The art of trackng corporate bond ndces Laurent Gouzlh, Marelle de Jong, Therry Lebeaupan and Hongwen Wu 1 Abstract The corporate bond ndces, bult by market ndex provders to serve as nvestment benchmarks, contan a great many securtes, and are for that reason dffcult to replcate. The art s to construct an nvestble portfolo that captures the general prce trend among the several thousands of securtes n the ndex, beng lmted to selectng few of them. Ths paper descrbes a practcal approach to ths, whch combnes a well-establshed portfolo constructon technque known as stratfed samplng wth a modern bond rsk measure named the Duraton Tmes Spread. The key dea s to dvde the ndex members nto samples related to dstnct sources of rsk that play n the corporate bond markets, and buld small subsamples that capture those rsks. As the Duraton Tmes Spread conveys lnear- as well as non-lnear bond prce behavour, t proves an effectve measure n the portfolo buldng process. Key words: stratfed samplng, ndex trackng, Duraton Tmes Spread 1. The stratfed samplng technque Stratfed samplng s a recognsed technque for constructng nvestment portfolos snce the early 1980s. By dvdng the unverse of assets nto strategc samples, or strata, and buldng sub-portfolos for each of them, the overall portfolo rsk can be controlled n a manageable way. The term for ths technque stems from the feld of statstcs, n partcular from the handlng of large surveys, see Neyman [1934], where n the same manner the task s made manageable by workng wth representatve sub-populatons. Stratfed samplng was ntroduced nto the nvestment professon by Rudd [1980] and Andrews et al. [1986] as a means to replcate and by that track market ndces n a tme when passve portfolo management frst became popular. 1 Respectvely fxed-ncome research analyst, head of fxed-ncome quant research, fund manager and consultant wth Amund. Marelle de Jong s correspondng author, marelle.dejong@amund.com. The vews expressed n ths paper are those of the authors and do not necessarly reflect those of Amund. 1
Stratfed samplng s compettve aganst the more habtual mean-varance optmsaton ntroduced by Markowtz [1952], when the nvestment unverse s large. As mean-varance optmsaton requres an estmaton of the prce covarance between all the assets, the number of parameters to estmate ncreases wth the sze of the unverse, makng the optmsaton problem unstable as a result. Stratfed samplng on the contrary gans from a large unverse. The more assets are avalable, the better are the condtons to buld representatve samples. For that reason the technque should be partcularly adept to the task of trackng corporate bond ndces, whch contan thousands of securtes. Fabozz [2008] and Martelln et al. [2005] descrbe how the technque s beng used n practce by nvestment managers. It proves effectve to use the Duraton Tmes Spread (DTS) measure developed by Lehman Brothers n 2007, n ths context. Here s where ths paper contrbutes to the lterature and to the standng practce. The DTS measure s bult on the nsght that bond spread varatons are not parallel but rather lnearly proportonal to the level of spread; see Ben Dor et al. [2007] for a complete dscusson. In ther artcle they show that ntegratng the spread level n the bond analyses gves a better sense of prce behavour than the standard measures do, based on duraton alone. We show n emprcal tests the DTS measure to be effectve for ndex-trackng purposes. We buld replcatve portfolos onto two leadng corporate bond ndces, whch are presented n Secton 2. These ndces are extensve and dsperse, and are for that matter suted for the test purposes. We descrbe how the portfolos are bult n Secton 3 and assess ther effectveness n terms of ndex-trackng capacty n Secton 4. Secton 5 concludes. 2. Data We test on two Merrll Lynch global corporate bond ndces, namely the Global Large Captalsaton Investment Grade ndex and the Global Hgh Yeld ndex, both hedged to US dollars. Our database contans the returns and the prncpal characterstcs of the bonds n the ndces on a monthly bass from June 2007 through to May 2014. The ndces are extensve the nvestment grade ndex (IG) conssts of 6718 bonds ssued by a total of 1201 frms as of May 2014 and the hgh yeld ndex (HY) of 3552 bonds ssued by 1687 frms and they are dsperse. Exhbt 1 shows the regons that are covered, the ndustral sectors and the credt ratngs, usng the Merrll Lynch classfcaton flags, whch are broken down n terms of market weght and n numbers of ssung frms. The bonds n the ndces are denomnated n fve dfferent currences n all and are domcled n more than eghty countres. The countres are n majorty developed economes, but there are advanced emergng economes as well ncludng the BRICS, so-called Fronter Markets (less advanced economes) and tax havens. The number of bonds ssued by the same frm vares. There are fve ssues per frm on average n the IG ndex, the record beng held by General Electrcs wth 99 bonds outstandng n May 2014, whle n the HY ndex two bonds are ssued on average per frm. The mplcatons of these market features for the portfolo constructon scheme are dscussed n next secton. 2
Exhbt 1 Index breakdown by regon, ndustral sector and credt ratng Industral sectors Auto Industry Basc Industry Captal Goods Consumer Cyclcals Consumer Non-Cyclcals Energy Healthcare Meda Servces Tech & Electroncs Telecom Utlty Bankng Insurance Real Estate Fnancal Servces Investment Grade ndex Hgh Yeld ndex weght ssuers weght ssuers 3.0% 4.9% 3.1% 3.0% 4.9% 10.4% 5.0% 3.1% 3.6% 3.0% 6.8% 8.1% 31.3% 4.5% 1.7% 3.5% 23 84 52 52 60 136 59 27 98 47 43 118 207 83 59 53 3.8% 12.2% 5.2% 4.0% 3.3% 11.8% 6.0% 7.6% 10.3% 3.5% 9.8% 3.8% 10.7% 0.9% 2.4% 4.6% 42 242 104 123 94 212 87 86 248 55 58 47 140 28 70 55 Ratng categores AAA BB AA B A CCC BBB CC / C /D 0.8% 14.1% 44.3% 40.8% 16 86 458 760 51.2% 35.8% 12.5% 0.6% 628 830 392 22 Regons Europe North-Amerca Latn-Amerca Asa-Pacfc and Afrca 37.2% 45.8% 3.6% 12.9% 363 490 73 263 29.3% 55.8% 5.8% 9.0% 353 1160 126 233 Data source: Merrll Lynch: the Global Large Captalsaton Investment Grade- and the Global Hgh Yeld Index as of May 2014. Calculatons made by the authors 3. The portfolo constructon procedure Gven the magntude and the complexty of the portfolo optmsaton problem at hand, we use a computer programmng algorthm to solve t. In ths secton we formulate the optmsaton problem n mathematcal terms and we descrbe the algorthm we have developed. The determnaton of the strata, the strategc buldng blocks n the portfolo buldng process, s dscussed separately. 3.1 The optmsaton problem The problem objectve s to buld a portfolo such that ts rsk wth respect to the benchmark s mnmal and certan mplementaton constrants hold. As s usual for ths type of problem we mpose () a postvty constrant, thus not allowng for short-sales, and () a cardnalty constrant, meanng that the number of holdngs n the portfolo should be restrcted. Formally we mnmse the trackng error, denoted as TE, between the benchmark b and the replcatve portfolo p, defned as the standard devaton of the return dfferentals, denoted R t b-p, over tme: 3
R t t b p mn TE = T ( ) 2 1 (1) At ths pont we ntroduce the stratfcaton structure. The return dfferentals are decomposed nto weghted strata returns, weghts denoted by w j b, whch are on ther turn decomposed n ndvdual bond returns pre-multpled by the portfolo weghts wth respect to the benchmark, w b w p. b b p 1 T ( w ( ) ) 2 j w w R 2 TE = (2) t j J The decson varables of the optmsaton problem appear, the portfolo weghts w p. We mpose them to sum up to the strata weghts, by that addng an auxlary set of constrants to the problem defnton. The purpose s to help lmt the portfolo rsk, the dea beng that the covarance terms across the strata are small enough to be gnored whle the covarances wthn the strata count. Whether ths assumpton holds n practce can be assessed ex post n backtests on the return performances. We do ths n Secton 4. The three sets of problem constrants are specfed as follows: () postvty : 0 (3) p w p p () cardnalty w w N () stratfcaton p w 0 j : J w = w The cardnalty constrant can be specfed n varous manners. Rather than mposng a maxmum number of nonzero holdngs N overall as s done above, the holdngs can be lmted per stratum. We have opted for the latter n our algorthm, to whch a mnmum buy-n threshold s added whch tends to reduce the number of holdngs as well. No matter how the cardnalty constrant s formulated, t s ths constrant that makes the problem partcularly dffcult to solve. The problem falls n the category of Quadratc Mxed-Integer Programmng problems (QMIP), whch are known to be NP-hard; see Jobst et al. [2001] for techncal detals. Such problems tend to be solved by means of combnatoral optmzaton heurstcs n practce, as dscuss Satchell and Scowcroft [2003], and ths s what we do n ths paper. Contnung wth the problem formulaton, we ntroduce the lnear approxmaton of the bond returns as suggested by Ben Dor et al. [2007], multplyng a Duraton Tmes Spread component wth a spread varaton component: p S 2 b b p S R d S so that ( ) S TE 1 T w t j j wds J (4) S The terms wds (weght x duraton x spread) are central n the portfolo optmsaton problem, the. We call them the bond betas n analogy wth equty portfolo theory n the sense that they defne the market senstvtes,.e. the exposures of the assets to market rsk or, n the case of bonds, to nterest rate rsk. We thus buld on Ben Dor s nsght that for credt rsk nstruments the senstvty to nterest rate rsk s not only determned by duraton but also by the spread level, as larger-spread bonds tend to have larger prce reactons to nterest rate movements. Our search algorthm reles on ths, t s set to pck the bonds wth the bggest bond betas wthn each stratum. b j 2 4
3.2 The optmsaton algorthm The optmsaton algorthm proceeds n two rounds, the frst takng place on a frm-aggregate level and the second on an ndvdual bond level. In the frst round the bond betas of those ssued by the same frm are aggregated to frm totals and sorted n descendng order wthn each stratum. The hghest k percent of frms are retaned -k beng a control varable- where after the weghts are reset so as to realgn wth those of the benchmark strata. Then a search procedure s appled that algns the stratum aggregates n terms of bond betas as well. The search procedure operates n a parwse fashon. Per par of two frms the weght of one s levered to the other n a way that the overall DTS algnment mproves. Bounds are set on the weghts and as soon as a frm hts the mnmum bound t s elmnated from the portfolo. The procedure handles the par wth the bggest DTS dfferental frst, n the assumpton that t entals the bggest potental for mprovement, proceeds n descendng order untl the algnment objectve s acheved or when all combnatons have been examned. In the second round a maxmum of two bonds are selected per frm. The two wth a duraton closest to that of the frm s overall debt structure are taken and the weghts are reset such that the frm s overall duraton tmes spread s met. If n ths process a weght drops below the mnmum bound, t s elmnated from the portfolo as well. The runnng tme of the algorthm s around one second per portfolo rebalancng for the IG ndex and about half a second for the HY ndex, when run on a Personal Computer wth a CPU at 3.2 GHz, a performance that s largely satsfactory for practcal use. 3.3 The strata As mentoned above, the optmsaton process reles on the fact that the bond return correlatons are low across the strata. In ths subsecton we explan how the strata have been desgned to acheve ths. As dscuss Martelln et al. [2005] t s usual practce to stratfy a bond nvestment unverse on the bass of certan bond characterstcs lke maturtes and coupon rates, or alternatvely on groupng defntons. We do the latter n ths paper, defnng the strata by a combnaton of geographcal- and economc sector groupngs that are gven n Exhbt 1. An mportant advantage of these groupngs s that they are stable over tme, whch makes them replcable by relatvely stable samples of bonds. The turnover n the portfolo can be kept low, whch s desrable n vew of keepng transactons costs down. Dsregardng whether the credt ratng groups would have low correlaton levels between them, the fact that they are not stable over tme -up to 5% of the bonds are re-graded each month- would make a portfolo management procedure based on ths crteron cost neffcent. Our portfolo buldng process reles thus on a geographcal- and an ndustry effect. It s ndeed ntutve that companes operatng n the same regon share common rsk factors and therefore have a smlar bond prce behavour whch s dstnctly dfferent from the other regons. In the same way companes operatng n the same ndustry tend to share certan rsks n common. We make the effect of that apparent by makng parwse comparsons between return correlatons measured over the test perod. In one such exercse we have aggregated the bond returns to a more refned sector level, level 4 n Merrll Lynch s defnton whch dstngushes 71 sub-categores among the 16 sectors. Among the correlatons measured between the sub-categores we observe that the correlaton s sgnfcantly hgher on average wthn the sectors than between. We measure 0.71 correlaton wthn sectors as opposed to 0.66 between them. Lkewse we compare correlatons geographcally over the contnents. The bond returns beng aggregated to country level, we measure the parwse correlaton between countres to be hgher wthn contnents than between them; they are 0.65 on average wthn contnents over 5
the test perod aganst 0.62 between them. Note that these correlaton numbers are lower n absolute terms than the sector correlatons gven above, ndcatng that the ndustry effect tends to be stronger than the geographcal effect. In other words, bond prce behavour appears to depend on the busness actvty of the ssung frm more than on where t s domcled. The mpact of that can be found back n the stratfcaton test results, as s shown n secton 4. We use the Merrll Lynch sector defnton (level 3) n unchanged format n the strata defnton except for the fnancal sectors, the bottom four lsted n Exhbt 1, whch we combne nto one. Ths decson s based on the observaton that there s a specfc rsk factor drvng the bond prces of fnancals over the test perod. Exhbt 2 makes ths factor apparent. A prncpal component analyss, see Jollffe [2002] for a general reference, has been run on the sxteen sector return seres, and the senstvtes to the two frst components -whch explan 87% of the return varance- are dsplayed n the Fgure. Note that the senstvtes of the four fnancal sectors (n the crcle) are dstnct n both dmensons defned by the components, makng the specfc factor apparent The prncpal component analyss has been run on the nvestment grade ndex results yet are smlar when run on the hgh yeld ndex. Exhbt 2 Prncpal components analyss results 1,00 2 nd component 0,60 0,20 1 st component 0,60 0,80 1,00 1,20-0,20-0,60 Data source: Merrll Lynch Global Large Captalsaton Investment Grade ndex. Calculatons made by the authors. As to the geographcal splt, we defne three regons: North-Amerca, Europe and the rest of the world. We do ths for practcal reasons, takng nto consderaton the market captalzaton as well as the maturtes of the corporate bond markets over the world. For the hgh-yeldng bonds especally, the Unted States have by and large the oldest and most establshed market, followed by Europe, whle n the rest of the world markets are ganng ground snce a few years. If these geographcal shfts contnue, the regonal splt wll need to be revsed accordngly n due course. As t stands the bond markets outsde North-Amerca and Europe do not add up to the crtcal mass whch allows a further full splt nto thrteen ndustry sectors. Instead a less refned splt s appled, dstngushng between fnancals versus non-fnancals only and defnng three country profles: developed economes, emergng-, and the so-called new fronter markets. We have used the market classfcaton defned by MSCI whch s gven n the appendx. Wth ths arguably haphazard splt of the rest-of-the-world regon we have managed to obtan reasonable ndex-trackng results over the test perod. Meanwhle ths regon ponts at the lmts of the stratfed samplng technque. It makes evdent that ths technque s suted for samples that are a) relatvely stable over tme and b) have a mnmum of nternal coherence. 6
The rest-of-the-world regon especally for the hgh-yeldng bond markets doesn t satsfy these condtons. 4. Emprcal tests The stratfed samplng technque s tested on two global corporate bond ndces presented n secton 2 over a seven-year perod from June 2007 through to May 2014. At regular tme ntervals the portfolo constructon procedure has been appled onto data that was avalable at the tme, thus wthout ntroducng foresght, and then held up to the next perod. In ths secton we frst present the prncpal back-test results and then make an nvestgaton on how the varous settngs n the portfolo constructon algorthm have contrbuted n achevng these results. 4.1. Prncpal back-test results Exhbt 3 gves the prncpal test results. For each ndex the realsed trackng error s gven of the portfolo measured over the entre test perod, the number of bond holdngs as well as the number of frms as of May 2014, and the average annual portfolo turnover over the perod. In ths back-test the portfolos have been rebalanced once a month. In the Exhbt the results have been splt by regon. The portfolos the algorthm produces seem nvestble and the foresght-free realsed trackng error s low. Especally for large funds t s reasonable n terms of portfolo management- and transactons costs to envsage holdng 165 or 184 postons out of several thousands to acheve a trackng error of 0.9% aganst the nvestment grade ndex, whch tself has a volatlty of 5.3%, and a trackng error of 2.6% aganst the hgh yeld ndex that has a volatlty of 14%. We make note that measures are taken over the partcularly volatle perod n 2008-2009 as well. Over the perod from June 2009 to date the average trackng errors are 0.5% and 1.5% respectvely. Exhbt 3 Prncpal test results corporate bond ndex Investment Grade North-Amerca & Europe Latn-Amerca, Afrca & Asa-Pacfc realsed trackng error 0.9% 1.0% 1.3% portfolo holdngs 165 129 (71 + 58) 36 (7+0+27) frms 120 90 30 turnover Hgh Yeld North-Amerca & Europe Latn-Amerca, Afrca & 2.6% 2.7% 4.2% 184 152 (107 + 45) 32 (8+ 1+ 23) 135 111 24 240% Asa-Pacfc Data source: Merrll Lynch: Global Large Captalsaton Investment Grade ndex and Global Hgh Yeld Index. The control varable k (see secton 3) has been set at 11% and the weght bounds at 0.3% to 3%. Turnover s n excess of the ntrnsc ndex turnover due to consttuent changes (whch amounts to ±84% for both ndces); the roundtrp mode s appled,.e. the ssues enterng and leavng the portfolo are both counted. Calculatons made by the authors. 7
4.2 Further analyss In order to understand more precsely how the results have been acheved we make four nvestgatons. Frst, we look what happens f the DTS measure s replaced by the more usual duraton measure. The same portfolo constructon algorthm s appled except that the matchng crteron s weghted duraton, not multpled by the spread. Second, we test the mpact of playng down the stratfcaton effort, takng out the geographcal dversfcaton and separately the sector dversfcaton. Thrd, we test the contrbuton of the parwse search procedure that tunes the weghts. It makes the effectveness of the local search method we have developed explct. And fourth, we test the mpact of reducng the portfolo rebalancng frequency. For practcal reasons we run these tests on a subset of our database, namely on the North- Amercan and European regon of the Hgh Yeld ndex. We have verfed that the conclusons that are drawn hold for the complementary regon and for the other ndex as well. The results are presented n Exhbt 4 n order and compared wth the optmal settng (settng 0) that was gven n Exhbt 3. Exhbt 4 Analyses results Algorthm settngs 0 The optmal settng 1 No DTS measure 2a No regonal stratfcaton 2b No sector stratfcaton 2c Reduced sector stratfcaton 3 No parwse fne-tunng 4 B-monthly rebalancng realsed trackng error 2.7% 4.7% 3.0% 4.1% 4.1% 3.1% 2.8% portfolo holdngs 152 152 150 153 155 236 152 Replacng the DTS measure by duratons (settng 1) deterorates the trackng performance of the portfolo constructon algorthm mportantly. Ths result confrms the fndngs of Ben Dor et al. [2007] that ntegratng the credt spread n the senstvty calculatons, or more generally n the rsk profle estmates of credt rsk nstruments, s an effectve means to control portfolo rsk. Ths s the man contrbuton of ths paper, to gve a practcal llustraton of the effectveness of the DTS measure n the management of portfolos contanng credt rsk nstruments. In settng 2a the regonal stratfcaton s played down, makng no dstncton between North- Amercan and European bonds. There are thus thrteen strata n ths settng nstead of twentysx. In settng 2b no dstncton s made between the ndustral sectors, thus resultng n two strata, one per regon. Note that n both cases the trackng performance deterorates. It cannot be excluded though that, especally for settng 2b, the result s an artefact due to the sharp reducton of strata. In order to check whether a numercal ssue s at stake or whether a dversfcaton opportunty s beng mssed, we have tested an addtonal settng, settng 2c, where the three sectors are set, dstngushng between fnancals, ndustrals and utltes (level 2 n the Merrll Lynch sector defnton). It can be seen n the Exhbt that t gves no mprovement wth respect to settng 2b, leadng to beleve that the deteroraton n trackng performance s due to the lesser dversfcaton. It s nterestng to note that the mpact of playng down the dversfcaton n terms of sectors s greater than n terms of regons. Ths result s n lne wth the data analyses dscussed n prevous secton where average return correlatons were compared. In settng 3 the parwse fne-tunng has been swtched off, resultng n a portfolo wth 236 holdngs. An extra 84 bonds are held compared to the optmal settng wthout gan n trackng 8
performance. It shows that ths module s effectve n pushng down the number of portfolo holdngs down to nvestble levels. The search procedure had been desgned on the bass of practcal portfolo management experence and the efforts made n formalsng the emprcal knowledge seems successful. Fourthly and lastly, we reduce the portfolo rebalancng frequency, settng t to once every two months (settng 4). Note that the trackng performance holds out whle the turnover drops from 240% to 120% per annum, whch s a bg wn. If we reduce the rebalancng frequency further to once every three- and four months, the turnover drops further to 72% and to 48% respectvely, however, n the meantme the trackng error goes up to 3.5% and 4.0%. In our test setup rebalancng once every two months seems the optmal settng. 5. Concluson Ths paper gves evdence that a passve nvestment strategy amng at trackng a global corporate bond ndex s actually feasble to mplement. It may seem a challenge to replcate the prce trend among the several thousands of bonds the global corporate bond ndces consst of, whle restrcted to selectng few of them. It proves successful to deploy stratfcaton technques whle usng the DTS measure as an estmate for bond rsk and enhance the outcome by carefully mprovng the portfolo buld-up n makng local adjustments. It s the combnaton of these three ngredents that leads to good results. We have not gven much attenton to tamng the rotaton n the portfolo. A turnover of 120% per annum that we attan n our tests, falls out rather hgh. The man reason for ths s that, snce the portfolos are bult over tme wthout gvng consderaton of the postons already held, efforts are beng put nto the portfolo optmsaton, not n controllng turnover. A more comprehensve test would be to nclude tradng costs and rules such that portfolo optmalty s weghed off aganst costs. We have not made such exploratons. The ntenton of our paper s to put forward the key elements of an effectve ndex-trackng technque for corporate bonds. References Andrews, C., Ford, D., Mallnson, K. (1986) The desgn of ndex funds and alternatve methods of replcaton, The Investment Analyst 82, pp. 16 23. Ben Dor, A., L. Dynkn, J. Hyman, P. Houwelng, E. van Leeuwen and O. Pennnga (2007) DTS SM (Duraton Tmes Spread), Journal of Portfolo Management 33 n 2, pp. 77-100. Fabozz, F. (2008) Handbook of fnance, fnancal markets and nstruments. John Wley & Sons. Jobst, N., M. Hornman, C. Lucas and G.Mtra (2001) Computatonal aspects of alternatve portfolo selecton models n the presence of dscrete asset choce constrants, Quanttatve Fnance 1, pp. 1-13. Jollffe, I. (2002) Prncpal Component Analyss, Sprnger Seres n Statstcs. Markowtz, H., (1952) Portfolo Selecton, Journal of Fnance 7, pp. 77-91. 9
Martelln, L., Praulet, P. and S. Praulet (2005) Fxed-ncome securtes valuaton, rsk management and portfolo strateges. Wley Fnance. Neyman, J. (1934) On the two dfferent aspects of the representatve method: the method of stratfed samplng and the method of purposve selecton, Journal of the Royal Statstcal Socety. Rudd A (1980) Optmal selecton of passve portfolos Fnancal Management 9, pp. 57 66. Satchell, S. and A. Scowcroft (2003) Advances n portfolo constructon and mplementaton. Butterworth Henemann. 10