Short answer
Macaulay duration is the weighted-average time to a bond’s cash flows; modified duration = Macaulay ÷ (1 + y) estimates price change per 1-point yield move. A 5% 10-year bond at 6% has modified duration 7.67.
1. Macaulay duration
Macaulay duration is the present-value-weighted average time, in years, until you receive the bond's cash flows. A zero-coupon bond's Macaulay duration equals its maturity.
2. Modified duration
Modified duration = Macaulay duration ÷ (1 + y), with y the yield per period. It estimates the percentage price change for a one-percentage-point change in yield: change ≈ −modified duration × yield change.
3. Convexity
Duration is a straight-line estimate. Real prices curve, so a yield fall raises the price by more than the estimate and a yield rise lowers it by less.
4. Example
$1,000 bond, 5% coupon, 10 years, semi-annual payments, 6% yield: price $925.61, Macaulay 7.89, modified 7.67.
| Yield change | Duration estimate | Actual price | Actual change |
|---|---|---|---|
| +1 point (7%) | −7.67% | $857.88 | −7.32% |
| −1 point (5%) | +7.67% | $1,000.00 | +8.04% |
5. Duration by maturity
For 4.5% coupon bonds at par with a 4.5% yield and semi-annual payments:
| Bond | Macaulay | Modified |
|---|---|---|
| 2-year | 1.93 | 1.89 |
| 5-year | 4.53 | 4.43 |
| 10-year | 8.16 | 7.98 |
| 30-year | 16.74 | 16.37 |
Longer maturities and lower coupons mean higher duration and bigger price swings.
6. Frequently asked questions
What is a good duration for a bond?expand_more
There is no single good number. Higher duration means more sensitivity to interest rate changes, so match it to your risk tolerance and time horizon.
How is modified duration different from Macaulay duration?expand_more
Macaulay is a time in years. Modified divides it by (1 + y) to give price sensitivity per point of yield.
Why is the real price change different from the estimate?expand_more
Because of convexity: price curves, so duration alone is only accurate for small yield moves.
Do zero-coupon bonds have higher duration?expand_more
Yes. With no coupons, all cash arrives at maturity, so their duration equals their maturity and they are the most rate-sensitive for their term.