Efficient Replication of Factor Returns

Transcription

1 Efficient Replication of Factor Returns To appear in te Journal of Portfolio Management June 009 Dimitris Melas Ragu Suryanarayanan Stefano Cavaglia 009 MSCI Barra. All rigts reserved. of 9 Electronic copy available at: ttp://ssrn.com/abstract=43765

2 Summary e present alternative metods for constructing factor-mimicking portfolios in practice. e illustrate ow portfolios wit a limited number of assets and relatively low turnover can be used to track pure factor returns. Tese portfolios provide an effective instrument to support te practice of investment management. e illustrate ow tey can be used to edge out unintended factor exposures of a passive bencmark, tus facilitating te optimal management of beta exposure. e illustrate ow tey can be used to edge out unintended factor exposures of an active strategy, tus isolating pure alpa and facilitating te management of alternative sources of alpa. Introduction Te empirical evidence of Cen, Roll, and Ross (986) and Fama and Frenc (993) suggests tat asset dynamics are caracterized best by a multifactor representation of asset returns. As in Sarpe (963), beta now captures multiple sources of systematic risk; exposures to eac systematic risk are compensated by teir respective risk premia. Passive investment management essentially aims to optimize exposures to various sources of beta. Active investment management aims to optimize sources of alpa tat are unrelated to systematic beta risks. In eiter context, investors can capture te premium or edge te risk associated wit a particular beta factor troug factor-mimicking portfolios. Tese portfolios ave unit exposure to a target factor, zero exposure to oter factors, and minimum portfolio risk. Te ability to transact in factor-mimicking portfolios may tus provide an efficient enancement to te practice of investment management. In tis paper, we examine different metods for constructing factor-mimicking portfolios. In particular, given a factor-based model of risk and return, we consider two types of factor replication. Full replication uses te underlying information embedded in te model to reengineer te targeted factor return, wile optimized replication uses constrained mean-variance optimization wit appropriately cosen expected asset returns and asset covariance matrix. In addition to imposing factor exposure constraints, in order to make factor portfolios easier to implement in practice, we examine metods tat impose limits on turnover, asset oldings, and te number of assets in te portfolio. Tis analysis is applied in te context of te Barra Global Equity Model (GEM). e present evidence tat te momentum and value factors can be replicated efficiently. Our tracking portfolios capture te risk and return properties of tese factors for te period analyzed (Jan 998 Jun 008) as well as periods of extreme market turbulence. Portfolios tat replicate factor returns efficiently can be utilized potentially to enance te risk and return profile of passive and active investment strategies, as illustrated by te evidence we present. e sow tat a manager wo ad attempted to track te MSCI orld Small Cap Index could ave enanced te risk-return profile of tis passive strategy by edging te factor exposures unrelated to size over te period 3 Dec 003 to 3 Dec 008. e also consider a perfect-foresigt active stock-picking strategy tat is long te top-50 performing US stocks and sort te bottom-50 performing US stocks. In tis context, factor-mimicking portfolios can be utilized to edge out te unintended factor exposures, leaving te manager wit pure stockselection alpa. Using te Barra USE3 model, we illustrate ow te risk and return profile of tis active investment strategy could ave been enanced by neutralizing some of te large factor exposures for te period 3 Dec 997 to 30 Jun 008. Tis paper is tus presented in two sections. In section one, we provide analytical considerations in te construction of factor-mimicking portfolios, along wit empirical evidence. In section two, we illustrate passive and active management applications of factor-mimicking portfolios. 009 MSCI Barra. All rigts reserved. of 9 Electronic copy available at: ttp://ssrn.com/abstract=43765

3 . Factor-Mimicking Portfolios Analytics Full Replication e consider te problem of constructing factor-mimicking portfolios corresponding to te factors of a fundamental multifactor model described by te following equations : r f e () V F D () Tese equations decompose return and risk into a systematic component (respectively f and F ) and a specific component (respectively e and D). Factor models often are estimated troug weigted cross-sectional regression. In tis regression, observations typically are weigted by te square root of market cap or by te inverse of specific volatility. Tis is done to ensure tat te estimated factor returns are not unduly influenced by very small or very volatile assets 3. In tis case, we can compute te weigts of factor-mimicking portfolios directly from te factor return estimation regression 4 below, were represents te weigt applied in te OLS estimation: r f e f ( ) r (3) Te last equation computes factor returns f as a weigted average of asset returns r. In tis equation, te rows of matrix ( ) - correspond to te weigts of te factor-mimicking portfolios. Te main advantage of tis metod is tat te resulting portfolios replicate exactly te factor returns estimated by te multifactor model. However, an important drawback of tis metod is tat it does not necessarily lead to factor-mimicking portfolios wit minimum ex-ante risk; in practice, tis is an important consideration, as te error term may represent systematic risks tat are not captured by te empirical factor model. Furtermore, te resulting portfolios ave long or sort positions in all te assets in te estimation universe, and tese positions can cange significantly from one period to te next. Terefore, tese teoretical portfolios tat replicate exactly te factor returns from multifactor models may be difficult or costly to implement in practice. Optimized Replication e can specify te factor-mimicking portfolio construction problem in general terms as follows. Given a factor model, we would like to construct portfolios tat ave maximum exposure to a target factor, zero exposure to all oter factors, and minimum portfolio risk. e can express tis problem as a general mean-variance optimization problem: max V (4) s. t. 0 (5) Here, and represent exposures to te target factor and to all oter factors. Tis constrained optimization problem can be solved analytically using te metod of Lagrange multipliers. Optimal portfolio weigts are given by te following expression: V V V * (6) 009 MSCI Barra. All rigts reserved. 3 of 9 Electronic copy available at: ttp://ssrn.com/abstract=43765

4 One interesting observation ere is tat if we substitute V - in expression (6) wit te weigts used in te factor return estimation regression, ten te factor portfolios we obtain troug expression (6) are exactly te same as te full replication portfolios obtained troug expression (3) 5. Terefore, te full replication metod can be viewed as a special case of te general optimization framework (4)-(6). Anoter interesting observation is tat, as we maximize exposure to a target factor and constrain exposure to all oter risk factors, we are effectively minimizing specific risk; terefore, we could substitute te total risk matrix V wit te specific risk matrix D in expression (4). In te Appendix, we sow analytically tat using specific risk instead of total risk in te optimization leads to te same optimal portfolios, up to a scaling parameter. In general, optimized replication leads to portfolios tat replicate closely (but not exactly) te factor return estimated by te model. In te special case were te inverse of specific variances were used as weigts in te model cross-sectional regression, te optimized replication metod captures exactly te factor return estimated by te model and leads to te same factor-mimicking portfolios as te full replication metod. Adding Investability Constraints Full replication and optimized replication portfolios ave long or sort positions in all te assets in te underlying model estimation universe; liquidity considerations or te availability of sorts may impact te ability to implement suc a strategy in practice. In addition, positions can cange significantly from one period to te next, possibly resulting in ig portfolio turnover and significant transactions costs. To make factor portfolios easier to implement and manage, investors may wis to impose additional constraints on tese portfolios. For example, internal risk management controls or regulatory requirements may impose limits on te leverage of te long/sort factor-replicating portfolio 6. Similar institutional requirements may impose constraints on te amount of capital tat can be allocated to a single asset or a group of assets 7 in te portfolio. Also, low liquidity, limited borrowing availability, increased operational complexity, and transaction cost considerations 8 make it cumbersome to manage portfolios tat require long or sort positions across many assets, especially medium- and small-capitalization assets. As a result, investors may wis to impose constraints on te turnover, maximum asset weigt, and number of assets in te factorreplicating portfolio. Our general optimization framework for constructing factor-mimicking portfolios can be extended to include different types of constraints and different bencmarks. Constructing Factor Portfolios Using Active Risk Optimization So far, we expressed te factor replication problem as a total-risk optimization problem: max V (7) However, in certain applications, minimizing tracking error relative to te target factor return is more important tan minimizing te total risk of te factor portfolio. In tis case, using F to represent te full replication portfolio weigts, we can express te factor replication problem as an active-risk optimization problem: max ( F ) V ( F ) (8) Tese two optimization approaces for constructing factor portfolios (total-risk optimization, active-risk optimization) may be appropriate for different applications. In general, investors can use factor portfolios to capture systematic returns or to edge common factor risks. In te former application, investors wising to capture risk premia may place more empasis on minimizing te 009 MSCI Barra. All rigts reserved. 4 of 9

5 total risk of te factor portfolio, in wic case te total-risk optimization approac would be more appropriate. On te oter and, for investors using factor portfolios to edge teir exposure to common factor risk, te active-risk approac may be more appropriate, as it ensures closer tracking and terefore better edging of te targeted factor. Empirical Results e use data from te Barra Global Equity Model (GEM) to analyze five different factor replication metods. First, we compare unconstrained full-replication and optimized-replication portfolios. Ten, we examine tree constrained-replication metods, wit constraints on turnover and on te maximum number of assets in te replicating portfolio. e test tese five replication metods over te period Jan 998 to Jun 008. At te beginning of eac mont, we form long/sort portfolios targeting te value factor and te momentum factor, respectively. Te weigts of te full-replication and optimized-replication portfolios are computed using te analytical formulas presented in te full-replication and optimized-replication sections. Te weigts of te oter replication portfolios are computed numerically 9. Te exposures and variance-covariance forecasts tat are used to construct te factor portfolios are given at te beginning of eac mont and are based solely on data available at te beginning of eac mont. Exibit Exibit presents te performance of te five replicating portfolios. Total-risk metods acieved lower predicted, as well as realized, portfolio risk, leading to a iger realized Sarpe ratio. For example, value portfolios based on total-risk optimization ad realized volatility of.97% and.98%, compared wit realized volatility of.4% and.45% for value portfolios based on activerisk optimization. On te oter and, active-risk metods experienced lower realized tracking error compared to total-risk metods. For example, momentum portfolios based on total-risk 009 MSCI Barra. All rigts reserved. 5 of 9

6 optimization experienced tracking error of.53% and.6%, compared wit tracking error of 0.86% and.07% for momentum portfolios based on active-risk optimization. Momentum portfolios were generally more volatile and ad iger tracking error and iger turnover compared to value portfolios. Specifically, momentum portfolios ad average montly one-way turnover of around 4%, compared to around 5% for value portfolios based on te same metods. Noneteless, even wen we allow for a very conservative transaction cost estimate of 5 bps, te replicating portfolios wit constrained turnover broadly succeed in capturing te riskadjusted performance (net of costs) of te target factor. Toug te first two sample moments of te target factor appear to be captured by our factormimicking portfolios, it is also important to examine tracking performance during periods of ig volatility or extreme market conditions. Tis is relevant particularly wen tese portfolios are utilized as edging instruments. Exibit reports te performance of te five replication metods during periods of extreme market conditions. Tese results igligt tat te edging portfolios, especially tose based on active risk optimization, track factor returns relatively well even during te observed periods of extreme market turmoil. 0 Exibit Factor portfolio leverage (gross exposure to net asset value) ranged between 89.7% and.0% for te value portfolios, and between 87.4% and 4.8% for te momentum portfolios. Tus, during te observed period, tese factor-mimicking portfolios did not require ig leverage in teir implementation. 009 MSCI Barra. All rigts reserved. 6 of 9

7 . Factor-Mimicking Portfolios Applications Passive Investment Strategies A simple way to capture factor returns would be to build a portfolio wit stocks tat ave ig exposure to a particular factor. For example, a value index, suc as te MSCI orld Value Index, tat contains only stocks tat ave been screened on different valuation ratios could be used as a proxy to capture te value risk premium. Also, a small cap index, suc as te MSCI orld Small Cap Index, could be used as a proxy to capture te size risk premium. However, tis simple approac based on screening typically leads to portfolios tat, in addition to te target factor, ave significant exposures to oter factors. Exibit 3 provides a snapsot of te factor exposures of tese passive bencmarks. Indeed, a passive allocation to te MSCI orld Value Index would result in small but nonzero exposures to oter factors and ig exposure to te Financial sector. Similarly, a passive allocation to te MSCI orld Small Cap Index would result in significant positive exposure to te volatility risk factor and somewat smaller negative exposure to te value and momentum factors. Plan sponsors may wis to recognize tese exposures explicitly wen building teir passive, optimal allocations to beta factors. Exibit 3 Factor-mimicking portfolios can be used as an overlay to bencmarks to isolate te risk premia investors aim to capture. In Exibit 4, we provide a snapsot of te risk caracteristics of te MSCI orld Small Cap Index bencmark. e ten provide te risk caracteristics of tis bencmark after sequentially edging eac of te unintended factor exposures using full replication. Te final, edged portfolio retains some of te key risk caracteristics of te bencmark: identical country, industry, and currency risk, and similar total risk; owever, by mitigating te value, success, and volatility style factors, we obtain a portfolio tat targets te global size premium. 009 MSCI Barra. All rigts reserved. 7 of 9

8 Exibit 4 Exibits 3 and 4 provide snapsots of risk. e tus considered te following experiment: wat if we edged te unintended style exposures to te MSCI orld Small Cap Index over time? Since te analysis is directed at passive investing, we utilize full replication on a montly basis for te period 3 Dec 003 to 3 Dec 008. Te results were somewat startling; over te period of analysis, te MSCI orld Small Cap Index returned.54% p.a. wit a volatility of 7.9%, wereas te edged MSCI orld Small Cap Index returned 6.5% p.a. wit a volatility of 7.6%. Te outperformance of te edged bencmark is attributable largely to eliminating exposure to te volatility risk factor, wic ad negative return. Toug tis outperformance cannot be guaranteed into te future, our experiment illustrates te importance of identifying and isolating te relevant beta premium wen constructing optimal, passive allocations. Active Investment Strategies Fundamental active investment strategies are often caracterized as being top down or bottom up. Bottom-up strategies empasize security selection. Analysts will rate te relative attractiveness of individual companies on te basis of balance seet fundamentals, expected future cas flows, and te quality of management in te context of te current environment. Portfolio managers of bottom-up investment processes build teir active exposures by going long te top analyst recommendations and going sort te dogs (or stocks tat are expected to underperform), wile trying to maintain balanced or minimal exposures to sectors and investment styles. 009 MSCI Barra. All rigts reserved. 8 of 9

9 Exibit 5 Factor-mimicking portfolios can be use to mitigate style exposures, enabling portfolio managers to capture pure alpa. To illustrate tis concept, we consider a ypotetical strategy were te investment manager as perfect foresigt over a -mont forecast orizon spanning US stocks. To exploit tese views, e constructs an absolute return investment strategy going long te 50 best expected performing stocks and sorting te 50 worst expected performers, on an equalweigted basis. Altoug tese portfolios will deliver large positive returns (by construction), tey may be exposed to large systematic risks. In Exibit 5, we provide summary statistics on te style exposures of tese portfolios for te period 3 Dec 997 to 30 Jun 008 using te Barra USE3 model. As sown in te table, te resulting portfolios ave nonzero exposure to systematic risk factors. For instance, te long-sort strategy maintains an average sort exposure to te Volatility factor in 00 of te 6 monts of te analysis. Tis would ave detracted from risk-adjusted performance, as tis factor earned about 60 bps/annum and contributed to systematic portfolio risk as te average factor volatility was about 7% over tis period. 009 MSCI Barra. All rigts reserved. 9 of 9

10 Exibit 6 Our portfolio manager migt ask weter te return from is strategy originated from stockspecific alpa reflecting te stock-selection skill of is analysts or from alpa derived from exposure to systematic risk. Alternatively, e migt ask weter te risk-adjusted performance of te strategy would be enanced by edging out te unwanted or incidental factor risk. In Exibit 5, we noted tat portfolios ad significant exposure to Volatility, Momentum, and Trading Activity. Tese tree factors were also among te most volatile during tis period. Unintended exposure to tese factors may reduce te risk-adjusted performance of te underlying stockpicking strategy by eroding te alpa and increasing te volatility of te portfolio. e tus consider te following experiment: at te beginning of eac mont, te manager will edge away te unintended systematic factor exposures to Volatility, Momentum, and Trading Activity using factor-overlay portfolios tat neutralize te portfolio exposure to tese factors. Clearly, one migt edge eac of te systematic exposures, and tis will be reviewed in future researc; our results, owever, will suffice to illustrate te potential benefits of edging commonfactor risk. Our ypotetical implementation is carried out wit tracking portfolios tat old 400 assets and tat ave a maximum montly turnover of 0%, as tis may reflect best ow te factor-mimicking portfolios migt be implemented in practice. Exibit 6 sows tat te edged perfect-foresigt portfolios significantly outperform te unedged portfolios in terms of percentage gain in risk-adjusted performance by about 40%. ile te return of te edged portfolios is sligtly decreased, tis is more tan compensated wit a substantial reduction in total risk from 8% to %, mainly coming from reduced common-factor risk, wile specific risk is only sligtly increased, if canged at all. Moreover, wen monitoring performance at te end of eac calendar year, Exibit 7 sows tat te edged portfolios outperform te perfect-foresigt strategy every year during te backtest. Interestingly, te gains in yearly cumulative returns to risk seem to be iger in periods of increased market volatility. 009 MSCI Barra. All rigts reserved. 0 of 9

11 Exibit 7 Indeed, wen market volatility peaked in 00, te unedged portfolio saw a significant portion of its risk coming from Volatility and Momentum, wit average contributions over te year to total risk from tese two factors being 58% and 35%, respectively. Hedging away exposure to tese two factors significantly improved te portfolio s risk-adjusted performance in 00. In fact, te substantial improvement in risk-adjusted performance from edging common-factor risk in 00 can be attributed largely to te monts of October and especially November of tat year. 009 MSCI Barra. All rigts reserved. of 9

12 Exibit 8 Exibit 8 plots unedged versus edged portfolio returns over te entire backtesting period. Tis exibit sows, despite aving perfect foresigt over -mont orizons, tat te manager of te unedged long/sort portfolio would still ave experienced different monts of negative montly returns and a maximum drawdown of approximately -35% during te period Oct - Nov 00. Even our super manager may not survive losing a tird of is assets over a two-mont period! On te oter and, te prudent manager wo used factor overlays to edge er portfolio s exposure to te most volatile sources of common factor risk would ave experienced only a -5% drawdown over te same two-mont period. Te option-like payoff structure of te edged portfolio illustrated in tis diagram is clearly most interesting. 009 MSCI Barra. All rigts reserved. of 9

13 Conclusions e ave presented alternative metods for constructing factor-replicating portfolios. e addressed te issue of implementation cost by recognizing tat portfolio turnover (and terefore transaction costs) is an important parameter to control in te construction of constrained-factor portfolios. Importantly, we demonstrated tat constrained-factor portfolios, wit a limited number of assets and relatively low turnover, track pure factor returns reasonably well and terefore can serve as an investment instrument for factor-based edging or to obtain beta exposure to a particular factor. e illustrate ow factor-mimicking portfolios can be applied to passive and active investment strategies. Factor-mimicking portfolios can be utilized to edge out te unintended factor exposures of conventional bencmarks aimed at targeting a particular beta factor; tis enables plan sponsors better to manage teir optimal allocations to beta factor risks. Factor-mimicking portfolios can be utilized to edge out te style exposures of active stock-picking strategies; tis enables te active manager to capture pure alpa. In brief, te practice of active and passive investment management can be enanced via factor-mimicking portfolios. 009 MSCI Barra. All rigts reserved. 3 of 9

14 Appendix In tis appendix, we sow tat specific-risk weigted and total-risk weigted factor portfolios are te same, up to a scaling parameter. Also, we sow ow factor portfolio weigts can be computed eiter troug GLS regression or constrained optimization. Defining Performance Criteria To assess te performance of optimized replication portfolios, we need to define appropriate performance criteria reflecting te objectives of te factor replication problem, namely, maximum exposure to te targeted factor and minimum portfolio risk. Tis reasoning leads us to consider two reward-to-variability performance criteria, similar to Sarpe (975), tat reflect tese objectives. Te two performance criteria we consider are te ratio of portfolio exposure to te targeted factor divided by specific risk (J S ) and te ratio of portfolio exposure to te targeted factor divided by total risk (J T ): J S Te Specific-Risk Approac ( ) D J T ( ) (A.) V First, we consider te specific-risk performance criterion J S. Te standard approac for constructing portfolios tat maximize tis criterion and ave zero exposure to te remaining risk factors involves constrained long/sort optimization. In tis optimization, asset exposures to te targeted factor play te role of expected returns, wile constraints are imposed to control portfolio exposure to te remaining risk factors. Using our general framework, te weigts tat maximize criterion J S are given by te following expression: Te Total-Risk Approac D D * S D D (A.) Next, we consider te problem of maximising te total-risk performance criterion J T, subject to aving zero exposure to all oter factors. As in te specific-risk case, tis problem can be solved troug constrained optimization tat maximizes portfolio exposure to te targeted factor for a given level of total risk, subject to constraints on all remaining risk factors. Te weigts tat maximize criterion J S are as follows: V V * T V V (A.3) Reconciling te Specific-Risk and Total-Risk Problems By construction, te common-factor risk CF of factor-replicating portfolios depends only on te exposure x of tese portfolios to te targeted factor and te risk of te targeted factor. In oter words, factor-replicating portfolios are not exposed to common-factor risk due to te variance and covariance of factors oter tan te targeted factor. As a result, we can express te common factor risk CF of factor replicating portfolios as follows: CF x (A.4) 009 MSCI Barra. All rigts reserved. 4 of 9

15 009 MSCI Barra. All rigts reserved. 5 of 9 Tis expression enables us to rewrite te total-risk performance criterion J T as follows: ) ( S a a CF T J x D x x D x V J (A.5) Te last equation demonstrates tat performance criteria J S and J T, constrained to =0, are maximized by te same optimal portfolio, up to a scaling factor. In oter words: * * S T (A.6) Equivalence between Constrained Optimization and GLS Regression Factor returns estimated troug te GLS regression () wit weigts - are given by: r f ) ( ˆ (A.7) Te component of factor returns f corresponding to te target factor can be estimated troug a two-step regression process. In te first step, we regress target-factor exposures on all oter factor exposures using te same weigting matrix - : b (A.8) Te residuals from tis weigted-least-squares regression can be estimated as follows: b ) ( ˆ ˆ (A.9) In te second step, we regress asset returns on te residuals from te first-step regression: f r ˆ (A.0) Te estimated target-factor-return component can ten be written as follows: r r f ˆ ˆ) ˆ ( ˆ (A.) e can see tat te factor-mimicking portfolio weigts are also te solution to te constrainedoptimization problem (4)-(5), wit a scaling parameter equal to - : ˆ ˆ) ˆ ( (A.)

16 References Cen, N., R. Roll, and S. Ross Economic Forces and te Stock Market. Journal of Business. Vol. 59, No 3, Fama, E. and J. MacBet Risk, Return, and Equilibrium: Empirical Tests. Journal of Political Economy, Vol. 8, Issue 3, Fama, E. and K. Frenc Common Risk Factors in te Returns on Stocks and Bonds. Journal of Financial Economics, Vol. 33, No., Grinold, R. and R. Kan Active Portfolio Management, Cicago: Irwin. Hansen, L.P. and S.F. Ricard Te Role of Conditioning Information in Deducing Testable Restrictions Implied by Dynamic Asset Pricing Models, Econometrica 55, Kandani, A. and A. Lo at Happened to te Quants in August 007? orking Paper, MIT (4 November): ttp://ssrn.com/abstract= Melas, D., Suryanarayanan, R., and S. Cavaglia On Hedging te Style Exposures of an Active Investment Strategy. orking Paper. Melas, D. and R. Suryanarayanan, /30 Implementation Callenges. Barra Horizon Newsletter, 83, 9-7. Sarpe, A Simplified Model for Portfolio Analysis. Management Science, January, Sarpe,. Adjusting for Risk in Portfolio Performance Measurement. Journal of Portfolio Management, inter 975, MSCI Barra. All rigts reserved. 6 of 9

17 Endnotes Dimitris Melas is Executive Director and Ragu Suryanarayanan is Senior Associate at MSCI Barra in London. Tis paper was completed wile Stefano Cavaglia was employed as Head of Quantitative Strategies at UBS OConnor LLC; te views presented in tis researc are tose of te autor and not tose of UBS OConnor LLC. In equation (), r is te n vector of asset excess returns, is te n k matrix containing asset exposures to fundamental factors, f is te k vector of factor returns, e is te n vector of specific returns, n is te number of assets in te estimation universe, and k is te number of factors in te model. In equation (), F is te k k factor return covariance matrix, D is te n n specific return covariance matrix, and V is te n n asset return covariance matrix. Asset exposures to fundamental factors are represented typically by dummy variables for industry factors and cross-sectional z-scores for style factors, for example, size, value, growt, momentum. Te specific-return covariance matrix D is usually assumed to be diagonal. However, some off-diagonal elements may be different from zero, for example, elements corresponding to securities issued by te same company. In our analysis. we do not need to impose any limiting assumptions on te structure of te specific-return covariance matrix. 3 From a teoretical perspective, regression coefficients estimated troug unweigted-least-squares regression are BLUE (best linear unbiased estimators) if te errors are uncorrelated wit eac oter and wit te independent variables and ave equal variance. However, if te errors ave different variance, ten weigted-least-squares regression using te inverse of te error variances as weigts leads to regression coefficients tat are BLUE. 4 For furter details on te full replication approac, see Grinold and Kan [995], page For a suitably cosen value of te scaling parameter. 6 Institutional constraints limit portfolio leverage for certain regulated mutual funds to :. For detailed analysis of te impact of leverage constraints on portfolio efficiency, see Melas and Suryanarayanan [008]. 7 For example, te 0/40 rule under te UCITS regulation in Europe restricts te amount of capital tat a fund can allocate to a single asset to less tan 0% and te total amount allocated to assets above 5% to less tan 40% of te net asset value of te portfolio. 8 In addition to standard transaction costs (commission and market impact) tat are a linear or power function of traded volume, tere are oter transaction costs, for example, ticket costs and custody fees, tat are a function of te number of securities traded or eld in te portfolio. 9 In te numerical simulations, we used Matlab and te Barra Open Optimizer. 0 Te statistics reported in Exibit igligt te performance of different replication metods over montly investment orizons. However, in certain edging applications, investors may also wis to ensure tat te edging portfolios over sorter orizons track te underlying factor returns well. In order to assess tracking error over daily investment orizons, we use data from te Barra US Equity Model (USE3) to assess te daily performance of different factor-replicating metods during te recent market turmoil of August 007. Te analysis, wic is available on request from te autors, suggests tat our broad conclusions remain uncanged at te daily frequency. It may seem surprising at first sigt tat in all cases leverage fluctuates around 00%. Tis is due to te fact tat factor portfolios maintain unit exposure to te target factor. In order to illustrate tis point analytically, we assume tat all assets ave te same specific risk. Ten optimal weigts of oterwise unconstrained optimized replication portfolios may be expressed as follows: * i i (9) Here, is te specific risk of asset i, and i is te residual from te regression of target factor exposures on exposures to all te oter factors. So, te scaling factor is given by: n n * i, i i, i (0) i i Ten, for large number of assets n, we can compute factor portfolio leverage as follows: L n i * i n n n n i E[ ] i, i E[ ] i i Here, we assumed tat te residuals are approximately normally distributed across assets, wit zero mean and standard deviation. Empirically, we find tat is equal to 0.78 on average for te Value and Momentum factors, wic leads to a leverage of 0%. () 009 MSCI Barra. All rigts reserved. 7 of 9

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