5 Fxed ncome rsk attrbuton Chthra Krshnamurth RskMetrcs Group chthra.krshnamurth@rskmetrcs.com We compare the rsk of the actve portfolo wth that of the benchmark and segment the dfference between the two to correspond to decsons made by the manager of the actve portfolo. There are three man decsons we consder: nterest rate, whch encompasses duraton, allocaton and selecton decsons; currency; and credt. 1 Introducton A fxed ncome portfolo manager may generate hgh returns compared to the benchmark, at least n the short term. But returns tell only half the story. For nstance, t mght turn out that the returns are not hgh enough to justfy the amount of rsk taken. So measurng and reportng portfolo rsk s crucal for judgng portfolo performance. Furthermore, understandng rsk s paramount to controllng t. So t s mportant to be able to measure the rsk contrbuton of each nvestment decson of the manager to the total relatve rsk of the portfolo wth respect to the benchmark. We compare the rsk of the actve portfolo wth respect to the benchmark and carve up the dfference between the two nto components that correspond to decsons made by the manager of the actve portfolo. Whle there s a potental for greater returns when devatng from benchmark postons by takng on more rsk, there s also a concurrent potental for underperformance relatve to the benchmark. Therefore, t s essental to be able to measure the rsk exposures of the portfolo vs-à-vs the benchmark. We provde a methodology for rsk attrbuton that s rgorous from a mathematcal standpont and relevant from an nvestment standpont. Although there s now an extensve body of research, both n academa and ndustry, pertanng to methodologes n fxed ncome attrbuton, there s no ndustry standard that may be used as a gude. Much of the work has been focused on return attrbuton and pad lttle attenton to rsk attrbuton for fxed ncome portfolos. Most of the fxed ncome return attrbuton methodologes
6 Fxed ncome rsk attrbuton Table 1 Components of return attrbuton Interest Rate Currency Credt Overall Duraton Sector Allocaton Securty Selecton that are used n the ndustry may be classfed nto two groups: duraton approach and prncpal component analyss. We have drawn on some of the general prncples of the duraton approach to come up wth our own framework for rsk attrbuton. In partcular, we have mounted our rsk attrbuton methodology on a return attrbuton system that s smlar to that n van Breukelen (2000). Ths return attrbuton system s also close n sprt to the one descrbed by Feser et al (2002). For the rsk attrbuton model, we have followed the methodology of Mna (2003). The broad scope of any performance and rsk attrbuton system s to compare the nvestment outcomes of the actve portfolo and the benchmark and to ascrbe the dfference between the two to the decsons made by the manager of the actve portfolo. Ths general prncple apples equally to equty and to fxed ncome attrbutons. However, wthn ths broad outlne, the nvestment processes and the sources of rsk for equty and fxed ncome portfolos dffer sgnfcantly. Equty returns are determned by market segments whle fxed ncome returns are ruled prmarly by duraton, yeld curve and credt qualty. Ths means that a fxed ncome attrbuton system cannot be obtaned by merely tweakng an equty attrbuton system. Nonetheless, we draw on the smlartes between the two systems to facltate our understandng. 2 Return attrbuton We start by establshng a return attrbuton system for fxed ncome portfolos and then use t to come up wth a rsk attrbuton system. The dfference n return between the benchmark and the portfolo are due to dfferent nvestment decsons made by the portfolo manager. Return attrbuton s a way of measurng the contrbuton of each of these decsons to the dfference n return between the benchmark and the portfolo. Table 1 lsts the return attrbuton components that we analyze. In the smplest case, a fxed ncome portfolo conssts of rskless government bonds whch may be denomnated n dfferent currences. For example, Central Banks usually nvest ther
Return attrbuton 7 nternatonal reserves n soveregn bonds denomnated n dfferent currences. 1 For such a portfolo whch has no credt rsk, there are two components of return attrbuton that we wll consder: nterest rate and currency. The nterest rate component may further be subdvded nto three components: overall duraton, market allocaton and securty selecton. One of the frst decsons s the overall duraton of the portfolo. If the manager has an overall postve vew of the bond markets, ths wll be reflected n a choce of a hgher total duraton for the portfolo compared to the total duraton of the benchmark. The next decson concerns the manager s vew about markets or sectors. The markets or sectors here could be countres, currency markets, maturty segments, duraton buckets or credt classes. A postve vew n a partcular market/sector s manfested as hgher allocaton n that market/sector compared to the benchmark; ths dfferental allocaton could be acheved by choosng ether a longer duraton than that of the benchmark or a larger weght compared to the benchmark or a combnaton of both. The fnal level of drll-down s the selecton of a partcular securty n each ndvdual market. Currency attrbuton s dealt wth separately by keepng nterest rates constant and shockng the currency rates to perform return and rsk attrbutons. If we consder a portfolo whch has corporate bonds n t as well, then we need to ncorporate return due to credt bets. And as wth the currency allocaton, the decson to nvest n corporate bonds, whch may offer a hgher rate of return, has to be consdered separately. In ths case, we shock the credt spreads and perform return and rsk attrbuton for ratng and/or sector allocaton and selecton. An mportant underpnnng of the fxed ncome nvestment process s the concept of duraton. Duraton s also the feature of a fxed ncome portfolo that dfferentates t from an equty portfolo. Duraton measures the senstvty of the bond s value to parallel changes n the yeld curve. We llustrate the role played by duraton n a fxed ncome portfolo wth the followng example. If the manager of an equty portfolo beleves that a partcular market s gong to perform well, he overweghts that market relatve to the benchmark. A fxed ncome portfolo manager wth the same belef about that market has two alternatves to express hs vew. He could choose to overweght that partcular market relatve to the benchmark, or he could choose the same total 1 de Almeda da Slva Junor (2004) provdes a specfc example.
8 Fxed ncome rsk attrbuton weght n that market as the benchmark but nvest n assets wth a longer duraton than the benchmark. A vald return attrbuton system should consder ths extra degree of freedom. The best way to understand fxed ncome return attrbuton s by drawng a parallel wth the more famlar structure of equty return attrbuton. The example above shows that allocaton decsons n a fxed ncome portfolo are made along two dmensons: weghts and duraton. The best way to understand fxed ncome return attrbuton s by drawng a parallel wth the more famlar structure of equty return attrbuton. The fxed ncome allocaton decson may be mplemented by changng ether the weght or the duraton or a combnaton of both, that s, by changng the overall value of weght tmes duraton whch s the term WD where W and D refer to the weght and duraton of a sector respectvely. So n a fxed ncome portfolo, WD s analogous to weght n an equty portfolo. Snce we would lke to make use of the product of weght and duraton as the equvalent of weght n an equty portfolo, we express the return the followng way: WD r, where r s D the securty return. Ths s a weghted average of the return where the weghts are the product of the true weghts and duratons. The term r/d denoted by R s analogous to return n an equty portfolo. Before we deal wth each of the components of return attrbuton n greater detal, we set up a general framework and establsh the notaton that wll be followed n the remander of ths document: T ={1, 2,..., N} denotes the unverse of securtes. These are the securtes present n the portfolo and the benchmark. W B s the benchmark poston n securty. If a securty has a weght of zero, t s not a part of the benchmark. s the portfolo poston n securty. If a securty has a weght of zero, t s not a part of the portfolo. W P W B A = A W B s the benchmark poston n sector A. W B T = T W B = 1. W P A = A W P s the portfolo poston n sector A. W P T = T W P = 1. r s the return of an ndvdual ssue n the portfolo or benchmark n local currency.
Return attrbuton 9 D s the duraton of securty. R = r D s the local return normalzed by duraton. D B A = A W B D W B A s the average duraton of benchmark ssues n sector A. D B T = T W B D W B T = T W B D s the total benchmark duraton. D P A = A W P D W P A s the average duraton of portfolo ssues n sector A. D P T = T W P D W P T = T W P D s the total portfolo duraton. R B A = A W B r W B A DB A R P A = A W P r W P A DP A R B T = T W B r D B T s the normalzed benchmark return for sector A. s the normalzed portfolo return for sector A. s the normalzed total benchmark return. R P T = T W P r D P T s the normalzed portfolo return for sector A. E c s the currency return for currency c. E s the currency return for securty. The return n the base currency of any gven fxed ncome portfolo consstng of the securtes 1, 2,...,N may be decomposed as Return of the portfolo = = W (r + E ) (1) W r + W E. (2) Equaton (1) states that the return of any sngle ssue n the base currency of the portfolo s the sum of the asset return n local currency and the currency return. Note that ths s a lnear approxmaton.
10 Fxed ncome rsk attrbuton Table 2 Return attrbuton Bench Reference port 1 Reference port 2 Actve port Overall duraton B P P P Sector allocaton B B P P Securty selecton B B B P Equaton (2) may be used to decompose the dfference n return between the actvely managed portfolo and the benchmark n the followng way: Excess return = + (W P (W P W B )D R W B )E. (3) The frst term on the rght hand sde of equaton (3) explans the contrbuton of actve fxed ncome management, ncludng choces of overall duraton, market allocaton and ssue selecton. The last term n the equaton explans the contrbuton of currency allocaton to the dfference n returns. We wll deal wth each of these two aspects ndvdually. In order to undertake a detaled analyss of the fxed ncome attrbuton, we now construct two portfolos: reference portfolo 1 and reference portfolo 2. Here we have used the termnology of van Breukelen (2000). Reference portfolo 1 dffers from the actual benchmark only n ts overall duraton, whch s that of the actve portfolo. Reference portfolo 2 s constructed such that t has the same overall duraton and the same sector allocaton as the actual portfolo. It dffers from the actual portfolo only n ssue selecton, that s R, at each level. It dffers from reference portfolo 1 only n allocaton across sectors or markets. Table 2 shows the dfferences and smlartes of the four portfolos. Specfcally, f we consder a sector A, benchmark and portfolo allocaton for A are gven by: WA BDB A and W A P DP A respectvely. Smlarly, by securty selecton for sector A, we mean: RA B and RP A. So we have the followng: (a) the dfference n return between reference portfolo 1 and the benchmark explans the overall duraton contrbuton, (b) the dfference n return between reference portfolo 2 and reference portfolo 1 gves the allocaton contrbuton, and (c) the dfference n
Return contrbuton 11 return between reference portfolo 2 and the actve portfolo lays out the selecton contrbuton. We now take a closer look at each of these three factors n turn. 3 Return contrbuton 3.1 Interest rate Overall duraton We can construct the reference portfolo 1 by assgnng the followng weghts to the securtes n T : W (1) = W B DT P DT B where W (1) are the weghts of the ssues n reference portfolo 1. Ths wll result n a portfolo whch assgns non-zero weghts only to those securtes whch have non-zero weghts n the benchmark (ths s equvalent to choosng the same subset of securtes from T as the benchmark does). Moreover, the non-zero weghts assgned to the securtes wll be n proporton to the benchmark weghts. The contrbuton of the overall duraton to the dfferental performances of the portfolo and the benchmark s gven by the dfference n returns between the benchmark and reference portfolo 1: Ths reduces to (W (1) D W B D )R = (W B DT P DT B D W B D )R. ( D P Overall duraton = T D B T ) N 1 W B D R. (4) Sector allocaton The sector allocaton contrbuton s determned by the dfference n return between reference portfolo 1 and reference portfolo 2. Reference portfolo 2 has to: (a) have securty weghts proportonal to those of the benchmark, and (b) the same overall duraton as the actve portfolo. These are crtera that are common wth reference portfolo 1. Addtonally, reference portfolo 2
12 Fxed ncome rsk attrbuton must have (c) the same sector allocaton as that of the actual portfolo. By ths we mean that for a gven sector A W (2) A DB A = W P A DP A. (5) Thus reference portfolo 2 s constructed by nvestng n the same securtes as the benchmark, but choosng the sector weghts to satsfy W (2) A = W P A DP A D B A At the ndvdual securty level, we may choose the weghts the same way as at the sector level. For each sector A W (2) = W B WA P DP A WA B, A, DB A so wthn a sector, securty weghts are proportonal to benchmark weghts. The sector allocaton contrbuton s the dfference n return between reference portfolo 2 and reference portfolo 1. Mathematcally, ths may be expressed as Sector allocaton = A W (2) A DB A RB A A W (1) A DB A RB A = A (WA P DP A W (1) A DB A )(RB A RB T ), (6) where R B A = A W B r W B A DB A RA B s the normalzed return for sector A. The term RB T s defned smlar to RB A taken over T nstead of over A. We also defne RA P and RP T analogously. wth the sum Securty selecton The actve portfolo and reference portfolo 2 have the same total duraton and the same allocaton across sectors and dffer only n the securty bets. Thus the securty selecton contrbuton s
Rsk attrbuton 13 determned by the dfference between reference portfolo 2 and the actve portfolo: Securty selecton = portfolo return reference portfolo 2 return (7) = A W P A DP A RP A A W P A DP A RB A (8) = A WA P DP A (RP A RB A ). (9) 3.2 Currency contrbuton Currency allocaton decsons, whch are dealt wth separately, can be wrtten as Currency performance = c W P c E c c W B c E c (10) = c (W P c W B c )(E c E B T ), (11) where W c s the weght of the portfolo or the benchmark aggregated over a partcular currency c and E B T = c W c BE c s the benchmark weghted average of the currency component. 4 Rsk attrbuton 4.1 Trackng error Wth a return attrbuton system defned, we proceed to defne rsk attrbuton n the same way as n Mna (2003). As n Mna (2003), we begn wth the relatonshp between the portfolo excess return and the forward lookng portfolo trackng error. The excess return s the sum over all securtes of the portfolo bet on each securty tmes the securty s realzed return. The forward lookng trackng error s defned smlarly, but wth the securty returns treated as yet-to-be-realzed random varables. Usng a model for future volatltes and correlatons of the securty returns, we may calculate the standard devaton of the future portfolo excess return. Ths s defned as the portfolo trackng error. Note that at the portfolo level, we are faced wth a decson of whether to use the actual securty returns n the base currency or the lnearzed returns as n Equaton (2). Usng the actual returns has the obvous attracton of makng our trackng error correspond to the true excess return, whle usng the lnearzed returns allow us to further decompose the rsk accordng to the return attrbuton system. Snce our goal s to attrbute the rsk along the same dmensons as the return, we proceed wth the lnearzed securty returns.
14 Fxed ncome rsk attrbuton For each level of decomposton, the return attrbuton specfes the amount of portfolo excess return that was due to a partcular part of the nvestment process. Analogously, our rsk attrbuton descrbes the amount of rsk, on a forward-lookng bass, nherent n the exstng nvestment choces. Thus, just as at the portfolo level, we defne rsk attrbuton at any level of drlldown by usng the expresson for return attrbuton, but treatng the returns as random varables. Agan, usng a model for return volatltes and correlatons, we defne the rsk attrbuton as the standard devaton of the return expresson. Note that the specfc model for volatltes and correlatons s not relevant here; we are concerned only wth defnng the quanttes for whch we wll compute standard devaton. We now treat the varous stages of the rsk attrbuton n detal. As explaned earler, fxed ncome return attrbuton conssts of two components: nterest rate and currency taken together and credt or spread. The nterest rate and currency components of return may be subdvded nto returns due to nterest rate, namely, overall duraton, sector or maturty allocaton, securty selecton and currency allocaton. For the nterest rate and currency component, we may defne the overall trackng error of the nterest rate and currency component, denoted by, as the standard devaton of excess returns due to the overall nterest rate and currency components: = std [ N W P D R n W B D R + W P E c W B E c ]. (12) At the next level of drll-down, we may splt the nterest rate rsk and currency rsk to deal wth them ndvdually. Currency rsk attrbuton may be dealt wth separately just as n the case of currency return attrbuton. From equaton (11), we get the total currency rsk [ ] std (Wc P Wc B )(E c E B T ). (13) c From ths we may nfer the next level of drlldown, stand-alone currency rsk for currency c: std[(w P c W B c )(E c E B T )]. (14) The nterest rate trackng error may correspondngly be defned as [ N ] n std W P D R W B D R. (15) The next drlldown level of the fxed ncome management component conssts n dervng the ndvdual trackng errors of the overall duraton, sector allocaton and securty selecton.
From the overall duraton return n equaton (4), we defne total duraton attrbuton rsk to be std [ (W B DT P A D B A T T From equaton(6), the sector allocaton rsk for sector A s Rsk attrbuton 15 ] DA B W A B DB A )RB A. (16) std[(wa P DP A W (1) A DB A )(RB A RB T )]. (17) Fnally, from equaton (9), the securty selecton rsk for securty n sector A s gven by std[r (W P W B WA P DP A WA B )]. (18) DB A 4.2 Incremental VaR We now consder the calculaton of the relatve ncremental VaR. In the case of an equty portfolo, relatve IVaR was ntroduced n order to overcome the obstacle presented by the trackng error (or relatve VaR) not beng addtve. 2 We use the same technque here for that very same reason, and such that t s n keepng wth the sprt of the rsk attrbuton methods for an equty portfolo. At the topmost level of the rsk attrbuton report we have the Overall Fxed Income Trackng Error whch, when t s based on the true returns, wll not be decomposed nto relatve IVaRs. At the next report level, the overall nterest rate and currency trackng errors taken together,.e.,, and overall credt trackng error may be decomposed nto relatve IVaRs. Suppose the excess return due to nterest rate and currency s denoted by ICR, then we have ICR = r IR + r CC, (19) where r IR and r CC refer to the nterest rate and the currency management rsk respectvely: r IR = r CC = W P D P RP W P E c n W B D B RB (20) W B E c. (21) 2 Mna (2003)
16 Fxed ncome rsk attrbuton Table 3 Weghts and duratons Sectors Benchmark Wts Portfolo Wts Actve Wts Benchmark Dur Portfolo Dur 1-3 0.299 0.262 0.038 0.596 0.419 3-5 0.234 0.452 0.218 0.922 1.718 5-10 0.297 0.194 0.103 1.994 1.357 10+ 0.171 0.093 0.078 2.251 1.160 Total 1.0 1.0 0.0 5.763 4.654 The overall nterest rate and currency rsk attrbuton may be made addtve n the followng way: = cov(icr,r IR) + cov(icr,r CC). (22) The overall nterest rate rsk may be decomposed n a smlar manner. Suppose the returns due to overall duraton, sector allocaton and securty selecton (equatons (4), (6), (9)) are denoted by r DD, r AA and r SS respectvely, and the overall nterest rate trackng error s denoted by ITE. Then we have ITE = cov(r IR,r DD ) ITE + cov(r IR,r AA ) ITE + cov(r IR,r SS ). (23) ITE 5 An example We present an example of a rsk attrbuton report for a fxed ncome portfolo and explan how to nterpret the statstcs. The benchmark conssts of a total of 233 soveregn bonds n seven dfferent currences wth U.S. dollars as the base currency. The duratons range from one to seventeen years. The actve portfolo s nvested n a subset of 25 of the benchmark bonds. In order to calculate the nterest rate rsk we choose the sectors to be the followng duraton buckets: 1-3 years, 3-5 years, 5-10 years and 10+ years. Table 3 shows weghts, bets and duratons of each sector. The portfolo s sgnfcantly overweght, that s 21.8%, n the 3-5 year duraton sector and the largest dfference n duratons between the portfolo and the benchmark s n the 10+ sector where there s a dfference of 1.09 years.
An example 17 Table 4 Fxed ncome weghts Sectors Benchmark Reference portfolo 1 Reference portfolo 2 Actve Portfolo 1-3 0.596 0.482 0.419 0.419 3-5 0.922 0.745 1.718 1.718 5-10 1.990 1.610 1.357 1.357 10+ 2.251 1.817 1.160 1.160 Total 5.763 4.654 4.654 4.654 Table 5 Interest rate trackng error Sectors Interest Rate Duraton Allocaton Selecton 1-3 13.017 7.754 1.174 5.736 3-5 56.874 13.077 16.596 9.568 5-10 43.509 24.866 1.533 10.277 10+ 58.930 23.316 9.417 11.531 Total 60.145 67.916 23.967 14.45 Table 4 shows the effectve weghts, that s WD, of the two reference portfolos. The row wth the Total WD numbers gves the overall duratons of the four portfolos and as we can see, both reference portfolo 1 and 2 have the same overall duraton as the actve portfolo. Notce that for reference portfolo 1, the allocaton across sectors s proportonal to the benchmark allocaton across sectors. For example, for sectors 1-3 and 3-5, the sector allocaton proportons for the benchmark and the reference portfolo 1 are 0.596/0.922 = 0.646 = 0.482/0.745. In Table 5, we have the total nterest rate trackng error along wth the duraton, allocaton and selecton trackng errors for each of the four duraton buckets. We see that duraton rsk s hghest n the 5-10 and 10+ duraton buckets. As mentoned earler, the dfference n duraton between the portfolo and the benchmark s hghest n the 10+ sector where t s 1.09 years. However, the portfolo s not sgnfcantly underweght n the 10+ sector compared to the underweght n the 5-10 sector where the duraton dfference s 0.64 years. Ths has the effect of magnfyng the
18 Fxed ncome rsk attrbuton Table 6 Interest rate IVaR Interest Rate Duraton Allocaton Selecton 60.145 58.772 8.239 9.285 duraton rsk n the 5-10 sector. As ponted out earler, the actve weght s hghest n the 3-5 sector whch explans the fact that the 3-5 s the sector wth the maxmum allocaton rsk. Wth rsk attrbuton, we not only get stand-alone numbers but we can also analyze correlaton effects across sectors and securtes. Whle stand-alone numbers are useful, the largest stand-alone trackng errors do not always contrbute the maxmum amount to the total trackng error. In Table 6, we have the nterest rate IVaR at the total portfolo level. The duraton, allocaton and selecton rsk contrbutons add up to the total nterest rate trackng error of 60.15 bp. We see that duraton rsk s larger than the allocaton and selecton rsks n almost all the sectors. Hence t s not surprsng that duraton rsk contrbuton to the total nterest rate trackng error s much hgher than the rsk contrbutons of ether allocaton or selecton. The ncremental rsk of allocaton and selecton are smaller than the trackng error for allocaton and selecton. Ths s explaned by correlaton effects whch come nto play across sectors. In partcular, the ncremental allocaton rsk beng negatve mples that one way to decrease the nterest rate trackng error would be to ncrease the allocaton bets n those sectors whch have negatve rsk contrbutons. 6 Concluson We developed a rsk attrbuton model for fxed ncome portfolos that measured the mpact of nvestment decsons on the dfference n rsk between an actvely managed portfolo and a gven benchmark. We dd ths by establshng a return attrbuton system and defnng a rsk statstc. We splt up the nvestment process nto a seres of decsons: nterest rate, encompassng duraton, allocaton and selecton decsons, and currency. We then measured the contrbuton of each of these components to the dfference n rsk between the benchmark and the portfolo.
Concluson 19 References de Almeda da Slva Junor, F. A. (2004). Performance attrbuton for fxed ncome portfolos n Central Bank of Brazl nternatonal reserves management, Rsk Management for Central Bank Foregn Reserves, European Central Bank. Feser, M., Rmaud, C., Wlson, M., and Fsher, S. (2002). Investment process and the sources of actve excess return, Investment Insght: Internatonal Fxed Income, J.P. Morgan Investment Management, Quarter 1. Mna, J. (2003). Rsk Attrbuton for Asset Managers, RskMetrcs Journal, 3(2): 33 57. Ramaswamy, S. (2001). Fxed Income Portfolo Management: Rsk Modellng, Portfolo Constructon and Performance Attrbuton, BIS Bankng Papers, Issue 6, Aprl 2001. van Breukelen, G. (2000). Fxed-Income Attrbuton, Journal of Performance Measurement, 4(4): 61 68.
20 Fxed ncome rsk attrbuton