Bond duration increases when:

Question

BlackTom AI · Accepted Answer

Macaulay duration is the weighted average time until a bond’s cash flows are received, where weights are the present values of those cash flows divided by the bond price. It captures interest-rate sensitivity: longer, later cash flows raise duration; larger earlier coupons reduce it. The formula: D = (1/P) * Σ[t * CF_t / (1+y)^t], where P = Σ[CF_t / (1+y)^t], y is yield, CF_t are cash flows, and t is time.

1) Relevant concepts
- Macaulay duration measures timing-weighted average of cash flows.
- For zero-coupon bonds, duration = maturity (all cash flow at maturity).
- For coupon bonds, higher coupons shift weight to earlier payments and lower duration.
- Higher yields generally reduce modified duration (and typically lower Macaulay duration slightly).

2) Step-by-step reasoning
- Duration increases when more weight is placed on later (higher t) cash flows.
- Extending maturity adds more or later payments (or pushes final large principal payment further out), increasing the weighted average time.

3) Numerical illustration (intermediate steps)
Compare a 5-year zero-coupon vs a 5-year coupon bond:
- Zero-coupon 5-year: single CF at t=5 → duration = 5.
- Coupon bond example: 5-year, annual coupon 8, y=5, face 100:
  PVs: year1 8/1.05 = 7.619; year2 7.256; year3 6.911; year4 6.582; year5 108/1.05^5 = 84.65.
  Price P ≈ 113.02. Weighted time numerator = 1*7.619 + 2*7.256 + 3*6.911 + 4*6.582 + 5*84.65 ≈ 492.44.
  Macaulay duration D ≈ 492.44 / 113.02 ≈ 4.36 years < 5. This shows higher coupons lower duration; pushing maturity out would increase D.

4) Why the correct answer is correct
- Option b (“maturity is longer”) directly moves significant cash flows farther into the future (especially principal), increasing the time-weighted average and thus duration.

5) Why other options are incorrect
- a. “Interest rates rise”: raising yields tends to reduce duration (modified duration falls; Macaulay also typically decreases slightly), so this does not increase duration.
- c. “Coupon rate is higher”: higher coupons shift weight to earlier payments and reduce duration.
- d. “Bond price increases”: price alone is not the driver; price typically rises when yields fall, but it is the timing of cash flows (maturity/coupon) that determines duration. A higher price does not necessarily increase duration.

6) Conclusion
Extending maturity increases the weighted average timing of cash flows and therefore increases bond duration. This confirms the correct choice is b.

Bond duration increases when:Single choice

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