Interest-rate risk
Macaulay and modified duration
Measure how far a bond's price moves when yields change — and why long, low-coupon bonds move most.
Learning objectives
- Calculate Macaulay duration as the present-value-weighted average time to a bond's cash flows
- Convert Macaulay duration to modified duration using the yield per period
- Estimate a bond's price change for a change in yield
- Explain how maturity, coupon and yield affect duration
Every bondholder is exposed to interest-rate risk: when yields rise, prices fall. Two bonds can react very differently to the same move, though, and a treasurer or portfolio manager needs one number that says how much. That number is duration.
Duration has two faces. Macaulay duration is a measure of time: the average time until you receive the bond's cash flows, weighting each payment by its present value. Modified duration is a measure of sensitivity: the approximate percentage change in price for a 1 percentage point (100 basis point) change in yield. The second is derived from the first.
Macaulay duration (years) = [Σ t × PV(CFₜ) ÷ price] ÷ f Modified duration = Macaulay duration ÷ (1 + y ÷ f) % change in price ≈ −modified duration × Δy
- t
- the period of each cash flow (1, 2, … N); dividing by f converts periods to years
- PV(CFₜ)
- the present value of the cash flow at the bond's yield
- y ÷ f
- the yield per period: y ÷ 2 for a semi-annual bond, y for an annual-pay bond
- Δy
- the change in yield, as a decimal (50 basis points = 0.005)
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