The discussion in the body of the chapter explained how increases in interest rates cause the prices of existing bonds to fall. Declining interest rates have the opposite effect; bond prices rise. For bonds with the same coupon, the amount of the price change varies with the term of the bond. The longer the term, the greater is the price fluctuation. The dis- cussion also indicated that longer duration is associated with greater price fluctuations.
Since both discussions were concerned with price fluctuations, both were also concerned with risk.
The two concepts, however, are different. Consider the two bonds in Exhibit 14A.1 and illustrated in Figure 14A.1. Both bonds have an 8 percent coupon rate of interest;
the difference between the bonds is the term, 10 years and 20 years. The exhibit and figure give the premium or discount and the duration for each bond at different inter- est rates. At a current rate of 3 percent, both bonds sell for a premium, but the 20-year bond sells for a larger premium. As the current interest rate rises, the prices of both
exhibit 14A.1
Premiums or (Discounts) and the Durations for 8 Percent Bonds with Terms to Maturity of 10 and 20 Years at Different Rates of Interest
10-Year Bond 20-Year Bond
Interest Rate Premium/(Discount) Duration Premium/(Discount) Duration
3% $429 7.60 $748 12.65
4 327 7.50 547 12.18
5 234 7.39 377 11.71
6 149 7.29 231 11.23
7 71 7.18 107 10.76
8 0 7.07 0 10.29
9 (65) 6.95 (92) 9.83
10 (125) 6.84 (172) 9.39
11 (179) 6.73 (241) 8.95
12 (229) 6.61 (301) 8.53
13 (275) 6.49 (354) 8.13
14 (318) 6.37 (400) 7.75
. . . . .
. . . . .
. . . . .
20 (511) 5.65 (587) 5.85
bonds fall. They sell for their face value when the coupon and the current rate of inter- est are equal at 8 percent. As the market rate of interest continues to rise, both bonds sell for a discount. Both the discount and the premium are larger for the bond with the longer term, but the difference between the discounts diminishes as the interest rate continues to rise. Since the maximum possible discount is the face amount of the bond ($1,000), the discount for the bond with the shorter term approaches the discount for the bond with the longer term.
figure 14A.1
Premiums, Discounts, and Durations for 8 Percent Coupon, 10-Year and 20-Year Bonds at Different Rates of Interest
15
10
5
14
10 12
8 4
2 6
0
Duration 20-Year Bond Duration 10-Year Bond Discount 10-Year Bond Discount 20-Year Bond Premium 20-Year Bond
Premium 10-Year Bond
Interest Rates (%) Interest Rates (%)
$800 600 400 200 0 2200 2400 2600
14
10 12
8 4
2 6
0
Years
Source: © Cengage Learning
6.37 years as interest rates rise from 3 to 14 percent). The decline in the duration of the 20-year bond is much larger (e.g., from 12.65 to 7.75 years as interest rates rise from 3 to 14 percent). The duration of the 20-year bond approaches the duration of the bond with the smaller term. At 21.5 percent the durations are equal, and for higher interest rates, the duration of the 20-year bond is actually less than the duration of the 10-year bond.
Exhibit 14A.2 and Figure 14A.2 present the premiums and discounts and the dura- tions for bonds with different coupons and 10- and 20-year terms to maturity when the current rate of interest is 8 percent. Exhibit 14A.1 held constant the bond’s coupon and varied the rate of interest; Exhibit 14A.2 holds constant the rate of interest and varies the coupon. The bonds with the lower coupons sell for a larger discount, and the lon- ger the term, the greater is the discount. The zero coupon 20-year bond’s price is $208 (a discount of $792), while the 10-year bond would sell for $446 and a $554 discount.
As the coupon increases, the discount decreases. When the coupon exceeds the current rate of interest, the bond sells for a premium, and the longer the term of the bond, the larger is the premium. For example, the 10- and 20-year bonds with the 12 percent coupons sell for premiums of $272 and $396, respectively.
The coupon also affects each bond’s duration. Higher coupons imply that cash is received faster, so the duration declines as the coupon increases. This relationship is also verified in Exhibit 14A.2. While the durations are equal to the term of the bond for the zero coupon bonds, the durations diminish as the coupons increase. As is also indicated
exhibit 14A.2
Premiums or (Discounts) and the Durations When Interest Rates are 8 Percent for Bonds with Terms to Maturity of 10 and 20 Years at Different Coupons
10-Year Bond 20-Year Bond
Interest Rate Premium/(Discount) Duration Premium/(Discount) Duration
0% ($554) 10.00 ($792) 20.00
2 (407) 8.76 (594) 14.03
3 (340) 8.33 (495) 12.79
4 (272) 7.99 (396) 12.00
5 (204) 7.70 (297) 11.38
6 (136) 7.45 (198) 10.92
7 (68) 7.25 (99) 10.57
8 0 7.07 0 10.29
9 68 6.91 99 10.06
10 136 6.78 198 9.87
11 204 6.65 297 9.71
12 272 6.61 396 9.57
13 340 6.44 495 9.45
14 407 6.35 594 9.34
. . . . .
. . . . .
. . . . .
20 815 5.96 1188 8.91
in the exhibit, the numerical value of the 20-year bonds’ duration declines more rapidly than the duration of the 10-year bonds.
Notice that in both exhibits as the numerical value of the left-hand column increases, that is, the interest rate or the coupon increases, the duration diminishes. Higher interest rates (i.e., lower bond prices and larger discounts) and higher coupons produce smaller durations. The investor is receiving more cash flow earlier, so the duration is smaller. This relation is different than the relationship between the values in the left-hand column and the premium/discount column. Higher interest rates and lower coupons produce larger discounts, while smaller interest rates and larger coupons generate larger premiums.
figure 14A.2
Discounts, Premiums, and Durations for 10- and 20-Year Bonds with Different Coupons (Interest Rate 5 8 Percent)
20
15
10
14
10 12
8 4
2 6
0
Duration 20-Year Bond Duration 10-Year Bond Premium 20-Year Bond Premium 10-Year Bond
Discount 10-Year Bond Discount 20-Year Bond
Coupon Rates (%) Coupon Rates (%)
$800 600 400 200 0
2200 2400 2600
14
10 12
8 4
2 6
0
Years
Appendix 14B