Diversification has been at the

Transcription

1 YVES CHOUEIFATY s the head of Quanttatve Asset Management, Europe, at Lehman Brothers n Pars, France. yves.chouefaty@gmal.com YVES COIGNARD s the co-deputy head of Quanttatve Asset Management, Europe, at Lehman Brothers n Pars, France. yves.cognard@gmal.com Toward Maxmum Dversfcaton YVES CHOUEIFATY AND YVES COIGNARD Dversfcaton has been at the center of fnance for over 50 years. And to paraphrase Markowtz [1952], dversfcaton s the only free lunch n fnance. Much effort has gone nto developng modern portfolo theory wthn the Markowtz mean-varance framework. Perhaps foremost amongst those efforts s the captal asset prcng model (CAPM) developed by Sharpe [1964]. Whle brllant n ts smplcty and clarty, years of examnaton have led to a vgorous debate about whether the assumptons upon whch the model depends reflect real market condtons and whether ts concluson can be transposed to actual portfolo management. A separate set of arguments concerns the dynamc aspects of portfolo constructon. Accountng for dynamc changes n the portfolo has led to an examnaton of these dynamc changes as a source of return. Fernholz and Shay [1982] stated that constant-proporton portfolos earned addtonal returns over the returns earned by buy-and-hold portfolos. Booth and Fama [1992] descrbed these addtonal returns as dversfcaton returns. Although they provde useful nsghts, the dfferent examnatons of multperod rebalancng effects are not drectly useful for portfolo constructon because they only deal wth portfolo dynamcs after the orgnal weghts have been decded. Although very useful n descrbng and understandng portfolo constructon ssues, the mean-varance framework has some practcal problems. For example, whle varance can be estmated wth a far level of confdence, returns are so much more dffcult to estmate that most popular models, such as CAPM and Black-Ltterman, have n one way or another completely put them asde. It s now ncreasngly popular to clam that the market captalzaton weghted ndces are not effcent. Several alternatve emprcal solutons have been suggested, such as fundamental ndexaton and equal weghts. In ths artcle, we nvestgate the theoretcal and emprcal propertes of dversfcaton as a crteron n portfolo constructon. We compare the behavor of the resultng portfolo to common, smple strateges, such as market cap weghted ndces, mnmum-varance portfolos, and equal-weght portfolos. DEFINITION OF THE DIVERSIFICATION RATIO AND MOST-DIVERSIFIED PORTFOLIO We begn by mathematcally defnng how we measure the dversfcaton of a portfolo. Let X 1, X 2,, X N be the rsky assets of unverse U. For smplfcaton, we wll consder X to be stocks. Let V be the covarance matrx of these assets and C the correlaton matrx. IT IS ILLEGAL TO REPRODUCE THIS ARTICLE IN ANY FORMAT 40 TOWARD MAXIMUM DIVERSIFICATION FALL 2008

2 1 2 Let Σ=. be the vector of asset volatltes.. N Any portfolo P wll be noted P = (w p1, w p2,, N w pn ), wth Σ w p = 1 = 1. We defne the dversfcaton rato of any portfolo P, denoted D(P), as the followng: The dversfcaton rato s the rato of the weghted average of volatltes dvded by the portfolo volatlty. Let Γ be a set of lnear constrants appled to the weghts of portfolo P. One usual set of constrants s the long-only constrant (.e., all weghts must be postve). The portfolo, whch under the set of constrants Γ maxmzes the dversfcaton rato n unverse U, s the Most- Dversfed Portfolo, denoted as M(Γ, U). An ntutve understandng of the way dversfcaton works n portfolo constructon can be ganed from the followng two examples. Example 1 Suppose we have an nvestment unverse of two stocks, A and B, wth a correlaton strctly lower than 1, and wth respectve volatltes of 15% and 30%. In ths case, dversfcaton means that we want both stocks to equally contrbute to portfolo volatlty. Ther respectve weghts n the Most-Dversfed Portfolo would thus be 66.6% for stock A and 33.3% for stock B (nversely proportonal to volatlty). Example 2 DP ( )= P Σ PVP Suppose we have an nvestment unverse of three stocks. Let us assume that two are bankng stocks wth a hgh correlaton of 0.9 and that the thrd stock, a pharmaceutcal stock, has a correlaton of 0.1 wth each of the two bankng stocks. Suppose, for smplcty, that the volatltes are all equal. The weghts of the Most-Dversfed Portfolo, accordng to the result just obtaned, are 25.7% (1) for each of the bankng stocks and 48.6% for the pharmaceutcal stock. THEORETICAL RESULTS The dversfcaton rato of any long-only portfolo wll be strctly hgher than 1 except when the portfolo s equvalent to a mono-asset portfolo, n whch case the dversfcaton rato wll be equal to 1. If the expected excess returns of assets are proportonal to ther rsks (volatltes), then ER(P) = kp Σ, where k s a constant, and maxmzng D(P) s equvalent to maxmzng, whch s the Sharpe rato of the portfolo. ER( P ) PVP In ths case, the Most-Dversfed Portfolo s also the tangency portfolo. To provde a better understandng of ths rato, and also to smplfy the math, we transpose the problem to a synthetc unverse n whch all the stocks have the same expected volatlty. Suppose that nvestors can lend and borrow cash at the same rate. We can then defne the synthetc assets Y 1, Y 2,, Y N by Y where $ s the rsk-free asset. We now have the unverse, U S, of the followng assets Y 1, Y 2,, Y N. In ths unverse, the volatlty S of Y s equal to 1, and S Σ X 1 = + 1 $ 1 1 Σ S =.. 1 S In DS ( )=, S s a portfolo composed of the SV SS synthetc assets, and V S s the covarance matrx of the synthetc assets. If we have S Σ S = 1, then maxmzng D(S) s equvalent to maxmzng 1 under constrants SV S S Γ S. Because all Y have a normalzed volatlty of 1 and because correlaton does not change wth leverage, V S s equal to the correlaton matrx C of our ntal assets, so FALL 2008 THE JOURNAL OF PORTFOLIO MANAGEMENT 41

3 that maxmzng the dversfcaton rato s equvalent to mnmzng Thus, n a unverse n whch all stocks have the same volatlty, we mnmze the varance, whch s ndeed the beneft we expect from dversfcaton. When buldng a real portfolo, we need to reconstruct synthetc assets by holdng real assets plus (or mnus) some cash. If S = (w S1, w S2,, w SN ) denotes the optmal weghts for the synthetc assets, then the optmal portfolo M of real assets wll be PROPERTIES M w w w N w S1 S2 SN S =,,...,, 1 $ = 1 2 N 1 If C s nvertble and Γ = Ø, then S = M(Γ S,U S ) s unque, and we have the followng analytcal results: The synthetc asset weghts, S, are proportonal to the nverse of the correlaton matrx C tmes 1, a vector of ones the same sze as the number of assets. Once agan, we can transform the synthetc assets back to the portfolo of orgnal assets by dvdng each synthetc portfolo weght by the volatlty of that asset and rescalng the portfolo to be 100% nvested. If we denote the vector of weghts of the orgnal assets as M, then we can wrte M S SCS C C 1 1 where s a dagonal matrx of the asset volatltes. Now, consder the propertes of asset correlaton n the context of the Most-Dversfed Portfolo. In a smlar manner, we can calculate the correlaton of an arbtrary portfolo P wth the Most-Dversfed Portfolo M. Because M s nversely proportonal to and C, we can wrte (2) (3) (4) M = κ 1 C 1 1 where κ s a constant factor. Thus, P CM P Cκ C ρ = = PM, P M P M w κ κ = = DP ( ) P Ths means that the correlaton of portfolo P wth the Most-Dversfed Portfolo M s proportonal to the dversfcaton rato of portfolo P, namely D(P). Now, consder the correlatons between sngle stocks and the Most-Dversfed Portfolo. The dversfcaton rato of a sngle stock s 1, because there s no dversfcaton. Usng Formula (5), we calculate the correlaton of asset, whch has a weght vector w, whose th asset s weght s 1 and other weghts are 0, wth the vector of the Most-Dversfed Portfolo holdngs M. We obtan ρ Remarkably, the correlaton of asset wth the Most- Dversfed Portfolo s the same for every one of the assets. Thus, we can dentfy the Most-Dversfed Portfolo as beng the one n whch all assets have the same postve correlaton to t. The specal case that P s the Most-Dversfed Portfolo M n Formula (5) leads us to the result that M κ = DM ( ) so that we have dentfed the constant κ. Thus, we can rewrte the correlaton between a general portfolo P and the Most-Dversfed Portfolo M as the rato of ther dversfcaton ratos, as follows: ρ PM, M = M, κ M ( P) = D DM ( ) M (5) (6) (7) (8) 42 TOWARD MAXIMUM DIVERSIFICATION FALL 2008

4 wth ths nformaton we can obtan a sngle (dversfcaton) factor model whch resembles the CAPM n form, but now dentfes the correlaton as the rato of the dversfcaton levels, R where R represents an excess return over cash, and α P and ε P are the constant and error terms, respectvely, normally assocated wth regresson. Long-Only Portfolos In the real world, Γ s usually not empty, and ncludes the constrant of havng postve weghts. As such, the propertes we descrbed for the unconstraned problem are stll true for the subunverse of securtes that s composed of the stocks selected by the constraned Most- Dversfed Portfolo. The subsequent results focus on the long-only Most- Dversfed Portfolo. Two consequences of ths are that 1) the postvty constrant wll reduce the potental mpact of estmaton errors, and 2) beng long-only ensures that the portfolo wll have a postve exposure to the equty rsk premum. Thus, all non-zero-weghted assets have the dentcal correlaton to the Most-Dversfed Portfolo. Zeroweghted assets, excluded from the Most-Dversfed Portfolo n the optmzaton, have correlatons to the Most-Dversfed Portfolo that are hgher than the nonzero-weghted assets n the Most-Dversfed Portfolo. Ths s consstent wth dentfyng the Most-Dversfed Portfolo subject to the constrants appled. Other Propertes P P DP ( ) = α + DM R + ε P M P ( ) M If all of the stocks n the unverse have the same volatlty, then the Most-Dversfed Portfolo s equal to the global mnmum-varance portfolo. Furthermore, f we contnuously rebalance the Most-Dversfed Portfolo, and because t s a market cap ndependent methodology, the Most-Dversfed Portfolo should get a sgnfcant part of the benefts from dversfcaton returns when compared to a pure buy-and-hold strategy (Booth and Fama [1992]). (9) EMPIRICAL RESULTS In ths secton, we explan the methodology used n our analyss and the results for Eurozone and U.S. equtes. Addtonally, we dscuss the bases n the methodology and present an analyss of the performance results. We conclude the secton wth a revew of the ssues relatng to stock selecton and the unqueness of the optmal portfolo. Methodology A number of steps need to be addressed before testng the Most-Dversfed Portfolo. Frst, a unverse of assets must be selected, and the returns data for these assets must be collected to cover at least a full market cycle. Care should be taken to establsh that the data are accurate, partcularly n regard to splts, dvdends, and, most sgnfcantly, survvorshp bas. Gven clean returns data, the covarance matrx must next be estmated. Because ths s the full nformaton set used to construct the portfolo, t s mportant to examne the mpact of estmaton errors on the resultng portfolo. A varety of ways exsts to estmate covarances, such as smple wndows, decayed weghtng, GARCH, and Bayesan update methodologes. Although estmaton errors often occur at the levels of volatlty and correlaton, the herarches of correlatons are more stable. Indeed, we fnd that portfolos bult usng dfferently estmated covarance matrces have smlar characterstcs. Changng the frequency of data and the estmaton perod has lttle mpact on the fnal results. Even portfolos bult on forward-lookng covarance matrces (havng perfect covarance foresght) have only slghtly dfferent results than when usng hstorcal covarances. We must also be aware that optmzers tend to allocate more rsk to factors whose volatlty has been underestmated (see Mchaud [1998]). Ths s especally true for long short portfolos bult from very large unverses where multcollnearty s lkely. A smple way to address ths ssue s to add postvty constrants to the optmzaton program. In the case of multcollnearty, the fact that the optmal portfolo mght not be unque s not a real problem for the portfolo manager t just provdes more choce, as n choosng between equvalent longonly portfolos. It s possble to further lmt the mpact of estmaton errors by addng upper weght lmts to the program. FALL 2008 THE JOURNAL OF PORTFOLIO MANAGEMENT 43

5 For purposes of comparson, we maxmze the dversfcaton rato, defned n Equaton (1), at every month-end for dfferent unverses of securtes. We compare the results for the long-only Most-Dversfed Portfolo wth the market cap weghted benchmark, mnmum-varance portfolo, and equal-weght portfolo. We analyze two dfferent regonal equty markets, the U.S. and the Eurozone. We use Standard & Poor s (S&P) 500 Index data from December 1990 to February 2008 as the daly performance seres for U.S. equtes, and the Dow Jones (DJ) Euro Stoxx Large Cap Index data from December 1990 to February 2008 as the daly performance seres for Eurozone equtes. The covarance matrx s computed usng 250 days of daly returns. The startng date for the emprcal test s December For computatonal reasons, we exclude from the unverse, for month-end computatons, all stocks havng less than a 250-day prce hstory. Because the portfolos are long-only, all weghts must be postve. We also lmt the contrbuton to rsk to 4% per asset. In order to conform to an asset managers framework, we also constran the month-end weghts to comply wth UCITS III rules (.e., the maxmum weght per securty s 10%, and the sum of weghts above 5% must be lower than 40%). Results for Eurozone and U.S. Equtes The results of the emprcal tests for the Most-Dversfed Portfolo are summarzed n Exhbts 1 and 2. The Most-Dversfed Portfolo consstently delvers superor rsk-adjusted returns n both regons. As expected, t s consstently less rsky than the market cap weghted ndces (.e., volatlty s 13.9% versus 17.9% for Eurozone equtes, and 12.7% versus 13.4% for U.S. equtes). The Most-Dversfed Portfolo shows a hgher Sharpe rato than the market cap weghted benchmark, mnmum-varance portfolo, and equal-weght portfolo over the entre perod. In order to further analyze the behavor durng dfferent market condtons, we splt the backtest results nto two subperods: Subperod to 2000 (.e., the end of the dot-com bubble) Subperod to 2008 Bases and Analyss of Performance It s clear that all market cap ndependent methodologes tend to be less based toward large captalzatons than market cap weghted ndces. Therefore, a comparson of the Most-Dversfed Portfolo to market captalzaton weghted ndces should always show a sze bas. Other bases (relatve to market captalzaton weghted ndces) can appear, even as the nverse of the ndex s bas. To measure the mportance of factor bas n the emprcal results, we performed a three-factor Fama French [1993, 1996] regresson of the performance of the portfolos versus the market, HML (hgh-mnus-low book value), and SMB (small-mnus-bg captalzaton) factors. The results are shown n Exhbts 3 and 4. Exhbt 3 shows that for the full perod the ntercept s sgnfcantly postve for the most-dversfed Eurozone portfolo, wth an annualzed excess return (ntercept) of 6.0% and a t-stat of These fgures compare to 5.1% and 3.51, respectvely, for the mnmum-varance portfolo, and 0.6% and 1.16, respectvely, for the equalweght portfolo. The herarchy of results s confrmed over the two subperods. Exhbt 4 shows results of the same nature for the most-dversfed U.S. portfolo, wth an annualzed excess return (ntercept) of 3.1% and a t-stat of These fgures compare to 2.2% and 1.40, respectvely, for the mnmum-varance portfolo, and 1.2% and 2.27, respectvely, for the equal-weght portfolo. More broadly, we analyzed the actve returns of the most-dversfed Eurozone portfolo wth the Lehman Brothers Equty Rsk Analyss (ERA) factor model for the perod (.e., the perod of avalablty for the factor model). The results, shown n Exhbt 5, ndcate that the domnant factor explanng the outperformance over the perod s specfc rsk, meanng that about 18% (out of 48%) of the outperformance cannot be explaned by the predefned factors of the model. Ths performance arses from real stockspecfc rsk, omtted common rsk factors, and changes n exposure to factors. Stock Selecton Issues and Unqueness of the Optmal Portfolo If the correlaton matrx s not nvertble, the soluton may not be unque. But because all possble portfolos brng maxmum dversfcaton, we are ndfferent to the soluton. We see that, n emprcal tests, runnng the Most-Dversfed Portfolo model on three subunverses (obtaned by randomly excludng one-thrd of the unverse each tme) gves very smlar results. 44 TOWARD MAXIMUM DIVERSIFICATION FALL 2008

6 E XHIBIT 1 Comparson of Eurozone Equty Portfolos, Note: MDP s the Most-Dversfed Portfolo. B represents the market cap weghted benchmark, the Dow Jones EuroStoxx Large Cap Total Return Index. MV represents a mnmum-varance portfolo, and EW represents an equally weghted portfolo, both on the same unverse as the benchmark B. The chart shows total return ndces. Exhbt 6 shows the performance of the Most- Dversfed Eurozone Portfolo and ts three subsets versus ts benchmark. The same test on the U.S. unverse produces smlar results. These results show that the Most- Dversfed Portfolos tend to allocate rsk to rsk factors much more than to specfc stocks or sectors, even though the average number of stocks n a Most-Dversfed Portfolo s relatvely low (between 30 and 60). Dversfcaton Rato Exhbt 7 shows the changes n dversfcaton ratos through tme for the Most-Dversfed Eurozone Portfolo and the Eurozone benchmark. We can see that although the levels of dversfcaton vary through tme as a result of the changes n the levels of correlaton, the FALL 2008 THE JOURNAL OF PORTFOLIO MANAGEMENT 45

7 E XHIBIT 2 Comparson of U.S. Equty Portfolos, Note: MDP s the Most-Dversfed Portfolo. B represents the market cap weghted benchmark, the Standard & Poor s (S&P) 500 Total Return Index. MV represents a mnmum-varance portfolo, and EW represents an equally weghted portfolo, both on the same unverse as the benchmark B. The exhbt shows total return ndces. Most-Dversfed Portfolo s always about 1.5 tmes as dversfed as the benchmark. CONDITIONS FOR OPTIMALITY Let us consder a world n whch nvestors can borrow and lend money at the same rsk-free rate. We wll assume that the nvestor s objectve utlty functon s to maxmze the Sharpe rato of ther portfolo of rsky assets, before leveragng or deleveragng t wth cash. What knd of expected returns would mply the optmalty (n terms of Sharpe rato) of the dfferent strateges? 46 TOWARD MAXIMUM DIVERSIFICATION FALL 2008

8 E XHIBIT 3 Fama-French Monthly Regresson Coeffcents, Eurozone Equtes, Monthly data are used. MKT s the benchmark s excess return over one-month LIBOR EUR; HML s the dfference n monthly performance between Dow Jones Euro Stoxx Large Cap Value and Growth Indces; SMB s the dfference n monthly performance between the smallest 30% and the bggest 30% of stocks n the ndex (n terms of weghts); and Intercept s a monthly excess return. Most-Dversfed Portfolo Recall that the Most-Dversfed Portfolo s optmal f the stocks expected returns are proportonal to ther volatltes; that s, E(R ) = k, where k s a constant factor and s the volatlty of stock. Market Cap Weghted Benchmark Consderng the CAPM assumptons, a market cap weghted benchmark wll be optmal, and we can state ER ( ) = βer ( ) = ρ ER ( ) B, B B (10) where E(R ) s the expected excess return of stock, E(R B ) s the expected excess return of the benchmark (a proxy for the market portfolo), and ρ,b s the correlaton between stock and the benchmark. For smplfcaton we assume the rsk-free rate s 0%. If we consder E(R B ) and B to be gven for the perod consdered, we have B E(R ) = Κρ,B (11) FALL 2008 THE JOURNAL OF PORTFOLIO MANAGEMENT 47

9 E XHIBIT 4 Fama-French Monthly Regresson Coeffcents, U.S. Equtes, Monthly data are used. MKT s the benchmark s excess return over one-month LIBOR USD; HML s the dfference n monthly performance between the S&P 500 Value and Growth Indces; SMB s the dfference n monthly performance between the smallest 30% and the bggest 30% of stocks n the ndex (n terms of weghts); and Intercept s a monthly excess return. where K s a constant. In other words, the stocks expected returns that are mpled by the optmalty of the market cap weghted benchmark are proportonal to ther total rsk (volatlty) and ther correlaton to the benchmark. Mnmum-Varance Portfolo In ths case, the expected returns that make the mnmum-varance portfolo optmal are equal for all assets, E(R ) = Κ (12) Economc Interpretaton of the Most- Dversfed Portfolo What assumptons would explan that the Most- Dversfed Portfolo s better (n terms of Sharpe rato) than the market cap weghted benchmark? We can see that, although the Most-Dversfed Portfolo s ultmately very dfferent from the benchmark, the mpled expected returns from the Most-Dversfed Portfolo are not very dfferent from those of the market cap weghted benchmark. Actually, the only dfference resdes n the correct prcng of ndvdual assets correlatons to the benchmark. In other words, we need correlatons (to the 48 TOWARD MAXIMUM DIVERSIFICATION FALL 2008

10 E XHIBIT 5 Actve Returns Factor Attrbuton for Eurozone Most-Dversfed Portfolo, Aprl 1999 February 2008 Note: The factor model used s the Lehman Brothers Equty Rsk Analyss (ERA) model. The returns are computed by cumulatng monthly actve returns, whch s a dfferent process from takng the dfference between cumulatve portfolo retrurns and cumulatve benchmark returns. market) to be only partally taken nto account by the market when securtes prces are determned. Why would ths be the case n the real world? The followng assumptons are consstent wth a market envronment that could explan the domnance of the Most-Dversfed Portfolo over market-cap ndces: Investors are ratonal (.e., all else beng equal, f a securty has a hgher volatlty, nvestors expect a hgher return). The market has enough effcency to prevent arbtrage opportuntes at the sngle-stock level (.e., securty prces reflect all publc nformaton; n other words, securtes are correctly prced on a stand-alone bass). Forecasts of volatltes are accurate. All other forecasts are ether naccurate or not taken nto account n the prcng of securtes. FALL 2008 THE JOURNAL OF PORTFOLIO MANAGEMENT 49

11 E XHIBIT 6 Emprcal Performance of Eurozone Most-Dversfed Portfolos, Full Unverse and Three Subsets, E XHIBIT 7 Comparson of Eurozone Most-Dversfed Portfolo and Benchmark Dversfcaton Ratos, TOWARD MAXIMUM DIVERSIFICATION FALL 2008

12 These assumptons alone, of course, are not enough for the Most-Dversfed Portfolo to be an equlbrum model. CONCLUSION In ths artcle, we provde a mathematcal defnton of dversfcaton and descrbe several mplcatons of dversfcaton as a goal. Most-Dversfed Portfolos have hgher Sharpe ratos than the market cap weghted ndces and have had both lower volatltes and hgher returns n the long run, whch can be nterpreted as capturng a bgger part of the rsk premum. Emprcal results tend to confrm the value of a theoretcal framework for dversfcaton. It s dffcult to determne f a portfolo was ex ante on the effcent fronter, but evdence tends to ndcate that the Most-Dversfed Portfolo s more effcent ex post than the market cap weghted benchmark, mnmum-varance portfolo, and equal-weght portfolo. Because the hypotheses n our analyss are not specfc to the equty market, the Most-Dversfed-Portfolo methodology can be adapted to other asset classes. And the dversfcaton rato can be vewed as a new measure of rsk that, when combned wth the performance of the Most-Dversfed Portfolo, has explanatory power for the performance of any portfolo wthn the same unverse of securtes. The goal of the Most-Dversfed Portfolo s not to be an equlbrum model. It can, however, potentally be transformed nto an equlbrum model ether by addng addtonal assumptons or by addng fundamental valuaton crtera, such as earnngs, sales, and so forth. Such addtons would allow the model to accommodate dfferent msprcngs. We have defned a portfolo constructon methodology that can be consdered an alternatve to other nonmarket-cap benchmarks (see, for example, Fernholz and Shay [1982] and Arnott, Hsu, and Moore [2005]), and, as such, s a new nvestment style that favors dversfcaton and avods bets based on return predcton or confdence n the mplct bets of market cap weghted benchmarks. In partcular, the authors would lke to thank Mchael Gran, Mchael E. Mura, Ayaaz Allymun, Davd Bellache, Trstan Frodure, Ynyan Huang, Ncolas Mejr, Nadejda Rakovska, Gullaume Toson and Dens Zhang for ther valued contrbutons to ths artcle. REFERENCES Arnott, R., J. Hsu, and P. Moore. Fundamental Indexaton. Fnancal Analysts Journal, Vol. 61, No. 2 (2005), pp Booth, D., and E. Fama. Dversfcaton and Asset Contrbutons. Fnancal Analysts Journal, Vol. 48, No. 3 (1992), pp Fama, E., and K. French. Common Rsk Factors n the Returns on Stocks and Bonds. Journal of Fnancal Economcs, 33 (February 1993), pp Multfactor Explanatons of Asset Prcng Anomales. Journal of Fnance, 51 (March 1996), pp Fernholz, R., and B. Shay. Stochastc Portfolo Theory and Stock Market Equlbrum. Journal of Fnance, Vol. 37, No. 2 (1982), pp Markowtz, H. Portfolo Selecton. Journal of Fnance, Vol. 7, No. 1 (1952), pp Mchaud, R. Effcent Asset Management: A Practcal Gude to Stock Portfolo Optmzaton and Asset Allocaton. Cambrdge, MA: Harvard Busness School Press, Sharpe, W. Captal Asset Prces: A Theory of Market Equlbrum under Condtons of Rsk. Journal of Fnance, Vol. 19, No. 3 (1964), pp Treynor, J. Why Market-Valuaton-Indfferent Indexng Works. Fnancal Analysts Journal, Vol. 61, No. 5 (2005), pp To order reprnts of ths artcle, please contact Dewey Palmer at dpalmer@journals.com or ENDNOTE The authors thank all the members of the European Quanttatve Team of the Lehman Brothers Investment Management Dvson for ther contrbutons and helpful comments. FALL 2008 THE JOURNAL OF PORTFOLIO MANAGEMENT 51