Question at position 18 All else equal, the ________ the coupon rate on a bond, the ________ the bondʹs duration.greater; longerhigher; shorterhigher; longerlower; shorter单项选择题
A
greater; longer
B
higher; shorter
C
higher; longer
D
lower; shorter

Question

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

BlackTom AI · Accepted Answer

Higher coupon payments return more cash earlier, which lowers a bond's Macaulay duration. Relevant concepts: Macaulay duration is the weighted average time to receive a bond’s cash flows (coupons and principal), with weights equal to the present value of each cash flow divided by the bond price. A bond that pays more early cash (higher coupons) shifts weight toward earlier times, reducing the weighted average time to receipt; lower coupons or a zero-coupon bond concentrate cash at maturity and raise duration.

Step-by-step reasoning with numbers (5-year example, yield = 5%, face = 1000):
1. Compute discount factors: 1/(1.05^t) for t=1..5:
   t1=0.952381, t2=0.907029, t3=0.863838, t4=0.822702, t5=0.783526.
2. Compare two coupon rates: 6% (coupon = 60) vs 2% (coupon = 20).
3. Price for 6% bond:
   PV coupons = 60*(sum discount factors) = 60*4.329476 = 259.769.
   PV principal = 1000*0.783526 = 783.526.
   Price = 259.769 + 783.526 = 1043.295.
4. Macaulay numerator (sum of t * PV(CF_t)) for 6%:
   1*60*0.952381 = 57.143
   2*60*0.907029 = 108.843
   3*60*0.863838 = 155.491
   4*60*0.822702 = 197.448
   5*1060*0.783526 = 4152.660
   Sum = 4671.585. Duration = 4671.585 / 1043.295 ≈ 4.48 years.
5. For 2% bond, repeating the steps yields Duration ≈ 4.79 years (higher than 4.48). A zero-coupon bond would have duration equal to maturity (5 years), the largest possible.

Why the correct answer is correct: higher coupons produce larger early cash flows, reducing the weighted average time to receipt and thus shortening duration. Quantitative examples above show duration decreases as coupon rises.

Why other options are incorrect:
- "greater; longer" and "higher; longer" assert the opposite relationship (coupon up → duration up), which contradicts the definition and numeric results.
- "lower; shorter" also contradicts the relationship: lowering coupons increases duration (approaching maturity for a zero coupon).

Conclusion: The correct choice is "higher; shorter," matching the provided answer.

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

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