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Impact of Covariance Cleaning Techniques on

Minimum Variance Portfolios

March, 20, 2012

WHITE PAPER

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Impact of Covariance Cleaning Techniques onMinimum Variance Portfolios

Pedro Jandrey-Natal and Ksenya Rulik

March, 20, 2012

Abstract

Pedro Jandrey-NatalResearch assistantpedro-fernando.jandrey-natal@

polytechnique.org

Ksenya Rulik, PhD, CFAHead of Quantitative [email protected]

Minimum variance portfolio is constructed using an optimiza-tion procedure that takes as an input a covariance matrix.Inevitable estimation errors in the covariance lead to distor-tions of optimal allocation and should be addressed in the op-timization. One solution is to use advanced estimation tech-niques that aim at reducing the error. We analyze here the im-pact of statistical covariance cleaning on the minimum varianceportfolio. We build several minimum variance portfolios usingdifferent covariance estimators: two shrinkage estimators andtwo estimators based on Random Matrix theory, and comparevolatility of these minimum variance portfolios with a portfo-lio constructed with a simple empirical covariance. Our resultsshow that in presence of basic portfolio constraints, such as along-only constraint, the effect of covariance cleaning on ex-post volatility is negligible. This result is in accordance withearlier research of Jagannathan, Ma [2003] and Roncalli [2011]who showed that portfolio constraints in their turn induce im-portant transformations on covariance matrix. It turns outthat an explicit cleaning of covariance matrix is dominated bythe implicit cleaning induced by portfolio constraints. We alsodiscuss a duality between the two-norm quadratic portfolio con-straint and the James-Stein covariance shrinkage estimator andfind and approximate relation between the degree of covarianceshrinkage and the target for the two-norm constraint.

Impact of Covariance Cleaning Techniques on Minimum Variance Portfolios

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Minimum variance investing, an approachthat significantly reduces equity portfolio risk,has recently gained particular attention. Onthe one hand, the uncertain market environ-ment poses questions on whether the equityinvestments can be accessed with less risk.On the other hand, over the past decades ev-idence was gathered of long-term outperfor-mance of low volatility stocks with respect tohigh volatility stocks ([1, 6, 7]) that under-mines the widespread belief that higher riskis necessarily rewarded with higher returns.

Minimum variance portfolio provides in thisregard an answer to both questions. It is a wayto invest in equities with significantly less risk(30% of volatility reduction is the number thatis often cited by diferent sources 1). Moreover,by construction, the minimum variance port-folio has important exposure to low-volatilitystocks that historically were shown to performmuch better than predicted by the Capital As-set Pricing Model.

Estimation of covariance matrix of stock re-turns is at the heart of the minimum varianceapproach. The information contained in single-stock volatilities and pairwise correlations istranslated into the portfolio positions via atechnique minimizing expected portfolio vari-ance. The resulting allocation gives the min-imal possible ex-ante volatility of the overallportfolio under constraints: being long-only,fully invested, and satisfying additional con-straints the investor might want to impose.

The quality of the covariance matrix estima-tion is thus an important challenge here. Themost straightforward way to accomplish this

1For example, the MSCI Minimum Volatility Worldindex backtested back to 1995 had around 30% re-duction in volatility with respect to the market-capitalization weighted index over the period 1995-2007, as documented by MSCI BARRA Research(Nielsen F., Aylursubramanian R., Far From theMadding Crowd - Volatility Efficient Indices, April2008).

task is to construct an empirical covariancematrix from the data on past returns. How-ever, it is well known that the simple histori-cal estimation contains errors coming from thefiniteness of historical samples used for estima-tion, as well as from the changing nature of themarket volatility. A number of advanced toolswere proposed by financial econometrics to im-prove estimation efficiency of covariance ma-trix. In this paper we discuss the effect of someadvanced statistical tools of covariance clean-ing: shrinkage techniques and Random Matrixtheory cleaning, and we assess whether thesemethods can improve the minimum varianceportfolio constuction.

In the first section we introduce four dif-ferent covariance matrix estimation techniquesand their basic properties. In the second sec-tion, we first define the minimum variance op-timization setup and then build a set of min-imum variance portfolios using advanced co-variance estimation methods introduced in thefirst section. To assess the impact of the co-variance cleaning, we then compare realizedvolatility of different minimum variance port-folios corresponding to the different covariancecleaning techniques to that of the benchmarkportfolio built with simple empirical covariancematrix. Covariance estimation errors alwaysincrease the ex-post portfolio variance, so acleaning technique has an added value if it al-lows to reduce the ex-post portfolio variance,giving an indication that the estimation errorafter application of the technique is lower. Inthe third section we review the connection be-tween the covariance matrix transformationsand the use of additional constraints in portfo-lio optimization process. We consider in par-ticular the equivalence between shrinkage tech-niques and 2-norm quadratic constraints, andfind the implicit degree of covariance shrinkagethat is implied by imposing quadratic equalityconstraints for a minimum variance portfolio.We conclude with the discussion of the results.


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Covariance Matrix Estimationand Cleaning

Consider a universe of N stocks, with observedpast returns over T days. We denote rti thearithmetic return of the i-th stock at date t2.The empirical covariance matrix of stock re-turns (S) is defined as follows:

Si,j =1

T − 1

T∑t=1

(rti − ri)(rtj − rj), (1)

where ri = 1T

∑Tt=1 r

ti .

When the number of observations is not verylarge compared to the number of stocks (i.e.when T is comparable to N), this estimator isknown to perform poorly (see e.g[10] or [2]).

Academic literature on efficient covarianceestimation and forecasting techniques is vast.A widespread approach is based on factormodels, re-constructing covariance matrix forstocks from a smaller covariance matrix amongrisk factors and the corresponding factor load-ings. Another family of methods is purely sta-tistical and aims at reducing the noise part ofthe estimated matrix. Finally, covariance fore-casting techniques are very popular, that usethe stylized facts such as volatility clusteringto model the evolution of covariance structurein time (GARCH models for example). In thispaper we focus on the statistical techniques ofefficient covariance matrix estimation: shrink-age and Random-Matrix-Theory cleaning.

Shrinkage Techniques

Although the estimator introduced in the Eq.1is unbiased, it may be very imprecise. Thishappens because the error of an estimator canbe explained not only by its bias, but also byits variance, and it is the latter that can be

2The arithmetic returns are defined as: rt =Pt/Pt−1 − 1, where Pt is stock price on the date t

very large for the empirical estimator. Theaim of the so-called shrinkage techniques is tointroduce an estimator that minimises estima-tion error as a whole, by making a combina-tion (shrinking) of the original estimator witha biased, though more efficient (with smallervariance) estimator.

The simplest example of a shrinkage esti-mator is the James-Stein shrinkage estimator(JSSE), defined as a combination of the em-pirical covariance and an identity matrix:

JSSE = [(1− α∗)S + α∗ν∗Id] , (2)

where Id is identity matrix, and α∗ and ν∗ areconsistent estimators of the constants resultingfrom the following minimisation program:

minα,ν

E[‖(1− α)S + ανId− Σtrue‖2

], (3)

where Σtrue is the true covariance matrixof the underlying statistical process and ‖A‖equals Trace(A2) for a symmetric matrix A.For the exact expression of these parameters,see [11].

The JSSE estimator uses an identity matrixas its shrinkage target. This target correspondsto a covariance matrix where all pairwise cor-relations are zero and variances of all stocksare equal, that is quite unrealistic. But thegreat advantage of the indentity matrix is thatit has no variance at all, leading to a signifi-cant reduction of the estimator variance. Theparameter α is called the degree of shrinkage.When α is zero, the shrinkage is absent and theoriginal empirical estimator is restored. Whenthe parameter α = 1 the weight is completelytransfered onto the identity matrix, with noinformation from the empirical covariance left.

More realistic shrinkage targets are possible.One way to get a non-trivial target matrix isto consider the one-factor model introduced bySharpe (1962), in which stock returns are gen-erated by:

r = α+ βr0 + ε, (4)


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where α is a constant intercept, r0 representsmarket return vector, β is market sensitivity(stock beta) and the ε is a white noise uncor-related with the market. In this model, thecovariance matrix of stock returns B can beeasily computed3. The Ledoit-Wolf shrinkageestimator (LWSE) consists in shrinking theempirical covariance matrix towards a consis-tent estimator of B:

LWSE =[(1− α∗)S + α∗B

], (5)

where α∗ is a consistent estimator of the con-stant resulting from the following minimisationprogram:

minα

E[‖(1− α)S + αB− Σtrue‖2

]. (6)

For more information on the exact expres-sion of these parameters, see [10].

Covariance Cleaning with RandomMatrix Theory

Another technique used in covariance matrixestimation is the noise filtering with RandomMatrix Theory (RMT). Although the mathe-matical tools necessary to achieve a completeunderstanding of this approach are quite tech-nical, its general idea is rather simple. Thinkabout a no-structure covariance matrix: allcorrelations are zero and all variances are equalto 1. If a sample of random variables hav-ing this covariance structure is generated alongwith estimations of the covariance matrix onthe finite sample, the resulting empirical co-variance matrix will not be a pure identity, asthe true covariance is. There will be non-zerodiagonal terms, corresponding to false nega-tive or positive correlations and the diagonalelements - variance estimations - will indicate

3the exact expression of this estimator is Bi,j =σ2mβiβj + Σε, where σ2

m is variance of market returnsand Σε is covariance of the white noise components εi

that some series are more volatile than others,that is not true. All this false structure comesfrom estimation noise, and the magnitude ofthe estimation noise can be quantified basedon the sample size and the number of obser-vations. The empirical covariance matrix canbe compared to a matrix generated randomlyand factor out the noise from the statisticallysignificant information. The specific amountof structure imposed on the random matrix iscomputed according to a certain prior beliefwe have for the true underlying statistical pro-cess. In other words, the greater the agreementbetween empirical data and model predictionsis, the more structure we impose. Obviously,this approach will yield good results only if themodel is accurate enough.

The simplest estimator of this family,Marcenko-Pastur estimator (MP), performsthe comparison of empirical covariance matrixto the simplest no-structure noisy covariance.More precisely, the cleaned covariance matrixis obtained by cutting off the eigenvalues ofthe empirical covariance that may be consid-ered as meaningless noise. Intuitively, withthis method, the structure of the true covari-ance matrix is very similar to the identity ma-trix, apart from few most significant eigenval-ues that correspond to major common risk fac-tors. The method is thoroughly explained in[3].

An evolution of the previous method, thePower-Law estimator (PL), is obtained by re-placing all the eigenvalues of the empirical co-variance matrix by a set of new, syntheticeigenvalues. These eigenvalues are distributedaccording to a Power Law distribution (alsoknown as Pareto distribution). For a completediscussion on RMT cleaning recipes, see [4].

The methods of Random Matrix Theory areapplied to correlation matrix rather than to thefull covariance matrix, as the tools designedin this field allow to quantify the noise in co-variance estimated on normalized random vari-


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ables. To construct a cleaned covariance ma-trix it is necessary first to decompose the em-pirical covariance matrix in a product of a vari-ance vector and correlation matrix, then cleanthe correlation matrix with an Random MatrixTheory technique, and finally reconstruct thefull covariance matrix using the original vari-ance vector and the cleaned correlation matrix.

Minimum variance setup

Minimum variance allocation is a solution ofthe following optimization problem:

minw∈C

w′Σw, (7)

where w denotes a vector of portfolio weightsand C represents a set of constraints. We im-pose here a budget constraint, i.e.

∑iwi = 1,

and a no-short-sale constraint, i.e. wi ≥ 0, i =1, 2, ..., N .

Note that the optimization scheme takes Σ,the covariance matrix of stock returns, as aninput. Different covariance matrices wouldyield different optimal allocations, each min-imizing expected (ex-ante) portfolio variancefor the given covariance among the stocks.There is no reference to expected stock returnsin this setup, that amounts to a silent assump-tion of equal expected returns for all stocks inthe selected group.

In order to test the estimation techniquesfor the covariance matrix presented above, weconstruct for every different covariance estima-tion method an associated minimum varianceportfolio. We also build a benchmark mini-mum variance portfolio using the empirical co-variance matrix (no cleaning applied). Thenwe analyse the ex-post volatility of these min-imum variance portfolios over a time periodsubsequent to that of the covariance estima-tion period. As was discussed above, if one es-timator happens to be more efficient than theothers, we expect a smaller ex-post variance of

the minimum variance portfolio associated tothis estimator.

We use a dataset of N = 350 stocks thatwere present in the STOXX Europe 600 in-dex between November 2006 and April 20114.The period considered (from January 2005 toApril 2011) consists of 1680 days, that we sub-divide in overlapping periods of T = 500 daysto run separate covariance estimations. Each500-day period is shifted by 100 days with re-spect to the period preceeding it. Then wetake 100 days after each covariance estima-tion period to compute realized volatility ofthe minimum variance portfolios. This test isequivalent to backtesting a set dynamic mini-mum variance strategies, one for each covari-ance estimation technique. The strategies arerebalanced every 100 days, with new covari-ance matrices estimated over 500-days windowpreceeding the rebalancing date. Finally, wecompute the volatilities of these strategies on100-day rolling window.

Fig.1 shows the results of this exercise. Forcomparison, we plot on the graph the 100-dayroling volatility of the STOXX Europe 600 in-dex over the same period. The comparison tothe STOXX 600 shows that all the minimumvariance strategies achieved significant volatil-ity reduction with respect to the market capi-talization weighted index. Still, it can be notedthat all covariance estimation techniques yieldvery similar ex-post volatilities and the moresophisticated techniques employing covariancecleaning produced no significant change withrespect to the empirical covariance matrix.

The failure of the covariance cleaning tech-

4The data is a courtesy of STOXX. The STOXX in-dices are the intellectual property (including registeredtrademarks) of STOXX Limited, Zurich, Switzerlandand/or its licensors (”Licensors”), which is used underlicense. None of the products based on those Indicesare sponsored, endorsed, sold or promoted by STOXXand its Licensors and neither of the Licensors shall haveany liability with respect thereto.


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niques to bring additional ex-post risk reduc-tion for minimum variance portfolio is surpris-ing. We know that all these cleaning recipesdepend on a parameter determining the de-gree of cleaning that needs to be estimated. Bycleaning parameter we mean here the shrinkingconstant α (see Eq.2 and Eq.5) in the shrinkagecase and the proportion of eigenvalues that arecut off from the empirical covariance matrix inthe Random Matrix cleaning case. What if thecause of the apparent inefficiency of cleaning isprecisely the misestimation of the cleaning pa-rameter? To answer this question, we analysethe changes in realized volatility that are pro-duced by varying the cleaning parameter value.

Fig.2 shows the results of this analysis. Thevertical axis represents the ratio of ex-postvolatility of a minimum variance portfolio cor-responding to a given cleaning technique tovolatility of a minimum variance portfolio cor-responding to the empirical covariance matrix.If the ratio is smaller than 1, it means that thecleaning technique in question yields lower ex-post volatility of the minimum variance portfo-lio than the empirical covariance matrix. Thehorizontal axis represents the value of cleaning

parameter used in covariance estimation withcleaning. Thus each line on the plot representsvolatility reduction potential of one cleaningtechnique, applied with different values of thecleaning parameter. The volatility ratios areestimated as averages over the 100-day non-overlapping subperiods defined above.

Figure 2: Ratio of ex-post volatility of mini-mum variance portfolios with cleaning to thevolatility of the portfolio without cleaning, asa function of cleaning parameter.

Figure 1: Out-of-sample annualized volatilityof minimum variance portfolios. Note thatall estimation techniques yield very similar re-sults.


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We denote by empty circles the theoreticalcleaning parameter values that are automat-ically determined by the respective cleaningschemes. On the left extreme of the graph,when the cleaning parameter α is equal to zero,all the covariance matrices coincide with theempirical covariance matrix. At the oppositeextreme, when the parameter goes towards 1,the cleaning techniques approach their targets(in the case of shrinkage methods) or noise pri-ors (in the case of Random Matrix cleaning).Three of the four techniques (JSSE, LWSE,MP) yield a volatility ratio that can be greaterthan 1 and grows rapidly when approachingthe right edge of the graph, indicating that theresulting allocation is not of the minimum vari-ance type anymore and important informationfrom the covariance matrix has been cleanedaway. The fourth method (PL) has more con-sistent behaviour, and the respective volatilityratio stays smaller than 1 over almost all therange of the values of cleaning parameter.

The maximal volatility reduction with re-spect to the empirical covariance matrix isachieved by the PL and JSSE estimators.The magnitude of the maximal reduction is of1.45% for PL with cleaning factor α = 0.78,and of 1.39% for the JSSE estimator withcleaning factor α = 0.31. The two remainingcleaning methods, LWSE and MP, bring vir-tually no volatility reduction across the wholerange of cleaning parameter.

This analysis shows that even changing thevalue of cleaning parameter, the covariancecleaning methods still do not lead to substan-tial ex-post volatility improvement of minimumvariance portfolio. For comparison, the mini-mum variance portfolio based on the empiricalcovariance matrix has the volatility reduced byroughly 30% with respect to the market cap-italization weighted portfolio. The additionalreduction of 1.45% is hardly a significant re-sult. The cause of this limited noise correctionpotential of cleaning techniques in the mini-

mum variance framework is thus not in the es-timator construction itself.

Role of Portfolio Constraints

In this section we argue that the surprising re-sult of the previous section, namely that thereis no additional ex-post volatility reduction inminimum variance portfolio when using covari-ance cleaning techniques, can be explained bythe presence of constraints in the portfolio op-timization problem.

First we review some recent results thatdemonstrate that imposing additional con-straints in the optimization problem describedby Eq.7 is equivalent to solving an uncon-strained problem with a modified covariancematrix. Further we discuss a direct correspon-dence between covariance shrinkage (of theJSSE type) and optimization with a quadratic(norm) constraint.

Duality between constraints and co-variance transformations

Equivalence between imposing constraints andtransforming the input covariance matrix wasfirst pointed out in [8], where the authors usedas example the non-negativity constraint. Arecent overview of the impact of different typesof constraints on covariance matrix in the op-timization framework can be found in [12].

First consider a problem of finding a globalminimum variance portfolio subject only to abudget constraint(i.e.

∑iwi = 1). Mathemat-

ically speaking, we look for a vector of weightsw that satisfies

minw∈RN ,1′w=1

w′Σw, (8)

where 1 is a vector of ones. Using Lagrangemultipliers, it can be easily demonstrated thatthe solution w∗(Σ) reads5 :

5In this section, we use the matrix notation to stress


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w∗(Σ) =Σ−11

1′Σ−11. (9)

The budget constraint is an essential ingre-dient of the minimum variance optimization.Omitting this constraint results in an emptysolution, corresponding to a zero variance ab-solute minimum. Therefore we will call theproblem given by the Eq.8 and the solutionEq.9 unconstrained. This solution in generalrepresents a long-short portfolio, with 100%net long exposure to stocks.

Now, what happens to the solution when weimpose additional constraints? For the sake ofsimplicity, we consider only lower bounds onweights as additional constraints. A particularcase of the lower bound constraint is a non-negativity, or no-short-sale constraint wherethe lower bound is equal to zero. Let’s de-note by w(Σ) the solution of the minimizationproblem in Eq. 7, where

C = {w ∈ RN : 1′w = 1 and ∀i, wi ≥ w+i }.

(10)

It was demonstrated that for a constrainedproblem of the type Eq.10 there exists an asso-ciated unconstrained problem of the type Eq.8with a transformed covariance matrix Σ, suchthat w∗(Σ) = w(Σ), i.e. the solution of Eq.10using Σ is the solution of Eq.8 using Σ.

It is possible to prove that such matrix Σexists, and moreover it can be related to theoriginal Σ matrix. Following [12], Σ can bedecomposed as:

Σ = Σ + ∆. (11)

The matrix ∆ is called perturbation matrix,and its elements depend both on the initial co-variance Σ and the vector of lower bounds. Noexplicit expression of the ∆ matrix has been

the fact that the solution of the minimisation problem(8) depends on the input matrix.

given in the literature, but we believe thatthis implicit transformation of covariance ma-trix is significant and strongly alters the ini-tial matrix. Actually, the transformation is soprofound that it converts a long-short alloca-tion into a long-only allocation, when added tothe minimum variance optimizer. The clean-ing techniques applied to the covariance matrixhave much weaker effect on the backgroundof this implicit constraints-implied transforma-tion, and thus do not lead to significant im-provements in the ex-post volatility of the op-timized portfolio.

Below we give an example of a portfolio con-straint that produces a transformation of co-variance matrix that can be given explicitly,in the form of a covariance shrinkage alreadydiscussed above.

The norm constraint

Following the same lines, De Miguel et al. in [5]showed that imposing quadratic equality con-straint to an optimization scheme is equivalentto modifying the input matrix Σ in a very spe-cific way. The quadratic constraint reads:∑

i

w2i =

1

H, (12)

where H is a constant. Using Lagrange mul-tipliers, it can be easily verified that imposingthe constraint Eq.12 amounts to replacing theoriginal covariance matrix by its shrinkage esti-mate that has as a shrinkage target an identitymatrix (in other words, the JSSE estimator):

Σ = Σ + αId. (13)

Thus the quadratic constraint has the sameeffect on the minimum variance allocation asthe application of the JSSE shrinkage estima-tor.

Now, for a fixed constant H, what degreeof implicit shrinkage are we imposing via this


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constraint? In other words, given the dual-ity between the covariance shrinkage and thequadratic constraint, what value of the shrink-age parameter α is equivalent to a given valueof the parameter H? To answer this questionwe run two optimization algorithms, one is along-only constrained problem given by Eq.7and Eq.10 with a covariance matrix shrunkwith the JSSE estimator, and a long-only con-strained problem with an additional quadraticconstraint 12 and a simple empirical covari-ance matrix (no shrinkage applied). Compar-ing distances among the allocations resultingfrom the two algorithms with varying values ofthe shrinkage parameter α and the quadraticconstraint right-hand-side H, we build an em-pirical correspondence among the different αand H values. This exercise was repeated onoverlapping 500-day periods of our sample. Wehave been using the same set of stocks and thesame period as in the previous section. Theresults are shown in the Fig.3.

Figure 3: Relation between the constantH andthe equivalent shrinkage coefficient α. Notethat there are different curves for different sam-ples.

The graph shows the correspondence curvesbetween the cleaning parameter values (α) andthe constant in the right-hand side of quadratic

portfolio constraint (H). Different curves areobtained by using different 500-days subsam-ples of our observation periods. The curves aregrouped quite close to each other, indicatingthat the correspondence is sufficiently stable,but depends on the covariance matrix detailsthat change from one subperiod to another.

Note that even modest values of the H-parameter imply a significant degree of shrink-age (alpha rapidly approaching 1). To givethe reader a better understanding about theabove result, we recall that the value of theH-parameter can be seen as a measure of port-folio diversification. For example, for an equi-weighted portfolio of N assets (wi = 1/N), theinverse of the sum of square weights (inverseof the Eq. 12) by construction yields H = N .Talking about a non-equal weighted portfolio,it could be said that a (generic) portfolio withthe inverse sum of square weights equal to His as diversified as an equiweighted portfolio ofH assets. This means that the values of Hthat are much smaller than the overall numberof stocks in the eligible universe can be associ-ated with quite concentrated portfolios.

With this interpretation in mind, by varyingthe value of H (or, quivalently, the shrinkageparameter α) the allocation starts from a veryconcentrated no-shrinkage one (H = 5, α = 0),and goes towards an equi-weighted portfolio(limit of H = 350, α = 1). This means that theinitial portfolio (no covariance shrinkage, or noquadratic constraint applied) is concentratedin just a few stocks and significant amounts ofshrinkage (or the quadratic norm constraint onthe weights with higher values of H) needs tobe imposed to make the allocation more diver-sified. Note that the combined results of Fig.2and Fig.3 indicate that an optimal value forH would be around 30, that gives an implicitshrinkage corresponding to α = 0.3.


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Conclusion

Given the difficulties arising in mean-varianceoptimization because of estimation errors, par-ticular attention should be paid to the designof the optimization problem. As the recent re-search shows, there might be little scope in si-multaneous application of both advanced noisecleaning of covariance matrix and stringentportfolio constraints. Indeed, the constraintsthat bound the optimized allocation may beseen as implicit transformations of covariancematrix, altering the latter in very profoundways.

We show here that in minimum variance op-timizaton with long-only constraints the use ofstatistical covariance cleaning techniques, suchas skrinkage and cleaning with Random Ma-trix Theory, have very marginal effect on theproperties of the resulting portfolios. Only twoof the four considered techniques, namely thePower Law cleaning and James-Stein covari-ance shrinkage, brought slight additional re-duction in the ex-post variance with respectto the use of empirical covariance matrix.The other two methods, Ledoit-Wolf covari-ance shrinkage and Marcenko-Pastur eigen-value clipping, had virtually no effect.

The very same cleaning methods are knownto be particularly efficient in the absence ofnon-negativity constraints. Ledoit and Wolf[10] showed that JSSE estimator can reducethe volatility of minimum variance long-shortportfolios by up to 17% compared to the empir-ical covariance matrix. Bouchaud and Pottersin [4] showed that PL noise cleaning techniquereduces realized volatility of minimum varianceLong-Short portfolios by up to 23%.

Why do the cleaning recipes yield so differ-ent results in the presence of non-negativityconstraints? Intuitively, unconstrained mini-mum variance portfolios (i.e., portfolios whoseweights are allowed to be negative) tend toallocate substantial weight to both long and

short positions, thus neutralizing the most im-portant risk factors and leaving the exposureto the weakest noise components of the covari-ance matrix. But this is exactly the part ofthe covariance matrix that tends to be moremisestimated. That is why noise filtering isextremely important when dealing with long-short unconstrained portfolios.

On the other hand, errors in covariance esti-mation do not have significant impact on con-strained minimum variance portfolios, where,for example, negative weights are not allowed,or where particular diversification targets aregiven. This comes from the fact that portfolioconstraints act as powerful implicit transfor-mations of the covariance matrix, thus dwarf-ing the effect of the additional cleaning applied.For example, imposing a non-negativity con-straint on the weights of the minimum vari-ance portfolio is strictly equivalent to solvingan unconstrained optimization problem witha transformed covariance matrix as an in-put. In other words, by simply imposing non-negativity constraints, we are implicitly imple-menting a cleaning procedure.

One particular case when explicit dual co-variance transformation can be established,is the case of quadratic constraint in portfo-lio optimization. This constraint is very use-ful because it allows to target a specific levelof portfolio diversification, expressed by thevalue of the constant in the right-hand-sideof the constraint. This type of constraint isstrictly equivalent to the well-known covari-ance shrinkage, and the shrinkage degree is di-rectly related to the diversification target setby the quadratic constraint.

References

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Expected Returns, The Journal of Finance,Vol. 61, No. 1, 2006, pp. 259-299.

[2] Bouchaud J.P. and Potters M., 2003, The-ory of Financial Risk and Derivative Pric-ing, Cambridge University Press.

[3] Bouchaud J.P. and Potters M., 2005, Fi-nancial Applications of Random MatrixTheory: Old Laces and New Pieces, ActaPhysica Polonica B 36 2767-2784.

[4] Bouchaud J.P. and Potters M., 2009,Financial Applications of Random Ma-trix Theory: a short review, arXiv-qfinn0910.1205.

[5] DeMiguel, V., Garlappi L., Nogales J.,Uppal R., 2009,A generalized approach toportfolio optimization: improving perfor-mance by constraining portfolio norms.Management Science, 55, pp. 798-812.

[6] Haugen R.. and Baker N., The EfficientMarket Inefficiency of Capitalization-Weighted Stock Portfolios, Journal ofPortfolio Management, Vol. 17, No. 3,1991, pp. 35-40

[7] Haugen R. and Baker N., Case Closed,Handbook of Portfolio Construction:Contemporary Applications of Markowitztechniques, Springer, 2010.

[8] Jagannathan J. and Ma T. (2003), Reduc-tion in Large Portfolios: Why Imposingthe Wrong Constraints Helps, Journal ofFinance, 58(4), pp. 1651-1683.

[9] DeMiguel V., Garlappi L. , NogalesF.J., and Uppal R., A Generalized Ap-proach to Portfolio Optimization: Improv-ing Performance by Constraining Portfo-lio Norms, Management Science, vol.55,pp.798-812, 2009.

[10] Ledoit, O. and Wolf, M., 2003, Improvedestimation of the covariance matrix ofstock returns with an application to portfo-lio selection. J. Empirical Finance10 603-621.

[11] Ledoit, O. and Wolf, M., 2004, Awell-conditioned estimator for large-dimensional covariance matrices. J.Multivariate Anal. 88 365-411.

[12] T. Roncalli, Understanding the Im-pact of Weights Constraints in Portfo-lio Theory, working paper at SSRN:http://ssrn.com/abstract=1761625, 2011.


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