# Duration

Source: https://www.yieldcurve.pro/learn/duration

**Duration** measures how much a bond's price changes when interest rates move. It is expressed in years but functions as a risk metric: a bond with a duration of 8 years will lose approximately 8% of its value for every 1 percentage point rise in yields.

Two related measures are commonly used:

- **Macaulay duration** is the weighted-average time to receive a bond's cash flows, where the weights are the present values of each payment. For a par bond with semiannual coupons, Macaulay duration approximates (1 + y/2) / (y/2) x (1 - 1/(1 + y/2)^(2n)), where y is the yield and n is the maturity in years.
- **Modified duration** adjusts Macaulay duration for the compounding frequency: modified duration = Macaulay / (1 + y/2). This gives the percentage price change per 1% yield change.

Duration increases with maturity and decreases with coupon rate. A 10-year Treasury at par has a modified duration around 8.2 years, while a 30-year bond has roughly 16.4 years.

Portfolio managers use duration to manage interest rate risk. A **duration-neutral** portfolio is constructed so that gains from one position offset losses from another when rates move. Understanding duration is essential for evaluating how yield curve shifts affect portfolio value.

## FAQ

### What is the difference between Macaulay and modified duration?

Macaulay duration is the present-value-weighted average time until a bond's cash flows are received, expressed in years. Modified duration adjusts that figure for compounding so it can be read directly as a percentage price change per 1% yield change. Modified duration = Macaulay / (1 + y/k), where k is the number of coupon payments per year.

### How do I estimate the price impact of a yield change using duration?

Multiply the bond's modified duration by the negative of the yield change. A bond with a modified duration of 7.5 will lose about 7.5% of its market value if yields rise by 1 percentage point, and gain a similar amount if yields fall by 1 point. The estimate is most accurate for small yield moves; for larger moves, convexity adds a meaningful second-order correction.

### Why does duration fall as coupon rate rises?

Higher coupons return more of the bond's cash flows earlier in its life, which shortens the present-value-weighted average maturity. A zero-coupon bond has the longest duration for its maturity (duration equals maturity), while a high-coupon bond of the same maturity has a noticeably shorter duration and therefore less interest rate sensitivity.