The mean-variance optimal portfolio

Transcription

1 ALEXANDRE S. DA SILVA is vice president in the Quntittive Investment Group t Neuberger ermn in New York, NY. lexndre.dsilv@nb.com WAI LEE is the chief investment officer nd hed of the Quntittive Investment Group t Neuberger ermn in New York, NY. wi.lee@nb.com OY PORNROJNANGKOOL is vice president in the Quntittive Investment Group t Neuberger ermn in New York, NY. bobby.pornrojnngkool@nb.com The lck Littermn Model for Active Portfolio Mngement ALEXANDRE S. DA SILVA, WAI LEE, AND OY PORNROJNANGKOOL The men-vrince optiml portfolio hs been criticized s counterintuitive. Often, smll chnges in expected returns inputted into n optimiztion solver cn led to big swings in portfolio positions, giving rise to extreme weightings in some ssets. Jobson nd Korkie [1981], Michud [1989, 1998], est nd Gruer [1991], Chopr nd Ziemb [1993], nd ritten-jones [1999], mong others, rgued tht the hypersensitivity of optiml portfolio weights is the result of the errormximizing nture of the men-vrince optimiztion. As remedy, constrints on positions re often imposed s n lterntive to prevent the optimiztion lgorithm from driving the result towrds some extreme corner solutions. This pproch, however, is often criticized s d hoc. Moreover, when enough constrints re imposed, one cn lmost pick ny desired portfolio without giving too much ttention to the optimiztion process itself. An interesting study by Jgnnthn nd M [003], however, suggested tht under some specil conditions, imposing constrints is equivlent to using yesin-shrunk covrince mtrix or expected return forecst in the optimiztion process. As n lterntive remedy, Michud [1998] proposes to interpret the efficient frontier s n uncertin sttisticl bnd rther thn s deterministic line in the men-vrince spce, introducing the resmpling technique s one potentil wy to derive more robust resulting portfolio. Scherer [00] points out, however, some of the pitflls of the resmpling methodology, such s the possibility of the resmpled frontier moving from concve to convex. Hrvey et l. [006] lso showed tht the resmpling methodology implicitly ssumes tht the investor hs bndoned the mximum expected utility frmework. In ddition, the resulting resmpled efficient frontier is shown to be suboptiml s dictted by Jensen s inequlity. In Hrvey, Liechty, nd Liechty [008], the yesin pproch to portfolio selection ws shown to be superior to the resmpling pproch. Since the lck Littermn (L) model ppered in the literture s lck nd Littermn [1991, 199], it hs received considerble interest from the investment mngement industry. Unlike the resmpling technique, which introduces noise into the efficient frontier, the L frmework tkes n entirely different route bsed on yesin nlysis in solving the error mximiztion problem. The L frmework points out tht, becuse ssets re correlted, chnges in some ssets expected excess returns due to ctive investment views should lso led to revisions of expected excess returns of ssets tht re not explicitly involved in the ctive investment views. Tke globl portfolio of stock nd bond mrkets s n exmple. If expected excess returns of the U.S. stock mrket re revised upwrd, then the expected excess returns of ll IT IS ILLEGAL TO REPRODUCE THIS ARTICLE IN ANY FORMAT WINTER 009 THE JOURNAL OF PORTFOLIO MANAGEMENT 61 Copyright 009

2 ssets nd portfolios of ssets tht re correlted with the U.S. stock mrket should lso be revised in direction tht is consistent with the covrince mtrix of the ssets. As such, the error in estimting expected excess return in one sset, if ny, will be extended to ll other correlted ssets, so tht robust optiml portfolio cn be derived when these revised inputs re fed into the optimiztion process. Mny studies inspired by this frmework further dvnce our understnding nd implementtion of the lck Littermn frmework. Lee [000] nd Stchell nd Scowcroft [000] further elborted nd expnded the theoreticl frmework, while others, such s evn nd Winkelmnn [1998], He nd Littermn [1999], Herold [003], Idzorek [004], nd Jones, Lim, nd Zngri [007] focused on implementtion. Mny prctitioners seem to suggest tht one of the key contributions of the L frmework is the derivtion of implied equilibrium excess returns from given portfolio through reverse optimiztion. For instnce, in the investment industry, mny clients strt with their strtegic predetermined benchmrk portfolio. Given the benchmrk portfolio weights nd scling prmeter, one cn esily derive the benchmrk implied expected excess returns of the ssets through reverse optimiztion, which my be interpreted s the equilibrium views employed s the strting point for subsequent ctive investment nlysis. To the best of our knowledge, however, Shrpe [1974] is the first to hve provided insights on this subject mtter. In our opinion, the key element of the L frmework is the combintion of ctive investment views nd equilibrium views through yesin pproch, which hs been shown to result in more robust portfolios tht re less sensitive to errors in expected excess return inputs. As ctive views re involved, by definition, the frmework hs to be nlyzed nd understood within the context of ctive mngement nmely, beting the benchmrk within certin trcking error. This rticle dds to the literture by first pointing out tht the lck Littermn frmework ws derived under the men-vrince portfolio efficiency prdigm, which is different from the common objective in ctive mngement, nmely, mximizing the ctive lph for the sme level of ctive risk. We show, by presenting nd nlyzing resulting portfolio sttistics, how the inconsistencies led to unintentionl trdes nd risks when the frmework is implemented t fce vlue. Finlly, we consider potentil remedies. REVIEW OF LACK LITTERMAN FRAMEWORK Suppose there re N ssets nd K ctive investment views. The originl lck Littermn model of expected excess returns in lck nd Littermn [1991, 199] ws expressed s μ = [( τσ) + P Ω P] [( τσ) Π + P Ω Q] where μ is n N vector of expected excess returns, τ is scling prmeter, Σ is n N N covrince mtrix, P is K N mtrix whose elements in ech row represent the weight of ech sset in ech of the K-view portfolios, Ω is the mtrix tht represents the confidence in ech view, nd Q is K 1 vector of expected returns of the K-view portfolios. A view portfolio my include one or more ssets through nonzero elements in the corresponding elements in the P mtrix. Severl ppers nd rticles discuss in detil how to formulte ctive investment views in the lck Littermn frmework; for exmples, see Lee [000], Idzorek [004], nd Jones, Lim, nd Zngri [007]. y pplying the Mtrix Inversion Lemm, the originl lck Littermn eqution cn be rewritten in more intuitive wy, s follows: Ω μ = Π + ΣP + PΣP τ ( Q PΠ) = Π + V (1) where V is term tht cptures ll devition of expected excess returns from the equilibrium due to ctive investment views, Ω V = ΣP + PΣP ( Q PΠ) τ Eqution (1) helps expose the intuition behind the lck Littermn frmework. Under the L frmework, the expected excess return of ssets is equl to the ssets equilibrium excess return, Π, plus term tht cptures the devition of our views of the K portfolio of ssets, Q, from the equilibrium implied views, PΠ. Therefore, the expected excess return will be different from the equilibrium excess () 6 THE LACK LITTERMAN MODEL FOR ACTIVE PORTFOLIO MANAGEMENT WINTER 009 Copyright 009

3 return if, nd only if, our investment views re not redundnt to, or implied by, the equilibrium views. OPTIMAL ACTIVE MANAGEMENT After expected excess returns re derived, risk trget needs to be defined in order to determine the finl ctive weights. In ctive mngement, ctive return, or wht is commonly known s lph, is typiclly defined s the return of the ctive portfolio in excess of the benchmrk portfolio. Active risk is defined s the stndrd devition of lph, lso known s trcking error. In nutshell, the objective of ctive mngement is to mximize lph for given level of trcking error. In other words, ctive mngement ttempts to mximize the informtion rtio, IR, defined s the rtio of lph to trcking error. For exmple, this objective is reflected in evn nd Winkelmnn [1998]: After finding expected returns, we then set trget risk levels. Since we construct our optiml portfolio reltive to benchmrk, we consider ll of our risk mesures s risks reltive to the benchmrk. The two risks tht we cre most bout re the trcking error nd the Mrket Exposure (p. 5). In this section, we provide n nlyticl frmework for determining the optiml ctive positions given the objective of mximizing the informtion rtio. Definitions: ω ω ω ω μ λ θ vector of benchmrk portfolio weights vector of ctive positions vector of ctive portfolio weights, which is the sum of ω nd ω vector of weights of the globl minimum vrince portfolio, expected excess return of the globl minimum vrince portfolio, scling prmeter ctive risk version prmeter Lgrngin multiplier Recll tht the objective function of ctive mngement in the presence of benchmrk is to mximize the totl return of the portfolio with penlty on the squre of trcking error; tht is, s.t. It is esy to show tht the previous solution lso mximizes the informtion rtio. Tking the first derivtive of the Lgrngin gives Substituting Eqution (4) into the budget constrint in the objective function gives tht is, mx ( + ) μ λ Σ λ ( μ θ Σ Σ ) = θ = 1 Σ μ = μ = μ 1 Σ 1 Combining with Eqution (4) gives the optiml vector of ctive positions s = 1 = 0 μ λσ θ1 = 0 = 1 Σ ( μ θ λ 1) 1 Σ ( μ μ λ 1) Alterntively, Eqution (5) cn be expressed s follows: 1 = Σ ( I 1 μ ) λ Eqution (5) offers intuitive economic menings. In optimizing the IR, the process mkes multiple pirwise comprisons of the return of ech sset ginst the return of the globl minimum vrince portfolio,. Long positions re tken for ssets tht re expected to outperform the portfolio, nd vice vers. The vector of optiml ctive weights is the result of the risk-djusted combintion of ll of these pir trdes. (3) (4) (5) (6) WINTER 009 THE JOURNAL OF PORTFOLIO MANAGEMENT 63 Copyright 009

4 E XHIIT 1 Summry Sttistics of Two-Asset Exmple: Stocks nd onds Exmple To put the discussion in context, consider the following oversimplified exmple in pplying the lck Littermn frmework. Suppose there re only two sset clsses in the benchmrk portfolio stocks nd bonds with benchmrk weights, voltilities, nd correltion s reported in Exhibit 1. To derive the equilibrium views, the literture, including evn nd Winkelmnn [1998], He nd Littermn [1999], Drobetz [001], Idzorek [004], nd Jones, Lim, nd Zngri [007], ssumes tht the benchmrk portfolio is men-vrince efficient portfolio. As result, implied equilibrium excess returns cn be derived by reverse optimiztion from the benchmrk weights ccording to Π = Σ where, in our exmple, Π is the 1 vector of equilibrium excess returns, is risk version prmeter, Σ is the covrince mtrix, nd ω is the 1 vector of mrket-cpitliztion benchmrk weights. 1 We set the vlue of such tht the resulting equilibrium excess returns will provide n expected Shrpe rtio of 0.5 for the portfolio. Given these prmeters, the equilibrium excess returns for stocks nd bonds re found to be 6.46% nd 1.0%, respectively. Next, we ssume tht there is only one ctive investment view stocks re expected to underperform bonds by 3%. In mtrix nottion, this view cn be expressed s Pμ = Q where P = [1 1] nd Q = 3%. To set the confidence of the view, we followed the suggestion of He nd Littermn [1999] by using (7) (8) Ω Σ τ = dig ( dig ( P P )) We then pplied Eqution (1) to derive the lck Littermn expected excess returns of stocks nd bonds t.39% nd 1.17%, respectively. All results re summrized in Exhibit 1. Note tht becuse the ctive view is berish on stocks reltive to bonds, the finl expected premium of stocks over bonds becomes 1.% (.39% 1.17%) versus the equilibrium premium of 5.44% (6.46% 1.0%). Lstly, for the ske of illustrtion, we set the vlue of λ in Eqution (6) so tht the resulting ctive positions give trcking error of %. The optiml ctive weights re determined to be +16% stocks nd 16% bonds, respectively. The results re interesting, if not surprising. The only ctive view in this exmple is berish view on stocks versus bonds. Why would the ctive positions overweight stocks nd underweight bonds? We explore this question more fully next. PROLEMS OF APPLYING LACK LITTERMAN IN ACTIVE MANAGEMENT The originl lck Littermn model ws derived under the men-vrince equilibrium frmework, which ttempts to mximize return for certin level of portfolio risk mesured by stndrd devition, or voltility. The objective of this section is to illustrte tht strict ppliction of the lck Littermn frmework in ctive mngement cn potentilly led to unintentionl trdes. As we previously discussed, the objective of ctive mngement is to mximize the informtion rtio (IR). Consider the cse where we do not hve ny investment views, such tht the vector of expected excess return is just the equilibrium. Presumbly, the only ction tht (9) 64 THE LACK LITTERMAN MODEL FOR ACTIVE PORTFOLIO MANAGEMENT WINTER 009 Copyright 009

5 mkes sense in this informtionless cse is to just hold the benchmrk portfolio nd mke no ctive trdes. However, the following nlysis will show tht, surprisingly, IR mximiztion will led to ctive trdes in this exmple. As suggested by the literture nd previously explined in this rticle, it hs become stndrd procedure to derive the benchmrk-implied equilibrium excess return, or Π, through reverse optimiztion, ccording to Eqution (7), s Σω. The optiml vector of ctive positions, given equilibrium ssumptions nd no ctive views in this cse, cn be derived by simply substituting Σω into the expected excess return in Eqution (6); tht is, 1 = Σ ( Σ ), Π Σ 1 λ λ Σ 1 Σ ) (10) Notice tht Σ is sclr equl to the covrince of the globl minimum vrince portfolio,, nd the benchmrk portfolio,. Together with the investment budget condition of we cn determine tht 1 1 = 1 1 Σ Π = 1 = 1 Σ 1 Σ Σ = = σ 1 Σ 1 1 Σ 1 (11) In fct, it cn be shown tht the covrince of ny given portfolio with the globl minimum vrince portfolio is equl to the vrince of (see Grinold nd Khn [1999] for detils). Substituting Eqution (11) into Eqution (10), the optiml vector of ctive positions under the no-lph informtion scenrio cn be given s Eqution (1) suggests tht unless the client chooses the s the benchmrk portfolio, such tht =, use of the lck Littermn model will generte set of ctive trdes, even in cse such s this of no investment informtion. In prticulr, the vector of ctive trdes is positive sclr multiple of the difference between the benchmrk portfolio nd the. This pprently counterintuitive result is relted to the mismtch in objective function between men-vrince portfolio efficiency, which ttempts to mximize the Shrpe rtio (SR), versus the lph trcking-error efficiency, which ttempts to mximize the informtion rtio insted. Some discussion on this topic ppers in Roll [199] nd Lee [000, Ch. ]. Recll tht the implied equilibrium excess return in Eqution (7) is the result of reverse-optimizing the benchmrk portfolio weights under the mximum-sr criteri. In generl, ll else equl, the higher the voltility of n sset, the higher will be the implied equilibrium excess return. The step in pproching the mximum-ir objective is where inconsistency emerges. Notice tht under the mximum-ir criteri, no ttention is pid to overll portfolio voltility. Insted, ny discrepncies in pirs of sset returns re perceived s lph opportunities nd, therefore, portion of the totl trcking error budget will be llocted to these opportunities. As result, wht seems to be t equilibrium under the mx-sr criteri is, by definition, t disequilibrium, nd ctive trdes re then initited to restore the portfolio to the mx-ir condition. In the previous exmple in which stocks nd bonds re the only portfolio ssets, the implied equilibrium return of stocks is higher thn for bonds under the mx-sr criteri. Therefore, even in the bsence of n ctive view, under the IR criteri, the long-stocks/short-bonds ctive trde is perceived s n lph-generting trde, even though this trde embeds bsolutely no lph informtion t ll. In the following sections, we further elborte on how the IR criteri perceive discrepncies in equilibrium returns s lph opportunities nd, furthermore, led to portfolio tht is more risky thn the benchmrk portfolio. λ Σ 1 = ( λ ), Π Σ = 1 1 (1) INFORMATIONLESS VIRTUAL ALPHA In this section, we derive some performnce sttistics for the ctive portfolio within this informtionless environment. Given no investment views, ll expected WINTER 009 THE JOURNAL OF PORTFOLIO MANAGEMENT 65 Copyright 009

6 excess returns re equl to the implied equilibrium expected excess returns from the benchmrk portfolio. Alph Perceived lph in this exmple cn be clculted s follows: α = Π Trcking Error λ ) Σ λ Σ Σ ) λ σ σ ), λ σ σ ) (13) Active risk, mesured by trcking error, in this informtionless exmple, is given by TE = Σ ) Σ( ) λ = + λ σ σ σ, = λ σ σ Informtion Rtio Therefore, the informtion rtio is given by α IR = = σ σ TE (14) (15) Eqution (15) revels the interesting result tht the perceived informtion rtio is proportionl to the squre root of the difference in the vrince of the benchmrk portfolio nd the globl minimum vrince portfolio. Consequently, the riskier the benchmrk portfolio, the lrger will be the virtul lph opportunity tht is perceived by pplying the L frmework, nd the more resulting ctive trdes re executed. et to enchmrk We first determine the covrince between the ctive weights nd the benchmrk s follows: (16) The ctive bet with respect to the benchmrk is then given by (17) Tht is, the informtionless ctive positions led to n unintentionl net exposure to the benchmrk. In other words, the set of ctive trdes together implies bullish view on the benchmrk portfolio so tht lph tends to be positive when the benchmrk portfolio delivers positive return. The portfolio bet cn be determined similrly, Voltility σ = Σ, λ ) Σ λ σ σ ) σ, β = σ λ ) Σ σ = 1 λ σ > 0 ( + ) Σ β = σ = 1 + β > 1 (18) The vrince of the ctive portfolio cn lso be derived s follows: 66 THE LACK LITTERMAN MODEL FOR ACTIVE PORTFOLIO MANAGEMENT WINTER 009 Copyright 009

7 Substituting Equtions (14), (16), nd (17) gives σ λ σ σ = + σ λ + ( ) = σ β + λ + 1 > σ + ) Σ( + ) σ = TE + σ + σ, (19) In summry, in this informtionless exmple, the mismtch of the objective function between mx-sr nd mx-ir leds the ctive mnger to believe tht positive informtion rtio is vilble nd, thus, the ctive mnger initites set of ctive trdes, which leds to n ctive portfolio tht is more voltile thn the benchmrk portfolio nd tht hs positive net-bet exposure. LACK LITTERMAN MODEL AND ACTIVE MANAGEMENT Implementing the lck Littermn model in ctive mngement simply requires substituting the expected excess returns from the lck Littermn frmework in Eqution (1) into the optiml ctive positions in Eqution (6), so tht 1 = Σ + V V + λ ( Π ) ( Π ) 1 which cn be grouped into two terms, s follows: 1 1 = Σ ( Π V V Π1) + Σ ( 1) λ λ = +, Π V, (0) The first term in Eqution (0) corresponds to the cse of using the equilibrium expected excess returns s inputs to chieve the mx-ir objective. Therefore, it is equivlent to the cse of the informtionless scenrio discussed erlier in which the mx-ir objective will still led to ctive trdes s given by Eqution (1). The second term in Eqution (0) hs similr functionl form, except tht the expected excess returns in the prentheses, V, re determined by the devition of ny investment views on portfolios P from the views which re implied by the equilibrium, PΠ. The detils re given in Eqution (). Consider the cse of no investment view, such tht Q nd PΠ re the sme, or, in other words, such tht V is zero. In this cse, the optiml vector of ctive trdes is the sme s the cse in which the equilibrium excess returns re used, s depicted in Eqution (1). In the presence of investment views, such tht Q is different from PΠ, the reltive importnce of the two terms in Eqution (0) lrgely depends on confidence in the investment views. For instnce, for investment views tht reflect very low confidence, such tht Ω, the second term in Eqution (0) pproches zero. The resulting optiml ctive positions once gin converge to the cse of no investment view. The more interesting nd relevnt cse is tht in which the investment views, lthough uncertin, come with some meningful degrees of confidence. The exct vlues of the two components in Eqution (0) depend on, mong other moving prts, how one specifies the confidence of views reltive to equilibrium. For exmple, Eqution (9), suggested by He nd Littermn [1999], gives rise to the following expression for V: 1 V = P Σ ( PΣP ) ( Q PΠ) (1) In generl, the equilibrium component, Π, is not negligible component of the vector of expected excess returns. As result, the unintentionl, informtionless component of ctive trdes, ω,π, plys nontrivil role in ctive mngement when the lck Littermn model is pplied. y now, it should be cler why, in the exmple using stocks nd bonds, the optiml tcticl trde is to go long stocks nd short bonds even when the only investment view is reltively berish on stocks. When the confidence ssigned to the ctive investment view is not prticulrly strong, the equilibrium reltive returns cn become so dominting tht they drive most of the tcticl positions in the portfolio, even when they do not represent ny relevnt investment informtion. WINTER 009 THE JOURNAL OF PORTFOLIO MANAGEMENT 67 Copyright 009

8 Remedies As the previous discussion highlighted, the root cuse of n inconsistency in pplying the lck Littermn frmework to ctive portfolio mngement is the mismtch between the optimiztion problem used in bcking out the equilibrium implied excess returns (i.e., n unconstrined SR optimiztion) nd the optimiztion problem used to construct n ctive portfolio (i.e., constrined IR optimiztion). The most obvious wy to fix this problem is to mke the two optimiztion problems consistent. Active mngement cnnot be chieved in n SR optimiztion frmework becuse the mnger s ctive bets will not be independent of the benchmrk portfolio when some constrints re binding (see Roll [199] for n exmple). Therefore, prcticl remedy is to bck out the equilibrium implied excess returns using the sme IR optimiztion problem employed in constructing n ctive portfolio. This is done by replcing Π tht solves the reverse SR optimiztion problem, s in Eqution (7), with Π tht implicitly solves the IR optimiztion problem intended to be used in building the ctive portfolio (i.e., reverseoptimizing the problem expressed in Eqution (3), with Π replcing μ nd imposing ny dditionl constrints pplicble to the portfolio). Formlly, Π implicitly solves 0 = rg mx ( + ) Π λ Σ s.t. ll other constrints () insted of Π, s suggested in lck Littermn [199], tht solves the following reverse SR optimiztion problem 0 = rgmx ( + ) Π λ( + ) Σ( + ) s.t. no constrint Solving Eqution () explicitly t first seems difficult in the presence of other constrints in the optimiztion. It turns out, however, tht if we choose ny Π whose elements re ll the sme, such s Π = Π i j i j then the first term in the objection function of Eqution () drops out of the optimiztion s it becomes constnt (recll tht we hve constrint 1 = 0), nd we re left with trcking error minimiztion problem. Clerly, we cn minimize trcking error by setting = 0 (i.e., tking no ctive weight). In other words, ny constnt vector of expected excess return Π lwys implicitly solves Eqution () regrdless of ny other constrints in the problem. This observtion is very intuitive. To ensure tht no unintentionl bet is mde in n ctive portfolio in the bsence of ny ctive view, the prior belief for expected excess returns of the ssets should be n uninformtive one tht is, ll ssets re expected to hve the sme excess returns. Of ll the possible vlues of prior expected return, the most intuitive one is to set Π equl to 0 where ll ssets re expected to yield risk-free rte of return s priors. Herold [003] is the only other study to our knowledge tht uses zeros s the equilibrium prior for ctive portfolio mngement. This choice of Π leds nturlly to the portble lph implementtion for ctive portfolio mngement. ecuse we cn drop the benchmrk weights from the IR optimiztion problem nd focus exclusively on optimizing ctive portfolio weights, mx μ λ Σ s.t. ll other constrints (3) where μ = V from setting Π =0 in Eqution (1), we cn build ny ctive portfolio by focusing only on constructing n lph overly portfolio. The totl portfolio will simply be the sum of the benchmrk nd the lph overly portfolio, = +. For exmple, suppose we re mnging long-only portfolio tht llows leverge through borrowing up to 5% of sset vlue nd with the S&P 500 s the portfolio s benchmrk. Suppose further tht we form ctive investment views of securities in the portfolio nd express them in the lck Littermn frmework in the view mtrix P, view expected excess returns vector Q, nd view confidence mtrix Ω. The portble lph implementtion using the lck Littermn frmework cn be chieved by first clculting expected excess returns of the ssets μ using Eqution (1) with Π set to zero, nd then using these returns s inputs to the IR optimiztion problem in Eqution (3) to solve for ctive weights ω with the constrints, 0 1 5% > (< 5% leverge) (long-only constrint) 68 THE LACK LITTERMAN MODEL FOR ACTIVE PORTFOLIO MANAGEMENT WINTER 009 Copyright 009

9 E XHIIT Revised Summry Sttistics of Two-Asset Exmple: Stocks nd onds Adding the benchmrk weight ω to the ctive weight ω will result in finl portfolio tht respects ll constrints. To further illustrte our point, we cn go bck to our originl exmple nd pply the remedies just proposed. In other words, we now set Π = 0, keeping ll other estimtes the sme, nd pply Eqution (1) to derive the expected ctive returns resulting from our berish investment view on stocks versus bonds. Exhibit reports the revised vlues. We cn now clculte the reverse-optimized ctive portfolio ssocited with the newly clculted ctive excess returns nd scle it so tht the finl portfolio meets the defined trget trcking error. It is esy to verify tht the finl ctive portfolio will be composed of short position in stocks ( 16.1%) nd long position in bonds (+16.1%) for totl trcking error of %. Consequently, the finl totl portfolio weights re +58.9% (75% 16.1%) nd +41.1% (5% %) on stocks nd bonds, respectively. As expected, the resulting portfolio underweights stocks nd overweights bonds reltive to the benchmrk nd is thus more intuitive given our initil investment view. CONCLUSION Of the vrious potentil remedies to the hypersensitivity of men-vrince optiml portfolio with respect to chnges in inputs, the lck Littermn frmework stnds out s the most theoreticlly sound nd elegnt of ll. In the erly dys fter this frmework ws introduced, it ws often interpreted s n sset lloction model, or s n expected-return forecsting model. In our view, the L frmework is portfolio construction process bsed on n elegnt ppliction of yesin nlysis in combining different sources of input estimtes. While we re fscinted by the strong theoreticl underpinning of this frmework, rooted in yesin updting nd equilibrium concepts in finncil economics, its implementtion my not be s strightforwrd. In prticulr, we hve provided both theoreticl nd empiricl results to shed light on how the stright ppliction of the L frmework in ctive investment mngement cn led to unintended trdes nd risk tking, which in turn leds to more risky portfolio thn desired. Focusing on the mismtch between the Shrpe rtio optimiztion behind the L frmework nd the informtion rtio optimiztion in the ctive investment industry, we propose remedy tht leds to portble lph implementtion of ctive portfolios. These resulting ctive portfolios reflect intentionl nd true investment insights of the investment process. ENDNOTE The uthors would like to thnk André Perold, Minh Trinh, Ping Zhou, nd members of the Quntittive Investment Group t Neuberger ermn for vluble comments. All opinions expressed in this rticle re solely personl opinions of the uthors nd should not be tken s investment dvice or the opinions of the uthors employer. 1 Formlly, reverse SR optimiztion problem sets Π tht solves 0 = rgmx ( + ) Π λ ( + ) Σ( + ). The reverse optimiztion solution in Eqution (7) is vlid only if none of the constrints, if ny, is binding in determining the benchmrk portfolio. sed on our necdotl observtions, mny others ignore this importnt point in ttempting to determine implied expected returns given set of portfolio weights. For exmple, in Jones, Lim, nd Zngri [007], the uthors derived individul stock lphs by reverse optimiztion of wht they lbeled s the optiml tile portfolio (OTP) (see their Eqution (6)). Note tht the reverse optimized lphs re vlid only if none of the constrints is binding. However, the OTP is result of constrined optimiztion ccording to their Eqution (5). As result, the derived lphs re distorted to n unknown extent, which my led to suboptiml ctive portfolios reltive to the originl informtion content embedded in the OTP. WINTER 009 THE JOURNAL OF PORTFOLIO MANAGEMENT 69 Copyright 009

10 REFERENCES est, M., nd R. Gruer. On the Sensitivity of Men-Vrince Efficient Portfolios to Chnges in Asset Mens: Some Anlyticl nd Computtionl Results. Review of Finncil Studies, Vol. 4, No. (Summer 1991), pp evn, A., nd K. Winkelmnn. Using the lck Littermn Globl Asset Alloction Model: Three Yers of Prcticl Experience. Working pper, Goldmn Schs, June lck, F., nd R. Littermn. Asset Alloction: Combining Investors Views with Mrket Equilibrium. Journl of Fixed Income, September 1991, pp Globl Portfolio Optimiztion. Finncil Anlysts Journl, September/October 199, pp ritten-jones, M. The Smpling Error in Estimtes of Men- Vrince Efficient Portfolios. Journl of Finnce, Vol. 54, No. (April 1999), pp Chopr, V., nd W. Ziemb. The Effects of Errors in Mens, Vrinces, nd Covrinces on Optiml Portfolio Choice. Journl of Portfolio Mngement, Vol. 19, No. (Winter 1993), pp Drobetz, W. How To Avoid The Pitflls in Portfolio Optimiztion? Putting The lck-littermn Approch At Work. Journl of Finncil Mrkets nd Portfolio Mngement, Vol. 15, No. 1 (001), pp Grinold, R.C., nd R.N. Khn. Active Portfolio Mngement: A Quntittive Approch for Producing Superior Returns nd Controlling Risk. New York: McGrw-Hill, Hrvey, C., J.C. Liechty, nd M.W. Liechty. yes vs. Resmpling: A Remtch. Journl of Investment Mngement, Vol. 6, No. 1 (008), pp Hrvey, C., J.C. Liechty, M.W. Liechty, nd P. Muller. Portfolio Selection with Higher Moments. Working pper, Duke University, 006. He, G., nd R. Littermn. The Intuition ehind lck Littermn Model Portfolios. Working pper, Goldmn Schs Quntittive Resources Group, Herold, U. Portfolio Construction with Qulittive Forecsts. Journl of Portfolio Mngement, Fll 003, pp Idzorek, T.M. A Step-by-Step Guide to the lck Littermn Model: Incorporting User-Specified Confidence Levels. Working pper, Zephyr Assocites, 004. Jgnnthn, R., nd T. M. Risk Reduction in Lrge Portfolios: Why Imposing the Wrong Constrints Helps. Journl of Finnce, Vol. 58, No. 4 (003), pp Jobson, J.D., nd. Korkie. Putting Mrkowitz Theory to Work. Journl of Portfolio Mngement, Vol. 7, No. 4 (1981), pp Jones, R., T. Lim, nd P.J. Zngri. The lck Littermn Model for Structured Equity Portfolios. Journl of Portfolio Mngement, Winter 007, pp Lee, W. Theory nd Methodology of Tcticl Asset Alloction. Hoboken, NJ: John Wiley & Sons, 000. Michud, R.O. The Mrkowitz Optimiztion Enigm: Is Optimized Optiml? Finncil Anlysts Journl, Vol. 45, No. 1, (Jnury/Februry 1989), pp Efficient Asset Mngement: A Prcticl Guide to Stock Portfolio Optimiztion nd Asset Alloction. New York: Oxford University Press, Roll, R. A Men/Vrince Anlysis of Trcking Error. Journl of Portfolio Mngement, Vol. 18, No. 4 (Summer 199), pp. 13. Stchell, S., nd A. Scowcroft. A Demystifiction of the lck Littermn Model: Mnging Quntittive nd Trditionl Portfolio Construction. Journl of Asset Mngement, Vol. 1, No. (000), pp Scherer,. Portfolio Resmpling: Review nd Critique. Finncil Anlysts Journl, November/December 00, pp Shrpe, W.F. Imputing Expected Security Returns from Portfolio Composition. Journl of Finncil nd Quntittive Anlysis, Vol. 9, No. 3 (June 1974), pp To order reprints of this rticle, plese contct Dewey Plmieri t dplmieri@iijournls.com or THE LACK LITTERMAN MODEL FOR ACTIVE PORTFOLIO MANAGEMENT WINTER 009 Copyright 009