52 Part One Value
bre44380_ch03_046-075.indd 52 09/30/15 12:47 PM
A change in interest rates has only a modest impact on the value of near-term cash flows
but a much greater impact on the value of distant cash flows. Thus the price of long-term
bonds is affected more by changing interest rates than the price of short-term bonds. For
example, compare the two curves in Figure 3.2. The brown line shows how the price of the
three-year 4.25% bond varies with the interest rate. The blue line shows how the price of a
30-year 4.25% bond varies. You can see that the 30-year bond is much more sensitive to inter-
est rate fluctuations than the three-year note.
Duration and Volatility
Changes in interest rates have a greater impact on the prices of long-term bonds than on
those of short-term bonds. But what do we mean by “long-term” and “short-term”? A coupon
bond that matures in year 30 makes payments in each of years 1 through 30. It’s misleading
to describe the bond as a 30-year bond; the average time to each cash payment is less than
30 years.
the months following the issue, the financial crisis reached its peak. Lehman Brothers filed for
bankruptcy with assets of $691 billion, and the government poured money into rescuing Fannie
Mae, Freddie Mac, AIG, and a host of banks. As investors rushed to the safety of Treasury
bonds, prices soared. By mid-December the price of the 4.375s of 2038 had reached 138.05%
of face value and the yield had fallen to 2.5%. Anyone fortunate enough to have bought the
bond at the issue price would have made a capital gain of $1,380.50 − $963.80 = $416.70. In
addition, on August 15 the bond made its first coupon payment of $21.875 (this is the semian-
nual payment on the 4.375% coupon bond with a face value of $1,000). Our lucky investor
would therefore have earned a seven-month rate of return of 45.5%:
Rate of return =
coupon income + price change
__________________________
investment
=
$21.875 + 416.70
_______________
$963.80
= .455, or 45.5%
Suddenly, government bonds did not seem quite so boring as before.
● ● ● ● ●
Table 3.2 calculates the prices of two seven-year bonds. We assume annual coupon payments
and a yield to maturity of 4% per year. Take a look at the time pattern of each bond’s cash
payments and review how the prices are calculated:
Which of these two bonds is the longer-term investment? They both have the same final
maturity, of course. But the timing of the bonds’ cash payments is not the same. In the case of
the 3s, the average time to each cash flow is longer, because a higher proportion of the cash
flows occurs at maturity, when the face value is paid off.
Suppose now that the yield to maturity on each bond falls to 3%. Which bond would you
most like to own? The 3s, of course. Since they have the longer effective maturity, they should
benefit most from a fall in yields. Table 3.3 confirms that this is indeed the case:
The 9s have the shorter average maturity, and therefore a shift in interest rates has a more
muted effect on the price. That much is clear. However, it would be useful to have a precise
EXAMPLE 3.2^ ●^ Which Is the Longer-Term Bond?
BEYOND THE PAGE
mhhe.com/brealey12e
Try It! Figure 3.2:
How changes in
interest rates
affect long- and
short-term bonds