FIXED INCOME PERFORMANCE ATTRIBUTION

Transcription

1 FIXED INCOME PERFORMANCE ATTRIBUTION ANALYSIS OF A MULTI-CURRENCY BOND PORTFOLIO Diploma hesis submied o Swiss Federal Insiue of Technology, Zürich Universiy of Zürich, Swiss Banking Insiue for he degree of Maser of Advanced Sudies in Finance presened by BLAISE RODUIT lic. sciences éco. Supervisors Dr. Nils Tuchschmid Dr. Anna Holzgang DECEMBER 2005

2 ABSTRACT Absrac The wo key asse classes available o invesmen managers are equiies and bonds. Equiy aribuion has been around for a while and well-esablished mehods of aribuion have been developed. I is herefore emping o generalize hese mehods o fixed income aribuion. However, in doing his he performance analys ignores essenial characerisics of fixed income invesmens. In many poins, risk facors in fixed income invesmens are fundamenally differen from hose in equiy. Some of hem do no even have an equivalen in he equiy aribuion universe hese include yield curves and credi spreads. Furhermore, he effec of yield curve moves and spread changes on bond value is non-rivial. This paper proposes in he firs par o review he differen facor decomposiions and mehodologies used in he fixed income indusry. A special emphasis is pu on he yield curve shif effecs (parallel, wis, buerfly, reshape) which play a cenral role in performance aribuion. In he second par we discuss he pracical problems of daa qualiy ha usually occur when implemening a fixed income performance aribuion. Then we will run a Fixed Income Performance Aribuion analysis (FIPA) on a real porfolio and inerpre he resuls obained. We finish by checking which FIPA facors are he main driver of excess reurns and if excess reurns idenified are sill presen under a risk-adjused basis. - I -

3 FIXED INCOME PERFORMANCE ATTRIBUTION CONTENTS 1. Inroducion Performance aribuion Fixed income performance aribuion Theoreical framework Fixed income reurn decomposiion Carry reurn Coupon income Carry reurn - Roll-down Marke reurn Yield curve Marke reurn Spread Marke reurn Volailiy Marke reurn FX rae Timing reurn Yield curve consrucion Yield o mauriy (YTM) curve Zero coupon yield curve Yield curve decomposiion Principal componen analysis mehod Empirical mehod Polynomial mehod Duraion based mehod Linking reurn effecs o muliple periods The arihmeic model The geomeric model Issues in pracice Daa qualiy Asses wihou price or wih an incorrec price Corporae acions Cash flows and managemen fees Managemen fees Accouning of reclaimable wihholding axes Reinvesmen of coupons Gross / Ne basis Replicaing he benchmark in general Characerisics of he porfolio analyzed Consrains on he porfolio Syle of he porfolio manager Se up of he fixed income performance analysis The yield curve The YC decomposiion facors Linking mehod The resuls The FIPA aribuion for he global porfolio Global reurn (TWR) Direc reurn Roll-down II -

4 CONTENTS YC shif 1 (parallel shif) YC reshape Secor spread reurn (credi spread) YC spread reurn (issue spread) Fixed income iming Fixed income currency reurn The key raios The Alpha The Bea Inerpreaion Excess reurns and FIPA facors Disribuion of excess reurns Ineracion beween FIPA facors and excess reurns Mulivariae analysis Performance on a risk-adjused basis Alpha and FIPA facors Excess reurns on a risk-adjused basis Conclusion Acknowledgmens References Appendix Appendix 1: US governmen yield curve principal componen analysis Appendix 2: Mulivariae analysis of he FIPA facors III -

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6 INTRODUCTION 1. INTRODUCTION 1.1. Performance aribuion A manager has a reurn of 8% for he year 2005 while he benchmark only performs 6%. How did he ge i? Wha could have caused an excess reurn of 2%? Hopefully i has somehing o do wih he manager s conscious decisions. Tha is, wih somehing he manager mean o do. Bu, in realiy, a whole lo of he reurn migh have o do wih hings he manager didn do, righ? Like, he effecs of he marke a large. The economy. The overall movemen of indusries relaive o acions of he Federal Reserve or oher bodies. Even some uninended consequences of he manager s acions! Performance aribuion ries o answer hese quesions. The purpose of performance aribuion is o undersand realized excess reurns and o relae his informaion o he acive decisions made in he invesmen organizaion, in order o undersand he sources of ou-performance and idenify he acive decisions ha have generaed he excess reurns. Aribuion models are designed o idenify he relevan facors ha impac performance and o asses he conribuion of each facor o he final resul. This informaion can hen be communicaed o cliens, managemen and (no leas) he porfolio managers ha conduced he acive bes. In doing so he performance analysis can over ime add value by assising in he idenificaion of he invesmen managemen paricular skills and of he areas where skills appear o be lagging Fixed income performance aribuion The slump in equiy markes during he las couple of years has changed many invesors aiude owards fixed income. From being a low reurning low volaile asse class bond invesmens are now considered more han jus a safe-haven. Measured on a risk-adjused basis he long-erm reurns from bond invesmens compare favorably wih equiy reurns. In order o undersand he acive decisions made during he invesmen process i is essenial o undersand he characerisics of he underlying asse classes and relevan risk facors ha drive he invesmens, since i is hese asses classes and risk facors ha he porfolio manager analyzes when designing porfolios. Two key asse classes available o invesmen managers are equiy and bonds. Equiy aribuion has been around for a while and well-esablished mehods of aribuion have been developed. I is herefore emping o generalize hese mehods o fixed income aribuion. However, in doing his he performance analys ignores essenial characerisics of fixed income invesmens. In many poins, risk facors in fixed income invesmens are fundamenally differen from hose in equiy. Some of hem do no even have an equivalen in he equiy aribuion universe hese include yield curves and credi spreads. Furhermore, he effec of yield curve moves and spread changes on bond value is non-rivial. For all hese reasons, Fixed Income Aribuion has been one of he key challenges in he porfolio managemen indusry; hough here is now an exensive se of research ino differing mehodologies, here is sill no agreed indusry sandard

7 FIXED INCOME PERFORMANCE ATTRIBUTION This paper proposes in is firs par o review he differen facor decomposiions and mehodologies used in he fixed income indusry. A special emphasis is pu on he yield curve shif effecs (parallel, wis, buerfly, reshape) which play a cenral role in performance aribuion. In he second par we will discuss briefly he differen problems ha usually occur in pracice when implemening he aribuion. Then we will run a Fixed Income Performance Aribuion analysis (FIPA) on a real porfolio and inerpre he resuls obained. We finish by checking which FIPA facors are he main driver of excess reurns and if he excess reurns idenified are sill presen under a risk-adjused basis. 2. THEORETICAL FRAMEWORK 2.1. Fixed income reurn decomposiion I is generally admied ha he value generaed by holding bonds is composed of hree differen componens. Unlike he case for equiies, he reurn generaed from periodic cash flows is significan. In addiion o he periodic reurn, bond reurns are sensiive o changes in he fundamenal marke variables or fixed income risk facors. Finally he reurn is affeced by iming of rades. These hree differen sources of reurn are usually denoed by carry, marke and iming reurn: r = r + r + r Toal Carry Marke Timing Fig. 1. Fixed income reurn componens Carry reurn Coupon income The carry reurn is composed of wo componens. The cenral componen is he (ypically annual) coupon being paid ou o he invesor we denoe his componen direc reurn. This componen is always posiive. This direc reurn is heoreically defined as: - 2 -

8 THEORETICAL FRAMEWORK C rdirec = = ycurren P where C is he annual coupon, is ime passed, P is he iniial price and y denoes yield. More generally a direc reurn is compued as follows wihin a end of he day cash-flow / geomeric model : r Direc 1, = Coupon ( N 1 ( P 1 + AI ) + C ) N ( P + AI ) X X where : ime, N: nominal amoun, AI: accrued ineres, P: price, C: coupon, X: FX rae. 1 1 days ineres earned Coupons Redempion + Coupon Fig. 2. Direc reurn Carry reurn - Roll-down A less pronounced componen of carry is he passage of ime. Bonds usually do no rade a par, bu hey are evenually redeemed a par, herefore a mauriy he marke price mus converge owards par. For longer-daed bonds his effec is minor, whereas i can be significan for shorer-daed bonds rading away from par. This reurn componen is called roll-down reurn. The effec is posiive for discoun bonds (he roll effec will pull he price up owards par) and negaive for premium bonds (he roll effec will pull he price down owards par). The roll-down reurn can be inerpreed as: r RollDown 1, = Coupon ( N 1 ( P ( YC 1, YCS 1) + AI 1) + C ) Coupon N ( P + AI ) + C X ( ) where : ime, N: nominal amoun, P: price, X: FX rae, YC: yield curve, YCS: yield curve spread. Remark: The arificial price P (.) is calculaed by a funcion of differen facors like yield curve, yield curve spread, volailiy for example (he number of facors depends on he model complexiy). Arificial prices are needed o sequenially calculae and decompose reurn effecs (see Fig. 1.). 1 X 1-3 -

9 FIXED INCOME PERFORMANCE ATTRIBUTION YC -1 Zero rae 1 Day Time Fig. 3. Roll-down reurn when he bond is overvalued Marke reurn Yield curve In conras o he carry reurn componens he marke reurn is less predicable. The marke reurn is driven by he marke variables on which bond value depends. In fixed income he yield curve is he cenral marke variable. Tradiionally he yield curve is based on bonds issued by governmen eniies. The raionale has been ha his provides a defaul free yield curve per counry. Therefore he marke value of governmen bonds are normally driven enirely by movemens in his curve. The basic approach o modeling yield curve movemens is o calculae he difference beween he final and he iniial yield curve for he period for which performance is measured. Fig. 4. Yield curve movemens Ofen porfolio managers decompose yield curve shifs furher ino basic movemens. Typically he number of basic movemens vary beween 2 and 5. This number is arbirarily chosen by he porfolio manager who consruced he porfolio and who did bes on yield curve moves. The number of basic movemens is consequenly a rade-off beween he explanaion power of he model and he complexiy of he inerpreaion. Recen sudies sugges ha mos of he yield curve shif can be explained prey well by essenially hree facors: parallel shif, slope (or wis) and curvaure (or buerfly). The unexplained shif lef is normally saisically small and pu in a residual facor called reshape

10 THEORETICAL FRAMEWORK Of course in marke crisis siuaions hese hree firs facors migh be insufficien o leave he reshape small and o do a good performance aribuion. a) Parallel shif A parallel shif appears when he raes a sandard mauriies move uniformly. Noe ha parallel shifs in yields are capured direcly by he bond duraion as rparallel D ycparallel where r denoes reurn, D is modified duraion and YC is he yield curve. b) Twis (seeping / flaening) We can see a wis effec when shor erm and long erm raes move in opposie direcion bu proporionaely in relaion o he disance from some pivo poin mauriy (usually defined a 5 years). c) Curvaure The curvaure or buerfly effec occurs when shor erm and long erm raes move in same direcion while medium erm raes move in an opposie direcion, sill proporionaely. By decomposing he yield curve movemens ino conribuions from hese shifs he bond fixed income porfolio reurn ha is due o he yield curve moves can be decomposed ino: r = r + r + r + r YieldCurve Parallel Twis Curvaure Reshape Fig. 5. Example of parallel, slope (seepness) and curvaure shifs The financial lieraure idenifies several mehods o exrac hese facors and quanify hem. Four, a leas, can be menioned: - 5 -

11 FIXED INCOME PERFORMANCE ATTRIBUTION Saisical Principal Componen Analysis (PCA) Empirically consruced user-defined facors Polynomial fi mimicking a Taylor decomposiion of he yield curve funcion Facor model based on duraion analysis. As he reurns ha are generaed by he yield curve shifs are he hear of a fixed income aribuion analysis, we are going o review hese four mehods in deail a bi laer in he paper Marke reurn Spread In addiion o he general yield levels, non-sovereign deb is also sensiive o credi risk. The marke measure of credi risk is he spread. This is he addiional yield ha an invesor will require in order o inves in such bonds. The Implied Yield Curve Spread (YCS), which is he discouning spread necessary o add o he yield curve in order o mach he marke price of he given bond. So he YCS of a bond is he soluion o he equaion: V ( 1+ y + YCS) ) = C where C denoes cash flow a ime, V he value of he bond and y is he zero yield for ime. The reurn resuling from he yield curve spread is fully defined by he following equaion: 1+ r YCS 1, = N N 1 1 ( P ( YC, YCS, Vol ) + AI ) ( P( YC, YCS, Vol ) + AI ) 1 where : ime, N: nominal amoun, P: price, X: FX rae, YC: yield curve, YCS: yield curve spread, Vol: volailiy. X X 1 1 The magniude of he spread reflecs he credi qualiy of he issue. The spread is ypically decomposed ino wo subcaegories he secor spread (indusry specifics and raing specifics) and he issue spread (issuer specifics). The firs subcaegory reflecs aspecs common across bonds issued by corporaions wih similar raings and in similar indusries; he second caegory reflecs issue/issuer specific consideraions. Therefore he spread can be decomposed as: r = r + r Spread Secor Issue The spread reurn componen is a marke reurn. Typically secor spreads vary subsanially over ime and he spread reurn can be sizeable - also in comparison o curve reurns. As an example he conagious spread of he Russian credi crisis ha occurred in he auumn of 1998 mean ha increases in spreads were of he same magniude as falls in governmen yields Marke reurn Volailiy For sandard domesic bonds he previous facors are he main drivers. For more complex insrumens oher marke variables can add value. An imporan caegory of bonds is bonds - 6 -

12 THEORETICAL FRAMEWORK wih embedded opions. Ofen asse-backed, morgage-backed and corporae bonds have buil-in opions in he form of pu, call or prepaymen opions. For such bonds changes o implied volailiy is an imporan facor behind marke value, since he volailiy drives he opion value. For mos vanilla bond porfolios he volailiy reurn is small compared o he direc, curve and spread reurn componens. However for porfolios wih large opions posiions or many morgage bonds he volailiy effec can be significan. Volailiy is compued as follows: 1+ r Vol 1, = Coupon ( N 1 ( P ( YC, YCS 1, Vol ) + AI 1) + C ) Coupon N ( P( YC, YCS ) + AI ) + C X ( ) where : ime, N: nominal amoun, P: price, C: coupon, X: FX rae, YC: yield curve, YCS: yield curve spread, Vol: volailiy. X Marke reurn FX rae For foreign invesmens he FX rae developmen is anoher key risk facor ha impacs he performance. The currency effec is generic (no specific o fixed income) and i is reaed exacly as for equiy porfolios. The FX effec is calculaed wih: 1+ r Currency 1, = Coupon ( N ( P + AI ) + C ) Coupon N ( P + AI ) + C ( ) X where : ime, N: nominal amoun, P: price, C: coupon, X: FX rae. X Timing reurn Timing reurn componen arises due o he rading aciviies in a porfolio. Performance measuremen is ypically done based on end of day prices. Usually rading is conduced during he rading hours and herefore some discrepancy will occur. The effec of his rading is compounded ino he iming reurn componen. In case he rader has execued on aracive levels relaive o end of day pricing he effec will show up as a posiive reurn componen. Timing reurn can be defined as: 1+ r Timing 1, = Coupon ( N ( P + AI ) + C ) X 1 Coupon ( N 1 ( P ( YC, YCS, Vol ) + AI 1) + C ) where : ime, N: nominal amoun, P: price, C: coupon, X: FX rae, YC: yield curve, YCS: yield curve spread, Vol: volailiy. Noe ha in his secion 2.1. all formulas come from an end of he day cash-flow / geomeric model. The formulas change a bi if we are in a beginning of he day and/or arihmeic X 1-7 -

13 FIXED INCOME PERFORMANCE ATTRIBUTION seing. However he logic behind remains he same. This concludes he sudy of fixed income reurn decomposiion as described in Fig. 1. The following secion is dedicaed o he mos imporan componen for fixed income aribuion he yield curve effec Yield curve consrucion The yield curves are exraced from bonds available on he markes. Basically here are wo main ypes of yield curves - he yield o mauriy curve and he zero coupon yield curve. The zero coupon yield curve is easier o model han he YTM curve. Therefore a consequen amoun of academic research has been done on he zero coupon yield curve where maybe he mos well known model is he sochasic model of Vasicek Yield o mauriy (YTM) curve The yield o mauriy (YTM) curve is compued wih he yield o mauriy, which is a securiy s inernal rae of reurn, or he anicipaed yield of he bond if held o mauriy. The YTM is he rae used when calculaing he presen value of all cash flows, so ha hey add up o he curren marke price. In oher words, i is he compounded rae of reurn ha invesors receive if he bond is held o mauriy and all cash flows are reinvesed a he same rae of ineres. If r is he curren yield o mauriy, hen a bond price is given by: C 1 2 n n ( 1+ r) + C( 1+ r) C( 1+ r) + B( 1+ r) = P where C is an annual coupon, n is he number of years o mauriy, B is he par value of he bond, P is he curren marke price of he bond Zero coupon yield curve The zero coupon yield curve is compued wih zero coupon yield, which is he reurn i would show if all coupons were sripped ou. Noe ha for securiies ha do no pay coupons, such as zero-coupon bonds or bills, here is only one repaymen cash flow a mauriy. In his case, he yield o mauriy is idenical o he zero coupon yield. Thanks o is simpliciy a lo of evaluaion mehods have been developed for he zero coupon yield curve. The following lis is no exhausive: Boosrapping Cubic spline Nelson Siegel Cox Ingersoll Ross Cox Ingersoll Ross (inflaion) Vasicek Longsaff Schwarz Maximum smoohness Naural spline - 8 -

14 THEORETICAL FRAMEWORK The evaluaion principles may be divided ino hree groups: boosrapping mehods, mahemaical mehods and erm srucure models. For all models, excep he boosrapping mehod, he underlying funcional form is esimaed using ordinary leas squares. Therefore, heoreical and observed prices on he bonds, which have provided daa for he yield curve will usually deviae. On he oher hand, in he boosrapping mehod heoreical and observed prices on he bonds ha have provided daa for he yield curve are always equal due o he calculaion principle. In he boosrapping mehod, he zero coupon yield curve is approximaed using a coninuous, piece-wise linear, funcion. The number of pieces are equal o he number of bonds (or money marke, FRA/IRF and/or swap quoes) wihin he segmen o provide daa for he yield curve. The break poins are defined by he ime o mauriy of he bonds. Therefore, if he segmen includes 18 bonds, he yield curve is defined by 18 parameers, i.e. he slopes of he 18 linear pieces. Wih he mahemaical mehods (cubic spline, naural spline, Nelson spline), esimaion echniques are used o creae yield curves. We invie he reader o refer o a saisical book for furher deails 1. Finally an alernaive is o use erm srucure models (Cox Ingersoll Ross, Cox Ingersoll Ross [inflaion], Vasicek, Longsaff Schwarz). They are descripions of changes o ineres raes over ime. Some of hese models are characerized by having closed form soluions o he price of zero bonds, which may be used in he yield curve esimaion. Basically he parameers in he ineres rae process are used as variables in he esimaion. By varying hese parameers, i is possible o find he process ha fis he prices of he insrumens used in he esimaion bes. Wih heses approaches he ineres rae models have good asympoic behavior (such as converging, as erm o mauriy is large) and someimes he parameers may have a financial inerpreaion. However, he approach is raher pragmaic. The models are simply used o produce zero curves wih ideal feaures and no furher inerpreaion is aemped. Vasicek and Longsaff-Schwarz are somewha more complex han he res of he models. In hese models he zero coupon yield curve is approximaed by an equaion ha is derived as a soluion o a sochasic differenial equaion. The change in ineres raes is decomposed ino a drif erm and a sochasic erm. The Longsaff-Schwarz model even includes wo sochasic differenial equaions. In hese models he underlying sochasic differenial equaions relae o so called facors, which are presumed o describe he pricing in he financial marke. As an example we presen here a brief model specificaion of he Vasicek model. Vasicek model 2 The change in ineres raes is modeled wih a sochasic differenial equaion: r a( r r) ( 0) = r0 dr = d + σdw 1 See for example [11] Kno G.D., Inerpolaing cubic splines, Birkhäuser, See for a complee specificaion [15] Vasicek O., An equilibrium characerisaion of he erm srucure, Journal of Financial Economics, 5: ,

15 FIXED INCOME PERFORMANCE ATTRIBUTION where r is he ineres rae. The oher parameers are defined as follows: a > 0 : speed of mean-reversion r > 0 : level of mean-reversion (he average value where he ineres rae converges) σ > 0 : absolue volailiy W : a sandard Brownian moion a ime Noe ha negaive ineres raes are possible wih posiive probabiliy wih hese seings, which is a weakness of he model. The soluion of he sochasic differenial equaion given above is for 0 s < : a( s) a( u ) () = r + e ( r() s r ) + e dw ( u) r σ s Given F s, he filraion a ime s, r() is normally disribued wih mean and variance: a( s) [ () Fs ] = r + e ( r() s r ) 2 σ 2a( s ) [ r() F ] = 1 e E r Var s 2a ( ) The zero coupon yield curve can hen be modeled wih r(). For a more deailed discussion concerning he characerisics of hese erm srucure models, please refer o relevan financial lieraure covering hese models Yield curve decomposiion As we saw in he secions above, he yield curve can be decomposed in 3 main facors in order o explain he global shif he facors being he parallel shif, wis, buerfly, plus a residual. Now we are going o review he four mehods ha people usually use in he indusry o decompose he yield shif. These mehods are generally applied on zero coupon yield curves Principal componen analysis mehod To explain all he possible disorions by using n mauriy poins o define he curve, n scenarios on each yield curve are required. PCA is a coordinae ransformaion ha reduces he redundancy conained wihin he daa by creaing a new series of componens in which he axes of he new coordinae sysems poin in he direcion of decreasing variance. The resuling componens are ofen more inerpreable han he original images. The mean of he original daa is he origin of he ransformed sysem wih he ransformed axes of each componen muually orhogonal. 3 See for example [9] Hughson L., Vasicek and Beyond: Approaches o Building and Applying Ineres Rae Models, Risk Books,

16 THEORETICAL FRAMEWORK The mehodology is as follows: 1. Impor he ineres rae series. For example daily yield curves of seleced mauriies up o 30 years. 2. Compue he saionary series of differences. 3. Compue he eigenvalues and eigenvecors of he series. The eigenvecors represen he facor loadings, while he eigenvalues represen he significance of he facors. Eigenvalues are repored in descending order. 4. The relaive weigh of he eigenvalues gives he explanaory power of he various facors. 5. The marix of componens is creaed. By consrucion because of orhogonaliy, he componens are muually uncorrelaed. 6. Every elemen of he original series can be reconsruced using he componens and he loading marix. 7. As repored in he ineres rae lieraure 4, he hree facors represen differen aspecs of ineres rae movemens. Typically, he firs facor is responsible for parallel shifs, he second one for wis changes and he hird one for buerfly adjusmens. Below you see a PCA analysis of he US Governmen yield curve. The ime period chosen goes from January 1997 o Augus We ook monhly daa. The Fig. 6. shows he differen indexes ha compose he US Governmen yield curve (1 monh, 3 monhs, 6 monhs, 9 monhs, 1 year, 2 years, 3 years, 5 years, 10 years and 30 years). USD Gov Yield Curve Indexes 8 % USD.INDEX.1M USD.INDEX.3M USD.INDEX.6M USD.INDEX.9M USD.INDEX.1Y USD.INDEX.2Y USD.INDEX.3Y USD.INDEX.5Y USD.INDEX.10Y USD.INDEX.30Y 0 01/ / / / / / / / / / / / / / / / / / / / / / / / / /2005 Fig. 6. Indexes ha compose he USD Governmen yield curve from Jan o Augus 2005 This period is paricularly ineresing o analyze because i encloses a yield curve reversion (shorer raes higher han longer raes) from May 2000 o January 2001, hen a seeping of he curve for 2001 and finally a flaening from May See for example [10] James J. & Webber N., Ineres Rae Modelling, Ed. J. Wiley,

17 FIXED INCOME PERFORMANCE ATTRIBUTION By applying a sandard PCA 5 analysis, we obain he following facor loadings and he cumulaive explained variance: Facor 1 loadings Facor 2 loadings Facor 3 loadings USD.INDEX.30Y USD.IND EX.30Y USD.INDEX.30Y USD.INDEX.10Y USD.IND EX.10Y USD.INDEX.10Y USD.INDEX.5Y USD.INDEX.5Y USD.INDEX.5Y USD.INDEX.3Y USD.INDEX.3Y USD.INDEX.3Y USD.INDEX.2Y USD.INDEX.2Y USD.INDEX.2Y USD.INDEX.1Y USD.INDEX.1Y USD.INDEX.1Y USD.INDEX.9M USD.INDEX.9M USD.INDEX.9M USD.INDEX.6M USD.INDEX.6M USD.INDEX.6M USD.INDEX.3M USD.INDEX.3M USD.INDEX.3M USD.INDEX.1M USD.INDEX.1M USD.INDEX.1M Fig. 7. Facor loadings of he firs hree principal componens Cumulaive variance explained Variances F.1 F.2 F.3 Fig. 8. Cumulaive variance explained by he firs hree facors The facor loadings of he firs principal componen are as expeced ypically large and similar for all variables. An upward shif in he firs principal componen herefore induces a roughly parallel shif in all variables. For his reason he firs principal componen is called parallel shif. Wih PCA mehod, he parallel componen is no sricly speaking a ranslaion of yield reurns bu raher a level change impacing he shor and long erm slighly differenly. Here for example he shorer raes are proporionally less affeced han he longer raes. The firs componen explains here 97% of he variaion during he daa period in consideraion. 5 The code is available in Appendix

18 THEORETICAL FRAMEWORK In his example, an upward movemen in he second principal componen induces a change in slope of he yield curve, where shor mauriies move up bu long mauriies move down wih an unchanged poin a approximaely 2 years. This second componen is called wis and explains abou 2.5% of he variaion. The hird principal componen influences he convexiy of he yield curve. The facor weighs are posiive for he shor raes, bu decreasing and becoming negaive for he medium erm raes and hen increasing and becoming posiive again for he longer mauriies. This is he buerfly effec. This effec explains 0.4% of he variaion. The unexplained variaion (less han 0.1%) is someimes called reshape and considered as he residual of he PCA decomposiion. Noe ha he PCA mehod is a pure saisical decomposiion and does no involve making srong assumpions on he magniude and direcion of yield changes occurring on a given period. The principal componens are perfecly uncorrelaed, making he performance numbers aached o each curve effec addiive and clearly definable and explain mos of he yield changes variance. PCA does no require ha funcional form of he parallel, wis and buerfly are defined a priori. We generally observe ha he firs componen idenified as he parallel shif is no even over he erm srucure of he yield curve and shows more movemen a he shor end han he long end. However a mehod exiss o force he firs componen o be sricly parallel by reprocessing he componens o orhogonalize hem. Furhermore, we sill have o make assumpions on he horizon lengh. There is a fine line beween saisical daa relevance and explanaory relevance. Saisically speaking, he longer he horizon he beer, however he changes in yield curve shape from a pas period may be less relevan han recen evens. For a performance aribuion, a ime window of 3 monhs seems o be appropriae Empirical mehod The empirical mehod decomposes reurns of he porfolio in a very similar manner o he PCA mehod. The difference is ha insead of using a saisical analysis o define he parallel, wis and buerfly componens, changes in zero-coupon yield a he beginning and end periods are measured empirically. I consiss in a decomposiion of he yield curve changes ino a combinaion of hree basic componens: parallel, wis and buerfly. Unlike he PCA, he componens are no saisically deermined hrough a se of axis roaions in he spo raes, bu raher by an empirical analysis of he yield curve. A mehod developed by Lehman brohers 6 uses a piecewise funcion wih 5 mauriy poins on he yield curve (2, 3, 5, 10, 30 years), he pivo poin being a 5 years. 6 See: [5] Dynkin L., Hyman J. & Konsaninovsky V., A reurn Aribuion Model for Fixed Income Securiies, Handbook of Porfolio Managemen,

19 FIXED INCOME PERFORMANCE ATTRIBUTION The mehod used by Lehman brohers is: 1. Firs define he beginning and end of he period and compue he changes in zerocoupon yields beween hose wo daes for he five reference mauriies. 2. The hree piecewise funcions for parallel, wis and buerfly are defined: Parallel shif called p is inuiively se o equal he average yield changes over he five references mauriies: p = 1 5 ( y + y + y + y + y ) Twis reurns are defined wih a pivo poin se a he 5-year mauriy poin. The 5- year poin is consequenly no affeced by he wis change. The wis magniude is defined as: = y 30 y 2 is applied such as he 30-year moves up by /2 and he 2-year poin moves down by /2 oo. Buerfly reurns is defined as: b = 1 2 ( y2 + y30 ) y5 b is applied such as he 2-year and 30-year move up by b/3 and he 5-year poin moves down by 2b/3. The empirical approach uses full revaluaion. The resuls are very consisen across all securiies and he whole porfolio. I is also ineresing o noice ha he shape effec (residual) is small, which means ha he empirically defined curve disorions explain a very imporan par of he reurns. The flexibiliy of he empirical mehod allows he porfolio manager o cusomize he aribuion relaively o his bes. For example he can move he pivo poin o beer mach his invesmen posiions. Such refinemens are an open discussion ha can lead o more accurae measures of performance aribuion. One way o exploi a bes his flexibiliy would be o calibrae he pivo poin by using a PCA. Hence we will keep he flexibiliy of he empirical mehod whils leveraging he PCA o describe he yield curve environmen in a perinen manner

20 THEORETICAL FRAMEWORK When he shif facors (parallel, wis, buerfly,...) are well defined, a facor loading is compued: = N N N N N R R F F I F F I YC YC YC YC Yield curve, Yield Curve, -1 Shif 1 Shif 5 Reshape (residual) where N represens he number of mauriy poins, F he facors and I he facor loadings. Wih hese facor loadings we can herefore quanify he yield curve shif explained by each facor. Two approaches are broadly used o define hese loadings - he firs is he sequenial OLS, he second one he sandard OLS. a) Sequenial OLS In he Sequenial Ordinary Leas Square procedure, he loadings are calculaed for each facor using simple algebra facor by facor. For example loading 1 is calculaed by maximizing: 1 1 1, 1 1 F F YC F I T T = where YC,-1 is he oal shif in he yield curve beween ime -1 and ime, I he facor loading and F he facor. Once he firs loading is calculaed he nex loading (I 2 ) is compued by maximizing: ( ) , 2 2 F F F I YC F I T T = This process coninues unil all loadings are calculaed (sequenial OLS), or he process sops a he desired number of facors and he remaining unexplained yield curve shif is he residual (reshape) change. The basic idea behind his mehod is ha he firs facor explains he mos yield curve variance as possible and leaves a residual. Then he second facor only explains he residual as bes as possible and gives anoher smaller residual and so on. By analyzing facors loadings compued wih a sequenial OLS, we have o be careful because a facor effec can offse anoher one. For example a yield curve shif can be parially offse by a wis effec. However in pracice, porfolio managers firs hink in erm of duraion (i.e. a parallel shif) and hen wih a wis for example. Therefore, even hough his mehod seems o be mahemaically less correc, i fis beer he mehodology of porfolio managers.

21 FIXED INCOME PERFORMANCE ATTRIBUTION b) Sandard OLS Alernaively all loadings can be calculaed direcly using an Ordinary Leas Square procedure. Here he vecor of loadings is he maximized soluion o he following problem: T [ F1,..., Fn ] [ I1 In] + YCReshape YC, 1 =,..., where YC is he oal shif in he yield curve beween ime -1 and ime, I he facor loading and F he facor. Here all facors maximize he variance explained in one sep. Mahemaically his resul is opimal bu less inerpreable han he sequenial OLS Polynomial mehod The polynomial mehod does no involve a muli-dimensional saisical analysis, bu raher uses a more inuiive of exracing he zero, firs and second order changes from polynomials ha fi he yield curve a he beginning and end of period and model he yield changes as he difference beween each polynomial of he same degree. The polynomial approach relies on defining coefficiens from fiing polynomials a he beginning and end of he period and using hem o esimae magniude changes in he porfolio yield. The mehodology: 1. Fi hree polynomials respecively of degree zero, one and wo o he yield curve a he beginning and end of period and exrac respecively hree ses of polynomials: Begin End ( α0, α0 ) Begin End Begin End ( β0, β0 ), ( β1, β1 ) Begin End Begin End Begin End ( γ, γ ), ( γ, γ ), ( γ, γ ) Compue for each yield curve componen he magniude of change due o each effec: a. The parallel magniude is defined as: p = α End Begin 0 α0 b. The wis magniude is defined as: s End Begin End Begin () = ( β β ) + ( β ) β1-16 -

22 THEORETICAL FRAMEWORK c. The buerfly magniude is defined as: b End Begin End Begin End Begin 2 () = ( γ γ ) + ( γ γ ) + ( γ γ ) where is he ime period on he yield curve erm srucure. These parameers are proxies for he parallel (p), wis (s) and buerfly (b) effecs a each mauriy poin of he zero-coupon yield curve. In pracice, relying on zero-degree polynomials o measure he parallel shif effecs leads o a poor oucome and a minuscule aribuion of he reurn o he parallel componen. This mehod aribues reurns mainly o he wis and more predominanly o he residual, emphasizing a redundancy or double-couning. To explain hese deficiencies of he polynomial decomposiion we have o undersand ha every one of he polynomial fis is independen from he ohers and explains as much of he variance as possible in a non-orhogonal space. We undersand hen ha wihou a correlaion effec correcion an over-esimaion of he oal reurn and a disproporionae residual is obained. The only applicable way o use his mehod is o measure he parallel shif wih he empirical mehod and hen apply he firs order polynomial for he wis and he second-order polynomial for he buerfly. This sequenial aribuion will ensure ha parallel shif explains mos of he reurn Duraion based mehod The duraion approach decomposes reurns of he porfolio based on is yield, duraion and convexiy. The calculaion can be applied a every level of he porfolio and is very inuiive and easy o implemen. Following he mehod deailed in Fong 7, he duraion mehod breaks down he yield curve movemen using he duraion and convexiy measures. The duraion componen explains he parallel effec and he convexiy componen capures he wis. The parallel shif can be simply calculaed as follows: Parallel = D R where R is he change in zero-coupon yield from he beginning of period o he end of period and D is he modified duraion. Similarly, he wis effec is measured as he second order erm of a Taylor expansion: 7 See [8] Fong G., Yoo D., & Zelaya Z.M., Global Performance Aribuion, Perspecives on Inernaional Fixed Income Invesing,

23 FIXED INCOME PERFORMANCE ATTRIBUTION Twis = 1 2 C i R 2 where C is he effecive convexiy. The advanage of he duraion approach is ha i does no require he definiion of erms and condiions of securiies. Secondly porfolio managers and raders have he inuiion for YTM, duraion and convexiy values, as hese measures are widely acceped and used in fixed income analyics. The key assumpion is ha he disribued cash flows of a fixed income insrumen are approximaed by a concenraed cash flow a he duraion of he securiy. Consequenly his mehod may no work well wih a bond feauring big disribued cash flows scaered abou he full erm srucure Linking reurn effecs o muliple periods Two broadly used mehods are available o link reurns over muliple ime period: he geomeric model and he arihmeic model The arihmeic model In arihmeic aribuions he daily excess reurn conribuion is simply obained by addiion of he differen facors: i r 1, = r 1, where i is used as indicaor for he facors, i={direc, Roll-down, YC shif 1,, Currency} We can compound his reurn ino muliple periods wih: i ( 1+ r 1, ) = ( 1+ r 1 ) 1 + r =, Compounding will however resul in cross producs of he differen reurn effecs as i is no possible o swap beween sums and producs. These cross producs creae residuals difficul o aribue and inerpre. Up o now, some mehods have been developed o handle his problem. For a complee descripion please refer for example o David Spaulding s book. We can shorly menion he mos used mehods: Arihmeic linking + a residual Geomeric linking + a residual Logarihmic linking + a residual disribued along each effec wih a repariion key Opimized approach (similar o he logarihmic one) 8 For a deeper analysis on muliple periods see [14] Spaulding D., Invesmen Performance Aribuion, McGraw-Hill,

24 THEORETICAL FRAMEWORK From he unpublished whie papers we read o wrie his hesis, i appears ha many aribuion vendors ypically use he geomeric mehod o link he sub-period effecs. Geomeric mehod has he meri o be simple and easy o comprehend The geomeric model Geomeric aribuion is no as linking challenged as arihmeic. In geomeric aribuions he daily reurn conribuion is obained by muliplicaion and he reurn can be decomposed as: 1 + i ( r 1 ) r 1, = 1+, where i is used as indicaor for he facors, i={direc, Roll-down, YC shif 1,, Currency} Compounding ino muliple periods is really sraigh forward: i i ( 1+ r 1, ) = ( 1+ r 1 ) 1 + r =, The big advanage of he geomeric mehod is ha i uses muliplicaion properies o link reurn effecs in muliple periods. This calculaion is consequenly very easy and wihou any residuals. Oher benefis of his mehod are he converibiliy and proporionaliy properies: a) Proporionaliy One advanage o geomeric excess reurn (relaive o a benchmark) is ha i akes ino consideraion he magniude of he individual reurns. Tha is, i provides some dimension o wha is going on. For example, le s say our porfolio had a reurn of 11% versus a benchmark of 10%. Arihmeically, we would have an excess reurn of 1%. Likewise, if our porfolio was 25% versus 24% benchmark, we would show an excess reurn of 1%. Geomerically, we ge differen numbers: = 0.991% and 1 = 0.81% The differences occur because he 1% addiion earned relaive o 10% couns a whole lo more han i does relaive o 24%. Make sense? b) Converibiliy Anoher benefi of he geomeric approach is ha i repors he same excess reurn, regardless of he currency

25 FIXED INCOME PERFORMANCE ATTRIBUTION For example, le s say on January 1, our porfolio sars ou a $100. On ha dae, he conversion rae o Euro was (i.e. for $1, we ge ). Our conversion o Pounds Serling is (i.e. for $1 we ge roughly 69 pence). Twelve monhs go by and our US porfolio has gone up 10%, o $110. The benchmark (in US dollars) has gone up 8% during his ime. The new FX raes are for Euro (i.e. for $1 we ge ) and for Pounds Serling (i.e. for 1$ we ge ). The following able shows he saring and ending values in he hree currencies for hee porfolio and benchmark. We also show he reurns and excess reurns. Saring values Ending values Reurn Excess reurn Porfolio Index Porfolio Index Porfolio Index Arihmeic Geomeric US $ % 8.00% 2.00% 1.85% Euro % 13.40% 2.10% 1.85% Pounds % 10.16% 2.04% 1.85% Fig. 9. Comparison of he arihmeic and geomeric reurns For example, he geomeric and arihmeic excess reurn for Euro is compued as: ER ER G, A, = 1 = 1.85% = = 2.10% As he able shows, he arihmeic excess reurn varies from counry o counry because of he exchange rae differences. The fac ha he geomeric excess reurn shows he same value regardless of he exchange is considered an advanage, especially for firms ha marke inernaionally. Wih hese words we are ending he heoreical pars of his maser hesis. Afer having reviewed he heoreical framework underlying a fixed income performance aribuion we would like o coninue his hesis by describing shorly he differen problems usually encounered in pracice. 3. ISSUES IN PRACTICE To perform a fixed income performance aribuion he enire porfolio has o be recalculaed each day in order o exrac he differen reurn effecs. Furhermore, a decomposiion is ypically done wih subcaegories like for example currency and mauriy. Consequenly index benchmarks provided by he marke are no sufficien, inernal benchmarks on securiy level have o be consruced as well o mach each subcaegory. These inernal benchmarks have o replicae exacly he index benchmark o which hey depend. We hen undersand ha he IT sysem, price sources, price qualiy, cash ou- and inflows mus be handled in a very rigorous way. A good performance aribuion wihou an excellen performance measuremen is nohing! Daabase mainenance is herefore he firs obligaory sep prior o any performance

26 ISSUES IN PRACTICE aribuion. And we would highly recommend people no o underesimae his daa qualiy issue! The nex paragraph gives a quick view of he principal issues ha will usually arise when implemening a fixed income performance analysis Daa qualiy Performance aribuion requires a very high daa qualiy, which is cerainly he mos sensible par of his kind of analysis because i is cosly and ime consuming o monior and mainain a high-qualiy daabase. To give a ime indicaion, i is no unusual ha firms inves more of a year for cleaning he daa hisory. For example daily prices coming in he sysem mus be closely moniored. Here you find a non-exhausive lis of inpus ha require a close monioring: The differen price sources ha feed he FIPA analysis The booking of non-sandard corporae acions The booking of managemen fees The reamen of wihholding axes The reinvesmen of coupon The dynamic changes in raings The dynamic changes of mauriy buckes (e.g. for muli-sep bonds or callable bonds) The dynamic changes of business classes You find nex a more complee descripion of some issues one will ge for sure by implemening a FIPA analysis: Asses wihou price or wih an incorrec price Generally daabases ge prices from differen sources like Morgan Sanley, Merrill Lynch, Pice, Lehman, JP Morgan, Prioriy liss are se up o prioriize he prices. A problem comes when delivered prices wih he highes prioriy are false. And his will happen for sure. To remedy his problem a daily process wih he back office should be pu in place o correc he wrong prices. Anoher source of incorrec prices can be caused by banking holidays abroad and no in he home counry of he porfolio. This causes an imporan number of asses o have no price alhough i was a working day in he home counry. Here again a close monioring has o be pu in place Corporae acions Oher minor price errors can be caused by special corporae acions (principally for socks) like splis, new issue righs, dividends in socks, bond converible issues. A iming error of he corporae acion booking is ofen responsible for he error. In fac, in mos cases he daabase ges 3 daes: he ex-dae, he recording dae and he payable dae which are no always sandardized and may cause errors

27 FIXED INCOME PERFORMANCE ATTRIBUTION 3.2. Cash flows and managemen fees Anoher issue is he booking of he differen cash in- and ouflows of he porfolio, which mus be performance neural. Here are lised he main sources of cash flows: Managemen fees Wheher you wan o compue a performance aribuion on a gross or ne basis, managemen fees have o be aken ino accoun or no. If you choose a ne performance, managemen fees have o be added and your relaive performance o he benchmark will be a bi less. This reflecs he poin of view of your clien. If, in he conrary, you choose a gross aribuion wihou managemen fees, your porfolio will be direcly comparable wih he benchmark porfolio, which is wha a porfolio manager wans. Bu by removing managemen fees, on a cumulaive basis, a linear rend will appear beween he porfolio value calculaed for your performance aribuion and he real accouning value Accouning of reclaimable wihholding axes Reurns should be calculaed ne of non-reclaimable wihholding axes. Reclaimable wihholding axes should consequenly be accrued. The main problem here is ha axes policy may differ wih he counry where he bond was issued, bu also wih he owner of he bond Reinvesmen of coupons The mehodology for he reinvesmen of coupon concern principally benchmark porfolios. In fac if one decides o creae benchmark porfolios on a securiy level, porfolios have o replicae exacly heir benchmark index. Unforunaely differen pracices are used by he main benchmark index providers, for example, Lehman Brohers records coupons on an accoun wihou ineres rae and reinvess hem every monh. Merrill Lynch records coupons on an accoun wih ineres rae and reinves every monh. JP Morgan and Morgan Sanley aggregae he coupon wih he daily reurns of he according securiy Gross / Ne basis Wha is reaed as a cash flow should be performance neural. Programs can generally calculae performance wih or wihou axes and fees, which means gross or ne. SPPS, which sands for Swiss Performance Presenaion Sandards is he Swiss version of he inernaional recognized Global Invesmen Performance Sandards (GIPS). The aim of hese sandards is o provide fair performance presenaions for cliens, which allows an objecive

28 ISSUES IN PRACTICE comparison beween invesmen fund companies. When a company is a member of SPPS, he company is asked o follow hese SPPS sandards. The differences beween gross and ne performance is summarized in he Fig. 7. Federal Direc Tax & anicipaory Tax Reclaimable Index Gross Div. reinvesed Index Ne Div. reinvesed Federal Samp Courage Fee Gross (SPPS) Mgm. Fee (inkl. MwS) Depo Fee Ne (SPPS) Fig. 10. Gross / Ne performance 3.4. Replicaing he benchmark in general The main problem wih securiy-level benchmark daa supplied by mos vendors is ha raes of reurn are no available on securiy-level. To calculae reurns of individual securiies, one needs o know heir prices and all heir cash flows. This in urn requires knowledge of he pricing formula used, he ex-coupon convenions, reamen of cash coming from a coupon paymen and non-sandard feaures such as non-uniform firs coupon paymen, muli-coupons, sep-up coupons, callable and so on. Perhaps surprisingly, he main problem in replicaion of fixed income benchmark reurns lies in he calculaion of coupon iming and amouns. The iming of coupons depends on he bond issuer, he ex-period convenion used and he frequency of coupon. In addiion, he amoun of coupon paid can depend on wheher he bond has a non-sandard firs coupon period, in which case he firs coupon may be more or less han he sandard paymen. The same consideraions may apply o oher coupon paymens. In principle, hese coupons may be recalculaed from firs principles if we know he incepion dae, firs and las coupon daes, mauriy dae, annual coupon paymen and coupon frequency and ex-dae convenion for each bond. In pracice, his imposes a subsanial burden on he index calculaor, who has o obain and verify large amouns of bond daa. In addiion, he exday convenions for many bonds are obscure. There is no easy answer o hese problems and he person who wans o implemen a FIPA analysis is srongly advised o consul and exper in his field. In our case, i ook us more han one year o replicae almos perfecly every benchmark used in he company. Bu we will spare he reader he explanaions of his edious work

29 FIXED INCOME PERFORMANCE ATTRIBUTION Finally, afer having solved all he echnical issues presened in his secion, we are finally ready o implemen a FIPA analysis of qualiy on a real porfolio. 4. CHARACTERISTICS OF THE PORTFOLIO ANALYZED 4.1. Consrains on he porfolio The porfolio we are going o analyze is a muli-currency bond porfolio wih reporing currency in CHF. Is size exceeds 1 billon CHF. The porfolio has a cusomized credi benchmark wih fixed weighs which are rebalanced every monh. The consiuens are indices from Morgan Sanley. The porfolio invess in invesmen grades credis (from AAA o BBB) and governmen bonds. There are no derivaives in he porfolio as well as in he benchmark Syle of he porfolio manager The porfolio is consruced o ake proacive bes on credi spread while being neural on he ineres rae and currency risk. The porfolio is hen claimed o generae reurns and alpha via he credi analysis abiliy of is manager. Fig. 11. The red color represens he ineres rae effec, he blue represens he credi spread and he green he currency effec. The porfolio is mainly acive in credi spread. Currency Disribuion Relaive o Benchmark Duraion Disribuion Relaive o Benchmark Raing Disribuion Relaive o Benchmark CAD AUD CAD AUD 20.0% USD USD 10.0% 0.0% EUR GBP EUR GBP -10.0% -20.0% AAA AA A BBB NR SEK SEK DKK DKK -0.2% -0.1% -0.1% 0.0% 0.1% 0.1% 0.2% Fig. 12. Graphs represening he syle of he porfolio manager. Noe he acive bes on credi (raing)