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UVA-F-1238 University of Virginia
The price of a bond is a function of the promised payments and the market required rateof return. Since the promised payments are fixed, bond prices change in response to changes inthe market determined required rate of return. For investor's who hold bonds, the issue of howsensitive a bond's price is to changes in the required rate of return is important. There are fourmeasures of bond price sensitivity that are commonly used. They are Simple Maturity,Macaulay Duration (effective maturity), Modified Duration, and Convexity. Each of theseprovides a more exact description of how a bond price changes relative to changes in therequired rate of return.
Maturity
Simple maturity is just the time left to maturity on a bond. We generally think of 5-yearbonds or 10-year bonds. It is straightforward and requires no calculation. The longer the time tomaturity 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 yearsand has a required rate of return of 10%. The price 1 of Bond A is $376.89, where
$1,000 PA = = $376.89 (1 + .10 / 2 )20Bond 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
1 By convention, zero coupon bonds are compounded on a semi -annual basis. Since almost all US bondshave 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 ofBusiness Administration, University of Virginia. Copyright 1998 by the University of Virginia Darden SchoolFoundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail todardencases@virginia.edu. No part of this publication may be reproduced, stored in a retrieval system, used in aspreadsheet or transmitted in any form or by any means-electronic, mechanical, photocopying, recording orotherwise without the permission of the Darden School Foundation. Rev. 7/02. -2- UVA-F-1238
If the required rate of return for each bond was to increase by 100 basis points to 11%, the priceswould then be $342.73 for Bond A and $585.43 for Bond B. This translates into a -9.1% changein 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
1200
1000
800
5-year 600 10-year
400
200
greater impact on Bond A than Bond B. This is reinforced in Figure 1, where the price curve forthe 10-year bond (Bond A) is much steaper than that for the 5-year bond (Bond B). Thus, forzero coupon bonds simple maturity can be used to compare price sensitivity.
The relationship between price and maturity is not as clear when yo u consider non-zerocoupon bonds. For a coupon-paying bond, many of the cash flows occur before the actualmaturity of the bond and the relative timing of these cash flows will affect the pricing of thebond. In order to deal with this, Frederick Macaulay2 in 1938 suggested that investors use theeffective maturity of a bond as a measure of interest rate sensitivity. He called this duration anddefined it as a value-weighted average of the timing of the cash flows. The easiest way to seethis 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 thebond is found by discounting each of the semi-annual payments. This is shown in Exhibit 1. 2 Fredick Macaulay, Some Theoretical Problems Suggested by the Movement of Interest Rates, Bond Yields,and Stock Prices in the United States since 1856 (New York: National Bureau of Economic Research, 1938). -3- UVA-F-1238
Exhibit 1
Macaulay Duration takes the present value of each payment and divides it by the totalbond price, P. By doing this, one has a percentage, wt , of the total bond value that is received ineach period, t.
Ct (1 + y ) t wt = P
The duration or effective maturity for the bond could then be estimated by multiplying theweight, wt , times the time, t and then summing all of the weighted values, or
Ct Duration = ∑ T (1 + y )t T ⋅ t = ∑ wt ⋅ t . t =1 P t =1
This measure takes into account the relative timing of the cash flows. Calculation of theMacaulay Duration measure is fairly straightforward but can be somewhat tedious 3 . Exhibit 2shows how a semi-annual duration for the example shown above would be calculated.
3 Excel offers a worksheet function DURATION(.), which calculates the Macaulay Duration. -4- UVA-F-1238
Exhibit 2
Semi-Annual Bond Value $ 827.17 Duration 10.014
The semi- annual duration for this bond is 10.014 six- month periods. We usually use annualduration and we annualize the semi-annual duration simply by dividing by 2 (the number of sixmonth periods in a year). In this case, the annualized duration would be 5.007 years. Note thatthe Macaulay Duration for a 5- year zero coupon bond is the same as the simple maturity, 5.0years. Hence, we can expect that the original 6-year, 6.1% coupon bond when interest rateschange to behave in a manner similar to a 5-year zero coupon bond, since their effective maturity(Macaulay Duration) is essentially the same.
Modified Duration
If we want a more direct measure of the relationship between changes in interest ratesand changes in bond prices, we can use Modified Duration. Modified Duration, D, is defined asthe following
1 ∆P D=− ⋅ P ∆y
where P is the bond price, ∆P is the change in bond price and ∆y is the change in the requiredrate of return (yield to maturity). For those with a math background, ∆P ∆y is the first derivativeof the bond price with respect to yield to maturity. The basic pricing formulation for bonds is -5- UVA-F-1238
T Ct P= ∑ t =1 (1 + y) t
where Ct is the cash payment received in time period t and y is the semi-annual yield to maturity.Taking the derivative of P with respect toy,
∆P = − 1 ⋅ T t ⋅ C t . ∑ ∆y (1 + y ) t = 1 (1 + y )t
1 1 T Ct D=− ⋅ − ⋅∑t ⋅ P (1 + y ) t =1 (1 + y )t
Comparing this to the definition of Macaulay Duration and using that definition we can writeModified Duration as
1 Modified Duration = D = Macaulay Duration (1 + y )While it is easy calculate Modified Duration once you have Macaulay Duration theinterpretations of the two are quite different. Macaulay Duration is an average or effectivematurity. Modified Duration really measures how small changes in the yield to maturity affectthe price of the bond. In fact, from the definition of Modified Duration we can write thefollowing relationship:
∆P = − D ⋅ ∆y P
or % change in bond price = - Modified Duration times the change in yield to maturity.
For example, the six- year 6.1% coupon bond above had a yield to maturity of 10% and asemi-annual Macaulay Duration of 10.014 (5.007 annual Macaulay Duration). The ModifiedDuration of this bond is 10.014 or 9.537 on a semi-annual basis or 9.537 = 4.77 years on an (1 + .05 ) 2 -6- UVA-F-1238
annual basis 4 . Assuming that the yield to maturity of 10% increases by 25 basis points to10.25%, based on the Modified Duration of 4.77 years the price of the bond should change by∆P = − D ⋅ ∆y = − 4.77 ⋅ (.25 %) = − 1.19% . The bond price should drop by 1.19% from $827.17 to $817.31 P($827.17*(1-.0119) = $817.31). The actual calculated price at a yield to maturity of 10.25% is$817.38.
Exhibit 3 shows the Modified Duration price change and the actual calculated pricechange for different changes in yield to maturity.
Exhibit 3 Bond Data Coupon = 6.1% Maturity= 6 years Face Value = $1,000 Yield to Maturity= 10% Price = $827.17 Modifed Duration= 4.770
Modified Duration assumes that the price changes are linear with respect to changes inthe yield to maturity. From Exhibit 3, the true relationship between the bond's price and theyield to maturity is not linear. The Column with the differences is always positive and increases
4 If the original compounding basis on the bond was semi -annual, the Modified Duration must first becalculated on a semi-annual basis and then annualized. You can not use the annual Macaulay Duration to calculatethe Modified Duration. -7- UVA-F-1238
as we move away from a yield to maturity of 10%. The actual relationship between the bondprice and the yield to maturity is shown in Figure 2. Figure 2. Bond Prices 6-year, 6.1% Coupon (paid semi-annually) Rate, Face value = $1,000
1,600.00
1,400.00
1,200.00
1,000.00
800.00
600.00
400.00
200.00
Yield to Maturity
The curved line is the actual price curve. The straight line is the price relationship usingModified Duration. Everywhere the actual price curve is above the Modified Durationrelationship. This is exactly what we saw in Exhibit 3. The difference was always positive, i.e.,actual calculated price was greater than the new price using the Modified Duration relationship.In addition, the percentage changes in price are not symmetric. The percentage decrease in pricefor a given increase in yield is always less than the percent increase for the same decrease inyield. This property is refered to as convexity. Note that the two prices are quite close for smallchanges in the yield to maturity but the difference grows as the change in yield to maturitybecomes bigger.
Convexity.
From Figure 2 it is clear that the Modified Duration relationship does not fully capturethe true relationship between bond prices and yield to maturity. In order to more fully capturethis, practitioners use Convexity. The definition of Convexity is
1 ∆2 P Convexity = CV = ⋅ P (∆y )2
Once again those with a math background will recognize the last term on the right as the secondderivative of price with respect to yield to maturity. The actual definition of Convexity that wecan use is -8- UVA-F-1238
Ct 1 ∆P 2 1 T (1 + y )t Convexity = CV = ⋅ = ⋅ ∑ t ⋅ (t + 1) ⋅ P (∆y ) 2 (1 + y )2 t=1 P
Exhibit 4 shows the calculation of the semi- annual convexity for the six-year 6.1% coupon bond.
Exhibit 4
Convexity Pres. 1 t Cash Value Pres. Value of Weight t x(t + 1) × weight × Flow Factor Cash Flow (1 + y) 2
1 % ∆ Price = - Modified Duration × ∆y + × Convexity × (∆y)2 2
5 Convexity is annualized by dividing the calculated Convexity by the number of payments per yearsquared. 6 For those with a math bent, this formula is based on using a Taylor series expansion to approximate thevalue of the percentage change in price. -9- UVA-F-1238
Adding the convexity adjustment corrects for the fact that Modified Duration understates the truebond price. For example, in Exhibit 3, at a yield of 12% the percentage price change using onlyModified Duration was -9.54%, while the actual was -9.01%. If we use the Convexity value wejust calculated, the predicted percentage price change would be
1 % ∆ Price = - 4.77 × (.02) + × 27.72 × (.02)2 = -.0954 + .00554 = -.0899. 2
This is -8.99%, which is much closer to the actual percentage price change of -9.01%.
The pricing aspect of Convexity is much less important now since most people have access tocalculators and computers that can do the pricing. The more important use of convexity is that itprovides insight into how a bond will react to yield changes. Again from Exhibit 3 and Figure 2we see that the price reaction to changes in yield is not symmetric. For a given change in yield,bond prices drop less for a given increase in yield and increase more for the same decreases inyield. The downside is less and the upside is more. This is clearly a desirable property. Thehigher the Convexity of a bond the more this is true. Thus, bonds with high convexity are moredesirable.
Summary
Each of the price sensitivity measures discussed in this note is part of the everydaylanguage and thinking of fixed income investors. They are relative risk measures that helpinvestment professionals think about the risks they face.
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