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Duration and Convexity The Actuary India January 2010 STUDENT’S COLUMN Duration and Convexity Saket Vasisth [email protected] 7KHSULFHRID¿[HGLQFRPHVHFXULW\QHJDWLYHO\ Coming back to bonds, Dollar duration for bonds is the negative of the relates to the interest rate i.e. its yield. An ¿UVWGHULYDWLYHRIWKHSULFH\LHOGIXQFWLRQLH'ROODU'XUDWLRQ G30 / upward movement in interest rate will cause a dy0 ¨3¨\ DSSUR[LPDWHO\ ,WLVWKH¿UVWGHULYDWLYHRIWKHSULFH±\LHOG downward movement in price and vice versa. function so it measures the slope of the tangent to the price – yield curve. Interest rate risk is one of the prime concerns 0RGL¿HG'XUDWLRQ 'ROODUGXUDWLRQ,QLWLDO3ULFHRIWKHERQGLQFOXGLQJ IRU D ¿[HG LQFRPH GHDOHU )RU PHDVXULQJ any accrued interest at time 0. LQWHUHVW UDWH ULVN LQ ¿[HG LQFRPH ERQGV approaches are used: (i) the full valuation EDVLVSRLQWVHQVLWLYLW\ 'ROODUGXUDWLRQ ĺURXJKO\DSSUR[LPDWHV DSSURDFK LH¿QGLQJYDOXHE\GLVFRXQWLQJWKH the change in bond price if yield changes by 0.01%. IXOOVWUHDPRIFDVKÀRZV DQG LL WKHGXUDWLRQ DQGFRQYH[LW\DSSURDFK7KHFRQFHSWRIGXUDWLRQFRQYH[LW\¿QGVZLGH This interpretation of duration can be used for analyzing change in price application in contemporary bond portfolio management. Primarily it is due to change in yield for a Zero Coupon Bond where price – yield curve is used to: simple and differentiable. t t t+1 Ɣ Quantify the impact of changes of interest rates on bond prices. P0 = CFt / (1+yt) => Dollar Duration = -t * CF / (1+yt) = (-t * P0 ) / (1+yt) Ɣ Asset – liability management of the banks, insurance companies, $QG0RGL¿HG'XUDWLRQ W \t). CFt is the redeemable amount at time t. mutual funds etc. Here we attempt to analyze the second approach. Duration measure assumes linear relationship between price and yield. In reality, the relationship between the changes in price and yield There are 3 schools of thought on how to interpret duration: LVFRQYH[,QWKH¿JXUHEHORZWKHFXUYHGOLQHUHSUHVHQWVWKHFKDQJH Ɣ 'XUDWLRQLVWKH¿UVWGHULYDWLYHRIWKHSULFH\LHOGIXQFWLRQ in prices when yield changes using the full valuation approach. The Ɣ 'XUDWLRQLVWKHGLVFRXQWHGPHDQWHUPRIFDVKÀRZVZHLJKWHGE\WKH straight line, tangent to the curve, represents the estimated change in present value of the bond. price calculated using the duration measure. The area between these Ɣ Duration is a measure of approximate sensitivity of the bond’s value two lines is the difference between the duration estimate and the actual to change in yield. price movement. As indicated, the larger the change in interest rates, the Ɣ Let us analyze these three interpretations one by one. larger the error in estimating the price change of the bond. >,@'XUDWLRQDVWKH¿UVWGHULYDWLYHRIWKHSULFH±\LHOGIXQFWLRQ As per this interpretation price of a bond is a function of its yield i.e. P0 = f (y0). Where P0 is the initial price of the bond and y0 is the initial yield of the bond. Curve of price - yield function is negatively sloped meaning as yield increases price decreases and when yield decreases price increases. Brook Taylor, an English mathematician in 1915 came RXWZLWKWKHIROORZLQJHTXDWLRQWRSUHGLFWWKH¿[HGLQWHUHVWERQG¶VYDOXH when its yield is changed and the relationship between yield and bond price is non-linear: -1 2 -1 P1 = P0 + (f ’(y0) * (y1 - y0)) + (2! * f ’’(y0) * (y1 - y0) ) + (3! * f ’’’(y0) * (y1 3 - y0) ) + - - - - - Where P1 is the revised price of the bond due to change in the yield, y1 is the revised yield and f’(y0 LVWKH¿UVWGHULYDWLYHRIWKHSULFH\LHOGIXQFWLRQ and f ’’(y0) is the second derivative of the price - yield function and so on. ,QFDVHRI¿[HGLQWHUHVWERQGVWKH¿UVWGHULYDWLYHLVXVHGIRUFDOFXODWLRQ Here we introduce convexity. Convexity is the second derivative of the of duration and the second derivative is used for calculation of convexity. price – yield function. Convexity is an approximation of the change in the Above result by Taylor can also be applied for approximating change in the bond’s duration resulting from interest rate change. It is a measure of the value of a derivative contract. For example: Option on any underlying asset: curvature of the changes in the price of a bond in relation to changes in Revised price of the option = interest rates and is used to address the above approximation error. One Current price of the option + (f’(current price of the underlying asset) * change in may wonder that why we do not use the third or higher order term in the the price of the underlying asset) + (2!-1* f ’’(current price of the underlying asset) Taylor expansion series. Well, additional complexity introduced by the * change in the price of the underlying asset2) + - - - - - - third or higher order terms do not contribute much to the improvement in +HUHWKH¿UVWGHULYDWLYHLVFDOOHGGHOWDDQGWKHVHFRQGGHULYDWLYHLVFDOOHGJDPPD WKH¿WWRWKHSULFH\LHOGFXUYH 16 Indian Actuarial Profession Serving the Cause of Public Interest The Actuary India January 2010 Convexity * P = d2P / dy2 0 Dollar 889.38 Dollar Duration 444.69 For a Zero Coupon Bond, convexity 2 2 Duration (in (in full years) =889.38/2 = (d P / dy ) / P0 2 half years) = ((t+1) * t * P0/ (1+y) ) / P0 = (t+1) * t / (1+y) 2 Convexity (in 859.84 Convexity (in 214.96 half years) =31*30/(1.04^2) years) =859.84/4 Note: Duration 4.4469 Convexity 0.3314 &RPSRXQGLQJĺ Annual Semi - Annual For conversion adjustment =(30*30.8319*0.01) adjustment to into years to initial price / (2*1.04) initial price Duration measure Years Half Years Duration measure when yield when yield is divided by 2 increases / increases / decreases decreases Convexity Years Half Years Convexity measure squared measure is Duration and Convexity Estimate of Market price is closer to the actual divided by 4 price using the full valuation approach as compared to the duration only estimate of the price. Hence, the approximation for estimating bond’s value because of change in yield can also be written as follows: Criticism of this interpretation: Revised Price This is an alternate approach for calculation of approximate change in ,QLWLDO3ULFH 0RGL¿HGGXUDWLRQ ,QLWLDOSULFH \1 – y0)) the value of a Zero Coupon Bond because of change in yield. However, -1 2 + (2! * Convexity * P0 * (y1 – y0) ) this is practically less useful interpretation as it does not contribute much in understanding the interest rate risk of a bond. (i) Actual price when yield changes (using full valuation approach) = 3UHVHQWYDOXHRIWKHFDVKÀRZVXVLQJFKDQJHG\LHOG >,,@'XUDWLRQDVDPHDVXUHRIFDVKÀRZVZHLJKWHGE\WKHSUHVHQWYDOXH (ii) Duration Estimate of actual price when yield changes (Ignoring of the bond: second and higher order terms) For a coupon paying bond price – yield function is not continuous. Many = P0 0RGL¿HGGXUDWLRQ 30 * (y1 – y0)) RIWKHFDVKÀRZVRFFXUEHIRUHDFWXDOPDWXULW\DQGWKHUHODWLYHWLPLQJRI (iii) Duration and Convexity Estimate of Market price when yield WKHVHFDVKÀRZVZLOODIIHFWWKHSULFLQJRIWKHERQG,QRUGHUWRGHDOZLWK changes (Ignoring third and higher order terms) this Frederick Macaulay came up with the effective maturity of a bond as 2 = P0 0RGL¿HGGXUDWLRQ 30 * (y1 – y0)) + (0.5 * convexity * P0*(y0 – y1) ) DPHDVXUHRIGXUDWLRQ'XUDWLRQJRWGH¿QHGDVWKHZHLJKWHGDYHUDJHRI SUHVHQWYDOXHRIFDVKÀRZVEDVHGRQWKHWLPLQJRIWKHFDVKÀRZV Example: Macaulay’s duration or Discounted Mean Term Zero Coupon Bond of 15 years redeemable at $100 - yield 8% with semiannual compounding n N &) N \ )-k Full Valuation Approach 0 k=1 Y0 Y0 + 0.01 Y0 - 0.01 = ----------------------- Yield p.a. 8% 9% 7% n -k Yield half 4% 4.50% 3.50% &) N \0) year k=1 Market Price $30.8319 = $26.7 = $35.6278 = 7KLVUHSUHVHQWVWKHDYHUDJHWLPHWRZDLWIRUDOOFDVKÀRZVWRRFFXU 100 / (1.04^(15*2)) 100 / 100 / Macaulay’s duration for Zero Coupon Bond = Term of that Zero Coupon Bond. (1.045^(15*2)) (1.035^(15*2)) 1RWH&DSIRUGXUDWLRQRIDQ\¿[HGLQWHUHVWERQGLVJLYHQE\ \HLOG \LHOG Duration and Convexity Calculations Convexity Duration Estimate of Market Price $ 26.3850 = $ 35.2788 = n -(k+2) $ 30.8319 - $ $ 30.8319 + $ N N &) N \0) 4.4469 4.4469 k=1 Duration and Convexity Estimate of $ 26.7163 = $ 35.6102 = = ----------------------------------------------- Market Price $ 30.8319 - $ $ 30.8319 + n &) N \ )-k 4.4469 + $ $ 4.4469 + $ 0 0.3314 0.3314 k=1 Workings: 0RGL¿HG 28.85 0RGL¿HG 14.42 Criticism of this interpretation: Duration (in =(15*2)/(1.04) Duration (in =28.85/2 (i) If a bond has duration of say 5 years, then it is not useful to think half years) full years, this measure in terms of time, but rather the bond has the price ignoring sensitivity to yield changes of a 5 year Zero Coupon bond. negative sign) (ii) For an interest only security say a Mortgage backed security duration Indian Actuarial Profession Serving the Cause of Public Interest 17 The Actuary India January 2010 LVQHJDWLYH$QGLWLVGLI¿FXOWWRLQWHUSUHWWLPHDVDQHJDWLYHTXDQWLW\ year)) (iii) Certain collateralized mortgage obligation (CMO) bond classes have Some observations: a greater duration than the underlying mortgage loans. That is, a CMO Ɣ &ƝWHUƯVSDULEXVWKHORQJHUWKHPDWXULW\JUHDWHUZLOOEHWKHPRGL¿HG bond class can have duration of 50 although the underlying mortgage duration. loans from which the CMO is created can have a maturity of 40 years. Ɣ For coupon paying bond, lower the coupon, generally the greater WKHPRGL¿HGDQG0DFDXOD\GXUDWLRQRIWKHERQG [III] Duration as a measure of approximate sensitivity of bond’s value to Ɣ ,IWKHPRGL¿HGGXUDWLRQLVJUHDWHUWKHSULFHYRODWLOLW\ZLOOEHJUHDWHU change in yield: Ɣ &ƝWHUƯVSDULEXVWKHKLJKHUWKH\LHOGOHYHOWKHORZHUWKHSULFHYRODWLOLW\ This is the interpretation which is commonly applied in practice. Duration 2 is the approximate percentage change in the value for a 100 basis
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