Set out how a bond's duration behaves and what drives it
The idea
Duration rises with maturity, falls as the coupon rises, and falls as the yield rises — each because the property shifts the weight of the present value earlier or later in the bond's life. A zero-coupon bond's Macaulay duration equals its maturity, which is the limiting case, and no coupon bond can have a duration longer than its maturity. The relationship with maturity is not strictly monotonic for deep-discount long bonds, where duration can peak before maturity, but for the ordinary cases the three rules hold and they explain most of what a portfolio manager needs about relative sensitivity.
A practice question
How does a higher coupon affect a bond's duration, holding maturity constant?
- It lowers duration — correct
- It raises durationWhy not: Backwards. More cash arriving early pulls the weighted average time to receipt forward.
- It leaves duration unchangedWhy not: The coupon is one of the two main determinants, which is why a zero has the longest duration of any bond of its maturity.
More of the present value arrives early, which pulls the weighted average time to receipt forward and reduces sensitivity to yield changes.
A harder one
What is the Macaulay duration of an eight-year zero-coupon bond?
- 4.0 yearsWhy not: Half the maturity, which is closer to what a heavily coupon-paying bond might show. A zero has no interim cash flows to pull the average forward.
- 7.6 yearsWhy not: Close to what the modified duration would be after dividing by one plus the yield. The Macaulay figure is the maturity itself.
- 8.0 years — correct
There is only one cash flow, so its weighted average time to receipt is simply its date.
The hardest tier
A manager wants less rate sensitivity without shortening the portfolio's maturity. What works?
- Buying bonds with lower couponsWhy not: That raises duration, which is the opposite of the objective.
- Buying bonds with lower yieldsWhy not: A lower yield raises duration by increasing the weight on distant cash flows — and yield is set by the market rather than chosen.
- Buying bonds with higher coupons — correct
A higher coupon pulls the weight of present value forward, shortening duration without changing maturity. The three properties are independent levers, so the one that is constrained need not be the one that is used.
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