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)

30 CHARACTERISTICS OF THE PORTFOLIO ANALYZED 16.03% 9.55% 18.00% 16.00% 14.00% 12.00% 10.00% 8.00% 6.00% 4.00% 2. 00% 0.00% AAA 2.17% AA A 8.30% 0.50% 1.37% BBB 12.35% 1.12% 2.16% 0.80% NR 8.48% 1.72% 2.53% 0.00% 1.13% % 0.00% 2.10% % 2.59% 0.82% 0.00% 1.11% 1.95% 0.00% 1.67% 1.84% 0.00% 0.00% % Fig. 13. Raing buckes per mauriy in he porfolio Bu is his saemen really rue? Is he porfolio manager s credi analysis abiliy really he main driver of he over-performance? Given ha he exposures oward ineres rae and currency risk are neural in any poin in ime, over-performances mus come from he credi par. An ineresing quesion o analyze is: does he performance come from an overweigh in A and BBB (asse allocaion and aking more risk) or is i due o selecion skills? 4.3. Se up of he fixed income performance analysis To answer his quesion we propose here o run a FIPA analysis in order o decompose he reurn of he porfolio. We hope ha his decomposiion will pu ligh on he main performance drivers of he porfolio and help us undersand beer where he performance come from. The followings seings have been used o analyze he porfolio: The yield curve Yield curves play a major role in fixed income aribuion analysis, because movemens in yield curves have a large effec on he pricing and hence he reurn of fixed income asses. A large and complex lieraure exiss on yield curves, reflecing he cenral par hey play in fixed income marke pricing. Many hundreds of research papers and several exbooks have been wrien on heir consrucion and modeling and yield curve expers coninue o devise ever-improved sofware sysems incorporaing boosrap echniques for consrucing curves, he dynamics of sochasic ineres rae modeling and sophisicaed echniques o mach bill srips o bond curves. So, he consrucion of yield curves and forecasing how hey behave, are deep and complex areas. Furhermore for a good FIPA analysis some consrains have o be pu on he yield curve behavior:

31 FIXED INCOME PERFORMANCE ATTRIBUTION The curve should be smooh, wih no gliches or disconinuiies. Oherwise, arbirage opporuniies will arise. Curves of differen credi raings should no cross or inersec. Oherwise, idenical bonds wih differen repaymen risks may show he same yield. The curve should inersec he cash rae a zero mauriy Firs we ried o generae yield curves from differen erm srucure models 9 wih all bonds available in our universe (someimes more han 300 bonds per yield curve), bu he ouliers produced biases in he compuaion. If you go his way, you will be ineviably forced o reduce he number of bonds (maybe beween 10 o 30 bonds) o remove he ouliers and make he model works. Bu hen occurs he sensible quesion of which bond should be seleced. Even if you do a meaningful selecion, i is likely ha your modeled curves will no always saisfy all he consrains menioned above. This yield curve generaion is herefore nonrivial. However, noe ha wha we are doing here is a Fixed Income Performance Aribuion (FIPA) and no a bond pricing. The differences are: In Pricing he absolue value of he curve plays a cenral role, while in FIPA only he daily relaive moves are imporan. In Pricing you ry o forecas he curve, while in FIPA you only need hisorical curves In Pricing he degree of precision has o be much higher han in FIPA. In Pricing he curves are no always compuable in illiquid markes. In FIPA if you use credi curves, you mus be able o generae a curve per day, per currency and per raing even if he marke is no very liquid. You canno afford o miss a curve for one day because he aribuion is done on a daily basis. To respec he consrains and o come up wih reasonable curves each day for each currency and each raing, we finally used a hybrid mehod o consruc he yield curve. We impored fair marke indices ha play he role of bonds. Then we compued he yield curve by cubic spline mehod. This mehod is simple, easy o implemen and we hink ha he curve reflecs a bes he mauriy erm srucure of he marke The YC decomposiion facors The yield curve decomposiion was done wih he empirical mehod (see ). A PCA has been used o calibrae he facors. Under he period ino consideraion, as he spread are prey ied, he buerfly effec is saisically non-significan. So we decided o remove his effec. We are lef wih: A parallel shif (more han 95% of he global shif) A reshape ha includes he wis effec (4%), he buerfly effec (0.5%) and a residual Linking mehod A geomeric approach has been chosen for diverse reasons. Firs, he converibiliy propery (see b) allows a consisency in reurns hrough differen currencies. For he porfolio 9 See poin

32 THE RESULTS analyzed, he reporing currency is in CHF bu he porfolio is invesed in USD, EUR, GBP, CAD, AUD, SEK and DKK. Second, by reading differen whie papers on his opic, i appears ha geomeric linking becomes he sandard in he indusry. 5. THE RESULTS 5.1. The FIPA aribuion for he global porfolio To compue ime weighed reurns (TWR), ransacion coss have been aken ino accoun bu no managemen fees and reurns generaed by cash accouns. In fac only he pure bond bucke of he porfolio is analyzed, which allows a direc and clean comparison wih he benchmark. The bond bucke is consolidaed in CHF. Reurns are given in basis poins. Dae TWR RC Direc reurn Roll down YC shif 1 YC reshape Aribuion Secor spread reurn YC spread reurn Fixed income iming Fixed income currency reurn 01/ / / / / / / / Fig. 14. Fixed income aribuion for he porfolio in bps per monh Dae TWR benchmark RC Direc reurn benchmark Roll down benchmark YC shif 1 benchmark YC reshape benchmark Aribuion Secor spread reurn benchmark YC spread reurn benchmark Fixed income iming benchmark Fixed income currency reurn benchmark 01/ / / / / / / / Fig. 15. Fixed income aribuion for he benchmark in bps per monh Dae Excess reurn RC Direc reurn excess Roll down excess YC shif 1 excess YC reshape excess Secor spread reurn excess YC spread reurn excess Fixed income iming excess Fixed income currency reurn excess 01/ / / / / / / / Fig. 16. Fixed income aribuion for excesses in bps per monh Aribuion

33 FIXED INCOME PERFORMANCE ATTRIBUTION Global reurn (TWR) The global reurn shows ha he porfolio over-performed regularly is benchmark over monhs wih an excepion in July. Therefore he porfolio seems on average o generae monhly 5 o 6 bps more han is benchmark. Noe ha all he reurns are compued in reporing currency ha is in CHF. Global reurn per monh Accumulaed global reurn 400 1' '000 Basis poins TWR RC TWR benchmark RC Excess reurn RC Basis poins Excess reurn RC TWR RC TWR benchmark RC / / / / / / / / / / / / / / / /2005 Fig. 17. Global reurn per monh and accumulaed in bps Direc reurn Carry reurn, also known as saic reurn, yield reurn or calendar reurn, measures he reurn of he securiy due o he passage of ime. This reurn may be decomposed furher in wo componens which are direc reurn and coupon reurn: Direc reurn is he par of reurn which originaes from ineres rae reurn, i.e. accrued ineres, coupon paymens and nex coupon value. This effec is always posiive. As he porfolio has on average bonds wih higher coupons han he ones in he benchmark, direc reurn generaed by he porfolio is slighly higher (around 10 bps per monh) han he direc reurn generaed by he benchmark. Direc reurn per monh Accumulaed direc reurn Basis poins Direc reurn Direc reurn benchmark Direc reurn excess Basis poins Direc reurn excess Direc reurn Direc reurn benchmark / / / / / / / / / / / / / / / /2005 Fig. 18. Direc reurn per monh and accumulaed in bps Roll-down Roll-down is he reurn, which originaes from changes in clean value due o ime changes, i.e. if he yield curve from yeserday is unchanged. To say ha in simply words: wih ime, each bond converges o par. As he ineres raes are hisorically low, mos of he bonds are raded wih a premium. As a resul, he roll-down effec is negaive. This effec is prey small

34 THE RESULTS and compensaes he relaive gain made wih direc reurn. As he porfolio has on average higher coupons han he benchmark, he bond prices in he porfolio are raded wih a higher premium han he ones in he benchmark and herefore he roll-down o par is higher oo. Roll down per monh Accumulaed roll down Basis poins Roll down Roll down benchmark Roll down excess Basis poins Roll down excess Roll down Roll down benchmark / / / / / / / / / / / / / / / /2005 Fig. 19. Roll-down per monh and accumulaed in bps YC shif 1 (parallel shif) As he porfolio was consruced o be duraion-neural, he resuls we obain are consisen wih he heory: he YC shif 1, which represens a parallel shif of he yield curve, can be approximaed by he duraion. This duraion-neural porfolio is herefore fully hedged agains he YC shif 1 effec and his effec is exacly he same for he porfolio and he benchmark. Yield curve shif 1 (parallel shif) per monh Accumulaed yield curve shif 1 (parallel shif) Basis poins YC shif 1 YC shif 1 benchmark YC shif 1 excess Basis poins YC shif 1 excess YC shif 1 YC shif 1 benchmark / / / / / / / / / / / / / / / /2005 Fig. 20. Yield curve shif 1 (parallel shif) per monh and accumulaed in bps YC reshape This effec regroups all he shifs of he yield curve oher han a parallel shif. The wis effec, he buerfly effec, plus a residual are included in he YC reshape. Of course a furher decomposiion ino sub-effecs would be possible. Bu noe ha he YC reshape effec is small relaive o he YC Shif 1. Ineres risk is managed no only by keeping he oal duraion close o he benchmark duraion bu also by managing he ineres risk exposure along he curve. Duraion is managed close o he benchmark for he buckes 0-3, 3-5, 5-7, 7-10 and 10+ (sraified sampling). The YC reshape effec (wis, buerfly, residual) is herefore relaively small (around 15 bps accumulaed unil Augus 2005) and for he analyzed period slighly posiive

35 FIXED INCOME PERFORMANCE ATTRIBUTION Yield curve reshape per monh Accumulaed yield curve reshape Basis poins YC reshape YC reshape benchmark YC reshape excess Basis poins YC reshape excess YC reshape YC reshape benchmark / / / / / / / / / / / / / / / /2005 Fig. 21. Yield curve reshape per monh and accumulaed in bps Secor spread reurn (credi spread) We coninue our analysis by puing he focus on credi analysis. The securiies can be grouped according o differen crierions. In our case, credi analysis is claimed o be he value driver of he porfolio invesmen approach. To calculae effecs for credi qualiy, we use curves for AAA, AA, A, BBB and BB qualiy. 10 Generally he yield curve shif is compued for a base curve (normally he governmen curves). There will be a residual curve shif for all oher curves han he base curve, as all curves do no change equally. The secor spread change, measured by our credi qualiy curves, explains his difference. This facor, hus, describes he yield curve shifs which canno be described by he base curve shifs (including reshape). Credi invesmens imply spread risk. Credi spreads show occasionally a subsanial widening. Such a widening usually occurs in anicipaion of a recession or in he conex of credi evens like LTCM, Enron or Ford/GM. Afer a cerain period credi spreads reurn o normal levels. Therefore here are always a period of over/under-performance due o spread volailiy. In spring 2005 such a spread widening was observable due o he problems in he US auo-secor. A he beginning of he year he main US auo-secors companies (Ford and GM) were raed BBB. The erosion of credi qualiy in hose issuers caused widening in credi spreads especially for he spread of he BBB segmen. In he porfolio analyzed, credi spread reurn from he porfolio is prey in line wih is benchmark. This would end o show ha credi spread risk in he porfolio and benchmark are almos he same. Noe ha an analysis under risk-adjused basis is given in chaper 6.2. Secor spread reurn per monh Accumulaed secor spread reurn Basis poins Secor spread reurn Secor spread reurn benchmark Secor spread reurn excess Basis poins Secor spread reurn excess Secor spread reurn Secor spread reurn benchmark / / / / / / / / / / / / / / / /2005 Fig. 22. Secor spread reurn (credi spread) per monh and accumulaed in bps 10 See Fig. 30 for more deailed explanaion

36 THE RESULTS YC spread reurn (issue spread) Issue spread is he par of he spread ha is no explained by credi curve shifs. I is he value added by picking specific bonds. The invesmen approach for he porfolio analyzed has a srong focus on credi analysis. The main driver of he over-performance relaive o he benchmark should be explained by issue spread, which is herefore closely relaed o he picking abiliy of he porfolio manager. The resuls of he FIPA analysis seem o confirm his view. The over-performance comes from he fac ha he porfolio akes bes on credi (issue spread) by picking he bes bonds, for example by buying cheap bonds and selling he expensive ones. On an accumulaed basis, 19bps have been creaed by issue spread from January o Augus 2005 which is prey high for a bond porfolio. A mulivariae analysis will be performed in he nex chaper o confirm or infirm his affirmaion. YC spread reurn (issue spread) per monh Accumulaed YC spread reurn (issue spread) Basis poins YC spread reurn YC spread reurn benchmark YC spread reurn excess Basis poins YC spread reurn excess YC spread reurn YC spread reurn benchmark / / / / / / / / / / / / / / / /2005 Fig. 23. YC spread reurn (issue spread) per monh and accumulaed in bps Fixed income iming This effec is explained by inraday flucuaions (rading price vs. closing price). Fixed income iming is he exra value added or subraced by dealing a prices ha are differen o end-of-day revaluaion raes. If a rader has he skill o choose advanageous levels a which o buy and sell securiies, hen his will add exra reurns o he porfolio. Addiionally, as iming effec is he las sep in he FIPA decomposiion hierarchy, i is also some kind of residual and herefore conains unexplained facors and/or oher errors of FIPA. As he model is well specified and as he inraday effec is close o zero, his effec is negligible (less han 3bps accumulaed unil Augus 2005). Fixed income iming per monh Accumulaed fixed income iming Basis poins Fixed income iming Fixed income iming benchmark Fixed income iming excess Basis poins Fixed income iming excess Fixed income iming Fixed income iming benchmark / / / / / / / / / / / / / / / /2005 Fig. 24. Fixed income iming per monh and accumulaed in bps

37 FIXED INCOME PERFORMANCE ATTRIBUTION Fixed income currency reurn This reurn is generaed by FX effec. The porfolio exposure is in any ime very close o he sraegic currency allocaion. Tha is he reason why he relaive accumulaed effec is very small for he porfolio analyzed. Fixed income currency reurn per monh Accumulaed fixed income currency reurn Basis poins Fixed income currency reurn Fixed income currency reurn benchmark Fixed income currency reurn excess Basis poins Fixed income currency reurn excess Fixed income currency reurn Fixed income currency reurn benchmark / / / / / / / / / / / / / / / /2005 Fig. 25. Fixed income currency reurn per monh and accumulaed in bps 5.2. The key raios We compued he alphas, beas and volailiy for he porfolio and he benchmark. The ime weighed reurns (TWR) were compued afer ransacion coss bu before managemen fees and bank accoun reurns, which are negligible on a more han one-billion porfolio. Dae TWR RC TWR benchmark RC Excess reurn RC Alpha RC Bea RC Volailiy RC Volailiy benchmark RC 01/ / / / / / / / Arih. average Fig. 26. The alpha, bea and volailiy per monh The Alpha TWR = α + β PF TWR BM Alpha is he inercep wih he y-axis in a linear regression of he periodic TWR reurn of he porfolio and he benchmark porfolio. Thus alpha measures he risk-adjused performance relaive o he benchmark. This is ofen referred o as a measure of sock selecion abiliy

38 THE RESULTS The Bea (, ) cov TWR TWR BM β = σpf 2 BM Bea is he slope of a linear regression of he periodic TWR reurn of he porfolio and he benchmark porfolio. Thus bea measures he sysemaic performance/risk relaive o he benchmark. Alpha per monh Bea per monh Basis poins Alpha RC Alpha RC average Basis poins Bea RC Bea RC average / / / / / / / / / / / / / / / /2005 Fig. 27. Alphas and beas in reporing currency (CHF) Monly TWR vs. Volailiy TWR Porfolio Benchmark Volailiy Fig. 28. Monhly ime weighing reurns vs. volailiy Wih heses words we are ending he FIPA analysis iself. We can observe on Fig. 28. ha he porfolio and benchmark are prey close o each oher. Bu he porfolio is a bi riskier han is benchmark and generaes on average 5bps per monh more han he benchmark. On an academic poin of view, i would be hen ineresing o find which are he main FIPA facors explaining his over-performance of 5bps. For insance, is i really he issue spread (he credi analysis abiliy of he porfolio manager) which drives he over-performance? In a second sep i would also be of high ineres o check if he porfolio excess reurn sill exiss on a risk adjused-basis. We answer hese quesions in he nex chaper called inerpreaion

39 FIXED INCOME PERFORMANCE ATTRIBUTION 6. INTERPRETATION In his secion, we propose a mulivariae analysis o beer undersand he ineracion beween he differen facors. In paricular we would like o undersand where exacly he performance comes from, if he acive rades of he porfolio manager have really creaed value and if he porfolio manager s credi analysis abiliy, which is measured by issue spread, is really he main driver of performance Excess reurns and FIPA facors The firs sep would be o examine if he FIPA facors ha decompose he excess reurns are independen from each oher or if some kind of co-lineariy eners in he model. So we will now concenrae our aenion on excess reurns Disribuion of excess reurns The daabase is composed of 174 business days from he o he The following graph represens he disribuion of he daily excess reurns. Frequency Excess.reurn Excess.reurn Fig. 29. Frequency and box plo of daily excess reurns A a firs view, he excess reurn disribuion is prey symmeric and remember us a sandard normal. The mean is slighly bigger han zero and no exreme days are presen in he ime period considered. This kind of disribuion is ypical for a bond porfolio. Unil here no surprise Ineracion beween FIPA facors and excess reurns Before proceeding wih a regression modeling of he daa i may be helpful o examine a scaer plo marix for he variables. The scaer plo will sugges he underlying relaionship 11 The code for he mulivariae analysis can be found in Appendix

40 INTERPRETATION beween he differen facors and will give us a firs appreciaion of he qualiy of he model we used Excess.reurn Direc.reurn Roll.down YC.shif.1 YC.reshape Secor.spread Issue.spread FI.iming FI.currency Fig. 30. Scaer plo of he FIPA facors To invesigae he underlying relaionships before underaking formal modeling of he daa, we ploed in each panel a simple linear regression (bold line) and a locally weighed regression fi 12. The locally weighed regression is approximaely he same han he linear regression which indicaes ha a linear model is he one o use. Here again no surprise because each facors are geomerically added o compose he excess reurns. If we have a look on he op righ hand corner of Fig. 30., we observe ha he slope of he linear regressions are around zero for mos facor pairs (direc reurn, roll-down, yield curve shif 1, yield curve reshape, secor spread, issue spread, fixed income iming and fixed income currency). This speaks for almos no correlaion beween facors which is again wha we would like o have in our model in order o be able o inerpre each aribuion facor clearly and wihou unwaned correlaion. An aenive reader would have however noiced a sligh negaive correlaion beween secor spread and issue spread. This is direcly caused by he qualiy of he credi yield curves we used in he model. Even wih he hybrid mehod implemened (see poin ) a small compensaion effec is sill presen beween he wo facors. The graph below illusraes how he wo facors inerac ogeher. 12 For comparison beween he wo regressions see [2] Chambers J.M. & Hasie T.J., Saisical Models in S, Wadsworh and Brook/Cole,

41 FIXED INCOME PERFORMANCE ATTRIBUTION Yield curve shif 9 8 Bond in 7 Yield in % Bond in -1 AAA in -1 AAA in BBB in -1 BBB in Shif 2 YC shif 1 1 Secor spread Year Issue spread Fig. 31. Ineracion beween secor spread and issue spread Le us assume ha he AAA curve is he base curve of our model. If he AAA curve moves up in one day of 1%, i is highly probable ha he BBB curve will move up of more han 1% especially in he long erm. The increase of one percen is explained by he yield curve shifs (here uniquely by he YC shif 1 because he shif is parallel). Now we ry o analyze he move of a bond represened by he black poin in he graph. In day -1, he bond was exacly on he BBB line. Because of he BBB curve move, he bond should sill lie on he BBB curve in (ligh blue curve). Bu as i is no a good bond, he issue spread pushes he bond a bi furher up. Finally we end up wih hree effecs: 1. A yield curve shif ha if fully explained by he move of he base curve (generally a AAA curve or a governmen curve) 2. A secor spread effec due o he credi spread of he bond. The moves of a BBB curve are ypically of higher inensiy ha he ones of a AAA curve 3. An issue spread effec which is specific o he bond. This effec is closely relaed o he picking effec. Therefore if he BBB curve is no perfecly esimaed a ime, a compensaion beween secor spread and issue spread migh appear. Unforunaely for he low raings (from BBB and below) he universe of bonds are very dispersed and i becomes difficul o build a yield curve reflecing perfecly he marke. Tha is he reason why we can see a sligh negaive correlaion beween secor spread and issue spread in Fig. 30. The qualiy of he yield curves is cerainly a poin we could improve in our model. If we go back o Fig. 30., we observe in he red box ha none of he effecs he issue spread exceped seem o significanly influence he excess reurn. This would end o speak in favor of he porfolio manager who played acive bes on credi. His picking abiliy (issue spread) seems o play a preponderan role for explaining excess reurn

42 INTERPRETATION Mulivariae analysis According o Fig. 30. only a few effecs should play an imporan role in he creaion of excess reurn. This is confirmed by a muliple regression: Call: lm(formula = Excess.reurn ~ Direc.reurn + Roll.down + YC.shif.1 + YC.reshape + Secor.spread + Issue.spread + FI.iming + FI.currency) Residuals: Min 1Q Median 3Q Max Coefficiens: Value Sd. Error value Pr(> ) (Inercep) Direc.reurn Roll.down YC.shif YC.reshape Secor.spread Issue.spread FI.iming FI.currency Residual sandard error: on 164 degrees of freedom Muliple R-Squared: 1 F-saisic: on 8 and 164 degrees of freedom, he p-value is 0 Correlaion of Coefficiens: (Inercep) Direc.reurn Roll.down YC.shif.1 YC.reshape Direc.reurn Roll.down YC.shif YC.reshape Secor.spread Issue.spread FI.iming FI.currency Secor.spread Issue.spread FI.iming Direc.reurn Roll.down YC.shif.1 YC.reshape Secor.spread Issue.spread FI.iming FI.currency Fig. 32. Resuls of he mulivariae analysis The firs observaion is ha he muliple R-squared sum o one indicaing ha no residuals were generaed. This is a nice propery of he geomeric model (see chaper ). The second observaion is ha he facor coefficiens are close o one. This is as well he value expeced as we obain excess reurn by adding geomerically each facor. As expeced he mos imporan coefficiens are: Issue spread (specific spread or picking abiliy) Secor spread (credi spread) Reshape Finally he impression we had wih Fig. 30. was correc: he correlaion beween coefficiens are prey low excep for issue spread and secor spread. If we add he 3 more imporan facors we obain a muliple R-squared of more han 84% wih all he coefficiens being highly significan. Issue.spread: Muliple R-Squared: Issue.spread + Secor.spread: Muliple R-Squared: Issue.spread + Secor.spread + YC.reshape: Muliple R-Squared:

43 FIXED INCOME PERFORMANCE ATTRIBUTION So we hink ha he mos appropriae model for his paricular porfolio would be he following: Call: lm(formula = Excess.reurn ~ Issue.spread + Secor.spread + YC.reshape) which consiss of a muliple linear regression of hree facors. The nex sage of he analysis should be an examinaion of he residuals from fiing he chosen model o check on he normaliy and consan variance assumpions. If he residuals are nicely disribued we will be able o validae his hree facor model. The analysis consiss of comparing he residual disribuion o a sandard normal. We made he analysis more informaive by supplemening i wih a confidence inerval, as suggesed in Akinson (1987) and Weisberg (1982). In essence, he procedure involves he generaion of m pseudo-residual vecors, e, from y i = β + β x β x + ε 0 1 i1 p ip i y = Xβ + ε H = X T T T ( X X ) X e k ( I H ) ε k =, k = 1,..., m and I being he ideniy marix On he following graph you can see he 95% (solid line) confidence inerval and he 90% confidence inerval (doed line): Fig. 33. Scaer plo of he FIPA facor residuals

44 INTERPRETATION The residuals pass he es of he 90 percen inerval, bu fail he es of he 95% inerval Poins ouside Toal Percenage Tes passed Confidence inerval 95% % No Confidence inerval 90% % Yes Fig % and 95% confidence inerval of he residuals We hen can argue ha for normal days (90% of he ime) our model wih hree facors describe very well he excess reurn. So we can conclude ha he issue spread effec, which is definiely he main facor in our model, plays a cenral role in he generaion of overperformance relaive o he benchmark. This proves ha, a leas for he period aken ino consideraion and afer ransacion coss bu before managemen fees and bank accoun reurns, i is he porfolio manger s credi analysis abiliy ha has generaed an excess reurn (on average 5 o 6bps per monh). We repeaed he same process afer managemen fees and bank accoun reurns which are boh very small compared o he porfolio size. Therefore he ne excess reurn gives similar resuls (on average an excess reurn of 4 o 5bps per monh). So he porfolio manager did really creae excess reurn for is clien. Tha s fine. Bu one could now ask his embarrassing quesion: Do you sill have he same picure under a risk-adjused basis?. We answer his quesion in he nex session Performance on a risk-adjused basis Alpha and FIPA facors We have shown ha he porfolio does produce excess reurn. More, we showed hanks o he FIPA decomposiion ha i is he manager s abiliy (issue spread) ha is he main driver of he excess reurn. Wih FIPA, we were able o decompose excess reurn ino differen facors. The ineresing propery of he porfolio we are analyzing is ha i only akes be on credi spread while being neural in ineres and currency risk. Therefore for his porfolio, he issue spread mus play he same role of he alpha because hey boh ry o measure he value added or subraced by he porfolio manager. We remind you ha alpha measures he difference beween a fund's acual reurns and is expeced performance, given is level of risk (as measured by bea). A posiive alpha figure indicaes he fund has performed beer han is bea would predic. In conras, a negaive alpha indicaes a fund has underperformed, given he expecaions esablished by he fund's bea. Some invesors see alpha as a measuremen of he value added or subraced by a fund's manager. Those invesors seeking higher reurn from heir porfolios can ake on more alpha risk and/or more bea risk. There are numerous alpha-seeking sraegies. In his paper, we focus on an alpha sraegy ha makes use of he issue spread beween bonds wih differen raings. I is generally admied ha he managemen of such a sraegy requires a significan amoun of fundamenal research on credi and specific risk analysis. Superior research is likely o generae greaer alpha. The hree largely independen sources of alpha are:

45 FIXED INCOME PERFORMANCE ATTRIBUTION higher expeced reurns from he sraegic allocaion ino he higher credi risk high acive managemen opporuniies wih bonds wih lower raings addiional reurns from he acive acical shifs across he differen fixed-income secors. The combinaion of muliple independen alpha sources creaes he opporuniy for higher reurn wih relaively small increase in risk. The skills required in he successful managemen of credi spread porfolios can be used o capure all hree sources of alpha described above. We would like here o es if our issue spread calculaed by FIPA is correlaed or no wih he alpha. Unforunaely, we only dispose of eigh monhs of complee daa. We are hen forced o do a weekly comparison in order o have enough daa o do an accepable correlaion analysis. Bu he qualiy of he alpha will of course suffer from he very shor period. Here are he resuls: Issue spread and alpha (by week) Pasis poins Issue spread Alpha RC / / / / / / / /2005 Fig. 35. Comparison beween issue spread and alpha on a weekly basis On a weekly basis he correlaion is 0.63 and on average he alpha is around 1.4bps per week while he issue spread is around 0.6bps per week. This indicaes ha a srong link beween alpha and issue spread exiss reflecing he manager s abiliy o analyze credis and pick up he righ bonds. Unforunaely he hisory is much oo shor o perform a robus analysis and we are suck here. We however hink ha i is a good way for idenifying he par of he risk-adjused reurn, measured by he alpha, which has been creaed by he porfolio manager. I would be ineresing o redo his analysis on a monhly basis o improve he alpha qualiy and wih 5 o 10 years of daa o improve he correlaion analysis. This being said, he Fig. 35. gives us a firs hin: i seems ha he porfolio does generae some alpha afer coss and ha a good par of i is capured by he issue spread Excess reurns on a risk-adjused basis Anoher way o adjus he excess reurns o risk would be o adap he benchmark in order o remove all he bea-reurns. We remind you ha he hree main facors ha explain he reurns of he porfolio analyzed are in order of imporance:

46 INTERPRETATION Issue spread (specific spread or picking abiliy) Secor spread (credi spread) Reshape wih a muliple R-squared of 84%. Furhermore, we know ha he porfolio only akes bes on credi spread and is neural on currency and ineres risk. Tha is he reason why, for his paricular porfolio, issue spread and secor spread play he cenral roles. Noe ha he porfolio is consruced o be neural on ineres risk bu only in he firs order (duraion). I is hen logic ha he reshape is he hird mos imporan facor. We know ha he porfolio is 10% underweighed in AAA and overweighed in A and BBB. The secor spread shows us how much reurn he porfolio manager has generaed by being a bi riskier han he benchmark, bu i does no ell us if he reurns were fairly compared o he risk aken. So how could we adjus excess reurns o risk? If we could arificially increase he benchmark credi risk and equalize i o he porfolio credi risk, we could ge rid of he reurn generaed by aking more risk on credis. Then, excess reurn could be compued on a riskadjused basis and compared wih he usual excess reurn. To implemen his process he necessary condiion is o be able o exrac he credi spread from he benchmark. An easy and diry way would be o ake a AAA benchmark and a BBB benchmark and compue he heoreical spread. Bu if we could spli up he benchmark ino differen raings and use he weighs of he porfolio o compue he overall performance, we could direcly adjus he benchmark risk o he porfolio risk. Unforunaely he main problem wih securiy-level benchmark daa supplied by mos vendors is ha raes of reurn are no available on securiy-level (see poin 3.4.). However, afer a huge work of daa qualiy improvemen, we are oally free wih FIPA o decompose a benchmark according o our wishes because he benchmark is reaed as a normal securiy-level porfolio. We propose here o decompose he benchmark by raings and weigh each bucke by he porfolio weighs o adjus he benchmark credi risk. This kind of decomposiion is prey similar o he well-esablished Brinson, Hood and Beebower (BHM) model 13 for op-down equiy performance aribuion. We have r r Excess Adj Excess = w = w PF PF r r PF PF w w BM PF r r BM BM Adj where r Excess is he excess reurn, r Excess is he risk-adjused excess reurn, w PF and wpm are respecively he weighs of he porfolio and he benchmark and finally we have he erms r and r which are respecively he reurn of he porfolio and he benchmark PF PM Then we can calculae: Adj Credirisk = rexcess rexcess = ( wpf rpf wbm rbm ) ( w r w r ) = w r w r = r ( w w ) PF PF PF BM PF BM BM BM BM PF BM 13 See for example [14] Splaulding D., Invesmen Performance Aribuion, McGraw-Hill,

47 FIXED INCOME PERFORMANCE ATTRIBUTION which is similar o he Asse Allocaion (AA) formula in he op-down BHM model. And, Selecion = r Adj Excess = r Excess Credi risk = w PF ( r r ) PF BM would be he reurn generaed by he porfolio manager s picking abiliy. In he BHM model his is described as he Sock Selecion effec (SS). As he weighs in he porfolio are larger in BBB bonds compared o he weighs of he BBB BBB benchmark ( w PF wpm > 0), we expec he formula of credi risk o be posiive because BBB bonds give on average higher reurns han AAA bonds. Credi risk explains he fair par of he excess reurn coming from he fac ha he porfolio is riskier han is benchmark. Here are he resuls for he porfolio analyzed: Dae 01/ / / / / / / /2005 Excess reurn calculaed by FIPA (r excess) Excess reurn on a risk adjused basis (Selecion) Reurn due o credi risk (Credi risk) Fig. 36. BHM decomposiion in bps. The resuls are a priori surprising bu in fac really ineresing. I seems ha we can spli he period in wo: he firs bucke would be January, February, May, June, July and Augus 2005 where he risk premium is posiive. The second bucke would be March and April 2005 wih a negaive risk premium. For he firs bucke an inerpreaion is sraighforward. The pure credi risk generaes abou 2bps per monh. So, in he excess reurn compued by FIPA we should remove abou 2bps per monh from he normal excess reurn o ge a reurn on a risk-adjused basis. So on he 5bps ha are on average generaed every monh by he porfolio (see ), 2bps are coming only from credi risk. These 2bps of pure credi risk are simply added along he decomposiion facors of he FIPA analysis (principally on credi spread and issue spread). The second bucke, March and April 2005, is rickier o analyze. I seems ha he premium o credi risk was negaive (-18bps in March)! In fac, he bond marke was prey shaky in March/April 2005 because of he US auo-secor issue. For example GM, one of he major auo-moive company, go downgraded from BBB- o BB in spring On he following graph you can see he disasrous effec of downgrading rumors on he GM Swap Relaive Value Curve. A very similar example could be given for Ford. Fig. 37. Effecs on he Swap Relaive Value of GM bonds afer he downgrading

48 INTERPRETATION The erosion of credi qualiy in US auo-secors companies caused a widening in credi spreads especially for he spread of he BBB segmen. I is he firs ime in he recen hisory a leas since 1997 ha only a segmen (BBB) widens so much while oher segmens remain fla. This widening was huge in March and induced large losses for people having GM bonds in heir porfolios. Addiionally, mos of hem were forced sellers. On his effec disappears because GM was downgraded o BB he and consequenly jumped ou of he BBB benchmark segmen Difference of yields - raing class vs. governmen bonds USD Bonds in basis poins, Augus Sep (Morgan Sanley) AAA AA A BBB Fig. 38. Effecs caused on he BBB yields by he downgrading of GM (in bps) Now i becomes easy o inerpre he negaive premium in March As we use he weighs of he porfolio o weigh he benchmark, he BBB segmen is consequenly overweighed. So he widening of he BBB curve in March induces large losses for he risk-adjused benchmark. In he opposie, all posiions in GM in he porfolio were already sold in December Consequenly, in he porfolio, we do no have such big losses due o he BBB curve widening. This reflecs a good picking abiliy of he porfolio manager who has been able o remove GM posiion soon enough. This is capured in our BHM model by he Selecion effec. And herefore his explains why we ge a negaive risk premium in March To summarize, if we remove he GM effec from he calculaion, he normal excess reurn should be correced of 2 o 3 bps per monh o be risk-adjused. Bu if we keep he GM effec in he calculaion, here is almos no differences beween he normal excess reurn compued by FIPA (4.57bps) and he risk-adjused excess reurn (5.01 bps). Mean wihou GM effec (wihou March Mean and April 2005) Excess reurn calculaed by FIPA (r excess) Excess reurn on a risk adjused basis (Selecion) Reurn due o credi risk (Credi risk) Fig. 39. Mean of he BHM decomposiion in bps (wih and wihou GM effec) To conclude, we would like o make he reader aenive ha normally he Brinson, Hood and Beebower (BHM) model is usually applied on equiy. The BHM model only splis he reurn

49 FIXED INCOME PERFORMANCE ATTRIBUTION ino Asse Allocaion and Sock Selecion effec. Wih bonds porfolios he BHM model is far oo simple because i does no ake ino accoun imporan effecs like he yield curves and he carry of he bonds. However we have been able o apply his BHM model on he porfolio analyzed because we proved in he former chapers ha he single main driver of he porfolio is he credi spread. So a BHM decomposiion on he credi spread only has been herefore possible for his paricular porfolio. 7. CONCLUSION The Fixed Income Performance Aribuion analysis (FIPA) is for sure an unbelievable powerful ool o decompose he performance of a bond porfolio. The advanages of his mehod are numerous and include: a sysemaic idenificaion of he basic sources of reurn in erms of bes and marke variables via he FIPA facors; a flexible and consisen inerpreaion of aribuion numbers; an easy measuremen of differen invesmen sraegies wihin he same framework. The main difficulies o perform such an analysis are cerainly he se up and he daa qualiy. The se up is raher complex, especially when defining accurae yield curves and replicaing he benchmark on a securiy-level. In addiion, daa qualiy is for sure he mos sensible poin of he whole process. Our experience has showed ha one single misake in a bond migh make he sysem collapse and make he aribuion difficul o inerpre because some facors are very sensiive o any misake. Wih he FIPA decomposiion we have also been able o idenify he main driver of he porfolio excess reurn in our case he issue spread. We also showed ha he porfolio generaes on average some alpha and a decen excess reurn of 5 o 6bps per monh afer coss bu before managemen fees and cash reurns (bank accouns). Afer fees and cash reurns, he excess reurn is beween 4 o 5bps a monh. Under risk-adjused basis, abou 2 basis poins should be removed from excess reurn. This being done, he porfolio always generaes a ne excess reurn of 2 o 3bps a monh for he clien which emphasis he porfolio manager s abiliy o analyze credis and generae some alpha. This could be inerpreed as he beginning of he proof ha alphas can be generaed afer all coss have been aken ino accoun. Unforunaely daa have been cleaned only back o , so we do no dispose of a sufficien ime period o have robus resuls. We can only conclude ha during he period analyzed he porfolio over-performed is benchmark on a risk-adjused basis. However we can say nohing abou he persisence over ime of he over-performance. For an academic poin of view, FIPA analysis brings a huge variey of new daa ha have no been analyzed ye. For insance, i would be ineresing o do he same analysis on a 10 or 15 year period o improve he robusness of he resuls and check he persisence of he overperformance. Unforunaely, he ime and he coss of a daa cleaning are enormous. Furhermore, we doub ha large banks or asse managemen companies will make he hear of heir fixed income sraegies available for an exernal analysis. In he near fuure, we hink ha major players in he bond marke will use FIPA analysis o analyze ex-pos heir bes and acive sraegies because a well se up FIPA analysis produces a mine of deailed and direcly analyzable informaion. Of course he same model could be easily exended for simulaing ex-ane scenarios like a yield curve shif or a change in FX

50 CONCLUSION raes. The FIPA analysis will herefore become also an acive managemen ool which will help in he consrucion of an efficien porfolio. When wriing his hesis, a he opposie of he equiy side, no sandard aribuion models for fixed income are well-esablished on he marke ye. This is cerainly due o he complexiy of he risk facors driving fixed income performance. We believe ha in he nex few years Fixed Income Performance Aribuion will remain a very ho opic in he bond indusry. Unforunaely we do no hink ha so sensible daa will be published very easily for academic researches and we are herefore a bi more reserved concerning he fuure benefis of FIPA for he academic world

51 FIXED INCOME PERFORMANCE ATTRIBUTION 8. ACKNOWLEDGMENTS This paper was sponsored by Swisscano Asse Managemen and conduced as a par of an inernal projec in he bond eam. I firs would like o hank my wo supervisors, Dr. Nils Tuchschmid for his guidance hroughou he work and Dr. Anna Holzgang for her precious suppor on he FIPA projec. I also would like o hank Mr. Alex Schöb for he confidence he gave me as well as all he bond eam, he reporing eam, he back office eam and he Business Applicaion eam of Swisscano for heir suppor hroughou he projec. I owe a special hank o Mr. Mahias Zimmermann who helped me se up he FIPA module. I would like also o hank Ms. Doreen Rodui for he spelling correcion. Finally I am graeful o my family for he suppor and encouragemens

52 REFERENCES 9. REFERENCES [1] Akinson A.C., Plos, Transformaions, and Regression, Oxord Saisical Science Series, 1988 [2] Chambers J.M. & Hasie T.J., Saisical Models in S, Wadsworh and Brook/Cole, 1988 [3] Colin A., Fixed Income Aribuion, Wiley Finance Series, 2005 [4] Cook R.D. & Weisberg S., Residuals and Influence in Regression ; CRC/Chapman & Hall, 1982 [5] Dynkin L., Hyman J. & Konsaninovsky V., A reurn Aribuion Model for Fixed Income Securiies, Handbook of Porfolio Managemen, 1998 [6] Esseghaier Z., Lal T., Cai P. & Hannay P., Yield Curve Decomposiion and Fixed Income Aribuion, DST inernaional, 2003 [7] Everi B.S., A Handbook on Saisical Analyses using S-Plus, Chapman & Hall/CRC, 2002 [8] Fong G., Yoo D., & Zelaya Z.M., Global Performance Aribuion, Perspecives on Inernaional Fixed Income Invesing, 1998 [9] Hughson L., Vasicek and Beyond: Approaches o Building and Applying Ineres Rae Models, Risk Books, 1997 [10] James J. & Webber N., Ineres Rae Modelling, Ed. J. Wiley, 2001 [11] Kno G.D., Inerpolaing cubic splines, Birkhäuser, 2000 [12] Søgaard-Andersen P., Fixed Income Performance Aribuion, a flexible approach, SimCorp Dimension, Knowledge-sharing, 2005 [13] Sørensen O., FIPA calculaions in SimCorp Dimension, SimCorp Dimension, 2005 [14] Spaulding D., Invesmen Performance Aribuion, McGraw-Hill, 2003 [15] Vasicek O., An equilibrium characerisaion of he erm srucure, Journal of Financial Economics, 5: ,

53 FIXED INCOME PERFORMANCE ATTRIBUTION 10. APPENDIX Appendix 1: US governmen yield curve principal componen analysis # # Fixed Income Performance Aribuion # Principal Componen Analysis # by Blaise Rodui # module(finmerics) # 1. Consruc appropriae reurn daa # # Loading daa in a ime series wih business days USD.GOV.YC<-read.able("D:/USD.GOV.YC2.x",header=T) USD.GOV.YC.s<-imeSeries(USD.GOV.YC, from="01/31/1997", by="monhs") # Selecing he 1 year period from 2004 o 2005 window2.sar <- imedae("01/31/1997") window2.end <- imedae("08/31/2005") sample2 <- (posiions(usd.gov.yc.s) > window2.sar & posiions(usd.gov.yc.s) < window2.end) USD.GOV.YC <- USD.GOV.YC.s[sample2,] Xdaa <- USD.GOV.YC Xdaa <- seriesdaa(xdaa) # 2. Plo he daa # plo(usd.gov.yc , plo.args = lis(col=1:10),plo.args = lis(ly=1:10)) ile("variaion of he indexes composing he USD GOVT Curve") legend(0,4,colids(usd.gov.yc.march.may),ly=1:10,col=1:10) #locaor(1) # 3. Specral Decomposiion # # Specral Decomposiion S <- var(xdaa,unbiased=f) mp.eigen <- eigen(s) names(mp.eigen) L <- diag(mp.eigen$values) G <- mp.eigen$vecors # Le S-Plus calculae principal componens ou <- princomp(xdaa) summary(ou) # 4. Graphical displays of explained variances and loadings # screeplo(ou)

54 APPENDIX plo(loadings(ou)) # Model wih hree facors ou3 <- mfacor(xdaa,k=3) ou3 loadings(ou3) plo(ou3,which.plos=c(1,3)) ile="cumulaive variance explained" # Closer look a facor loadings par(mfrow=c(1,3)) barplo(loadings(ou3)[1,],names=colids(loadings(ou3)),horiz=t,main="facor 1 loadings") barplo(loadings(ou3)[2,],names=colids(loadings(ou3)),horiz=t,main="facor 2 loadings") barplo(loadings(ou3)[3,],names=colids(loadings(ou3)),horiz=t,main="facor 3 loadings") # Sandardise loadings o give porfolio weighs par(mfrow=c(1,1)) plo(summary(mimic(ou3))) # Correlaion srucures implied by facor model S.pcafac <- vcov(ou) S.pcafac2 <- vcov(ou2) S.pcafac2 <- vcov(ou3) round(s.pcafac,2) round(s.pcafac2,2) round(s.pcafac2,2) round(s,2) Appendix 2: Mulivariae analysis of he FIPA facors # # Fixed Income Performance Aribuion # Mulivariae analyis Regression # by Blaise Rodui # module(finmerics) # 1. Consruc appropriae reurn daa # # Loading daa in a ime series wih business days XXX.BND<-read.able("C:/Documens and Seings/brodui/My Documens/Blaise Rodui/MASTER THESIS FOLDER/XXX.Bond_2.x",header=T) XXX.BND.s<-imeSeries(XXX.BND, from="12/31/2004", by="bizdays") # Selecing he 1 year period from 2004 o 2005 window2.sar <- imedae("12/31/2004") window2.end <- imedae("10/01/2005") sample2 <- (posiions(xxx.bnd.s) > window2.sar & posiions(xxx.bnd.s) < window2.end) XXX.BND.2005 <- XXX.BND.s[sample2,]

55 FIXED INCOME PERFORMANCE ATTRIBUTION # 2. Analysis # Xdaa <- XXX.BND.2005 Xdaa <- seriesdaa(xdaa) aach(xdaa) par(mfrow=c(1,2)) his(excess.reurn, xlab=names(xdaa)[1],ylab="frequency") boxplo(excess.reurn, ylab=names(xdaa)[1]) # 3. Confidence inerval (Cook and Wisberg 1982) # Excess.one <- cbind(rep(1,nrow(xdaa)), Xdaa[, c("issue.spread", "Secor.spread", "YC.reshape")]) Excess.one <- as.marix(excess.one) Excess.ha <- Excess.one%*%solve((Excess.one)%*%Excess.one)%*%(Excess.one) Iden <- diag(nrow(xdaa)) se.seed(547) epsilon <- marix(rnorm(100*nrow(xdaa),0,1),ncol=100) e <- (Iden-Excess.ha)%*% epsilon e <- (e) e <- apply(e,2,sor) e.bis <- e e <- e[5:95,] E <- apply(e,2,range) e.bis <- e.bis[10:90,] E.bis <- apply(e.bis,2,range) win.graph() ylim <- range(excess.res, E[1,], E[2,],E.bis[1,], E.bis[2,]) qqnorm(excess.res,ylim=ylim, ylab="sandardised residuals") par(new=t) qqnorm(sor(e[1,]), ype="l", axes=f, ylim=ylim, xlab="", ylab="") par(new=t) qqnorm(sor(e[2,]),ype="l", axes=f, ylim=ylim, xlab="", ylab="") par(new=t) qqnorm(sor(e.bis[1,]), ype="l",ly=2, axes=f, ylim=ylim, xlab="", ylab="") par(new=t) qqnorm(sor(e.bis[2,]),ype="l",ly=2, axes=f, ylim=ylim, xlab="", ylab="") qqline(excess.res) pairs(xdaa,panel=funcion(x,y){ poins(x,y,col=2) abline(lm(y~x),lwd=2) lines(lowess(x,y),ly=2,lwd=2)}) aach(xdaa) Excess.fi <- lm(excess.reurn ~ Direc.reurn + Roll.down + YC.shif.1 + YC.reshape + Secor.spread + Issue.spread + FI.iming + FI.currency) summary(excess.fi)

56 APPENDIX Excess.fi.1 <-lm(excess.reurn ~ Issue.spread) Excess.fi.2 <-lm(excess.reurn ~ Issue.spread + Secor.spread) Excess.fi.3 <-lm(excess.reurn ~ Issue.spread + Secor.spread + YC.reshape) Excess.fi.4 <-lm(excess.reurn ~ Issue.spread + Secor.spread + YC.reshape + Roll.down) Excess.fi.5 <-lm(excess.reurn ~ Issue.spread + Secor.spread + YC.reshape + Roll.down + FI.currency) summary(excess.fi.1) summary(excess.fi.2) summary(excess.fi.3) summary(excess.fi.5) anova(excess.fi.1, Excess.fi.2, Excess.fi.3, Excess.fi.4) # 4. Check he normaliy of residuals # s <- summary(excess.fi.3)$sigma h <- lm.influence(excess.fi.3)$ha Excess.res <- residuals(excess.fi.3)/(s*sqr(1-h)) qqnorm(excess.res) qqline(excess.res)