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
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.
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 email@example.com. No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet or transmitted in any form or by any means-electronic, mechanical, photocopying, recording or otherwise without the permission of the Darden School Foundation. Rev. 7/02.
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. Just from the pricing formulation, it is clear that any change in interest rates will have a much Figure 1. Comparison of 5-year and 10-year Bonds
Required Rate of Return
greater impact on Bond A than Bond B. This is reinforced in Figure 1, where the price curve for the 10-year bond (Bond A) is much steaper than that for the 5-year bond (Bond B). Thus, for zero coupon bonds simple maturity can be used to compare price sensitivity. Macaulay Duration (Effective Maturity) The relationship between price and maturity is not as clear when yo u consider non-zero coupon bonds. For a coupon-paying bond, many of the cash flows occur before the actual maturity of the bond and the relative timing of these cash flows will affect the pricing of the bond. In order to deal with this, Frederick Macaulay2 in 1938 suggested that investors use the effective maturity of a bond as a measure of interest rate sensitivity. He called this duration and defined it as a value-weighted average of the timing of the cash flows. The easiest way to see this is to use an example. Consider a six- year bond with face value of $1,000, and a 6.1% coupon rate (semi-annual payments). If the current yield to maturity is 10%, the value of the bond is found by discounting each of the semi-annual payments. This is shown in Exhibit 1. Fredick Macaulay, Some Theoretical Problems Suggested by the Movement of Interest Rates, Bond Yields,...