Bond Duration

Bond Duration

Duration measures a bond's sensitivity to interest rate changes and its weighted-average time to receive cash flows.

Macaulay Duration

$D_{Mac} = \frac{\sum_{t=1}^{n} t \cdot \frac{CF_t}{(1+y)^t}}{P}$

This is the weighted average of when cash flows arrive, weighted by their present values.

Modified Duration

$D_{Mod} = \frac{D_{Mac}}{1 + y}$

Modified duration approximates the percentage price change for a 1% change in yield:

$\frac{\Delta P}{P} \approx -D_{Mod} \cdot \Delta y$

Example

A 10-year, 5% coupon bond at 5% YTM has:

• Macaulay Duration ≈ 8.11 years
• Modified Duration ≈ 7.72

So if yields rise by 1%, the bond price falls by approximately 7.72%.

Key Insights

• Zero-coupon bonds have duration = maturity
• Higher coupon → shorter duration (more early cash flows)
• Higher yield → shorter duration