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a current yield of 9.6 percent, and it sells at a yield to maturity of 9.25 percent. The firm
wishes to issue additional bonds to the public at par value. What coupon rate must the new
bonds offer in order to sell at par?
8. Financial Pages. Refer to Figure 3.2. What is the current yield of the 61„4 percent, August
2002 maturity bond? What was the closing ask price of the bond on the previous day?

9. Bond Prices and Returns. One bond has a coupon rate of 8 percent, another a coupon rate
Practice of 12 percent. Both bonds have 10-year maturities and sell at a yield to maturity of 10 per-
Problems cent. If their yields to maturity next year are still 10 percent, what is the rate of return on
each bond? Does the higher coupon bond give a higher rate of return?
10. Bond Returns.

a. If the AT&T bond in problem 6 has a yield to maturity of 8 percent 1 year from now, what
will its price be?
b. What will be the rate of return on the bond?
c. If the inflation rate during the year is 3 percent, what is the real rate of return on the bond?

11. Bond Pricing. A General Motors bond carries a coupon rate of 8 percent, has 9 years until
maturity, and sells at a yield to maturity of 9 percent.

a. What interest payments do bondholders receive each year?
b. At what price does the bond sell? (Assume annual interest payments.)
c. What will happen to the bond price if the yield to maturity falls to 7 percent?
12. Bond Pricing. A 30-year maturity bond with face value $1,000 makes annual coupon pay-
ments and has a coupon rate of 8 percent. What is the bond™s yield to maturity if the bond is
selling for

a. $900
b. $1,000
c. $1,100

13. Bond Pricing. Repeat the previous problem if the bond makes semiannual coupon pay-

14. Bond Pricing. Fill in the table below for the following zero-coupon bonds. The face value
of each bond is $1,000.

Price Maturity (Years) Yield to Maturity
$300 30 __
$300 __ 8%
__ 10 10%
15. Consol Bonds. Perpetual Life Corp. has issued consol bonds with coupon payments of $80.
(Consols pay interest forever, and never mature. They are perpetuities.) If the required rate
of return on these bonds at the time they were issued was 8 percent, at what price were they
sold to the public? If the required return today is 12 percent, at what price do the consols
16. Bond Pricing. Sure Tea Co. has issued 9 percent annual coupon bonds which are now sell-
ing at a yield to maturity of 10 percent and current yield of 9.8375 percent. What is the re-
maining maturity of these bonds?
17. Bond Pricing. Large Industries bonds sell for $1,065.15. The bond life is 9 years, and the
yield to maturity is 7 percent. What must be the coupon rate on the bonds?
18. Bond Prices and Yields.

a. Several years ago, Castles in the Sand, Inc., issued bonds at face value at a yield to ma-
turity of 8 percent. Now, with 8 years left until the maturity of the bonds, the company
has run into hard times and the yield to maturity on the bonds has increased to 14 per-
cent. What has happened to the price of the bond?
b. Suppose that investors believe that Castles can make good on the promised coupon pay-
ments, but that the company will go bankrupt when the bond matures and the principal
comes due. The expectation is that investors will receive only 80 percent of face value at
maturity. If they buy the bond today, what yield to maturity do they expect to receive?

19. Bond Returns. You buy an 8 percent coupon, 10-year maturity bond for $980. A year later,
the bond price is $1,050.

a. What is the new yield to maturity on the bond?
b. What is your rate of return over the year?

20. Bond Returns. You buy an 8 percent coupon, 10-year maturity bond when its yield to ma-
turity is 9 percent. A year later, the yield to maturity is 10 percent. What is your rate of re-
turn over the year?
21. Interest Rate Risk. Consider three bonds with 8 percent coupon rates, all selling at face
value. The short-term bond has a maturity of 4 years, the intermediate-term bond has matu-
rity 8 years, and the long-term bond has maturity 30 years.

a. What will happen to the price of each bond if their yields increase to 9 percent?
b. What will happen to the price of each bond if their yields decrease to 7 percent?
c. What do you conclude about the relationship between time to maturity and the sensitiv-
ity of bond prices to interest rates?
22. Rate of Return. A 2-year maturity bond with face value $1,000 makes annual coupon pay-
ments of $80 and is selling at face value. What will be the rate of return on the bond if its
yield to maturity at the end of the year is

a. 6 percent
b. 8 percent
c. 10 percent
Valuing Bonds 277

23. Rate of Return. A bond that pays coupons annually is issued with a coupon rate of 4 per-
cent, maturity of 30 years, and a yield to maturity of 8 percent. What rate of return will be
earned by an investor who purchases the bond and holds it for 1 year if the bond™s yield to
maturity at the end of the year is 9 percent?
24. Bond Risk. A bond™s credit rating provides a guide to its risk. Long-term bonds rated Aa
currently offer yields to maturity of 8.5 percent. A-rated bonds sell at yields of 8.8 percent.
If a 10-year bond with a coupon rate of 8 percent is downgraded by Moody™s from Aa to A
rating, what is the likely effect on the bond price?
25. Real Returns. Suppose that you buy a 1-year maturity bond for $1,000 that will pay you
back $1,000 plus a coupon payment of $60 at the end of the year. What real rate of return
will you earn if the inflation rate is

a. 2 percent
b. 4 percent
c. 6 percent
d. 8 percent

26. Real Returns. Now suppose that the bond in the previous problem is a TIPS (inflation-in-
dexed) bond with a coupon rate of 4 percent. What will the cash flow provided by the bond
be for each of the four inflation rates? What will be the real and nominal rates of return on
the bond in each scenario?
27. Real Returns. Now suppose the TIPS bond in the previous problem is a 2-year maturity bond.
What will be the bondholder™s cash flows in each year in each of the inflation scenarios?

28. Interest Rate Risk. Suppose interest rates increase from 8 percent to 9 percent. Which bond
Challenge will suffer the greater percentage decline in price: a 30-year bond paying annual coupons of
8 percent, or a 30-year zero coupon bond? Can you explain intuitively why the zero exhibits
greater interest rate risk even though it has the same maturity as the coupon bond?

1 a. The ask price is 101 23/32 = 101.71875 percent of face value, or $1,017.1875.
Solutions to b. The bid price is 101 21/32 = 101.65625 percent of face value, or $1,016.5625.
c. The price increased by 1/32 = .03125 percent of face value, or $.3125.
d. The annual coupon is 6 1/4 percent of face value, or $62.50, paid in two semiannual in-
Questions stallments.
e. The yield to maturity, based on the ask price, is given as 5.64 percent.

2 The coupon is 9 percent of $1,000, or $90 a year. First value the 6-year annuity of coupons:
PV = $90 — (6-year annuity factor)

[ ]
1 1
= $90 — “
.12 .12(1.12)6
= $90 — 4.11 = $370.03

Then value the final payment and add up:
PV = = $506.63
PV of bond = $370.03 + $506.63 = $876.66

3 The yield to maturity is about 8 percent, because the present value of the bond™s cash returns
is $1,199 when discounted at 8 percent:

PV = PV (coupons) + PV (final payment)
= (coupon — annuity factor) + (face value — discount factor)

[ ]
1 1 1
= $140 — + $1,000 —

.08 .08(1.08)4 1.084
= $463.70 + $735.03 = $1,199
4 The 6 percent coupon bond with maturity 2002 starts with 3 years left until maturity and
sells for $1,010.77. At the end of the year, the bond has only 2 years to maturity and in-
vestors demand an interest rate of 7 percent. Therefore, the value of the bond becomes
$60 $1,060
PV at 7% = + = $981.92
(1.07) (1.07)2

You invested $1,010.77. At the end of the year you receive a coupon payment of $60 and
have a bond worth $981.92. Your rate of return is therefore
$60 + ($981.92 “ $1,010.77)
Rate of return = = .0308, or 3.08%
The yield to maturity at the start of the year was 5.6 percent. However, because interest rates
rose during the year, the bond price fell and the rate of return was below the yield to matu-
5 By the end of this year, the bond will have only 1 year left until maturity. It will make only
one more payment of coupon plus face value, so its price will be $1,060/1.056 = $1,003.79.
The rate of return is therefore
$60 + ($1,003.79 “ $1,007.37)
= .056, or 5.6%

6 At an interest rate of 5.6 percent, the 3-year bond sells for $1,010.77. If the interest rate
jumps to 10 percent, the bond price falls to $900.53, a decline of 10.9 percent. The 30-year
bond sells for $1,057.50 when the interest rate is 5.6 percent, but its price falls to $622.92
at an interest rate of 10 percent, a much larger percentage decline of 41.1 percent.
Stocks and the Stock Market
Reading the Stock Market Listings

Book Values, Liquidation Values, and Market Values
Valuing Common Stocks
Today™s Price and Tomorrow™s Price
The Dividend Discount Model

Simplifying the Dividend Discount Model
The Dividend Discount Model with No Growth
The Constant-Growth Dividend Discount Model
Estimating Expected Rates of Return
Nonconstant Growth

Growth Stocks and Income Stocks
The Price-Earnings Ratio
What Do Earnings Mean?
Valuing Entire Businesses


nstead of borrowing cash to pay for its investments, a firm can sell new

I shares of common stock to investors. Whereas bond issues commit the
firm to make a series of specified interest payments to the lenders, stock
issues are more like taking on new partners. The stockholders all share in the
fortunes of the firm according to the number of shares they hold. We will take a first
look at stocks, the stock market, and principles of stock valuation.
We start by looking at how stocks are bought and sold. Then we look at what deter-
mines stock prices and how stock valuation formulas can be used to infer the rate of re-
turn that investors are expecting. We will see how the firm™s investment opportunities
are reflected in the stock price and why stock market analysts focus so much attention
on the price-earnings, or P/E ratio of the company.
Why should you care how stocks are valued? After all, if you want to know the value
of a firm™s stock, you can look up the stock price in The Wall Street Journal. But you
need to know what determines prices for at least two reasons. First, you may wish to
check that any shares that you own are fairly priced and to gauge your beliefs against
the rest of the market. Second, corporations need to have some understanding of how
the market values firms in order to make good capital budgeting decisions. A project is
attractive if it increases shareholder wealth. But you can™t judge that unless you know
how shares are valued.
After studying this material you should be able to
Understand the stock trading reports in the financial pages of the newspaper.
Calculate the present value of a stock given forecasts of future dividends and future
stock price.
Use stock valuation formulas to infer the expected rate of return on a common stock.
Interpret price-earnings ratios.

Stocks and the Stock Market
A shareholder is a part-owner of the firm. For example, there were 1,471 million shares
of PepsiCo outstanding at the beginning of 1999, so if you held 1,000 shares of Pepsi,
you would have owned 1,000/1,471,000,000 = .00007 percent of the firm. You would
have received .00007 percent of any dividends paid by the company and you would
be entitled to .00007 percent of the votes that could be cast at the company™s annual
Firms issue shares of common stock to the public when they need to raise money.1
Ownership shares in a
publicly held corporation.
1We use the terms “shares,” “stock,” and “common stock” interchangeably, as we do “shareholders” and

Valuing Stocks 281

They typically engage investment banking firms such as Merrill Lynch or Goldman
Sachs to help them market these shares. Sales of new stock by the firm are said to occur
in the primary market. There are two types of primary market issues. In an initial
public offering, or IPO, a company that has been privately owned sells stock to the


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