Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.

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1 Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised paymens are fixed, bond prices change in response o changes in he marke deermined required rae of reurn. For invesor's who hold bonds, he issue of how sensiive a bond's price is o changes in he required rae of reurn is imporan. here are four measures of bond price sensiiviy ha are commonly used. hey are Simple Mauriy, Macaulay Duraion (effecive mauri, Modified Duraion, and Convexiy. Each of hese provides a more exac descripion of how a bond price changes relaive o changes in he required rae of reurn. Mauriy Simple mauriy is jus he ime lef o mauriy on a bond. We generally hink of 5-year bonds or 0-year bonds. I is sraighforward and requires no calculaion. he longer he ime o mauriy he more sensiive a paricular bond is o changes in he required rae of reurn. Consider wo zero coupon bonds, each wih a face value of $,000. Bond A maures in 0 years and has a required rae of reurn of 0%. he price of Bond A is $376.89, where A = $,000 ( +.0 / ) 0 = $ Bond B has a mauriy of 5 years and also has a required rae of reurn of 0%. Is price is $63.9 or $,000 = $ / = B ( ) By convenion, zero coupon bonds are compounded on a semi -annual basis. Since almos all US bonds have semi-annual coupon paymens, his noe will always assume semi-annual compounding unless oherwise noed. his noe was wrien by Rober M. Conroy, rofessor of Business Adminisraion, Darden Graduae School of Business Adminisraion, Universiy of Virginia. Copyrigh 998 by he Universiy of Virginia Darden School Foundaion, Charloesville, VA. All righs reserved. o order copies, send an o dardencases@virginia.edu. No par of his publicaion may be reproduced, sored in a rerieval sysem, used in a spreadshee or ransmied in any form or by any means-elecronic, mechanical, phoocopying, recording or oherwise wihou he permission of he Darden School Foundaion. Rev. 7/0.

2 -- UVA-F-38 If he required rae of reurn for each bond was o increase by 00 basis poins o %, he prices would hen be $34.73 for Bond A and $ for Bond B. his ranslaes ino a -9.% change in price for Bond A and -4.6% for Bond B. Jus from he pricing formulaion, i is clear ha any change in ineres raes will have a much Figure. Comparison of 5-year and 0-year Bonds year 0-year greaer impac on Bond A han Bond B. his is reinforced in Figure, where he price curve for he 0-year bond (Bond A) is much seaper han ha for he 5-year bond (Bond B). hus, for zero coupon bonds simple mauriy can be used o compare price sensiiviy. Macaulay Duraion (Effecive Mauri Required Rae of Reurn he relaionship beween price and mauriy is no as clear when you consider non-zero coupon bonds. For a coupon-paying bond, many of he cash flows occur before he acual mauriy of he bond and he relaive iming of hese cash flows will affec he pricing of he bond. In order o deal wih his, Frederick Macaulay in 938 suggesed ha invesors use he effecive mauriy of a bond as a measure of ineres rae sensiiviy. He called his duraion and defined i as a value-weighed average of he iming of he cash flows. he easies way o see his is o use an example. Consider a six-year bond wih face value of $,000, and a 6.% coupon rae (semi-annual paymens). If he curren yield o mauriy is 0%, he value of he bond is found by discouning each of he semi-annual paymens. his is shown in Exhibi. Fredick Macaulay, Some heoreical roblems Suggesed by he Movemen of Ineres Raes, Bond Yields, and Sock rices in he Unied Saes since 856 (New York: Naional Bureau of Economic Research, 938).

3 -3- UVA-F-38 Exhibi Cash Flow res. Value Facor res. Value of Cash Flow $ $ , Macaulay Duraion akes he presen value of each paymen and divides i by he oal bond price,. By doing his, one has a percenage, w, of he oal bond value ha is received in each period,. w C ( + = he duraion or effecive mauriy for he bond could hen be esimaed by muliplying he weigh, w, imes he ime, and hen summing all of he weighed values, or Duraion = = C ( + = = w. his measure akes ino accoun he relaive iming of he cash flows. Calculaion of he Macaulay Duraion measure is fairly sraighforward bu can be somewha edious 3. Exhibi shows how a semi-annual duraion for he example shown above would be calculaed. 3 Excel offers a workshee funcion DURAION(.), which calculaes he Macaulay Duraion.

4 -4- UVA-F-38 Exhibi Cash Flow res. Value Facor res. Value of Cash Flow Weigh x weigh $ $ % % % % % % % % % % % 0.37, % Bond Value $ 87.7 Semi-Annual Duraion 0.04 he semi-annual duraion for his bond is 0.04 six-monh periods. We usually use annual duraion and we annualize he semi-annual duraion simply by dividing by (he number of six monh periods in a year). In his case, he annualized duraion would be years. Noe ha he Macaulay Duraion for a 5-year zero coupon bond is he same as he simple mauriy, 5.0 years. Hence, we can expec ha he original 6-year, 6.% coupon bond when ineres raes change o behave in a manner similar o a 5-year zero coupon bond, since heir effecive mauriy (Macaulay Duraion) is essenially he same. Modified Duraion If we wan a more direc measure of he relaionship beween changes in ineres raes and changes in bond prices, we can use Modified Duraion. Modified Duraion, D, is defined as he following D = y where is he bond price, is he change in bond price and y is he change in he required rae of reurn (yield o mauri. For hose wih a mah background, y is he firs derivaive of he bond price wih respec o yield o mauriy. he basic pricing formulaion for bonds is

5 -5- UVA-F-38 = C = + ( where C is he cash paymen received in ime period and y is he semi-annual yield o mauriy. aking he derivaive of wih respec oy, y = ( + C = ( +. Insering his ino he formula for Modified Duraion yields, D = C ( + ( + ) = y Rearranging he above slighly, D = ( + = C ( +. Comparing his o he definiion of Macaulay Duraion and using ha definiion we can wrie Modified Duraion as Modified Duraion = D = ( + Macaulay Duraion While i is easy calculae Modified Duraion once you have Macaulay Duraion he inerpreaions of he wo are quie differen. Macaulay Duraion is an average or effecive mauriy. Modified Duraion really measures how small changes in he yield o mauriy affec he price of he bond. In fac, from he definiion of Modified Duraion we can wrie he following relaionship: = D y or % change in bond price = - Modified Duraion imes he change in yield o mauriy. For example, he six-year 6.% coupon bond above had a yield o mauriy of 0% and a semi-annual Macaulay Duraion of 0.04 (5.007 annual Macaulay Duraion). he Modified Duraion of his bond is or on a semi-annual basis or = 4.77 years on an ( +.05)

6 -6- UVA-F-38 annual basis 4. Assuming ha he yield o mauriy of 0% increases by 5 basis poins o 0.5%, based on he Modified Duraion of 4.77 years he price of he bond should change by. he bond price should drop by.9% from $87.7 o $87.3 = D y = 4.77 (.5%) =.9% ($87.7*(-.09) = $87.3). he acual calculaed price a a yield o mauriy of 0.5% is $ Exhibi 3 shows he Modified Duraion price change and he acual calculaed price change for differen changes in yield o mauriy. Bond Daa Coupon = 6.% Mauriy= 6 years Face Value = $,000 Yield o Mauriy= 0% rice = $87.7 Modifed Duraion= Exhibi 3 New Yield o Mauriy Change in Yield -D*Change in Yield rediced rice Acual % change* Acual rice** Difference***.00%.00% -9.54% % %.75% -8.35% % %.50% -7.6% % %.5% -5.96% % %.00% -4.77% % % 0.75% -3.58% % % 0.50% -.39% % % 0.5% -.9% % % 0.00% 0.00% % % -0.5%.9% % % -0.50%.39% % % -0.75% 3.58% % % -.00% 4.77% % % -.5% 5.96% % % -.50% 7.6% % % -.75% 8.35% % % -.00% 9.54% % * Acual % change is based on he calculaed price relaive o he price of $87.7. ** Acual price is he calculaed price based on he yield o mauriy. *** Difference is Acual rice - rediced rice. Modified Duraion assumes ha he price changes are linear wih respec o changes in he yield o mauriy. From Exhibi 3, he rue relaionship beween he bond's price and he yield o mauriy is no linear. he Column wih he differences is always posiive and increases 4 If he original compounding basis on he bond was semi -annual, he Modified Duraion mus firs be calculaed on a semi-annual basis and hen annualized. You can no use he annual Macaulay Duraion o calculae he Modified Duraion.

7 -7- UVA-F-38 as we move away from a yield o mauriy of 0%. he acual relaionship beween he bond price and he yield o mauriy is shown in Figure. Figure. Bond rices 6-year, 6.% Coupon (paid semi-annuall Rae, Face value = $,000,600.00,400.00,00.00, Yield o Mauriy he curved line is he acual price curve. he sraigh line is he price relaionship using Modified Duraion. Everywhere he acual price curve is above he Modified Duraion relaionship. his is exacly wha we saw in Exhibi 3. he difference was always posiive, i.e., acual calculaed price was greaer han he new price using he Modified Duraion relaionship. In addiion, he percenage changes in price are no symmeric. he percenage decrease in price for a given increase in yield is always less han he percen increase for he same decrease in yield. his propery is refered o as convexiy. Noe ha he wo prices are quie close for small changes in he yield o mauriy bu he difference grows as he change in yield o mauriy becomes bigger. Convexiy. From Figure i is clear ha he Modified Duraion relaionship does no fully capure he rue relaionship beween bond prices and yield o mauriy. In order o more fully capure his, praciioners use Convexiy. he definiion of Convexiy is Convexiy= CV = ( Once again hose wih a mah background will recognize he las erm on he righ as he second derivaive of price wih respec o yield o mauriy. he acual definiion of Convexiy ha we can use is

8 -8- UVA-F-38 Convexiy= CV = = ( ( + = ( + ) C ( + y ) Exhibi 4 shows he calculaion of he semi-annual convexiy for he six-year 6.% coupon bond. Exhibi 4 Convexiy Cash Flow res. Value Facor res. Value of Cash Flow Weigh x( + ) weigh ( + $ $ % % % % % % % % % % %.58, % Bond Value $87.7 Semi-annual Convexiy 0.88 We can annualize he semi-annual convexiy of 0.88 by dividing 5 i by or 4. Here i would be 7.7. Convexiy is useful o praciioners in a number of ways. Firs i can be used in conjuncion wih duraion o ge a more accurae esimae of he percenage price change resuling from a change in he yield. he formula 6 is % rice = - Modified Duraion y + Convexiy ( 5 Convexiy is annualized by dividing he calculaed Convexiy by he number of paymens per year squared. 6 For hose wih a mah ben, his formula is based on using a aylor series expansion o approximae he value of he percenage change in price.

9 -9- UVA-F-38 Adding he convexiy adjusmen correcs for he fac ha Modified Duraion undersaes he rue bond price. For example, in Exhibi 3, a a yield of % he percenage price change using only Modified Duraion was -9.54%, while he acual was -9.0%. If we use he Convexiy value we jus calculaed, he prediced percenage price change would be % rice = (.0) (.0) = = his is -8.99%, which is much closer o he acual percenage price change of -9.0%. he pricing aspec of Convexiy is much less imporan now since mos people have access o calculaors and compuers ha can do he pricing. he more imporan use of convexiy is ha i provides insigh ino how a bond will reac o yield changes. Again from Exhibi 3 and Figure we see ha he price reacion o changes in yield is no symmeric. For a given change in yield, bond prices drop less for a given increase in yield and increase more for he same decreases in yield. he downside is less and he upside is more. his is clearly a desirable propery. he higher he Convexiy of a bond he more his is rue. hus, bonds wih high convexiy are more desirable. Summary Each of he price sensiiviy measures discussed in his noe is par of he everyday language and hinking of fixed income invesors. hey are relaive risk measures ha help invesmen professionals hink abou he risks hey face.