Question14 Assume the term structure of the interest rate is flat at 5%. There are two zero-coupon bonds available:[table]
| Time to maturity | Face value | Coupon rate Bond A | 3 years | $100 | 0% Bond B | 9 years | $100 | 0%
[/table] Suppose you have a liability that requires you to pay $100 in year 4 and another $100 in year 8, and you want to immunize your positions by using Bond A and Bond B. What is your portfolio weight in Bond A and Bond B, respectively? (Hint: You need to compute the duration of your liability and match it with the duration of your portfolio of bonds.) Select one alternative: A. 17%,83 % B. 40%,60% C. 53%,47% D. 83%,17% ResetMaximum marks: 2 Flag question undefined

Question

Question14 Assume the term structure of the interest rate is flat at 5%. There are two zero-coupon bonds available:[table]
| Time to maturity | Face value | Coupon rate Bond A | 3 years | $100 | 0% Bond B | 9 years | $100 | 0%
[/table] Suppose you have a liability that requires you to pay $100 in year 4 and another $100 in year 8, and you want to immunize your positions by using Bond A and Bond B. What is your portfolio weight in Bond A and Bond B, respectively? (Hint: You need to compute the duration of your liability and match it with the duration of your portfolio of bonds.) Select one alternative: A. 17%,83 %  B. 40%,60% C. 53%,47% D. 83%,17% ResetMaximum marks: 2 Flag question undefined

BlackTom AI · Accepted Answer

Macaulay duration of the liability is the PV-weighted average payment time. Compute present values at 5% and the liability duration first.

1) PVs of liabilities
- PV1 (year 4) = 100 / 1.05^4. 1.05^2 = 1.1025, 1.05^4 = 1.21550625, so PV1 = 100 / 1.21550625 = 82.27027.
- PV2 (year 8) = 100 / 1.05^8. 1.05^8 = 1.4774554437890625, so PV2 = 100 / 1.4774554437890625 = 67.68394.
- Total PV of liabilities = 82.27027 + 67.68394 = 149.95421.

2) Liability Macaulay duration
- Numerator = 4 * 82.27027 + 8 * 67.68394 = 329.08108 + 541.47152 = 870.55260.
- Duration = 870.55260 / 149.95421 ≈ 5.80546 years.

3) Use zero-coupon bonds (duration = maturity). Bond A: 3 years, Bond B: 9 years. Let wA and wB be market-value weights (wA + wB = 1). Portfolio duration = wA*3 + wB*9 must equal 5.80546.

4) Solve for wA:
3 wA + 9(1 − wA) = 5.80546
=> 3 wA + 9 − 9 wA = 5.80546
=> 9 − 6 wA = 5.80546
=> 6 wA = 3.19454
=> wA = 3.19454 / 6 = 0.532424 ≈ 53.24%.
Thus wB = 1 − wA ≈ 46.76% ≈ 47%.

Why this is correct: matching the Macaulay duration of the asset portfolio to the liability duration ensures small parallel shifts in the yield curve produce offsetting price and reinvestment effects, immunizing the position. For zero-coupon bonds the duration equals maturity, which simplifies the linear weight equation above.

Why other options are incorrect:
- A (17%,83%): gives portfolio duration 0.17*3 + 0.83*9 = 7.98 (too high).
- B (40%,60%): duration = 0.4*3 + 0.6*9 = 6.6 (too high).
- D (83%,17%): duration = 0.83*3 + 0.17*9 = 4.02 (too low).

Conclusion: The correct portfolio weights are approximately 53% in Bond A and 47% in Bond B, matching answer C.

类似问题

A zero-coupon bond has three years until maturity. The bond’s face value is $100 and the price today is $80. The bond’s duration is:

Consider the following statements. Choose the correct statement.

A 6% coupon bond paying interest annually has a modified duration of 10 years, sells for $800, and is priced at a yield to maturity of 8%. If the YTM increases to 9%, what is the predicted dollar change in price based on the bond’s duration? (Round to 2 decimal places.)

Question at position 18 Which bond is most sensitive to interest rate changes?All equal5-year, 8% coupon3-year, 10% coupon10-year, 3% coupon

Which of the following two semiannual Treasury bonds is more price sensitive to changes in interest rates? 1) A par value bond X, with 5-year-to-maturity and 10% coupon rate. 2) A zero-coupon bond Y, with 5-year-to-maturity and 10% yield to maturity.

An investor buys a 6% annual payment bond with three years to maturity. The bond has a yield-to-maturity of 8% and is currently priced at 94.845806 per 100 of par. The bond’s Macaulay duration is closest to:

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