Duration and Convexity
Graduate School of Business Administration University of Virginia
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. Since the promised payments are fixed, bond prices change in response to changes in the market determined required rate of return. For investor's 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. They are Simple Maturity, Macaulay Duration (effective maturity), Modified Duration, and Convexity. Each of these 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 5-year bonds or 10-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 price 1 of Bond A is $376.89, where
PA =
(1 + .10 / 2 )20
$1,000
= $376.89
Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.91 or $1,000 PB = = $613.91 (1 + .10 / 2 )10
By convention, zero coupon bonds are compounded on a semi -annual basis. Since almost all US bonds have semi-annual coupon payments, this note will always assume semi-annual compounding unless otherwise noted.
1
This note was written by Robert M. Conroy, Professor of Business Administration, Darden Graduate School of Business Administration, University of Virginia. Copyright © 1998 by the University of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail to dardencases@virginia.edu. No part of
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. Since the promised payments are fixed, bond prices change in response to changes in the market determined required rate of return. For investor's 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. They are Simple Maturity, Macaulay Duration (effective maturity), Modified Duration, and Convexity. Each of these 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 5-year bonds or 10-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 price 1 of Bond A is $376.89, where
PA =
(1 + .10 / 2 )20
$1,000
= $376.89
Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.91 or $1,000 PB = = $613.91 (1 + .10 / 2 )10
By convention, zero coupon bonds are compounded on a semi -annual basis. Since almost all US bonds have semi-annual coupon payments, this note will always assume semi-annual compounding unless otherwise noted.
1
This note was written by Robert M. Conroy, Professor of Business Administration, Darden Graduate School of Business Administration, University of Virginia. Copyright © 1998 by the University of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail to dardencases@virginia.edu. No part of