
Current Marketable Accounts
3. = Cash + + + Inventories
assets Securities receivable
$810,000 = $120,000 + Accounts receivable + $432,000
Accounts receivable = $258,000.
Sales Sales
4. = 6.0× = 6.0×
Inventory $432,000
Sales = $2,592,000.
Accounts receivable $258,000
5. DSO = = = 36.33 days.
Sales/365 $2,592,000/ 365
138 TIE = EBIT/INT, so find EBIT and INT.
Interest = $500,000 × 0.1 = $50,000.
Net income = $2,000,000 × 0.05 = $100,000.
Pretax income = $100,000/(1  T) = $100,000/0.7 = $142,857.
EBIT = $142,857 + $50,000 = $192,857.
TIE = $192,857/$50,000 = 3.86×.
139 a. (Dollar amounts in thousands.)
Industry
Answers and Solutions: 13  280 Firm Average
Current assets $655,000
= = 1.98×
Current liabilities $330,000 2.0×
Accounts receivable $336,000
DSO = = = 76 days 35
Sales/ 365 $4,404.11 days
Sales $1,607,500
= = 6.66×
Inventory $241,500 6.7×
Sales $1,607,500
= = 5.50×
Fixed assets $292,500 12.1×
Sales $1,607,500
= =
Total assets $947,500 1.70× 3.0×
Net income $27,300
= =
Sales $1,607,500 1.7% 1.2%
Net income $27,300
= = 2.9%
Total assets $947,500 3.6%
Industry
Firm Average
Net income $27,300
= = 7.6%
Common equity $361,000 9.0%
Total debt $586,500
= = 61.9% 60.0%
Total assets $947,500
Answers and Solutions: 13  281 b. For the firm,
$947,500
ROE = PM × T.A. turnover × EM = 1.7% × 1.7 × = 7.6%.
$361,000
For the industry, ROE = 1.2% × 3 × 2.5 = 9%.
Note: To find the industry ratio of assets to common equity, recognize that 1  (total debt/total
assets) = common equity/total assets. So, common equity/total assets = 40%, and 1/0.40 = 2.5 =
total assets/common equity.
c. The firm's days sales outstanding is more than twice as long as the industry average, indicating
that the firm should tighten credit or enforce a more stringent collection policy. The total assets
turnover ratio is well below the industry average so sales should be increased, assets decreased, or
both. While the company's profit margin is higher than the industry average, its other
profitability ratios are low compared to the industrynet income should be higher given the
amount of equity and assets. However, the company seems to be in an average liquidity position
and financial leverage is similar to others in the industry.
d. If 2004 represents a period of supernormal growth for the firm, ratios based on this year will be
distorted and a comparison between them and industry averages will have little meaning.
Potential investors who look only at 2003 ratios will be misled, and a return to normal conditions
in 2005 could hurt the firm's stock price.
1310 1. Debt = (0.50)(Total assets) = (0.50)($300,000) = $150,000.
2. Accounts payable = Debt Longterm debt = $150,000  $60,000
= $90,000
Total liabilities
3. Common stock =  Debt  Retained earnings
and equity
= $300,000  $150,000  $97,500 = $52,500.
4. Sales = (1.5)(Total assets) = (1.5)($300,000) = $450,000.
5. Inventory = Sales/5 = $450,000/5 = $90,000.
6. Accounts receivable = (Sales/365)(DSO) = ($450,000/365)(36.5)
= $45,000.
7. Cash + Accounts receivable = (0.80)(Accounts payable)
Cash + $45,000 = (0.80)($90,000)
Cash = $72,000  $45,000 = $27,000.
8. Fixed assets = Total assets  (Cash + Accts rec. + Inventories)
= $300,000  ($27,000 + $45,000 + $90,000) = $138,000.
9. Cost of goods sold = (Sales)(1  0.25) = ($450,000)(0.75)
= $337,500.
Answers and Solutions: 13  282 1311 a. Here are the firm's base case ratios and other data as compared to the industry:
Firm Industry Comment
Quick 0.8× 1.0×
Weak
Current 2.3 2.7
Weak
Inventory turnover 4.8 7.0 Poor
Days sales outstanding 37 days 32 days Poor
Fixed assets turnover 10.0× 13.0× Poor
Total assets turnover 2.3 2.6 Poor
Return on assets 5.9% 9.1% Bad
Return on equity 13.1 18.2 Bad
Debt ratio 54.8 50.0 High
Profit margin on sales 2.5 3.5 Bad
EPS $4.71 n.a. 
Stock Price $23.57 n.a. 
P/E ratio 5.0× 6.0× Poor
P/CF ratio 2.0× 3.5× Poor
M/B ratio 0.65 n.a. 
The firm appears to be badly managedall of its ratios are worse than the industry averages, and
the result is low earnings, a low P/E, P/CF ratio, a low stock price, and a low M/B ratio. The
company needs to do something to improve.
b. A decrease in the inventory level would improve the inventory turnover, total assets turnover, and
ROA, all of which are too low. It would have some impact on the current ratio, but it is difficult
to say precisely how that ratio would be affected. If the lower inventory level allowed the
company to reduce its current liabilities, then the current ratio would improve. The lower cost of
goods sold would improve all of the profitability ratios and, if dividends were not increased,
would lower the debt ratio through increased retained earnings. All of this should lead to a
higher market/book ratio and a higher stock price.
Answers and Solutions: 13  283 SOLUTION TO SPREADSHEET PROBLEM
1312 The detailed solution for the
problem is available both on the instructor's resource CDROM (in the file Solution to FM11 Ch 13
P12 Build a Model.xls) and on the instructor's side of the web site, http://brigham.swcollege.com.
Answers and Solutions: 13  284 MINI CASE
The first part of the case, presented in chapter 3, discussed the situation that Computron Industries was in after an expansion program. Thus far, sales have not been up to the forecasted level, costs have been higher than were projected, and a large loss occurred in 2004, rather than the expected profit. As a result, its managers, directors, and investors are concerned about the firm's survival.
Donna Jamison was brought in as assistant to Fred Campo, Computron's chairman, who had the task of getting the company back into a sound financial position. Computron's 2003 and 2004 balance sheets and income statements, together with projections for 2005, are shown in the following tables. Also, the tables show the 2003 and 2004 financial ratios, along with industry average data. The 2005 projected financial statement data represent Jamison's and Campo's best guess for 2005 results, assuming that some new financing is arranged to get the company "over the hump."
Jamison examined monthly data for 2004 (not given in the case), and she detected an improving pattern during the year. Monthly sales were rising, costs were falling, and large losses in the early months had turned to a small profit by December. Thus, the annual data looked somewhat worse than final monthly data. Also, it appears to be taking longer for the advertising program to get the message across, for the new sales offices to generate sales, and for the new manufacturing facilities to operate efficiently. In other words, the lags between spending money and deriving benefits were longer than Computron's managers had anticipated. For these reasons, Jamison and Campo see hope for the companyprovided it can survive in the short run.
Jamison must prepare an analysis of where the company is now, what it must do to regain its financial health, and what actions should be taken. Your assignment is to help her answer the following questions. Provide clear explanations, not yes or no answers.
Answers and Solutions: 14  285 Balance Sheets
Assets 2003 2004 2005e Cash $ 9,000 $ 7,282 $ 14,000 ShortTerm Investments. 48,600 20,000 71,632 Accounts Receivable 351,200 632,160 878,000 Inventories 715,200 1,287,360 1,716,480
Total Current Assets $ 1,124,000 $ 1,946,802 $ 2,680,112 Gross Fixed Assets 491,000 1,202,950 1,220,000 Less: Accumulated Depreciation 146,200 263,160 383,160
Net Fixed Assets $ 344,800 $ 939,790 $ 836,840 Total Assets $ 1,468,800 $ 2,886,592 $ 3,516,952
Liabilities And Equity 2003 2004 2005e Accounts Payable $ 145,600 $ 324,000 $ 359,800 Notes Payable 200,000 720,000 300,000 Accruals 136,000 284,960 380,000
Total Current Liabilities $ 481,600 $ 1,328,960 $ 1,039,800 LongTerm Debt 323,432 1,000,000 500,000 Common Stock (100,000 Shares) 460,000 460,000 1,680,936 Retained Earnings 203,768 97,632 296,216
Total Equity $ 663,768 $ 557,632 $ 1,977,152 Total Liabilities And Equity $ 1,468,800 $ 2,886,592 $ 3,516,952
Income Statements
2003 2004 2005e Sales $ 3,432,000 $ 5,834,400 $ 7,035,600 Cost Of Goods Sold 2,864,000 4,980,000 5,800,000 Other Expenses 340,000 720,000 612,960 Depreciation 18,900 116,960 120,000
Total Operating Costs $ 3,222,900 $ 5,816,960 $ 6,532,960
EBIT $ 209,100 $ 17,440 $ 502,640 Interest Expense 62,500 176,000 80,000
EBT $ 146,600 $ (158,560) $ 422,640 Taxes (40%) 58,640 (63,424) 169,056 Net Income $ 87,960 $ (95,136) $ 253,584
Other Data 2003 2004 2005e Stock Price $ 8.50 $ 6.00 $ 12.17 Shares Outstanding 100,000 100,000 250,000 EPS $ 0.880 $ (0.951) $ 1.014 DPS $ 0.220 $ 0.110 $ 0.220 Tax Rate 40% 40% 40% Book Value Per Share $ 6.638 $ 5.576 $ 7.909 Lease Payments $ 40,000 $ 40,000 $ 40,000
Answers and Solutions: 14  286 Ratio Analysis 2003 2004 2005e Industry Average Current 2.3 1.5 2.58 2.7 Quick 0.8 0.5 0.93 1.0 Inventory Turnover 4.8 4.5 4.10 6.1 Days Sales 37.4 39.5 45.5 32.0 Outstanding Fixed Assets Turnover 10.0
6.2 8.41 7.0 Total Assets Turnover
2.3 2.0 2.00 2.5 Debt Ratio 54.8% 80.7% 43.8% 50.0% TIE 3.3 0.1 6.3 6.2 EBITDA Coverage 2.6 0.8 5.5 8.0 Profit Margin 2.6% 1.6% 3.6% 3.6% Basic Earning Power 14.2% 0.6% 14.3% 17.8% ROA 6.0% 3.3% 7.2% 9.0% ROE 13.3% 17.1% 12.8% 17.9% Price/Earnings (P/E) 9.7 6.3 12.0 16.2 Price/Cash Flow 8.0 27.5 8.1 7.6 Market/Book 1.3 1.1 1.5 2.9
a. Why are ratios useful? What are the five major categories of ratios?
Answer: Ratios are used by managers to help improve the firm's performance, by lenders to help evaluate
the firm's likelihood of repaying debts, and by stockholders to help forecast future earnings and
dividends. The five major categories of ratios are: liquidity, asset management, debt
management, profitability, and market value.
Answers and Solutions: 14  287 b. Calculate the 2005 current and quick ratios based on the projected balance sheet
and income statement data. What can you say about the company's liquidity
position in 2003, 2004, and as projected for 2005? We often think of ratios as
being useful (1) to managers to help run the business, (2) to bankers for credit
analysis, and (3) to stockholders for stock valuation. Would these different
types of analysts have an equal interest in the liquidity ratios?
Answer: Current Ratio05 = Current Assets/Current Liabilities
= $2,680,112/$1,039,800 = 2.58×.
Quick Ratio05 = (Current Assets Inventory)/Current Liabilities
= ($2,680,112  $1,716,480)/$1,039,800 = 0.93×.
The company's current and quick ratios are higher relative to its 2003 current and quick ratios;
they have improved from their 2004 levels. Both ratios are below the industry
average, however.
c. Calculate the 2005 inventory turnover, days sales outstanding (DSO), fixed
assets turnover, and total assets turnover. How does Computron's utilization
of assets stack up against other firms in its industry?
Answer: Inventory Turnover05 = Sales/Inventory
= $7,035,600/$1,716,480 = 4.10×.
DSO05 = Receivables/(Sales/365)
= $878,000/($7,035,600/365) = 45.5 Days.
Fixed Assets Turnover05 = Sales/Net Fixed Assets
= $7,035,600/$836,840 = 8.41×.
Total Assets Turnover05 = Sales/Total Assets
= $7,035,600/$3,516,952 = 2.0×.
The firm's inventory turnover ratio has been steadily declining, while its days sales
outstanding has been steadily increasing. While the firm's fixed assets turnover ratio is below its
2003 level, it is above the 2004 level. The firm's total assets turnover ratio is below its 2003
level and equal to its 2004 level.
The firm's inventory turnover and total assets turnover are below the industry average. The
firm's days sales outstanding is above the industry average (which is bad); however, the firm's
fixed assets turnover is above the industry average. (This might be due to the fact that
Computron is an older firm than most other firms in the industry, in which case, its fixed assets
are older and thus have been depreciated more, or that Computron's cost of fixed assets were
lower than most firms in the industry.)
Answers and Solutions: 14  288 d. Calculate the 2005 debt, timesinterestearned, and EBITDA coverage ratios.
How does Computron compare with the industry with respect to financial
leverage? What can you conclude from these ratios?
Answer: Debt Ratio05 = Total Liabilities/Total Assets
= ($1,039,800 + $500,000)/$3,516,952 = 43.8%.
Tie05 = EBIT/Interest = $502,640/$80,000 = 6.3×.
Lease Loan Lease
EBITDA Coverage05 = EBITDA +
/ Interest +
+
Payments Repayments Payments
= ($502,640 + $120,000 + $40,000)/($80,000 + $40,000) = 5.5×.
The firm's debt ratio is much improved from 2004, and is still lower than its 2002 level and the
industry average. The firm's TIE and EBITDA coverage ratios are much improved
from their 2003 and 2004 levels. The firm's TIE is better than the industry average,
but the EBITDA coverage is lower, reflecting the firm's higher lease obligations.
e. Calculate the 2005 profit margin, basic earning power (BEP), return on assets
(ROA), and return on equity (ROE). What can you say about these ratios?
Answer: Profit Margin05 = Net Income/Sales = $253,584/$7,035,600 = 3.6%.
Basic Earning Power05 = EBIT/Total Assets = $502,640/$3,516,952
= 14.3%.
ROA05 = Net Income/Total Assets = $253,584/$3,516,952 = 7.2%.
ROE05 = Net Income/Common Equity = $253,584/$1,977,152 = 12.8%.
The firm's profit margin is above 2003 and 2004 levels and is at the industry average. The
basic earning power, ROA, and ROE ratios are above both 2003 and 2004 levels, but below the
industry average due to poor asset utilization.
Answers and Solutions: 14  289 f. Calculate the 2005 price/earnings ratio, price/cash flow ratios, and market/book
ratio. Do these ratios indicate that investors are expected to have a high or low
opinion of the company?
Answer: EPS = Net Income/Shares Outstanding = $253,584/250,000 = $1.0143.
Price/Earnings05 = Price Per Share/Earnings Per Share
= $12.17/$1.0143 = 12.0×.
Check: Price = EPS × P/E = $1.0143(12) = $12.17.
Cash Flow/Share05 = (NI + DEP)/Shares
= ($253,584 + $120,000)/250,000
= $1.49.
Price/Cash Flow = $12.17/$1.49 = 8.2×.
BVPS = Common Equity/Shares Outstanding
= $1,977,152/250,000 = $7.91.
Market/Book = Market Price Per Share/Book Value Per Share
= $12.17/$7.91 = 1.54x.
Both the P/E ratio and BVPS are above the 2003 and 2004 levels but below the industry average.
g. Perform a common size analysis and percent change analysis. What do these analyses tell
you about Computron?
Answer: For the common size balance sheets, divide all items in a year by the total assets for that year.
For the common size income statements, divide all items in a year by the sales in that year.
Answers and Solutions: 14  290 Common Size Balance Sheets
Assets
2003 2004 2005e Ind.
Cash 0.6% 0.3% 0.4% 0.3%
Short Term Investments 3.3% 0.7% 2.0% 0.3%
Accounts Receivable 23.9% 21.9% 25.0% 22.4%
Inventories 48.7% 44.6% 48.8% 41.2%
Total Current Assets 76.5% 67.4% 76.2% 64.1%
Gross Fixed Assets 33.4% 41.7% 34.7% 53.9%
Less Accumulated Depreciation 10.0% 9.1% 10.9% 18.0%
Net Fixed Assets 23.5% 32.6% 23.8% 35.9%
Total Assets 100.0% 100.0% 100.0% 100.0%
Liabilities And Equity 2003 2004 2005e Ind.
Accounts Payable 9.9% 11.2% 10.2% 11.9%
Notes Payable 13.6% 24.9% 8.5% 2.4%
Accruals 9.3% 9.9% 10.8% 9.5%
Total Current Liabilities 32.8% 46.0% 29.6% 23.7%
LongTerm Debt 22.0% 34.6% 14.2% 26.3%
Common Stock (100,000 Shares) 31.3% 15.9% 47.8% 20.0%
Retained Earnings 13.9% 3.4% 8.4% 30.0%
Total Equity 45.2% 19.3% 56.2% 50.0%
Total Liabilities And Equity 100.0% 100.0% 100.0% 100.0%
Common Size Income Statement 2003 2004 2005e Ind.
Sales 100.0% 100.0% 100.0% 100.0%
Cost Of Goods Sold 83.4% 85.4% 82.4% 84.5%
Other Expenses 9.9% 12.3% 8.7% 4.4%
Depreciation 0.6% 2.0% 1.7% 4.0%
Total Operating Costs 93.9% 99.7% 92.9% 92.9%
EBIT 6.1% 0.3% 7.1% 7.1%
Interest Expense 1.8% 3.0% 1.1% 1.1%
EBT 4.3% 2.7% 6.0% 5.9%
Taxes (40%) 1.7% 1.1% 2.4% 2.4%
Net Income 2.6% 1.6% 3.6% 3.6%
Computron has higher proportion of inventory and current assets than industry.
Computron has slightly more equity (which means less debt) than industry.
Computron has more shortterm debt than industry, but less longterm debt than
industry. Computron has lower COGS than industry, but higher other expenses.
Result is that Computron has similar EBIT as industry.
Answers and Solutions: 14  291 For the percent change analysis, divide all items in a row by the value in the first year of the analysis.
Percent Change Balance Sheets
Assets
2003 2004 2005e
Cash 0.0% 19.1% 55.6%
Short Term Investments 0.0% 58.8% 47.4%
Accounts Receivable 0.0% 80.0% 150.0%
Inventories 0.0% 80.0% 140.0%
Total Current Assets 0.0% 73.2% 138.4%
Gross Fixed Assets 0.0% 145.0% 148.5%
Less Accumulated Depreciation 0.0% 80.0% 162.1%
Net Fixed Assets 0.0% 172.6% 142.7%
Total Assets 0.0% 96.5% 139.4%
Liabilities And Equity 2003 2004 2005e
Accounts Payable 0.0% 122.5% 147.1%
Notes Payable 0.0% 260.0% 50.0%
Accruals 0.0% 109.5% 179.4%
Total Current Liabilities 0.0% 175.9% 115.9%
LongTerm Debt 0.0% 209.2% 54.6%
Common Stock (100,000 Shares) 0.0% 0.0% 265.4%
Retained Earnings 0.0% 52.1% 45.4%
Total Equity 0.0% 16.0% 197.9%
Total Liabilities And Equity 0.0% 96.5% 139.4%
Percent Change Income Statement 2003 2004 2005e
Sales 0.0% 70.0% 105.0%
Cost Of Goods Sold 0.0% 73.9% 102.5%
Other Expenses 0.0% 111.8% 80.3%
Depreciation 0.0% 518.8% 534.9%
Total Operating Costs 0.0% 80.5% 102.7%
EBIT 0.0% 91.7% 140.4%
Interest Expense 0.0% 181.6% 28.0%
EBT 0.0% 208.2% 188.3%
Taxes (40%) 0.0% 208.2% 188.3%
Net Income 0.0% 208.2% 188.3%
We see that 2005 sales grew 105% from 2002, and that NI grew 188% from 2003. So
Computron has become more profitable. We see that total assets grew at a rate of 139%, while
sales grew at a rate of only 105%. So asset utilization remains a problem.
Answers and Solutions: 14  292 h. Use the extended Du Pont equation to provide a summary and overview of
Computron's financial condition as projected for 2005. What are the firm's
major strengths and weaknesses?
Profit Total Assets Equity Answer: Du Pont Equation = × ×
Margin Turnover Multiplier
= 3.6% × 2.0 × ($3,516,952/$1,977,152)
= 3.6% × 2.0 × 1.8 = 13.0%.
Strengths: The firm's fixed assets turnover was above the industry average. However, if the
firm's assets were older than other firms in its industry this could possibly account for the higher
ratio. (Computron's fixed assets would have a lower historical cost and would have been
depreciated for longer periods of time.) The firm's profit margin is slightly above the industry
average, despite its higher debt ratio. This would indicate that the firm has kept costs down, but,
again, this could be related to lower depreciation costs.
Weaknesses: The firm's liquidity ratios are low; most of its asset management ratios are poor
(except fixed assets turnover); its debt management ratios are poor, most of its profitability ratios
are low (except profit margin); and its market value ratios are low.
i. What are some potential problems and limitations of financial ratio analysis?
Answer: Some potential problems are listed below:
1. Comparison with industry averages is difficult if the firm operates many different divisions.
2. Different operating and accounting practices distort comparisons.
3. Sometimes hard to tell if a ratio is "good" or "bad."
4. Difficult to tell whether company is, on balance, in a strong or weak position.
5. "Average" performance is not necessarily good.
6. Seasonal factors can distort ratios.
7. "Window dressing" techniques can make statements and ratios look better.
Answers and Solutions: 14  293 j. What are some qualitative factors analysts should consider when evaluating a
company's likely future financial performance?
Answer: Top analysts recognize that certain qualitative factors must be considered when evaluating a
company. These factors, as summarized by the American Association Of Individual Investors
(AAII), are as follows:
1. Are the company's revenues tied to one key customer?
2. To what extent are the company's revenues tied to one key product?
3. To what extent does the company rely on a single supplier?
4. What percentage of the company's business is generated overseas?
5. Competition
6. Future prospects
7. Legal and regulatory environment
Answers and Solutions: 14  294 Chapter 14
Financial Planning and Forecasting Pro Forma Financial
Statements
ANSWERS TO ENDOFCHAPTER QUESTIONS
141 a. The operating plan provides detailed implementation guidance designed to accomplish corporate
objectives. It details who is responsible for what particular function, and when specific tasks are
to be accomplished. The financial plan details the financial aspects of the corporation's
operating plan. In addition to an analysis of the firm's current financial condition, the financial
plan normally includes a sales forecast, the capital budget, the cash budget, pro forma financial
statements, and the external financing plan. A sales forecast is merely the forecast of unit and
dollar sales for some future period. Of course, a lot of work is required to produce a good sales
forecast. Generally, sales forecasts are based on the recent trend in sales plus forecasts of the
economic prospects for the nation, industry, region, and so forth. The sales forecast is critical to
good financial planning.
b. A pro forma financial statement shows how an actual statement would look if certain assumptions
are realized. With the percent of sales forecasting method, many items on the income statement
and balance sheets are assumed to increase proportionally with sales. As sales increase, these
items that are tied to sales also increase, and the values of these items for a particular year are
estimated as percentages of the forecasted sales for that year.
c. Funds are spontaneously generated if a liability account increases spontaneously (automatically)
as sales increase. An increase in a liability account is a source of funds, thus funds have been
generated. Two examples of spontaneous liability accounts are accounts payable and accrued
wages. Note that notes payable, although a current liability account, is not a spontaneous source
of funds since an increase in notes payable requires a specific action between the firm and a
creditor.
Answers and Solutions: 14  295 d. Additional funds needed (AFN) are those funds required from external sources to increase the
firm's assets to support a sales increase. A sales increase will normally require an increase in
assets. However, some of this increase is usually offset by a spontaneous increase in liabilities
as well as by earnings retained in the firm. Those funds that are required but not generated
internally must be obtained from external sources. Although most firms' forecasts of capital
requirements are made by constructing pro forma income statements and balance sheets, the AFN
formula is sometimes used to forecast financial requirements. It is written as follows:
Additional Required Spontaneous Increase in
funds = increase  increase in  retained
needed in assets liabilities earnings
A
AFN = S  L S  MS1 (1  d ).
S S
Capital intensity is the dollar amount of assets required to produce a dollar of
sales. The capital intensity ratio is the reciprocal of the total assets turnover
ratio.
e. "Lumpy" assets are those assets that cannot be acquired smoothly, but require large, discrete
additions. For example, an electric utility that is operating at full capacity cannot add a small
amount of generating capacity, at least not economically.
142 Accounts payable, accrued wages, and accrued taxes increase spontaneously and proportionately with
sales. Retained earnings increase, but not proportionately.
143 The equation gives good forecasts of financial requirements if the ratios A*/S and L*/S, as well as M
and d, are stable. Otherwise, another forecasting technique should be used.
145 a. +.
b. +. It reduces spontaneous funds; however, it may eventually increase retained earnings.
c. +.
d. +.
Answers and Solutions: 14  296 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
141 AFN = (A*/S0)S  (L*/S0)S  MS1(1  d)
$3,000,000 $500,000
= $1,000,000  $1,000,000  0.05($6,000,000)(1  0.7)
$5,000,000 $5,000,000
= (0.6)($1,000,000)  (0.1)($1,000,000)  ($300,000)(0.3)
= $600,000  $100,000  $90,000
= $410,000.
$4,000,000 142 AFN = $1,000,000 (0.1)($1,000,000) ($300,000)(0.3)
$5,000,000
= (0.8)($1,000,000)  $100,000  $90,000
= $800,000  $190,000
= $610,000.
The capital intensity ratio is measured as A*/S0. This firm's capital intensity ratio is higher than
that of the firm in Problem 141; therefore, this firm is more capital intensiveit would require a
large increase in total assets to support the increase in sales.
143 AFN = (0.6)($1,000,000)  (0.1)($1,000,000)  0.05($6,000,000)(1  0)
= $600,000  $100,000  $300,000
= $200,000.
Under this scenario the company would have a higher level of retained earnings which would
reduce the amount of additional funds needed.
Answers and Solutions: 14  297 144 S2004 = $2,000,000; A2004 = $1,500,000; CL2004 = $500,000;
NP2004 = $200,000; A/P2004 = $200,000; Accruals2004 = $100,000;
PM = 5%; d = 60%; A*/S0 = 0.75.
AFN = (A*/S0)S  (L*/S0)S  MS1(1  d)
$300,000
= (0.75)S  S (0.05)(S1)(1  0.6)
$2,000,000
= (0.75)S  (0.15)S  (0.02)S1
= (0.6)S  (0.02)S1
= 0.6(S1  S0)  (0.02)S1
= 0.6(S1  $2,000,000)  (0.02)S1
= 0.6S1  $1,200,000  0.02S1
$1,200,000 = 0.58S1
$2,068,965.52 = S1.
Sales can increase by $2,068,965.52  $2,000,000 = $68,965.52 without additional funds being
needed.
145 a. AFN = (A*/S)(S) (L*/S)(S) MS1(1 d)
$122.5 $17.5 $10.5
= ($70)  ($70)  ($420)(0.6) = $13.44 million.
$350 $350 $350
Answers and Solutions: 14  298 b. Upton Computers
Pro Forma Balance Sheet
December 31, 2005
(Millions of Dollars)
Forecast Pro Forma
Basis %
after
2004 2005 Sales Additions Pro Forma Financing Financing
Cash $ 3.5 0.0100 $ 4.20 $ 4.20
Receivables 26.0 0.7430 31.20 31.20
Inventories 58.0 0.1660 69.60 69.60
Total current
assets $ 87.5 $105.00 $105.00
Net fixed assets 35.0 0.100 42.00 42.00
Total assets $122.5 $147.00 $147.00
Accounts payable $ 9.0 0.0257 $ 10.80 $ 10.80
Notes payable 18.0 18.00 +13.44 31.44
Accruals 8.5 0.0243 10.20 10.20
Total current
liabilities $ 35.5 $ 39.00 $ 52.44
Mortgage loan 6.0 6.00 6.00
Common stock 15.0 15.00 15.00
Retained earnings 66.0 7.56* 73.56 73.56
Total liab.
and equity $122.5 $133.56 $147.00
AFN = $ 13.44
*PM = $10.5/$350 = 3%.
Payout = $4.2/$10.5 = 40%.
NI = $350 × 1.2 × 0.03 = $12.6.
Addition to RE = NI  DIV = $12.6  0.4($12.6) = 0.6($12.6) = $7.56.
Answers and Solutions: 14  299 146 a. Stevens Textiles
Pro Forma Income Statement
December 31, 2005
(Thousands of Dollars)
Forecast Pro Forma
2004 Basis
2005
Sales $36,000 1.15 × Sales04 $41,400
Operating costs $32,440 0.9011 × Sales05 37,306
EBIT $ 3,560 $ 4,094
Interest 460 0.10 × Debt04 560
EBT $ 3,100 $ 3,534
Taxes (40%) 1,240 1,414
Net income $ 1,860 $ 2,120
Dividends (45%) $ 837 $ 954
Addition to RE $ 1,023 $ 1,166
Stevens Textiles
Pro Forma Balance Sheet
December 31, 2005
(Thousands of Dollars)
Forecast Pro Forma
Basis % after
2004 2005 Sales Additions Pro Forma Financing Financing
Cash $ 1,0800 0.0300 $ 1,242 $ 1,242
Accts receivable 6,480 0.1883 7,452 7,452
Inventories 9,000 0.2005 10,350 10,350
Total curr.
assets $16,560 $19,044 $19,044
Fixed assets 12,600 0.3500 14,490 14,490
Total assets $29,160 $33,534 $33,534
Accounts payable $ 4,320 0.1200 $ 4,968 $ 4,968
Accruals 2,880 0.0800 3,312 3,312
Notes payable 2,100 2,100 +2,128 4,228
Total current
liabilities $ 9,300 $10,380 $12,508
Longterm debt 3,500 3,500 3,500
Total debt $12,800 $13,880 $16,008
Common stock 3,500 3,500 3,500
Retained earnings 12,860 1,166* 14,026 14,026
Total liabilities
and equity $29,160 $31,406 $33,534
Answers and Solutions: 14  300 AFN = $ 2,128
*From income statement.
147 a. & b. Garlington Technologies Inc.
Pro Forma Income Statement
December 31, 2005
Forecast
2004
Basis Additions 2005
Sales $3,600,000 1.10 × Sales04 $3,960,000
Operating costs 3,279,720 0.911 × Sales05 3,607,692
EBIT $ 320,280 $
352,308
Interest 18,280 0.13 × Debt04 20,280
EBT $ 302,000 $ 332,028
Taxes (40%) 120,800 132,811
Net income $ 181,200 $ 199,217
Dividends: $ 108,000 Set by management $ 112,000
Addition to RE: $ 73,200 $ 87,217
Answers and Solutions: 14  301 Garlington Technologies Inc.
Pro Forma Balance Statement
December 31, 2005
Forecast
Basis % AFN With AFN
2004 2005 Sales Additions 2005 Effects 2005 Cash $ 180,000 0.05 $ 198,000 $ 198,000 Receivables 360,000 0.10 396,000 396,000 Inventories 720,000 0.20 792,000 792,000
Total current
assets $1,260,000 $1,386,000 $1,386,000 Fixed assets 1,440,000 0.40 1,584,000 1,584,000
Total assets $2,700,000 $2,970,000 $2,970,000
Accounts payable $ 360,000 0.10 $ 396,000 $ 396,000 Notes payable 156,000 156,000 +128,783 284,783 Accruals 180,000 0.05 198,000 198,000
Total current
liabilities $ 696,000 $ 750,000 $ 878,783 Common stock 1,800,000 1,800,000 1,800,000 Retained earnings 204,000 87,217* 291,217 291,217
Total liab. and equity $2,700,000 $2,841,217 $2,970,000
AFN = $ 128,783
Cumulative AFN = $ 128,783
*See income statement.
Answers and Solutions: 14  302 Total liabilities Accounts Long  term Common Retained 148 a. = + + + .
and equity Payable debt stock earnings
$1,200,000 = $375,000 + Longterm debt + $425,000 + $295,000
Longterm debt = $105,000.
Total debt = Accounts payable + Longterm debt
= $375,000 + $105,000 = $480,000.
Alternatively,
Total
Total debt = liabilities  Common stock  Retained earnings
and equity
= $1,200,000  $425,000  $295,000 = $480,000.
b. Assets/Sales (A*/S) = $1,200,000/$2,500,000 = 48%.
L*/Sales = $375,000/$2,500,000 = 15%.
2002 Sales = (1.25)($2,500,000) = $3,125,000.
AFN = (A*/S)(S)  (L*/S)(S)  MS1(1  d)  New common stock
= (0.48)($625,000)  (0.15)($625,000)  (0.06)($3,125,000)(0.6)  $75,000
= $300,000  $93,750  $112,500  $75,000 = $18,750.
Alternatively, using the percentage of sales method:
Forecast
Basis % Additions (New
2004 2005 Sales Financing, R/E) Pro Forma
Total assets $1,200,000 0.48 $1,500,000
Current liabilities $ 375,000 0.15 $ 468,750
Longterm debt 105,000 105,000
Total debt $ 480,000 $ 573,750
Common stock 425,000 75,000* 500,000
Retained earnings 295,000 112,500** 407,500
Total common equity $ 720,000 $ 907,500
Total liabilities
and equity $1,200,000 $1,481,250
AFN = Longterm debt = $ 18,750
*Given in problem that firm will sell new common stock = $75,000.
**PM = 6%; Payout = 40%; NI2005 = $2,500,000 x 1.25 x 0.06 = $187,500.
Addition to RE = NI x (1  Payout) = $187,500 x 0.6 = $112,500.
Answers and Solutions: 14  303 149 Cash $ 100.00 × 2 = $ 200.00
Accounts receivable 200.00 × 2 = 400.00
Inventories 200.00 × 2 = 400.00
Net fixed assets 500.00 + 0.0 = 500.00
Total assets $1,000.00 $1,500.00
Accounts payable $ 50.00 × 2 = $ 100.00
Notes payable 150.00 150.00 + 360.00 = 510.00
Accruals 50.00 × 2 = 1 00.00
Longterm debt 400.00
400.00
Common stock 100.00
100.00
Retained earnings 250.00 + 40 = 290.00
Total liabilities
and equity $1,000.00
$1,140.00
AFN
$ 360.00
Capacity sales = Sales/0.5 = $1,000/0.5 = $2,000.
Target FA/S ratio = $500/$2,000 = 0.25.
Target FA = 0.25($2,000) = $500 = Required FA. Since the firm currently has $500 of
fixed assets, no new fixed assets will be required.
Addition to RE = M(S1)(1  Payout ratio) = 0.05($2,000)(0.4) = $40.
Answers and Solutions: 14  304 SOLUTION TO SPREADSHEET PROBLEM
1410 The detailed solution for the spreadsheet problem is available both on the instructor's resource
CDROM (in the file Solution to FM11 Ch 14 10 Build a Model.xls) and on the instructor's side of
the web site, http://brigham.swcollege.com.
Answers and Solutions: 14  305 MINI CASE
Betty Simmons, the new financial manager of Southeast Chemicals (SEC), a Georgia producer of specialized chemicals for use in fruit orchards, must prepare a financial forecast for 2005. SEC's 2004 sales were $2 billion, and the marketing department is forecasting a 25 percent increase for 2005. Simmons thinks the company was operating at full capacity in 2004, but she is not sure about this. The 2004 financial statements, plus some other data, are shown below.
Assume that you were recently hired as Simmons' assistant, and your first major task is to help her develop the forecast. She asked you to begin by answering the following set of questions.
Financial Statements And Other Data On SEC
(Millions Of Dollars) A. 2004 Balance Sheet % of % of
sales sales
Cash & Securities $ 20 1% Accounts Payable
And Accruals $ 100 5%
Accounts Receivable 240 12 Notes Payable 100
Inventory 240 12 Total Current Liabilities $ 200
Total Current Assets $ 500 LongTerm Debt 100
Net Fixed Assets 500 25 Common Stock 500
Retained Earnings 200
Total Assets $1,000 Total Liabilities And Equity $1,000
B. 2004 Income Statement % of
sales
Sales $2,000.00
Cost Of Goods Sold (COGS) 1,200.00 60%
Sales, General, And Administrative Costs 700.00 35
Earnings Before Interest And Taxes $ 100.00
Interest 10.00
Earnings Before Taxes $ 90.00
Taxes (40%) 36.00
Net Income $ 54.00
Dividends (40%) $ 21.60
Addition To Retained Earnings $ 32.40
Answers and Solutions: 15  306 C. Key Ratios Sec Industry
Profit Margin 2.70 4.00
Return On Equity 7.71 15.60
Days Sales Outstanding (365 Days) 43.80 Days 32.00 Days
Inventory Turnover 8.33× 11.00×
Fixed Assets Turnover 4.00 5.00
Debt/Assets 30.00% 36.00%
Times Interest Earned 10.00× 9.40×
Current Ratio 2.50 3.00
Return On Invested Capital
(NOPAT/Operating Capital) 6.67% 14.00%
a. Describe three ways that pro forma statements are used in financial planning.
Answer: Three important uses: (1) forecast the amount of external financing that will be required, (2)
evaluate the impact that changes in the operating plan have on the value of the firm, (3) set
appropriate targets for compensation plans
b. Explain the steps in financial forecasting.
Answer: (1) forecast sales, (2) project the assets needed to support sales, (3) project internally generated
funds, (4) project outside funds needed, (5) decide how to raise funds, and (6) see effects of plan
on ratios and stock price.
c. Assume (1) that SEC was operating at full capacity in 2004 with respect to all assets, (2) that
all assets must grow proportionally with sales, (3) that accounts payable and accruals will
also grow in proportion to sales, and (4) that the 2004 profit margin and dividend payout
will be maintained. Under these conditions, what will the company's financial
requirements be for the coming year? Use the AFN equation to answer this question.
Answer: SEC will need $184.5 million. Here is the AFN equation:
AFN = (A*/S0)S  (L*/S0)S  M(S1)(RR)
= (A*/S0)(g)(S0)  (L*/S0)(g)(S0)  M(S0)(1 + g)(1  payout)
= ($1,000/$2,000)(0.25)($2,000)  ($100/$2,000)(0.25)($2,000)
 0.0270($2,000)(1.25)(0.6)
= $250  $25  $40.5 = $184.5 million.
d. How would changes in these items affect the AFN? (1) sales increase, (2) the dividend
payout ratio increases, (3) the profit margin increases, (4) the capital intensity ratio
increases, and (5) SEC begins paying its suppliers sooner. (Consider each item separately
and hold all other things constant.)
Answer: 1. If sales increase, more assets are required, which increases the AFN.
Answers and Solutions: 15  307 2. If the payout ratio were reduced, then more earnings would be retained, and this would reduce
the need for external financing, or AFN. Note that if the firm is profitable and has any
payout ratio less than 100 percent, it will have some retained earnings, so if the growth rate
were zero, AFN would be negative, i.e., the firm would have surplus funds. As the growth
rate rose above zero, these surplus funds would be used to finance growth. At some point,
i.e., at some growth rate, the surplus AFN would be exactly used up. This growth rate where
AFN = $0 is called the "sustainable growth rate," and it is the maximum growth rate which
can be financed without outside funds, holding the debt ratio and other ratios constant.
3. If the profit margin goes up, then both total and retained earnings will increase, and this will
reduce the amount of AFN.
4. The capital intensity ratio is defined as the ratio of required assets to total sales, or a*/s0. Put
another way, it represents the dollars of assets required per dollar of sales. The higher the
capital intensity ratio, the more new money will be required to support an additional dollar of
sales. Thus, the higher the capital intensity ratio, the greater the AFN, other things held
constant.
5. If SEC begins paying sooner, this reduces spontaneous liabilities, leading to a higher AFN.
e. Briefly explain how to forecast financial statements using the percent of sales approach. Be
sure to explain how to forecast interest expenses.
Answer: Project sales based on forecasted growth rate in sales. Forecast some items as a percent of the
forecasted sales, such as costs, cash, accounts receivable, inventories, net fixed assets, accounts
payable, and accruals. Choose other items according to the company's financial policy: debt,
dividend policy (which determines retained earnings), common stock. Given the previous
assumptions and choices, we can estimate the required assets to support sales and the specified
sources of financing. The additional funds needed (AFN) is: required assets minus specified
sources of financing. If AFN is positive, then you must secure additional financing. If AFN is
negative, then you have more financing than is needed and you can pay off debt, buy back stock,
or buy shortterm investments.
Interest expense is actually based on the daily balance of debt during the year. There are three
ways to approximate interest expense. You can base it on: (1) debt at end of year, (2) debt at
beginning of year, or (3) average of beginning and ending debt.
Basing interest expense on debt at end of year will overestimate interest expense if debt is added
throughout the year instead of all on January 1. It also causes circularity called financial
feedback: more debt causes more interest, which reduces net income, which reduces retained
earnings, which causes more debt, etc.
Basing interest expense on debt at beginning of year will underestimate interest expense if debt
is added throughout the year instead of all on December 31. But it doesn't cause problem of
circularity.
Basing interest expense on average of beginning and ending debt will accurately estimate the
interest payments if debt is added smoothly throughout the year. But it has the problem of
circularity.
Answers and Solutions: 15  308 A solution that balances accuracy and complexity is to base interest expense on beginning debt,
but use a slightly higher interest rate. This is easy to implement and is reasonably accurate.
See FM11 Ch 14 Mini Case Feedback.xls for an example basing interest expense on average
debt.
f. Now estimate the 2005 financial requirements using the percent of sales approach. Assume
(1) that each type of asset, as well as payables, accruals, and fixed and variable costs, will be
the same percent of sales in 2005 as in 2004; (2) that the payout ratio is held constant at 40
percent; (3) that external funds needed are financed 50 percent by notes payable and 50
percent by longterm debt (no new common stock will be issued); (4) that all debt carries an
interest rate of 10 percent; and (5) interest expenses should be based on the balance of debt
at the beginning of the year.
Answer: See the completed worksheet. The problem is not difficult to do "by hand," but we used a
spreadsheet model for the flexibility such a model provides.
Income Statement (In Millions Of Dollars) Actual Forecast
2004 Forecast Basis 2005 Sales $ 2,000.0 Growth 1.25 $ 2,500.0 COGS $ 1,200.0 % Of Sales 60.00% $ 1,500.0 SGA Expenses $ 700.0 % Of Sales 35.00% $ 875.0 EBIT $ 100.0 $ 125.0 Less Interest $ 10.0 Interest Rate X Debt04 $ 20.0 EBT $ 90.0 $ 105.0 Taxes (40%) $ 36.0 $ 42.0 Net Income $ 54.0 $ 63.0 Dividends $ 21.6 $ 25.2 Add. To Retained Earnings $ 32.4 $ 37.8
Answers and Solutions: 15  309 2005 2005 Balance Sheet Forecast Forecast (In Millions Of Dollars) Without With
Forecast
2004 Basis AFN AFN AFN Assets 0 Cash $ 20.0 % Of Sales 1.00% $ 25.0 $ 25.0 Accounts Receivable $240.0 % Of Sales 12.00% $300.0 $300.0 Inventories $240.0 % Of Sales 12.00% $300.0 $300.0 Total Current Assets $500.0 $625.0 $625.0 Net Plant And Equipment $500.0 % Of Sales 25.00% $625.0 $625.0 Total Assets $1,000.0 $1,250.0 $1,250.0
Liabilities And Equity Accounts Payable & Accruals $100.0 % Of Sales 5.00% $125.0 $125.0 Notes Payable $100.0 CarryOver $100.0 $93.6 $193.6 Total Current Liabilities $200.0 $225.0 $318.6 LongTerm Bonds $100.0 CarryOver $100.0 $93.6 $193.6 Total Liabilities $300.0 $325.0 $512.2 Common Stock $500.0 CarryOver $500.0 $500.0
RE04 + Retained Earnings $200.0 RE04 $237.8 $237.8 Total Common Equity $700.0 $737.8 $737.8 Total Liabilities And Equity $1,000.0 $1,062.8 $1,250.0
Required Assets = $1,250.0 Specified Sources Of Financing = $1,062.8 Additional Funds Needed (AFN) $187.20
g. Why does the percent of sales approach produce a somewhat different AFN than the
equation approach? Which method provides the more accurate forecast?
Answer: The difference occurs because the AFN equation method assumes that the profit margin remains
constant, while the forecasted balance sheet method permits the profit margin to vary. The
balance sheet method is somewhat more accurate, but in this case the difference is not very large.
The real advantage of the balance sheet method is that it can be used when everything does not
increase proportionately with sales. In addition, forecasters generally want to see the resulting
ratios, and the balance sheet method is necessary to develop the ratios.
In practice, the only time we have ever seen the AFN equation used is to provide (1) a "quick
and dirty" forecast prior to developing the balance sheet forecast and (2) a rough check on the
balance sheet forecast.
Answers and Solutions: 15  310 h. Calculate SEC's forecasted ratios, and compare them with the company's 2004 ratios and
with the industry averages. Calculate SEC's forecasted free cash flow and return on
invested capital (ROIC).
Answer:
Actual Forecast
Key Ratios 2004 2005 Industry
Profit Margin 2.70% 2.52% 4.00%
ROE 7.71% 8.54% 15.60%
DSO 43.80 43.80 32.00
Inventory Turnover 8.33 8.33 11.00
Fixed Asset Turnover 4.00 4.00 5.00
Debt/Assets 30.00% 40.98% 36.00%
TIE 10.00 6.25 9.40
Current Ratio 2.50 1.96 3.00
Free Operating Gross Investment in
= 
Cash Flow Cash Flow Operating Capital
= NOPAT  Net Investment In Operating Capital
FCF = NOPAT  (Operating Capital2005  Operating Capital2004)
= $125(1  0.4) + [($625  $125 + $625)  ($500  $100 + $500)
= $75  ($1,125  $900) = $75  $225 = $150.
Note: Operating Capital = Net Operating Working Capital + Net Fixed Assets.
ROIC = NOPAT / Capital = $75 / $1,125 = 0.067 = 6.67%.
Answers and Solutions: 15  311 i. Based on comparisons between SEC's days sales outstanding (DSO) and inventory turnover
ratios with the industry average figures, does it appear that SEC is operating efficiently with
respect to its inventory and accounts receivable? Suppose SEC was able to bring these
ratios into line with the industry averages and reduce its SGA/sales ratio to 33%. What
effect would this have on its AFN and its financial ratios? What effect would this have on
free cash flow and ROIC?
Answer: The DSO and inventory turnover ratio indicate that SEC has excessive inventories and receivables.
The effect of improvements here would reduce asset requirements and AFN. See the results
below based on the spreadsheet FM11 Ch 14 Mini Case.xls.
Inputs Before After
DSO 43.20 32.01
Accounts Receivable/Sales 12.0% 8.77%
Inventory Turnover 8.33 11.00
Inventory/Sales 12.0% 9.09%
SGA/Sales 35.0% 33.0%
Outputs
AFN $187.2 $15.7
FCF $150.0 $33.5
ROIC 6.7% 10.8%
ROE 8.5% 12.3%
j. Suppose you now learn that SEC's 2004 receivables and inventories were in line with
required levels, given the firm's credit and inventory policies, but that excess capacity
existed with regard to fixed assets. Specifically, fixed assets were operated at only 75 percent
of capacity.
j. 1. What level of sales could have existed in 2004 with the available fixed assets?
Actual sales $2,000 Answer: Full Capacity Sales = = = $2,667.
% of capacity at which 0.75
fixed assets were operated
Since the firm started with excess fixed asset capacity, it will not have to add as much fixed assets
during 2005 as was originally forecasted:
j. 2. How would the existence of excess capacity in fixed assets affect the additional funds needed
during 2005?
Answer: We had previously found an AFN of $184.5 using the balance sheet method. The fixed assets
increase was 0.25($500) = $125. Therefore, the funds needed will decline by $125.
k. The relationship between sales and the various types of assets is important in financial
forecasting. The percent of sales approach, under the assumption that each asset item
Answers and Solutions: 15  312 grows at the same rate as sales, leads to an AFN forecast that is reasonably close to the
forecast using the AFN equation. Explain how each of the following factors would affect
the accuracy of financial forecasts based on the AFN equation: (1) economies of scale in
the use of assets, and (2) lumpy assets.
Answer: 1. Economies of scale in the use of assets mean that the asset item in question must increase less
than proportionately with sales; hence it will grow less rapidly than sales. Cash and
inventory are common examples, with possible relationship to sales as shown below:
Inventories
Cash
0 Sales
0 Sales
Inventories
Base
Stock
0 Sales
2. Lumpy assets would cause the relationship between assets and sales to look as shown below.
This situation is common with fixed assets.
Answers and Solutions: 15  313 Fixed assets
0 Sales
Answers and Solutions: 15  314 Chapter 15
Corporate Valuation, ValueBased Management, and
Corporate Governance
ANSWERS TO ENDOFCHAPTER QUESTIONS
151 a. Assetsinplace, also known as operating assets, include the land, buildings, machines, and
inventory that the firm uses in its operations to produce its products and services. Growth
options are not tangible. They include items such as R&D and customer relationships.
Financial, or nonoperating, assets include investments in marketable securities and
noncontrolling interests in the stock of other companies.
b. Operating current assets are the current assets used to support operations, such
as cash, accounts receivable, and inventory. It does not include shortterm
investments. Operating current liabilities are the current liabilities that are a
natural consequence of the firm's operations, such as accounts payable and
accruals. It does not include notes payable or any other shortterm debt that
charges interest. Net operating working capital is operating current assets
minus operating current liabilities. Operating capital is sum of net operating
working capital and operating longterm assets, such as net plant and equipment.
Operating capital also is equal to the net amount of capital raised from investors.
This is the amount of interestbearing debt plus preferred stock plus common
equity minus shortterm investments. NOPAT is the amount of net income a
company would generate if it had no debt and held no financial assets. NOPAT
is a better measure of the performance of a company's operations because debt
lowers income. In order to get a true reflection of a company's operating
performance, one would want to take out debt to get a clearer picture of the
situation. Free cash flow is the cash flow actually available for distribution to
investors after the company has made all the investments in fixed assets and
working capital necessary to sustain ongoing operations. It is the most
important measure of cash flows because it shows the exact amount available to
all investors.
c. The value of operations is the present value of all the future free cash flows that
are expected from current assetsinplace and the expected growth of
assetsinplace when discounted at the weighted average cost of capital:
FCFt
Vop(at time 0) = .
t =1 (1 + WACC)
t
The terminal, or horizon value, is the value of operations at the end of the
explicit forecast period. It is equal to the present value of all free cash flows
beyond the forecast period, discounted back to the end of the forecast period at
the weighted average cost of capital:
FCFN +1 FCFN (1 + g)
Vop(at time N) = = .
WACC  g WACC  g The corporate valuation model defines the total value of a company as the value of
operations plus the value of nonoperating assets plus the value of growth
options.
Answers and Solutions: 15  315 d. Valuebased management is the systematic application of the corporate value
model to a company's decisions. The four value drivers are the growth rate in
sales (g), operating profitability (OP=NOPAT/Sales), capital requirements
(CR=Capital/Sales), and the weighted average cost of capital (WACC). Return
on Invested Capital (ROIC) is NOPAT divided by the amount of capital that is
available at the beginning of the year.
e. Managerial entrenchment occurs when a company has such a weak board of
directors and has such strong antitakeover provisions in its corporate charter
that senior managers feel there is very little chance that they will be removed.
Nonpecuniary benefits are perks that are not actual cash payments, such as
lavish offices, memberships at country clubs, corporate jets, and excessively
large staffs.
f. Targeted share repurchases, also known as greenmail, occur when a company
buys back stock from a potential acquiror at a higher than fairmarket price.
In return, the potential acquiror agrees not to attempt to take over the company.
Shareholder rights provisions, also known as poison pills, allow existing
shareholders in a company to purchase additional shares of stock at a lower than
market value if a potential acquiror purchases a controlling stake in the
company. A restricted voting rights provision automatically deprives a
shareholder of voting rights if the shareholder owns more than a specified
amount of stock.
g. A stock option allows its owner to purchase a share of stock at a fixed price,
called the exercise price, no matter what the actual price of the stock is. Stock
options always have an expiration date, after which they cannot be exercised.
A restricted stock grant allows an employee to buy shares of stock at a large
discount from the current stock price, but the employee is restricted from selling
the stock for a specified number of years. An Employee Stock Ownership Plan,
often called an ESOP, is a type of retirement plan in which employees own stock
in the company. 152 The first step is to find the value of operations by discounting all expected future free cash flows at the
weighted average cost of capital. The second step is to find the total corporate value by summing the
value of operations, the value of nonoperating assets, and the value of growth options. The third step
is to find the value of equity by subtracting the value of debt and preferred stock from the total value
of the corporation. The last step is to divide the value of equity by the number of shares of common
stock.
153 A company can be profitable and yet have an ROIC that is less than the WACC if the company has
large capital requirements. If ROIC is less than the WACC, then the company is not earning enough
on its capital to satisfy its investors. Growth adds even more capital that is not satisfying investors,
hence, growth decreases value.
154 Entrenched managers consume to many perquisites, such as lavish offices, excessive staffs, country
club memberships, and corporate jets. They also invest in projects or acquisitions that make the firm
larger, even if they don't make the firm more valuable.
Answers and Solutions: 15  316 155 Stock options in compensation plans usually are issued with an exercise price equal to the current
stock price. As long as the stock price increases, the option will become valuable, even if the stock
price doesn't increase as much as investors expect.
SOLUTIONS TO ENDOFCHAPTER PROBLEMS
151 NOPAT = EBIT(1  T)
= 100(1  0.4) = $60.
Net operating WC04 = ($27 + $80 + $106)  ($52 + $28)
= $213  $80 = $133.
Operating capital04 = $133 + $265 = $398.
Net operating WC05 = ($28 + $84 + $112)  ($56 + $28)
= $224  $84 = $140.
Operating capital05 = $140 + $281 = $421.
FCF = NOPAT  Net investment in operating capital
= $100(0.6)  ($421  $398)
= $37.0.
152 Value of operations = Vop = PV of expected future free cash flow
FCF (1 + g) $400,000(1.05)
Vop = = = $6,000,000.
WACC  g 0.12  0.05
$108 ,000 153 a. Vop2 = = $2,700,000.
0.12  0.08
b. 0 1 2 3
WACC = 12% g = 8%
N
    ··· 
$80,000 $100,000 $108,000
$ 71,428.57
79,719.39
2,152,423.47
$2,303,571.43
Answers and Solutions: 15  317 $40 (1.07) 154 a. Vop 3 = = $713.33.
0.13  0.07
b. 0 1 2 3
4 WACC = 13% N
g = 7%
     ··· 
20 30 40
($ 17.70)
23.49 Vop 3 = 713.33
522.10 753.33
$527.89
c. Total valuet=0 = $527.89 + $10.0 = $537.89.
Value of common equity = $537.89  $100 = $437.89.
$437.89
Price per share = = $43.79.
10.0 155 The growth rate in FCF from 2006 to 2007 is g=($750.00$707.55)/$707.50 = 0.06.
$707.55 (1.06)
VOp at 2006 = = $15,000.
0.11  0.06
$200,000,000 156 Vop = $200,000,000 + [0.09  0.10]
0.098  0.05
=$200,000,000 + ($40,000,000)= $160,000,000.
MVA = $160,000,000  $200,000,000 = 40,000,000.
157 Capital2008 = Sales2008 (0.43)= $129,000,000.
$300,000,000 (1 + 0.05) 0.43
VOp at 2008 = $129,000,000 + 0.06  (0.098) 1 + 0.05
0.098  0.05
= $129,000,000 + [$6,562,500,000][0.020]
= $129,000,000 + $130,375,000 = $259,375,000.
158 Total corporate value = Value of operations + marketable securities
= $756 + $77 = $833 million.
Value of equity = Total corporate value debt Preferred stock
= $833 ($151 + $190)  $76 = $416 million.
159 Total corporate value = Value of operations + marketable securities
= $651 + $47 = $698 million.
Value of equity = Total corporate value debt Preferred stock
= $698 ($65 + $131)  $33 = $469 million.
Price per share = $469 / 10 = $46.90.
Answers and Solutions: 15  318 1510 a. NOPAT2005 = $108.6(10.4) = $65.16
NOWC2005 = ($5.6 + $56.2 + $112.4) ($11.2 + $28.1) = $134.9 million.
Capital2005 = $134.9 + $397.5 = $532.4 million.
FCF2005 = NOPAT Investment in Capital = $65.16 ($532.4  $502.2)
= $65.16  $30.2 = $34.96 million.
b. HV2005 = [$34.96(1.06)]/(0.110.06) = $741.152 million.
c. VOp at 12/31/2004 = [$34.96 + $741.152]/(1+0.11) = $699.20 million.
d. Total corporate value = $699.20 + $49.9 = $749.10 million.
e. Value of equity = $749.10 ($69.9 + $140.8)  $35.0 = $503.4 million.
Price per share = $503.4 / 10 = $50.34.
Answers and Solutions: 15  319 SOLUTION TO SPREADSHEET PROBLEM
1511 The detailed solution for the problem is available both on the instructor's resource CDROM (in the
file "Solution for FM11 Ch 1511 Build a Model.xls") and on the instructor's side of the web site,
http://brigham.swcollege.com.
Answers and Solutions: 15  320 MINI CASE
You have been hired as a consultant to Kulpa Fishing Supplies (KFS), a company that is
seeking to increase its value. KFS has asked you to estimate the value of two privately
held companies that KFS is considering acquiring. But first, the senior management of
KFS would like for you to explain how to value companies that don't pay any dividends.
You have structured your presentation around the following questions.
a. List the three types of assets that companies own.
Answer: Assetsinplace, growth options, and nonoperating, or financial, assets.
b. What are assetsinplace? How can their value be estimated?
Answer: Assetsinplace are tangible, such as buildings, machines, inventory. Usually they are expected
to grow. They generate free cash flows. The PV of their expected future free cash flows,
discounted at the WACC, is the value of operations.
c. What are growth options? How can their value be estimated?
Answer: Growth options are not tangible. They include R&D, such as at drug companies and genetic
engineering companies, and building customer relationships, such as at amazon.com. Growth
options are valued using option pricing techniques in Chapter 17.
d. What are nonoperating assets? How can their value be estimated?
Answer: Nonoperating assets are marketable securities and ownership of noncontrolling interest in another
company. The value of nonoperating assets usually is very close to figure that is reported on
balance sheets.
e. What is the total value of a corporation? Who has claims on this value?
Answer: Total corporate value is sum of value of operations, value of nonoperating assets, and value of
growth options. (No examples in this chapter have a growth option this is deferred until
chapter 17). Debt holders have first claim. Preferred stockholders have the next claim. Any
remaining value belongs to stockholders.
. Answers and Solutions: 16  321 f. 1. The first acquisition target is a privately held company in a mature industry. The
company currently has free cash flow of $20 million. Its WACC is 10% and it is
expected to grow at a constant rate of 5%. The company has marketable securities
of $100 million. It is financed with $200 million of debt, $50 million of preferred
stock, and $210 million of book equity. What is its value of operations?
Answer:
FCF0 (1 + g )
VOp =
(WACC  g )
20 (1 + 0.05)
VOp = = 420
(0.10  0.05)
f. 2. What is its total corporate value? What is its value of equity?
Answer: Total Corporate Value = VOP + MKT. SEC.
= $420 + $100
= $520 million
Value Of Equity = Total  Debt  Pref.
= $520  $200  $50
= $270 million
f. 3. What is its MVA (MVA = total corporate value total book value)?
Answer: MVA = total corporate value of firm minus total book value of firm
total book value of firm = book value of equity + book value of debt
+ book value of preferred stock
MVA = $520  ($210 + $200 + $50)
= $60 million
. Answers and Solutions: 16  322 g. 1. The second acquisition target is a privately held company in a growing industry.
The target has recently borrowed $40 million to finance its expansion; it has no other
debt or preferred stock. It pays no dividends and currently has no marketable
securities. KFS expects the company to produce free cash flows of $5 million in
one year, $10 million in two years, and $20 million in three years. After three years,
free cash flow will grow at a rate of 6%. Its WACC is 10% and it currently has 10
million shares of stock. What is its horizon value (i.e., its value of operations at year
three)? What is its current value of operations (i.e., at time zero)?
Answer: 0 rc = 10% 1 g = 6% 2 3
4 N
     ··· 
5 10 20
$ 4.545
8.264
15.026
398.197 Vop 3 = 530 = 20 (1 + 0.06)
0.10  0.06 $416.942 = Value Of Operations
g. 2. What is its value of equity on a price per share basis?
Answer:
Value of equity = value of operations  debt
= $416.94  $40 = $376.94 million.
price per share = $376.94/10 = $37.69.
h. KFS is also interested in applying valuebased management to its own divisions.
Explain what valuebased management is.
Answer: VBM is the systematic application of the corporate valuation model to all corporate decisions and
strategic initiatives. The objective of VBM is to increase market value added (MVA).
i. What are the four value drivers? How does each of them affect value?
Answer: MVA is determined by four drivers: sales growth, operating profitability (OP=NOPAT/sales),
capital requirements (CR=operating capital / sales, and the weighted average cost of capital.
MVA will improve if WACC is reduced, operating profitability (OP) increases, or the capital
requirement (CR) decreases. See the next question for an explanation of the impact of growth.
. Answers and Solutions: 16  323 j. What is return on invested capital (ROIC)? Why is the spread between ROIC and
WACC so important?
Answer: ROIC is the return on the capital that is in place at the beginning of the period:
NOPATt +1
ROIC =
t +1 Capital t
If the spread between the expected return, ROICt+1, and the required return,
WACC, is positive, then MVA is positive and growth makes MVA larger. The
opposite is true if the spread is negative.
k. KFS has two divisions. Both have current sales of $1,000, current expected growth
of 5%, and a WACC of 10%. Division A has high profitability (OP=6%) but high
capital requirements (CR=78%). Division B has low profitability (OP=4%) but low
capital requirements (CR=27%). What is the MVA of each division, based on the
current growth of 5%? What is the MVA of each division if growth is 6%?
Answer:
Sales t (1 + g) CR
MVA t = OP  WACC (1 + g )
WACC  g
Division A Division B
OP 6% 6% 4% 4%
CR 78% 78% 27% 27%
Growth 5% 6% 5% 6%
MVA (300.0) (360.0) 300.0 385.0
l. What is the ROIC of each division for 5% growth and for 6% growth? How is this
related to MVA?
Answer:
Division A Division B
Capital0 $780 $780 $270 $270
Growth 5% 6% 5% 6%
Sales1 $1,050 $1,060 $1,050 $1,060
Nopat1 $63 $63.6 $42 $42.4
Roic1 8.1% 8.2% 15.6% 15.7%
Mva (300.0) (360.0) 300.0 385.0
. Answers and Solutions: 16  324 The expected ROIC of division A is less than the WACC, so the division should
postpone growth efforts until it improves ROIC by reducing capital requirements (e.g.,
reducing inventory) and/or improving profitability.
The expected ROIC of division b is greater than the WACC, so the division should continue with
its growth plans.
m. The managers at KFS have heard that corporate governance can affect shareholder
value. List for them the three mechanisms of corporate governance.
Answer: The three mechanisms are provisions in the charter that affect takeovers, composition of the board
of directors, and compensation plans.
n. Why is entrenched management potentially harmful to shareholders?
Answer: Entrenchment occurs when there is little chance that poorly performing managers will be replaced.
There are two causes: antitakeover provisions in the charter and a weak board of directors. Management consumes perks: lavish offices, corporate jets, excessively large staffs,
memberships at country clubs Management accepts projects (or acquisitions) to make firm larger, even if MVA goes
down. This is because salary and prestige are highly correlated with size.
o. List three provisions in the corporate charter that affect takeovers.
Answer: These include targeted share repurchases (i.e., greenmail), shareholder rights provisions (i.e.,
poison pills), and restricted voting rights plans.
p. Explain the difference between insiders and outsiders on the board of directors.
What are interlocking boards?
Answer: Weak boards have many insiders (i.e., those who also have another position in the company)
compared with outsiders. Interlocking boards are weaker (CEO of company A sits on board of
company B, CEO of B sits on board of A).
q. What is a stock option in a compensation plan?
Answer: Gives owner of option the right to buy a share of the company's stock at a specified price (called
the exercise price) even if the actual stock price is higher. Usually can't exercise the option for
several years (called the vesting period). Can't exercise the option after a certain number of years
(called the expiration, or maturity, date).
. Answers and Solutions: 16  325 Chapter 16
Capital Structure Decisions: The Basics
ANSWERS TO ENDOFCHAPTER QUESTIONS
161 a. Capital structure is the manner in which a firm's assets are financed; that is, the righthand side of
the balance sheet. Capital structure is normally expressed as the percentage of each type of
capital used by the firmdebt, preferred stock, and common equity. Business risk is the risk
inherent in the operations of the firm, prior to the financing decision. Thus, business risk is the
uncertainty inherent in a total risk sense, future operating income, or earnings before interest and
taxes (EBIT). Business risk is caused by many factors. Two of the most important are sales
variability and operating leverage. Financial risk is the risk added by the use of debt financing.
Debt financing increases the variability of earnings before taxes (but after interest); thus, along
with business risk, it contributes to the uncertainty of net income and earnings per share.
Business risk plus financial risk equals total corporate risk.
b. Operating leverage is the extent to which fixed costs are used in a firm's operations. If a high
percentage of a firm's total costs are fixed costs, then the firm is said to have a high degree of
operating leverage. Operating leverage is a measure of one element of business risk, but does
not include the second major element, sales variability. Financial leverage is the extent to which
fixedincome securities (debt and preferred stock) are used in a firm's capital structure. If a high
percentage of a firm's capital structure is in the form of debt and preferred stock, then the firm is
said to have a high degree of financial leverage. The breakeven point is that level of unit sales at
which costs equal revenues. Breakeven analysis may be performed with or without the inclusion
of financial costs. If financial costs are not included, breakeven occurs when EBIT equals zero.
If financial costs are included, breakeven occurs when EBT equals zero.
c. Reserve borrowing capacity exists when a firm uses less debt under "normal" conditions than
called for by the tradeoff theory. This allows the firm some flexibility to use debt in the future
when additional capital is needed.
162 Business risk refers to the uncertainty inherent in projections of future ROEU.
163 Firms with relatively high nonfinancial fixed costs are said to have a high degree of operating
leverage.
164 Operating leverage affects EBIT and, through EBIT, EPS. Financial leverage has no effect on
EBITit only affects EPS, given EBIT.
165 If sales tend to fluctuate widely, then cash flows and the ability to service fixed charges will also vary.
Such a firm is said to have high business risk. Consequently, there is a relatively large risk that the
firm will be unable to meet its fixed charges, and interest payments are fixed charges. As a result,
firms in unstable industries tend to use less debt than those whose sales are subject to only moderate
fluctuations.
166 Public utilities place greater emphasis on longterm debt because they have more stable sales and
profits as well as more fixed assets. Also, utilities have fixed assets which can be pledged as
collateral. Further, trade firms use retained earnings to a greater extent, probably because these firms
. Answers and Solutions: 16  326 are generally smaller and, hence, have less access to capital markets. Public utilities have lower
retained earnings because they have high dividend payout ratios and a set of stockholders who want
dividends.
167 EBIT depends on sales and operating costs. Interest is deducted from EBIT. At high debt levels,
firms lose business, employees worry, and operations are not continuous because of financing
difficulties. Thus, financial leverage can influence sales and costs, and hence EBIT, if excessive
leverage is used.
168 The tax benefits from debt increase linearly, which causes a continuous increase in the firm's value
and stock price. However, financial distress costs get higher and higher as more and more debt is
employed, and these costs eventually offset and begin to outweigh the benefits of debt.
. Answers and Solutions: 16  327 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
161 a. Here are the steps involved:
(1) Determine the variable cost per unit at present, V:
Profit = P(Q)  FC  V(Q)
$500,000 = ($100,000)(50)  $2,000,000  V(50)
50(V) = $2,500,000
V = $50,000.
(2) Determine the new profit level if the change is made:
New profit = P2(Q2)  FC2  V2(Q2)
= $95,000(70)  $2,500,000  ($50,000  $10,000)(70)
= $1,350,000.
(3) Determine the incremental profit:
Profit = $1,350,000  $500,000 = $850,000.
(4) Estimate the approximate rate of return on new investment:
ROI = Profit/Investment = $850,000/$4,000,000 = 21.25%.
Since the ROI exceeds the 15 percent cost of capital, this analysis suggests that the firm should go
ahead with the change.
. Answers and Solutions: 16  328 b. If we measure operating leverage by the ratio of fixed costs to total costs at the expected output,
then the change would increase operating leverage:
FC $2,000,000
Old: = = 44.44%.
FC + V(Q) $2,000,000 + $2,500,000
FC 2 $2,500,000
New: = = 47.17%.
FC 2 + V2 (Q 2 ) $2,500,000 + $2,800,000
The change would also increase the breakeven point:
F $2,000,000
Old: QBE = = = 40 units.
PV $100,000  $50,000
$2,500,000
New: QBE = = 45.45 units.
$95,000  $40,000
However, one could measure operating leverage in other ways, say by degree of operating
leverage:
Q( P  V ) 50($50,000)
Old: DOL = = = 5.0.
Q( P  V )  F 50($50,000)  $2,000,000
New: The new DOL, at the expected sales level of 70, is
70($95,000  $40,000)
= 2.85.
70($55,000)  $2,500,000
The problem here is that we have changed both output and sales price, so the DOLs are not really
comparable.
c. It is impossible to state unequivocally whether the new situation would have more or less business
risk than the old one. We would need information on both the sales probability distribution and
the uncertainty about variable input cost in order to make this determination. However, since a
higher breakeven point, other things held constant, is more risky, the change in breakeven
pointsand also the higher percentage of fixed costssuggests that the new situation is more
risky.
. Answers and Solutions: 16  329 162 a. Expected ROE for Firm C:
ROEC = (0.1)(5.0%) + (0.2)(5.0%) + (0.4)(15.0%)
+ (0.2)(25.0%) + (0.1)(35.0%) = 15.0%.
Note: The distribution of ROEC is symmetrical. Thus, the answer to this problem could have
been obtained by simple inspection.
Standard deviation of ROE for Firm C:
0.1( 5.0  15.0) 2 + 0.2(5.0  15.0) 2 + 0.4(15.0  15.0) 2 +
C =
0.2( 25.0  15.0) 2 + 0.1(35.0  15.0) 2
= 0.1( 20) 2 + 0.2( 10) 2 + 0.4(0) 2 + 0.2(10) 2 + 0.1( 20) 2
= 40 + 20 + 0 + 20 + 40 = 120 = 11.0%.
b. According to the standard deviations of ROE, Firm A is the least risky, while C is the most risky.
However, this analysis does not take into account portfolio effectsif C's ROE goes up when
most other companies' ROEs decline (that is, its beta is negative), its apparent riskiness would be
reduced.
c. Firm A's ROE = BEP = 5.5%. Therefore, Firm A uses no financial leverage and has no financial
risk. Firm B and Firm C have ROE > BEP, and hence both use leverage. Firm C uses the most
leverage because it has the highest ROE  BEP = measure of financial risk. However, Firm C's
stockholders also have the highest expected ROE.
163 a. Original value of the firm (D = $0):
V = D + S = 0 + ($15)(200,000) = $3,000,000.
Original cost of capital:
WACC = wd rd(1T) + wers
= 0 + (1.0)(10%) = 10%.
With financial leverage (wd=30%):
WACC = wd rd(1T) + wers
= (0.3)(7%)(10.40) + (0.7)(11%) = 8.96%.
Because growth is zero, the value of the company is:
FCF ( EBIT)(1  T ) ($500,000)(1  0.40)
V= = = = $3,348,214.286. .
WACC WACC 0.0896
Increasing the financial leverage by adding $900,000 of debt results in an increase in the firm's
value from $3,000,000 to $3,348,214.286.
. Answers and Solutions: 16  330 b. Using its target capital structure of 30% debt, the company must have debt of:
D = wd V = 0.30($3,348,214.286) = $1,004,464.286.
Therefore, its debt value of equity is:
S = V D = $2,343,750.
Alternatively, S = (1wd)V = 0.7($3,348,214.286) = $2,343,750.
The new price per share, P, is:
P = [S + (D D0)]/n0 = [$2,343,750 + ($1,004,464.286 0)]/200,000
= $16.741.
c. The number of shares repurchased, X, is:
X = (D D0)/P = $1,004,464.286 / $16.741 = 60,000.256 60,000.
The number of remaining shares, n, is:
n = 200,000 60,000 = 140,000.
Initial position:
EPS = [($500,000 0)(10.40)] / 200,000 = $1.50.
With financial leverage:
EPS = [($500,000 0.07($1,004,464.286))(10.40)] / 140,000
= [($500,000 $70,312.5)(10.40)] / 140,000
= $257,812.5 / 140,000 = $1.842.
Thus, by adding debt, the firm increased its EPS by $0.342.
EBIT EBIT
d. 30% debt: TIE = = .
I $70,312.5
Probability TIE
0.10 ( 1.42)
0.20 2.84
0.40 7.11
0.20 11.38
0.10 15.64
The interest payment is not covered when TIE < 1.0. The probability of this occurring is 0.10, or
10 percent.
. Answers and Solutions: 16  331 164 a. Present situation (50% debt):
WACC = wd rd(1T) + wers
= (0.5)(10%)(10.15) + (0.5)(14%) = 11.25%.
FCF ( EBIT)(1  T ) ($13.24)(1  0.15)
V= = = = $100 million.
WACC WACC 0.1125
70 percent debt:
WACC = wd rd(1T) + wers
= (0.7)(12%)(10.15) + (0.3)(16%) = 11.94%.
FCF ( EBIT)(1  T ) ($13.24)(1  0.15)
V= = = = $94.255 million.
WACC WACC 0.1194
30 percent debt:
WACC = wd rd(1T) + wers
= (0.3)(8%)(10.15) + (0.7)(13%) = 11.14%.
FCF ( EBIT)(1  T ) ($13.24)(1  0.15)
V= = = = $101.023 million.
WACC WACC 0.1114
165 a. BEA's unlevered beta is bU=bL/(1+ (1T)(D/S))=1.0/(1+(10.40)(20/80)) = 0.870.
b. bL = bU (1 + (1T)(D/S)).
At 40 percent debt: bL = 0.87 (1 + 0.6(40%/60%)) = 1.218.
rS = 6 + 1.218(4) = 10.872%
c. WACC = wd rd(1T) + wers
= (0.4)(9%)(10.4) + (0.6)(10.872%) = 8.683%.
FCF ( EBIT)(1  T ) ($14.933)(1  0.4)
V= = = = $103.188 million.
WACC WACC 0.08683
. Answers and Solutions: 16  332 166 Tax rate = 40% rRF = 5.0%
bU = 1.2 rM rRF = 6.0%
From data given in the problem and table we can develop the following table:
D/A E/A D/E rd rd(1 T) Leveraged r sb WACCc
betaa
0.00 1.00 0.0000 7.00% 4.20% 1.20 12.20% 12.20%
0.20 0.80 0.2500 8.00 4.80 1.38 13.28 11.58
0.40 0.60 0.6667 10.00 6.00 1.68 15.08 11.45
0.60 0.40 1.5000 12.00 7.20 2.28 18.68 11.79
0.80 0.20 4.0000 15.00 9.00 4.08 29.48 13.10
Notes:
a
These beta estimates were calculated using the Hamada equation,
b = bU[1 + (1 T)(D/E)].
b
These rs estimates were calculated using the CAPM, rs = rRF + (rM rRF)b.
c
These WACC estimates were calculated with the following equation:
WACC = wd(rd)(1 T) + (wc)(rs).
The firm's optimal capital structure is that capital structure which minimizes the firm's
WACC. Elliott's WACC is minimized at a capital structure consisting of 40% debt and
60% equity. At that capital structure, the firm's WACC is 11.45%.
. Answers and Solutions: 16  333 SOLUTION TO SPREADSHEET PROBLEM
167 The detailed solution for the problem is available both on the instructor's resource CDROM (in the
file Solution for FM11 Ch 16 P7 Build a Model.xls) and on the instructor's side of the web site,
http://brigham.swcollege.com.
. Answers and Solutions: 16  334 MINI CASE
Assume you have just been hired as business manager of PizzaPalace, a pizza restaurant located adjacent to campus. The company's EBIT was $500,000 last year, and since the university's enrollment is capped, EBIT is expected to remain constant (in real terms) over time. Since no expansion capital will be required, PizzaPalace plans to pay out all earnings as dividends. The management group owns about 50 percent of the stock, and the stock is traded in the overthecounter market.
The firm is currently financed with all equity; it has 100,000 shares outstanding; and P0 = $25 per share. When you took your MBA Corporate Finance course, your instructor stated that most firms' owners would be financially better off if the firms used some debt. When you suggested this to your new boss, he encouraged you to pursue the idea. As a first step, assume that you obtained from the firm's investment banker the following estimated costs of debt for the firm at different capital structures:
% Financed With Debt Rd
0% 
20 8.0%
30 8.5
40 10.0
50 12.0
If the company were to recapitalize, debt would be issued, and the funds received would be used to repurchase stock. PizzaPalace is in the 40 percent stateplusfederal corporate tax bracket, its beta is 1.0, the riskfree rate is 6 percent, and the market risk premium is 6 percent.
a. Provide a brief overview of capital structure effects. Be sure to identify the ways in which
capital structure can affect the weighted average cost of capital and free cash flows.
Answer: The basic definitions are:
(1) V = Value Of Firm
(2) FCF = Free Cash Flow
(3) WACC = Weighted Average Cost Of Capital
(4) Rs And Rd are costs of stock and debt
(5) We And Wd are percentages of the firm that are financed with stock and debt.
The impact of capital structure on value depends upon the effect of debt on: WACC and/or FCF.
Debt holders have a prior claim on cash flows relative to stockholders. Debt holders' "fixed"
claim increases risk of stockholders' "residual" claim, so the cost of stock, rs, goes up.
Firm's can deduct interest expenses. This reduces the taxes paid, frees up more cash for
payments to investors, and reduces aftertax cost of debt
Answers and Solutions: 17  335 Debt increases the risk of bankruptcy, causing pretax cost of debt, rd, to increase.
Adding debt increase the percent of firm financed with lowcost debt (wd) and decreases the
percent financed with highcost equity (we).
The net effect on WACC is uncertain, since some of these effects tend to increase WACC and
some tend to decrease WACC.
Additional debt can affect FCF. The additional debt increases the probability of bankruptcy.
The direct costs of financial distress are legal fees, "fire" sales, etc. The indirect costs are lost
customers, reductions in productivity of managers and line workers, reductions in credit (i.e.,
accounts payable) offered by suppliers. Indirect costs cause NOPAT to go down due to lost
customers and drop in productivity and causes the investment in capital to go up due to increases
in net operating working capital (accounts payable goes up as suppliers tighten credit).
Additional debt can affect the behavior of managers. It can cause reductions in agency costs,
because debt "precommits," or "bonds," free cash flow for use in making interest payments.
Thus, managers are less likely to waste FCF on perquisites or nonvalue adding acquisitions.
But it can cause increases in other agency costs. Debt can make managers too riskaverse,
causing "underinvestment" in risky but positive NPV projects.
There are also effects due to asymmetric information and signaling. Managers know the firm's
future prospects better than investors. Thus, managers would not issue additional equity if they
thought the current stock price was less than the true value of the stock (given their inside
information). Hence, investors often perceive an additional issuance of stock as a negative
signal, and the stock price falls.
b. (1) What is business risk? What factors influence a firm's business risk?
Answer: Businsess risk is uncertainty about EBIT. Factors that influence business risk include:
uncertainty about demand (unit sales); uncertainty about output prices; uncertainty about input
costs; product and other types of liability; degree of operating leverage (DOL). b. (2) what is operating leverage, and how does it affect a firm's business risk? Show the
operating break even point if a company has fixed costs of $200, a sales price of $15, and
variables costs of $10.
Answer: Operating leverage is the change in EBIT caused by a change in quantity sold. The higher the
proportion of fixed costs within a firm's overall cost structure, the greater the operating leverage.
Higher operating leverage leads to more business risk, because a small sales decline causes a
larger EBIT decline.
Q is quantity sold, F is fixed cost, V is variable cost, TC is total cost, and P is price per unit.
Operating Breakeven = QBE
QBE = F / (P V)
Example: F=$200, P=$15, AND V=$10:
Answers and Solutions: 17  336 QBE = $200 / ($15 $10) = 40.
c. Now, to develop an example which can be presented to PizzaPalace's management to
illustrate the effects of financial leverage, consider two hypothetical firms: firm U, which
uses no debt financing, and firm L, which uses $10,000 of 12 percent debt. Both firms have
$20,000 in assets, a 40 percent tax rate, and an expected EBIT of $3,000.
1. Construct partial income statements, which start with EBIT, for the two firms.
Answer: Here are the fully completed statements:
Firm U Firm L
Assets $20,000 $20,000
Equity $20,000 $10,000
EBIT $ 3,000 $ 3,000
INT (12%) 0 1,200
EBT $ 3,000 $ 1,800
Taxes (40%) 1,200 720
NI $ 1,800 $ 1,080
c. 2. Now calculate roe for both firms.
Answer: Firm U Firm L
BEP 15.0% 15.0%
ROI 9.0% 11.4%
ROE 9.0% 10.8%
TIE 2.5×
c. 3. What does this example illustrate about the impact of financial leverage on ROE?
Answer: Conclusions from the analysis:
· The firm's basic earning power, BEP = EBIT/total assets, is unaffected by financial
leverage.
· Firm L has the higher expected ROI because of the tax savings effect:
o ROIU = 9.0%.
o ROIL = 11.4%.
Answers and Solutions: 17  337 · Firm L has the higher expected roe:
o ROEU = 9.0%.
o ROEL = 10.8%.
Therefore, the use of financial leverage has increased the expected profitability to
shareholders. The higher roe results in part from the tax savings and also
because the stock is riskier if the firm uses debt.
· At the expected level of EBIT, ROEL > ROEU.
· The use of debt will increase roe only if ROA exceeds the aftertax cost of debt.
Here ROA = unleveraged roe = 9.0% > rd(1  t) = 12%(0.6) = 7.2%, so the use of debt
raises roe.
· Finally, note that the TIE ratio is huge (undefined, or infinitely large) if no debt is
used, but it is relatively low if 50 percent debt is used. The expected tie would be
larger than 2.5× if less debt were used, but smaller if leverage were increased.
d. Explain the difference between financial risk and business risk.
Answer: Business risk increases the uncertainty in future EBIT. It depends on business factors such as
competition, operating leverage, etc. Financial risk is the additional business risk concentrated
on common stockholders when financial leverage is used. It depends on the amount of debt and
preferred stock financing.
e. Now consider the fact that EBIT is not known with certainty, but rather has the following
probability distribution:
Economic State Probability EBIT
Bad 0.25 $2,000
Average 0.50 3,000
Good 0.25 4,000
Redo the part A analysis for firms U and L, but add basic earning power (BEP), return
on investment (ROI), [defined as (net income + interest)/(debt + equity)], and the
timesinterestearned (TIE) ratio to the outcome measures. Find the values for each
firm in each state of the economy, and then calculate the expected values. Finally,
calculate the standard deviation and coefficient of variation of ROE. What does this
example illustrate about the impact of debt financing on risk and return?
Answer: Here are the pro forma income statements:
Answers and Solutions: 17  338 Firm U Firm L
Bad Avg. Good Bad Avg. Good
Prob. 0.25 0.50 0.25 0.25 0.50 0.25
EBIT $2,000 $3,000 $4,000 $2,000 $3,000 $4,000
Interest 0 0 0 1,200 1,200 1,200
EBT $2,000 $3,000 $4,000 $ 800 $1,800 $2,800
Taxes (40%) 800 1,200 1,600 320 720 1,120
NI $1,200 $1,800 $2,400 $ 480 $1,080 $1,680
BEP 10.0% 15.0% 20.0% 10.0% 15.0% 20.0%
ROIC 6.0% 9.0% 12.0% 6.0% 9.0% 12.0%
ROE 6.0% 9.0% 12.0% 4.8% 10.8% 16.8%
TIE 1.7× 2.5× 3.3×
E(BEP) 15.0% 15.0%
E(ROIC) 9.0% 9.0%
E(ROE) 9.0% 10.8%
ROIC 2.12% 2.12%
ROE 2.12% 4.24%
This example illustrates that financial leverage can increase the expected return to
stockholders. But, at the same time, it increases their risk.
· Firm L has a wider range of ROEs and a higher standard deviation of ROE,
indicating that its higher expected return is accompanied by higher risk. To be
precise:
ROE (Unleveraged) = 2.12%, and ROE (Leveraged) = 4.24%.
Thus, in a standalone risk sense, firm L is twice as risky as firm Uits business risk is 2.12
percent, but its standalone risk is 4.24 percent, so its financial risk is 4.24%  2.12% =
2.12%.
f. What does capital structure theory attempt to do? What lessons can be learned from
capital structure theory? Be sure to address the MM models.
Answer: MM theory begins with the assumption of zero taxes. MM prove, under a very restrictive set of
assumptions, that a firm's value is unaffected by its financing mix:
VL = VU.
Therefore, capital structure is irrelevant. Any increase in roe resulting from financial leverage is
exactly offset by the increase in risk (i.e., rs), so WACC is constant.
MM theory later includes corporate taxes. Corporate tax laws favor debt financing over equity
financing. With corporate taxes, the benefits of financial leverage exceed the risks because more
EBIT goes to investors and less to taxes when leverage is used. MM show that:
VL = VU + TD.
If T=40%, then every dollar of debt adds 40 cents of extra value to firm.
Miller later included personal taxes. Personal taxes lessen the advantage of corporate debt.
Answers and Solutions: 17  339 Corporate taxes favor debt financing since corporations can deduct interest expenses, but personal
taxes favor equity financing, since no gain is reported until stock is sold, and longterm gains are
taxed at a lower rate. Miller's conclusions with personal taxes are that the use of debt financing
remains advantageous, but benefits are less than under only corporate taxes. Firms should still
use 100% debt. Note: however, miller argued that in equilibrium, the tax rates of marginal
investors would adjust until there was no advantage to debt.
MM theory ignores bankruptcy (financial distress) costs, which increase as more leverage is used.
At low leverage levels, tax benefits outweigh bankruptcy costs. At high levels, bankruptcy costs
outweigh tax benefits. An optimal capital structure exists that balances these costs and benefits.
This is the tradeoff theory.
MM assumed that investors and managers have the same information. But managers often have
better information. Thus, they would sell stock if stock is overvalued, and sell bonds if stock is
undervalued. Investors understand this, so view new stock sales as a negative signal. This is
signaling theory.
One agency problem is that managers can use corporate funds for nonvalue maximizing purposes.
The use of financial leverage bonds "free cash flow," and forces discipline on managers to avoid
perks and nonvalue adding acquisitions.
A second agency problem is the potential for "underinvestment". Debt increases risk of financial
distress. Therefore, managers may avoid risky projects even if they have positive NPVs.
g. With the above points in mind, now consider the optimal capital structure for PizzaPalace.
g. (1) For each capital structure under consideration, calculate the levered beta, the cost of equity,
and the WACC.
Answer: MM theory implies that beta changes with leverage. Bu is the beta of a firm when it has no debt
(the unlevered beta.) Hamada's equation provides the beta of a levered firm: BL = BU [1 + (1 
T)(D/S)]. For example, to find the cost of equity for wd = 20%, we first use Hamada's equation
to find beta:
BL = BU [1 + (1  T)(D/S)]
= 1.0 [1 + (10.4) (20% / 80%)]
= 1.15
Then use CAPM to find the cost of equity:
RS = RRF + BL (RPM)
= 6% + 1.15 (6%) = 12.9%
We can repeat this for the capital structures under consideration.
WD D/S BL RS
0% 0.00 1.000 12.00%
20% 0.25 1.150 12.90%
30% 0.43 1.257 13.54%
40% 0.67 1.400 14.40%
Answers and Solutions: 17  340 50% 1.00 1.600 15.60%
Next, find the WACC. For example, the WACC for wd = 20% is:
WACC = wd (1T) rd + we rs
WACC = 0.2 (1 0.4) (8%) + 0.8 (12.9%)
WACC = 11.28%
Then repeat this for all capital structures under consideration.
wd rd rs WACC
0% 0.0% 12.00% 12.00%
20% 8.0% 12.90% 11.28%
30% 8.5% 13.54% 11.01%
40% 10.0% 14.40% 11.04%
50% 12.0% 15.60% 11.40%
g. (2) Now calculate the corporate value, the value of the debt that will be issued, and the resulting
market value of equity.
Answer: For example the corporate value for wd = 20% is:
V = FCF / (WACCG)
G=0, so investment in capital is zero; so FCF = NOPAT = EBIT (1T). In this example, NOPAT
= ($500,000)(10.40) = $300,000.
Using these values, V = $300,000 / 0.1128 = $2,659,574.
Repeating this for all capital structures gives the following table:
wd WACC Corp. Value
0% 12.00% $2,500,000
20% 11.28% $2,659,574
30% 11.01% $2,724,796
40% 11.04% $2,717,391
50% 11.40% $2,631,579
As this shows, value is maximized at a capital structure with 30% debt.
g. (3) Calculate the resulting price per share, the number of shares repurchased, and the
remaining shares.
Answers and Solutions: 17  341 Answer: First, find the dollar value of debt and equity. For example, for wd = 20%, the dollar value of
debt is:
d = wd V = 0.2 ($2,659,574) = $531,915.
We can then find the dollar value of equity:
S=VD
S = $2,659,574  $531,915 = $2,127,659.
We repeat this process for all the capital structures.
wd Debt, D Stock Value, S
0% $0 $2,500,000
20% $531,915 $2,127,660
30% $817,439 $1,907,357
40% $1,086,957 $1,630,435
50% $1,315,789 $1,315,789
Note: these are rounded; see FM11 Ch 16 mini case.xls for full calculations.
Notice that the value of the equity declines as more debt is issued, because debt is used to
repurchase stock. But the total wealth of shareholders is the value of stock after the recap plus
the cash received in repurchase, and this total goes up (it is equal to corporate value on earlier
slide).
The firm issues debt, which changes its WACC, which changes value. The firm then uses debt
proceeds to repurchase stock. The stock price changes after debt is issued, but does not change
during actual repurchase (or arbitrage is possible). The stock price after debt is issued but before
stock is repurchased reflects shareholder wealth, which is the sum of the stock and the cash paid
in repurchase.
For example, to find the stock price for wd = 20%, let D0 and N0 denote debt and outstanding
shares before the recap. D  D0 is equal to cash that will be used to repurchase stock. S + (D 
D0) is the wealth of shareholders' after the debt is issued but immediately before the repurchase.
We can express the stock price per share prior to the repurchase, P, for wd = 20%, as:
P = [S + (D D0)]/N0.
P = [$2,127,660 + ($531,915 0)] / 100,000
P = $26.596 per share.
The number of shares repurchased is:
# repurchased = (D  D0) / P
# rep. = ($531,915 0) / $26.596
= 20,000.
The number of remaining shares after the repurchase is:
# remaining = N = S / P
Answers and Solutions: 17  342 N = $2,127,660 / $26.596
= 80,000.
We can apply this same procedure to all the capital structures under consideration.
# Shares # Shares
Wd P Repurch. Remaining
0% $25.00 0 100,000
20% $26.60 20,000 80,000
30% $27.25 30,000 70,000
40% $27.17 40,000 60,000
50% $26.32 50,000 50,000
h. Considering only the capital structures under analysis, what is PizzaPalace's optimal capital
structure?
Answer: The optimal capital structure is for wd = 30%. This gives the highest corporate value, the lowest
WACC, and the highest stock price per share. But notice that wd = 40% is very similar to the
optimal solution; in other words, the optimal range is pretty flat.
i. What other factors should managers consider when setting the target capital structure?
Answer: Managers should also consider the debt ratios of other firms in the industry, pro forma coverage
ratios at different capital structures under different economic scenarios, lender and rating agency
attitudes (i.e., the impact on bond ratings), reserve borrowing capacity, the effects on control (i.e.,
does the capital structure make it easier of harder for an outsider to take over the firm), the firm's
types of assets (i.e., are they tangible, and hence suitable as collateral?, and the firm's projected
tax rates.
Answers and Solutions: 17  343 Chapter 17
Capital Structure Decisions: Extensions
ANSWERS TO ENDOFCHAPTER QUESTIONS
171 a. MM Proposition I states the relationship between leverage and firm value. Proposition I without
taxes is V = EBIT/rsU. Since both EBIT and rsU are constant, firm value is also constant and
capital structure is irrelevant. With corporate taxes, Proposition I becomes V = Vu + TD. Thus,
firm value increases with leverage and the optimal capital structure is virtually all debt.
b. MM Proposition II states the relationship between leverage and cost of equity. Without taxes,
Proposition II is rsL = rsU + (rsU rd)(D/S). Thus, rs increases in a precise way as leverage
increases. In fact, this increase is just sufficient to offset the increased use of lower cost debt.
When corporate taxes are added, Proposition II becomes rsL = rsU + (rsU rd)(1 T)(D/S).
Here the increase in equity costs is less than the zerotax case, and the increasing use of lower cost
debt causes the firm's cost of capital to decrease, and again, the optimal capital structure is
virtually all debt.
c. The Miller model introduces personal taxes. The effect of personal taxes is, essentially, to
reduce the advantage of corporate debt financing.
d. Financial distress costs are incurred when a leveraged firm facing a decline in earnings is forced
to take actions to avoid bankruptcy. These costs may be the result of delays in the liquidation of
assets, legal fees, the effects on product quality from cutting costs, and evasive actions by
suppliers and customers.
e. Agency costs arise from lost efficiency and the expense of monitoring management to ensure that
debtholders' rights are protected.
f. The addition of financial distress and agency costs to either the MM tax model or the Miller
model results in a tradeoff model of capital structure. In this model, the optimal capital
structure can be visualized as a tradeoff between the benefit of debt (the interest tax shelter) and
the costs of debt (financial distress and agency costs).
g. The value of the debt tax shield is the present value of the tax savings from the interest payments.
In the MM model with taxes, this is just interest x tax rate / discount rate = iDT/r, and since i = r
in the MM model, this is just TD. If a firm grows and the discount rate isn't r, then the value of
this growing tax shield is rdTDg/(1+rTS) where rd is the interest rate on the debt and rTS is the
discount rate for the tax shield.
Answers and Solutions: 17  344 h. When a firm has debt outstanding it can choose to default if the firm is not worth more than the
face value of the debt. This decision to default when the value of the firm is low is like the
decision not to exercise a call option when the stock price is low. If management (and hence
the stockholders) make the debt payment, they get to keep the company. This makes equity like
an option on the underlying value of the entire firm, with a strike price equal to the face value of
the debt. If D is the face value of debt maturing in one year and S is the value of the entire firm
(the firm's debt plus equity) then the payoff to the stockholder when the debt matures is: Payoff
= max(SD, 0). This is the same payoff as a call option on S with a strike, or exercise, price of
D.
172 Modigliani and Miller show that the value of a leveraged firm must be equal to the value of an
unleveraged firm. If this is not the case, investors in the leveraged firm will sell their shares
(assume they owned 10%). They will then borrow an amount equal to 10% of the debt of the
leveraged firm. Using these proceeds, they will purchase 10% of the stock of the unleveraged
firm (which provides the same return as the leveraged firm) with a surplus left to be invested
elsewhere. This arbitrage process will drive the price of the stock of the leveraged firm down
and drive up the price of the stock of the unleveraged firm. This will continue until the value of
both stocks are equal.
The assumptions of the MM model are:
· Firms must be in a homogeneous business risk class. If the firms have varying
degrees of risk, the market will value the firms at different rates. The earnings
of the firms will be capitalized at different costs of capital.
· Investors have homogeneous expectations about expected future EBIT. If
investors have different expectations about future EBIT then individual investors
will assign different values to the firms. Therefore, the arbitrage process will not
be effective.
· Stocks and bonds are traded in perfect capital markets. Therefore, (a) there are
no brokerage costs and (b) individuals can borrow at the same rate as corporations.
Brokerage fees and varying interest rates will, in effect, lower the surplus
available for alternative investment.
· Investors are rational. If by chance, investors were irrational, then they would
not go through the entire arbitrage process in order to achieve a higher return.
They would be satisfied with the return provided by the leveraged firm.
· There are no corporate taxes. With the existence of corporate taxes the value of
the leveraged firm (VL) must be equal to the value of the unleveraged firm (VU)
plus the tax shield provided by debt (TD). 173 MM without taxes would support AT&T, although if AT&T really believed MM, they should not
object to Gordon's 50 percent debt ratio. MM with taxes would lead ultimately to 100 percent
debt, which neither Gordon nor AT&T accepted. In effect, Gordon and AT&T seemed to be
taking a "traditional" or perhaps a "compromise" view, but with different conclusions about the
optimal debt ratio. We might note, in a postscript, that AT&T did raise its debt ratio, but not to
the extent that Gordon recommended.
Answers and Solutions: 17  345 174 The value of a growing tax shield is greater than the value of a constant tax shield. This means
that for a given initial level of debt a growing firm will have more value from the debt tax shield
than a nongrowing firm. Thus for a given face value of debt, D, and unlevered value of equity,
U, a growing firm will have a smaller wD, a larger levered cost of equity, reL, and a larger WACC.
So the MM model will underestimate the value of the levered firm and its cost of equity and
WACC.
175 If equity is viewed as an option on the total value of the firm with a strike price equal to the face
value of debt then the equity value should be affected by risk in the same way that an option is
affected by risk. An option is worth more if the underlying asset is more risky, so a manager
wanting to maximize the option value of the firm might want to switch investment decisions to
make the firm more risky. Of course bondholders will not like this, since the increase in equity
value comes at their expense. They will write covenants in to the bonds specifying how the
proceeds can be used, and if management still manages to engage in this "bait and switch" tactic,
the firm will find it difficult to raise capital through bond issues in the future.
Answers and Solutions: 17  346 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
171 a. bL = bU[1 + (1  T)(D/S)].
bL 1.8 1.8
bU = = = = 1.125.
1 + (1  T )( D / S) 1 + (1  0.4)(0.5 / 0.5) 1.6
b. rsU = rRF + (rM  rRF)bU = 10% + (5%)1.125 = 10% + 5.625% = 15.625%.
c. $2 Million Debt: VL = VU + TD = $10 + 0.25($2) = $10.5 million.
rsL = rsU + (rsU  rRF)bU(1  T)(D/S)
= 15.625% + (15.625%  10%)(0.75)($2/$8.5)
= 15.625 + 5.625% (0.75)($2/$8.5) = 16.62%.
$4 Million Debt: VL = $10 + 0.25($4) = $11.0 million.
rsL = 15.625% + 5.625%(0.75)($4/$7) = 18.04%.
$6 Million Debt: VL = $10 + 0.25($6) = $11.5 million.
rsL = 15.625% + 5.625% (0.75)($6/$5.5) = 20.23%.
d. $6 Million Debt: VL = $8.0 + 0.40($6) = $10.4 million.
rsL = 15.625% + 5.625%(0.60)($6/$4.4) = 20.23%.
The mathematics of MM result in the required return, and, thus, the same financial risk premium.
However, the market value debt ratio has increased from $6/$11.5 = 52% to $6/$10.4 = 58% at
the higher tax rate. Hence, a higher tax rate reduces the financial risk premium at a given market
value debt/equity ratio. This is because a higher tax rate increases the relative benefits of debt
financing.
Answers and Solutions: 17  347 EBIT $2 million 172 a. VU = = = $20 million.
rsU 0.10
b. rsU = 10.0%. (Given)
rsL = rsU + (rsU  rd)(D/S) = 10% + (10%  5%)($10/$10) = 15.0%.
EBIT  rd D $2  0.05($10)
c. SL = = = $10 million.
rsL 0.15
SL + D = VL = VU + TD.
$10 + $10 = $20 = VL = $20 + (0)$10 = $20 million.
d. WACCU = rsU = 10%.
For Firm L, we know that WACC must equal rsU = 10% according to Proposition I. But, we can
demonstrate this as follows:
WACCL = (D/V)rd + (S/V)rs = ($10/$20)5% + ($10/$20)15%
= 2.5% + 7.5% = 10.0%.
e. VL = $22 million is not an equilibrium value according to MM. Here's why. Suppose you
owned 10 percent of Firm L's equity, worth 0.10($22 million  $10 million) = $1.2 million.
You could (1) sell your stock, (2) borrow an amount (at 5%) equal to 10 percent of Firm L's debt,
or 0.10($10 million) = $1 million, and (3) end up with $1.2 million + $1 million = $2.2 million.
You could spend $2 million to buy 10% of Firm U's stock, and invest $200,000 in riskfree debt.
Your cash stream would now be 10 percent of Firm U's flow, or 0.10(EBITU) = 0.10($2 million) =
$200,000, plus the return on the $200,000 of riskfree debt, minus the 0.05($1 million) = $50,000
interest expense for $150,000 plus the return on the extra $200,000. Before the arbitrage, your
return was 10 percent of the $2 million  0.05($10 million) = $1.5 million, or $150,000.
Investors would do this arbitrage until VL = VU = $20 million.
EBIT(1  T ) $2(1  0.4) 173 a. VU = = = $12 million.
rsU 0.10
VL = VU + TD = $12 + (0.4)$10 = $16 million.
b. rsU = 0.10 = 10.0%.
rsL = rsU + (rsU  rd)(1  T)(D/S)
= 10% + (10%  5%)(0.6)($10/$6) = 10% + 5% = 15.0%.
Answers and Solutions: 17  348 ( EBIT  rd D)(1  T ) [$2  0.05($10)]0.6
c. SL = = = $6 million.
rsL 0.15
VL = SL + D = $6 + $10 = $16 million.
d. WACCU = rsU = 10.00%.
WACCL = (D/V)rd(1  T) + (S/V)rs = ($10/$16)5%(0.6) + ($6/$16)15%
= 7.50%.
EBIT(1  TC )(1  Ts ) EBIT(1  TC ) $2(0.6) 174 a. VU = = = = $12 million.
rsU (1  Ts ) rsU 0.10
(1  TC )(1  Ts )
b. VL = VU + 1  D = $12 +
(1  Td )
(0.6)(0.8) 1  (0.72) $10
= $12 + [1  0.67]$10 = $12 + 0.33($10)
= $15.33 million.
VL = $15.93 million. Gain from leverage = $3.33 million.
c. The gain from leverage under Miller is 0.33($10) = $3.33 million. The gain from leverage in
Problem 173 is 0.4($10) = $4 million. Thus, the addition of personal tax rates reduced the value
of the debt financing.
d. VU = VL = $20 million. Gain from leverage = $0.00.
e. VU = $12 million. VL = $16 million. Gain from leverage = $4 million.
f. VU = $12 million. VL = $16 million.
Gain from leverage = $4.0 million. Note that the gain from leverage is the same as in Part (e)
and will be the same value, as long as Td = Ts.
Answers and Solutions: 17  349 175 a. VU = $500,000/(rsU g) = $500,000/(0.13  0.09) = 12,500,000.
0.07 x 0.40 x 5 million
b. VL = $12.5 million + = $16.0 million . So since
0.13  0.09
D = 5, S = 16 5 = $11.0 million.
5
rsL = 0.13 + (0.13  0.07) = 15.7%
11
c. Under MM, VL = VU + TD = $12.5 million + (0.40)(5 million)
= $14.5 million. S = $14.5 5 = $9.5 million. rsL = 0.13+(0.130.07)(1.40)(5/9.5) = 14.9%
d. VL is greater under the extension that incorporates growth than under MM because MM assumes
0 growth. A positive growth rate gives a larger value to the tax shield. In this case the value of
the tax shield under MM is 2.0 million and is $3.5 million if growth is included. The cost of
capital when growth is included is higher because the relative weight of equity is higher and the
relative weight of debt is lower than when growth is ignored.
EBIT $1,600,000 176 a. VU = SU = = = $14,545,455.
rsU 0.11
Answers and Solutions: 17  350 VL = VU = $14,545,455.
b. At D = $0:
rs = 11.0%; WACC = 11.0%
At D = $6 million:
rsL = rsU + (rsU rd)(D/S)
$6,000,000
= 11% + (11%  6%) = 11% + 3.51% = 14.51%.
$8,545,455
WACC = (D/V)rd + (S/V)rs
$6,000,000 $8,545,455
= 6% + 14.51%
$14,545,455 $14,545,455
= 11.0%.
At D = $10 million:
$10,000,000
rsL = 11% + 5% = 22.00%.
$4,545,455
$10,000,000 $4,545,455
WACC = 6% + 22%
$14,545,455 $14,545,455
= 11.0%.
Leverage has no effect on firm value, which is a constant $14,545,455 since WACC is a constant
11%. This is because the cost of equity is increasing with leverage, and this increase exactly offsets
the advantage of using lower cost debt financing.
c. VU = [(EBIT  I)(1  T)]/rsU = [($1,600,000  0)(0.6)]/0.11 = $8,727,273.
VL = VU + TD = $8,727,273 + 0.4($6,000,000) = $11,127,273
Answers and Solutions: 17  351 d. At D = $0:
rs = 11.0%. WACC = 11.0%.
At D = $6 million:
VL = VU + TD = $8,727,273 + 0.4($6,000,000) = $11,127,273.
rsL = rsU + (rsU  rd)(1  T)(D/S)
= 11% + (11%  6%)(0.6)($6,000,000/$5,127,273) = 14.51%.
WACC = (D/V)rd(1  T) + (S/V)rs
=($6,000,000/$11,127,273)(6%)(0.6) + ($5,127,273/$11,127,273)(14.51%)
= 8.63%.
At D = $10 million:
VL = $8,727,273 + 0.4($10,000,000) = $12,727,273.
rsL = 11% + 5%(0.6)($10,000,000/$2,727,273) = 22.00%.
WACC = ($10,000,000/$12,727,273)(6%)(0.6) + ($2,727,273/$12,727,273)(22%)
=7.54%.
Summary: (in millions)
D V D/V rs WACC
$0 $ 8.73 0% 11.0% 11.0%
6 11.13 53.9 14.5 8.6
10 12.73 78.6 22.0 7.5
Value (Millions of Dollars)
15
14
13
12
11
10
9
8
25 50 75 100
D/V (%)
Answers and Solutions: 17  352 e. The maximum amount of debt financing is 100 percent. At this level
D = V, and hence
VL = VU + TD = D
$8,727,273 + 0.4D = D
D  0.4D = $8,727,273
0.6D = $8,727,273
D= $8,727,273/0.6 = $14,545,455 = V.
Since the bondholders are bearing the same risk as the equity holders of the unleveraged firm, rd is
now 11 percent. Now, the total interest payment is $14,545,455(0.11) = $1.6 million, and the
entire $1.6 million of EBIT would be paid out as interest. Thus, the investors (bondholders)
would get $1.6 million per year, and it would be capitalized at 11 percent:
$1,600,000
VL = = $14,545,455.
0.11
Cost of Capital (%)
25 ks
rS
20
15
10
WACC
5
kd(1T)
r (1T)
d
25 50 75 100
D/V (%)
f. (1) Rising interest rates would cause rd and hence rd(1  T) to increase, pulling up WACC.
These changes would cause V to rise less steeply, or even to decline.
(2) Increased riskiness causes rs to rise faster than predicted by MM. Thus, WACC would
increase and V would decrease.
Answers and Solutions: 17  353 177 a.The inputs to the Black and Scholes option pricing model are P = 5, X = 2, rRF =
6%,
= 50%, and t = 2 years. Given these inputs, the value of a call option is calculated as:
ln( P / X ) + [ rRF + 2 / 2]t ln(5 / 2) + [0.06 + 0.52 / 2]2
d1 = = = 1.819
t 0.5 2
d 2 = d1  t = 1.819  0.5 2 = 1.112
Using Excel's Normsdist function N(d1) = 0.9656, and N(d2) = 0.8669. This gives a
value of the call option equal to:
V = P[N(d 1)]  Xe rRF t [N(d 2 )] = 5[1.819]  2e 0.06( 2) [1.112] = 3.29 .
b. The debt must therefore be worth 53.29 = $1.71 million. Its yield is
2.0 / 1.71  1 = 0.81 = 8.1% .
c. At a volatility of 30% d1 = 2.566 and N(d1) = 0.996. d2 = 2.230 and N(d2) = 0.987.
This gives an option value of $3.32 million. The debt value is then 5.0 3.23 =
$1.77 million. Its yield is 6.8%. The value of the stock goes down and the
value of the debt goes up because with lower risk, Fethe has less of a chance of a
"home run."
Answers and Solutions: 17  354 SOLUTION TO SPREADSHEET PROBLEM
178 The detailed solution for the problem is available both on the instructor's resource CDROM (in the
file Solution for FM11 Ch 17 P8 Build a Model.xls) and on the instructor's side of the web site,
http://brigham.swcollege.com.
Solution to Spreadsheet Problem: 15  355 MINI CASE
David Lyons, CEO of Lyons Solar Technologies, is concerned about his firm's level of debt financing. The company uses shortterm debt to finance its temporary working capital needs, but it does not use any permanent (longterm) debt. Other solar technology companies average about 30 percent debt, and Mr. Lyons wonders why they use so much more debt, and what its effects are on stock prices. To gain some insights into the matter, he poses the following questions to you, his recently hired assistant:
a. Business Week recently ran an article on companies' debt policies, and the names Modigliani
and Miller (MM) were mentioned several times as leading researchers on the theory of
capital structure. Briefly, who are MM, and what assumptions are embedded in the MM
and Miller models?
Answer: Modigliani and Miller (MM) published their first paper on capital structure (which assumed zero
taxes) in 1958, and they added corporate taxes in their 1963 paper. Modigliani won the Nobel
Prize in economics in part because of this work, and most subsequent work on capital structure
theory stems from MM. Here are their assumptions:
· Firms' business risk can be measured by EBIT, and firms with the same degree of
risk can be grouped into homogeneous business risk classes.
· All investors have identical (homogeneous) expectations about all firms' future
earnings.
· There are no transactions (brokerage) costs, either to individuals or to firms.
· All debt is riskless, and both individuals and corporations can borrow unlimited
amounts of money at the same riskfree rate.
· All cash flows are perpetuities. This implies that firms and individuals issue
perpetual debt, and also that firms pay out all earnings as dividends, hence have
zero growth. Additionally, this implies that expected EBIT is constant over time,
although realized EBIT may turn out to be higher or lower than was expected.
· In their first paper (1958), MM also assumed that there are no corporate or
personal taxes.
Mini Case: 17  356 These assumptionsall of themwere necessary in order for MM to use the arbitrage
argument to develop and prove their equations. If the assumptions are unrealistic,
then the results of the model are not guaranteed to hold in the real world.
b. Assume that firms U and L are in the same risk class, and that both have EBIT = $500,000.
Firm U uses no debt financing, and its cost of equity is rsU = 14%. Firm L has $1 million of
debt outstanding at a cost of rd = 8%. There are no taxes. Assume that the MM
assumptions hold, and then:
1. Find v, s, rs, and WACC for firms U and L.
Answer: First, we find Vu and VL:
EBIT $500,000
VU = = = $3,571,429.
rsU 0.14
VL = VU = $3,571,429.
To find rsL, it is necessary first to find the market values of firm L's debt and equity. The value
of its debt is stated to be $1,000,000. Therefore, we can find s as follows:
D + SL = VL
SL = VL  D = $3,571,429  $1,000,000 = $2,571,429.
Mini Case: 17  357 Now we can find L's cost of equity, rsL:
rsL = rsU + (rsU  rd)(D/S)
= 14.0% + (14.0%  8.0%)($1,000,000/$2,571,429)
= 14.0% + 2.33% = 16.33%.
We know from Proposition I that the WACC must be WACC = rsU = 14.0% for all firms in this
risk class, regardless of leverage, but this can be verified using the WACC formula:
WACC = wdrd + wcers = (D/V)rd + (S/V)rs
= ($1,000/$3,571)(8.0%) + ($2,571/$3,571)(16.33%)
= 2.24% + 11.76% = 14.0%.
b. 2. Graph (a) the relationships between capital costs and leverage as measured by D/V,
and (b) the relationship between value and D.
Answer: Figure 1 plots capital costs against leverage as measured by the debt/value ratio. Note that,
under the MM notax assumption, rd is a constant 8 percent, but rs increases with leverage.
Further, the increase in rs is exactly sufficient to keep the WACC constantthe more debt the firm
adds to its capital structure, the riskier the equity and thus the higher its cost. Figure 2 plots the
firm's value against leverage (debt). With zero taxes, MM argue that value is unaffected by
leverage, and thus the plot is a horizontal line. (Note that we should not really extend the graphs
to D/V = 100% or D = $2.5 million, because at this amount of leverage the debtholders become
the firm's owners, and thus a discontinuity exists.)
Mini Case: 17  358 Figure 1
25% Without Taxes
20%
Cost of Capital
15%
rs
10% WACC
rd
5%
0%
0% 20% 40% 60%
Debt/Value Ratio
Figure 2
Value of Firm, V
($)
4
VU VL
3
Firm Value ($3.6 Million)
2
1
0 0.5 1.0 1.5 2.0 2.5
Debt ($)
(Millions of $)
c. Using the data given in part B, but now assuming that firms L and U are both subject
to a 40 percent corporate tax rate, repeat the analysis called for in B(1) and B(2)
under the MM withtax model.
Answer: With corporate taxes added, the MM propositions become:
Proposition I: VL = VU + TD.
Proposition II: rsL = rsU + (rsU rd)(1  T)(D/S).
Mini Case: 17  359 There are two very important differences between these propositions and the zerotax propositions:
(1) when corporate taxes are added, VL does not equal VU; rather, V L increases as debt is added to
the capital structure, and the greater the debt usage, the higher the value of the firm. (2) rsL
increases less rapidly when corporate taxes are considered. This is seen by noting that the
Proposition II slope coefficient changes from (rsU rd) to (rsU rd)(1 t), so at any positive T, the
slope coefficient is smaller.
Note also that with corporate taxes considered, VU changes to
EBIT(1  T ) $500,000(0.6)
VU = = = $2,142,857 versus $3,571,429.
rsU 0.14
This represents a 40% decline in value, and it is logical, because the 40% tax rate takes
away 40% of the income and hence 40% of the firm's value.
Looking at VL, we see that:
VL = VU + TD = $2,142,857 + 0.4($1,000,000)
VL = $2,142,857 + $400,000  $2,542,857 versus $2,142,857 for VU.
Thus, the use of $1,000,000 of debt financing increases firm value by T(D) = $400,000
over its leveragefree value.
To find rsL, it is first necessary to find the market value of the equity:
D + SL = VL
$1,000,000 + SL = $2,542,857
SL = $1,542,857.
now,
rsL = rsU + (rsU  rd)(1  T)(D/S)
= 14.0% + (14.0%  8.0%)(0.6)($1,000/$1,543)
= 14.0% + 2.33% = 16.33%.
Mini Case: 17  360 Firm L's WACC is 11.8 percent:
WACCL = (D/V)rd(1  T) + (S/V)rs
= ($1,000/$2,543)(8%)(0.6) + (1,543/$2,543)(16.33%)
= 1.89% + 9.91% = 11.8%.
The WACC is lower for the leveraged firm than for the unleveraged firm when corporate taxes are considered.
Figure 3 below plots capital costs at different D/V ratios under the MM model with corporate taxes. Here the WACC declines continuously as the firm uses more and more debt, whereas the WACC was constant in the withouttax model. This result occurs because of the tax deductibility of debt financing (interest payments), which impacts the graph in two ways: (1) the cost of debt is lowered by (1  T), and (2) the cost of equity increases at a slower rate when corporate taxes are considered because of the (1  T) term in Proposition II. The combined effect produces the downwardsloping WACC curve.
Figure 4 shows that, when corporate taxes are considered, the firm's value increases continuously as more and more debt is used.
Figure 3
rs
With Taxes
WACC
50%
rd x (1T)
45%
40%
Cost of Capital
35%
30%
25%
20%
15%
10%
5%
0%
0% 20% 40% 60% 80% 100%
Debt/Value Ratio
Mini Case: 17  361 Figure 4
Value of Firm, V
($)
4
VL
3
TD
2 VU
1
0 0.5 1.0 1.5 2.0 2.5
Debt ($)
(Millions of $)
d. Now suppose investors are subject to the following tax rates:
TD = 30% and TS = 12%.
1. What is the gain from leverage according to the miller model?
Answer: To begin, note that Miller's Proposition I is stated as follows:
(1  TC )(1  TS )
VL = VU + 1  D.
(1  TD )
Here the bracketed term replaces T in the earlier MM tax model, and
Tc = corporate tax rate, Td = personal tax rate on debt income, and
Ts = personal tax rate on stock income.
If there are no personal or corporate taxes, then Tc = Ts = Td = 0, and Miller's model
simplifies to
VL = VU,
Which is the same as in MM's 1958 model, which assumed zero taxes.
If there are corporate taxes, but no personal taxes, then Ts = Td = 0, and Miller's model
simplifies to
VL = VU + TCD,
Which is the same as MM obtained in their 1963 article, which considered only corporate taxes.
Mini Case: 17  362 We can now analyze the firm's value numerically, using Miller's model: if Tc = 40%, Td =
30%, and Ts = 12%, then Miller's model becomes
(1  TC )(1  TS )
VL = VU + 1  D
(1  TD
(1  0.40)(1  0.12
= VU + 1  D = VU + (1  0.75) D = VU + 0.25D.
(1  0.30)
d. 2. How does this gain compare to the gain in the MM model with corporate taxes?
Answer: If only corporate taxes were considered, then
VL = VU + TCD = VU + 0.40D.
The net effect depends on the relative effective tax rates on income from stocks and bonds, and on
corporate tax rates. The tax rate on stock income is reduced visàvis the tax rate on debt income
if the company retains more of its income and thus provides more capital gains. If Ts declines,
while Tc and Td remain constant, the slope coefficient, which shows the benefit of debt in a graph
like figure 4, is increased. Thus, a company with a low payout ratio gets greater benefits under
the miller model than a company with a high payout.
Note that the effects of leverage as computed by Miller's model were much more important
before 1987, because in earlier years capital gains were taxed at only 40 percent of the rate
imposed on dividends (Ts 20% and Td 50%). Now the advantages of capital gains are (1) the
fact that taxes on them are deferred, and (2) individuals in the higher tax brackets obtain an
advantage because the tax rate imposed on longterm capital gains is 20 percent.
Mini Case: 17  363 d. 3. What does the Miller model imply about the effect of corporate debt on the value of
the firm, that is, how do personal taxes affect the situation?
Answer: The addition of personal taxes lowers the value of debt financing to the firm. The underlying
rationale can be explained as follows: the U.S. corporate tax laws favor debt financing over
equity financing, because interest expense is tax deductible while dividends are not. This
provides an incentive for firms to use debt financing, and this was the message of the mm 1963
paper. At the same time, though, the U.S. personal tax laws favor investment in equity securities
over debt securities, because equity income is effectively taxed at a lower rate. Thus, investors
require higher riskadjusted beforetax returns on debt to be induced to buy debt rather than equity,
and this reduces the advantage to issuing debt.
The bottom line conclusion we reach from an analysis of the Miller model is that personal
taxes lower, but do not eliminate, the value of debt financing.
e. What capital structure policy recommendations do the three theories (MM without
taxes, MM with corporate taxes, and Miller) suggest to financial managers?
Empirically, do firms appear to follow any one of these guidelines?
Answer: In a zero tax world, MM theory says that capital structure is irrelevantit has no impact on firm
value. Thus, one capital structure is as good as another. With corporate but not personal taxes
considered, the MM model states that firm value increases continuously with financial leverage,
and hence firms should use (almost) 100 percent debt financing. Miller added personal taxes to
the analysis, and the value of debt financing is seen to be reduced but not eliminated, so again
firms should use (almost) 100 percent debt financing.
The Miller model is the most realistic of the three, but if it were really correct, we would
expect to see firms using almost all debt financing. However, on average, firms use only about
40 percent debt. Note, though, that debt ratios increased all during the 1980s, so companies
were moving toward the miller position. However, in the 1990s we see firms reducing their
debt.
f. How is the analysis in part C different if firms U and L are growing? Assume that
both firms are growing at a rate of 7 percent and that the investment in net operating
assets required to support this growth is 10 percent of EBIT.
Answers and Solutions: 18  364 Answer: If a firm is growing, the assumptions that MM made are violated. The extension to
the MM model shows how growth affects the value of the debt tax shield and the cost
of capital. The first difference in this situation is that the appropriate discount rate
for the debt tax shield is the unlevered cost of equity, not the cost of debt. The
second difference is that a growing debt tax shield is more valuable than a constant
debt tax shield.
First, calculate expected free cash flow:
NOPAT = EBIT X (1T) = 500,000 X (1 0.40) = $300,000
Investment In Net Operating Assets = 0.10 X EBIT = $50,000
Free Cash Flow = NOPAT Investment In Net Operating Assets
= $300,000  $50,000 = $250,000
(Note that this is an expected value for the coming year since EBIT is an expected value
for the coming year.)
Next, note that WACC = unlevered cost of equity if there is no debt so
WACC = rsU = 14%
The Value Of U = Expected FCF/(WACC g)
= 250,000/(0.14 0.07) = $3,571,429
Which is greater than in part C because the firm is growing.
If there is $1,000,000 in debt then:
The value of l = the value of U + value of debt tax shield
The value of the (growing) debt tax shield = rdTD/(rsU g)
= 0.08(0.40)(1,000,000)/(0.14 0.07)
= $457,143
Therefore, the value of the firm = $3,571,429 + $457,143 = $4,028,571.
The value of the equity is the value of the firm less the value of the
debt = $4,028,571  $1,000,000 = $3,028,571.
In this case the increase in the firm's value due to the debt tax shield as a percent of its
zero debt value is $457,143/$3,571,429 = 12.80%
This is less than the increase in the nongrowing firm's value as calculated using the MM
model: $400,000/$2,142,857 = 18.7%.
To calculate the new levered cost of equity:
rsL = rsU + (rsU rd)(D/S)
= 14% + (14%  8%)(1,000,000/3,028,571)
= 15.98%
And the new levered WACC:
WACCL = (D/V)rd(1  T) + (S/V)rs
= (1,000,000/4,028,571)8%(1.40)
+ ($3,028,571/4,028,571)15.98%
= 13.2%.
Answers and Solutions: 18  365 g. What if L's debt is risky? For the purpose of this example, assume that the value of L's
operations is $4 millionwhich is the value of its debt plus equity. Assume also that its
debt consists of 1year zero coupon bonds with a face value of $2 million. Finally, assume
that L's volatility is 0.60 ( = 0.60) and that the risk free rate is 6 percent.
Answer: L's equity can be considered as a call option on the total value of l with an exercise price of $2
million, and an expiration date in one year. If the value of L's operations is less than $2 million
in a year, then L's management will not be able to make its required payment on the debt, and the
firm will be bankrupt. The debtholders will take over the firm and the equity holders will
receive nothing. If L's value is greater than $2 million in one year, then management will repay
the debt and the stockholders will keep the company.
This option can be valued with the BlackScholes Option Pricing Model:
V = PN(D1) XeRTN(D2)
where
D1 = [ln(P/X) + (r + 0.52)T]/[T0.5]
D2 = D1  T0.5
And n() is the cumulative normal distribution function, from either appendix a in the back of the
text, or the NORMSDIST() function in excel.
in this case, P = $4
X = $2
= 0.60
T = 1.0
R = 0.06
and calculating,
D1 = 1.552
D2 = 0.9552
N(D1) = 0.9491
N(D2) = 0.8303
Answers and Solutions: 18  366 and V = $2.1964 million.
This leaves debt value of $4 million  $2.1964 million = $1.8036 million.
The yield on this debt is calculated as
Price = (Face Value)/(1+Yield)N
so that
Yield = [Face Value/Price]1/N 1.0
= [2.0/1.8036] 1.0
= 10.89%
In this case, the value of the debt must be $1.8036 million, and it is yielding 10.89%. The value
of the equity is $2.1964 million.
h. What is the value of L's stock for volatilites between 0.20 and 0.95? What
incentives might the manager of L have if she understands this relationship? What
might debtholders do in response?
Answer: The mini case model shows the calculations for the table below.
Value of Stock and Debt for
Different Volatilities
Volatility Equity Debt
0.20 2.12 1.88
0.25 2.12 1.88
0.30 2.12 1.88
0.35 2.12 1.88
0.40 2.13 1.87
0.45 2.14 1.86
0.50 2.16 1.84
0.55 2.17 1.83
0.60 2.20 1.80
0.65 2.22 1.78
0.70 2.25 1.75
0.75 2.28 1.72
0.80 2.31 1.69
0.85 2.34 1.66
0.90 2.38 1.62
0.95 2.41 1.59
Answers and Solutions: 18  367 The value of the equity increases as the volatility increasesand the value of the debt decreases as well. A manager who knows this may choose to invest the proceeds from borrowing in assets that are riskier than usual. This is called "bait and switch." This action decreases the value of the debt, because now its claim is riskier. It increases the value of equity because the worse the stockholders can do is default on the bonds, but the best they can do is potentially unlimited.
Bondholders who face this possibility will write covenants into their bond contracts limiting management's ability to invest in assets other than originally planned. If this isn't possible, then bondholders will demand a higher rate of return in order to compensate them for the possibility that management will switch investments.
Answers and Solutions: 18  368 Chapter 18
Distributions to Shareholders:
Dividends and Repurchases
ANSWERS TO ENDOFCHAPTER QUESTIONS
181 a. The optimal distribution policy is one that strikes a balance between dividend yield and capital
gains so that the firm's stock price is maximized.
b. The dividend irrelevance theory holds that dividend policy has no effect on either the price of a
firm's stock or its cost of capital. The principal proponents of this view are Merton Miller and
Franco Modigliani (MM). They prove their position in a theoretical sense, but only under strict
assumptions, some of which are clearly not true in the real world. The "birdinthehand" theory
assumes that investors value a dollar of dividends more highly than a dollar of expected capital
gains because the dividend yield component, D1/P0, is less risky than the g component in the total
expected return equation rS = D1/P0 + g. The tax preference theory proposes that investors
prefer capital gains over dividends, because capital gains taxes can be deferred into the future, but
taxes on dividends must be paid as the dividends are received.
c. The information content of dividends is a theory which holds that investors regard dividend
changes as "signals" of management forecasts.
Thus, when dividends are raised, this is viewed by investors as recognition by
management of future earnings increases. Therefore, if a firm's stock price
increases with a dividend increase, the reason may not be investor preference for
dividends, but expectations of higher future earnings. Conversely, a dividend
reduction may signal that management is forecasting poor earnings in the future.
The clientele effect is the attraction of companies with specific dividend policies
to those investors whose needs are best served by those policies. Thus,
companies with high dividends will have a clientele of investors with low
marginal tax rates and strong desires for current income. Similarly, companies
with low dividends will attract a clientele with little need for current income, and
who often have high marginal tax rates. d. The residual distribution model states that firms should make distributions only
when more earnings are available than needed to support the optimal capital
budget. An extra dividend is a dividend paid, in addition to the regular
dividend, when earnings permit. Firms with volatile earnings may have a low
regular dividend that can be maintained even in lowprofit (or high capital
investment) years, and then supplement it with an extra dividend when excess
funds are available.
Answers and Solutions: 18  369 e. The declaration date is the date on which a firm's directors issue a statement declaring a dividend.
If a company lists the stockholder as an owner on the holderofrecord date, then the stockholder
receives the dividend. The exdividend date is the date when the right to the dividend leaves the
stock. This date was established by stockbrokers to avoid confusion and is 2 business days prior
to the holder of record date. If the stock sale is made prior to the exdividend date, the dividend
is paid to the buyer.
If the stock is bought on or after the exdividend date, the dividend is paid to the seller. The
date on which a firm actually mails dividend checks is known as the payment date.
f. Dividend reinvestment plans allow stockholders to automatically purchase shares of common
stock of the paying corporation in lieu of receiving cash dividends. There are two types of
plansone involves only stock that is already outstanding, while the other involves newly issued
stock. In the first type, the dividends of all participants are pooled and the stock is purchased on
the open market. Participants benefit from lower transaction costs. In the second type, the
company issues new shares to the participants. Thus, the company issues stock in lieu of the
cash dividend.
g. In a stock split, current shareholders are given some number (or fraction) of shares for each stock
owned. Thus, in a 3for1 split, each shareholder would receive 3 new shares in exchange for
each old share, thereby tripling the number of shares outstanding. Stock splits usually occur
when the stock price is outside of the optimal trading range. Stock dividends also increase the
number of shares outstanding, but at a slower rate than splits. In a stock dividend, current
shareholders receive additional shares on some proportional basis. Thus, a holder of 100 shares
would receive 5 additional shares at no cost if a 5 percent stock dividend were declared. Stock
repurchases occur when a firm repurchases its own stock. These shares of stock are then
referred to as treasury stock. The higher EPS on the now decreased number of shares
outstanding will cause the price of the stock to rise and thus capital gains are substituted for cash
dividends.
Answers and Solutions: 18  370 182 a. From the stockholders' point of view, an increase in the personal income tax rate would make it
more desirable for a firm to retain and reinvest earnings. Consequently, an increase in personal
tax rates should lower the aggregate payout ratio.
b. If the depreciation allowances were raised, cash flows would increase. With higher cash flows,
payout ratios would tend to increase. On the other hand, the change in taxallowed depreciation
charges would increase rates of return on investment, other things being equal, and this might
stimulate investment, and consequently reduce payout ratios. On balance, it is likely that
aggregate payout ratios would rise, and this has in fact been the case.
c. If interest rates were to increase, the increase would make retained earnings a relatively attractive
way of financing new investment. Consequently, the payout ratio might be expected to decline.
On the other hand, higher interest rates would cause rd, rs, and firm's MCCs to risethat would
mean that fewer projects would qualify for capital budgeting and the residual would increase
(other things constant), hence the payout ratio might increase.
d. A permanent increase in profits would probably lead to an increase in dividends, but not
necessarily to an increase in the payout ratio. If the aggregate profit increase were a cyclical
increase that could be expected to be followed by a decline, then the payout ratio might fall,
because firms do not generally raise dividends in response to a shortrun profit increase.
e. If investment opportunities for firms declined while cash inflows remained relatively constant, an
increase would be expected in the payout ratio.
f. Dividends are currently paid out of aftertax dollars, and interest charges from beforetax dollars.
Permission for firms to deduct dividends as they do interest charges would make dividends less
costly to pay than before and would thus tend to increase the payout ratio.
g. This change would make capital gains less attractive and would lead to an increase in the payout
ratio.
183 The difference is largely one of accounting. In the case of a split, the firm simply increases the
number of shares and simultaneously reduces the par or stated value per share. In the case of a stock
dividend, there must be a transfer from retained earnings to capital stock. For most firms, a 100
percent stock dividend and a 2for1 split accomplish exactly the same thing; hence, investors may
choose either one.
184 a. The residual distribution policy is based on the premise that, since new common stock is more
costly than retained earnings, a firm should use all the retained earnings it can to satisfy its
common equity requirement. Thus, the distribution under this policy is a function of the firm's
investment opportunities.
b. Yes. A more shallow plot implies that changes from the optimal capital structure have little
effect on the firm's cost of capital, hence value. In this situation, dividend policy is less critical
than if the plot were Vshaped.
Answers and Solutions: 18  371 185 a. True. When investors sell their stock they are subject to capital gains taxes.
b. True. If a company's stock splits 2 for 1, and you own 100 shares, then after the
split you will own 200 shares.
c. True. Dividend reinvestment plans that involve newly issued stock will increase the amount of
equity capital available to the firm.
d. False. The tax code, through the tax deductibility of interest, encourages firms to use debt and
thus pay interest to investors rather than dividends, which are not tax deductible. In addition,
due to a lower capital gains tax rate than the highest personal tax rate, the tax code encourages
investors in high tax brackets to prefer firms who retain earnings rather than those that pay large
dividends.
e. True. If a company's clientele prefers large dividends, the firm is unlikely to adopt a residual
dividend policy. A residual dividend policy could mean low or zero dividends in some years
which would upset the company's developed clientele.
f. False. If a firm follows a residual dividend policy, all else constant, its dividend payout will tend
to decline whenever the firm's investment opportunities improve.
SOLUTIONS TO ENDOFCHAPTER PROBLEMS
181 70% Debt; 30% Equity; Capital Budget = $3,000,000; NI = $2,000,000;
PO = ?
Equity retained = 0.3($3,000,000) = $900,000.
NI $2,000,000
Additions 900,000
Earnings Remaining $1,100,000
$1,100,000
Payout = = 55%.
$2,000,000
182 P0 = $90; Split = 3 for 2; New P0 = ?
$90
P0 New = = $60.
3/ 2
183 Retained earnings = Net income (1  Payout ratio)
Answers and Solutions: 18  372 = $5,000,000(0.55) = $2,750,000.
External equity needed:
Total equity required = (New investment)(1  Debt ratio)
= $10,000,000(0.60) = $6,000,000.
New external equity needed = $6,000,000  $2,750,000 = $3,250,000.
184 The company requires 0.40($1,200,000) = $480,000 of equity financing. If the
company follows a residual dividend policy it will retain $480,000 for its capital
budget and pay out the $120,000 "residual" to its shareholders as a dividend. The
payout ratio would therefore be $120,000/$600,000 = 0.20 = 20%.
185 Equity financing = $12,000,000(0.60) = $7,200,000.
Dividends = Net income  Equity financing
= $15,000,000  $7,200,000 = $7,800,000.
Dividend payout ratio = Dividends/Net income
= $7,800,000/$15,000,000 = 52%.
186 DPS after split = $0.75.
Equivalent presplit dividend = $0.75(5) = $3.75.
New equivalent dividend = Last year's dividend(1.09)
$3.75 = Last year's dividend(1.09)
Last year's dividend = $3.75/1.09 = $3.44.
187 Capital budget should be $10 million. We know that 50% of the $10 million should be equity.
Therefore, the company should pay dividends of:
Dividends = Net income  needed equity
= $7,287,500  $5,000,000 = $2,287,500.
Payout ratio = $2,287,500/$7,287,500 = 0.3139 = 31.39%.
Answers and Solutions: 18  373 188 a. 1. 2005 Dividends = (1.10)(2004 Dividends)
= (1.10)($3,600,000) = $3,960,000.
2. 2004 Payout = $3,600,000/$10,800,000 = 0.33 = 33%.
2005 Dividends = (0.33)(2005 Net income)
= (0.33)($14,400,000) = $4,800,000.
(Note: If the payout ratio is rounded off to 33%, 2005 dividends are then calculated as
$4,752,000.)
3. Equity financing = $8,400,000(0.60) = $5,040,000.
2005 Dividends = Net income  Equity financing
= $14,400,000  $5,040,000 = $9,360,000.
All of the equity financing is done with retained earnings as long as they are available.
4. The regular dividends would be 10% above the 2004 dividends:
Regular dividends = (1.10)($3,600,000) = $3,960,000.
The residual policy calls for dividends of $9,360,000. Therefore, the extra dividend, which
would be stated as such, would be
Extra dividend = $9,360,000  $3,960,000 = $5,400,000.
An even better use of the surplus funds might be a stock repurchase.
b. Policy 4, based on the regular dividend with an extra, seems most logical. Implemented properly,
it would lead to the correct capital budget and the correct financing of that budget, and it would
give correct signals to investors.
189 a. Capital Budget = $10,000,000; Capital structure = 60% equity, 40% debt.
Retained Earnings Needed = $10,000,000 (0.6) = $6,000,000.
b. According to the residual dividend model, only $2 million is available for dividends.
NI  Retained earnings needed for cap. projects = Residual dividend.
$8,000,000  $6,000,000 = $2,000,000.
DPS = $2,000,000/1,000,000 = $2.00.
Payout ratio = $2,000,000/$8,000,000 = 25%.
c. Retained Earnings Available = $8,000,000  $3.00 (1,000,000)
Retained Earnings Available = $8,000,000  $3,000,000
Retained Earnings Available = $5,000,000.
d. No. If the company maintains its $3.00 DPS, only $5 million of retained earnings will be
available for capital projects. However, if the firm is to maintain its current capital structure, $6
million of equity is required. This would necessitate the company having to issue $1 million of
Answers and Solutions: 18  374 new common stock.
e. Capital Budget = $10 million; Dividends = $3 million; NI = $8 million.
Capital Structure = ?
RE Available = $8,000,000  $3,000,000
= $5,000,000.
$5,000,000
Percentage of Cap. Budget Financed with RE = = 50%.
$10,000,000
$5,000,000
Percentage of Cap. Budget Financed with Debt = = 50%.
$10,000,000
f. Dividends = $3 million; Capital Budget = $10 million; 60% equity, 40% debt; NI = $8 million.
Equity Needed = $10,000,000(0.6) = $6,000,000.
RE Available = $8,000,000  $3.00(1,000,000)
= $8,000,000  $3,000,000
= $5,000,000.
External (New) Equity Needed = $6,000,000  $5,000,000
= $1,000,000.
g. Dividends = $3 million; NI = $8 million; Capital structure = 60% equity, 40% debt.
RE Available = $8,000,000  $3,000,000
= $5,000,000.
We're forcing the RE Available = Required Equity to find the new capital budget.
Required Equity = Capital Budget (Target Equity Ratio)
$5,000,000 = Capital Budget(0.6)
Capital Budget = $8,333,333.
Therefore, if Buena Terra cuts its capital budget from $10 million to $8.33 million, it can maintain
its $3.00 DPS, its current capital structure, and still follow the residual dividend policy.
h. The firm can do one of four things:
(1) Cut dividends.
(2) Change capital structure, that is, use more debt.
(3) Cut its capital budget.
(4) Issue new common stock.
Realize that each of these actions is not without consequences to the company's cost of
capital, stock price, or both.
Answers and Solutions: 18  375 Answers and Solutions: 18  376 SPREADSHEET PROBLEM
1810 The detailed solution for the problem is available both on the instructor's resource CDROM (in the
file Solution for FM11 Ch 18 P10 Build a Model.xls) and on the instructor's side of the web site,
http://brigham.swcollege.com.
Answers and Solutions: 18  377 MINI CASE
Southeastern Steel Company (SSC) was formed 5 years ago to exploit a new continuouscasting process. SSC's founders, Donald Brown and Margo Valencia, had been employed in the research department of a major integratedsteel company, but when that company decided against using the new process (which Brown and Valencia had developed), they decided to strike out on their own. One advantage of the new process was that it required relatively little capital in comparison with the typical steel company, so Brown and Valencia have been able to avoid issuing new stock, and thus they own all of the shares. However, SSC has now reached the stage where outside equity capital is necessary if the firm is to achieve its growth targets yet still maintain its target capital structure of 60 percent equity and 40 percent debt. Therefore, Brown and Valencia have decided to take the company public. Until now, Brown and Valencia have paid themselves reasonable salaries but routinely reinvested all aftertax earnings in the firm, so dividend policy has not been an issue. However, before talking with potential outside investors, they must decide on a dividend policy.
Assume that you were recently hired by Arthur Adamson & Company (AA), a national consulting firm, which has been asked to help SSC prepare for its public offering. Martha Millon, the senior AA consultant in your group, has asked you to make a presentation to Brown and Valencia in which you review the theory of dividend policy and discuss the following questions.
a. 1. What is meant by the term "distribution policy"?
Answer: Distribution policy is defined as the firm's policy with regard to (1) the level of distributions, (2)
the form of distributions (dividends or stock repurchases), and (3) the stability of distributions.
a. 2. The terms "irrelevance," "birdinthehand," and "tax preference" have been
used to describe three major theories regarding the way dividend payouts affect a
firm's value. Explain what these terms mean, and briefly describe each theory.
Answer: Dividend irrelevance refers to the theory that investors are indifferent between dividends and
Answers and Solutions: 19  378 capital gains, making dividend policy irrelevant with regard to its effect on the value of the firm.
"Birdinthehand" refers to the theory that a dollar of dividends in the hand is preferred by
investors to a dollar retained in the business, in which case dividend policy would affect a firm's
value.
The dividend irrelevance theory was proposed by MM, but they had to make some very
restrictive assumptions to "prove" it (zero taxes, no flotation or transactions costs). MM argued
that paying out a dollar per share of dividends reduces the growth rate in earnings and dividends,
because new stock will have to be sold to replace the capital paid out as dividends. Under their
assumptions, a dollar of dividends will reduce the stock price by exactly $1. Therefore,
according to MM, stockholders should be indifferent between dividends and capital gains.
The "birdinthehand" theory is identified with Myron Gordon and John Lintner, who argued
that investors perceive a dollar of dividends in the hand to be less risky than a dollar of potential
future capital gains in the bush; hence, stockholders prefer a dollar of actual dividends to a dollar
of retained earnings. If the birdinthehand theory is true, then investors would regard a firm
with a high payout ratio as being less risky than one with a low payout ratio, all other things equal;
hence, firms with high payout ratios would have higher values than those with low payout ratios.
MM opposed the GordonLintner theory, arguing that a firm's risk is dependent only on the
riskiness of its cash flows from assets and its capital structure, not by how its earnings are
distributed to investors.
The tax preference theory recognizes that there are two taxrelated reasons for believing that
investors might prefer a low dividend payout to a high payout: (1) taxes are not paid on capital
gains until the stock is sold. (2) if a stock is held by someone until he or she dies, no capital
gains tax is due at allthe beneficiaries who receive the stock can use the stock's value on the
death day as their cost basis and thus escape the capital gains tax.
a. 3. What do the three theories indicate regarding the actions management should
take with respect to dividend payout?
Answer: If the dividend irrelevance theory is correct, then dividend payout is of no consequence, and the
Answers and Solutions: 19  379 firm may pursue any dividend payout. If the birdinthehand theory is correct, the firm should
set a high payout if it is to maximize its stock price. If the tax preference theory is correct, the
firm should set a low payout if it is to maximize its stock price. Therefore, the theories are in
total conflict with one another.
a. 4. What results have empirical studies of the dividend theories produced? How
does all this affect what we can tell managers about dividend payouts?
Answer: Unfortunately, empirical tests of the theories have been inconclusive (because firms don't differ
just with respect to payout), so we cannot tell managers whether investors prefer dividends or
capital gains. Even though we cannot determine what the optimal dividend policy is, managers
can use the types of analyses discussed in this chapter to help develop a rational and reasonable, if
not completely optimal, dividend policy.
b. Discuss (1) the information content, or signaling, hypothesis, (2) the clientele effect,
and (3) their effects on distribution policy.
Answer: 1. Different groups, or clienteles, of stockholders prefer different dividend payout policies.
For example, many retirees, pension funds, and university endowment funds are in a low (or
zero) tax bracket, and they have a need for current cash income. Therefore, this group of
stockholders might prefer high payout stocks. These investors could, of course, sell some of
their stock, but this would be inconvenient, transactions costs would be incurred, and the sale
might have to be made in a down market. Conversely, investors in their peak earnings years
who are in high tax brackets and who have no need for current cash income should prefer low
payout stocks.
Answers and Solutions: 19  380 2. Clienteles do exist, but the real question is whether there are more members of one clientele
than another, which would affect what a change in its dividend policy would do to the
demand for the firm's stock. There are also costs (taxes and brokerage) to stockholders who
would be forced to switch from one stock to another if a firm changes its policy. Therefore,
we cannot say whether a policy change to appeal to one particular clientele or another would
lower or raise a firm's cost of equity. MM argued that one clientele is as good as another, so
in their view the existence of clienteles does not imply that one dividend policy is better than
another. Still, no one has offered convincing proof that firms can disregard clientele effects.
We know that stockholder shifts will occur if policy is changed, and since such shifts result in
transaction costs and capital gains taxes, policy changes should not be taken lightly. Further,
dividend policy should be changed slowly, rather than abruptly, in order to give stockholders
time to adjust.
3. It has long been recognized that the announcement of a dividend increase often results in an
increase in the stock price, while an announcement of a dividend cut typically causes the
stock price to fall. One could argue that this observation supports the premise that investors
prefer dividends to capital gains. However, MM argued that dividend announcements are
signals through which management conveys information to investors. Information
asymmetries existmanagers know more about their firms' prospects than do investors.
Further, managers tend to raise dividends only when they believe that future earnings can
comfortably support a higher dividend level, and they cut dividends only as a last resort.
Therefore, (1) a largerthannormal dividend increase "signals" that management believes the
future is bright, (2) a smallerthanexpected increase, or a dividend cut, is a negative signal,
and (3) if dividends are increased by a "normal" amount, this is a neutral signal.
Answers and Solutions: 19  381 c. 1. Assume that SSC has an $800,000 capital budget planned for the coming year.
You have determined that its present capital structure (60 percent equity and 40
percent debt) is optimal, and its net income is forecasted at $600,000. Use the
residual distribution model approach to determine SSC's total dollar distribution.
Assume for now that the distribution is in the form of a dividend. Then, explain
what would happen if net income were forecasted at $400,000, or at $800,000.
Answer: We make the following points:
a. Given the optimal capital budget and the target capital structure, we must now determine the
amount of equity needed to finance the projects. Of the $800,000 required for the capital
budget, 0.6($800,000) = $480,000 must be raised as equity and 0.4($800,000) = $320,000
must be raised as debt if we are to maintain the optimal capital structure:
Debt $320,000 40%
Equity 480,000 60%
$800,000 100%
b. If a residual existsthat is, if net income exceeds the amount of equity the company
needsthen it should distribute the residual amount out as either dividends or stock
repurchases. For now, we assume all payouts are in the form of dividends. Since $600,000
of earnings is available, and only $480,000 is needed, the residual is $600,000  $480,000 =
$120,000, so this is the amount which should be paid out as dividends. Thus, the payout
ratio would be $120,000/$600,000 = 0.20 = 20%.
c. If only $400,000 of earnings were available, the theoretical break point would occur at BP =
$400,000/0.6 = $666,667. Assuming the intersection of the investment opportunity set and
marginal cost of capital was still at $800,000, the firm would still need $480,000 of equity.
It should then retain all of its earnings and also sell $80,000 of new stock. The residual
policy would call for a zero payment.
d. If $800,000 of earnings was available, the dividend would be increased to $800,000 
$480,000 = $320,000, and the payout ratio would rise to $320,000/$800,000 = 40%.
c. 2. In general terms, how would a change in investment opportunities affect the
payout ratio under the residual payment policy?
Answer: A change in investment opportunities would lead to an increase (if investment opportunities were
good) or a decrease (if investment opportunities were not good) in the amount of equity needed,
Answers and Solutions: 19  382 hence in the residual dividend payout.
c. 3. What are the advantages and disadvantages of the residual policy? (Hint:
don't neglect signaling and clientele effects.)
Answer: The primary advantage of the residual policy is that under it the firm makes maximum use of
lower cost retained earnings, thus minimizing flotation costs and hence the cost of capital.
Also, whatever negative signals are associated with stock issues would be avoided.
However, if it were applied exactly, the residual model would result in dividend payments
which fluctuated significantly from year to year as capital requirements and internal cash flows
fluctuated. This would (1) send investors conflicting signals over time regarding the firm's
future prospects, and (2) since no specific clientele would be attracted to the firm, it would be an
"orphan." These signaling and clientele effects would lead to a higher required return on equity
which would more than offset the effects of lower flotation costs. Because of these factors, few if
any publicly owned firms follow the residual model on a yeartoyear basis.
Even though the residual approach is not used to set the annual dividend, it is used when
firms establish their longrun dividend policy. If "normalized" cost of capital and investment
opportunity conditions suggest that in a "normal" year the company should pay out about 60
percent of its earnings, this fact will be noted and used to help determine the longrun policy.
d. What are stock repurchases? Discuss the advantages and disadvantages of a firm's
repurchasing its own shares.
Answer: A firm may distribute cash to stockholders by repurchasing its own stock rather than paying out
cash dividends. Stock repurchases can be used (1) somewhat routinely as an alternative to
regular dividends, (2) to dispose of excess (nonrecurring) cash that came from asset sales or from
temporarily high earnings, and (3) in connection with a capital structure change in which debt is
sold and the proceeds are used to buy back and retire shares.
Advantages of repurchases:
1. A repurchase announcement may be viewed as a positive signal that management believes the
shares are undervalued.
2. Stockholders have a choiceif they want cash, they can tender their shares, receive the cash,
and pay the taxes, or they can keep their shares and avoid taxes. On the other hand, one
must accept a cash dividend and pay taxes on it.
Answers and Solutions: 19  383 3. If the company raises the dividend to dispose of excess cash, this higher dividend must be
maintained to avoid adverse stock price reactions. A stock repurchase, on the other hand,
does not obligate management to future repurchases.
4. Repurchased stock, called treasury stock, can be used later in mergers, when employees
exercise stock options, when convertible bonds are converted, and when warrants are
exercised. Treasury stock can also be resold in the open market if the firm needs cash.
Repurchases can remove a large block of stock that is "overhanging" the market and keeping
the price per share down.
5. Repurchases can be varied from year to year without giving off adverse signals, while
dividends may not.
6. Repurchases can be used to produce largescale changes in capital structure.
Disadvantages of repurchases:
1. A repurchase could lower the stock's price if it is taken as a signal that the firm has relatively
few good investment opportunities. On the other hand, though, a repurchase can signal
stockholders that managers are not engaged in "empire building," where they invest funds in
lowreturn projects.
2. If the IRS establishes that the repurchase was primarily to avoid taxes on dividends, then
penalties could be imposed. Such actions have been brought against closely held firms, but
to our knowledge charges have never been brought against publicly held firms.
3. Selling shareholders may not be fully informed about the repurchase; hence they may make
an uninformed decision and may later sue the company. To avoid this, firms generally
announce repurchase programs in advance.
4. The firm may bid the stock price up and end up paying too high a price for the shares. In
this situation, the selling shareholders would gain at the expense of the remaining
shareholders. This could occur if a tender offer were made and the price was set too high,
or if the repurchase was made in the open market and buying pressure drove the price above
its equilibrium level.
e. Describe the series of steps that most firms take in setting dividend policy in
practice.
Answers and Solutions: 19  384 Answer: Firms establish dividend policy within the framework of their overall financial plans. The steps
in setting policy are listed below:
1. The firm forecasts its annual capital budgets and its annual sales, along with its working
capital needs, for a relatively longterm planning horizon, often 5 years.
2. The target capital structure, presumably the one which minimizes the WACC while retaining
sufficient reserve borrowing capacity to provide "financing flexibility," will also be
established.
3. With its capital structure and investment requirements in mind, the firm can estimate the
approximate amount of debt and equity financing required during each year over the planning
horizon.
4. A longterm target payout ratio is then determined, based on the residual model concept.
Because of flotation costs and potential negative signaling, the firm will not want to issue
common stock unless this is absolutely necessary. At the same time, due to the clientele
effect, the firm will move cautiously from its past dividend policy, if a new policy appears to
be warranted, and it will move toward any new policy gradually rather than in one giant step.
5. An actual dollar dividend, say $2 per year, will be decided upon. The size of this dividend
will reflect (1) the longrun target payout ratio and (2) the probability that the dividend, once
set, will have to be lowered, or, worse yet, omitted. If there is a great deal of uncertainty
about cash flows and capital needs, then a relatively low initial dollar dividend will be set, for
this will minimize the probability that the firm will have to either reduce the dividend or sell
new common stock. The firm will run its corporate planning model so that management can
see what is likely to happen with different initial dividends and projected growth rates under
different economic scenarios.
f. What are stock dividends and stock splits? What are the advantages and
disadvantages of stock dividends and stock splits?
Answer: When it uses a stock dividend, a firm issues new shares in lieu of paying a cash dividend. For
example, in a 5 percent stock dividend, the holder of 100 shares would receive an additional 5
shares. In a stock split, the number of shares outstanding is increased (or decreased in a reverse
split) in an action unrelated to a dividend payment. For example, in a 2for1 split, the number
of shares outstanding is doubled. A 100% stock dividend and a 2for1 stock split would
produce the same effect, but there would be differences in the accounting treatments of the two
Answers and Solutions: 19  385 actions.
Both stock dividends and stock splits increase the number of shares outstanding and, in effect,
cut the pie into more, but smaller, pieces. If the dividend or split does not occur at the same time
as some other event which would alter perceptions about future cash flows, such as an
announcement of higher earnings, then one would expect the price of the stock to adjust such that
each investor's wealth remains unchanged. For example, a 2for1 split of a stock selling for $50
would result in the stock price being cut in half, to $25.
It is hard to come up with a convincing rationale for small stock dividends, like 5 percent or
10 percent. No economic value is being created or distributed, yet stockholders have to bear the
administrative costs of the distribution. Further, it is inconvenient to own an odd number of
shares as may result after a small stock dividend. Thus, most companies today avoid small stock
dividends.
On the other hand, there is a good reason for stock splits or large stock dividends.
Specifically, there is a widespread belief that an optimal price range exists for stocks. The
argument goes as follows: if a stock sells for about $20$80, then it can be purchased in round lots,
hence at reduced commissions, by most investors. A higher price would put round lots out of the
price range of many small investors, while a stock price lower than about $20 would convey the
image of a stock that is doing poorly. Thus, most firms try to keep their stock prices within the
$20 to $80 range. If the company prospers, it will split its stock occasionally to hold the price
down. (Also, companies that are doing poorly occasionally use reverse splits to raise their price.)
Many companies do operate outside the $20 to $80 range, but most stay within it.
Another factor that may influence stock splits and dividends is the belief that they signal
management's belief that the future is bright. If a firm's management would be inclined to split
the stock or pay a stock dividend only if it anticipated improvements in earnings and dividends,
then a split/dividend action could provide a positive signal and thus boost the stock price.
However, if earnings and cash dividends did not subsequently rise, the price of the stock would
fall back to its old level, or even lower, because managers would lose credibility.
Interestingly, one of the most astute investors of the 20th century, Warren Buffett, chairman of
BerkshireHathaway, has never split his firm's stock. Berkshire currently sells for over $34,000
per share, and its performance over the years has been absolutely spectacular. It may be that
Berkshire's market value would be higher if it had a 425:1 stock split, or it may be that the
conventional wisdom is wrong.
g. What is a dividend reinvestment plan (drip), and how does it work?
Answers and Solutions: 19  386 Answer: Under a dividend reinvestment plan (DRIP), shareholders have the option of automatically
reinvesting their dividends in shares of the firm's common stock. In an open market purchase
plan, a trustee pools all the dividends to be reinvested and then buys shares on the open market.
Shareholders use the drip for three reasons: (1) brokerage costs are reduced by the volume
purchases, (2) the drip is a convenient way to invest excess funds, and (3) the company generally
pays all administrative costs associated with the operation.
In a new stock plan, the firm issues new stock to the DRIP members in lieu of cash dividends.
No fees are charged, and many companies even offer the stock at a 5 percent discount from the
market price on the dividend date on the grounds that the firm avoids flotation costs that would
otherwise be incurred. Only firms that need new equity capital use new stock plans, while firms
with no need for new stock use an open market purchase plan.
Answers and Solutions: 19  387 Chapter 19
Initial Public Offerings, Investment Banking, and Financial
Restructuring
ANSWERS TO ENDOFCHAPTER QUESTIONS
191 a. A closely held corporation goes public when it sells stock to the general public. Going public
increases the liquidity of the stock, establishes a market value, facilitates raising new equity, and
allows the original owners to diversify. However, going public increases business costs, requires
disclosure of operating data, and reduces the control of the original owners. The new issue
market is the market for stock of companies that go public, and the issue is called an initial public
offering (IPO).
b. A public offering is an offer of new common stock to the general public; in other words, an offer
in which the existing shareholders are not given any preemptive right to purchase the new shares.
A private placement is the sale of stock to only one or a few investors, usually institutional
investors. The advantages of private placements are lower flotation costs and greater speed,
since the shares issued are not subject to SEC registration.
c. A venture capitalist is the manager of a venture capital fund. The fund raises most of its capital
from institutional investors and invests in startup companies in exchange for equity. The
venture capitalist gets a seat on the companies' boards of directors. Before an IPO, the senior
management team and the investment banker make presentations to potential investors. They
make presentations in tent to twenty cities, with three to five presentations per day, over a two
week period. The spread is the difference between the price at which an underwriter sells the
stock in an IPO and the proceeds that the underwriter passes on to the issuing firm. In other
words, it is the fee collected by the underwriter, and it usually is seven percent of the offering
price.
d. The Securities and Exchange Commission (SEC) is a government agency which regulates the
sales of new securities and the operations of securities exchanges. The SEC, along with other
government agencies and selfregulation, helps ensure stable markets, sound brokerage firms, and
the absence of stock manipulation. Registration of securities is required of companies by the
SEC before the securities can be offered to the public. The registration statement is used to
summarize various financial and legal information about the company. Frequently, companies will
file a master registration statement and then update it with a shortform statement just before an
offering. This procedure is termed shelf registration because companies put new securities "on the
shelf" and then later sell them when the market is right. Blue sky laws are laws that prevent the
sale of securities that have little or no asset backing. The margin is the percentage of a stock's
price that an investor has borrowed in order to purchase the stock. The SEC sets margin
requirements, which is the maximum percentage of debt that can be used to purchase a stock.
The SEC also controls trading by corporate insiders, who are the officers, directors, and major
stockholders of the firm.
e. A prospectus summarizes information about a new security issue and the issuing company. A
"red herring," or preliminary prospectus, may be distributed to potential buyers prior to approval
of the registration statement by the SEC. After the registration has become effective, the
securities, accompanied by the prospectus, may be offered for sale.
Answers and Solutions: 19  388 f. The National Association of Securities Dealers (NASD) is an industry group primarily concerned
with the operation of the overthecounter (OTC) market.
g. A best efforts arrangement versus an underwritten sale refers to two methods of selling new stock
issues. In a best efforts sale, the investment banker is only committed to making every effort to
sell the stock at the offering price. In this case, the issuing firm bears the risk that the new issue
will not be fully subscribed. If the issue is underwritten, the investment banker agrees to buy
the entire issue at a set price, and then resells the stock at the offering price. Thus, the risk of
selling the issue rests with the investment banker.
h. Refunding occurs when a company issues debt at current low rates and uses the proceeds to
repurchase one of its existing high coupon rate debt issues. Often these are callable issues,
which means the company can purchase the debt at a lowerthanmarket price. Project
financings are arrangements used to finance mainly large capital projects such as energy
explorations, oil tankers, refineries, utility power plants, and so on. Usually, one or more firms
(sponsors) will provide the equity capital required by the project, while the rest of the project's
capital is supplied by lenders and lessors. The most important aspect of project financing is that
the lenders and lessors do not have recourse against the sponsors; they must be repaid from the
project's cash flows and the equity cushion provided by the sponsors. Securitization is the
process whereby financial instruments that were previously thinly traded are converted to a form
that creates greater liquidity. Securitization also applies to the situation where specific assets are
pledged as collateral for securities, and hence assetbacked securities are created. One example
of the former is junk bonds; an example of the latter is mortgagebacked securities. Maturity
matching refers to matching the maturities of debt used to finance assets with the lives of the
assets themselves. The debt would be amortized such that the outstanding amount declined as
the asset lost value due to depreciation.
192 No. The role of the investment banker is more important if the stock demand curve has a steep slope
and the negative signaling effect is substantial. Under such conditions, the investment banker will
have a harder time holding up the stock price.
193 No. The real value of a security is determined by the equilibrium forces of an efficient market.
Assuming that the information provided on newly issued securities is accurate, the market will
establish the value of a security regardless of the opinions rendered by the SEC, or, for that matter,
opinions offered by any advisory service or analyst.
194 a. Going public would tend to make attracting capital easier and to decrease flotation costs.
b. The increasing institutionalization of the "buy side" of the stock and bond markets should
increase a firm's ability to attract capital and should reduce flotation costs.
c. Financial conglomerates can offer a variety of financial services and types of investments, thus it
seems a company's ability to attract capital would increase and flotation costs would decrease.
Answers and Solutions: 19  389 d. Elimination of the preemptive right would likely not affect a large company where percentage
ownership is not as important. Indeed, the trend today seems to be for companies to eliminate
the preemptive right.
f. The introduction of shelf registration tended to speed up SEC review time and lower the costs of
floating each new issue. Thus, the company's ability to attract new capital was increased.
195 Investment bankers must investigate the firms whose securities they sell, simply because, if an issue is
overvalued and suffers marked price declines after the issue, the banker will find it increasingly
difficult to dispose of the new issue. In other words, reputation is highly important in the investment
banking industry.
Answers and Solutions: 19  390 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
191 a. $5 per share
Gross proceeds = (3,000,000)($5) = $15,000,000.
Net profit = $15,000,000  $14,000,000  $300,000 = $700,000.
b. $6 per share
Gross proceeds = (3,000,000)($6) = $18,000,000.
Net profit = $18,000,000  $14,000,000  $300,000 = $3,700,000.
c. $4 per share
Gross proceeds = (3,000,000)($4) = $12,000,000.
Net profit = $12,000,000  $14,000,000  $300,000 = $2,300,000.
192 Net proceeds per share = $22(1  0.05) = $20.90.
Number of shares to be sold = ($20,000,000 + 150,000)/$20.90 = 964,115 shares.
193 a. If 100 shares are outstanding, then we have the following for Edelman:
1999 2004
Earnings per share $8,160 $12,000
Dividends per share 4,200 6,000
Book value per share 90,000
b. Using the following two equations, the growth rate for EPS and DPS can be
determined.
(1 + gEPS)5 EPS99 = EPS04.
(1 + gDPS)5 DPS99 = DPS04.
gEPS
gDPS
Kennedy 8.4% 8.4%
Strasburg 6.4 6.4
Edelman 8.0 7.4
c. Based on the figures in Part a, it is obvious that Edelman's stock would not sell in the range of
$25 to $100 per share. The small number of shares outstanding has greatly inflated EPS, DPS,
and book value per share. Should Edelman attempt to sell its stock based on the EPS and DPS
above, it would have difficulty finding investors at the economically justified price.
Answers and Solutions: 19  391 d. Edelman's management would probably be wise to split the stock so that EPS, DPS, and book
value were closer to those of Kennedy and Strasburg. This would bring the price of the stock into
a more reasonable range.
e. A 4,000for1 split would result in 400,000 shares outstanding. If Edelman has 400,000 shares
outstanding, then we would have the following:
1999 2004
Earnings per share $2.04 $ 3.00
Dividends per share 1.05 1.50
Book value per share 22.50
f. ROE
Kennedy 15.00%
Strasburg 13.64
Edelman 13.33
g. Payout Ratio
1999 2004
Kennedy 50% 50%
Strasburg 50 50
Edelman 51 50
All three companies seem to be following similar dividend policies, paying out about 50 percent
of their earnings.
h. D/A is 43 percent for Kennedy, 37 percent for Strasburg, and 55 percent for Edelman. This
suggests that Edelman is more risky, hence should sell at relatively low multiples.
i. P/E
Kennedy $36/$4.50 = 8.00×
Strasburg $65/$7.50 = 8.67
These ratios are not consistent with g and ROE; based on gs and ROEs, Kennedy should have
the higher P/E. Probably size, listing status, and debt ratios are offsetting g and ROE.
Answers and Solutions: 19  392 j. The market prices of Kennedy and Strasburg yield the following multiples:
Multiple of Multiple of Multiple of Book
EPS, 2004 DPS, 2004 Value per Share, 2004
Kennedy 8.00× 16.00× 1.20×
Strasburg 8.67 17.33 1.18
Applying these multiples to the data in Part e, we obtain the following market prices:
Indicated Market Price
for Edelman Stock
Based on Data of:
Kennedy Strasburg
Based on earnings, 2003 $24.00 $26.01
Based on dividends, 2003 24.00 26.00
Based on book value per share 27.00 26.55
D 0 (1 + g)
k. r = + g.
P0
$2.25(1.084)
Kennedy r= + 8.4% = 15.18.
$36
$3.75(1.064)
Strasburg r= + 6.4% = 12.54%.
$65
^ D1
Edelman P0 = .
rg
1.50(1.077)
Based on Kennedy: P0 = = $21.54.
0.152  0.077
1.50(1.077)
Based on Strasburg: P0 = = $33.66.
0.125  0.077 l. The potential range, based on these data, is between $21.54 and $33.66 a share.
The data suggest that the price would be set toward the low end of the range:
(1) Edelman has a high debt ratio, (2) Edelman is relatively small, and (3)
Edelman is new and will not be traded on an exchange. The actual price would
be based on negotiations between the underwriter and Edelman; we cannot say
what the exact price would be, but the price would probably be set below $21.54,
with $20 being a reasonable guess.
Answers and Solutions: 19  393 194 a. Since the call premium is 11 percent, the total premium is 0.11($40,000,000) = $4,400,000.
However, this is a tax deductible expense, so the relevant aftertax cost is $4,400,000(1  T) =
$4,400,000(0.60) = $2,640,000.
b. The dollar flotation cost on the new issue is 0.04($40,000,000) = $1,600,000. This cost is
not immediately tax deductible, and hence the aftertax cost is also $1,600,000. (Note that
the flotation cost can be amortized and expensed over the life of the issue. The value of this
tax savings will be calculated in Part e.)
c. The flotation costs on the old issue were 0.06($40,000,000) = $2,400,000. These costs were
deferred and are being amortized over the 25year life of the issue, and hence $2,400,000/25
= $96,000 are being expensed each year, or $48,000 each 6 months. Since the bonds were
issued 5 years ago, (5/25)($2,400,000) = $480,000 of the flotation costs have already been
expensed, and (20/25)($2,400,000) = $1,920,000 remain unexpensed.
If the issue is refunded, the unexpensed portion of the flotation costs can be immediately
expensed, and this would result in a tax savings of T($1,920,000) = 0.40($1,920,000) =
$768,000.
d. The net aftertax cash outlay is $3,472,000, as shown below:
Old issue call premium $2,640,000
New issue flotation cost 1,600,000
Tax savings on old issue
flotation costs (768,000)
Net cash outlay $3,472,000
e. The new issue flotation costs of $1,600,000 would be amortized over the 20year life of the
issue. Thus, $1,600,000/20 = $80,000 would be expensed each year, or $40,000 each 6
months. The tax savings from this tax deduction is (0.40)$40,000 = $16,000 per semiannual
period.
By refunding the old issue and immediately expensing the remaining old issue flotation
costs, the firm forgoes the opportunity to continue to expense the old flotation costs over
time. Specifically, $2,400,000/25 = $96,000 each year, or $48,000 semiannually. The
value of each $48,000 deduction forgone is 0.40($48,000) = $19,200.
f. The interest on the old issue is 0.11($40,000,000) = $4,400,000 annually, or $2,200,000
semiannually. Since interest payments are tax deductible, the aftertax semiannual amount is
0.6($2,200,000) = $1,320,000.
The new issue carries an 8 percent coupon rate. Therefore, the annual interest would be
0.08($40,000,000) = $3,200,000, or $1,600,000 semiannually. The aftertax cost is thus
0.6($1,600,000) = $960,000. Thus, the aftertax net interest savings if refunding takes place
would be $1,320,000 $960,000 = $360,000 semiannually.
g. The net amortization tax effects are $3,200 per year for 20 years, while the net interest
savings are $360,000 per year for 20 years. Thus, the net semiannual cash flow is $356,800,
as shown below.
Answers and Solutions: 19  394 Semiannual Flotation Cost Tax Effects:
Semiannual tax savings on new flotation: $16,000
Tax benefits lost on old flotation: (19,200)
Net amortization tax effects ($ 3,200)
Semiannual Interest Savings Due To Refunding:
Semiannual interest on old bond: $1,320,000
Semiannual interest on new bond: (960,000)
Net interest savings $ 360,000
Semiannual cash flow: $ 356,800
The cash flows are based on contractual obligations, and hence have about the same amount
of risk as the firm's debt. Further, the cash flows are already net of taxes. Thus, the
appropriate interest rate is GST's aftertax cost of debt. (The source of the cash to fund the
net investment outlay also influences the discount rate, but most firms use debt to finance this
outlay, and, in this case, the discount rate should be the aftertax cost of debt.) Finally,
since we are valuing future flows, the appropriate debt cost is today's cost, or the cost of the
new issue, and not the cost of debt floated 5 years ago. Thus, the appropriate discount rate is
0.6(8%) = 4.8% annually, or 2.4 percent per semiannual period.
At this discount rate, the present value of the semiannual net cash flows is $9,109,425:
PV = $356,800(PVIFA2.4%,40) = $9,109,425.
Alternatively, using a financial calculator, input N = 40, I = 2.4, PMT = 356800, FV = 0, PV
= ? PV = $9,109,413.
h. The bond refunding would require a $3,472,000 net cash outlay, but it would produce
$9,109,413 in net savings on a present value basis. Thus, the NPV of refunding is
$5,637,413:
PV of net benefits $9,109,413
Cost (3,472,000)
Refunding NPV $5,637,413
The decision to refund now rather than wait till later is much more difficult than finding
the NPV of refunding now. If interest rates were expected to fall, and hence GST would be
able to issue debt in the future below today's 8 percent rate, then it might pay to wait.
However, interest rate movements are very difficult, if not impossible, to forecast, and hence
most financial managers would probably take the "birdinthehand" and refund now with
such a large NPV. Note, though, that if the NPV had been quite small, say $1,000,
management would have undoubtedly waited, hoping that interest rates would fall further,
and the cost of waiting ($1,000) would not have been high enough to worry about.
195 a. Investment outlay required to refund the issue:
Call premium on old issue: $5,400,000
New flotation cost: 5,000,000
Answers and Solutions: 19  395 Tax savings on old flotation: (1,666,667)
Additional interest on old issue: 450,000
Interest earned on investment: (225,000)
Total investment outlay: $8,958,333
Annual Flotation Cost Tax Effects:
Annual tax savings on new flotation: $ 80,000
Tax benefits lost on old flotation: (66,667)
Amortization tax effects $ 13,333
Annual Interest Savings Due to Refunding:
Annual interest on old bond: $5,400,000
Annual interest on new bond: (4,500,000)
Net interest savings $ 900,000
Annual cash flows: $ 913,333
NPV of refunding decision: $2,717,128
Using a financial calculator, enter the cash flows into the cash flow register, I = 6, NPV = ?
NPV = $2,717,128.
b. The company should consider what interest rates might be next year. If there is a high
probability that rates will drop below the current rate, it may be more advantageous to refund
later versus now. If there is a high probability that rates will increase, the firm should act
now to refund the old issue. Also, the company should consider how much ill will is created
with investors if the issue is called. If Tarpon is highly dependent on a small group of
investors, it would want to avoid future difficulty in obtaining financing. However, bond
issues are callable after a certain time and investors expect them to be called if rates drop
considerably.
Answers and Solutions: 19  396 SOLUTION TO SPREADSHEET PROBLEM
196 The detailed solution for the problem is available both on the instructor's resource CDROM (in the
file Solution for FM11 Ch 19 P6 Build a Model.xls) and on the instructor's side of the web site,
http://brigham.swcollege.com.
Answers and Solutions: 19  397 MINI CASE
Randy's, a familyowned restaurant chain operating in Alabama, has grown to the point where expansion throughout the entire southeast is feasible. The proposed expansion would require the firm to raise about $15 million in new capital. Because Randy's currently has a debt ratio of 50 percent, and also because the family members already have all their personal wealth invested in the company, the family would like to sell common stock to the public to raise the $15 million. However, the family does want to retain voting control. You have been asked to brief the family members on the issues involved by answering the following questions:
a. What agencies regulate securities markets?
Answer: The main agency that regulates the securities market is the Securities And Exchange Commission.
Some of the responsibilities of the SEC include: regulation of all national stock
exchangescompanies whose securities are listed on an exchange must file annual reports with
the SEC; prohibiting manipulation by pools or wash sales; controls over trading by corporate
insiders; and control over the proxy statement and how it is used to solicit votes.
The Federal Reserve Board controls flow of credit into security transactions through
margin requirements. States also have some control over the issuance of new
securities within their boundaries. The securities industry itself realizes the
importance of stable markets, therefore, the various exchanges work closely with the
sec to police transactions and to maintain the integrity and credibility of the system.
Mini Case: 19  398 b. How are startup firms usually financed?
Answer: The first financing comes from the founders. The first external financing comes from angels,
who are wealthy individuals. The next external financing comes from a venture capital fund.
The fund raises capital from institutional investors, usually around $70 to $80 million. The
managers of the fund are called venture capitalists. The fund invests in ten to twelve companies,
and the venture capitalist sits on their boards.
c. Differentiate between a private placement and a public offering.
Answer: In a private placement stock is sold directly to one or a small group of investors rather than being
distributed to the public at large. A private placement has the advantage of lower flotation costs;
however, since the stock would be bought by a small number of outsiders, it would not be actively
traded, and a liquid market would not exist. Further, since it would not have gone through the
SEC registration process, the holders would be unable to sell it except to a restricted set of
"sophisticated" investors. Further, it might be difficult to find investors willing to invest large
sums in the company and yet be minority stockholders. Thus, many of the advantages listed
above would not be obtained. For these reasons, a public placement makes more sense in
Randy's situation.
d. Why would a company consider going public? What are some advantages and
disadvantages?
Answer: A firm is said to be "going public" when it sells stock to the public for the first time. A
company's first stock offering to the public is called an "initial public offering (IPO)." Thus,
Randy's will go public if it goes through with its planned IPO. There are several advantages and
disadvantages to going public:
Mini Case: 19  399 Advantages to going public:
· Going public will allow the family members to diversify their assets and reduce the
riskiness of their personal portfolios.
· It will increase the liquidity of the firm's stock, allowing the family stockholders to
sell some stock if they need to raise cash.
· It will make it easier for the firm to raise funds. The firm would have a difficult
time trying to sell stock privately to an investor who was not a family member.
Outside investors would be more willing to purchase the stock of a publicly held
corporation which must file financial reports with the sec.
· Going public will establish a value for the firm.
Disadvantages to going public:
· The firm will have to file financial reports with the SEC and perhaps with state
officials. There is a cost involved in preparing these reports.
· The firm will have to disclose operating data to the public. Many small firms do
not like having to do this, because such information is available to competitors.
Also, some of the firm's officers, directors, and major stockholders will have to
disclose their stock holdings, making it easy for others to estimate their net worth.
· Managers of publiclyowned corporations have a more difficult time engaging in
deals which benefit them personally, such as paying themselves high salaries,
hiring family members, and enjoying notstrictlynecessary, but taxdeductible,
fringe benefits.
· If the company is very small, its stock may not be traded actively and the market
price may not reflect the stock's true value.
The advantages of public ownership would be recognized by key employees, who would most
likely be granted stock options, which would certainly be more valuable if the stock
were publicly traded.
Mini Case: 19  400 e. What are the steps of an initial public offering?
Answer: Select an investment banker, file the S1 registration document with the SEC, choose a price range
for the preliminary, or "red herring," prospectus, go on a roadshow, set final price on final
prospectus.
f. What criteria are important in choosing an investment banker?
Answer: (1) reputation and experience in the industry. (2) existing mix of institutional and retail (i.e.,
individual) clients. (3) support in the postIPO secondary market, especially the reputation of
the analyst who will cover the stock.
g. Would companies going public use a negotiated deal or a competitive bid?
Answer: The firm would almost certainly use a negotiated deal. The competitive bid process for setting
investment bankers' fees is feasible only for large, wellestablished firms on large issues, and even
here the use of bids is rare for equity issues. This is because the process of making a bid is
costly (mainly for the research necessary to establish the price, but also because of the need for
sec registration), and investment bankers simply would not incur these costs unless they were
assured of getting the deal or the issue was so large that a huge fee awaited the winner.
h. Would the sale be on an underwritten or best efforts basis?
Answer: Most stock offerings are done on an underwritten basis, but the price is not set until the
investment banker has checked investors for interest in the stock, and has received oral assurances
of commitments at a price that will virtually guarantee the success of the offering barring a major
stock market collapse. So, there is little effective difference between a best efforts and an
underwritten deal.
Mini Case: 19  401 i. Without actually doing any calculations, describe how the preliminary offering range for the
price of an IPO would be determined?
Answer: Since the firm is going public for the first time, there is no established price for its stock. The
firm and its investment banker would project future earnings and free cash flows. The banker
would then compare the firm with other restaurant firms of similar size. The banker would try to
determine a price range for the firm's stock by applying the price/earnings ratios, price/dividends
ratios, and price/book value ratios of similar firms to the firm's earnings, dividends, and book
value data. The banker would also determine the price at which the firm's stock would have to
sell to earn the same rate of return as other firms in its industry. On the basis of all these factors,
the investment bankers would determine a ballpark price. They would then specify a range (i.e.,
$10 to $12) in the preliminary prospectus.
j. What is a roadshow? What is bookbuilding?
Answer: The senior management team, the investment banker, and the lawyer make presentations to
potential institutional investors. They usually visit ten to twenty cities, and make three to five
presentations in each city. Management can't say anything that is not in the registration
statement, because the SEC imposes a "quiet period" from the time it makes the registration
effective until 25 days after the stock begins trading. The purpose is to prevent select investors
from getting information that is not available to other investors.
During the roadshow, the investment bankers asks the investors to indicate how many shares
they plan on buying. The banker records this in his book. The banker hopes for
oversubscription. Based on demand, the banker sets the final offer price on the evening before
the stock is issued.
Mini Case: 19  402 k. Describe the typical firstday returns of an IPO and the longterm returns to IPO investors.
Answer: Firstday returns average 14.1%, with many stocks having much higher returns. The investment
banker has an incentive to set a low price, both to make its brokerage customers happy and to
make it easy to sell the issue, whereas the firm would like to set as high a price as possible.
Returns over the twoyear period following the IPO are generally lower than for comparable firms,
indicating that the offering price is too low, but that the firstday runup is too high.
l. What are the direct and indirect costs of an IPO?
Answer: The underwriter usually charges a 7% fee, based on the offer price. In addition, there are direct
costs to lawyers, accountants, printers, etc. That can easily total $400,000.
Indirect costs include the money left on the table, which is equal to the difference between the
offer price and endoffirstday price, multiplied by the number of shares. Also, much of
management's time and attention is consumed by the IPO in the months preceding the IPO.
m. What are equity carveouts?
Answer: Equity carveouts are a special type of IPO in which a public company creates a new public
company from one of its subsidiaries by issuing public stock in the subsidiary. The parent
usually retains a controlling interest.
n. In what other ways are investment banks involved in issuing securities?
Answer: Investment bankers help companies with shelf registration (SEC rule 451) in which securities are
registered but not all of the issue is sold at once. Instead, the company sells a percentage of the
issue each time it needs to raise capital.
Investment bankers also help place private and public debt issues. They also help in seasoned
equity offers, in which a public firms issues additional shares of stock.
o. What is meant by going private? What are some advantages and disadvantages?
Answer: Going private is the reverse of going public. Typically, the managers of a firm team up with a
small group of outside investors, who furnish most of the equity capital, and purchase all of the
Answers and Solutions: 20  403 publicly held shares of the company. The new equity holders usually use a large amount of debt
financing, up to 90 percent, to complete the purchase. Such a transaction is called a "leveraged
buyout (LBO)."
Going private gives the managers greater incentives and more flexibility in running the
company. It also removes the burden of sec filings, stockholder relations, annual reports, analyst
meetings, and so on. The major disadvantage of going private is that it limits significantly the
availability of new capital. Since the stock is not publicly traded, a new stock issue would not be
practical, and, since such firms are normally leveraged to the hilt, it is tough to find additional
debt financing. For this reason, it is common for firms that have recently gone private to sell off
some assets to quickly reduce the debt burden to more conventional levels to give added financial
flexibility.
After several years of operating the business as a private firm, the owners typically go
public again. At this time, the firm is presumably operating at its peak, and it will
command top dollar compared to when it went private. In this way, the equity
investors of the private firm are able to recover their investment and, hopefully, make
a tidy profit. So far LBOs have, on average, been extremely profitablesince the
1970s, when LBO firms such as Kohlberg Kravis Roberts (KKR) began operating,
their annual rates of return are reported to have averaged over 50 percent annually.
However, Wall Street is becoming increasingly concerned about the use of debt, and
in the beginning of the nineties the number of new LBOs has fallen dramatically and
some old ones have had major financial difficulties.
p. How do companies manage the maturity structure of their debt?
Answer: In discussing this question, we emphasize that, if markets are truly efficient and conditions are
stable, the type of debt instrument will be immaterial, as the cost of each will be commensurate
with its risk. However, if markets are not totally efficient (perhaps because management has
information which investors do not have), if the company's tax position changes, if some new
security innovation is developed, or the like, then some types of securities might truly be less
expensive, on a riskadjusted basis, than others.
Answers and Solutions: 20  404 Factors that influence the decision to issue longterm bonds rather than shortterm debt:
· Maturity matching (assets to be financed)
· Information asymmetries. If managers know that the firm has strong prospects,
they will issue shortterm debt and refinance later when the market recognizes
their prospects.
q. Under what conditions would a firm exercise a bond's call provision?
Answer: Refunding decisions involve two separate questions: (1) is it profitable to call an outstanding
issue in the current period and replace it with a new issue; and (2) if refunding is currently
profitable, would the value of the firm be increased even more if the refunding were postponed to
a later date? If these two conditions are true, a company would exercise their bond's call
provision.
r. Explain how firms manage the risk structure of their debt with: (1) project financing,
and (2) securitization.
Answer: 1. Project financings are arrangements used to finance mainly large capital projects such as
energy explorations, oil tankers, refineries, utility power plants, and so on. Usually, one or
more firms (sponsors) will provide the equity capital required by the project, while the rest of
the project's capital is supplied by lenders and lessors. The most important aspect of project
financing is that the lenders and lessors do not have recourse against the sponsors; they must
be repaid from the project's cash flows and the equity cushion provided by the sponsors.
2. Securitization is the process whereby financial instruments that were previously thinly traded
are converted to a form that creates greater liquidity. Securitization also applies to the
situation where specific assets are pledged as collateral for securities, and hence assetbacked
securities are created. One example of the former is junk bonds; an example of the latter is
mortgagebacked securities.
Chapter 20
Lease Financing
ANSWERS TO ENDOFCHAPTER QUESTIONS
Answers and Solutions: 20  405 201 a. The lessee is the party leasing the property. The party receiving the payments from the lease
(that is, the owner of the property) is the lessor.
b. An operating lease, sometimes called a service lease, provides for both financing and
maintenance. Generally, the operating lease contract is written for a period
considerably shorter than the expected life of the leased equipment, and contains
a cancellation clause. A financial lease does not provide for maintenance
service, is not cancelable, and is fully amortized; that is, the lease covers the
entire expected life of the equipment. In a sale and leaseback arrangement, the
firm owning the property sells it to another firm, often a financial institution,
while simultaneously entering into an agreement to lease the property back from
the firm. A sale and leaseback can be thought of as a type of financial lease.
A combination lease combines some aspects of both operating and financial
leases. For example, a financial lease that contains a cancellation
clausenormally associated with operating leasesis a combination lease. A
synthetic lease is an arrangement between a company and a special purpose
entity that it creates to borrow money and purchase equipment. Although the
"lease" amounts to actually borrowing money guaranteed by the lessee, it
doesn't appear on the company's books as an obligation. A special purpose
entity (SPE) is a company set up to facilitate the creation of a synthetic lease. It
borrows money that is guaranteed by the lessee, purchases equipment, and
leases it to the lessee. Its purpose is keep the lessee from having to capitalize
the lease and carry its payments on its books as a liability.
c. Offbalance sheet financing refers to the fact that for many years neither leased assets nor the
liabilities under lease contracts appeared on the lessees' balance sheets. To correct this problem,
the Financial Accounting Standards Board issued FASB Statement 13. Capitalizing means
incorporating the lease provisions into the balance sheet by reporting the leased asset under fixed
assets and reporting the present value of future lease payments as debt.
d. FASB Statement 13 is the Financial Accounting Standards Board statement (November 1976) that
spells out in detail the conditions under which a lease must be capitalized, and the specific
procedures to follow.
e. A guideline lease is a lease that meets all of the IRS requirements for a genuine lease. A
guideline lease is often called a taxoriented lease. If a lease meets the IRS guidelines, the IRS
allows the lessor to deduct the asset's depreciation and allows the lessee to deduct the lease
payments.
f. The residual value is the market value of the leased property at the expiration of the lease. The
estimate of the residual value is one of the key elements in lease analysis.
g. The lessee's analysis involves determining whether leasing an asset is less costly than
buying the asset. The lessee will compare the present value cost of leasing the
asset with the present value cost of purchasing the asset (assuming the funds to
purchase the asset are obtained through a loan). If the present value cost of the
lease is less than the present value cost of purchasing, the asset should be leased.
The lessee can also analyze the lease using the IRR approach. The IRR of the
incremental cash flows of leasing versus purchasing represents the aftertax cost
Answers and Solutions: 20  406 rate implied in the lease contract. If this rate is lower than the aftertax cost of
debt, there is an advantage to leasing. Finally, the lessee might evaluate the
lease using the equivalent loan method, which involves comparing the net
savings at Time 0 if the asset is leased with the present value of the incremental
costs of leasing over the term of the lease. If the Time 0 savings is greater than
the present value of the incremental costs, there is an advantage to leasing.
The lessor's analysis involves determining the rate of return on the proposed lease. If the
rate of return (or IRR) of the lease cash flows exceeds the lessor's opportunity cost of capital, the
lease is a good investment. This is equivalent to analyzing whether the NPV of the lease is
positive.
h. The net advantage to leasing (NAL) gives the dollar value of the lease to the lessee. It is, in a
sense, the NPV of leasing versus owning.
i. The alternative minimum tax (AMT), which is figured at about 20 percent of the profits reported
to stockholders, is a provision of the tax code that requires profitable firms to pay at least some
taxes if such taxes are greater than the amount due under standard tax accounting. The AMT has
provided a stimulus to leasing for those firms paying the AMT because leasing lowers profits
reported to stockholders.
202 An operating lease is usually cancelable and includes maintenance. Operating leases are, frequently,
for a period significantly shorter than the economic life of the asset, so the lessor often does not
recover his full investment during the period of the basic lease. A financial lease, on the other hand,
is fully amortized and generally does not include maintenance provisions. An operating lease would
probably be used for a fleet of trucks, while a financial lease would be used for a manufacturing plant.
203 You would expect to find that lessees, in general, are in relatively low incometax brackets, while
lessors tend to be in high tax brackets. The reason for this is that owning tends to provide tax
shelters in the early years of a project's life. These tax shelters are more valuable to taxpayers in
high brackets. However, current tax laws (1998) have reduced the depreciation benefits of owning,
so tax rate differentials are less important now than in the past.
204 The banks, when they initially went into leasing, were paying relatively high tax rates. However,
since municipal bonds are taxexempt, their heavy investments in municipals lowered the banks'
effective tax rates. Similarly, when the REIT loans began to sour, this further reduced the bank's
income, and consequently cut the effective tax rate even further. Since the lease investments were
predicated on obtaining tax shelters, and since the value of these tax shelters is dependent on the
banks' tax rates, when the effective tax rates were lowered, this reduced the value of the tax shelters
and consequently reduced the profitability of the lease investments.
205 a. Pros:
Answers and Solutions: 20  407 · The use of the leased premises or equipment is actually an exclusive right, and the
payment for the premises is a liability that often must be met. Therefore, leases
should be treated as both assets and liabilities.
· A fixed policy of capitalizing leases among all companies would add to the
comparability of different firms. For example, Safeway Stores' leases should be
capitalized to make the company comparable to A&P, which owns its stores
through a subsidiary.
· The capitalization highlights the contractual nature of the leased property.
· Capitalizing of leases could help management make useful comparisons of
operating results; that is, return on investment data.
b. Cons:
· Because the firm does not actually own the leased property, the legal aspect can
be cited as an argument against capitalization.
· Capitalizing leases worsens some key credit ratios; that is, the debttoequity ratio
and the debttototal capital ratio. This may hamper the future acquisition of
funds.
· There is a question of choosing the proper discount rate at which to capitalize the
leases.
· Some argue that other items should be listed on the balance sheet before leases;
for example, service contracts, property taxes, and so on.
· Capitalizing leases violates the principle that liabilities should be recorded only
when assets are purchased. 206 Lease payments, like depreciation, are deductible for tax purposes. If a 20year asset were
depreciated over a 20year life, depreciation charges would be 1/20 per year (more if MACRS were
used). However, if the asset were leased for, say, 3 years, tax deductions would be 1/3 each year for
3 years. Thus, the tax deductions would be greatly accelerated. The same total taxes would be
paid over the 20 years, but because of the high deductions in the early years, taxes would be deferred
more under the lease, and the PV of the future taxes would be reduced under the lease.
207 In fact, Congress did this in 1981. Depreciable lives were shorter than before; corporate tax rates
were essentially unchanged (they were lowered very slightly on income below $50,000); and the
investment tax credit had been improved a bit by the easing of recapture if the asset was held for a
short period. As a result, companies that were either investing at a very high rate or else were only
marginally profitable were generating more depreciation and/or investment tax credits than they could
use. These companies were able to "sell" their tax shelters through a leasing arrangement, being
"paid" in the form of lower lease charges. A highbracket lessor could earn a given aftertax return
with lower rental charges, after the 1981 tax law changes, than previously because the lessor would
get (1) the larger tax credits and (2) faster depreciation writeoffs.
208 A cancellation clause would reduce the risk to the lessee since the firm would be allowed to terminate
the lease at any point. Since the lease is less risky than a standard financial lease, and less risky than
Answers and Solutions: 20  408 straight debt, which cannot usually be prepaid without a prepayment charge, the discount rate on the cost of leasing might be adjusted to reflect lower risk. (Note that this requires increasing the discount rate since cash outflows are being discounted.) The effect on the lessor is just the oppositerisk is increased. (Note that this would also require an increase in the lessor's discount rate.)
Answers and Solutions: 20  409 SOLUTIONS TO ENDOFCHAPTER PROBLEMS
201 a. (1) Reynolds' current debt ratio is $400/$800 = 50%.
(2) If the company purchased the equipment its balance sheet would look like:
Current assets $300 Debt (including lease) $600
Fixed assets 500
Leased equipment 200 Equity $400
Total assets $1,000 Total claims $1,000
Therefore, the company's debt ratio = $600/$1,000 = 60%.
(3) If the company leases the asset and does not capitalize the lease, its debt ratio = $400/$800 =
50%.
b. The company's financial risk (assuming the implied interest rate on the lease is equivalent to the
loan) is no different whether the equipment is leased or purchased.
202 Cost of owning:
0 1 2
  
Cost (200)
Depreciation shield 40 40
(200) 40 40
PV at 6% = $127.
Cost of leasing:
0 1 2
  
Aftertax lease payment (66) (66)
PV at 6% = $128.
Reynolds should buy the equipment, because the cost of owning is less than the cost of leasing.
Answers and Solutions: 20  410 203 Year_________________
0 1 2 3 4____
I. Cost of Owning:
Net purchase price ($1,500,000)
Depr. tax savingsa $198,000 $270,000 $ 90,000 $ 42,000
Net cash flow ($1,500,000) $198,000 $270,000 $ 90,000 $ 42,000
PV cost of owning at 9% ($ 991,845)
II. Cost of Leasing:
Lease payment (AT) (240,000) (240,000) (240,000) (240,000)
Purch. option priceb (250,000)
Net cash flow $ 0 ($240,000) ($240,000) ($240,000) ($490,000)
PV cost of leasing at 9% ($ 954,639)
III. Cost Comparison
Net advantage to leasing (NAL)= PV cost of owning  PV cost of leasing
= $991,845  $954,639
= $37,206.
a
Cost of new machinery: $1,500,000.
MACRS Deprec. Tax Savings
Year Allowance Factor Depreciation T (Depreciation)
1 0.33 $495,000 $198,000
2 0.45 675,000 270,000
3 0.15 225,000 90,000
4 0.07 105,000 42,000
b
Cost of purchasing the machinery after the lease expires.
Note that the maintenance expense is excluded from the analysis since Big Sky Mining will have to
bear the cost whether it buys or leases the machinery. Since the cost of leasing the machinery is
less than the cost of owning it, Big Sky Mining should lease the equipment.
Answers and Solutions: 20  411 204 a. Balance sheets before lease is capitalized:
Energen
Balance Sheet
(Thousands of Dollars)
Debt $100
Equity 100
Total assets $200 Total claims $200
Debt/assets ratio = $100/$200 = 50%.
Hastings Corporation
Balance Sheet
(Thousands of Dollars)
Debt $ 50
Equity 100
Total assets $150 Total claims $150
Debt/assets ratio = $50/$150 = 33%.
b. Balance sheet after lease is capitalized:
Hastings Corporation
Balance Sheet
(Thousands of Dollars)
Assets $150 Debt $ 50
Value of leased asset 50 PV of lease payments 50
Equity 100
Total assets $200 Total claims $200
Debt/assets ratio = $100/$200 = 50%.
c. Yes. Net income, as reported, would probably be less under leasing because the
lease payment would be larger than the interest expense, both of which are
income statement expenses. Additionally, total assets are significantly less
under leasing without capitalization.
The net result is difficult to predict, but we can state positively that both ROA
Answers and Solutions: 20  412 and ROE are affected by the choice of financing.
205 a. Borrow and buy analysis:
Year 0 Year 1 Year 2 Year 3
Loan payments (430,731) (430,731) (430,731)
Interest tax savings 47,600 33,761 17,985
Depreciation tax savings 112,200 153,000 51,000
Net cash flow $ 0 $270,931 $243,970 $361,746
PV cost of owning @ 9.24%a = ($729,956)
Depreciation Scheduleb
Year Allowance Depreciation
1 0.33 $330,000
2 0.45 450,000
3 0.15 150,000
$930,000
Loan Amortization Schedule
Repayment
Beginning of
Remaining
Year Amount Payment Interest Principal
Balance
1 $1,000,000 $430,731 $140,000 $290,731
$709,269
2 709,269 430,731 99,298
331,433 $377,836
3 377,836 430,731 52,897
377,834 2*
*Difference due to rounding.
Lease analysis:
Year 0 Year 1 Year 2 Year 3
Lease payment ($320,000) ($320,000) ($320,000)
Answers and Solutions: 20  413 Payment tax savings 108,800 108,800 108,800
Mkt Value Machine ( 200,000)c
Net cash flow $ 0 ($211,200) ($211,200) ($411,200)
PV cost of leasing @ 9.24% = ($685,752)
Notes:
a
Discount rate = 14% x (1  T) = 14% x (1  0.34) = 9.24%.
b
Depreciable basis = Cost = $1,000,000. MACRS allowances = 33%, 45%, 15%. Depreciation
tax savings = T(Depreciation).
c
Cost of purchasing the machinery after the lease expires. Note that since the firm is purchasing
the machine at the end of the lease, there are no tax effects due to the residual value (purchase
price) being greater than the book value. If we were to assume that the firm would not want to
keep the machine beyond the lease term, then we would show the residual value of selling the
machine as an inflow under the purchase alternative, and there would be no residual value flow
under the lease alternative. In that situation, there would be tax on the residual value from
selling the machine: ($200,000  $70,000)0.34 = $44,200.
Note that the maintenance expense is excluded from the analysis since the firm will have to
bear the cost whether it buys or leases the machinery. Since the cost of leasing the machinery is
less than the cost of owning it ($729,956  $685,752 = $44,204), the firm should lease the
equipment.
b. We assume that the company will buy the equipment at the end of 3 years if the lease plan is used;
hence, the $200,000 is an added cost under leasing. We discounted it at 9.24 percent, but it is
risky, so should we use a higher rate? If we do, leasing looks even better. However, it really
makes more sense in this instance to use a lower rate to discount the residual value so as to
penalize the lease decision, because the residual value uncertainty increases the uncertainty of</