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

Question

BlackTom AI · Accepted Answer

Macaulay duration is the present-value-weighted average time to receive an asset’s cash flows: D = Σ[t * (CF_t / (1+y)^t)] / Price. Key implications: larger early cash flows (higher coupons) increase weights at short times and lower duration; a zero-coupon bond has a single cash flow at maturity, so its duration equals its maturity; weights are PV-based, not equal.

1) Relevant knowledge points and concepts
- Macaulay duration formula and interpretation as weighted average time to cash flows.
- Present-value weights: weight for time t = PV(CF_t) / Price.
- Effect of coupon size and yield on duration: higher coupons → earlier cash flows relatively more important → lower duration; higher yield generally reduces the PV of later payments relatively more, so duration falls as yield rises.

2) Complete reasoning process (with numeric example)
Compare a 2-year zero-coupon bond and a 2-year 10% coupon bond, both face = 100, YTM = 5%:
- Zero-coupon: single CF at t=2 of 100.
  Price = 100 / (1.05)^2 = 100 / 1.1025 = 90.7029.
  Macaulay duration = 2 * (PV of CF) / Price = 2 * 90.7029 / 90.7029 = 2.00 years.
- Coupon bond (10% annual): CF at t1 = 10, t2 = 110.
  PV t1 = 10 / 1.05 = 9.5238.
  PV t2 = 110 / 1.1025 = 99.7740.
  Price = 9.5238 + 99.7740 = 109.2978.
  Macaulay duration = (1*9.5238 + 2*99.7740) / 109.2978
                    = (9.5238 + 199.5480) / 109.2978
                    = 209.0718 / 109.2978 = 1.9128 years.
This shows increasing coupon (from 0% to 10%) reduces duration (2.00 → 1.913), illustrating the general rule.

3) Intermediate steps shown above for PVs, price, and weighted sum.

4) Why the correct answer is correct
Because duration is a PV-weighted average time to cash flows, raising the coupon rate shifts cash flow weight toward earlier payments (larger coupons earlier), lowering the weighted average time and therefore reducing duration (all else equal).

5) Why other options are incorrect
- "Zero-coupon duration different from maturity": false — for a zero-coupon bond, the only cash flow is at maturity, so duration = maturity.
- "Use equal weights": false — weights are present-value-based, not equal across payments.
- "If maturity 4, duration 2": false in general — duration depends on coupon pattern and yield; only specific cash-flow structures give duration = half maturity.
- "Duration should increase when YTM increases": false — higher yields typically reduce duration because later cash flows are discounted relatively more, lowering the weighted average time.

6) Conclusion
The correct statement is: "The duration of a bond decreases when there is an increase in the coupon rate (all else equal)." This matches the provided answer.

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

类似问题

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]

Choose the most correct answer. Ceteris paribus, the duration of a bond is positively correlated with the bond's

Consider the following two Treasury securities: Bond A is sold at $120 with modified duration = 5, and Bond B is sold at $80 with modified duration = 7. Which bond will have greater dollar price change for a 50-basis-point change in interest rates?

Question at position 3 For bonds, price sensitivity to a given change in interest rates is generally greater the longer before the bond matures.TrueFalse

If an investor is convinced that all YTMs in the market will decrease by 1% in the next month, then, they should invest in the bond with the longest maturity available, as the price of this bond will increase the most due to the change in the YTM.

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

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单