The Black-Litterman Model For Active Portfolio Management Forthcoming in Journal of Portfolio Management Winter 2009

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1 The lck-littermn Model For Active Portfolio Mngement Forthcoming in Journl of Portfolio Mngement Winter 009 Alexndre Schutel D Silv Senior Vice President, Quntittive Investment Group Neuberger ermn Wi Lee, Ph.D. Mnging Director, Quntittive Investment Group Neuberger ermn obby Pornrojnngkool, Ph.D. Senior Vice President, Quntittive Investment Group Neuberger ermn One of the chllenges of portfolio optimiztion is the tendency of smll differences in expected return inputs to crete mjor swings in portfolio weightings, sometimes leding to extreme portfolio lloctions. In our view, the lck-littermn (L) model my ply highly constructive role in lleviting this problem, due to its combintion of ctive investment views nd equilibrium views through yesin pproch. This rticle explores the development of the L model within the men-vrince portfolio efficiency prdigm, looks t the phenomenon of unintentionl trdes nd dditionl risks relted to the trditionl implementtion of L nd suggests potentil remedies to get the most out of this importnt investment model. 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 [98], Michud [989, 998], est nd Gruer [99], Chopr nd Ziemb [993] nd ritten-jones [999], mong others, rgue tht the hypersensitivity of optiml portfolio weights is the result of the error-mximizing 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, suggests 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 [998] 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. However, Scherer [00] points out some of the pitflls of the resmpling methodology, such s the possibility of the resmpled frontier moving from concve to convex. Hrvey, Liechty, Liechty nd Muller [006] lso show 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 is shown to be superior to the resmpling pproch. For Informtion Only. This document is not for use with retil clients in Europe nd Asi.

2 The lck-littermn Model For Active Portfolio Mngement (continued) 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, which re less sensitive to errors in expected excess return inputs. The L frmework ws derived under the menvrince 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. Since the lck-littermn (L) model ppered in literture s lck nd Littermn [99, 99], 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 expected excess returns in some ssets due to some ctive investment views should lso led to revisions of expected excess returns of those ssets which 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 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 L frmework. Lee [000] nd Stchell nd Scowcroft [000] further elborte nd expnd the theoreticl frmework, while others such s evn nd Winkelmnn [998], He nd Littermn [999], Herold [003], Idzorek [004], nd Jones, Lim nd Zngri [007] focus 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 [974] 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, which 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 L 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 how the inconsistencies led to unintentionl trdes nd risks when the frmework is implemented t fce vlue, presenting nd nlyzing resulting portfolio sttistics. Finlly, we consider potentil remedies. REVIEW OF LACK-LITTERMAN FRAMEWORK Suppose there re N ssets nd K ctive investment views. The originl L model of expected excess returns in lck nd Littermn [99, 99] is expressed s P' P P' Q See disclosures on lst pge which re n importnt prt of this document.

3 The lck-littermn Model For Active Portfolio Mngement (continued) where µ is n N x vector of expected excess returns, τ is scling prmeter, Σ is n N x N covrince mtrix, P is K x 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 x vector of expected returns of the K view portfolios. A view portfolio my include one or more ssets through non-zero elements in the corresponding elements in the P-mtrix. Severl ppers discuss in detil how to formulte ctive investment views in the L frmework; see Lee [000], Idzorek [004] nd Jones, Lim nd Zngri [007] for exmples. y pplying the Mtrix Inversion Lemm, the eqution bove cn be rewritten in more intuitive wy s follows: Ω µ = Π ΣP' () P ΣP' Q PΠ= Π V τ 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 () helps expose the intuition behind the L frmework. Under the L frmework, the expected excess return of ssets re equl to their 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 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, one needs to define risk trget 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, or wht is 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 [998, p.5]: 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. In this section, we provide n nlyticl frmework for determining the optiml ctive positions given the objective of mximizing the IR. Definitions: ω vector of benchmrk portfolio weights ω vector of ctive positions ω vector of ctive portfolio weights, which is the sum of ω nd ω vector of weights of the globl minimum vrince portfolio, ω See disclosures on lst pge which re n importnt prt of this document. 3

4 The lck-littermn Model For Active Portfolio Mngement (continued) µ expected excess return of the globl minimum vrince portfolio, γ λ θ scling prmeter ctive risk version prmeter Lgrngin multiplier Recll tht the objective function of ctive mngement in the presence of benchmrk is to mximize totl return of the portfolio with penlty on the squre of trcking error. Tht is, mx ϖ ϖ ' µ λϖ' Σϖ (3) s.t. ϖ ' = 0 It is esy to show tht the solution bove lso mximizes the IR. Tking the first derivtive of the Lgrngin gives µ λσϖ θ = 0 ϖ = Σ λ µ θ (4) Substituting (4) into the budget constrint in the objective function gives Tht is, λ µ ' Σ θ' Σ = 0 ' Σ θ = µ = ϖ' µ = µ ' Σ Combining with eqution (4) gives the optiml vector of ctive positions s ϖ = Σ µ µ λ Alterntively, eqution (5) cn be expressed s follows: (5) µ ϖ ϖ' = Σ I (6) 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. Exmple To put the discussion in context, consider the following oversimplified exmple in pplying the L 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 on pge. To derive the equilibrium views, the literture, including evn nd Winkelmnn See disclosures on lst pge which re n importnt prt of this document. 4

5 The lck-littermn Model For Active Portfolio Mngement (continued) Strict ppliction of the L frmework in ctive mngement cn potentilly led to unintentionl trdes. [998], He nd Littermn [999], Drobetz [00], 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 = γ Σ ϖ (7) where, in our exmple, Π is the x vector of equilibrium excess returns, γ is risk version prmeter, Σ is the x covrince mtrix, nd ω is the x vector of mrket cpitliztion benchmrk weights. We set the vlue of γ such tht the resulting equilibrium excess returns will provide n expected Shrpe rtio (SR) of 0.5 for the portfolio. Given these prmeters, the equilibrium excess returns for stocks nd bonds re found to be 6.46% nd.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 P µ = Q (8) where P = [ -] nd Q = -3%. To set the confidence of the view, we follow the suggestion of He nd Littermn [999] by using Ω = dig ( dig ( PΣP' )) (9) τ We then pply eqution () to derive the L expected excess returns of stocks nd bonds t.39% nd.7%, respectively. All results re summrized in Exhibit. Note tht becuse the ctive view is berish one on stocks reltive to bonds, the finl expected premium of stocks over bonds becomes.% (.39%.7%) versus the equilibrium premium of 5.44% (6.46%.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 +6% stocks nd -6% 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 L 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 L frmework in ctive mngement cn potentilly led to unintentionl trdes.. Formlly, reverse SR optimiztion problem sets Π tht solves 0 = rg mx ϖ ϖ ϖ ' Π λ( ϖ ϖ ) Σ ( ϖ ϖ). 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 derive individul stock lphs by reverse optimiztion of wht they lbel s the optiml tile portfolio (OTP); see their eqution (6). Note tht the reverse optimized lphs re vlid only if none of the constrints re 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. See disclosures on lst pge which re n importnt prt of this document. 5

6 The lck-littermn Model For Active Portfolio Mngement (continued) As we previously discussed, the objective of ctive mngement is to mximize 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 mkes sense in this informtion-less cse is to just hold the benchmrk portfolio nd mke no ctive trdes. However, the nlysis below will show tht, surprisingly, IR mximiztion will led to ctive trdes in this exmple. As suggested by the literture nd explined bove, it hs become the stndrd procedure to derive the benchmrk-implied equilibrium excess return, or Π, through reverse optimiztion, ccording to eqution (7) s γσω. The optiml vector of ctive positions given equilibrium ssumptions nd no ctive views in this cse cn be derived by simply substituting γσω into the expected excess returns in eqution (6). Tht is, ϖ, Π = Σ γ Σ ϖ ' ϖ γ Σϖ λ γ = ϖ ' Σ ϖ Σ ϖ λ Notice tht ϖ' Σ ϖ is sclr equl to the covrince of the globl minimum vrince portfolio,, nd the benchmrk portfolio,. Together with the investment budget condition of ' ϖ = ' Σ γ we cn determine tht Π = ' Σ ϖ ' Σϖ = Σ ϖ = = () ' Σ ' Σ 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 [999] for detils. Substituting () for (0), the optiml vector of ctive positions under the no-lph informtion scenrio cn be given s γ Σ γ ϖ ϖ ϖ ϖ () λ, Π = = ' Σ λ Eqution () suggests tht unless the client chooses the s the benchmrk portfolio such tht ϖ = ϖ, using the L model will generte set of ctive trdes even in this cse 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 SR, versus the lph-trcking error efficiency, which ttempts to mximize the IR insted. Some discussion on this topic ppers in Roll [99] nd Lee [000, Chpter ]. 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 the implied equilibrium excess return. (0) See disclosures on lst pge which re n importnt prt of this document. 6

7 The lck-littermn Model For Active Portfolio Mngement (continued) 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 it. As result, wht seems to be t equilibrium under the mximum-sr criteri is, by definition, t disequilibrium, nd ctive trdes re then initited to restore the portfolio to the mximum-ir condition. In the prior exmple, where stocks nd bonds re the only portfolio ssets, the implied equilibrium return of stocks is higher thn for bonds under the mximum-sr criteri. Therefore, even in the bsence of ny ctive view, the long-stocks/short-bonds ctive trde is perceived s n lph-generting trde under the IR-criteri, even though this trde embeds bsolutely no lph informtion t ll. In the following sections, we further elborte on how the IR criteri perceives discrepncies in equilibrium returns s lph opportunities nd, furthermore, leds to portfolio tht is more risky thn the benchmrk portfolio. INFORMATION LESS VIRTUAL ALPHA In this section, we derive some performnce sttistics for the ctive portfolio within this informtion-less environment. Given no investment views, ll expected 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: ' ' ' ', (3) Trcking Error Active risk, mesured by trcking error, in this informtion-less exmple, is given by TE ' ' Informtion Rtio Therefore, IR is given by IR TE, (4) (5) See disclosures on lst pge which re n importnt prt of this document. 7

8 The lck-littermn Model For Active Portfolio Mngement (continued) 8 See disclosures on lst pge which re n importnt prt of this document. Eqution (5) revels the interesting result tht the perceived IR is proportionl to the squre root of the difference in vrince of the benchmrk portfolio nd the globl minimum vrince portfolio. As result, the riskier the benchmrk portfolio, the lrger 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 below: ', ' (6) The ctive bet with respect to the benchmrk is then given by 0 ', (7) Tht is, the informtion-less 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: ' (8) Voltility The vrince of the ctive portfolio cn lso be derived s follows: TE, ' Substituting equtions (4), (6), nd (7) gives (9)

9 The lck-littermn Model For Active Portfolio Mngement (continued) The more interesting nd relevnt cse is where the investment views, lthough uncertin, come with some meningful degrees of confidence. In summry, in this informtion-less exmple, the mismtch of objective function between mximum-sr nd mximum-ir leds the ctive mnger to believe tht positive IR is vilble nd, therefore, to initite set of ctive trdes tht leds to n ctive portfolio tht is more voltile thn the benchmrk portfolio nd with positive net bet exposure. LACK-LITTERMAN MODEL AND ACTIVE MANAGEMENT Implementing the L model in ctive mngement simply requires substituting the expected excess returns from thel frmework in eqution () for the optiml ctive positions in eqution (6). V ' V which cn be grouped into two terms s follows:, ' V ' V, V (0) The first term in eqution (0) corresponds to the cse of using the equilibrium expected excess returns s inputs to chieve the mximum-ir objective. Therefore, it is equivlent to the cse of informtion-less scenrio discussed erlier in which the mximum-ir objective will still led to ctive trdes s given by eqution (). The second term in eqution (0) hs similr functionl form, except tht the expected excess returns within 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 when the equilibrium excess returns re used, s depicted in eqution (). 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, 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 where 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 [999] will give rise to the following expression for V: V P' P P' Q P In generl, the equilibrium component, Π, is not negligible component of the vector of expected excess returns. As result, the unintentionl, informtion-less component of ctive trdes, ω,π, plys nontrivil role in ctive mngement when the L model is pplied. () See disclosures on lst pge which re n importnt prt of this document. 9

10 The lck-littermn Model For Active Portfolio Mngement (continued) 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. y now, it should hve become cler why, in the exmple using stocks nd bonds, the optiml tcticl trde is to 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. R emedies As cn be seen from previous discussion, the root cuse of n inconsistency in pplying the L frmework to ctive portfolio mngement is the mismtch between the optimiztion problem used to bck out the equilibrium implied excess returns (i.e., n unconstrined SR optimiztion) nd the one used to construct n ctive portfolio (i.e., constrined IR optimiztion). The most obvious wy to fix this problem is to mke both 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 [99] for n exmple. Therefore, prcticl remedy is to bck out the equilibrium implied excess returns using the sme IR optimiztion problem used to construct n ctive portfolio. This is done by replcing Π tht solves the reverse SR optimiztion problem s in (7) with Π tht implicitly solves the IR optimiztion problem one intends to use to build n ctive portfolio, i.e., reverse optimizing problem (3) with Π replcing µ nd imposing dditionl constrints one my hve for the portfolio. Formlly, Π implicitly solves: ϖ ϖ ϖ' Π λϖ Σ ϖ 0 = rg mx ' s.t. ll other constrints, () insted of Π, s suggested in lck-littermn [99], tht solves the following reverse SR optimiztion problem: ϖ ϖ ' Π λ( ϖ ϖ ) Σ ( ϖ ϖ ) 0 = rg mxϖ s.t. no constrint. Solving () explicitly t first seems difficult in the presence of other constrints in the optimiztion. However, it turns out tht if we choose ny Π whose elements re ll the sme, i.e., Π i = Π j i j, then the first term in the objection function of () drops out of the optimiztion s it becomes constnt (recll tht we hve constrint ϖ ' = 0 ), nd we re left with trcking error minimiztion problem. Clerly, we my minimize trcking error by setting ϖ = 0, i.e., tking no ctive weight. In other words, ny constnt vector of expected excess return, Π, lwys implicitly solves () 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 returns, 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 See disclosures on lst pge which re n importnt prt of this document. 0

11 The lck-littermn Model For Active Portfolio Mngement (continued) implementtion for ctive portfolio mngement. Since we cn drop benchmrk weight from the IR optimiztion problem nd focus exclusively on optimizing ctive portfolio weights mx ' ' s.t. to ll other constrints, (3) where V from setting Π = 0 in (), we cn build ny ctive portfolio by only focusing on constructing n lph overly portfolio. The totl portfolio will simply be the sum of the benchmrk nd lph overly portfolio:. For exmple, suppose we re mnging long-only portfolio with the S&P 500 s its benchmrk tht llows leverge through borrowing up to 5% of sset vlue. Suppose further tht we form ctive investment views of securities in the portfolio nd express them in the L frmework in the view mtrix P, view expected excess returns vector Q, nd view confidence mtrix Ω. The portble lph implementtion using the L frmework cn be chieved by first clculting expected excess returns of the ssets µ using () with Π set to zero, which re then used s inputs to the IR optimiztion problem in (3) to solve for ctive weights, ω, with the constrints: 0 ' 5% (less thn 5% leverge), (long-only constrint). Adding benchmrk weight, ω, to ctive weight, ω, results in our finl portfolio tht will respect ll constrints. To further illustrte our point, let us go bck to our originl exmple nd pply the remedies proposed bove. In other words, we now set 0, keeping ll other estimtes the sme, nd pply eqution () to derive the expected ctive returns resulting from our berish investment view on stocks versus bonds. Exhibit on the next pge reports the revised vlues. We cn now clculte the reverse-optimized ctive portfolio ssocited with the bove 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 comprised of short position in stocks (-6.%) nd long position in bonds (+6.%) for totl trcking error of %. Consequently, the finl totl portfolio weights re +58.9% (75%-6.%) nd +4.% (5%+6.%) on stocks nd bonds, respectively. As expected, the resulting portfolio underweight stocks nd overweight bonds reltive to the benchmrk thus being 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 L 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. To us, 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. See disclosures on lst pge which re n importnt prt of this document.

12 The lck-littermn Model For Active Portfolio Mngement (continued) 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 SR optimiztion behind the L frmework nd the IR optimiztion in the ctive investment industry, we hve proposed remedy tht leds to portble lph implementtion of ctive portfolios. These resulting ctive portfolios reflect intentionl nd true investment insights of the investment process. EXHIIT Summry Sttistics of Two Assets Exmple: Stocks nd onds Mrket Cp Weight Voltility Correltion Equilibrium Excess Return lck-littermn Excess Return Stocks 75% 3% %.39% onds 5% 5%.0%.7% EXHIIT Revised Summry Sttistics of Two Assets Exmple: Stocks nd onds Mrket Cp Equilibrium lck-littermn Weight Excess Return Active Excess (enchmrk) Voltility Correltion ( ) Return Stocks 75% 3% % -.45% onds 5% 5% 0.0% +0.05% For illustrtive purposes only. See disclosures on lst pge which re n importnt prt of this document.

13 The lck-littermn Model For Active Portfolio Mngement (continued) REFERENCES est, Michel J., nd Robert 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 99: evn, Andrew, nd Kurt Winkelmnn, Using the lck-littermn Globl Asset Alloction Model: Three Yers of Prcticl Experience, Goldmn Schs working pper, June 998. lck, Fischer, nd Robert Littermn, Asset Alloction: Combining Investors Views with Mrket Equilibrium, Journl of Fixed Income, September 99: 7 8. lck, Fischer, nd Robert Littermn, Globl Portfolio Optimiztion, Finncil Anlysts Journl, September/October 99: ritten-jones, Mrk, The Smpling Error in Estimtes of Men-Vrince Efficient Portfolios, Journl of Finnce, Vol. 54, No., April 999: Chopr, V. K., nd Willim T. Ziemb, The Effects of Errors in Mens, Vrinces, nd Covrinces on Optiml Portfolio Choice, Journl of Portfolio Mngement, Vol. 9, No., Winter 993: 6. Drobetz, Wolfgng, How to Avoid The Pitflls in Portfolio Optimiztion? Putting The lck-littermn Approch At Work, Journl of Finncil Mrkets nd Portfolio Mngement, Vol. 5, No., 00: Grinold, Richrd C., nd Ronld N. Khn, Active Portfolio Mngement: A Quntittive Approch for Producing Superior Returns nd Controlling Risk, McGrw-Hill Professionl, 999. Hrvey, Cmpbell, John C. Liechty, Merrill W. Liechty, nd Peter Muller, Portfolio Selection With Higher Moments, Duke University working pper, 006. Hrvey, Cmpbell, John C. Liechty, nd Merrill W. Liechty, yes Vs. Resmpling: A Remtch, Journl of Investment Mngement, Vol. 6, No., 008: 7. He, Gungling, nd Robert Littermn, The Intuition ehind lck-littermn Model Portfolios, Goldmn Schs Quntittive Resources Group working pper,999. Herold, Ulf, Portfolio Construction with Qulittive Forecsts, Journl of Portfolio Mngement, Fll 003: 6 7. Idzorek, Thoms M., A Step-by-Step Guide to the lck-littermn Model: Incorporting User-Specified Confidence Levels, Zephyr Assocites working pper, 004. Jgnnthn, Rvi, nd Tongshu M, Risk Reduction in Lrge Portfolios: Why Imposing the Wrong Constrints Helps, Journl of Finnce, Americn Finnce Assocition, Vol. 58(4), 003: Jobson, J.D. nd ob Korkie, Putting Mrkowitz Theory to Work, Journl of Portfolio Mngement, Vol. 7, No. 4, 98: Jones, Robert, Terence Lim, nd Peter J. Zngri, The lck-littermn Model for Structured Equity Portfolios, Journl of Portfolio Mngement, Winter 007: Lee, Wi, Theory nd Methodology of Tcticl Asset Alloction, John Wiley & Sons, 000. Michud, Richrd O., The Mrkowitz Optimiztion Enigm: Is Optimized Optiml? Finncil Anlysts Journl, Vol. 45, No., Jnury/Februry 989: 3 4. Michud, Richrd O., Efficient Asset Mngement: A Prcticl Guide to Stock Portfolio Optimiztion nd Asset Alloction, Oxford University Press, 998. Roll, Richrd, A Men/Vrince Anlysis of Trcking Error, Journl of Portfolio Mngement, Vol. 8, No. 4, Summer 99: 3. Stchell, Stephen, nd Aln Scowcroft, A Demystifiction of the lck-littermn Model: Mnging Quntittive nd Trditionl Portfolio Construction, Journl of Asset Mngement, Vol.,, 000: Scherer, ernd, Portfolio Resmpling: Review nd Critique, Finncil Anlysts Journl, November/December 00: Shrpe, Willim F., Imputing Expected Security Returns from Portfolio Composition, Journl of Finncil nd Quntittive Anlysis, Vol. 9, No. 3, June 974: See disclosures on lst pge which re n importnt prt of this document. 3

14 The lck-littermn Model For Active Portfolio Mngement (continued) CONTACT INFORMATION Wi Lee, Ph.D Alexndre Schutel D Silv lexndre.dsilv@nb.com obby Pornrojnngkool, Ph.D bobby.pornrojnngkool@nb.com Disclosures This mteril is presented solely for informtionl purposes nd nothing herein constitutes investment, legl, ccounting or tx dvice, or recommendtion to buy, sell or hold security. No recommendtion or dvice is being given s to whether ny investment or strtegy is suitble for prticulr investor. Reders should not ssume tht ny investments in securities, compnies, sectors or mrkets identified nd described were or will be profitble. Informtion is obtined from sources deemed relible, but there is no representtion or wrrnty s to its ccurcy, completeness or relibility. All informtion is current s of the dte of this mteril nd is subject to chnge without notice. Any views or opinions expressed my not reflect those of the Firm s whole. Third-prty economic or mrket estimtes discussed herein my or my not be relized nd no opinion or representtion is being given regrding such estimtes. Certin products nd services my not be vilble in ll jurisdictions or to ll client types. Expected returns re shown for illustrtive purposes only nd should not be viewed s prediction of returns. Investing entils risks, including possible loss of principl. Pst performnce is no gurntee of future results. While hedge funds offer you the potentil for ttrctive returns nd diversifiction for your portfolio, they lso pose greter risks thn more trditionl investments. An investment in hedge funds is only intended for sophisticted investors. Investors my lose ll or substntil portion of their investment. When equity cpitl is mde vilble to compnies or investors, but not quoted on stock mrket. The funds rised through privte equity cn be used to develop new products nd technologies, to expnd working cpitl, to mke cquisitions, or to strengthen compny s blnce sheet. The investments to be mde by the Fund re specultive in nture nd the possibility of prtil or totl loss of cpitl will exist. The Neuberger ermn Group is comprised of vrious subsidiries including but not limited to Neuberger ermn LLC, Neuberger ermn Mngement LLC, Neuberger ermn Fixed Income LLC, N Alterntive Fund Mngement LLC, N Alterntive Investment Mngement LLC, N Alterntives GP Holdings LLC nd N Alterntives Advisers LLC. Neuberger ermn nd N Alterntives re mrketing nmes used by Neuberger ermn Group nd its subsidiries. The specific investment dviser for prticulr product or service is identified in the product offering mterils nd or pplicble investment dvisory greement. N Alterntive Fund Mngement LLC is registered investment dviser. This document is pproved by Neuberger ermn Europe Limited which is uthorized nd regulted by the UK Finncil Services Authority nd is registered in Englnd nd Wles, 5 nk Street, London, E4 5LE. J / N Alterntive Fund Mngement LLC. All rights reserved. 4