The Foundation
5.5 Interest Rate Risk
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Based on the expected price appreciation for discount bonds shown in Figure 5.7 and Table 5.1, can we logically conclude that investors will prefer discount bonds to premium bonds? Ignoring differences in the taxation of long-term capital gains and ordinary income, investors would not have a preference. Recall that a bond sells at a discount from par value when the coupon rate is less than the bond’s required rate of return (bond pricing property
#3). Thus, discount bonds pay interest income that is less than what is required for the bond to sell at par value. In order for an investor to earn the required rate of return on a discount bond, the expected price appreciation must augment the income yield (the return earned from the interest payments) so that the investor can expect to earn the required rate of return.
The opposite occurs for a premium bond. These bonds pay more interest income than investors require. As a result, such bonds sell at a premium to par value. The decrease in price over the bond’s life will reduce the higher income yield offered on premium bonds such that an investor can expect to earn the required rate of return. The upshot of this argument is that both discount and premium bonds will offer investors an expected rate of return equal to the required rate of return.
Concept Check 5.4
1 Should bond prices rise or fall as the general level of interest rates in the economy rise?
2 If a bond has a coupon rate that exceeds its required rate of return, should the bond sell at a discount or a premium? Why?
3 What does the term positive convexity mean?
4 What happens to the price of a discount and a premium bond as it approaches maturity?
Practical Financial Tip 5.6
Bond price volatility depends on more than market yields. For bonds without embedded options, the speed or rate of change in bond prices is a function of a bond’s remaining term to maturity and coupon rate. All else equal, bonds with longer maturities and lower coupons have more price volatility than bonds with shorter maturities and higher coupons.
Example 5.9 Maturity and Interest Rate Risk
To illustrate how the remaining term to maturity affects a bond’s interest rate risk, consider two bonds, Bond S(hort) and Bond L(ong). Both have 8 percent coupon rates, pay interest semiannually, and have initial required returns of 8 percent. The bonds are identical except Bond S has 1 year and Bond L has 20 years remaining until maturity.
With initial required returns equal to their coupon rates, both bonds will sell for par value (bond pricing property #2). Now assume that interest rates in the economy rise immediately (no time elapses) and the required returns on both bonds increase to 10 percent. What are the new prices of the bonds?
Solution: Instead of using Equations 5.7 or 5.8 to answer this question, let’s use the BA II PLUS® financial calculator as follows:
Bond Inputs Output
S 2 N; 5 I/YR; 40 +/− PMT; FV +/− 1,000; CPT PV 981.41 L 40 N; 5 I/YR; 40 +/− PMT; FV +/− 1,000; CPT PV 828.41
The price of the long-term bond, Bond L, will decline to $828.41 while the price of Bond S will decline only to $981.41. On the other hand, if the required returns were to immediately decline to 6 percent, the prices of Bonds S and L would increase to
$1,019.13 and $1,231.15, respectively. Table 5.2 summarizes these results.
The price of the long-term bond (Bond L) experiences greater changes in price than the short-term bond (Bond S) as the required rate of return varies from 6 percent to 10 percent.
Table 5.2 Price changes for 8% bonds with different terms to maturity Required rate of return (yield) Bond Term to maturity
(years) 6% 8% 10%
S 1 $1,019.13 $1,000.00 $981.41
L 20 1,231.15 1,000.00 828.41
128 THEFOUNDATION
Example 5.10 Coupon Rate and Interest Rate Risk
To illustrate how the coupon rate affects a bond’s interest rate risk, consider two semiannual-pay bonds. Bond LC (low coupon) has a 6 percent coupon and Bond HC (high coupon) has a 10 percent coupon. Both bonds have 15 years to maturity and are currently selling to yield 8 percent. Table 5.3 summarizes the price changes for these two bonds.
Table 5.3 Price changes for 6% and 10% coupon bonds with 25 years to maturity currently priced to sell at 8%
Required rates of return (yield)
Bond Coupon rate (%) 6% 8% 10%
LC 6 $1,000.00 $785.18 $634.88
HC 10 $1,514.60 $1,214.82 $1,000.00
As Table 5.3 shows, if investors required an 8 percent yield, the price of the LC bond would be $785.18 and the price of the HC bond would be $1,214.82. Consider the following interest rate changes:
• If the yield required by investors increases by 2 percentage point to 10 percent, the price of the LC bond would fall to $634.88 and the price of the HC bond would fall to $1,000.00. These would represent percentage decreases of 19.1 percent for the LC bond and 17.7 percent for the HC bond.
• If required yields fall by 2 percentage points to 6 percent, the price of the LC bond would increase by 27.4 percent compared with a 24.7 percent increase for the HC bond.
Thus, the lower coupon bond exhibits greater price sensitivity to changes in interest rates than the higher coupon bond.
Bond investors pay close attention to the interest rate risk of their bond portfolios. When these investors expect increases in interest rates, they can reduce the interest rate risk of their portfolios by selling low-coupon, long-term bonds and buying higher-coupon bonds with shorter terms to maturity.
Concept Check 5.5
1 What does interest rate risk mean?
2 What relationship exists between changes in interest rates and bond prices?
3 How do the term to maturity and coupon rate affect the interest rate risk of a bond?
4 Two 30-year bonds are alike in all respects except Bond A’s coupon rate is 6 per- cent and Bond B’s coupon rate is 12 percent. Which of the two bonds will have the greater relative market price increase if interest rates decrease sharply? Why?