Properties of Macaulay Duration

Duration varies with maturity, coupon, and yield. Broadly, it increases with maturity. A bonds duration is generally shorter than its maturity. This is because the cash flows received in the early years of the bond s life have the greatest present values and therefore are given the greatest weight. That shortens the avetj e time in which cash flows are received. A zero-coupon bond s cash flows are all received at redemption, so there is no present-value weighting. Therefore, a zero-coupon bond s duration is equal to its term to maturity. 

Duration increases as coupon and yield decrease. The lower the coupon, the greater the relative weight of the cash flows received on the maturity date, and this causes duration to rise. T ong the non—plain vanilla types of bonds are some whose coupon rate varies according to an index, usually the consumer price index. Index-linked bonds generally have much lower coupons than vanilla bonds with similar maturities. This is true because they are inflation-protected, causing the real yield required to be lower than the nominal yield, but their durations tend to be higher. 

Yield s relationship to duration is a function of its role in discounting future cash flows. As yield increases, the present values of all future cash flows fall, but those of the more distant cash flows fall relatively more. This has the effect of increasing the relative weight of the earlier cash flows and hence of reducing duration. 

Although newcomers to the market commonly consider duration, much as Macaulay did, a proxy for a bond s time to maturity, this interpretation misses the main point of duration, which is to measure price volatility, or interest rate risk. Using the Macaulay duration can derive a measure of a bond s interest rate price sensitivity, i.e., how sensitive a bond s price is to changes in its yield. This measure is obtained by applying a mathematical property known as a Taylor expansion to the basic equation. 

The relationship between price volatility and duration can be made clearer if the bond price equation, viewed as a function of r, is expanded as a Taylor series (see Butler, pp. 112—114 for an accessible explanation of Taylor expansions). Using the first term of this series, the relationship can be expressed as (2.13). 

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