Consider two bonds with the same maturity and yield - Bond M: 10-year zero-coupon bond, and Bond N: 10-year bond with an 8% annual coupon. If market interest rates increase unexpectedly, which bond will experience a larger percentage price decline?[Fill in the blank]单项选择题
A
a. Bond N
B
b. Both will decline by the same percentage
C
c. It cannot be determined without knowing the face value
D
d. Bond M

Question

Consider two bonds with the same maturity and yield - Bond M: 10-year zero-coupon bond, and Bond N: 10-year bond with an 8% annual coupon. If market interest rates increase unexpectedly, which bond will experience a larger percentage price decline?[Fill in the blank]单项选择题

A

a. Bond N

B

b. Both will decline by the same percentage

C

c. It cannot be determined without knowing the face value

D

d. Bond M

BlackTom AI · Accepted Answer

A zero-coupon bond has Macaulay duration equal to its maturity (10 years), whereas a coupon-paying bond has a shorter duration because interim coupon payments return part of the investment earlier. Duration measures interest-rate sensitivity: the approximate percentage price change ≈ −(modified duration) × (change in yield).

1) Relevant concepts
- Macaulay duration = weighted average time of cash flows.
- Modified duration = Macaulay duration / (1 + y) and gives approximate % price change per 1.0 (100%) change in yield; for small Δy, %ΔP ≈ −(modified duration)·Δy.
- Higher duration → larger percentage price sensitivity.

2) Step-by-step reasoning
Because Bond M (zero) pays nothing until year 10, its Macaulay duration = 10 years. Bond N (8% annual coupon) receives many earlier cash flows, so its duration is lower. With the same yield, the bond with higher modified duration (the zero) will fall more in percentage terms when yields rise.

3) Calculations (assuming face = 100 and yield y = 8% since yields are the same and coupon = 8%)
- Zero-coupon: Macaulay duration = 10. Modified duration = 10 / 1.08 = 9.2593.
- Coupon bond price at y=8%: equals par = 100. Compute Macaulay duration:
  v = 1 / 1.08 = 0.9259259; v^10 ≈ 0.463193.
  Sum t·v^t for t=1..10 using the formula gives ≈ 33.00.
  Weighted numerator = 8*(sum t·v^t) + 100*10*v^10 ≈ 8*33.00 + 463.193 = 727.193.
  Macaulay duration = 727.193 / 100 = 7.2719 years.
  Modified duration = 7.2719 / 1.08 ≈ 6.73.
- Comparison: Modified duration of Bond M ≈ 9.26 > Bond N ≈ 6.73, so Bond M has greater % sensitivity.

4) Why the correct answer is correct
Because percentage price change ≈ −(modified duration)×Δy, the zero-coupon (higher modified duration) will experience the larger percentage price decline for the same unexpected rise in rates.

5) Why other choices are incorrect
- a (Bond N): incorrect because coupon payments reduce duration and percent sensitivity.
- b (both same): incorrect because durations differ.
- c (cannot determine without face value): incorrect because percentage changes depend on duration, not absolute face value; using a normalized face (e.g., 100) yields valid comparison.

Conclusion: Bond M (the zero-coupon) will experience the larger percentage price decline — answer d.

Consider two bonds with the same maturity and yield - Bond M: 10-year zero-coupon bond, and Bond N: 10-year bond with an 8% annual coupon. If market interest rates increase unexpectedly, which bond will experience a larger percentage price decline?[Fill in the blank]单项选择题

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Consider two bonds with the same maturity and yield - Bond M: 10-year zero-coupon bond, and Bond N: 10-year bond with an 8% annual coupon. If market interest rates increase unexpectedly, which bond will experience a larger percentage price decline?[Fill in the blank]单项选择题

类似问题

Consider the following statements about the duration of loans and bonds. Which of these statements is correct?

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希望你的学习变得更简单

Question at position 18 All else equal, the the coupon rate on a bond, the the bondʹs duration.greater; longerhigher; shorterhigher; longerlower; shorter