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

Question

BlackTom AI · Accepted Answer

Macaulay duration equals the weighted-average time until a bond’s cash flows are received, with weights equal to each cash flow’s present value divided by the bond price: D = Σ[t * PV(CF_t)] / Price, where PV(CF_t) = CF_t / (1 + y)^t.

1) Relevant knowledge points and concepts
- Macaulay duration definition and interpretation (time-weighted PV of cash flows).
- How maturity, coupon rate, and yield affect the relative size and timing of discounted cash flows.
- Modified duration (sensitivity) relates to Macaulay duration but the sign of correlations is the same.

2) Step-by-step reasoning
- The numerator of duration sums t times each discounted cash flow. Adding later cash flows (increasing time to maturity) introduces terms with larger t, increasing the numerator relative to the denominator, so D rises.
- If coupons are larger, a greater share of PV comes from earlier payments, shifting weight to small t and reducing D.
- If yield (y) increases, later cash flows are discounted more heavily than earlier ones, reducing their weight and lowering D.

3) Example / intermediate steps
- Formula restated: D = Σ[t * CF_t / (1+y)^t] / Σ[CF_t / (1+y)^t].
- Zero-coupon bond: only CF at t = T, so D = T (duration equals maturity). For a coupon bond with same T, earlier coupons reduce D < T. Hence increasing maturity increases D toward T.
- Numerical intuition: a 5-year zero-coupon has D = 5. A 5-year bond with coupons might have D ≈ 3–4 depending on coupon and yield — shorter than maturity.

4) Why the correct answer is correct
- Time to maturity directly increases the time weights (t) in the duration formula and thus raises duration ceteris paribus.

5) Why other options are incorrect
- B (coupon rate): higher coupon lowers duration (negative correlation).
- C (yield to maturity): higher yield lowers duration (negative correlation).
- D (both B and C): incorrect because both are negatively correlated, not positively.
- E: false because time to maturity is indeed positively correlated.

6) Conclusion
Therefore, ceteris paribus, duration is positively correlated with the bond’s time to maturity. Final answer: A. time to maturity.

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

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