Bond Duration

Bond Duration

In finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received. When an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage change in price for a parallel shift in yields.

Since cash flows for bonds are usually fixed, a price change can come from two sources:

1. The passage of time (convergence towards par). This is of course totally predictable, and hence not a risk.
2. A change in the yield. This can be due to a change in the benchmark yield, and/or change in the yield spread.

The yield-price relationship is inverse, and we would like to have a measure of how sensitive the bond price is to yield changes. A good approximation for bond price changes due to yield is the duration, a measure for interest rate risk. For large yield changes convexity can be added to improve the performance of the duration. A more important use of convexity is that it measures the sensitivity of duration to yield changes. Similar risk measures used in the options markets are the delta and gamma.

The dual use of the word "duration", as both the weighted average time until repayment and as the percentage change in price, often causes confusion. Strictly speaking, Macaulay duration is the name given to the weighted average time until cash flows are received, and is measured in years. Modified duration is the name given to the price sensitivity and is the percentage change in price for a unit change in yield. When yields are continuously compounded Macaulay duration and modified duration will be numerically equal. When yields are periodically compounded Macaulay and modified duration will differ slightly, and in this case there is a simple relation between the two. Modified duration is used more than Macaulay duration.

Macaulay duration and modified duration are both termed "duration" and have the same (or close to the same) numerical value, but it is important to keep in mind the conceptual distinctions between them. Macaulay duration is a time measure with units in years, and really makes sense only for an instrument with fixed cash flows. For a standard bond the Macaulay duration will be between 0 and the maturity of the bond. It is equal to the maturity if and only if the bond is a zero-coupon bond.

Modified duration, on the other hand, is a derivative (rate of change) or price sensitivity and measures the percentage rate of change of price with respect to yield. (Price sensitivity with respect to yields can also be measured in absolute (dollar) terms, and the absolute sensitivity is often referred to as dollar duration, DV01, PV01, or delta (δ or Δ) risk). The concept of modified duration can be applied to interest-rate sensitive instruments with non-fixed cash flows, and can thus be applied to a wider range of instruments than can Macaulay duration.

For every-day use, the equality (or near-equality) of the values for Macaulay and modified duration can be a useful aid to intuition. For example a standard ten-year coupon bond will have Macaulay duration somewhat but not dramatically less than 10 years and from this we can infer that the modified duration (price sensitivity) will also be somewhat but not dramatically less than 10%. Similarly, a two-year coupon bond will have Macaulay duration somewhat below 2 years, and modified duration somewhat below 2%. (For example a ten-year 5% par bond has a modified duration of 7.8% while a two-year 5% par bond has a modified duration of 1.9%.)