The relevant risk of an individual asset is its contribution to the riskiness of a

•

well-diversi¬ed portfolio, which is the asset™s market risk. This risk is measured

by beta. Since market risk cannot be eliminated by diversi¬cation, investors must

be compensated for bearing it.

The Security Market Line (SML) equation shows the relationship between a

•

security™s risk and its required rate of return. The return required on any security

is equal to the risk-free rate plus the market risk premium times the security™s

beta:

ri rRF bi(rM rRF).

The Capital Market Line describes the risk/return relationship for optimal port-

•

folios, that is, for portfolios that consist of a mix of the market portfolio and a

riskless asset.

The beta coef¬cient is measured by the slope of the stock™s characteristic line,

•

which is found by regressing historical returns on the stock versus historical

returns on the market.

The value of a share of stock is calculated as the present value of the stream of

•

dividends the stock is expected to provide in the future.

The equation used to ¬nd the value of a constant growth stock is

•

D0(1 g) D1

ˆ

P0 .

rg r g

ˆ

The equation for rs, the expected rate of return on a constant growth stock, can

•

ˆ

be expressed as follows: rs D1/P0 g.

To ¬nd the present value of a supernormal growth stock (1) ¬nd the dividends

•

expected during the supernormal growth period, (2) ¬nd the price of the stock at

the end of the supernormal growth period, (3) discount the dividends and the

projected price back to the present, and (4) sum these PVs to ¬nd the stock™s

ˆ

value, P0.

The value of a perpetuity can be found using the constant growth formula with

•

g 0.

Questions

(28-1) De¬ne each of the following terms:

a. PV; i or I; FVn; PMT; m; inom

b. FVIFi,n; PVIFi,n; FVIFAi,n; PVIFAi,n

c. Equivalent Annual Rate (EAR); nominal (quoted) interest rate

d. Amortization schedule; principal component versus interest component of a

payment

e. Par value; maturity date; coupon payment; coupon interest rate

f. Premium bond; discount bond

g. Stand-alone risk; risk; probability distribution

ˆ

h. Expected rate of return, r

i. Risk Premium for Stock i, RPi; market risk premium, RPM

j. Capital Asset Pricing Model, CAPM

k. Market risk; diversi¬able risk; relevant risk

l. Beta coef¬cient, b

m. Security Market Line, SML

n. Optimal (or ef¬cient) portfolio

28-26 Basic Financial Tools: A Review

Chapter 28

o. Capital Market Line, CML

p. Characteristic line

ˆ

q. Intrinsic value, P0; market price, P0

ˆ

r. Required rate of return, rS; expected rate of return, rS; actual, or realized, rate

¯ S.

of return, r

s. Normal, or constant, growth; supernormal, or nonconstant, growth

(28-2) Would you rather have a savings account that pays 5 percent interest compounded

semiannually, or one that pays 5 percent interest compounded daily? Explain.

(28-3) The rate of return you would earn if you bought a bond and held it to its maturity

date is called the bond™s yield to maturity. How is the yield to maturity related to

overall interest rates in the economy? If interest rates in the economy fall after a

bond has been issued, what will happen to the bond™s yield to maturity and price?

Will the size of any changes be affected by the bond™s maturity?

(28-4) Security X has an expected return of 6 percent, a standard deviation of expected

returns of 40 percent, a correlation coef¬cient with the market of 0.20, and a

beta coef¬cient of 0.4. Security Y has an expected return of 13 percent, a stan-

dard deviation of returns of 25 percent, a correlation with the market of 0.8, and

a beta coef¬cient of 1.0. Which security is riskier? Why?

(28-5) If a stock™s beta were to drop to half of its former level, would its expected return

also drop by half?

(28-6) What is the difference between the SML and the CML?

(28-7) Two investors are evaluating GE™s stock for possible purchase. They agree on the

expected value of D1 and also on the expected future dividend growth rate.

Further, they agree on the riskiness of the stock. However, one investor normally

holds stocks for 2 years, while the other normally holds stocks for 10 years. On

the basis of the type of analysis done in this chapter, should they be willing to pay

the same price for GE™s stock? Explain.

Problems

(28-1) Find the future value of the following annuities. The ¬rst payment is made at the

end of Year 1, so they are ordinary annuities:

Future Value of an

Annuity

a. $500 per year for 8 years at 9%.

b. $300 per year for 6 years at 4%.

c. $500 per year for 6 years at 0%.

d. Now rework parts a, b, and c assuming that payments are made at the begin-

ning of each year; that is, they are annuities due.

(28-2) Find the present value of the following ordinary annuities:

Present Value of an a. $500 per year for 8 years at 9%.

Annuity

b. $300 per year for 6 years at 4%.

c. $500 per year for 6 years at 0%.

d. Now rework parts a, b, and c assuming that payments are made at the begin-

ning of each year; that is, they are annuities due.

(28-3) Find the interest rates, or rates of return, on each of the following:

Effective Rate of

a. You borrow $900 and promise to pay back $972 at the end of 1 year.

Interest

b. You lend $900 and receive a promise to be paid $972 at the end of 1 year.

c. You borrow $65,000 and promise to pay back $310,998 at the end of 15

years.

d. You borrow $11,000 and promise to make payments of $2,487.22 per year for

7 years.

28-27

Problems

(28-4) Your grandmother has asked you to evaluate two alternative investments for her.

PV and Effective The ¬rst is a security that pays $50 at the end of each of the next 3 years, with a

Annual Yield ¬nal payment of $1,050 at the end of Year 4. This security costs $900. The second

investment involves simply putting the same amount of money in a bank savings

account that pays an 8 percent nominal (quoted) interest rate, but with quarterly

compounding. Your grandmother regards the two investments as being equally

safe and liquid, so the required effective annual rate of return on the security is

the same as that on the bank deposit. She has asked you to calculate the value of

the security to help her decide whether it is a good investment. What is its value

relative to the bank deposit?

(28-5) Assume that your father is now 55 years old, that he plans to retire in 12 years,

Required Annuity and that he expects to live for 20 years after he retires, that is, until he is 87. He

Payments wants a ¬xed retirement income that has the same purchasing power at the time

he retires as $60,000 has today (he realizes that the real value of his retirement

income will decline year by year after he retires, but he wants level payments dur-

ing retirement anyway). His retirement income will begin the day he retires, 12 years

from today, and he will receive 20 annual payments. In¬‚ation is expected to be 5

percent per year from today forward. He currently has $100,000 in savings, and

he expects to earn a return on his savings of 8 percent per year, annual compound-

ing. To the nearest dollar, how much must he save during each of the next 12 years

(with deposits being made at the end of each year) to meet his retirement goal?

(28-6) Cargill Diggs™ bonds have 15 years remaining to maturity. Interest is paid annually,

Yield to Maturity the bonds have a $1,000 par value, and the coupon interest rate is 9.5 percent.

The bonds sell at a price of $850. What is their yield to maturity?

(28-7) The Peabody Company has two bond issues outstanding. Both pay $110 annual

Bond Valuation interest plus $1,000 at maturity. Bond H has a maturity of 14 years and Bond K a

maturity of 2 years.

a. What is the value of each of these bonds if the going rate of interest is (1) 5

percent, (2) 8 percent, and (3) 12 percent? Assume that two more interest pay-

ments will be made on Bond K and 14 more on Bond H.

b. Why does the longer-term (14-year) bond ¬‚uctuate more when interest rates

change than does the shorter-term bond (2-year)?

(28-8) Suppose Integon Inc. sold an issue of bonds with a 10-year maturity, a $1,000 par

Bond Valuation value, a 9 percent coupon rate, and semiannual interest payments.

a. Two years after the bonds were issued, the going rate of interest on bonds such

as these fell to 7 percent. At what price would the bonds sell?

b. Suppose that, 2 years after the initial offering, the going interest rate had risen

to 11 percent. At what price would the bonds sell?

c. Suppose that the conditions in part a existed”that is, interest rates fell to 7 per-

cent 2 years after the issue date. Suppose further that the interest rate remained

at 7 percent for the next 8 years. What would happen to the price of the Integon

bonds over time? (Hint: How much should one of these bonds sell for just before

maturity?)

(28-9) A bond trader purchased each of the following bonds at a yield to maturity of 9

Interest Rate Risk percent. Immediately after she purchased the bonds, interest rates fell to 8 percent.

To show the price sensitivity of each bond to changes in interest rates, ¬ll in the

blanks in the following table:

Price @ 9% Price @ 8% Percentage Change

10-year 10% annual coupon ““““““ ““““““

10-year zero coupon ““““““ ““““““

5-year zero coupon ““““““ ““““““

30-year zero coupon ““““““ ““““““

$100 perpetuity ““““““ ““““““

28-28 Basic Financial Tools: A Review

Chapter 28

Could you use the percentage change as a measure of the bonds™ interest rate

risk? Would you be making any assumptions about the shape of the yield curve if

you did this? Might changes in the shape of the yield curve affect changes in the

bonds™ prices in the real world? Discuss how the yield curve might affect a bond™s

price variability over time.

(28-10) An investor has two bonds in his portfolio. Each bond matures in 3 years, has a

Bond Valuation face value of $1,000, and has a yield to maturity of 11.1 percent. Bond A pays an

annual coupon of 13 percent. Bond Z is a zero coupon bond. Assuming the yield

to maturity of each bond remains at 11.1 percent over the next 4 years, what will

be the price of each of the bonds at the following time periods? Fill in the following

table:

t Price of Bond A Price of Bond Z

0 ““““““ ““““““

1 ““““““ ““““““

2 ““““““ ““““““

3 ““““““ ““““““

(28-11) A stock™s expected return has the following distribution:

Expected Return

Rate of Return

Demand for the Probability of This if This Demand

Company™s Products Demand Occurring Occurs

Weak 0.2 (40%)

Below average 0.2 (10)

Average 0.1 8

Above average 0.3 20

Strong 0.2 50

Calculate the stock™s expected return, standard deviation, and coef¬cient of variation.

(28-12) An individual has $60,000 invested in a stock that has a beta of 1.3 and $15,000

invested in a stock with a beta of 0.6. If these are the only two investments in her

Portfolio Beta

portfolio, what is her portfolio™s beta?

(28-13) Suppose rRF 9%, rM 14%, and rA 20%.

Required Rate of

a. Calculate Stock A™s beta.

Return

b. If Stock A™s beta changed to 1.5, how would the required rate of return change?

(28-14) Suppose rRF 6%, rM 11%, and bi 1.3.

Required Rate of

a. What is ri, the required rate of return on Stock i?

Return

b. Now suppose rRF (1) increases to 7 percent or (2) decreases to 5 percent. The

slope of the SML remains constant. How would this affect rM and ri?

c. Now assume that rRF remains at 6 percent but rM (1) increases to 13 percent or

(2) falls to 10 percent. The slope of the SML does not remain constant. How

would these changes affect ri?

(28-15) You have a $3 million portfolio consisting of a $150,000 investment in each of 20

different stocks. The portfolio has a beta of 1.2. You are considering selling

Portfolio Beta

$150,000 worth of one stock that has a beta equal to 0.8 and using the proceeds

to purchase a stock that has a beta equal to 1.4. What will be the new beta of

your portfolio following this transaction?

28-29

Problems

(28-16) You are given the following set of data:

Characteristic Line

and SML

HISTORICAL RATES OF RETURN

Year NYSE Stock ABC

1 (26.5%) (14.5%)