1. CALC: N = 2 X 2 = 4 R = 9% / 2 = 4.5% PV = ? PMT = 7.25% X 1000 / 2 = $36.25

QUESTION

B15. (Interest-rate risk) A quick look at bond quotes will tell you that GMAC has many different issues of bonds outstanding. Suppose that four of them have identical coupon rates of 7.25% but mature on four different dates. One matures in 2 years, one in 5 years, one in 10 years, and the last in 20 years. Assume that they all made coupon payments yesterday. a. If the yield curve were flat and all four bonds had the same yield to maturity of 9%, what would be the fair price of each bond today? b. Suppose that during the first hour of operation of the capital markets today, the term structure shifts and the yield to maturity of all these bonds changes to 10%. What is the fair price of each bond now? c. Suppose that in the second hour of trading, the yield to maturity of all these bonds changes once more to 8%. Now what is the fair price of each bond? d. Based on the price changes in response to the changes in yield to maturity, how is interest-rate risk a function of the bond’s maturity? That is, is interest-rate risk the same for all four bonds, or does it depend on the bond’s maturity?

SOLUTION

1. CALC: n = 2 x 2 = 4 r = 9% / 2 = 4.5% PV = ? PMT = 7.25% x 1000 / 2 = $36.25

FV = $1000 PV = -$968.61

2. CALC: n = 5 x 2 = 10 r = 9% / 2 = 4.5% PV = ? PMT = 7.25% x 1000 / 2 = $36.25

FV = $1000 PV = -$930.76

3. CALC: n = 10 x 2 = 20 r = 9% / 2 = 4.5% PV = ? PMT = 7.25% x 1000 / 2 = $36.25

FV = $1000 PV = -$886.18

4. CALC: n = 20 x 2 = 40 r = 9% / 2 = 4.5% PV = ? PMT = 7.25% x 1000 / 2 = $36.25

FV = $1000 PV = -$838.99

1. CALC: n = 2 x 2 = 4 r = 10% / 2 = 5% PV = ? PMT = 7.25% x 1000 / 2 = $36.25

FV = $1000 PV = -$951.24

2. CALC: n = 5 x 2 = 10 r = 10% / 2 = 5% PV = ? PMT = 7.25% x 1000 / 2 = $36.25

FV = $1000 PV = -$893.83

3. CALC: n = 10 x 2 = 20 r = 10% / 2 = 5% PV = ? PMT = 7.25% x 1000 / 2 = $36.25

FV = $1000 PV = -$828.64

4. CALC: n = 20 x 2 = 40 r = 10% / 2 = 5% PV = ? PMT = 7.25% x 1000 / 2 = $36.25

FV = $1000 PV = -$764.06

1. CALC: n = 2 x 2 = 4 r = 8% / 2 = 4% PV = ? PMT = 7.25% x 1000 / 2 = $36.25

FV = $1000 PV = -$986.39

2. CALC: n = 5 x 2 = 10 r = 8% / 2 = 4% PV = ? PMT = 7.25% x 1000 / 2 = $36.25

FV = $1000 PV = -$969.58

3. CALC: n = 10 x 2 = 20 r = 8% / 2 = 4% PV = ? PMT = 7.25% x 1000 / 2 = $36.25

FV = $1000 PV = -$949.04

4. CALC: n = 20 x 2 = 40 r = 8% / 2 = 4% PV = ? PMT = 7.25% x 1000 / 2 = $36.25

FV = $1000 PV = -$925.78

d. Interest rate risk varies directly with maturity. The longer maturity bonds have higher price changes when interest rates change.

QUESTION

B17. (Default risk) You buy a very risky bond that promises an 8.8% coupon and return of the $1,000 principal in 10 years. You pay only $500 for the bond. a.You receive the coupon payments for two years and the bond defaults. After liquidating the firm, the bondholders receive a distribution of $150 per bond at the end of 2.5 years. What is the realized return on your investment? b. The firm does far better than expected and bondholders receive all of the promised interest and principal payments. What is the realized return on your investment?

SOLUTION

a. CALC: n = 2.5x2 =5 r = ? PV = -$500 PMT = 8.8%x1,000/2 = $44 FV = $150 - 44 = $106 r = -10.7200%

APY = (1 + r)m – 1 = (1 - 0.1072)2 – 1 = -20.2908%

YTM = 2x10.7200 = 21.4400% APR

b. CALC: n = 10 x 2 =20 r = ? PV = -$500 PMT = 8.8% x 1000 / 2 = $44 FV = $1,000 r = 10.4574%

APY = (1 + r)m – 1 = (1 + 0.104574)2 – 1 = 22.0083%

YTM = 2x10.4574 = 20.9147% APR