6

Long-term

4 Treasury

2

0

1954 1958 1962 1966 1970 1974 1978 1982 1986 1990 1994 1998 2002

>

8. What is the expected yield to maturity if the firm in Example 9.9 is in even worse

Concept

condition and investors expect a final payment of only $600?

CHECK

To compensate for the possibility of default, corporate bonds must offer a default

default premium premium. The default premium is the difference between the promised yield on a corporate

bond and the yield of an otherwise identical government bond that is riskless in terms of de-

The increment to

fault. If the firm remains solvent and actually pays the investor all of the promised cash flows,

promised yield that

the investor will realize a higher yield to maturity than would be realized from the government

compensates the

investor for bond. If, however, the firm goes bankrupt, the corporate bond is likely to provide a lower re-

default risk. turn than the government bond. The corporate bond has the potential for both better and worse

performance than the default-free Treasury bond. In other words, it is riskier.

The pattern of default premiums offered on risky bonds is sometimes called the risk struc-

ture of interest rates. The greater the default risk, the higher the default premium. Figure 9.10

shows the yield to maturity of bonds of different risk classes since 1954 and the yields on junk

bonds since 1984. You can see here clear evidence of default-risk premiums on promised

yields.

9.6 THE YIELD CURVE

Return to Figure 9.1 again, and you will see that while yields to maturity on bonds of various

maturities are reasonably similar, yields do differ. Bonds with shorter maturities generally of-

yield curve

fer lower yields to maturity than longer term bonds. The graphical relationship between the

A graph of yield to yield to maturity and the term to maturity is called the yield curve. The relationship also is

maturity as a function called the term structure of interest rates because it relates yields to maturity to the term

of term to maturity.

(maturity) of each bond. The yield curve is published regularly in The Wall Street Journal;

four such sets of curves are reproduced in Figure 9.11. Figure 9.11 illustrates that a wide range

term structure of of yield curves may be observed in practice. Panel A is a downward sloping, or “inverted”

interest rates yield curve. Panel B is an upward sloping curve, and Panel C is a hump-shaped curve, first ris-

ing and then falling. Finally the yield curve in Panel D is essentially flat. Rising yield curves

The relationship

between yields are most commonly observed. We will see why momentarily.

to maturity and

Why should bonds of differing maturity offer different yields? The two most plausible pos-

terms to maturity

sibilities have to do with expectations of future rates and risk premiums. We will consider each

across bonds.

of these arguments in turn.

Bodie’Kane’Marcus: III. Debt Securities 9. Bond Prices and Yields © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

323

9 Bond Prices and Yields

Treasury yield curve Treasury yield curve Treasury yield curve Treasury yield curve

Yields as of 4:30 P.M. Eastern time Yields as of 4:30 P.M. Eastern time Yields as of 4:30 P.M. Eastern time Yields as of 4:30 P.M. Eastern time

Percent Percent Percent Percent

6.50 6.50 8.60 9.00

8.40

6.25 8.60

6.00

8.20

6.00 8.20

5.50 8.00

5.75 7.80

7.80

Yesterday Yesterday Yesterday Yesterday

5.00

5.50 7.40

1 week ago 1 week ago 1 week ago 1 week ago

7.60

4 weeks ago 4 weeks ago 4 weeks ago 4 weeks ago

5.25 4.50 7.40 7.00

3 6 1 2 3 5 10 30 3 6 1 2 3 5 10 30 3 6 1 2 3 45710 30 3 6 1 2 3 45 710 30

Months Year Maturities Months Year Maturities Months Year Maturities Months Year Maturities

A. (September 11, 2000) B. (November 18, 1997) D. (October 17, 1989)

C. (October 4, 1989)

Falling yield curve Rising yield curve Flat yield curve

Hump-shaped yield curve

F I G U R E 9.11

Treasury yield curves

Source: Various editions of The Wall Street Journal. Reprinted by permission of The Wall Street Journal, © 1989, 1997, 2000 Dow Jones & Company, Inc.

All Rights Reserved Worldwide.

The Expectations Theory

Suppose everyone in the market believes firmly that while the current one-year interest rate is

8%, the interest rate on one-year bonds next year will rise to 10%. What would this belief im-

ply about the proper yield to maturity on two-year bonds issued today?

It is easy to see that an investor who buys the one-year bond and rolls the proceeds into an-

other one-year bond in the following year will earn, on average, about 9% per year. This value

WEBMA STER

Term Structure of Interest Rates

The bond section of the Smart Money website has a section called the Living Yield Curve.

It has a graph that allows you to compare the shape of the yield curve at different points

in time. Go to Smart Money™s website at http://www.smartmoney.com/onebond/

index.cfm?story yieldcurve. Then, use the site to answer the following questions:

1. What is considered a normal yield curve?

2. Compare the yield curve for December 2001 with the average yield curve.

According to their explanations what would the market be expecting with a steep

upward sloping yield curve?

3. What type of yield curve was present in March 1980? How does that curve

compare with the typical yield curve?

Bodie’Kane’Marcus: III. Debt Securities 9. Bond Prices and Yields © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

324 Part THREE Debt Securities

is just the average of the 8% earned this year and the 10% expected for next year. More pre-

cisely, the investment will grow by a factor of 1.08 in the first year and 1.10 in the second

year, for a total two-year growth factor of 1.08 1.10 1.188. This corresponds to an annual

growth rate of 8.995% (because 1.089952 1.188).

For investments in two-year bonds to be competitive with the strategy of rolling over one-

year bonds, these two-year bonds also must offer an average annual return of 8.995% over the

two-year holding period. This is illustrated in Figure 9.12. The current short-term rate of 8%

and the expected value of next year™s short-term rate are depicted above the time line. The

two-year rate that provides the same expected two-year total return is below the time line. In

this example, therefore, the yield curve will be upward sloping; while one-year bonds offer an

8% yield to maturity, two-year bonds offer an 8.995% yield.

This notion is the essence of the expectations hypothesis of the yield curve, which asserts

expectations

that the slope of the yield curve is attributable to expectations of changes in short-term rates.

hypothesis

Relatively high yields on long-term bonds are attributed to expectations of future increases in

The theory that yields

rates, while relatively low yields on long-term bonds (a downward-sloping or inverted yield

to maturity are

curve) are attributed to expectations of falling short-term rates.

determined solely by

expectations of One of the implications of the expectations hypothesis is that expected holding-period re-

future short-term turns on bonds of all maturities ought to be about equal. Even if the yield curve is upward

interest rates.

sloping (so that two-year bonds offer higher yields to maturity than one-year bonds), this does

not necessarily mean investors expect higher rates of return on the two-year bonds. As we™ve

seen, the higher initial yield to maturity on the two-year bond is necessary to compensate in-

vestors for the fact that interest rates the next year will be even higher. Over the two-year pe-

riod, and indeed over any holding period, this theory predicts that holding-period returns will

be equalized across bonds of all maturities.

Suppose we buy the one-year zero-coupon bond with a current yield to maturity of 8%. If its

face value is $1,000, its price will be $925.93, providing an 8% rate of return over the com-

9.10 EXAMPLE ing year. Suppose instead that we buy the two-year zero-coupon bond at its yield of 8.995%.

Its price today is $1,000/(1.08995)2 $841.76. After a year passes, the zero will have a re-

Holding-Period

maining maturity of only one year; based on the forecast that the one-year yield next year will

Returns

be 10%, it then will sell for $1,000/1.10 $909.09. The expected rate of return over the

year is thus ($909.09 $841.76)/$841.76 .08, or 8%, precisely the same return provided

by the one-year bond. This makes sense: If risk considerations are ignored when pricing the

two bonds, they ought to provide equal expected rates of return.

In fact, advocates of the expectations hypothesis commonly invert this analysis to infer the

market™s expectation of future short-term rates. They note that we do not directly observe the

expectation of next year™s rate, but we can observe yields on bonds of different maturities.

F I G U R E 9.12 2-year cumulative

expected returns

Returns to two 2-year

investment strategies

E(r2) 10 1.08 1.10 1.188

r1 8%

0 1 2

2-year investment,

1.089952

y2 8.995% 1.188

Bodie’Kane’Marcus: III. Debt Securities 9. Bond Prices and Yields © The McGraw’Hill

Essentials of Investments, Companies, 2003

Fifth Edition

325

9 Bond Prices and Yields

Suppose, as in this example, we see that one-year bonds offer yields of 8% and two-year

bonds offer yields of 8.995%. Each dollar invested in the two-year zero would grow after two

years to $1 1.089952 $1.188. A dollar invested in the one-year zero would grow by a fac-

tor of 1.08 in the first year and, then, if reinvested or “rolled over” into another one-year zero

in the second year, would grow by an additional factor of 1 r2. Final proceeds would be

$1 1.08 (1 r2).

The final proceeds of the rollover strategy depend on the interest rate that actually tran-

spires in year 2. However, we can solve for the second-year interest rate that makes the ex-

pected payoff of these two strategies equal. This “breakeven” value is called the forward rate forward rate

for the second year, f2, and is derived as follows: The inferred short-

term rate of interest

1.089952 1.08 (1 f2)

for a future period

that makes the

which implies that f2 .10, or 10%. Notice that the forward rate equals the market™s expecta-

expected total return

tion of the year 2 short rate. Hence, we conclude that when the expected total return of a long-

of a long-term bond

term bond equals that of a rolling over a short-term bond, the forward rate equals the expected equal to that of

short-term interest rate. This is why the theory is called the expectations hypothesis. rolling over short-

More generally, we obtain the forward rate by equating the return on an n-period zero- term bonds.

coupon bond with that of an (n 1)-period zero-coupon bond rolled over into a one-year

bond in year n:

yn)n yn 1)n 1

(1 (1 (1 fn)

The actual total returns on the two n-year strategies will be equal if the short-term interest rate

in year n turns out to equal fn.

Suppose that two-year maturity bonds offer yields to maturity of 6%, and three-year bonds

have yields of 7%. What is the forward rate for the third year? We could compare these two

EXAMPLE 9.11

strategies as follows:

Forward Rates

1. Buy a three-year bond. Total proceeds per dollar invested will be