Modified Duration
Macaulay duration scaled by 1/(1 + y/m). Equals the negative percentage price change per unit yield change — the standard first-order rate-sensitivity coefficient for a bond.
When to use: Use to estimate how much a bond's price will move for a small yield change: ΔP/P ≈ −ModDur × Δy. Used everywhere in fixed-income risk management — DV01, duration matching, hedge ratios.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| ModDur | Modified Duration | Price sensitivity coefficient: −(1/P)(dP/dy), in years | years |
| F | Face Value | Par value paid at maturity | $ |
| CR | Coupon Rate | Annual coupon rate as a decimal (e.g. 0.05 for 5%) | % |
| y | Yield | Annual yield as a decimal; periodic yield is y/m | % |
| N | Years to Maturity | Years remaining until the bond matures | years |
| m | Coupons per Year | Number of coupons paid per year (e.g. 2 for semi-annual) | integer |
Real-Life Examples
Example 1: Par Bond, 5y, 5%, Semi-Annual
Same 5-year par bond as the Macaulay example (D_Mac = 4.485 years).
Given
Step-by-Step
A 1% yield rise (100 bps) drops the bond's price by approximately 4.38%. For a $1,000 par bond, that's about $43.80 of price loss — a useful rule of thumb for first-order rate risk.
Frequently Asked Questions
Macaulay duration measures time. Modified duration measures sensitivity. The factor 1/(1 + y/m) emerges from differentiating the bond price with respect to y: dP/dy = −D_Mac/(1+y/m) × P, so −(1/P)(dP/dy) = D_Mac/(1+y/m) = D_Mod.
For large yield moves (>~50 bps) or for bonds with embedded options. Add the convexity term — 0.5 × Conv × (Δy)² — for second-order accuracy.
Less well. The price function is non-smooth around the call boundary, so analytical modified duration overstates sensitivity. Use effective duration (numerical) instead.