The duration of a coupon bond increases when:

Question

BlackTom AI · Accepted Answer

Macaulay duration equals the weighted average time to receive a bond’s cash flows, so D = (Σ t·PV(CF_t)) / Price, where PV(CF_t) = CF_t/(1+y)^t. Each cash flow’s time t is weighted by its present value; anything that shifts weight toward later payments raises D.

Step-by-step reasoning:
1. Write the Macaulay-duration formula:
   D = (Σ_{t=1}^n t · CF_t/(1+y)^t ) / (Σ_{t=1}^n CF_t/(1+y)^t).
   The numerator is the time-weighted present values, the denominator is the bond price.
2. Effect of longer maturity (option a):
   - Increasing maturity from n to n+1 adds an extra positive term (t = n+1) in the numerator and a corresponding PV in the denominator. Because the added cash flow occurs at a later time, it raises the weighted average time (numerator gains relatively more weight toward later t), so D increases. In the limit, a zero-coupon bond’s duration = maturity, showing duration grows as maturity lengthens.
3. Why higher coupon (option b) is incorrect:
   - A higher coupon increases earlier cash flows’ PVs relative to the terminal principal, shifting weight to earlier t values and thus lowering the weighted average time. Therefore higher coupon reduces duration.
4. Why "bond price is higher" (option c) is incorrect:
   - If price is directly scaled up (multiply all PVs by same factor), both numerator and denominator scale equally so D is unchanged. Price alone is not a causal driver; price increases caused by lower yield could increase duration, but the option statement is too general and not universally true.
   - Algebraic illustration: if every PV is multiplied by k, numerator and denominator both multiply by k ⇒ D unchanged.
5. Why interest rates increase (option d) is incorrect:
   - Higher yields (y) discount distant cash flows more heavily, lowering their PVs and shifting weight toward earlier payments, so duration falls when rates rise.

Conclusion: Because adding later cash flows (longer maturity) shifts the weighted average time to the right, duration increases. This confirms the correct answer: a. maturity is longer.

The duration of a coupon bond increases when:单项选择题

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