Fixed Income Securities: Valuation, Risk, And Risk Management, 1st Edition Solution Manual

Solutions to Chapter 3 Exercise 1. a. 3; equal to the maturity of the zero bond b. 2.9542; the duration of the coupon bond is the weighted average of the coupon payment times c. 0.9850 d. 0.5; equal to the time left to the next coupon payment e. 0.5111; obtain the price PF R of the floating rate bond (see Chapter 2, equation (2.39)). In analogy to a coupon bond, the duration is computed as: 0.5s × 100 × 0.5 + DF R = PF R P3 t=0.5 Z(0, t) × t PF R . (14) f. 0.2855; proceed as in point e. above but recognize that the valuation is outside the reset date. Assume that the coupon applying to the next reset date has been set at r2 (0) = 6.4%. Exercise 2. Portfolio A: Security Duration, D Weight, w D×w 4.5yr @ 5% semi 3.8660 40% 1.55 7yr @ 2.5% semi 6.4049 25% 1.60 1.75 fl + 30bps semi 0.2540 20% 0.05 1yr zero 1.0000 10% 0.10 2yr @ 3% quart 1.9530 5% 0.10 Port. D 3.40 Portfolio B: Security Duration, D Weight, w D×w 7yr @ 10% semi 5.4262 40% 2.17 4.25yr @ 3% quart 3.9838 25% 1.00 90 day zero 0.2500 20% 0.05 2yr fl semi 0.5000 10% 0.05 1.5yr @ 6% semi 1.4564 5% 0.07 Port. D 3.34 The investors would select the shorter duration portfolio B. 6 Exercise 3. Obtain yield to maturity y for each security. Compute modified and Macaulay duration accodring to equation (3.19) and (3.20) in the book. Yield Duration Modified Macaulay a. 6.95% 3 3 2.8993 b. 6.28% 2.9542 2.9974 2.9061 c. 6.66% 0.9850 0.9850 0.9689 d. 0.00% 0.5 0.5 0.5 e. 6.82% 0.5111 0.5111 0.4943 f. 6.76% 0.2855 0.2855 0.2761 Exercise 4. Compute the duration of each asset and use the fact that the dollar duration is the bond price times its duration. Price Duration $ Duration a. $89.56 4.55 $407.88 b. $67.63 -7.00 ($473.39) c. $79.46 3.50 $277.74 d. $100.00 0.5 $50.00 e. $100.00 -0.25 ($25.00) f. $102.70 -0.2763 ($28.38) Exercise 5. a. Compute number of units of each security (N ) in the portfolio and apply Fact 3.5. Portfolio A: Security Price Duration Weight N D×P ×N 4.5yr @ 5% semi 94.03 3.8660 40% 0.43 154.64 7yr @ 2.5% semi 81.56 6.4049 25% 0.307 160.12 1.75 fl + 30bps semi 102.09 0.2540 20% 0.20 5.08 1yr zero 93.61 1.0000 10% 0.11 10.00 2yr @ 3% quart 92.54 1.9530 5% 0.05 9.76 $D 339.61 7 Portfolio B : Security Price Duration Weight N D×P ×N 7yr @ 10% semi 123.36 5.4262 40% 0.32 217.05 4.25yr @ 3% quart 86.83 3.9838 25% 0.29 99.59 90 day zero 98.45 0.2500 20% 0.20 5.00 2yr fl semi 100.00 0.5000 10% 0.10 5.00 1.5yr @ 6% semi 98.83 1.4564 5% 0.05 7.28 $D 333.92 b. For 1 bps increase, we have (see Definition 3.5): Portfolio A: 339.61 × 0.01/100 = −$0.0340 Portfolio B: 333.92 × 0.01/100 = −$0.0334 c. Yes. Exercise 6. After the reshuffling of the portfolio, its value becomes $50 mn. a. Short -0.307 units of long bond in portfolio A, and -0.081 units in portfolio B. b. New dollar durations are: 19.36 and 62.61 for portfolio A and B, respectively. c. The conclusion reverses. N LT bond New weight LT bond New $D Port. A -0.307 -25% 19.36 Port. B -0.081 -10% 62.61 Exercise 7. a. $10 mn b. Compute the dollar duration of the cash flows in each bond, and then the dollar duration of the portfolio: Security Position $ (mn) Price N $D $D × N 6yr IF @ 20% – fl quart Long 20.00 146.48 0.137 1,140.28 155.69 4yr fl 45bps semi Long 20.00 101.62 0.197 53.54 10.54 5yr zero Short (30.00) 76.41 -0.393 382.052 -150.00 Port. value $10.00 mn Port. $D 16.23 Exercise 8. a. The price of the 3yr @ 5% semi bond is $97.82. You want the duration of the hedged portfolio to be zero. You need to short 0.058 units of the 3-year bond, i.e. the short position is -$5.69. 8 b. The total value of the portfolio is: $4.31 mn. Exerxise 9. Compute the new value of the portfolio assuming the term structure of interest rates as of May 15, 1994. Original Now ∆ value Unhedged port. $10.00 $8.97 ($1.03) Hedge ($5.69) ($5.44) $0.25 Total $4.31 $3.53 ($0.78) a. $8.97 mn b. $3.53 mn c. The immunization covered part of the loss. The change in the value of the portfolio is both due to (i) the passage of time (coupon) and (ii) the increase in interest rates. Exercise 10. Use the curve given on May 15, 1994, but keep the times to maturity unchanged from the initial ones. The change in value is due to the change in interest rates only. Original Now ∆ value Unhedged port. $10.00 $9.97 ($0.03) Hedge ($5.69) ($5.43) $0.26 Total $4.31 $4.54 $0.23 Exercise 11. Use the curve given on February 15, 1994, but change the times to maturity to those on May 15, 1994. The change in value is due to coupon only. Original Now ∆ value Unhedged port. $10.00 $9.12 ($0.88) Hedge ($5.69) ($5.68) $0.01 Total $4.31 $3.45 ($0.87) Exercise 12. a.,b. Loss of $0.87 mn. c. Gain of $0.08 mn. 9 Original Now ∆C ∆r Total Ex.8 Ex.9 Ex.11 (9)-(8)-(11) (9)-(8) Unhedged port. $10.00 $8.97 ($0.88) ($0.15) ($1.03) Hedge ($5.69) ($5.44) $0.01 $0.24 $0.25 Total $4.31 $3.53 ($0.87) $0.08 ($0.78) 10