Chapter 7 Bonds and Their Valuation 211Interest rate risk is higher on bonds that have long maturities than on bonds
that will mature in the near future.^14 This follows because the longer the maturity,
the longer before the bond will be paid off and the bondholder can replace it with
another bond with a higher coupon. This point can be demonstrated by showing
how the value of a 1-year bond with a 10% annual coupon " uctuates with changes
in rd and then comparing those changes with changes on a 15-year bond. The
1-year bond’s values at different interest rates are shown here:
Value of a 1-year bond at
N I/YR PV PMT FV5 100 1000–1,047.621Output (Bond Value):rd = 5%: Inputs:
N I/YR PV PMT FV10 100 1000–1,000.001Output (Bond Value):rd = 10%: Inputs:
N I/YR PV PMT FV15 100 1000–956.521Output (Bond Value):rd = 15%: Inputs:
You would obtain the! rst value with a! nancial calculator by entering N! 1,
I/YR! 5, PMT! 100, and FV! 1000 and then pressing PV to get $1,047.62. With
everything still in your calculator, enter I/YR! 10 to override the old I/YR! 5
and press PV to! nd the bond’s value at a 10% rate; it drops to $1,000. Then enter
I/YR! 15 and press the PV key to! nd the last bond value, $956.52.
The effects of increasing rates on the 15-year bond as found earlier can be com-
pared with the just-calculated effects for the 1-year bond. This comparison is
shown in Figure 7-3, where we show bond prices at several rates and then plot
those prices on the graph. Compared to the 1-year bond, the 15-year bond is far
more sensitive to changes in rates. At a 10% interest rate, both the 15-year and
1-year bonds are valued at $1,000. When rates rise to 15%, the 15-year bond falls to
$707.63, but the 1-year bond falls only to $956.52. The price decline for the 1-year
bond is only 4.35%, while that for the 15-year bond is 29.24%.
For bonds with similar coupons, this differential interest rate sensitivity always holds
true—the longer its maturity, the more its price changes in response to a given change in
interest rates. Thus, even if the risk of default on two bonds is exactly the same, the
(^14) Actually, a bond’s maturity and coupon rate both a! ect interest rate risk. Low coupons mean that most of the
bond’s return will come from repayment of principal, whereas on a high-coupon bond with the same maturity,
more of the cash # ows will come in during the early years due to the relatively large coupon payments. A
measurement called duration, which " nds the average number of years the bond’s PV of cash # ows remain
outstanding, has been developed to combine maturity and coupons. A zero coupon bond, which has no interest
payments and whose payments all come at maturity, has a duration equal to its maturity. All coupon bonds have
durations that are shorter than their maturity; and the higher the coupon rate, the shorter the duration. Bonds
with longer duration are exposed to more interest rate risk. A discussion of duration would go beyond the scope
of this book, but see any investments text for a discussion of the concept.