How bond prices change as interest rates change. Note that longer-term bonds are more sensitive to interest rate changes.
Proportion of
Total Value
Proportion of
Year
Ct
PV(Ct ) at 4.9%
[PV(Ct)/V]
Total Value Time
1
68.75
65.54
0.060
0.060
2
68.75
62.48
0.058
0.115
3
68.75
59.56
0.055
0.165
4
68.75
56.78
0.052
0.209
5
1068.75
841.39
0.775
3.875
V 1085.74
1.000
Duration 4.424 years
T A B L E 2 4 . 2
The first four columns show that the cash flow in year 5 accounts for only 77.5 percent of the present value of the
6 7/8s of 2006. The final column shows how to calculate a weighted average of the time to each cash flow. This average is the bond’s duration.
dard deviation of annual returns on a portfolio of long-term bonds was 9.4 percent compared with a standard deviation of 3.2 percent for bills.
Figure 24.4 illustrates why long-term bonds are more variable. Each line shows how the price of a 5-percent bond changes with the level of interest rates. You can see that the price of a longer-term bond is more sensitive to interest rate fluctuations than that of a shorter bond.
But what do we mean by long-term and short-term bonds? It is obvious in the case of strips that make payments in only one year. However, a coupon bond that matures in year 10 makes payments in each of years 1 through 10. Therefore, it is somewhat misleading to describe the bond as a 10-year bond; the average time to each cash flow is less than 10 years.
Consider the Treasury 6 7/8s of 2006. In mid-2001 these bonds had a present value of 108.57 percent of face value and yielded 4.9 percent. The third and fourth columns in Table 24.2 show where this present value comes from. Notice that the cash flow in year 5 accounts for only 77.5 percent of the bond’s value. The remaining 22.5 percent of the value comes from the earlier cash flows.
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Bond analysts often use the term duration to describe the average time to each payment. If we call the total value of the bond V, then duration is calculated as follows:15
The Treasury 4 5/8s of 2006 have the same maturity as the 6 7/8s, but the first four years’ coupon payments account for a smaller fraction of the bond’s value. In this sense the 4 5/8s are longer bonds than the 6 7/8s. The duration of the 4 5/8s is 4.574 years.
Consider now what happens to the prices of our two bonds as interest rates change:
6 7/8s of 2006
4 5/8s of 2006
New Price
Change
New Price
Change
Yield falls .5%
1108.96
2.14%
1009.91
2.21%
Yield rises .5%
1063.16
2.08%
966.81
2.15%
Difference
4.22%
4.36%
Thus, a 1 percentage-point variation in yield causes the price of the 6 7/8s to change by 4.22 percent. We can say that the 6 7/8s have a volatility of 4.22 percent, while the 4 5/8s have a volatility of 4.36 percent.
Notice that the 4 5/8 percent bonds have the greater volatility and that they also have the longer duration. In fact, a bond’s volatility is directly related to its duration:16
Volatility 1 percent 2 duration 1 yield
In the case of the 6 7/8s,
4.424 Volatility 1 percent 2 1.049 4.22
In Figure 24.4 we showed how bond prices vary with the level of interest rates. Each bond’s volatility is simply the slope of the line relating the bond price to the interest rate. You can see this more clearly in Figure 24.5, where the convex curve shows the price of the 5 percent 30-year bond for different interest rates. The bond’s volatility is measured by the slope of a tangent to this curve. For example, the dotted line in the figure shows that, if the interest rate is 5 percent, the curve has a slope of 15.4. At this point the change in bond price is 15.4 times a change in the interest rate. Notice that the bond’s volatility changes as the interest rate changes. Volatility is higher at lower interest rates (the curve is steeper), and it is lower at higher rates (the curve is flatter).
15This measure is also known as Macaulay duration after its inventor. See F. Macaulay, Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields, and Stock Prices in the United States since 1856, National Bureau of Economic Research, New York, 1938.
16For this reason volatility is also called modified duration.
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250
percent
230
210
190
170
price,
150
130
Bond
110
90
70
50
0
1
2
3
4
5
6
7
8
9
10
Interest rate, percent
F I G U R E 2 4 . 5
Volatility is the slope of the curve relating the bond price to the interest rate. For example, a 5 percent 30-year bond has a volatility of 15.4 when the interest rate is 5 percent. At this point the change in price is 15.4 times the change in the interest rate. Its volatility is higher at lower interest rates (the curve is steeper) and lower at higher rates (the curve is flatter).
Managing Interest Rate Risk
Volatility is a useful, summary measure of the likely effect of a change in interest rates on the value of a bond. The longer a bond’s duration, the greater is its volatility. In Chapter 27 we will make use of this relationship between duration and volatility to describe how firms can protect themselves against interest rate changes. Here is an example that should give you a flavor of things to come.
Suppose your firm has promised to make pension payments to retired employees. The discounted value of these pension payments is $1 million; therefore, the firm puts aside $1 million in the pension fund and invests the money in government bonds. So the firm has a liability of $1 million and (through the pension fund) an offsetting asset of $1 million. But, as interest rates fluctuate, the value of the pension liability will change and so will the value of the bonds in the pension fund. How can the firm ensure that the value of the bonds in the fund is always sufficient to meet the liabilities? Answer: By making sure that the duration of the bonds is always the same as the duration of the pension liability.
A Cautionary Note
Bond volatility measures the effect on bond prices of a shift in interest rates. For example, we calculated that the 6 7/8s had a volatility of 4.22. This means a 1 percentagepoint change in interest rates leads to a 4.22 percent change in bond price:
Change in bond price 4.22 change in interest rates
This relationship is sometimes called a one-factor model of bond returns; it tells us how each bond’s price changes in response to one factor—a change in the overall level of interest rates. One-factor models have proved very useful in helping firms to understand how they are affected by interest-rate changes and how they can protect themselves against these risks.
If the yields on all Treasury bonds moved in precise lockstep, then changes in the price of each bond would be exactly proportional to the bond’s duration. For example, the price of a long-term bond with a duration of 20 years would always rise or fall twice as much as the price of a medium-term bond with a duration of 10 years. However, Figure 24.6 illustrates that shortand long-term interest rates do
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F I G U R E 2 4 . 6
Short-term and long-term interest rates do not always move in parallel. Between September 1992 and April 2000 short-term rates rose sharply while long-term rates declined.
7.5
percent
7
6.5
September 1992
6
5.5
April 2000
Yield,
5
4.5
4
3.5
2 3
5
7
10
30
Bond maturity, years
not always move in perfect unison. Between 1992 and 2000 short-term interest rates nearly doubled while long-term rates declined. As a result, the term structure, which initially sloped steeply upward, shifted to a downward slope. Because shortand long-term yields do not move in parallel, one-factor models cannot be the whole story, and managers need to worry not just about the risks of an overall change in interest rates but also about shifts in the term structure.
24.4 EXPLAINING THE TERM STRUCTURE
The term structure that we showed in Figure 24.3 was upward-sloping. In other words, long rates of interest are higher than short rates. This is the more common pattern but sometimes it is the other way around, with short rates higher than long rates. Why do we get these shifts in term structure?
Let us look at a simple example. Figure 24.3 showed that in the summer of 2001 the one-year spot rate 1r1 2 was about 3.5 percent. The two-year spot rate 1 r2 2 was higher at 4 percent. Suppose that in 2001 you invest in a one-year U.S. Treasury strip. You would earn the one-year spot rate of interest and by the end of the year each dollar that you invested would have grown to $1 1 r1 2 $1.035. If instead you were prepared to invest for two years, you would earn the two-year spot rate of r2 and by the end of the two years each dollar would have grown to $1 1 r2 2 2 $1.042 $1.0816. By keeping your money invested for a further year, your savings grow from $1.0350 to $1.0816, an increase of 4.5 percent. This extra 4.5 percent that you earn by keeping your money invested for two years rather than one is termed the forward interest rate or f2 .
Notice how we calculated the forward rate. When you invest for one year, each dollar grows to $1 1 r1 2 . When you invest for two years, each dollar grows to
$1 1 r2 2 2. Therefore,
the extra return that you earn for that second year is
If you twist this equation around, you obtain an expression for the two-year spot rate, r2, in terms of the one-year spot rate, r1, and the forward rate, f2 :
1 1 r2 2 2 1 1 r1 2 1 1 f2 2
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(a ) The future value of $1 invested in a two-year loan
Period 0 Period 2 (1 + r2)2 = (1 + r1) (1 + f2)
(b ) The future value of $1 invested in two successive one-year loans
Period 0
Period 1
Period 2
(1 + r1)
(1 + 1r2)
F I G U R E 2 4 . 7
An investor can invest either in a two-year loan [a] or in two successive one-year loans [b]. The expectations theory says that in equilibrium the expected payoffs from these two strategies must be equal. In other words, the forward rate, f2, must equal the expected spot rate, 1r2.
In other words, you can think of the two-year investment as earning the one-year spot rate for the first year and the extra return, or forward rate, for the second year.
The Expectations Theory
Would you have been happy in the summer of 2001 to earn an extra 4.5 percent for investing for two years rather than one? The answer depends on how you expected interest rates to change over the coming year. Suppose, for example, that you were confident that interest rates would rise sharply, so that at the end of the year the one-year rate would be 5 percent. In that case rather than investing in a two-year bond and earning the extra 4.5 percent for the second year, you would do better to invest in a one-year bond and, when that matured, to reinvest the money for a further year at 5 percent. If other investors shared your view, no one would be prepared to hold the two-year bond and its price would fall. It would stop falling only when the extra return from holding the two-year bond equalled the expected future one-year rate. Let us call this expected rate 1r2—that is, the spot rate of interest at year 1 on a loan maturing at the end of year 2.17 Figure 24.7 shows that at that point investors would earn the same expected return from investing in a two-year loan as from investing in two successive one-year loans.
This is known as the expectations theory of term structure.18 It states that in equilibrium the forward interest rate, f2, must equal the expected one-year spot rate, 1r2. The expectations theory implies that the only reason for an upward-sloping term structure, such as existed in the summer of 2001, is that investors expect short-term interest rates to rise; the only reason for a declining term structure is that investors expect short-term rates to fall.19 The expectations theory also implies that investing in a succession of short-term bonds gives exactly the same expected return as investing in long-term bonds.
If short-term interest rates are significantly lower than long-term rates, it is often tempting to borrow short-term rather than long-term. The expectations theory
17Be careful to distinguish 1r2 from r2 , the spot interest rate on a two-year bond held from time 0 to time 2. The quantity 1r2 is a one-year spot rate established at time 1.
18The expectations theory is usually attributed to Lutz and Lutz. See F. A. Lutz and V. C. Lutz, The Theory of Investment in the Firm, Princeton University Press, Princeton, NJ, 1951.
19This follows from our example. If the two-year spot rate, r2, exceeds the one-year rate, r1, then the forward rate, f2, also exceeds r1. If the forward rate equals the expected spot rate, 1r2 then 1r2 must also exceed r1. The converse is likewise true.
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implies that such naïve strategies won’t work. If short rates are lower than long rates, then investors must be expecting interest rates to rise. When the term structure is upward-sloping, you are likely to make money by borrowing short only if investors are overestimating future increases in interest rates.
Even on a casual glance the expectations theory does not seem to be the complete explanation of term structure. For example, if we look back over the period 1926–2000, we find that the return on long-term U.S. Treasury bonds was on average 1.9 percent higher than the return on short-term Treasury bills. Perhaps shortterm interest rates did not go up as much as investors expected, but it seems more likely that investors wanted a higher expected return for holding long bonds and that on the average they got it. If so, the expectations theory is wrong.
The expectations theory has few strict adherents, but most economists believe that expectations about future interest rates have an important effect on term structure. For example, the expectations theory implies that if the forward rate of interest is 1 percent above the spot rate of interest, then your best estimate is that the spot rate of interest will rise by 1 percent. In a study of the U.S. Treasury bill market between 1959 and 1982, Eugene Fama found that a forward premium does on average precede a rise in the spot rate but the rise is less than the expectations theory would predict.20
The Liquidity-Preference Theory
What does the expectations theory leave out? The most obvious answer is “risk.” If you are confident about the future level of interest rates, you will simply choose the strategy that offers the highest return. But, if you are not sure of your forecast, you may well opt for the less risky strategy even if it offers a lower expected return.
Remember that the prices of long-duration bonds are more volatile than those of short-term bonds. For some investors this extra volatility may not be a concern. For example, pension funds and life insurance companies with long-term liabilities may prefer to lock in future returns by investing in long-term bonds. However, the volatility of long-term bonds does create extra risk for investors who do not have such long-term fixed obligations.
Here we have the basis for the liquidity-preferencetheory of the term structure.21 If investors incur extra risk from holding long-term bonds, they will demand the compensation of a higher expected return. In this case the forward rate must be higher than the expected spot rate. This difference between the forward rate and the expected spot rate is usually called the liquidity premium. If the liquidity-preference theory is right, the term structure should be upward-sloping more often than not. Of course, if future spot rates are expected to fall, the term structure could be downward-sloping and still reward investors for lending long. But the liquidity-preference theory would predict a less dramatic downward slope than the expectations theory.
20See E. F. Fama, “The Information in the Term Structure,” Journal of Financial Economics 13 (December 1984), pp. 509–528. Evidence from the Treasury bond market that the forward premium has some power to predict changes in spot rates is provided in J. Y. Campbell, A. W. Lo, and A. C. MacKinlay, The Econometrics of Financial Markets, Princeton University Press, Princeton, NJ, 1997, pp. 421–422.
21The liquidity-preference hypothesis is usually attributed to Hicks. See J. R. Hicks, Value and Capital: An Inquiry into Some Fundamental Principles of Economic Theory, 2nd ed., Oxford University Press, Oxford, 1946. For a theoretical development, see R. Roll, The Behavior of Interest Rates: An Application of the Efficient-Market Model to U.S. Treasury Bills, Basic Books, Inc., New York, 1970.
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Introducing Inflation
The money cash flows on a U.S. Treasury bond are certain, but the real cash flows are not. In other words, Treasury bonds are still subject to inflation risk. Let us look therefore at how uncertainty about inflation affects the risk of bonds with different maturities.22
Suppose that Irving Fisher is right and short rates of interest always incorporate fully the market’s latest views about inflation. Suppose also that the market learns more as time passes about the likely inflation rate in a particular year. Perhaps today investors have only a very hazy idea about inflation in year 2, but in a year’s time they expect to be able to make a much better prediction. Since investors expect to learn a good deal about the inflation rate in year 2 from experience in year 1, next year they will be in a much better position to judge the appropriate interest rate in year 2.
You are saving for your retirement. Which of the following strategies is the more risky? Invest in a succession of one-year Treasury bonds or invest in a 20-year bond?
If you buy the 20-year bond, you know what money you will have at the end of 20 years, but you will be making a long-term bet on inflation. Inflation may seem benign now, but who knows what it will be in 10 or 20 years? This uncertainty about inflation makes it more risky for you to fix today the rates at which you will lend in the distant future.
You can reduce this uncertainty by investing in successive short-term bonds. You do not know the interest rate at which you will be able to reinvest your money at the end of each year, but at least you know that it will incorporate the latest information about inflation in the coming year. So, if the prospects for inflation deteriorate, it is likely that you will be able to reinvest your money at a higher interest rate.
Inflation uncertainty may help to explain why long-term bonds provide a liquidity premium. If inflation creates additional risks for long-term lenders, borrowers must offer some incentive if they want investors to lend long. Therefore, the forward rate of interest f2 must be greater than the expected spot rate E1 1r2 2 by an amount that compensates investors for the extra risk of inflation.
Relationships between Bond Returns
These term structure theories tell us how bond prices may be determined at a point in time. More recently, financial economists have proposed some important theories of how price movements are related. These theories take advantage of the fact that the returns on bonds with different maturities tend to move together. For example, if short-term interest rates are high, it is a good bet that long-term rates will also be high. If short-term rates fall, long-term rates usually keep them company. Such linkages between interest rate movements can tell us something about relationships between bond prices.
The models that bond traders use to exploit these relationships can be quite complex and we can’t get deeply into the subject here. However, the following example will give you a flavor of how the models work.
Suppose that you can invest in three possible government loans: a threemonth Treasury bill, a medium-term bond, and a long-term bond. The return on
22See R. A. Brealey and S. M. Schaefer, “Term Structure and Uncertain Inflation,” Journal of Finance 32 (May 1977), pp. 277–290.
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T A B L E 2 4 . 3
Illustrative payoffs from three government securities. Note the wider range of outcomes from the longer-duration loans. We don’t know what the medium-term bond sells for; we need to figure it out from how its value changes when interest rates rise or fall.
Change in Value
Beginning
If Interest
If Interest
Ending
Price
Rates Rise
Rates Fall
Value
Treasury bill
98
2
2
100
Medium-term bond
?
6.5
10
?
Long-term bond
105
15
18
90 or 123
the Treasury bill over the next three months is certain; we will assume it yields a 2 percent quarterly rate. The return on each of the other bonds depends on what happens to interest rates. Suppose that you foresee only two possible outcomes— a sharp rise in interest rates or a sharp fall. Table 24.3 summarizes how the prices of the three investments would be affected. Notice that the long-term bond has a longer duration and therefore a wider range of possible outcomes.
Here’s the puzzle. You know the price of the Treasury bill and the long-term bond. But can you get rid of the two question marks in Table 24.3 and figure out what the medium-term bond should sell for?
Suppose that you start with $100. You invest half of this money in the Treasury bill and half in the long-term bond. In this case the change in the value of your portfolio will be 1 .5 2 2 3 .5 1 15 2 4 $6.5 if interest rates rise and 1 .5 2 2 1 .5 18 2 $10 if interest rates fall. Thus, regardless of whether interest rates rise or fall, your portfolio will provide exactly the same payoffs as an investment in the medium-term bond. Since the two investments provide identical payoffs, they must sell for the same price or there will be a money machine. So, the value of the medium-term bond must be halfway between the value of a three-month bill and that of the long-term bond, that is, 1 98 105 2 /2 101.5. Knowing this, you can calculate what the yield to maturity on the medium-term bond has to be. You can also calculate its value next year, either 101.5 6.5 95 or 101.5 10 111.5.
Everything now checks; regardless of whether interest rates rise or fall, the medium-term bond will provide the same payoff as the package of Treasury bill and long-term bond and therefore it must cost the same:
Ending Value
Initial
If Interest
If Interest
Outlay
Rates Rise
Rates Fall
Equal holdings
(.5 98)
(.5
(.5 100) (.5
(.5 100) (.5
of Treasury bill
105)
101.5
90) 95
123) 111.5
& long-term
bond
Medium-term
101.5
101.5 6.5 95
101.5 10
bond
111.5
Our example is grossly oversimplified, but you have probably already noticed that the basic idea is the same that we used when valuing an option. To value an option on a share, we constructed a portfolio of a risk-free loan and the common stock that would exactly replicate the payoffs from the option. That allowed us to
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price the option given the price of the risk-free loan and the share. Here we value a bond by constructing a portfolio of two or more other bonds that will provide exactly the same payoffs.23 That allows us to value one bond given the prices of the other bonds.
Our example carries three messages. First, bond traders focus on changes in bond prices and on how the changes for different bonds are linked. Second, changes in bond prices can be related to a small number of factors (in our example, the change in the overall level of interest rates completely defined the change in the price of each bond). Third, once the linkages between bond prices can be pinned down, then each bond can be priced relative to a package of other bonds.
24.5 ALLOWING FOR THE RISK OF DEFAULT
You should by now be familiar with some of the basic ideas about why interest rates change and why short rates may differ from long rates. It only remains to consider our third question: Why do some borrowers have to pay a higher rate of interest than others?
The answer is obvious: Bond prices go down, and interest rates go up, when the probability of default increases. But when we say “interest rates go up,” we mean promised interest rates. If the borrower defaults, the actual interest rate paid to the lender is less than the promised rate. The expected interest rate may go up with increasing probability of default, but this is not a logical necessity.
These points can be illustrated by a simple numerical example. Suppose that the interest rate on one-year, risk-freebonds is 9 percent. Backwoods Chemical Company has issued 9 percent notes with face values of $1,000, maturing in one year. What will the Backwoods notes sell for?
The answer is easy—if the notes are risk-free, just discount principal ($1,000) and interest ($90) at 9 percent:
PV of notes $1,000 90 $1, 000 1.09
Suppose instead that there is a 20 percent chance that Backwoods will default and that, if default does occur, holders of its notes receive nothing. In this case, the possible payoffs to the noteholders are
Payoff
Probability
Full payment
$1,090
.8
No payment
0
.2
The expected payment is .81 $1, 090 2
.21 $0 2
$872.
We can value the Backwoods notes like any other risky asset, by discounting their expected payoff ($872) at the appropriate opportunity cost of capital. We might discount at the risk-free interest rate (9 percent) if Backwoods’s possible default is
23Two early examples of models that use no-arbitrage conditions to model the term structure are O. Vasicek, “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics 5 (November 1977), pp. 177–188; and J. C. Cox, J. E. Ingersoll, and S. A. Ross, “A Theory of the Term Structure of Interest Rates,” Econometrica 53 (May 1985), pp. 385–407.
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T A B L E 2 4 . 4
Key to Moody’s and Standard and Poor’s bond ratings. The highest quality bonds are rated triple-A. Then come double-A bonds, and so on. Investmentgrade bonds have to be Baa or higher. Bonds that don’t make this cut are called junk bonds.
Moody’s Ratings
Standard and Poor’s Ratings
Investment-grade:
Aaa
AAA
Aa
AA
A
A
Baa
BBB
Junk bonds:
Ba
BB
B
B
Caa
CCC
Ca
CC
C
C
totally unrelated to other events in the economy. In this case the default risk is wholly diversifiable, and the beta of the notes is zero. The notes would sell for
PV of notes $1.09872 $800
An investor who purchased these notes for $800 would receive a promised yield of about 36 percent:
$1, 090
Promised yield $800 1 .363
That is, an investor who purchased the notes for $800 would earn a 36.3 percent rate of return if Backwoods does not default. Bond traders therefore might say that the Backwoods notes “yield 36 percent.” But the smart investor would realize that the notes’ expected yield is only 9 percent, the same as on risk-free bonds.
This of course assumes that risk of default with these notes is wholly diversifiable, so that they have no market risk. In general, risky bonds do have market risk (that is, positive betas) because default is more likely to occur in recessions when all businesses are doing poorly. Suppose that investors demand a 2 percent risk premium and an 11 percent expected rate of return. Then the Backwoods notes will sell for 872/1.11 $785.59 and offer a promised yield of 1 1, 090/785.59 2 1 .388, or about 39 percent.
You rarely see traded bonds offering 39 percent yields, although we will soon encounter an example of one company’s bonds that had a promised yield of 50 percent.
Bond Ratings
The relative quality of most traded bonds can be judged from bond ratings given by Moody’s and Standard and Poor’s. Table 24.4 summarizes these ratings. For example, the highest quality bonds are rated triple-A (Aaa) by Moody’s, then come double-A (Aa) bonds, and so on. Bonds rated Baa or above are known as investment-gradebonds. Commercial banks, many pension funds, and other financial institutions are not allowed to invest in bonds unless they are investment-grade.24
24Investment-grade bonds can usually be entered at face value on the books of banks and life insurance companies.
Period 2 (1 + 
