Dde 321 - Solutions Exercise 6

Advanced Corporate Finance

Leonidas Rompolis

EXERCISES -6 (SOLUTIONS) 1. Using the general relationship between spot and forward rates we have: 2 (1 + r2 ) = (1 + r1 )(1 +1 f1 ) = 1.06 × 1.064 ⇒ r2 = 6.2%

(1 + r3 ) = (1 + r2 ) (1 +1 f 2 ) = 1.0622 × 1.071 ⇒ r2 = 6.5% 4 3 (1 + r4 ) = (1 + r3 ) (1 +1 f3 ) = 1.0653 ×1.073 ⇒ r2 = 6.7% 5 4 (1 + r5 ) = (1 + r4 ) (1 +1 f 4 ) = 1.067 4 ×1.082 ⇒ r5 = 7% 3

2

spot rates

Yield curve 7% 7% 7% 7% 6% 6% 6% 6% 6% 5% 1

2

3

4

5

maturity

If the pure expectation theory holds we can infer from the fact that the forwards rates are increasing, that the future spot rates are expected to increase.

2. The company will receive $100 million at t = 4. It can lock this amount by signing a contract today to invest this amount in four years for a time period of one at today’s forward rate 1 f 4 = 8.2% . It will receive $108.2 million at t = 5, thus covering the payment of $107 million.

3. a. PA =

50 1, 050 + = $992.79 1.05 1.0542

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Advanced Corporate Finance

Leonidas Rompolis

50 50 50 50 1, 050 + + + + = $959.34 2 3 4 1.05 1.054 1.057 1.059 1.065 100 100 100 100 1,100 PC = + + + + = $1,171.43 2 3 4 1.05 1.054 1.057 1.059 1.065 b. We must first calculate the YTM: 5 50 1, 000 959.34 = ∑ + ⇒ r = 5.96% t (1 + r)5 t =1 (1 + r) 5 100 1, 000 1,171.43 = ∑ + ⇒ r = 5.93% t (1 + r)5 t =1 (1 + r) The YTM depends upon both the coupon payment and the spot rate at the time of the coupon payment. The 10% bond has a slightly greater proportion of its total payments coming earlier, when interest rates are low, compared to the 5% bond. Thus the YTM of the 10% bond is slightly lower. c. Coupon payments: 100 × 5 = $500 Capital gain: 1,000 - 1,171.43 = -$171.43 ⎡1 − 1.0595 ⎤ − 5⎥ = $62.58 Reinvestment income: 100 ⎢ ⎣ 1 − 1.059 ⎦ If the bond will be sold at year 3 the total dollar return for this period would be: 3 1,171.43 (1 + 0.059 ) − 1,171.43 = $219.82 PB =

d. The YTM of a 5-year zero-coupon bond should be 6%.

9 109 + = $101.86 1.07 1.082 9 109 b. $101.86 = + ⇒ r = 0.079 1 + r (1 + r) 2

4. a. P0 =

(1 + r2 ) 2 − 1 = 0.090 c. E ( 1 r1 ) =1 f1 = 1 + r1 109 Therefore, P1 = = $99.99 1.09 d. If the liquidity premium is 1% then E ( 1 r1 ) + 1% =1 f1 ⇒ E ( 1 r1 ) =1 f1 − 0.01 = 0.08 Then, P1 =

109 = $100.97 1.08

5. a. Since the bond is sold at the face value the YTM is equal the coupon rate. The duration of the bond is: 1 + y (1 + y) + T(c − y) D= − = 7.51 y c ⎡⎣ (1 + y)T − 1⎤⎦ + y where y = c = 7% and T = 10. The convexity is: 2

Advanced Corporate Finance

Leonidas Rompolis

10 CFt 1 (t 2 + t) = 64.93 2 ∑ t 1, 000(1.07) t =1 (1.07) b. The actual price of the bond if YTM increases would be: 10 70 1, 000 + = $932.89 P0 = ∑ t (1.08)10 t =1 (1.08) c. The duration rule imply that: ΔP D 7.51 0.01 = −0.07 = − D*Δy = − Δy = − P 1+ y 1.07 P − 1, 000 = −0.07 ⇒ P0 = $930 Thus, 0 1, 000 932.89 − 930 The percentage error of the rule is: = 0.3% 932.89 d. The convexity rules predicts that: ΔP 1 1 = − D*Δy + Con(Δy) 2 = −0.07 + 64.93(0.01) 2 = −0.066 P 2 2 P0 − 1, 000 Thus, = −0.066 ⇒ P0 = $934 1, 000 932.89 − 934 The percentage error of the rule is: = −0.11% 932.89

Con =

6. We must first calculate the modified duration and convexity of the two bonds. 13 The modified duration is D1* = = 12.03 1.08 100 The price is P01 = = $36.79 1.0813 1 100 The convexity is Con1 = (132 + 13) = 155.95 2 36.79(1.08) (1.08)13 30 6 100 The price of the coupon bond is P02 = ∑ + = $77.48 t (1.08)30 t =1 (1.08) 1 + y (1 + y) + T(c − y) The duration is D 2 = − = 12.73 , thus the modified duration is y c ⎡⎣ (1 + y)T − 1⎤⎦ + y D*2 =

12.73 = 11.78 1.08

The convexity is Con 2 =

1 77.48(1.08) 2

30

CFt

∑ (1.08) t =1

t

(t 2 + t) = 231.2

ΔP1 1 1 a. = − D1*Δy + Con1 ( Δy) 2 = −12.03(0.01) + 155.95(0.01) 2 = −0.112 P1 2 2 Therefore if the YTM increases to 9% the price of the zero-coupon will fall by 11.2%.

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Advanced Corporate Finance

Leonidas Rompolis

ΔP2 1 1 = − D*2 Δy + Con 2 ( Δy) 2 = −11.78(0.01) + 231.2(0.01) 2 = −0.106 P2 2 2 Therefore if the YTM increases to 9% the price of the coupon bond will fall by 10.6%. ΔP1 1 b. = 12.03(0.01) + 155.95(0.01) 2 = 0.128 P1 2 Therefore if the YTM decreases to 7% the price of the zero-coupon bond will increase by 12.8%. ΔP2 1 = 11.78(0.01) + 231.2(0.01) 2 = 0.129 P2 2 Therefore if the YTM decreases to 7% the price of the coupon bond will increase by 12.9%. c. The coupon bond has a more attractive performance in both scenarios. When interest rates increase its decrease in price is smaller compared to the zero-coupon bond. On the other hand, when interest rates decrease the rise in its price is larger. This is due to convexity. d. It would not be possible for these two bonds to be priced in the same yield, because they are not exposed equally to interest rate risk since they have different convexities.

7. a. The present value of the liability is: 10, 000 P0L = = $8,534.9 1.028 The prices of the two bonds are: 5 10 1, 000 P0(5) = ∑ + = $952.865 t 1.025 t =1 1.02 1, 000 P0(10) = = $820.348 1.0210 The duration of the liability is DL = 8. The duration of the 10-year zero coupon bond is D10 = 10. The duration of the 5-year coupon bond is: 1.02 1.02 + 5(0.01 − 0.02) D5 = − = 4.89 0.02 0.01 ⎡⎣(1.02)5 − 1⎤⎦ + 0.02 The first rule defines that the present value of the liability should be equal to the present value of the portfolio: 7 X P0 L = x1P0 (5) + x 2 P0 (10) + ∑ t t = 0 (1 + y) which is written as: 7 X (1) 8,534.9 = x1 952.865 + x 2 820.348 + ∑ t t = 0 (1.02) The second rule defines that the duration of the liability should be equal to the duration of the portfolio:

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Advanced Corporate Finance

Leonidas Rompolis

X D L = w1D5 + w 2 D10 + L P0

7

t

∑ (1 + y) t =1

t

which is written as: 7 952.865x1 820.348x 2 t X (2) 8= 4.89 + 10 + ∑ 8,534.9 8,534.9 8,534.9 t =1 (1.02) t We set X = $200 and we solve (1) and (2) and we obtain: x1 = 1.48, x2 = 6.86. If the market interest rate increases at 3% the prices of the two bonds would be: 5 10 1, 000 P0(5) = ∑ + = $908.406 t 1.035 t =1 1.03 1, 000 P0(10) = = $744.094 1.0310 The total income would be: 7

1.48 × 908.406 (1.03) + 6.86 × 744.094 (1.03) + ∑ 200 (1.03) 8

8

8− t

= $10, 004.3

t =0

If the market interest rate decreases at 1% the prices of the two bonds would be: P0(5) = $1, 000 P0(10) =

1, 000 = $905.287 1.0110

The total income would be: 7

1.48 × 1, 000 (1.01) + 6.86 × 905.287 (1.01) + ∑ 200 (1.01) 8

8

8− t

= $10, 004.3

t =0

b. By setting X = $670 we obtain x1 = -3.29, x2 = 8.12, by solving (1) and (2). The initial investment is: −3.29 × 952.865 + 8.12 × 820.348 = $3, 971.22 which satisfies the constraint. c. We add to equations (1) and (2) a third one, namely: 952.86x1 + 820.348x 2 = 0 Thus we have a system of 3 equations with 3 unknowns. The system has a unique solution: x1 = -8.08, x2 = 9.39 and X = $1,142.5.

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