Duration and Convexity

UVA-F-1238

DURATION AND CONVEXITY

The price of a bond is a function of the promised payments and the market-required rate

of return. Because the promised payments are fixed, bond prices change in response to changes

in the market-required rate of return. For investors who hold bonds, the issue of how sensitive a

bond’s price is to changes in the required rate of return is important. There are four measures of

bond-price sensitivity that are commonly used: Simple Maturity, Macaulay Duration (effective

maturity), Modified Duration, and Convexity. Each of these measures provides a more exact

description of how a bond price changes relative to changes in the required rate of return.

Maturity

Simple maturity is just the time left to maturity on a bond. We generally think of fiveyear bonds or ten-year bonds. It is straightforward and requires no calculation. The longer the

time to maturity, the more sensitive a particular bond is to changes in the required rate of return.

Consider two zero-coupon bonds, each with a face value of $1,000. Bond A matures in 10 years,

and has a required rate of return of 10%. The price1 of Bond A is $376.89, where

PA =

$1,000

= $376.89 .

(1 + .10 / 2)20

Bond B has a maturity of five years and has a required rate of return of 10%. Its price is $613.91,

or

$1,000

PB =

= $613.91 .

(1 + .10 / 2)10

If the required rate of return for each bond was to increase by 100 basis points, to 11%,

the prices would then be $342.73 for Bond A and $585.43 for Bond B. This translates into a

−9.1% change in price for Bond A and −4.6% for Bond B.

1

By convention, zero-coupon bonds are compounded on a semiannual basis. Because almost all U.S. bonds

have semiannual coupon payments, this note will always assume semiannual compounding unless otherwise noted.

This note was written by Robert M. Conroy, Professor of Business Administration. Copyright  1998 by the

University of Virginia Darden School Foundation,...