Fixed Income Attribution. Remco van Eeuwijk, Managing Director Wilshire Associates Incorporated 15 February 2006

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1 Fxed Incoe Arbuon eco van Eeuwk Managng Drecor Wlshre Assocaes Incorporaed 5 February 2006

2 Agenda Inroducon Goal of Perforance Arbuon Invesen Processes and Arbuon Mehodologes Facor-based Perforance Arbuon Facor Models Theory behnd Wlshre s Facor Model Esang he Wlshre s Facor Model Concluson P A G E

3 Wlshre Assocaes Inc. Wlshre Assocaes Inc. Esablshed n 973 Four dvsons: Analycs Consulng Fund Manageen Prvae Equy Analycs Dvson Sofware syses Invesen sraegy research porfolo consrucon rsk easureen coplance and perforance easureen Equy fxed ncoe alernave asses and balanced porfolos 60 saff globally: Sana Monca New York London and Sngapore Analycs EMEA London offce opened n 996; 25 saff currenly > 0 clens P A G E 2

4 Purpose of Perforance Arbuon To quanfy he value added by he decsons n he nvesen process Decsons be nenonal or unnenonal Invesen process s leadng P A G E 3

5 Invesen Processes: 3 Types Herarchcal Independen Mxed P A G E 4

6 Invesen Processes: Herarchcal Porfolo eurn or sk egon A egon B Secor Secor 2 Secor Secor 2 Sock I Sock II Sock III Sock IV Sock V Sock VI Sock VII Sock VIII Three levels of decsons egon allocaon decson Secor allocaon decson whn each regon Sock selecon decson whn each secor n each regon Can be op-down or boo-up Coon n equy porfolos P A G E 5

7 Invesen Processes: Independen Porfolo eurn or sk Currency A Duraon Issuer I Issuer V Currency B Duraon 2 Issuer II Issuer III Issuer VI Issuer VII Secures are decoposed no facors Facor exposures are anaged ndependenly equres exensve se of dervaves Especally.forwards/fuures/swaps Coon n fxed ncoe porfolos Issuer IV Issuer VIII P A G E 6

8 Arbuon Mehodologes () Group-based Porfolo and benchark are dvded no groups of secures Weghng decsons whn and beween groups: selecon and allocaon Exaples: Brnson Apples o herarchcal nvesen processes Facor-based Secures are decoposed no facor exposures ha are anaged ndependenly Exposure o each facor n he porfolo and benchark are easured Exaples: Currency/Local eurn Mul-facor odels Apples o ndependen nvesen processes P A G E 7

9 Arbuon Mehodologes (2) Mxed Secures are grouped or spl no facors a varous levels n he nvesen decson ree Exaple: Any ul-currency arbuon (eg. Karnosky-Snger) Appled o xed nvesen processes P A G E 8

10 Facor-Based Arbuon Four seps: Decopose secures no facors Esae reurn for each facor Measure facor exposures for porfolo and benchark Arbue perforance P A G E 9

11 Facor Models: Decoposng Secures no Facors Facor odels a o explan secury reurns by decoposng secures no facors: K β f ε eurn of secury durng perod B Exposure of secury o facor a he sar of perod f eurn of facor durng perod E esdual reurn of secury durng perod P A G E 0

12 Types of Facor Models There are hee ypes of facor odels Macro-econoc facor odels Mcro-econoc or fundaenal facor odels Sascal facor odels P A G E

13 Macro-econoc Facor Models Conan facors such as ndex reurns GDP nflaon neres raes ol prces Facors can relae well o nvesen process Facor reurns are observed; facor exposures are esaed va a e-seres regresson Praccal use led especally n fxed ncoe equres hsorcal daa on all secures o esae exposures Assues facor exposures are consan Facor reurns are observed nfrequenly wh grea lags and easureen error P A G E 2

14 Sascal Facor Models Facors are denfed usng a sascal opzaon echnque such as prncpal coponens analyss Facor exposures and reurns are esaed sulaneously Maxzes he n-saple explanaory power of he facor odel Praccal use led eurns are no explaned n ers of real observable econoc facors Facors change over e equres hsorcal daa on all secures P A G E 3

15 Mcro-econoc Facor Models Includes facors ha are specfc o each secury such as duraon secor ec. Facors herefore ofen relae well o he nvesen process Exposures are observed facor reurns are esaed va crossseconal regresson. Praccally useful Facor exposures are allowed o change over e No hsorcal daa needed on secures n porfolo and benchark P A G E 4

16 P A G E 5 Selecon of Facors K Currency Facors (K2 K) Ineres ae Facors (K K2) Cred Spread Facors esdual Model K K f K K f K f 2 2 ε β β β

17 P A G E 6 Currency Facors One facor for each currency Each secury has an exposure of o s currency of denonaon and an exposure of 0 o all ohers Currency facor reurn s observed fro arke daa I.e. no esaon c eurn of he currency n whch secury s denonaed durng perod K K f K K f c 2 2 ε β β

18 P A G E 7 Ineres ae and Cred Spread Facors () Dscouned cash flow odel P Prce of secury a e n local currency CF Cash flow of secury I a e r Treasury spo rae a e s Spread over he Treasury spo rae a e s r J e CF P

19 Ineres ae and Cred Spread Facors (2) If cash flows are known hen he change n he prce of he bond s a funcon of Passage of e Change n he yeld curve Change n he spread p p p r d dr ds p p p p p s ε P A G E 8

20 Passage of Te Includes yeld and roll down Can be calculaed analycally for any fxed ncoe secury Yeld deerned by curren yeld and passage of e oll-down deerned by revalung he secury agans he sae yeld curve bu reducng he e o aury by he passage of e P A G E 9

21 P A G E 20 Changes n he Yeld Curve () Nelson and Segel (987) s 3 facor odel [] Δ e x e x x r

22 Changes n he Yeld Curve (2) We can herefore rewre: p r p x p x2 p dr dx dx2 p p p p x 3 dx 3 eason for dong hs s o reduce he nuber of facors and hus o oban ore relable esaes P A G E 2

23 P A G E 22 Changes n he Yeld Curve (2) Sensvy wh respec o parallel shfs Ths s he usual expresson for Macauley duraon Sensvy wh respec o slope changes Sensvy wh respec o curvaure changes [ ] s r p e CF p x p D [ ] s r p e CF e p x p D [ ] s r p e CF e p x p D 7 2 2

24 Changes n he Yeld Curve (3) We decopose he effec of yeld curve changes no hree facors Parallel shfs Slope changes Curvaure changes The exposures o each of hese shfs s calculaed by applyng a 00 bp shf for each facor and revaluang he secury usng s prcng forula E.g. bnoal lace for ebedded opons Tha s we use effecve duraon P A G E 23

25 Spread Facors () Three ypes of spread facors Secor spreads Qualy spreads: ang buckes Oher spreads Secor Spreads Merrll Lynch secor classfcaon Each bond s exposed o only one secor P A G E 24

26 Spread Facors (2) Qualy Spreads Moody s rang buckes Granulary depends on daa avalably for he arke Three buckes n os arkes: >Aa <Aa and >Baa <Baa If Moody s rang ssng ap he S&P rang Each bond has an exposure o one qualy facor Oher Spreads Euro counry spreads Inflaon-lnked spreads MBS spreads P A G E 25

27 Spread Facors (3) aher han a 3-facor odel we use a -facor odel o approxae changes n he spread: parallel shf: ds ds The cred spread reurn can herefore be rewren as: p s ds p SD ds SD The spread duraon of secury a e We calculae hs by shockng he spread by 00 bp and revaluang he secury usng s valuaon forula Tha s we use effecve spread duraon P A G E 26

28 Oher Facors Prepayen Capures uncerany of cash flows for asse-backed secures Volaly Capures sensvy o changes n pled volaly for secures wh ebedded opons Convexy Capures he 2 nd order effec of yeld and spread changes P A G E 27

29 The Coplee Model c D dr SD ds k 3 K c D dr SD dsk ε k eurn of secury durng perod n base currency Currency reurn for secury durng perod Yeld and roll-down reurn for secury durng perod Exposure of secury o yeld curve changes of ype a he begnnng of perod Change n he yeld durng perod for a yeld curve change of ype Exposure of secury o a spread change a he begnnng of perod Change n he spread for spread facor k durng perod ε esdual reurn for secury durng perod P A G E 28

30 Esang he Model egresson Mehod Daly cross-seconal regresson on a unverse of bonds snce 3/2/98 Currency yeld and roll down are reoved. Frs-sage regresson on a unverse of soveregn bonds deernes yeld curve and Euro counry spread facors. Second-sage regresson on a unverse of cred bonds deernes secor qualy and oher facors fro he resdual of he frs-sage regresson Unverse Depends on arke Increasngly shfng o Boxx o add curves (Asa) and o reove prcng nose P A G E 29

31 egresson Sascs: Average -Squared fro Frs-Sage Average P A G E 30

32 egresson Sascs: Facor Sgnfcance fro Frs-Sage P A G E 3

33 egresson Sascs: Average -Squared fro Second-Sage Average Varable over e bu ncreasng o abou 0.25 ore recenly P A G E 32

34 egresson Sascs: Facor Sgnfcance fro Second-Sage P A G E 33

35 P A G E 34 Facor-Based Perforance Arbuon () eurn of Porfolo eurn of Benchark N N K k N k N N c N p w ds SD w dr D w w w w 3 ε N N K k N k N N c N b b ds SD b dr D b b b b 3 ε

36 P A G E 35 Facor-Based Perforance Arbuon (2) elave eurn ( ) ( ) ( ) ( ) ( ) Selecon Secury/Issuer Secor and Qualy Allocaon Duraon and Ter Srucure Down Yeld and oll Currency Allocaon 3 N K k N k N N N c b p b w ds SD b w dr D b w b w b w ε

37 The Power of Facor Models Facor-based Arbuon Ensures arbuon o he decsons ade n fxed ncoe nvesen processes sk Measureen and Arbuon Sae facor odel can also be used o easure and arbue rsk sk-adused Perforance Arbuon Usng he facor odel o arbue rsk and reurn P A G E 36

38 Challenges and Fuure Developen Challenges: Daa daa daa Prcng source aers! Fuure Developen Add ore curves Use of swap curve daa Invesgae counry of ssuer facor (EMD) More granulary of qualy facors Decopose convexy no hree facors lke duraon Enhance nflaon-lnked odellng Enhance MBS odellng P A G E 37

39 Concluson Invesen process deernes approprae arbuon ehodology Fxed ncoe porfolos ofen requre facor-based arbuon Fxed ncoe aheacs allows for powerful facor-based arbuon odels o capure Effecs of he passage of e Effecs of changes n he yeld curve Effecs of changes n spreads Wlshre suppors boh group-based and facor-based arbuon P A G E 38

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