Multi-Alpha Equity Portfolios: An Integrated Risk Budgeting Approach for Constrained Robust Portfolios WHITE PAPER

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1 WHITE PAPER Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos For professonal nvestors - MAY 2013

2 2 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 About the authors Raul Leote de Carvalho s head of Quanttatve Strateges and Research n the Fnancal Engneerng team at BNP Parbas Investment Partners n Pars, France. raul.leotedecarvalho@bnpparbas.com Tel. +33 (0) Xao Lu s a quanttatve analyst n the Fnancal Engneerng team at BNP Parbas Investment Partners n Pars, France. xao.lu@bnpparbas.com Tel. +33 (0) Perre Mouln s head of Fnancal Engneerng at BNP Parbas Investment Partners n Pars, France. perre.mouln@bnpparbas.com, Tel. +33 (0)

3 3 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 FOREWORD AND NON-TECHNICAL SUMMARY By Raul Leote de Carvalho - Head of Quanttatve Strateges & Research It was n October 2010 n Berln at the post-conference dnner of the Insttute for Quanttatve Investment Research that I had the opportunty to dscuss the hot topc of smart beta strateges wth Bob Ltterman, now executve edtor of the Fnancal Analysts Journal, who sat next to me. I remember askng hm what he thought about all these new, so-called smart beta strateges. What n partcular dd he thnk could be the source of ther excess returns and rsk? Hs answer was clear: they must all be explaned by factor exposures! Shortly afterwards, back at the offce n Pars, I came across the preprnt of a paper by Bernd Scherer, professor at the Edhec Insttute n London, whch, n the case of mnmum varance, made exactly ths pont: the excess returns of tradtonal mnmum varance portfolos relatve to the market captalsaton ndex can be explaned by a small number of factor tlts, n partcular the factor tltng the portfolo towards low-rsk stocks. Bernd wrote: "my conjecture s that the portfolo constructon process behnd mnmum varance nvestng mplctly pcks up rsk-based prcng anomales. If that s true, mnmum varance nvestng wll be a clumsy and ndrect process to beneft from. Investors would be better advsed to drectly decde f, when and to what degree they want to nvest n long/short anomaly portfolos on top of a market-weghted benchmark." He went on to prove t n the paper (publshed n the Journal of Emprcal Fnance n 2011, vol. 18, ssue 4). In the course of 2011, wth my colleagues Xao and Perre, we extended ths lne of research to other smart beta strateges. Besdes mnmum varance, we also consdered maxmum dversfcaton, two forms of rsk party (whether correlatons are taken nto account or not) and equally weghtng. We appled these strateges to the stocks n the MSCI World ndex, the MSCI USA ndex, the MSCI Europe ndex and the MSCI Japan ndex. We were staggered by the results. In a paper publshed n The Journal of Portfolo Management (sprng 2012 ssue, vol. 39, no. 3) enttled Demystfyng Rsk-Based Strateges: a Smple Alpha Plus Beta Descrpton ), we demonstrated that all these smart beta strateges do ndeed have pronounced factor tlts. Performance of mnmum varance and maxmum dversfcaton ndces can be almost fully explaned by tlts towards low-rsk stocks. Lkewse, the performance of equally weghted ndces can be explaned almost entrely by tlts towards smaller captalsaton stocks, whlst performance of rsk party s the result of a blend of exposures to smaller captalsaton and low-rsk stocks. Our research confrmed that not only mnmum varance but also maxmum dversfcaton, rsk party and equally weghtng strateges are ndeed clumsy approaches whch can be almost fully explaned by ther exposures to smple factor tlts. If ths s true for rsk-based smart beta strateges, then the same apples to fundamentally weghted smart beta ndces where stocks are weghted n proporton to fundamental metrcs (as opposed to ther market captalsaton) such as book-value, earnngs or dvdend yelds. In a paper whch appeared n the Fnancal Analysts Journal n 2011, (vol. 67, no. 5), by Jason Hsu, Tzee-man Chow and Vtal Kalesnk from Research Afflates (who created fundamental ndexng) and co-authored wth Bryce Lttle from Cornell Unversty, they acknowledge that value factors almost entrely explan the performances and rsk of ther fundamentally weghted strateges. Smart beta strateges are therefore nothng more than a new form of systematc actve strateges where stock selecton s based on a formula that determnes how the underlyng portfolos wll devate from the market captalsaton portfolo. They are transparent n the sense that the formula tends to be made publc but they can be qute complex, mnmum varance beng the typcal example of a complex smart beta strategy. Ther complexty, or sometmes apparent smplcty, hdes a number of ptfalls. Mnmum varance and maxmum dversfcaton, whch rely on optmsaton algorthms, suffer from oversenstvty to error estmaton n rsk models and, as a consequence, can result n large portfolo turnover, concentraton n just a few stocks, sector bases and other shortcomngs. Rsk party ndces and equally weghted strateges are vulnerable to problems of mplementaton as they are overexposed to smaller captalsaton stocks. Ths rases the queston of the sustanablty of performance n such strateges should ther assets under management grow and market (negatve prce) mpact of ther mplementaton become even more sgnfcant. Even fundamentally ndexng can be too exposed to dstressed stocks whch tend to underperform. In fact, when lookng n more detal at the formulas used n most smart beta ndces, we apprecate that rsk s n general poorly controlled. The trackng error rsk relatve to the market captalsaton ndex s often at the market s mercy and can swng wldly over tme, wth, n some cases, the change beng extremely large. Smlarly, the factor exposures can fluctuate and are generally poorly controlled. Fnally, not all known factor-based market anomales are ncluded n the current smart beta ndces. We often come across the value, low rsk and smaller captalsaton factors, but momentum, for example, s much less ubqutous. Factor-based anomales based on analyst earnngs revsons, proftablty or earnngs qualty are seen even less, f at all.

4 4 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 What then s the alternatve to smart beta ndexng? The answer s to go back to behavoural fnance, to the lterature on factor ms-prcngs, to dentfy the sources of alpha whch can be generated from systematc factor exposures, to mprove those strateges desgned to capture alpha and to fnd more effcent ways of combnng them nto portfolos. Academc lterature offers a good startng pont when t comes to factor ms-prcng. Research on ms-prcng probably began n 1972 when Bob Haugen and A. James Sten showed n ther workng paper (avalable from the lbrary of the Unversty of Wsconsn- Madson) that rsk tself had been ms-prced by the US market between 1926 and 1969, wth low-rsk stocks typcally havng hgher returns than expected from ther low beta and hghrsk stocks havng lower returns than expected from ther hgh beta. Ths fndng has been confrmed by many authors snce and apples not only to US stocks, but also to stocks n almost all markets. Bob Haugen hmself demonstrated ths agan n a 2012 preprnt, co-authored by Nardn Baker and enttled Low Rsk Stocks Outperform wthn All Observable Markets of the World. Other ms-prcngs are well known. The value anomaly can be traced back to at least 1977 n a paper by Sanjoy Basu n the Journal of Fnance, the earnngs revson anomaly to 1979 n a paper by Josef Lakonshok and Dan Gvoly n the Journal of Accountng and Economcs, the small captalsaton stock anomaly to 1981 by Rolf Banz n the Journal of Fnancal Economcs, the short-term reversal anomaly to 1990 by Narasmhan Jagadeesh n the Journal of Fnance, the momentum anomaly by Narasmhan Jagadeesh and Sherdan Ttman to 1993 n the Journal of Fnance and the accruals anomaly n 1996 by Rchard Sloan n the Accountng Revew. Just to cte a few. In ths paper, we go back to bascs proposng that the constructon of equty portfolos should start wth allocatng rsk to sources of alpha rather than undertakng unnecessarly complex and clumsy portfolo constructs. Ganng exposure to the anomales mentoned above s a queston of confdence: to what extent does one beleve that those sources of actve rsk wll contnue to delver alpha n the future? In most cases, we expect they wll contnue to do so, but ths s for each nvestor to decde. Allocatng rsk requres a forecast n terms of the amount of alpha per unt of rsk and cross-correlatons of these alphas. If these are avalable, then mean-varance could be used to buld an optmal rsk allocaton. However, t s unlkely that these data are known wth suffcent accuracy. The good news s that we can show emprcally that smply allocatng the same rsk budget to a selecton of those anomales would have generated a smlar result to that obtaned from n-sample mean-varance optmsaton. A smple sensble allocaton to a mult-alpha combnaton s therefore lkely to suffce. We also dscuss the possblty of addng other sources of alpha than just systematc stock tlts derved from factor exposures. In the paper s frst example, we show how to combne alpha expected from a forward-lookng fundamental stock-pckng nvestment process where company returns are forecast based on company analyss wth alpha generated systematcally from value stocks, small-cap stocks, momentum drven stocks and a momentum alpha capture strategy appled to sectors. We use ths same example to show n detal how the framework can be appled. Our am s to demonstrate that once rsk has been allocated to alpha sources, the steps to buld the portfolo are straghtforward. The portfolo constructon framework proposed can be dvded nto three steps. The frst s the allocaton of actve rsk to sources of alpha, as just dscussed. The second s the constructon of an unconstraned actve allocaton from those alpha sources and rsk budgetng and the thrd s the applcaton of constrants to generate the optmal constraned allocaton whch s the least mpacted by the constrants. Regulatory constrants, the longonly constrant, lqudty constrants and lmts to the maxmum allocaton to a stock are often unavodable. The second step requres only basc arthmetc. The target unconstraned actve allocaton s just the rsk weghted average of the actve allocaton determned by each strategy desgned to capture alpha at a gven pont n tme. For example, n the case of a systematc alpha capture strategy such as value, the actve allocaton can be a smple tlt towards stocks wth the lowest prce-to-book rato and away from ether the market ndex or from stocks wth the hghest prce-to-book rato. The thrd step s less straghtforward, but we show that the fnal result can be easly understood and remans transparent. We propose that mpled returns are calculated from the unconstraned target allocaton and then used n meanvarance optmsaton where constrants are appled. Impled returns are the returns of each stock n the nvestment unverse whch render the unconstraned allocaton effcent,.e. the soluton of the mean-varance optmsaton problem n the absence of constrants. It happens that these are the returns whch accurately reflect our vews and therefore can be used n mean-varance constraned optmsaton wthout expectng the usual problems of, for example, corner solutons. In fact we show that when optmsng from mpled returns, the fnal constraned portfolo remans as close as possble to the targeted unconstraned allocaton. We demonstrate, wth numercal examples, that constraned portfolos retan as much of the rsk budget allocaton n actve rsk as the constrants allow. We also show that the exposure to systematc rsk factors n our target unconstraned portfolo s retaned as much as possble. It s the exposure to stock specfc rsk whch pays for the mpact of constrants and flattens the effcent fronter.

5 5 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 To our knowledge, ths s the frst report that dscusses comprehensvely the problem of portfolo constructon startng from a rsk allocaton to alpha sources and shows how to buld constraned portfolos whch retan as much as possble those chosen actve rsk exposures, wth a partcular focus on analysng the mpact of constrants on the fnal portfolo stock and rsk allocaton. Although nspred by the well-known Black- Ltterman model, the approach proposed here does contan a number of mportant dfferences whch we endeavour to set out n the appendces. We strongly beleve that our fndngs are presented at an mportant tme as nvestors seek to keep sght of the key aspects of smart beta strateges. We hope that our work contrbutes to the buldng of better portfolos and mproves navgaton n the world of smart beta.

6 Abstract We propose a robust optmzaton approach to construct realstc constraned multstrategy portfolos whch starts wth the dentfcaton of dfferent sources of alpha and the rsk-budgetng exercse to optmally combne these sources. We show how systematc alpha-capture strateges can be combned wth judgmental strateges and how bottom-up based strateges for stock pckng can be combned wth top-down sector and country allocaton strateges. The approach s shown to be fully transparent for both unconstraned and constraned portfolos wth a dscusson of how constrants mpact the fnal optmal portfolo allocaton. In partcular we show that the constraned portfolos retan the exposures to systematc rsk factors n the unconstraned target soluton as much as possble, and that specfc rsk takes the toll of portfolo constrants. Through a realstc back-tested example combnng dfferent well-known alpha capture strateges we demonstrate the robustness and transparency of the approach. Fnally we also dscuss the advantages of ths approach over the alternatve process based on selectng and nvestng n a mx of dfferent ndex-funds mplementng offthe-shelf actve strateges for alpha capture. We beleve that our approach s partcularly suted for nsttutonal nvestors nterested n rsk budgetng the alpha n ther portfolos whle fully understandng the fnal allocaton n ther constraned portfolos.

7 7 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 Emprcal evdence that the Captal Asset Prcng Model (CAPM) does not gve an accurate descrpton of markets has been growng for years. For equtes, examples nclude the alpha of smaller captalzaton stocks over large captalzaton stocks (Banz [1981]), of value stocks over growth stocks (Fama and French [1992]), of longer-term wnners over losers (Jegadeesh and Ttman [1993]), of short-term losers over wnners (Lehmann [1990]) or of low-rsk over hgh-rsk stocks (Haugen and Hens [1972], Black et al. [1972]). Although the focus of the lterature has been manly on alpha at bottom-up stock level, there s also emprcal evdence of volatons of the CAPM model at top-down sector and country level. Evdence of momentum and sze anomales at sector and ndustry level have been reported (Moskowtz and Grnblatt [1999], Capaul [1999]). Abnormal seasonal effects n global sector returns have been observed (Doeswjk [2008]). Emprcal evdence of momentum and value anomales at country level have been reported (Desrosers et al. [2004]). These alphas tend to be explaned by structural ms-prcngs due to behavoural bases of nvestors whch are not accounted for n the smplstc CAPM. The emprcal demonstraton of alphas s usually done by showng how a gven actve systematc strategy whch tlts some stocks away from ther market captalzaton weght would have generated abnormally hgher returns that are not explaned by the portfolo exposure to the market as descrbed by ts beta. The tlts are determned by the exposure of stocks to factors beleved to predct nvestor behavour. The actve strateges desgned to capture the underlyng alphas may use dfferent factors. For example, prce-to-book, prce-to-earnngs or dvdend yeld are typcal factors used to capture value alpha. But t s dffcult to know whch best predcts nvestor behavour. It s even possble that several factors could be at play. For nstance, hgher demand for rsky stocks s sometmes attrbuted to nvestor preference for rsk (stocks as lottery tckets) and to overconfdence n forecastng hgher returns for rsky stocks. But t can also be explaned by the fact that most nvestors cannot easly leverage low-beta stocks (Baker et al. [2011]). It s therefore not surprsng that both volatlty of returns and beta are factors consdered n strateges desgned to capture the alpha from low-rsk stocks. The emprcal evdence that systematc strateges can be used to capture alpha opens the queston of whether markets are really nformaton effcent. Investors spendng the extra tme researchng company statements and foreseeng the mpact of management behavour and decsons may be able to use such nformaton to generate alpha from ther forecasts of stock returns. Although t wll always be much more dffcult to prove the ablty to generate alpha n such a way, evdence s avalable (e.g. Gergaud and Zemba [2012]). Alpha s not equally avalable to nvestors. The larger the nvestor the more dffcult t s to mplement actve strateges wthout leavng a market mpact whch wll detract from performance, n partcular for strateges whch requre hgh turnover. Momentum alpha, and n partcular short-term momentum alpha, wll be much more dffcult to capture than value or low-rsk alpha whch requre less turnover. The optmal capture of alpha should therefore take nto account nvestor sze and constrants. The sze of the nvestor caps turnover and lmts the access to some alpha strateges lke short-term momentum. Other typcal constrants are ) a long-only constrant, whch makes t mpossble to sell-short stocks ) lqudty constrants, n partcular for larger nvestors who may fnd t dffcult to nvest at all n some stocks, ) stock excluson lst constrants, where nvestng n some stocks s not authorzed due for example to non socally-responsble behavor of companes, and fnally, v) restrcted access to dervatves nstruments, n partcular over-the-counter. In fact, many nvestors are restrcted to construct ther portfolos wth only a small number of suffcently lqud stocks, typcally avodng those wth the smaller market captalzatons and volume, and are subject to pre-defned excluson lsts. Passvely managed ndex funds mplementng off-the-shelf actve strateges exposed to some of these alphas are now avalable. For example, ndex-funds desgned to capture value alpha have been shown to be fully explaned by ther loadngs to value factors (Bltz and Swnkels [2008]). Whle ndex funds offer easy access to systematc alphas at low management fees, nvestors should be aware of some drawbacks. The transparency and proftablty requrements n many such approaches tend to lead to over-smplfcaton, unnecessary constrants and consequently to sub-optmal capture of the underlyng alphas (Bltz [2012]). Rsk management s sometmes sacrfced for smplcty wth no clear defnton of the target rsk budget. Constrants on country and sector maxmum devatons aganst the market captalzaton ndex are also over-smplfcatons of rsk controls whch n many cases are not needed. The fact that ndex funds need to grow for proftablty reasons also means that ther strateges may constran turnover to sub-optmal levels for a gven nvestor. And some ndex-funds may present a hgher rsk of crowdng as they grow. Increasng assets nto sometmes relatvely concentrated portfolos can lead to consderable devaton from the more lqud market captalzaton allocaton. Fnally, the hstory of most actve ndexes avalable today s, for the most part, not more than a hstorcal back-test constructed wth the beneft of hndsght, free of transacton costs and market mpact. The beneft of lower management fees can qute easly be offset by poorer returns resultng from such drawbacks. Non-ndexed actve funds clearly have more choces to adapt to snce they are not slaves of an ndex-strategy that s dffcult to change once the ndex-fund attracts assets.

8 8 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 If many ndex-funds are very clear about ther objectve to capture alpha there are also examples of ncreased complexty beyond what seems necessary, obscurng the real sources of alpha n the fnal ndex. Carvalho et al. [2012] have recently shown that equty rsk-based strateges lke Mnmum Varance, Maxmum Dversfcaton or Rsk Party are examples of unnecessarly complex and sub-optmal portfolo constructons capturng a mx of low-rsk, small-cap and value alphas whle offerng a defensve beta ( < 1). Ths has mportant mplcatons for the constructon of a portfolo amng at beng exposed to a number of such alpha sources. The fact that ndex-funds are mplementng alpha strateges whch may be sub-optmal, that many are already exposed to more than one alpha source and that not all known alpha sources are readly avalable are strong constrants to a proper rsk-budgetng exercse. defnes how alpha sources are combned to form an optmal unconstraned target portfolo allocaton. In the absence of any constrants the portfolo constructon would end here. But ths s never the case. We show how the stock mpled returns derved from ths optmal unconstraned allocaton can be used to generate mean-varance effcent constraned portfolos, whch are robust n the sense that they retan the exposures to the alpha sources as much as constrants allow. In the second part of the paper, we present two examples to llustrate the mplementaton of the approach. The propertes of constraned portfolos bult usng ths approach are dscussed n detal. In the annex, we dscuss the dfferences between ths approach and the well-know Black-Ltterman model [Black and Ltterman (1992)] and nclude a number of analytcal nsght regardng the mpact of constrants. The alternatve s to use an ntegrated approach to construct one sngle portfolo combnng the dfferent strateges to capture alpha. In our vew, constructng such a portfolo starts wth a rsk-budgetng exercse to combne unconstraned systematc or judgmental portfolos desgned for alpha capture, whether for stock pckng or top-down sector or country allocaton approaches. The optmal unconstraned portfolo whch combnes all avalable alpha sources s then smply a rsk-weghted average of the unconstraned portfolos gven by the strateges desgned to capture each ndvdual source of alpha. A robust constraned optmal portfolo can be derved from the unconstraned optmal allocaton by estmatng ts mpled stock returns and usng them as nputs n constraned meanvarance optmzaton. The mpled returns are, by defnton, the stock returns whch render the underlyng allocaton optmal n the absence of constrants. As we shall show, constraned portfolos generated from mpled returns do compare well wth the startng underlyng unconstraned portfolo; constrants mpact essentally the specfc rsk of the portfolo whle systematc rsk exposures reman represented at a comparable level to the extent that t s possble. We show that for as long as constrants are not too bndng the fnal resultng constraned stock allocaton remans close to that n the unconstraned portfolo. The more the portfolo reles on specfc rsk the more constrants are lkely to flatten the effcent fronter. Portfolos relyng more on systematc rsk wll handle constrants more easly as the optmzer wll be able to fnd smlar exposures to systematc rsk wth small changes n the weghts of other stocks. In ths paper we frst show how to buld an unconstraned alpha capture portfolo usng both systematc and judgmental approaches. We dscuss the mportant exercse of allocatng a rsk budget to each alpha source n the portfolo, whch

9 9 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 Framework Below we present n detal the framework startng wth ) a dscusson of how to buld alpha capture portfolos followed by ) the rsk budgetng exercse the each alpha capture strategy and fnally ) the step of portfolo constructon, unconstraned and wth constrants. If we can forecast the expected rsk-adjusted returns for each strategy (nformaton rato) and measure the expected level of correlaton between each par of strateges (e.g. hstorcally), then, the mean-varance optmal rsk budget s gven by 1 : 1 1 RB IR (1) Alpha capture portfolos Whether a judgmental or systematc approach s used, unconstraned strateges to capture alpha can be bult from a lst of selected stocks whch are expected to generate alpha. Systematc alpha capture strateges select stocks on the bass of ther exposure to a gven factor lke value or momentum. Judgmental approaches wll select stocks on the bass of fundamental analyss. Alpha can be created ether by tltng the portfolo n favour of selected stocks expected to generate postve alpha, away from those expected to generate negatve alpha or both. The actve portfolo representng devatons to the market captalzaton portfolo can be represented by a zero-sum long-short portfolo. The actve weghts allocated to each selected stock can be nversely proportonal to ther volatlty, n whch case the actve allocaton s mean-varance effcent f all par-wse correlatons were equal and the expected nformaton rato for each stock wth a non-zero actve weght s equal n magntude, wth the sgn changng to negatve for those wth a negatve actve weght. An equally-weghted long-short portfolo representng actve weghts s more often used for smplcty and because t wll generate less turnover. Ths would be the optmal soluton f addtonal stocks had the same volatlty. In general, when a suffcently large number of stocks are retaned, the dfference between equally-weghtng or rsk weghtng can be small. At each re-balancng, each alpha capture strategy wll thus generate an equally-weghted longshort portfolo representng actve stock weghts. Unconstraned portfolos for sectors and countres can be constructed usng smlar approaches. Rsk budgetng allocaton to strateges Mean-varance optmzaton offers a good startng pont for alpha rsk budgetng. Krtzman [2006] has shown that meanvarance optmzaton allocates sensbly when correlatons are not too large, but other more robust approaches lke resamplng optmzaton could be consdered (Scherer [2004]) when relable data s avalable. where RB s the vector of optmal rsk budget allocated to each strategy, s the par-wse correlaton matrx for the strateges and IR s the vector wth the expected nformaton rato for each strategy. Fnally, s a parameter whch measures the overall rsk averson and can be scaled so that the ex-ante rsk reaches a gven target level. The total rsk budget allocaton s nversely proportonal to the decrease n the level of overall rsk averson. The nformaton rato of the mult-strategy portfolo s the same rrespectve of the level of rsk averson and s the largest possble for any combnaton of the strateges,.e. the mean-varance optmzaton maxmzes the nformaton rato of the unconstraned mult-strategy portfolo. Uncorrelated strateges: f the strateges are uncorrelated then the unconstraned optmal mean-varance rsk budget allocaton to each strategy s smply proportonal to the nformaton rato for each ndvdual strategy. The equaton above smplfes to: 1 RB IR Equal rsk-adjusted returns: f there s no reason to expect a dfferent rsk-adjusted return from each strategy (no vew on the nformaton rato of the strateges, n whch case t can makes sense to assume they wll delver the same rskadjusted return), then equaton (1) smplfes to: IR RB 1 1 IR s the nformaton rato of all strateges and 1 s unt vector. Now the rsk budget depends only on correlatons: the optmal rsk budget allocaton mnmzes correlaton allocatng hgher rsk budget to strateges wth the lowest correlatons and lower rsk budget to those more correlated wth others whle scalng wth IR/. Ths strategy has been called maxmum dversfcaton n a dfferent context (Chouefaty and Cognard [2008]). Absence of any pror nformaton: n the absence of any pror nformaton about the expected future performance of the strateges and ther correlaton, the most obvous soluton s to allocate an equal rsk budget to each strategy. Ths s the optmal mean-varance allocaton f we expect all strateges to delver exactly the same rsk-adjusted return and to be equally correlated. In ths case, equaton (1) smplfes to: IR RB (4) 1 Ths can be derved from w 1 1 R, wth w the vector of weghts allocated to each strategy and R the vector of expected returns for each strategy, by decomposng the varance-covarance matrx of strategy returns,, nto correlatons and varances 2 (2) (3)

10 10 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 Portfolo constructon Ths follows three steps: ) the constructon of the unconstraned target allocaton at stock level from the rsk budget allocaton to strateges, ) the estmaton of the stock mpled returns for that unconstraned target allocaton and ) a mean-varance optmzaton step startng from the stock mpled returns and applyng constrants. Aggregaton of strateges nto an optmal unconstraned target allocaton: the optmal vector of unconstraned actve stock weghts P A can be determned from the rsk budget allocaton to strateges and, from ther underlyng portfolos, resolved at stock level. The vector P A can be bult from a matrx P S wth the actve weght of each stock n each strategy, wth strateges n rows and stocks n columns. The vector P A s then gven by the weghted average of the actve stock weght n each strategy: P A P S w where the vector takes nto account the ex-ante volatlty of the stock level allocaton representng the vew of strategy at that pont n tme and the rsk budget allocaton to each strategy, RB : w 1 RB Rsk model: as usual when dealng wth large stock unverses, we consder a lnear factor model. We assume a set of stock exposures to rsk factors and stock specfc rsks, the rsk model s then: ' wth the varance-covarance matrx of factor returns. The framework s ndependent of the rsk model. Commercal rsk models lke Barra, Northfeld, Axoma or APT can be easly employed. Alternatvely, a statstcal rsk model based on prncpal component rsk factor decomposton could be selected. Full covarance matrx models lke Bayesan approaches can be equally consdered although the results concernng separaton nto systematc and specfc rsk dscussed below do not apply. Impled actve returns for each stock: once the target unconstraned actve allocaton P A has been bult and the rsk model chosen, we can estmate the mpled excess returns R I for each stock from R I P A (5) (6) (8) (7) s related to the nformaton rato of the portfolo P A by IRP PA' R I / P PA ' PA / P A A Therefore, wth T = 12, 52 or 260 for monthly, weekly or daly tradng data, respectvely, IRP T A PA. The vector of mpled stock excess returns s by defnton the set of stock excess returns that renders the unconstraned * actve portfolo y P A effcent, where y* s the soluton to the unconstraned mean-varance optmzaton: y * 1 1 R I for a level of rsk averson. The mpled excess returns translate at stock level the rsk budgetng allocaton to each alpha capture strategy and are robust wth regard to the rsk model. In the annex we dscuss the dfferences between the mpled returns estmated n ths way and the stock returns estmated wth a Black-Ltterman model. Handlng constrants: the problem of portfolo constrants can now be addressed by runnng a constraned mean-varance optmzaton usng the mpled excess returns whch effcently represent the aggregaton of vews gven by the dfferent strateges. The mean-varance optmzaton problem n (9) under k lnear constrants, ( v ) 1 k, can be translated nto: * y arg mn y' y y' R v y u k (10) I u.c. ', 1 2 wth the soluton y* the optmal constraned portfolo of actve weghts at rsk averson. RI s defned n equaton (8) as the vector of mpled excess return for the unconstraned portfolo of actve weghts P A. When k=0.e. no constrants, the soluton s smply A as seen n the prevous secton. Usng equaton (8) n equaton (10) t s then relatvely easy to show that the mnmzaton s equvalent to: * y arg mn( y PA )' ( y PA) A u.c. y v ' y u, 1 k * (9) P.e. mnmzng the trackng error rsk of a long-short allocaton representng the actve weghts of the unconstraned portfolo relatve to the long-short allocaton representng the actve weghts of the constraned portfolo at the same rsk averson. If nstead we look for the optmal constraned portfolo at a gven target trackng error rsk, then n equaton (10) the frst term s constant and we can re-wrte t usng equaton (8) for the mpled returns as: (11) * y arg max y' P A u.c. v ' y u, 1 k (12)

11 11 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 whch s the maxmzaton of the covarance of the constraned actve allocaton y* wth the unconstraned actve portfolo P A. Mean-varance optmzaton can handle a number of tradtonal lnear constrants, for example constrants on the weght of ndvdual stocks or portfolos of stocks. In the annex we dscuss analytcally the mpact of the most typcal lnear constrants on the fnal constraned portfolo. Turnover constrants can also be mposed but we do not nclude a dscusson of those. From a practcal pont of vew, the mpact of constrants n the fnal stock allocaton can be montored by comparng not only the resultng constraned actve weghts wth the ntal target unconstraned actve weghts but also the factor rsk exposures n the ntal unconstraned target allocaton to rsk exposures n the constraned allocaton. As demonstrated n the annex, the optmal mean-varance unconstraned and constraned portfolos exhbt smlar exposures to the rsk factors n the rsk model unless constrants become too bndng. The mpact of constrants wll be felt essentally n the exposure to stock specfc rsks. We can also montor how much return s detracted by the constrants when comparng the expected return of the constraned portfolo wth that of the unconstraned target portfolo. The percentage of return lost to constrants s a useful ndcator to fnd the best workng range of ex-ante rsk. If rsk s too hgh, there s the danger that the hgher rsk s no longer compensated by hgher returns when constrants flatten the effcent fronter too much. Constrants can also have a major mpact at lower rsk levels and, as we shall see later, takng too lttle actve rsk can lead to a less optmal representaton of vews too. Examples Benchmark ndex: for the sake of llustraton we chose a smplfed example startng from a market captalzaton ndex wth a relatvely small number of stocks. We bult an actve portfolo based on the 50 European stocks n the Eurostoxx 50 ndex overlayng actve strateges on ths market captalzaton ndex on 16 th August Three systematc sources of alpha at stock level were taken nto account: small-cap, value and momentum. We also consdered a systematc momentum approach at sector level as an addtonal systematc source of alpha. Fnally we nclude a typfed example of judgmental vews. Alpha capture portfolos: for the sake of transparency, the alpha capture strateges here consdered have been delberately kept smple and by no means do we pretend that they represent the most effcent approach to capture the underlyng alpha. In exhbt one we show an unconstraned actve portfolo for each alpha capture strategy on ths date. The stocks have been screened by market captalzaton for small-cap, book-to-prce for value and the average 11 month returns endng the 18 th July 2012 (excludng the last four weeks) for momentum. The portfolos are long (short) the smaller (larger) captalzaton stocks, those wth the larger (lowest) book-to-prce and wth the strongest (weakest) momentum. For smplfcaton we chose to equally-weght stocks n each long-short portfolo. For sector momentum, the strategy portfolo s long sectors wth the strongest momentum measured by the last 12 month return and short sectors wth the weakest momentum. Fnally, we assumed that a judgmental process had selected a number of stocks based on ther hgher expected returns. As t s not our purpose to buld nvestment cases for companes but just to show how these could have been taken nto account, we arbtrarly selected all companes startng wth the letter A. The judgmental strategy can also be represented usng an equally-weghted long-short portfolo, long all stocks selected and sellng-short all other stocks n the ndex. Portfolo allocaton at a gven date for European stocks In ths frst example we want to llustrate step-by-step how the methodology can be mplemented and dscuss n detal the mpact of dfferent constrants on the fnal portfolo. Rsk model: we used a prncpal components analyss (PCA) rsk model based on two years of weekly returns followng the methodology proposed by Plerou et al. [2002] that consders results from random matrx theory showng that the egenvalues of a T N random matrx wth varance 2 are capped asymptotcally at max= 2 (1 + N / T + 2 (N / T) 1/2 ) wth T the number of perods and N the number of stocks. Thus we dscard all egenvalues smaller than max, consdered to be statstcal nose.

12 12 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 Exhbt 1 Portfolos combnng fve alpha capture strateges wth the same rsk budget: Small, Value, Momentum, Judgmental and Sector Momentum. On the left, the actve weghts for each alpha strategy represented by long-short portfolos. The szng of the actve weghts has been set so that ther ex-ante volatlty s 5%. The average of these portfolos, Unconstraned, has been scaled so that ts ex-ante volatlty s also 5%. The portfolo weghts of the unconstraned and constraned portfolos are shown on the rght. Long-only and lqudty constrants were appled. (contnued) Stocks Sectors Actve Weghts (%) Small Value Momentum Judgmental Sector Momentum Ex-ante trackng error rsk (%) Damler AG LVMH Moet Hennessy Lous Vutton Volkswagen AG (Pfd Non-Vtg) Consumer dscretonary Bayersche Motorenwerke AG BMW Industra de Dseno Textl S.A Anheuser-Busch InBev Unlever N.V Danone S.A. Consumer staples L'Oreal S.A Carrefour S.A Total S.A ENI S.p.A. Energy Repsol S.A Banco Santander S.A Allanz SE BNP Parbas S.A Banco Blbao Vzcaya Argentara S.A Deutsche Bank AG ING Groep N.V AXA S.A. Fnancals Muenchener Rueckverscherungs-Gesellschaft AG UnCredt S.p.A Socete Generale S.A. (France) Intesa Sanpaolo S.p.A Unbal-Rodamco SE Asscurazon General S.p.A

13 13 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 Unconstraned Impled Excess Returns (%) Unconstraned Portfolo Eurostoxx Index Portfolo Weghts (%) Unconstraned Long-only 1 Long-only + Lq Constrants Long-only

14 14 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 Exhbt 1 Portfolos combnng fve alpha capture strateges wth the same rsk budget: Small, Value, Momentum, Judgmental and Sector Momentum. On the left, the actve weghts for each alpha strategy represented by long-short portfolos. The szng of the actve weghts has been set so that ther ex-ante volatlty s 5%. The average of these portfolos, Unconstraned, has been scaled so that ts ex-ante volatlty s also 5%. The portfolo weghts of the unconstraned and constraned portfolos are shown on the rght. Long-only and lqudty constrants were appled. Stocks Sectors Actve Weghts (%) Small Value Momentum Judgmental Sector Momentum Sanof S.A Bayer AG Health care Esslor Internatonal S.A Semens AG Schneder Electrc S.A Vnc S.A. Industrals Konnkljke Phlps Electroncs N.V Compagne de Sant-Goban S.A SAP AG ASML Holdng N.V. Informaton technology Noka Corp BASF SE Ar Lqude S.A Materals 0.00 ArcelorMttal SA CRH PLC Telefonca S.A Deutsche Telekom AG Telecom Svs 0.00 France Telecom Vvend E.ON AG GDF Suez S.A RWE AG Utltes Enel S.p.A Iberdrola S.A

15 15 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 Unconstraned Impled Excess Returns (%) Unconstraned Portfolo Eurostoxx Index Portfolo Weghts (%) Unconstraned Long-only 1 Long-only + Lq Constrants Long-only Datasource: BNPP IP, FactSet

16 16 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 Rsk budget: to keep the exercse smple we assumed that all alpha capture strateges should delver the same nformaton rato of 0.3 and that they are fully uncorrelated. Hence, an equal rsk budgetng approach apples and nformaton rato of the combnaton of strateges s IR P A. Snce all alpha capture portfolos have the same ex-ante rsk, 5%, n equaton (6) s the same for all strateges. We target P A 5% ex-ante trackng error for the unconstraned portfolo. Then, n equaton (8) = 697.7, obtaned from IRP T A PA wth T=52 weeks, as defned before. Unconstraned portfolo: the fnal unconstraned actve allocaton s then smply the weghted average of the stock weghts n each alpha capture portfolo. In exhbt one we also show the mpled excess returns calculated from the product of the varance-covarance matrx wth these actve weghts. The fully nvested actve portfolo can be obtaned by addng the unconstraned actve allocaton to the benchmark ndex. Constraned portfolo: the mpled excess returns were used as nputs n the optmzer to maxmze excess returns aganst trackng error rsk. A budget constrant statng that the sum of stock actve weghts must be zero apples. In exhbt one we show two portfolos optmzed under long-only constrants and lmtng the absolute weght of each stock to 10%. The latter can be expressed n terms of stock actve weghts as w < 10% - w MC, wth w MC the stock weght n the market captalzaton ndex, whereas the long-only constrant can be expressed as w > -w MC. The frst of these two portfolos, Long-only 1, comes at the same rsk averson as the unconstraned portfolo,, where as the second, Long-only 2, comes at the same * ex-ante trackng error rsk of 5% whch requred We also nclude a portfolo optmzed at the same rsk averson,, wth an addtonal constrant for lqudty not allowng t to nvest n stocks wth the lowest market captalzatons,.e. MC w = -w, for any of the 10 stocks n the ndex wth the lowest market captalzaton. Analyss of portfolos: the portfolo weghts of the Long-only 1 portfolo reman close to those n the unconstraned portfolo but no short postons or stock weghts above 10% are found. As seen n exhbt two, the trackng error rsk s lower for the same rsk averson as a result of the applcaton of constrants. Exposure to systematc rsk s not too dfferent but exposure to specfc rsk was reduced as expected (see annex). Even after applyng the lqudty constrant, the portfolo stll remans relatvely close to the startng ntal unconstraned portfolo. Nevertheless, the lqudty constrants mpact two stocks whch were before assgned large postve weghts. The portfolo wll not nvest n those stocks and reallocates ther weght, ncreasng the dfference wth the orgnally unconstraned portfolo. In exhbt two, we can see that the contrbuton from systematc rsk to trackng error rsk s slghtly lower but stll remans comparable to that n the unconstraned portfolo. The contrbuton from specfc rsk s now even smaller, and so s the ex-ante trackng error. Exhbt 2 The trackng error rsk, expected excess return and nformaton rato for the unconstraned and constraned portfolos shown n exhbt 1. The decomposton of trackng error rsk nto systematc and specfc components s also shown. Portfolos: Unconstraned Long-only 1 Long-only + Lq Constrants Long-only 2 Ex-ante trackng error rsk (%) Systematc trackng error rsk (%) Specfc trackng error rsk (%) Expected excess return (%) Expected nformaton rato Datasource: BNPP IP

17 17 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 If we target the same trackng error rsk as n the unconstraned portfolo, then (see annex) the contrbuton of the systematc rsk to trackng error rsk ncreases n the constraned portfolo. Thus the portfolo dstances tself more from the unconstraned soluton n terms of stock weghts. It would be better compared to an unconstraned portfolo of hgher trackng error rsk * * obtaned at the same level of rsk averson,. In exhbt two we also show that the expected nformaton rato for the unconstraned portfolo s hgher than that for the constraned portfolos. The more we constran the portfolo the lower the expected nformaton rato. The dfferent portfolos obtaned wth the same rsk averson can be more easly compared n exhbt three where we plot the stock weghts of the unconstraned portfolo on the horzontal axs aganst those of the constraned portfolos on the vertcal axs. It s clear that constraned allocatons reman relatve close to those n the unconstraned portfolo. The more constraned the portfolo s, the more t devates from the unconstraned allocaton. Negatve vews are more dffcult to express than postve vews snce short-sellng s not possble. The constrant lmtng the maxmum stock weght at 10% also has some mpact. Addng lqudty constrants also mpacts stocks for whch the unconstraned portfolo has a postve weght and whch are excluded from the constraned portfolo. Exhbt 3 The stocks weghts of constraned portfolos plotted aganst the stock weghts of the unconstraned portfolo as shown n Exhbt 1. All portfolos have been optmzed wth the same level of rsk averson. Optmsed Portfolo Weghts 16% 14% 12% 10% Unconstraned Long-only 1 Long-only + Lqudty constrants 8% 6% 4% 2% 0% -6% -4% -2% 0% 2% 4% 6% 8% 10% 12% 14% 16% -2% Unconstraned Portfolo Weghts -4% -6% Datasource: BNPP IP

18 18 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 In exhbt four we see how constrants mpact exposures to the rsk factors n the rsk model. There s no sgnfcant exposure n the unconstraned portfolo to the frst factor (whch typcally corresponds to the market factor). Ths s also observed n the constraned portfolos. Exposures to other factors are smaller n the constraned portfolos obtaned wth the same level of rsk averson but always n lne wth those n the unconstraned portfolo. The Long-only 2 constraned portfolo whch comes wth the same trackng error rsk as the unconstraned portfolo shows larger exposures to the systematc factors, sensbly 2.39 tmes larger than those found n Long-only 1 (see annex). Exhbt 4 Exposures to the frst fve rsk factors n the PCA rsk model for unconstraned portfolo, the two long-only portfolos and for the portfolo wth long-only and lqudty constrants. Exposures are gven n terms of contrbuton to trackng error rsk Factor Factor 1 2 Factor 3 Factor 4 Factor Unconstraned Long-only 1 Long-only + Lqudty constrant Long-only 2 Datasource: BNPP IP

19 19 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 In exhbt fve, we show the mpact of addng constrants on the effcent fronter whch n the absence of any constrants s smply a straght lne (sold lne) wth excess return ncreasng proportonally to trackng error rsk. The lne goes to nfnty. The dashed lne represents the effcent fronter for portfolos optmzed wth long-only constrants and maxmum 10% for each stock weght. The effcent fronter overlaps the unconstraned one untl the frst constrant s ht (portfolo C). Beyond that pont the effcent fronter starts flattenng and the expected nformaton rato decreases. The Exhbt 5 Effcent fronters wth expected excess returns aganst ex-ante trackng error rsk for the unconstraned and constraned portfolos. Portfolo B s the long-only portfolo wth lqudty constrants wth the largest nformaton rato and pont A s the mnmum trackng error rsk portfolo for the same set of constrants. Portfolo C s the rsker long-only portfolo wth the largest nformaton rato but all effcent long-only portfolos wth lower trackng error rsk have the same nformaton rato. 8% 7% 6% Expected Excess Returns 5% 4% 3% Unconstraned Long-only Ex-ante trackng error rsk 2% C Long-only + Lqudty Constrants 1% B 0% 0% A 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% more constrants are bndng, the more the effcent fronter flattens. All portfolos above the nflecton pont C have lower ex-ante nformaton ratos than the unconstraned portfolo. The fronter wll be capped at the portfolo wth the maxmum expected excess return. When lqudty constrants are added the effcent fronter (dotted lne) no longer starts at the orgn. The startng pont s now portfolo A wth the lowest ex-ante trackng error rsk, a portfolo whch s ndependent of expected excess returns that optmally mmcs the market captalzaton ndex when some stocks are removed from the ndex. The optmal portfolo wth the largest nformaton rato can be found at the tangent pont B. The lower the trackng error rsk, the hgher the danger of smply replcatng the ndex whle excludng llqud stocks, hardly explotng the alpha Datasource: BNPP IP capture actve strateges. A too large trackng error rsk s also not compensatng rsk wth hgher excess returns, as the constrants flatten the effcent fronter and the nformaton rato falls beyond B. Even f the optmzer stll fnds optmal solutons beyond B, from a practcal pont there s lttle sense n chasng margnally hgher returns at an ncreasngly large ncremental trackng error rsk. The mpact of constrants can be measured by comparng the ex-ante nformaton ratos of the constraned portfolos wth that of an unconstraned portfolo at the same ex-ante trackng error rsk. When the dfference s too large then constrants are eatng too much performance and the constraned portfolo reflects poorly the combnaton of alpha capture strateges.

20 20 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 Back-test of an example for global equtes We now turn to an example desgned to show how the approach would have behaved over a gven perod of tme. Benchmark ndex: we chose to buld portfolos benchmarked aganst the MSCI World Index 2. Alpha capture portfolos: we consdered four alpha capture strateges whch agan have been delberately kept smple and are not necessarly the most effcent to capture the underlyng alpha. The frst nvests n small-cap stocks, the second n value stocks usng the book-to-prce as defnton of value, the thrd n wnners defned by the momentum of monthly returns estmated over 11 months endng one month pror to estmaton date, and the fourth nvests n low-rsk stocks wth the lowest two year volatlty of weekly returns. All these alpha capture strateges are represented by long-short portfolos rebalanced at the start of each month, nvestng n the top ranked 300 stocks and sellng short the MSCI World Index 2. In exhbt sx we show how these ndvdual strateges would have performed n the absence of constrants and ther respectve par-wse correlatons. The nformaton rato n the perod Jan 95 through Dec 11 was postve. The leverage of each alpha capture portfolo was chosen so that the ex-ante rsk s equal to 5% for all when they were re-balanced at the start of each month. Ex-ante rsk s estmated from a statstcal rsk model based on a prncpal component approach, as descrbed n the prevous secton, usng two years of rollng hstorcal weekly data. Rsk budget: we consdered three approaches to rsk budgetng. For the frst, mean-varance, we appled equaton (1) and used the results n exhbt sx, for the second, maxmum dversfcaton, we appled equaton (3), and fnally, for equal rsk budgetng we used equaton (4). The rsk averson was set so that the ex-ante rsk budget of ther combnaton s 5% at each monthly re-balancng. The rsk budget allocaton to each alpha factor strategy can be found n exhbt seven. Naturally, mean-varance tlts n favor of momentum wth the hghest nformaton rato and beng the most uncorrelated of them all. Small-cap and value are qute correlated. Despte the lower nformaton rato value gets a larger rsk budget n meanvarance because t s negatvely correlated wth momentum. In maxmum dversfcaton momentum gets the largest rsk budget because t s the least correlated and small-cap gets an even smaller rsk budget than n mean-varance because ths approach assumes all strateges have the same expected nformaton rato. The portfolos combnng the dfferent alpha capture strateges for each set of rsk budgets were re-balanced each month applyng two sets of constrants. The rsk averson parameter was set n order that the trackng error rsk of the unconstraned allocaton was 5% each month at the tme of re-balancng. The rsk averson used for the optmal constraned allocatons was set so that all reach 5% ex-ante trackng error rsk aganst the MSCI World ndex, and s therefore lower than that n the respectve unconstraned portfolos on the same date. Exhbt 6 Informaton ratos for the four alpha capture strateges and ther respectve par-wse correlatons n the perod Jan-95 through Dec- 11. The MSCI World ndex defnes the unverse of stocks and smulatons were performed n USD. Informaton Rato Correlaton (%) Value Momentum Low Volatlty Small Value Momentum Low Volatlty 0.47 Datasource: BNPP IP 2 Due to lcensng constrants we have used the consttuents of our propretary replcaton of the MSCI World ndex pror to The trackng error rsk of the replcaton pror to 2005 s very small

21 21 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 Analyss of portfolos: n exhbt seven we also show the ex-post observed exposures to the dfferent alpha capture strateges obtaned from regresson of the excess return aganst the MSCI World ndex for each combned portfolo strategy. Portfolo drft durng the month and uncertanty n rsk and correlaton forecasts, whch are not constant, explan the dfferences between ex-post and ex-ante target rsk budgets. Nevertheless, n lght of that, the dfferences are remarkably small: the unconstraned portfolo strateges reman very close to the target rsk budgets for each alpha strategy and capture more than 96% of the expected varaton of the combnaton of alphas. Two sets of constrants were consdered. The frst set, Longonly + Lqudty constrants, ncludes the long-only constrant and a cap on the weght of each stock at the lowest of ether between ether 5% or 20 tmes ts market captalzaton weght. The second set, Long-only + Lqudty + # stocks constrants, ncludes an addtonal constrant to cap the number of stocks 3 n the fnal portfolo at 250. As shown n exhbt seven, the exposure to the dfferent alpha strateges n the constraned portfolos remans close to target rsk budgets but exposure to the alpha from small captalzaton stocks n partcular, tends to be more dffcult to retan. When the number of stocks s capped, the exposure to small-cap alpha dsappears. Nevertheless, the excess returns of the most constraned optmal portfolos stll show r-squares of about 70% when regressed aganst the excess returns of the unconstraned alpha capture strateges. If the constrant on the number of stocks s removed, then the r-squares jump to 90%. The long-only and lqudty constrants are much less bndng n ths example. Indeed, the constrant to reduce the number of stocks can be qute damagng. Ths s a constrant could have been justfed on an operatonal bass n the past but makes lttle sense today snce the cost of mplementng portfolos wth large numbers of stocks has fallen substantally n recent years. Puttng more stocks n the portfolo allows for better rsk control (some stocks may be needed just for trackng error rsk control) and for a better representaton of the orgnally ntended unconstraned allocaton. It s mportant to constran portfolos n sensble ways, ether to comply wth regulatons, to better control rsks or to ease operatonal burdens, but not more than that. The dea that concentrated portfolos carry stronger convctons and may delver more alpha s not true. Exhbt 7 The rsk budget for each alpha capture strategy estmated from three dfferent approaches. The ex-ante trackng error rsk s 5%. Below the factor exposures obtaned from regresson of the excess returns of each aggregate strategy, unconstraned and constraned, aganst the factor returns. The MSCI World ndex defnes the unverse of stocks and smulatons were performed n USD from Jan-95 through Dec-11. Mean Varance Ex-ante rsk budget (%) Maxmum Dversfcaton Equal Rsk Budget Small Value Momentum Low Volatlty Ex-post rsk budget (%) Unconstraned Intercept Small Value Momentum Low Volatlty R-square 96% 96% 98% Long-only + Lqudty constrants Intercept Small Value Momentum Low Volatlty R-square 91% 90% 89% Long-only + Lqudty + # stocks constrants Intercept Small Value Momentum Low Volatlty R-square 73% 66% 70% Datasource: BNPP IP 3 We use an teratve procedure to fnd the optmal portfolo whch respects the constrant of maxmum number of stocks

22 22 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 In exhbt eght, we show the smulaton results for these strateges. The unconstraned portfolos are long-portfolos wth a short-extenson. It turns out that the number of stocks n that short-extenson s slghtly hgher than that n the long-leg. If the number of stocks n the portfolo s not constraned, the fnal optmal allocaton stll nvests n a large number of stocks, nearly on average. The unverse average n the perod was nearly Exhbt 8 Smulaton results for the dfferent strateges, unconstraned and constraned. The MSCI World ndex defnes the unverse of stocks and smulatons were performed n USD from Jan-95 through Dec-11. MSCI World Index Unconstraned Mean-Varance Rsk Budget Long-only + Lqudty constrants Long-only + Lqudty + # of stocks constrants Average return (%) Volatlty (%) Sharpe rato Excess return (%) Trackng error rsk (%) Informaton rato Average number of stocks (long / short) 839 / / / 0 MSCI World Index Unconstraned Maxmum Dversfcaton Rsk Budget Long-only + Lqudty constrants Long-only + Lqudty + # of stocks constrants Average return (%) Volatlty (%) Sharpe rato Excess return (%) Trackng error rsk (%) Informaton rato Average number of stocks (long / short) 837 / / / 0 MSCI World Index Unconstraned Equal - Rsk Budget Long-only + Lqudty constrants Long-only + Lqudty + # of stocks constrants Average return (%) Volatlty (%) Sharpe rato Datasource: BNPP IP, MSCI Excess return (%) Trackng error rsk (%) Informaton rato Average number of stocks (long / short) 825 / / / 0 Datasource: BNPP IP, MSCI

23 23 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 The most strkng concluson from the analyss of exhbt eght s that all portfolos delver qute comparable performances and generate levels of rsk not too dfferent from one another, even when strongly constraned. The ex-post level of trackng error rsk also comes out sensbly n lne wth ex-ante target, demonstratng that the rsk model performed relatvely well. Ths proves the robustness of the methodology consdered here, as an effcent approach to buld constraned portfolos. Even f we were helped n the example by the fact that the alpha from small captalzaton stocks, the most dffcult to pass on to the constraned portfolo, was qute correlated wth the alpha from value, the approach provdes the means to detect whch alphas are beng eroded by constrants as demonstrated n exhbt seven. It s also nterestng to see that both maxmum dversfcaton and equal-rsk budgetng approaches acheve comparable results to those from mean-varance. That the latter comes wth the hghest Sharpe rato and nformaton rato should come as no surprse: the rsk budgets are optmzed n-sample, wth the beneft of hndsght. But t s comfortng to realze that smplfed approaches requrng less nputs, reach only margnally worse results. It s also nterestng that the long-only portfolo delvers results comparable to those for the unconstraned portfolo. From an operatonal pont of vew t s clearly much smpler, cheaper and less rsky snce there are no short postons, no leverage and t requres no counter-party rsk. That even rases the queston of whether the so called 130/30 portfolos, or portfolos wth short-extensons, are really needed when the sources of alpha are systematc factor strateges lke those used here. In the same sprt, alpha-beta separaton acheved through an nvestment n the market captalzaton ndex-fund complemented wth hghly leveraged portable alpha strateges, provded through long-short equty hedged funds or absolute returns funds, can be questoned when ther excess returns are derved from smlar systematc alpha capture strateges. Conclusons In a world where markets are not nformaton effcent there are opportuntes to generate alpha by nvestng away from the market captalzaton ndex. Several systematc actve strateges are known to generate abnormal returns not explaned by CAPM. The alpha generated by these strateges s beleved to arse from ms-prcngs due to nvestor behavor that s not taken nto account by CAPM. The dea that dlgent nvestors dedcatng tme to analyze nformaton not necessarly easy to cast n a systematc fashon may generate alpha from ther nformaton advantage can therefore not be dscarded. The queston then s how to buld a portfolo devatng from the market captalzaton ndex whle tryng to effcently explot dfferent ndependent alphas. We suggest that alphas should be captured usng constrantfree strateges and that portfolo constrants should be dealt wth at the level of portfolo constructon. The portfolo constructon starts wth a rsk budgetng exercse to allocate rsk to dfferent sources of alpha at an aggregate level. Rsk budgetng should take nto account any avalable nformaton about the expected rsk-adjusted performance of each alpha capture strategy and ther nteractons. Alpha strateges are not restrcted to bottom-up but can also nclude top-down sector and country decsons, and can be desgned to systematcally handle nformaton or be based on judgmental analyss of nformaton that s not easy to cast n a systematc fashon. We show that constraned portfolos bult from the stock mpled returns of the unconstraned allocaton retan much the same exposures to systematc rsk factors than the unconstraned portfolo. We also show that the constraned portfolo remans very close to the unconstraned target portfolo when constrants are not too bndng. When constrants are too bndng, the rsk-adjusted returns of the constraned portfolo can be too low to justfy takng actve rsk. It s also shown that when constrants force stocks out of the portfolo, takng a too low trackng error rsk may be sub-optmal wth the effcent portfolo smply mmckng the market captalzaton ndex. It s suggested that constraned portfolos are consdered at the levels of trackng error whch maxmze the nformaton rato. At very hgh trackng error levels, the remuneraton of each unt of trackng error rsk wll be smaller and therefore suboptmal; there s no pont n seekng levels of trackng error rsk for whch the nformaton rato of the constraned portfolo s far below that of the underlyng unconstraned portfolo as constrants eat too much alpha. Fnally, we show wth back-tested examples how robust the approach proposed here s when appled to the example of combnng four well known alpha capture strateges appled to global stocks and workng wth realstc sets of constrants. The results suggest that n partcular when the alpha capture portfolos are exposed to systematc rsk factors the fnal constraned portfolos manage to delver rsk-adjusted performances comparable to those from the unconstraned portfolo and that ther excess returns over the market captalzaton ndex are largely correlated. Insttutonal nvestors seems to recognze more and more that there are ndeed some systematc ms-prcngs n the market whch justfy tltng away from the market-captalzaton portfolos and takng actve postons (Chambers et al. [2012]). We propose a robust framework to budget the rsk allocated to each alpha and to buld sensbly constraned portfolos n an extremely transparent fashon. The portfolos retan as much as possble the alphas derved from systematc or judgmental approaches. We beleve that ths type of framework s partcularly suted to those nsttutonal mandates where

24 24 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 nvestors wll be able to decde on ther own rsk budget allocaton as a functon of ther convcton on the sources of alpha, as represented by ther expected nformaton ratos for the dfferent alpha capture strateges. The framework can also be easly appled to mult-asset and fxed ncome portfolos. Annex The Black-Ltterman approach The unconstraned soluton to the Black-Ltterman (BL) model can also be wrtten n the form of a market captalzaton portfolo plus a weghted sum of portfolos wth actve weghts (He and Ltterman [1999]). Usng our notaton, the optmal BL actve weghts are: * -1 y P w where w ( / P ' P ) ( R/ P ' ) S wth the varance-covarance matrx of strategy returns, w MC s proportonal to the market captalzaton portfolo mpled stock returns, s the rsk averson n terms of absolute rsk of the portfolo and a level of confdence n the strateges. The BL portfolos are obtaned from an optmzaton n a space of returns over the rsk free rate and volatlty rather than excess returns and trackng error rsk aganst the market captalzaton portfolo. For a gven, a smaller ncreases the trackng error rsk allocaton n BL. In our framework the choce of trackng error rsk s left to the nvestor. Rather than use a parameter lke to decde on a relatve allocaton between CAPM and strateges smply focus on the problem of budgetng the strateges and leave the choce of trackng error rsk to the nvestor who can decde n a much smpler manner what to use e.g. pck the maxmum nformaton rato portfolo. Equaton (A1) s more complex than our framework, where the weght of the strateges 1 s just w 1 1 R. It shows that the rsk budget of a strategy n BL wll be postve only f ts expected return gven n R, adjusted for the rsk averson, s larger than the strategy expected return as estmated from today s underlyng allocaton n P S and the market mpled returns for each stock (CAPM). The strategy weghts depend not just on -1 but on the nverse of the average of weghted by wth today s strategy varances determned by ther underlyng stock allocaton P S and the rsk model.the fnal result s therefore more complex than n our framework and requres an approach to estmate for whch there s no establshed procedure. The BL approach needs to consder the CAPM mpled returns n the fnal rsk budgetng of strateges wth a weght determned by ths parameter because t was conceved for an optmzaton n the space of absolute performance and rsk. The problem wth the BL model s that t assumes zero correlaton between the returns to the market captalzaton portfolo and the alpha capture strateges (or S S S (A1) vews), whch s often not the case n practcal applcatons. The resdual correlaton wth the market captalzaton portfolo returns of some well-known smple alpha capture strateges lke those here dscussed, e.g. small captalzaton stock alpha or low-volatlty alpha, s therefore not properly accounted for. A recent extenson of the BL model, called the non-orthogonal Black-Ltterman model, has been proposed by Oglaro et al. [2012] to take those correlatons (couplng) nto account showng that they are the key to determne. Nevertheless, the framework here presented stll has the advantage of beng smpler than the non-orthogonal BL approach. Impact of long-only and cash neutral constrants We shall analyze the mpact of two common constrants: ) that the portfolo must be long-only and ) fully nvested n equtes. We shall frst look at the mpact of constrants on the trackng error and on the exposure to systematc rsk for a gven level of rsk averson. We shall also compare the unconstraned portfolo wth the constraned one when a smlar trackng error exsts. In a unverse of n stocks the cash neutral and long-only constrants translate nto n MC y 0 MC and y w, wth w the weght of stock n the market captalzaton ndex. Applyng the Kuhn-Tucker condtons to (10): y ( ) P * 1 A where * ( y w 0 MC ) 0 Multplyng ths equaton by y * ', usng the fact that y* s a zero sum portfolo and fnally applyng ( * MC y w ) 0, leads to * * * * * * y ' y y ' ( PA) y ' y ' y ' w' * * y ' y * y ' y * * * y ' y PA' P A MC * * * PA' PA y ' y y ' ( PA) 0 0 showng that for the same level of rsk averson the cash neutral and long-only constraned portfolo y* has a lower trackng error rsk than the unconstraned portfolo PA. Now we look at the mpact of the cash neutral and long-only constrants on the systematc rsk exposure of the constraned soluton at same rsk averson. The rsk model can be separated nto ts systematc and specfc rsk terms, The systematc term s defned from the m egenvectors 1m and the egenvalues m of the nxn matrx ' The specfc term s smply a nxn dagonal matrx of resdual 2 varances 1n (A2) (A3)

25 25 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 If we complete the ortho-normal vector base ( v ) 1 m ( v ) 1 n we can then wrte the dfference between the constraned and unconstraned soluton as: y * n P x A 1 v ( x ) 1 m are the dfferences n exposure to systematc factor between the constraned portfolo and the unconstraned portfolo. If we use ths result n (11) then m 2 ( y PA)' ( y PA) x 1 m x 1 m x n 2 2 ( y PA) 1 n 2 2 ( y PA) 1 n x (A4) n (A5) where n 1 In CAPM, m=1 wth v 1 and also 1 n. We can then 2 2 show that 1 n market 2 2. Consequently, n CAPM the optmzer wll essentally mnmze the dfference between the constraned portfolo s actve exposure to the market factor and that of the unconstraned portfolo. More generally, under such constrans, the optmzer tends to preserve the actve exposures to all systematc factors when compared to those n the unconstraned portfolo. 1 wth the unconstraned mnmum varance portfolo 1 ', the equty portfolo wth the smallest possble ex-ante rsk. The soluton wll sell the mnmum varance portfolo f the cash exposure n the unconstraned portfolo s postve and buy otherwse. Ths makes sense ntutvely as the mnmum varance portfolo s the closest you can get to cash whle nvestng n equtes. The second case of nterest s the neutralzaton of the exposure to the market as measured by. In ths case k=1 and u=0 and the constrant vector V s equal to the matrx : * y 1 ' PA ' PA 1 From the defnton of cov( r, rmc ) var( rmc, rmc ) and wth w MC the vector wth market captalzaton weghts, 1 1 w MC w' MC w MC w MC '. Thus, the beta neutral constraned soluton s exactly equal to the unconstraned soluton PA mnus the beta exposure of ths unconstrant soluton tmes the market captalzaton portfolo. The soluton wll sell the market captalzaton portfolo when the beta s postve and buy otherwse. (A7) We now consder the constraned portfolo wth the same trackng error rsk of an unconstraned portfolo PA. When * the former exsts then there s a rsk averson such that ts trackng error rsk s equal to that of P A. (A4) tells us that * 1. As shown n (A3), the systematc rsk exposures after * the optmzaton are then most lkely larger than that of the unconstraned actve allocaton P A and consequently the exposure to systematc rsk s larger n the constraned soluton y* of same trackng error rsk. Cash neutral and beta neutral constrants In a unverse of n stocks, wth V the n x k constrant matrx ( v ) 1 k and u the k x 1 vector ( u ) 1 k we can show by usng Lagrange multplers that when all constrants are based on equaltes, v' y = u, then the soluton to (10) s: * y PA V ( V ' V ) ( u V ' PA) There are two specal cases of partcular nterest. The frst s mposng that the soluton y* s cash neutral. In ths case k=1 and u=0 and every sngle coeffcent v n V s 1, noted as a matrx I. Then (A5) can be smplfed to: (A6) 1 * y PA ' PA 1 ' (A6) The fnal soluton s equal to the unconstraned soluton PA mnus the product of the cash exposure n P A, whch s I' P A,

26 26 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 References Arnott, R.D., J. Hsu, and P. Moore Fundamental Indexaton. Fnancal Analysts Journal, Vol. 61, No. 2 (2005), p Banz, R.W. The Relatonshp Between Return and Market Value of Common Stocks. Journal of Fnancal Economcs, Vol. 9, No. 1 (1981), pp Baker, M.P., B. Bradley, and J.A. Wurgler. Benchmarks as Lmts to Arbtrage: Understandng the Low Volatlty Anomaly. Fnancal Analysts' Journal, Vol. 67, No. 1 (2011), pp Black, F., M.C. Jensen, and M. Scholes The Captal Asset Prcng Model: Some Emprcal Tests, Studes n the Theory of Captal Markets, Praeger Publshers Inc, Black F., and R. Ltterman Global Portfolo Optmzaton. Fnancal Analysts Journal, vol. 48, no. 5 (1992), pp Bltz, D. Strategc Allocaton to Premums n the Equty Market. The Journal of Index Investng, Vol. 2, No. 4 (2012), pp Bltz, D.C. and L.A.P. Swnkels, Fundamental Indexaton: an Actve Value Strategy n Dsguse. Journal of Asset Management, Vol. 9, No. 4 (2008), pp Carvalho, R.L.de, X. Lu, and P. Mouln Demystfyng Equty Rsk Based Strateges: A Smple Alpha plus Beta Descrpton. Journal of Portfolo Management, Vol. 38, No. 3 (2012), pp Chambers, D., E. Dmson, and A. Ilmanen The Norway Model. Journal of Portfolo Management, Vol. 38, No. 2 (2012), pp Chouefaty, Y., and Y. Cognard Towards Maxmum Dversfcaton. The Journal of Portfolo Management, Vol. 34, No. 4 (2008), pp Desrosers, S., J.-F. L Her, and J.-F. Plante Style Management n Equty Country Allocaton. Fnancal Analysts Journal, Vol. 60, No. 6 (2004), pp Doeswjk, R., and P. van Vlet Global Tactcal Sector Allocaton: A Quanttatve Approach. Journal of Portfolo Management, Vol. 38, No. 1 (2011), pp Doeswjk, R.Q. The Optmsm Cycle: Sell n May. De Economst, Vol. 156, No. 2 (2008), pp Gergaud, O., and W.T. Zemba Great Investors: Ther Methods, Results, and Evalutaton. Journal of Portfolo Management, Vol. 28, No. 4 (2012), pp Haugen, R.A., and A.J. Hens On the Evdence Supportng the Exstence of Rsk Premums n the Captal Markets. Workng Paper, Wsconsn Unversty, December He, G., and R. Ltterman The Intuton Behnd Black-Ltterman Model Portfolos. Workng Paper, Goldman Sachs Investment Management, Jegadeesh, N. Evdence of Predctable Behavor of Securty Returns. Journal of Fnance, Vol. 45, (1990), pp Krtzman, M. Are Optmzers Error Maxmzers? Journal of Portfolo Management, Vol. 32, No.4 (2006), pp Lehmann, B.N. Fads, martngales, and market effcency. Quarterly Journal of Economcs, Vol. 105 (1990), pp Oglaro, F., R.K. Rce, S. Becker, and R.L. de Carvalho Explct couplng of nformatve pror and lkelhood functons n a Bayesan multvarate framework and applcaton to a new non-orthogonal formulaton of the Black Ltterman model. Journal of Asset Management, vol. 13 (2012), pp Moskowtz, T.J., and M. Grnblatt Do Industres Explan Momentum? Journal of Fnance, Vol. 54, No. 4 (1999), pp Plerou, V., P. Gopkrshnan, B. Rosenow, L. Amaral, T. Guhr, and H. Stanley A Random Matrx Approach to Cross-Correlatons n Fnancal Data. Physcal Revew E, Vol. 65, No. 6 (2002), pp Sherer, B. Portfolo Constructon and Rsk Budgetng, 2 nd ed. Rsk Books, 2004.

27 27 - Mult-Alpha Equty Portfolos: An Integrated Rsk Budgetng Approach for Constraned Robust Portfolos - BNP Parbas Investment Partners May 2013 Thought Leadershp at BNP Parbas Investment Partners At BNP Parbas Investment Partners we see the nvestment world as dynamc and drven by multple agents of change. Perodcally our clents our confronted wth ssues to resolve. Developments wthn the nvestment world mean that new solutons are possble. We am to be ahead of the pack n dentfyng where agents of change are forcng a reconfguraton of the nvestment paradgm. We strve to be an nnovaton leader n developng the approprate strateges and products to enable our clents to meet these challenges. Each year a number of whte papers are publshed by the Thought Leadershp group at BNP Parbas Investment Partners. These whte papers artculate the vews and research of teams from across our nvestment partners on ssues that are shapng nvestment thnkng. Our whte papers seek to set out the ntellectual bass of our approach to nvestng. They underpn much of the ratonal wthn our nvestment processes. Please fnd below access to a number of whte papers recently publshed wthn the framework of our Thought Leadershp group. Whte papers prevously produced by the Thought Leadershp Group at BNP Parbas Investment Partners nclude: If you wsh to receve a copy of a partcular whte paper n ths seres please send an emal wth your name, full postal address and occupaton to: contact-thoughtleadershp@bnpparbas.com

28 BNP Parbas Investment Partners s the source for all data n ths document as at end Aprl 2013, unless otherwse specfed. 1. an offer to buy nor a solctaton to sell, nor shall t form the bass of or be reled upon n connecton wth any contract or commtment whatsoever 2. any nvestment advce Opnons ncluded n ths materal consttute the judgment of BNPP AM at the tme specfed and may be subject to change wthout notce. BNPP AM s not oblged to update or alter the nformaton or opnons contaned wthn ths materal. Investors should consult n order to make an ndependent determnaton of the sutablty and consequences of an nvestment theren, f permtted. Please note that dfferent types of nvestments, f contaned wthn ths materal, nvolve varyng degrees of rsk and there can be no assurance that any specfc nvestment may ether be sutable, approprate or proftable for a clent or prospectve clent s nvestment portfolo. Gven the economc and market rsks, there can be no assurance that any nvestment strategy or strateges mentoned heren wll acheve ts/ther nvestment objectves. Returns may be affected by, amongst other thngs, nvestment strateges or objectves of the condtons. The dfferent strateges appled to the Fnancal Instruments may have a sgnfcant effect on the results portrayed n ths materal. The value of an nvestment account may declne as well as rse. Investors may not get back the amount they orgnally nvested. The performance data, as applcable, reflected n ths materal, do not take nto account the commssons, costs ncurred on the ssue and redempton and taxes. Pars asset management enttes wthn BNP Parbas Investment Partners f specfed heren, are specfed for nformaton only and do not necessarly carry on busness n your jursdcton. For further nformaton, please contact your locally lcensed Investment Partner. 14, rue Bergère Pars - France MAY Desgn: P