Journal of Banking & Finance 37 (2013) 1232–1242

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Journal of Banking & Finance

journal homepage: www.elsevier .com/locate / jbf

On portfolio optimization: Imposing the right constraints

Patrick Behr a, Andre Guettler b,⇑, Felix Miebs b

a Brazilian School of Public and Business Administration, Getulio Vargas Foundation, Rio de Janeiro, Brazilb Department of Finance, Accounting and Real Estate, EBS Business School, Wiesbaden, Germany

a r t i c l e i n f o

Article history:Received 6 December 2011Accepted 26 November 2012Available online 28 December 2012

JEL classification:G11

Keywords:Portfolio optimizationShrinkageMean squared errorBootstrap

0378-4266/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.jbankfin.2012.11.020

⇑ Corresponding author. Tel.: +49 611 7102 1236; fE-mail addresses: patrick.behr@fgv.br (P. Behr),

Guettler), felix.miebs@ebs.edu (F. Miebs).1 See Frost and Savarino (1986), Jorion (1986), Mi

Litterman (1992), among others.

a b s t r a c t

We reassess the recent finding that no established portfolio strategy outperforms the naively diversifiedportfolio, 1/N, by developing a constrained minimum-variance portfolio strategy on a shrinkage theorybased framework. Our results show that our constrained minimum-variance portfolio yields significantlylower out-of-sample variances than many established minimum-variance portfolio strategies. Further,we observe that our portfolio strategy achieves higher Sharpe ratios than 1/N, amounting to an averageSharpe ratio increase of 32.5% across our six empirical datasets. We find that our constrained minimum-variance strategy is the only strategy that achieves the goal of improving the Sharpe ratio of 1/N consis-tently and significantly. At the same time, our developed portfolio strategy achieves a comparatively lowturnover and exhibits no excessive short interest.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction turnover and short interest that do not hamper the practical imple-

Since the foundation of modern portfolio theory by Markowitz(1952), the development of new portfolio strategies has becomea horserace-like challenge among researchers. The sobering findingthat theoretically optimal, utility maximizing portfolios performpoorly out-of-sample1 can be attributed to the error prone estima-tion of expected returns, leading to unbalanced optimization results.This result directed researchers’ attention to the minimum-varianceportfolio, the only portfolio on the efficient frontier that simply re-quires the variance–covariance matrix as input parameter for theoptimization. For instance, Merton (1980), Jorion (1985), and Nelson(1992) remark that variance–covariance estimates are relatively sta-ble over time and can, hence, be predicted more reliably than ex-pected returns. Nevertheless, DeMiguel et al. (2009b) have arguedthat no single portfolio strategy from the existing portfolio selectionliterature outperforms the naively diversified portfolio, 1/N, consis-tently in terms of out-of-sample Sharpe ratio. Similar to Fletcher(2009), we evaluate in this paper a broader range of minimum-var-iance portfolios to challenge the findings of DeMiguel et al. (2009b).Additionally, we develop a constrained minimum-variance portfoliostrategy that outperforms 1/N in terms of a lower out-of-sample var-iance and a higher Sharpe ratio while, at the same time, yielding a

ll rights reserved.

ax: +49 611 7102 101236.andre.guettler@ebs.edu (A.

chaud (1989), and Black and

mentation of this portfolio strategy.We propose a minimum-variance portfolio strategy with flexi-

ble upper and lower portfolio weight constraints. Incorporatingthese constraints into the portfolio optimization process tradesoff the reduction of sampling error and loss of sample information(Jagannathan and Ma, 2003). On the one hand, weight constraintsensure that portfolio weights are not too heavily driven by thesampling error inherent in parameter estimates based on historicaldata, which often leads to highly concentrated portfolios.2 On theother hand, portfolio weight constraints cause a misspecification ofthe optimization problem because the resulting portfolio weightsare less driven by potentially useful sample information (Greenand Hollifield, 1992). Consequently, incorporating portfolio weightconstraints into the portfolio optimization problem is promising ifthe input parameters are error prone.

We calibrate portfolio weight constraints such that the desiredreduction of sampling error and the concomitant loss of sampleinformation is traded-off. Using this shrinkage theory based frame-work, we introduce portfolio weight constraints which depend onthe error inherent in the empirical variance–covariance matrixestimate. In particular, we impose the set of lower and upper port-folio weight constraints that minimizes the sum of the meansquared errors (MSE) of the covariance matrix entries. The latterserves as a loss function, quantifying the trade-off between thereduction of sampling error and loss of sample information.

2 See Green and Hollifield (1992), Chopra (1993), and Chopra and Ziemba (1993)for evidence concerning this point.

3 We solve the constrained minimum-variance portfolio optimization problemusing the Mosek quadratic programming solver quadprog for MATLAB.

P. Behr et al. / Journal of Banking & Finance 37 (2013) 1232–1242 1233

Our empirical results show that our constrained minimum-var-iance portfolio with lower and upper portfolio weight constraintsachieves substantial out-of-sample variance reductions in compar-ison to various minimum-variance portfolios. We observe that thevariance of our portfolio strategy achieves the lowest variancesamong all twelve considered portfolio strategies. In terms of riskadjusted performance, we observe that our portfolio strategy gen-erates a 32.5% higher Sharpe ratio than 1/N. This Sharpe ratio in-crease is statistically significant on five out of six datasets.Further, we observe that our portfolio strategy achieves on averagea higher Sharpe ratio than every other benchmark strategy. Thisfinding is robust with respect to the estimation window period,which we vary from 120 to 240 months.

Concerning the importance of weight constraints for the out-of-sample portfolio performance, we observe that imposing solitarylower or lower and upper weight constraints results in an equallyeffective risk reduction. The risk adjusted performance of bothportfolios is on average equally good. However, we find that theconstrained minimum-variance portfolio with lower and upperportfolio weight constraints achieves a less volatile Sharpe ratioover the various datasets than the constrained minimum-varianceportfolio with solitary lower portfolio weight constraints. This isreflected in the statistical significance of the outperformance over1/N. While imposing lower and upper weight constraints yields onfive out of six datasets a significantly higher Sharpe ratio than 1/N,imposing solitary lower weight constraints yields only on twodatasets a significantly higher Sharpe ratio. Further, we observethat the constrained minimum-variance portfolio with lower andupper portfolio weight constraints yields a lower turnover andshort interest than the constrained minimum-variance portfoliowith solitary lower weight constraints. Hence, we find that impos-ing lower and upper portfolio weight constraints is beneficial withrespect to the resulting out-of-sample performance and the practi-cal implementation of the portfolio strategy.

The impact of our ex ante calibrated weight constraints varieswith respect to the size of the investment universe. While the im-posed constraints are loose for small portfolios, they are compara-tively tight for larger investment universes. Specifically, weobserve for portfolios comprising 30 or more assets, that the ex antecalibrated lower portfolio weight constraints are close to zero, i.e. ashort-sale constraint. While the lower portfolio weight constraintsof our minimum-variance strategies with solitary lower, respec-tively lower and upper weight constraints are similar across allinvestment universes, we find that the tightness of the additionalupper constraint is particularly pronounced for larger universes.

Our paper contributes to three lines of literature. First, we addto the prevalent discussion whether optimized portfolios representa preferable investment vehicle over 1/N by amending the empir-ical evidence of DeMiguel et al. (2009b) and Fletcher (2009). Con-trary to recent contributions to this ongoing discussion by Pflugand Pichler (2012) and Kritzman et al. (2010) that assess the con-ditions rendering 1/N an optimal strategy, i.e. the reasons for theunsatisfying performance of optimal portfolios, we develop a port-folio strategy that outperforms 1/N consistently and significantly.Thus, our paper relates to Chevrier and McCulloch (2008), DeMig-uel et al. (2009a), and Tu and Zhou (2011), who claim to developportfolio strategies achieving consistently higher Sharpe ratiosthan 1/N. However, neither Chevrier and McCulloch (2008) norTu and Zhou (2011) provide statistical inference for their results.

Second, we extend previous work on portfolio optimization inpresence of constraints. Alexander and Baptista (2006) and Alexan-der et al. (2007) evaluate the imposition of drawdown constraints,while DeMiguel et al. (2009a) and Gotoh and Takeda (2011) assessthe impact of norm constraints on the portfolio optimization pro-cess. Our paper relates more closely to Frost and Savarino (1988),Grauer and Shen (2000), and Jagannathan and Ma (2003), evaluating

the role of portfolio weight constraints. While the aforementionedwork on weight constraints is concerned with arbitrarily chosenor ex post determined upper and/or lower constraints, we postulatea framework to determine these constraints ex ante. Our new ap-proach should thus perform better out-of-sample given that flexibleex ante constraints are better able to suit the data at hand.

Third, our framework represents a new approach to the estima-tion of the variance–covariance matrix for portfolio optimization.Similar to the Ledoit and Wolf (2003, 2004a,b) shrinkage strategies,our approach imposes a data-dependent structure on the variance–covariance matrix. Our approach, however, requires fewer assump-tions than the aforementioned shrinkage estimators. In particular,our framework requires neither any distributional assumptions,such as iid returns, nor the identification of a shrinkage target,which may have a significant impact on the out-of-sample portfo-lio performance.

The remainder of the paper is organized as follows. Section 2outlines our methodology and data, while Section 3 contains theempirical results. Section 4 reports the robustness checks, Section 5concludes.

2. Methodology and data

2.1. Calibrating portfolio weight constraints

We consider the standard myopic constrained minimum-varianceportfolio optimization problem. In particular, the objective is theminimization of the portfolio variance, w0Rw, where w denotes thecolumn vector of optimal portfolio weights and R the population var-iance–covariance matrix. Since R is not observable, an estimate basedon the available sample information has to be used instead. For thepurpose of our constrained minimum-variance portfolio strategy,we use the sample variance–covariance matrix, S ¼ 1

s�1 R0R, as an esti-mator of R, where R denotes the s� N matrix of de-meaned (in-sam-ple) returns, s the number of in-sample returns, and N the number ofassets. Accordingly, the sample based estimate of the portfolio vari-ance is given by w0Sw, where w represents the sample estimate ofw. Formally, we may express the constrained minimum-varianceportfolio optimization problem as follows3:

wmin

w0Sw ð1Þs:t: w01N ¼ 1; ð2Þ

wi P wmin; for i ¼ 1;2; . . . ;N; ð3Þwi 6 wmax; for i ¼ 1;2; . . . ;N: ð4Þ

The Kuhn–Tucker conditions (necessary and sufficient) areaccordingly:X

j

bSi;jwj � ki þ di ¼ k0 P 0; for i ¼ 1;2; . . . ;N;

ki P 0 and ki ¼ 0 if wi > wmin; for i ¼ 1;2; . . . ;N;

di P 0 anddi ¼ 0 if wi < wmax; for i ¼ 1;2 . . . ;N:

The notation is as follows: 1N denotes the column vector of onesof appropriate size, k the column vector of Lagrange multipliers forthe lower portfolio weight constraint, d the column vector of multi-pliers for the upper portfolio weight constraints, k0 the multiplier forthe portfolio weights to sum up to unity, and wmax as well as wmin

denote column vectors with uniform elements such that every assethas the same upper and lower weight constraint. Let wþþ denote thesolution to the constrained minimum-variance portfolio optimiza-tion problem in (1)–(4) and 1N a conformable column vector of ones.We may then state the following proposition:

1234 P. Behr et al. / Journal of Banking & Finance 37 (2013) 1232–1242

Proposition 1 Jagannathan and Ma, 2003. LeteS ¼ Sþ ðd10N þ 1Nd0Þ � k10N þ 1Nk0� �

; ð5Þ

then eS is symmetric and positive semi-definite, and wþþðeSÞ is one of itsglobal minimum-variance portfolios.

In particular, the proposition states that a global, i.e. uncon-strained, minimum-variance optimization using eS yields the sameresults as the constrained optimization on S. Jagannathan and Ma(2003) interpret the modified matrix eS as a variance–covariancematrix shrunk towards zero. The imposed portfolio weight con-straints shrink the extremely large and small elements of the var-iance–covariance matrix, which are most likely affected by theestimation error, towards zero (Jagannathan and Ma, 2003). Effec-tively, each element of the sample variance–covariance matrix, Si,j,is adjusted by the quantity (ki + kj) + (di + dj) whenever the portfolioweight constraints are binding. Hence, the imposed portfolioweight constraints impose a certain structure on the variance–covariance matrix estimate which (potentially) reduces the impactof sampling variation and thus the variance of the new variance–covariance matrix estimator, eS. Critically, the reduction of theestimation variance comes at a cost: the shrunk variance–covariance matrix, eS, is no longer unbiased. This stems from theaddition of the matrix spanned by the Lagrange multipliers,d10N þ 1Nd0� �

� k10N þ 1Nk0� �

, to the unbiased variance–covariancematrix estimate S, which causes the new estimator eS to be biased.This underpins the trade-off between the reduction of sampling er-ror and loss of sample information (Jagannathan and Ma, 2003). Onthe one hand, tighter portfolio weight constraints prevent the var-iance–covariance matrix estimate and the based upon portfolioweights from being too heavily driven by sampling error. On theother hand, they cause a misspecification of the optimization prob-lem and thus introduce a bias to the variance–covariance matrixand portfolio weight estimate (Green and Hollifield, 1992).

Since the minimum-variance portfolio depends solely on thevariance–covariance matrix, the estimation error inherent in port-folio weights stems solely from the error in the variance–covari-ance matrix estimate. Following this, we may assess theusefulness of portfolio weight constraints by exploiting the link be-tween the imposed constraints and the variance–covariance ma-trix. Accordingly, we aim to determine the set of portfolio weightconstraints (and corresponding Lagrange multipliers) that yield aminimum estimation error for the modified variance–covariancematrix, eS, and the based upon portfolio weights.

Following Ledoit and Wolf (2003, 2004a,b), we utilize a quadraticmeasure of distance between the modified variance–covariance ma-trix, eS, and the true but unknown variance–covariance matrix, R,under the Frobenius norm to quantify the estimation error:

Lðwmin;wmaxÞ ¼ Sþ d10N þ 1Nd0� �

� k10N þ 1Nk0� �

� R�� ��2

F : ð6Þ

Using the definition of the Frobenius norm that kZk2F ¼PN

i¼1

PNj¼1z2

i;j and taking expectations of (6) we arrive at the follow-ing risk function:

RFðwmin;wmaxÞ ¼ EðLðwmin;wmaxÞÞ

¼XN

i¼1

XN

j¼1

Eðsi;j þ ðdi þ djÞ � ðki þ kjÞ � ri;jÞ2

¼XN

i¼1

XN

j¼1

Varðsi;j þ ðdi þ djÞ � ðki þ kjÞ � ri;jÞ

þ ½Eðsi;j þ ðdi þ djÞ � ðki þ kjÞ � ri;jÞ�2

¼XN

i¼1

XN

j¼1

Varðsi;j þ ðdi þ djÞ � ðki þ kjÞÞ

þ ½Eððdi þ djÞ � ðki þ kjÞÞ�2: ð7Þ

Obviously, the risk function in (7) corresponds to the sum of theMSEs of the entries in the modified variance–covariance matrix, eS.Consequently, we observe that the sum of the MSEs of thevariance–covariance matrix entries consists of the variance andthe squared bias of the entries in eS. In order to find the minimumsum of the MSEs, we may thus be willing to deviate from the samplevariance–covariance matrix and accept the bias from portfolioweight constraints if the reduction in estimation varianceovercompensates the introduced bias. The idea of obtaining anMSE minimizing parameter estimate is closely connected toshrinkage estimation theory. It proposes finding a combination ofan unbiased but estimation error prone and a biased but estimationerror free parameter estimate to minimize the error of the resultingestimator.

Contrary to standard shrinkage estimation, however, we do notaim to find the linear combination of the empirical estimate with apredefined shrinkage target. We rather use a fixed shrinkage inten-sity since the new variance–covariance matrix eS is constructed bysimply adding S and d10N þ 1Nd0

� �� k10N þ 1Nk0� �

. The flexibility ofour approach stems from the variability of the matrixd10N þ 1Nd0� �

� k10N þ 1Nk0� �

which is determined by the tightnessof the imposed portfolio weight constraints on the portfolio opti-mization process. Thus, a major difference between our approachand standard shrinkage estimators is the necessity of defining ashrinkage target. While standard shrinkage estimators require adefinition of a shrinkage target, our proposed approach is free ofany such requirement.

Unfortunately, the quantities in the risk function (7) are not eas-ily obtainable since they depend on the vector of Lagrange multi-pliers d and k. As the distribution of the Lagrange multiplierscannot be determined in closed form, we resort to bootstrappingtechniques to get consistent estimates for Var (si,j + (di + dj) � (ki +kj)) and E ((di + dj) � (ki + kj)). In order to capture the time series ef-fects in returns, we employ the stationary bootstrap by Politis andRomano (1994). Hence, we randomly draw from the sample timeseries of returns, which is wrapped into a circle, a certain block,bi, of consecutive cross-sectional returns. The block size, bi, followsa geometric distribution with parameter 0 < q < 1. In empiricalapplications we set q = 0.2, resulting in an average block size of�b ¼ 1

q ¼ 5. The drawn blocks are concatenated as {b1,b2, . . .} untilthe resampled time series contains s or more returns, in which casethe resampled time series is truncated after the sth observation.Repeating the aforementioned steps K times leaves us with K boot-strap samples, each comprising a time series of s � N returns. Foreach bootstrap sample, k, we estimate the sample variance–covari-ance matrix and compute the constrained minimum-varianceportfolio for a given set of portfolio weight constraints. From thisprocedure, we obtain for the kth bootstrap sample the sample var-iance–covariance matrix estimate, Sk, as well as the vector of La-grange multipliers for the imposed upper and lower portfolioweight constraints, dk and kk. This finally enables us to estimateVar(si,j + (di + dj) � (ki + kj)) as:dVarBSðsi;j þ ðdi þ djÞ � ðki þ kjÞÞ

¼ 1K � 1

XK

k¼1

sk

i;j þ dki þ dk

j

� �� kk

i þ kkj

� �� �

� 1K

XK

k¼1

ski;j þ dk

i þ dkj

� �� kk

i þ kkj

� �� �!2

; ð8Þ

and E((di + dj) � (ki + kj)) as:

bEBSððdi þ djÞ � ðki þ kjÞÞ ¼1K

XK

k¼1

dki þ dk

j

� �� kk

i þ kkj

� �� �; ð9Þ

which are the standard bootstrap estimators of the variance andmean. Inserting these estimators into (7), we estimate the value

P. Behr et al. / Journal of Banking & Finance 37 (2013) 1232–1242 1235

of the risk function for any particular set of portfolio weight con-straints as:

bEðLðwmin;wmaxÞÞ ¼XN

i¼1

XN

j¼1

dVarBSðsi;j þ ðdi þ djÞ � ðki þ kjÞÞ

þ ½bEBSððdi þ djÞ � ðki þ kjÞÞ�2: ð10Þ

Since (7) quantifies the sum of the MSEs of the entries in thevariance–covariance matrix after the imposition of portfolioweight constraints, we calibrate the set of constraints to minimizethe consistent estimate of (7) given in (10). Accordingly, we formu-late the task of estimating the upper, w�max, and lower, w�min, portfo-lio weight constraints for the minimum-variance portfolio as thefollowing nonlinear optimization problem:4

w�min;w�max

min bE L w�min;w�max

� �� �¼XN

i¼1

XN

j¼1

dVarBSðsi;j þ ðdi þ djÞ � ðki þ kjÞÞ

þ ½bEBSððdi þ djÞ � ðki þ kjÞÞ�2; ð11Þ

s:t: �1 6 w�min 61N; ð12Þ

1N6 w�max 61; ð13Þ

which we solve by direct search.5 The bounds of the optimizationproblem in Eqs. (12) and (13) ensure that the portfolio weight con-straints do not violate the adding up constraint (2) of the constrainedminimum-variance portfolio optimization problem. Practitionersmay, of course, find it necessary to choose tighter intervals for thelower and upper bound in Eqs. (12) and (13) due to regulatory orinvestor specific constraints. Note that the bounds on the upperand lower portfolio weight constraints in (12) and (13) allow ourportfolio strategy to nest (i) the unconstrained minimum-varianceportfolio (if w�min and w�max are not binding), (ii) the short-sale con-strained minimum-variance portfolio (if w�min ¼ 0 and w�max is notbinding or not imposed), and (iii) the naively diversified portfolio(if w�min ¼ 1

N and/or w�max ¼ 1N).

The solution to (11)–(13) yields the portfolio weight constraintsthat we propose. We denote the resulting portfolio strategies byMin-L and Min-B for the minimum-variance portfolio with flexiblelower and lower and upper portfolio weight constraints. In order toget reliable estimates of the portfolio weight constraints, we setthe number of bootstrap iterations to K = 1,000.

2.2. Benchmark portfolio strategies

This section provides a brief overview of the portfolio selectionliterature from which we choose our benchmark portfolio strate-gies. We consider two groups of benchmark portfolios in our anal-ysis: Simple portfolio strategies which do not, or only sparsely, relyon historical data and minimum-variance portfolio strategies. Thefirst group includes the equally-weighted (1/N) and value-weighted (VW) portfolio strategy. The second group comprisesseveral minimum-variance portfolio strategies. We follow Fletcher

4 Note that we also consider the solitary imposition of lower portfolio weightconstraints in the latter such that the optimization problem is formulated in thesecases without constraint (13). We also considered the solitary imposition of upperweight constraints. However, we observed that solitary lower weight constraintsachieve a better out-of-sample performance.

5 See Marcucci and Quagliariello (2009) for an application of direct search to solvenonlinear optimization problems. We perform the direct search optimization usingJohn d’Errico’s fminsearchbnd for MATLAB. The function builds on MATLAB’sfminsearch optimization routine, extending it by variable transformation to allowthe imposition of bound constraints. To enforce convergence of the optimizer, werestart it from the last solution found, until no improvement is reached in thefunction value (using an error tolerance of 1e�8) and the variable (using an errortolerance of 1e�4).

(2009) and extend the selection of strategies used by DeMiguelet al. (2009b). The comprehensive selection of minimum-varianceportfolios allows us to amend the findings of DeMiguel et al.(2009b) and to benchmark our portfolio strategy to a wide rangeof established portfolio strategies (see Table 1).

The various minimum-variance strategies to which we bench-mark our approach can be attributed to the different variance–covariance matrix estimators, R, employed in these strategies. Allconsidered estimators differ in the way they address the problemof noisy data in estimating the variance–covariance matrix reli-ably. All portfolio weights are computed by solving the followingoptimization problem:

wmin

w0Rw; ð14Þs:t: w01N ¼ 1: ð15Þ

The optimization problem solves as:

w ¼ R�11N 10N bR�11N

� ��1; ð16Þ

which represents the estimate of the global minimum-varianceportfolio weights. For our analysis we focus on the most prominentvariance–covariance matrix estimators and corresponding mini-mum-variance portfolios in the portfolio selection literature. First,we evaluate the minimum-variance portfolio based on the samplevariance–covariance matrix, Min-U, given by:bRSample ¼ S: ð17Þ

A major drawback of this estimator is that it is highly error prone ifthe number of assets is not substantially lower than the number ofobservations for the variance–covariance matrix estimation. Theimpact of the estimation error can, however, be alleviated byimposing structure on the empirical variance–covariance matrixestimate.

The short-sale constrained minimum-variance portfolio, Min-C,aims to reduce the estimation error by imposing structure on theempirical variance–covariance matrix estimator via the prohibitionof short-sales. In order to compute the optimal weights of theshort-sale constrained minimum-variance portfolio, the optimiza-tion problem in (14) and (15) is extended by the following addi-tional weight constraint:

wi P 0; i ¼ 1;2; . . . ;N: ð18Þ

Jagannathan and Ma (2003) show that the portfolio weights of theshort-sale constrained minimum-variance portfolio correspond tothose of the unconstrained minimum-variance portfolio with thefollowing structured variance–covariance matrix estimate:bRC ¼ S� kC10 þ 1k0C

� �; ð19Þ

where kC denotes the column vector of Lagrange multipliers for theshort-sale constraint in the minimum-variance portfolio optimiza-tion problem.

The intuition of imposing structure on the variance–covariancematrix estimate also underlies the factor model based approach.Following the evidence from Chan et al. (1999), that consideringmore factors than the market for the estimation of the variance–covariance matrix does not notably affect the resulting out-of-sam-ple portfolio performance, we use the variance–covariance matrixestimate implied from Sharpe’s (1963) single-index model asbenchmark:bR1F ¼ s2

Mbb0 þW; ð20Þ

where s2M denotes the sample variance of market returns, b a col-

umn vector of slope coefficients, and W a N � N diagonal matrixof residual variances. The factor model based approach structures

Table 1List of considered portfolio strategies. The table lists the developed and the benchmark portfolio strategies. Panel A lists our minimum-variance portfolio strategies with flexibleportfolio weight constraints. Panel B shows the simple benchmark strategies which are considered following DeMiguel et al. (2009b). Panel C lists minimum-variance portfoliostrategies proposed in the literature.

# Description Abbreviation

Panel A. Developed portfolio strategies1 Minimum-variance portfolio with solitary lower portfolio weight constraints Min-L2 Minimum-variance portfolio with lower and upper portfolio weight constraints Min-B

Panel B. Simple benchmark strategies which do not require optimization3 Equally weighted (1/N) portfolio 1/N4 Value weighted (market) portfolio VW

Panel C. Minimum-variance portfolios5 Minimum-variance portfolio without constraints Min-U6 Minimum-variance portfolio with short-sale constraints Min-C7 Minimum-variance portfolio with the market as single factor Min-1F8 Minimum-variance portfolio with the variance–covariance matrix as weighted average between the sample variance–covariance and the single

factor variance–covariance matrixLW1F

9 Minimum-variance portfolio with the variance–covariance matrix as weighted average between the sample variance–covariance and a scalarmultiple of the identity matrix

LWID

10 Minimum-variance portfolio with the variance–covariance matrix as weighted average between the sample variance–covariance and a constantcorrelation covariance matrix

LWCC

11 Minimum-variance portfolio with a 1-norm constraint calibrated to minimize the out-of-sample variance NC112 Minimum-variance portfolio with a 2-norm constraint calibrated to minimize the out-of-sample variance NC2

1236 P. Behr et al. / Journal of Banking & Finance 37 (2013) 1232–1242

the empirical variance–covariance matrix estimator by eliminatingall idiosyncratic components in the off-diagonal elements of thevariance–covariance matrix, thereby reducing all covariances totheir systematic value. The resulting strategy is denoted by Min-1F.

Further, we include the Ledoit and Wolf (2003, 2004a,b) shrink-age estimators, which are an optimally weighted average of theempirical variance–covariance matrix and a shrinkage target, bF :bRLW ¼ abF þ ð1� aÞS: ð21Þ

The empirical estimator of the optimal combination intensity, a,proposed by Ledoit and Wolf (2003, 2004a,b) is thereby given by:

a ¼max 0;min1s

p� qw

;1

! !;

with p denoting a consistent estimator of the sum of the asymptoticvariances of the entries of the sample covariance matrix, q repre-senting a consistent estimator of the sum of asymptotic covariancesof the entries of the shrinkage target with the entries of the samplecovariance matrix and w denoting a consistent estimator of the mis-specification of the shrinkage target. From this, one can see that theoptimal shrinkage intensity constitutes a trade-off between the var-iance of the empirical estimate and the bias of the shrinkage targetand is restricted to taking values between 0 and 1.6

Following the authors, we consider three different candidatesfor the shrinkage target: The variance–covariance matrix impliedby the single-factor model (Ledoit and Wolf, 2003), a multiple ofthe identity matrix (Ledoit and Wolf, 2004a), and the constant cor-relation model implied variance–covariance matrix (Ledoit andWolf, 2004b). The three strategies are labeled LW1F, LWID, andLWCC.7

DeMiguel et al. (2009b) exclude the Ledoit and Wolf (2003,2004a,b) strategies explicitly, arguing that Jagannathan and Ma(2003) find no differences in the performance of a short-sale con-strained minimum-variance portfolio and the aforementionedstrategies. However, Jagannathan and Ma (2003) base their argu-mentation solely on the N > s case, i.e. the case where the number

6 We refer to Ledoit and Wolf (2003, 2004a,b) for a derivation of the optimalshrinkage intensity and an in-depth discussion of the estimators and the variousshrinkage targets.

7 The values for the shrinkage intensities used in the study are available from theauthors upon request.

of assets, N, exceeds the number of observations for the parameterestimation, s. In the opposite case, where N < s, we find that theperformance of the short-sale constrained minimum-varianceportfolio is mostly worse than the performance of the Ledoit andWolf (2003, 2004a,b) shrinkage strategies. This may be attributedto the strong structure imposed by short-sale constraints. Whilethe strong structure may be advantageous in cases where the esti-mation error is particularly severe, i.e. the ratio of N/s is large, itmay hamper the portfolio performance whenever N/s is small.The disregard of the aforementioned shrinkage strategies is note-worthy since the Ledoit and Wolf (2004a) strategy outperforms1/N on all of our considered datasets at least empirically.

Finally, we include the 1- and 2-norm constrained minimum-variance portfolios calibrated to minimize the out-of-sample vari-ance as suggested by DeMiguel et al. (2009a).8 We denote the 1-and 2-norm constrained portfolio strategies by NC1 and NC2. In orderto compute the optimal weights of the norm constrained minimum-variance portfolios, the optimization problem in (14) and (15) isextended by the following 1- respectively 2-norm constraints on w:

kwk1 6 /1; ð22Þkwk2 6 /2; ð23Þ

where the tuning parameters, /1 and /2, are determined via cross-validation to minimize the expected out-of-sample variance as de-scribed in DeMiguel et al. (2009a). The cross-validation procedureworks as follows. We first drop the tth cross-section of return obser-vations, rt, from the in-sample period and compute the resultingcovariance matrix based on the remaining s � 1 returns, denotedby SCV

t . Next, we determine the corresponding optimal portfolioweights, wCV

NC1;t and wCVNC2;t , based on SCV

t under the respective normconstraints (22) and (23) for given values for /1 and /2, respec-tively. Finally, we compute the returns, rCV

NC1;t and rCVNC2;t , that would

have been achieved from holding wCVNC1;t and wCV

NC2;t in period t, givenby wCV 0

NC1;trt and wCV 0NC2;trt . Repeating the aforementioned process for all

t = {1, . . . ,s} observations leaves us with s cross-validation returnsfor NC1 and NC2. We then compute the variance for each of the

8 We limit our analysis to the 1- and 2-norm constrained portfolios and do notinclude the F-norm and partial minimum-variance portfolios suggested by DeMiguelet al. (2009a). We omit the aforementioned portfolio strategies because the empiricalresults in DeMiguel et al. (2009a) show that the 1- and 2-norm constrained portfoliosachieve a similar out-of-sample performance as the omitted strategies.

P. Behr et al. / Journal of Banking & Finance 37 (2013) 1232–1242 1237

two portfolio strategies based on the s cross-validation returns, cor-responding to the expected out-of-sample variance of the respec-tive strategies.9 We repeat the outlined procedure until we obtainthe values for /1 and /2 minimizing the expected out-of-samplevariance.10

2.3. Portfolio optimization and performance measurement

We adopt the well established rolling sample approach to testthe performance of our constrained minimum-variance strategyin comparison to the considered benchmark portfolio strategies.Specifically, we revise the portfolios on an annual frequency, takingthe returns of the last s = 120 months as input for the variance–covariance matrix estimation. We then compute the optimal port-folio weights for all portfolio strategies. After the optimization pro-cess, we hold the portfolio weights constant for twelve months andcompute the resulting out-of-sample return. Repeating the afore-mentioned steps over the whole sample period, T, gives us a timeseries of T � s � 1 monthly out-of-sample returns for all portfoliostrategies, with which we assess the performance of the strategies.

We evaluate the out-of-sample performance of the portfoliostrategies using four measures: The portfolio variance, r2, theSharpe ratio, cSR, the turnover, TRN, and the average short interest,SI, of each portfolio strategy:

r2 ¼ 1T � s� 1

XT�1

t¼sðrtþ1wt � lÞ2; with l ¼ 1

T � sXT�1

t¼srtþ1wt; ð24Þ

cSR ¼ l� rf

r; ð25Þ

TRN ¼ 1T � s� 1

XT�1

t¼skwtþ1 � wtþk1; ð26Þ

SI ¼ 1T � s� 1

XT�1

t¼s

kwtk1 � 12

: ð27Þ

Here, rt+1 denotes a column vector containing the cross-section ofreturns at time t + 1, wt the sample estimate of optimal portfolioweights at time t, and kwtk1 the 1-norm of portfolio weights at timet; wtþ denotes the portfolio weights at the end of period t, includingthe shift in portfolio weights attributable to the returns over thetime period t, rt.

The out-of-sample variance, as a measure of how well theobjective of risk minimization is achieved, is of major importancefor the assessment of our proposed portfolio strategy. Second, weevaluate the risk adjusted performance of each portfolio strategyby means of the out-of-sample Sharpe ratio in order to reassessand augment the evidence whether no established portfolio strat-egy outperforms 1/N consistently. Finally, we evaluate the turn-over, i.e. the fraction of wealth that is traded each period, andshort interest, i.e. the amount of wealth that is held in short posi-tions, of each portfolio strategy to investigate how well suited eachportfolio strategy is for practical implementation. Both the turn-over and short interest may be associated with nonnegligible costsif it comes to the practical implementation of a portfolio strategy.Thus, the lower the turnover and short interest of a strategy, theeasier is that particular strategy implementable.

In order to compare the various benchmark strategies to ourportfolio strategy in terms of variance minimization, we testwhether the pairwise differences between the out-of-sample vari-ance of our portfolio strategy and each benchmark strategy are sta-tistically different from zero. Further, we add to the findings by

9 See Basak et al. (2009) for the point that the variance of the cross-validationreturns corresponds to the expected out-of-sample variance.

10 The values for /1 and /2 used in the study are available from the authors uponrequest.

DeMiguel et al. (2009b) by testing whether none of the consideredminimum-variance portfolio strategies outperforms 1/N signifi-cantly. Hence, we test whether the pairwise differences betweenthe out-of-sample Sharpe ratio of 1/N and each minimum-varianceportfolio strategy are statistically different from zero. Since stan-dard hypothesis tests do not control for time series characteristicsin portfolio returns (e.g., autocorrelation, volatility clustering, anddeparture from normally distributed returns), we employ boot-strapping techniques to overcome these problems.

We follow Ledoit and Wolf (2011) and employ a studentizedversion of the circular block bootstrap by Politis and Romano(1992) to test whether the difference between the variance ofour constrained minimum-variance portfolio with flexible lowerand upper weight constraints (Min-B) and a particular portfoliostrategy, i, is significantly different from zero. Accordingly, we con-struct a two-sided confidence interval for the difference of the (log)variances and then compute the resulting p-value for the afore-mentioned null hypothesis: H0 : log r2

i

� �� log r2

Min-B

� �¼ 0 using Re-

mark 3.2 by Ledoit and Wolf (2008).11

For the Sharpe ratio, we use Ledoit and Wolf’s (2008) studen-tized circular block bootstrap. In particular, we report thetwo-sided p-value for the null hypothesis that the Sharpe ratio ofportfolio strategy i is equal to that of 1/N: H0 : cSRi � cSR1=N ¼ 0.

For all bootstraps, we use a block length of b = 5 and base ourreported p-values on K = 1000 bootstrap iterations. We interpretdifferences between portfolio strategies as statistically significantif the two-sided p-value is below 0.1.

2.4. Data

We use six datasets which have been used extensively in theportfolio optimization literature to analyze the performance ofour proposed portfolio strategy. Table 2 lists the selected datasets.The ten and thirty industry portfolios, 10Ind and 30Ind, as well asthe six and twenty-five Fama French portfolios, 6FF and 25FF, offirms sorted by size and book-to-market ratio represent differentcuts of the US stock market and are taken from Ken French’swebsite.12

The single stock datasets, 50Stock and 100Stock, are con-structed from the CRSP database. The datasets are constructed ina similar fashion as in Chan et al. (1999), Jagannathan and Ma(2003), and DeMiguel et al. (2009a). Accordingly, each year in Julywe select the 50 respectively 100, in terms of market capitaliza-tion, largest stocks from the CRSP database that fulfill certain filtercriteria at that point in time. In particular, we filter out stocks thathave a price of less than $5 or exhibit missing returns in the pre-ceding 120 months and subsequent 12 months after the selectiondate. The drawn subset of 50 respectively 100 stocks then servesfor the next 12 months as investment universe and is reshuffledin July of the following year. For the risk free rate we take the30-day T-bill rate, which we also extract from Ken French’swebsite.

3. Empirical results

3.1. Performance analysis

Table 3 shows the out-of-sample variances of all consideredportfolio strategies and the corresponding p-values for the test thatthe variance of a particular portfolio strategy is different from that

11 The log-transformation follows Ledoit and Wolf (2011) for the purpose ofconducing better finite-sample properties.

12 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

Table 2List of the considered datasets. The table lists the various datasets used for the evaluation of the portfolio performance, their abbreviations, the number of assets that each datasetcomprises, the time period over which we use data from each particular dataset, and the data sources. All datasets comprise monthly data and apply in case of portfolio data thevalue weighting scheme to the respective constituents. The datasets from Ken French are taken from his website http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html and represent different cuts of the US stock market. The 50Stock and 100Stock dataset contain the 50 respectively 100 largest single stocks in terms of marketcapitalization on July each year after filtering out stocks that have a price of less than $5 or exhibit missing returns in the preceding 120 months and subsequent 12 months to theselection date.

# Dataset Abbreviation N Time period Source

1 6 Fama and French portfolios of firms sorted by size and book-to-market 6FF 6 07/1963–12/2008 K. French2 25 Fama and French portfolios of firms sorted by size and book-to-market 25FF 25 07/1963–12/2008 K. French3 10 industry portfolios representing the US stock market 10IND 10 07/1963–12/2008 K. French4 30 industry portfolios representing the US stock market 30IND 30 07/1963–12/2008 K. French5 50 Stocks with the largest market capitalization from the CRSP database 50Stock 50 07/1963–12/2008 CRSP6 100 Stocks with the largest market capitalization from the CRSP database 100Stock 100 07/1963–12/2008 CRSP

Table 3Variances of the portfolio strategies. The table reports the monthly out-of-samplevariances of all considered portfolio strategies. The p-values (in italics) are computedfor the null hypothesis that the (log) portfolio variance of a particular portfoliostrategy i is equal to that of our constrained minimum-variance portfolio withflexible upper and lower portfolio weight constraints (denoted by Min-B):H0 : log r2

i

� �� log r2

Min�B

� �¼ 0. We follow Ledoit and Wolf (2011) and employ a

studentized version of the circular block bootstrap by Politis and Romano (1992) tocompute the p-values, using a block length of 5 and 1,000 iterations.

6FF 25FF 10Ind 30Ind 50Stock 100Stock

Min-L 0.00176 0.00147 0.00135 0.00147 0.00157 0.001470.876 0.216 0.894 0.199 0.285 0.088

Min-B 0.00175 0.00149 0.00135 0.00142 0.00163 0.001341.000 1.000 1.000 1.000 1.000 1.000

VW 0.00212 0.00212 0.00213 0.00213 0.00206 0.002030.004 0.000 0.000 0.000 0.003 0.003

1/N 0.00240 0.00259 0.00188 0.00231 0.00212 0.002110.002 0.000 0.000 0.000 0.002 0.000

Min-U 0.00174 0.00156 0.00136 0.00167 0.00219 0.004830.670 0.303 0.701 0.002 0.000 0.000

Min-C 0.00202 0.00192 0.00139 0.00139 0.00140 0.001360.017 0.001 0.311 0.684 0.013 0.822

Min-1F 0.00243 0.00352 0.00144 0.00158 0.00168 0.001510.036 0.018 0.244 0.092 0.689 0.136

LW1F 0.00176 0.00146 0.00134 0.00141 0.00152 0.001260.906 0.611 0.412 0.750 0.104 0.089

LWID 0.00171 0.00140 0.00131 0.00143 0.00164 0.001420.201 0.069 0.053 0.869 0.814 0.128

LWCC 0.00214 0.00224 0.00136 0.00146 0.00148 0.001340.024 0.000 0.789 0.479 0.059 0.955

NC1 0.00182 0.00173 0.00132 0.00137 0.00154 0.001360.417 0.041 0.531 0.219 0.157 0.737

NC2 0.00183 0.00169 0.00133 0.00138 0.00156 0.001910.423 0.099 0.697 0.347 0.191 0.000

1238 P. Behr et al. / Journal of Banking & Finance 37 (2013) 1232–1242

of our constrained minimum-variance portfolio with upper andlower weight constraints (Min-B).13

Assessing the variances within the group of our developed strat-egies (Min-L and Min-B), we find that imposing both upper andlower, or solely lower, portfolio weight constraints does not resultin a significantly different out-of-sample variance except for the100Stock dataset. Here we find, that the imposition of both upperand lower weight constraints yields a significantly lower variance.

Vis-á-vis the sample and the factor model based minimum-variance portfolios, Min-U and Min-1F, we observe on average19.4% and 19.2% lower variances than for Min-B. The variancereductions in comparison to both benchmark strategies arestatistically significant on three out of the six considered datasets.In comparison to the short sale constrained minimum-varianceportfolio, Min-C, we note a variance reduction of 3.4% on average,with the variance of Min-B being significantly lower on the 6FF and25FF datasets. Yet, we find that Min-C achieves a significantlylower variance than Min-B on the 50Stock dataset.

Compared to the Ledoit and Wolf (2003, 2004a,b) shrinkagestrategies, we observe similar variances for Min-B and the strate-gies shrinking towards the factor model based variance–covariancematrix, LW1F, and a multiple of the identity matrix, LWID. Vis-á-vis the shrinkage towards the constant correlation model impliedvariance–covariance matrix, LWCC, we find that Min-B reducesthe variance by 7.3% on average. The heterogeneity in the portfoliovariances between the three shrinkage strategies indicates that thechoice of the shrinkage target may have a substantial impact onthe resulting out-of-sample portfolio performance.

Comparing Min-B to the norm constrained portfolios suggestedby DeMiguel et al. (2009a), we find on average similar variances forMin-B and the 1-norm constrained portfolio, NC1. Compared to the2-norm constrained portfolio, NC2, we note a 6.1% lower variancefor Min-B, with the variance of Min-B being significantly lower ontwo datasets.

With respect to the simple portfolio strategies that do not aimto minimize the portfolio variance, i.e. 1/N and VW, we find thatour constrained minimum-variance portfolios as well as almostany other minimum-variance portfolio strategy achieves signifi-cantly lower variances on all datasets.

Table 4 displays the out-of-sample Sharpe ratios of the variousportfolio strategies and the corresponding p-values for the test thatthe Sharpe ratio of a particular portfolio strategy is different fromthat of 1/N.

Comparing Min-B to the simple benchmark strategies, we notethat Min-B achieves on all datasets an empirically higher Sharperatio than 1/N and VW. The Sharpe ratio increase amounts on aver-age to 32.5% in comparison to 1/N and is significant on five out of

13 We also computed all pairwise p-values and base the interpretation concerningthe significance of our findings partly on these values. The detailed results are omittedto save space and are available upon request.

six datasets. In comparison to VW, we note a Sharpe ratio increaseof 64.1% on average, with the observed increase being significanton all considered datasets. The Min-L strategy achieves a similarlystrong Sharpe ratio increase, outperforming 1/N and VW on eachdataset. Yet, we find that the Sharpe ratio increase in comparisonto 1/N is only on two out of six datasets statistically significant,which may be attributed to a less pronounced increase on the30Ind, 50Stock, and 100Stock datasets.

The only portfolio strategies beside our constrained minimum-variance portfolios achieving a consistently higher Sharpe ratiothan 1/N are LWID and NC1. Yet, we observe that LWID and NC1achieve only on one respectively three datasets a significantlyhigher Sharpe ratio than 1/N.

Comparing Min-B directly to the various minimum-variancestrategies, we observe on average consistently higher Sharpe ratiosfor our portfolio strategy. Specifically, we find on average Sharperatio increases ranging from 2.2% (LW1F) to 20.1% (NC2). TheSharpe ratio increase of Min-B vis-á-vis Min-C and Min-1F is statis-tically significant on the 6FF and 25FF datasets in comparison to

Table 4Sharpe ratios of the portfolio strategies. The table reports the monthly out-of-sampleSharpe ratio of the portfolio strategies. We use the 30-day T-bill rate as the risk freerate. The p-values (in italics) are computed for the null hypothesis that the Sharperatio of a particular portfolio strategy i is significantly different from that of 1/N:H0 : cSRi � cSR1=N ¼ 0. The p-values are computed using the studentized circular blockbootstrap of Ledoit and Wolf (2008) with a block length of 5 and 1,000 iterations.

6FF 25FF 10Ind 30Ind 50Stock 100Stock

Min-L 0.1844 0.1811 0.1300 0.1223 0.1093 0.10890.041 0.040 0.280 0.282 0.197 0.221

Min-B 0.1772 0.1725 0.1299 0.1333 0.1162 0.13320.044 0.070 0.284 0.090 0.078 0.046

VW 0.0941 0.0941 0.0931 0.0932 0.0756 0.07540.079 0.111 0.033 0.448 0.259 0.483

1/N 0.1291 0.1352 0.1113 0.1100 0.0865 0.08421.000 1.000 1.000 1.000 1.000 1.000

Min-U 0.2075 0.2504 0.1376 0.0887 0.0863 0.09160.015 0.002 0.245 0.853 0.977 0.642

Min-C 0.1201 0.1200 0.1258 0.1195 0.1251 0.12650.999 0.996 0.274 0.267 0.017 0.028

Min-1F 0.1009 0.1199 0.1579 0.1183 0.1387 0.17060.632 0.677 0.224 0.525 0.063 0.104

LW1F 0.1940 0.2321 0.1396 0.1096 0.1075 0.10660.034 0.006 0.219 0.467 0.216 0.234

LWID 0.1607 0.2102 0.1408 0.1187 0.1016 0.09100.125 0.009 0.131 0.318 0.351 0.489

LWCC 0.1238 0.1756 0.1353 0.1112 0.1155 0.11690.991 0.299 0.259 0.556 0.143 0.251

NC1 0.1299 0.1844 0.1518 0.1150 0.1217 0.11450.675 0.088 0.056 0.283 0.046 0.207

NC2 0.1255 0.1934 0.1519 0.1340 0.1137 0.06570.763 0.060 0.054 0.093 0.111 0.811

Table 5Turnover and short interest of the portfolio strategies. The table reports the averagemonthly turnover (Panel A) and short interest (Panel B) of the portfolio strategies.Turnover is measured by the average percentage of total wealth traded in each period,while short interest is measured by the average amount of wealth that is held in shortpositions.

6FF 25FF 10Ind 30Ind 50Stock 100Stock

Panel A: TurnoverMin-L 0.157 0.274 0.087 0.165 0.221 0.202Min-B 0.149 0.269 0.087 0.162 0.167 0.140

VW 0.010 0.017 0.005 0.007 0.000 0.0001/N 0.015 0.017 0.023 0.029 0.048 0.050Min-U 0.172 0.476 0.102 0.302 0.386 0.856Min-C 0.026 0.047 0.034 0.043 0.087 0.047Min-1F 0.116 0.231 0.082 0.141 0.176 0.126LW1F 0.154 0.361 0.093 0.205 0.220 0.175LWID 0.076 0.218 0.074 0.189 0.221 0.217LWCC 0.093 0.231 0.070 0.162 0.186 0.154NC1 0.051 0.214 0.050 0.153 0.193 0.166NC2 0.056 0.215 0.050 0.154 0.241 0.320

Panel B: Short interestMin-L 1.087 2.031 0.402 0.792 0.828 1.253Min-B 0.985 1.913 0.391 0.750 0.612 0.807

VW 0.000 0.000 0.000 0.000 0.000 0.0001/N 0.000 0.000 0.000 0.000 0.000 0.000Min-U 1.380 3.224 0.675 1.686 1.579 4.586Min-C 0.000 0.000 0.000 0.000 0.000 0.000Min-1F 0.819 1.536 0.531 0.894 0.593 0.661LW1F 1.172 2.575 0.625 1.216 0.834 1.049LWID 0.554 1.635 0.440 1.052 0.890 1.323LWCC 0.613 1.514 0.389 0.916 0.626 0.914NC1 0.397 1.247 0.174 0.677 0.656 0.845NC2 0.426 1.331 0.179 0.793 0.696 2.010

P. Behr et al. / Journal of Banking & Finance 37 (2013) 1232–1242 1239

both benchmark strategies as well as on the 100Stock dataset incomparison to Min-C. Besides, we note significantly higher Sharperatios for Min-B in comparison to NC2 on the 6FF and 100Stockdataset. Further statistical significance of the Sharpe ratio increaseis obtained on the 6FF and 30Ind datasets with respect to LWCC aswell as on the 30Ind, 50Stock and 100Stock datasets vis-á-visMin-U.

Table 5 shows the turnover and short interest associated witheach portfolio strategy. Unsurprisingly, the long only portfoliostrategies 1/N, VW, and Min-C exhibit the lowest turnover of allportfolio strategies. The turnover of Min-L and Min-B is compara-tively similar, amounting on average to 18.1% and 16.7%, respec-tively. We find that the turnover of Min-L and Min-B is in therange of the turnover generated by the shrinkage strategies andthe norm constrained portfolio, ranging between 13.8% (NC1)and 20.7% (LW1F) on average. The by far highest average turnoveris generated by Min-U, amounting to 38.2%.

The high turnover of Min-U is reflected in the enormous averageshort interest of over 218.8% across the six datasets. The secondhighest average short interest is achieved by LW1F, amounting to124.5%, which is above the short interest of Min-L (106.5%). Theshort interest of the remaining portfolio strategies to which webenchmark our constrained minimum-variance strategies rangesbetween 66.6% (NC1) and 98.2% (LWID), which is comparable tothe average short interest of Min-B (91.0%). Given the moderateturnover and short interest of Min-B, we argue that our con-strained minimum-variance strategy with upper and lower portfo-lio weight constraints represents a practically implementableapproach to outperforming 1/N consistently and significantly.

3.2. Behavior of portfolio weight constraints

Fig. 1 depicts the behavior of the lower and upper portfolioweights of Min-L and Min-B in comparison to the lower andupper portfolio weights of the unconstrained minimum-varianceportfolio (Min-U) over the sample period. We report the minimum

and maximum portfolio weights for all three strategies eventhough the Min-U and Min-L strategies do not impose any, respec-tively only lower portfolio weight constraints. We do so because itallows us to evaluate the tightness of the imposed lower (Min-L),respectively upper and lower (Min-B) portfolio weight constraintsin comparison to the unconstrained solution (Min-U).

We find that the calibrated constraints behave as expected. Inparticular, we note that the constraints are mild on datasets wherethe number of assets is small and the variance–covariance matrixcan be estimated more reliably. Hence, we note only little differ-ences between the lower and upper portfolio weights of Min-Land Min-U on the 6FF and 10IND datasets. The same pattern ariseswhen comparing Min-B to Min-U. Accordingly, we observe thatMin-L and Min-B impose a similarly mild structure.

With an increasing number of assets and a concomitant in-crease of estimation error in the empirical variance–covariancematrix estimator, we note a tightening of the imposed portfolioweight constraints. Specifically, we find on the 25FF and 30INDdatasets that the lower portfolio weight constraints of both, Min-L and Min-B, vary around the �10% level, limiting the maximumshort position in a single asset to �10%. Contrary to the 6FF and10IND datasets, we note that the upper portfolio weight of Min-Bis notably below the upper weight of Min-U and Min-L. This showsthat Min-B imposes an even stronger structure on the empiricalvariance–covariance matrix and based upon portfolio weights thanMin-L on larger datasets.

The effect of tighter constraints is most pronounced on the50Stock and 100Stock datasets for which the estimation errorproblem is severest. Both, Min-L and Min-B, impose tight restric-tions on portfolio weights, resulting in sizable differences to Min-U. Specifically, we observe that the lower portfolio weight con-straint of Min-L and Min-B is close to zero, i.e. a short-sale con-straint, while the lower portfolio weight of Min-U varies around�20% on the 50Stock and between �20% and �60% on the

01/1980 01/1990 01/2000

−1

0

1

2

Min

./ m

ax. w

Min−L 6FF

01/1980 01/1990 01/2000

−1

0

1

2

Min

./ m

ax. w

Min−B 6FF

01/1980 01/1990 01/2000

−0.50

0.51

Min

./ m

ax. w

Min−L 25FF

01/1980 01/1990 01/2000

−0.50

0.51

Min

./ m

ax. w

Min−B 25FF

01/1980 01/1990 01/2000

−0.5

0

0.5

Min

./ m

ax. w

Min−L 10IND

01/1980 01/1990 01/2000

−0.5

0

0.5

Min

./ m

ax. w

Min−B 10IND

01/1980 01/1990 01/2000−0.5

0

0.5

Min

./ m

ax. w

Min−L 30IND

01/1980 01/1990 01/2000−0.5

0

0.5

Min

./ m

ax. w

Min−B 30IND

01/1980 01/1990 01/2000

−0.5

0

0.5

Min

./ m

ax. w

Min−L 50Stock

01/1980 01/1990 01/2000

−0.5

0

0.5

Min

./ m

ax. w

Min−B 50Stock

01/1980 01/1990 01/2000

−0.5

0

0.5

1

Min

./ m

ax. w

Min−L 100Stock

01/1980 01/1990 01/2000

−0.5

0

0.5

1

Min

./ m

ax. w

Min−B 100Stock

Fig. 1. Lower and upper portfolio weights of the developed constrained minimum-variance portfolios. The six figures on the left show the lower and upper portfolio weightsof our constrained minimum-variance portfolio with solitary lower portfolio weight constraints (Min-L). The six figures on the right show lower and upper portfolio weightconstraints of our constrained minimum-variance portfolio with lower and upper portfolio weight constraints (Min-B). Each figure also contains lower and upper portfolioweights of the unconstrained minimum-variance portfolio (Min-U), serving as a benchmark. The figures show the behavior of the portfolio weights for the six datasets overthe out-of-sample period 07/1973–12/2008. The portfolio weights of our two strategies (Min-L and Min-B) are depicted by the solid lines, while the portfolio weights of thebenchmark are represented by the dotted lines.

1240 P. Behr et al. / Journal of Banking & Finance 37 (2013) 1232–1242

100Stock dataset. Similar to the 25FF and 30IND datasets, we findthat the calibrated upper weight constraints of Min-B limit theportfolio weight constraint in comparison to Min-U and Min-L sub-stantially. In particular, we note that the upper portfolio weight ofMin-B varies around 7%, while the corresponding upper weights ofMin-L and Min-U vary around 26% and 42%.

4. Robustness checks

We check the robustness of our baseline empirical results intwo ways. First, we check whether our results depend on the cho-sen estimation period length by doubling the window size. Second,we test whether our numerical solutions are stable. Due to space

constraints, we briefly comment on these results but do not reportthem in additional tables. All results discussed, but not displayedhere, are available from the authors on request.

4.1. Estimation period length

In view of the findings in Kritzman et al. (2010), we double theestimation period to 20 years in this subsection. Similar to Kritz-man et al. (2010), we observe that the risk adjusted performanceof all considered minimum-variance portfolios increases such thatall but the Min-U strategy outperform 1/N empirically. The averageSharpe ratio increase for the various minimum-variance portfoliostrategies over 1/N amounts to 51.1%. We find similarly high

P. Behr et al. / Journal of Banking & Finance 37 (2013) 1232–1242 1241

Sharpe ratio increases for the Min-L and Min-B strategy. Accord-ingly, we note that the Min-B strategy outperforms 1/N in termsof Sharpe ratio by 66.8% on average. The respective increase forMin-L amounts to 62.9% on average. Vis-á-vis the various mini-mum-variance benchmark strategies, we observe Sharpe ratio in-creases for Min-B of between 4.4% (LWID) and 37.1% (Min-C) onaverage. As an exception, we note a 2.5% higher Sharpe ratio forLW1F on average. Overall we find that the Sharpe ratio improve-ments of Min-B and Min-L in comparison to the various mini-mum-variance benchmark strategies are in a similar range as forthe shorter estimation period comprising 120 months.

Concerning the out-of-sample variances, we find a similar pic-ture as for the shorter estimation period. Min-B achieves in com-parison to all benchmark strategies variance decreases rangingfrom 0.3% (LW1F) to 28.8% (Min-C), with the average decreaseamounting to 9.8%. Similar to the overall picture concerning thevariance and Sharpe ratio, we find that the turnover and shortinterest of Min-B and Min-L relative to the various benchmarkstrategies is in the same range as for the shorter estimation period.The absolute values of the turnover and short interest are therebyslightly below the values for the shorter estimation period.

4.2. Numerical stability

Our approach to determining portfolio weight constraints re-quires the numerical solution of the optimization problem in(11) and (12). To validate the numerical stability of our approach,we vary the starting points for the optimization problem in (11)and (12). As starting points, we choose the minimum (and maxi-mum) weights of (i) Min-U, (ii) Min-C and (iii) 1/N for determiningthe minimum (and maximum) portfolio weight constraints of Min-L (and Min-B). Additionally, we perform the optimization with (iv)random minimum and maximum portfolio weight constraintsdrawn from a uniform distribution with support [min(wMin�U),max(wMin�U)].

Our results show that the numerical solution of (11) and (12) andthe resulting minimum and maximum portfolio weights reported inFig. 1 are not driven by the choice of the start values for the opti-mizer. We find an average maximum deviation between the deter-mined portfolio weight constraints resulting from choosing differentstart values of between 1.67% (50CRSP) and 2.77% (30Ind) for Min-L.The average maximum deviation is computed as the average differ-ence between the largest and smallest lower respectively upperportfolio weight constraint resulting from the choice of differentstart values. The respective average maximum deviation for thelower portfolio weight constraint of Min-B amounts to values be-tween 1.72% (50CRSP) and 2.68% (30Ind) for Min-B. For the maxi-mum portfolio weight constraint of Min-B we observe valuesbetween 1.73% (50CRSP) and 2.49% (10Ind). Accordingly, we con-clude that the impact of choosing different start values is neglect-able for determining portfolio weight constraints.

5. Conclusion

In this paper, we reassess the finding by DeMiguel et al. (2009b)that no existing portfolio strategy outperforms 1/N in terms ofSharpe ratio out-of-sample. Evaluating a broader range of mini-mum-variance portfolios than DeMiguel et al. (2009b), we find thatthe Ledoit and Wolf (2004a) strategy and the 1-norm constrainedminimum-variance portfolio calibrated to minimize the out-of-sample variance, suggested by DeMiguel et al. (2009a), deliverempirically higher Sharpe ratios than 1/N. Yet, we find that theSharpe ratio increases of the aforementioned strategies are onlyon one, respectively three out of the six considered datasets signif-icant. Hence, our results corroborate the findings by DeMiguel et al.

(2009b) that no established portfolio strategy outperforms 1/Nsignificantly.

Contrary to the established portfolio strategies, our constrainedminimum-variance portfolio strategy developed in this paperachieves the goal of generating statistically significantly higherSharpe ratios than 1/N. Our portfolio strategy builds on shrinkageestimation theory, trading-off the reduction of sampling errorand loss of sample information by imposing a data dependentstructure on the empirical variance–covariance matrix estimate.The advantage of our approach over the established shrinkage ap-proaches to estimating the variance–covariance matrix is that weneither require any distributional assumptions such as iid returns,nor do we require the identification of a shrinkage target, whichhas a significant impact on the out-of-sample portfolio perfor-mance as our empirical results suggest.

The findings document that our constrained minimum-varianceportfolio achieves sizable out-of-sample variance reductions vis-á-vis various established minimum-variance portfolio strategies, i.e.the sample based, the short sale constrained, and the factor modelbased minimum-variance portfolio. Compared to the Ledoit andWolf (2003), Ledoit and Wolf (2004a) shrinkage strategies andnorm constrained minimum-variance portfolios suggested byDeMiguel et al. (2009a), we observe a similar out-of sample vari-ance. In terms of risk adjusted performance, we find that our port-folio strategy outperforms the value weighted portfolio and 1/Nconsistently in terms of Sharpe ratio. On average, we observe thatour portfolio strategy achieves across the six considered datasets a32.5% higher Sharpe ratio than 1/N. We find that the Sharpe ratioincrease is significant on five out of six datasets.

The outperformance of our constrained minimum-varianceportfolio is not plagued by excessive turnovers or short interest.We rather observe that our portfolio strategy achieves a compara-tively low turnover and short interest of 16.7% and 91.0%, respec-tively. This is in the range of the short interest and turnover ofthe established Min-1F, Ledoit and Wolf (2003), Ledoit and Wolf(2004a), Ledoit and Wolf (2004b) and norm constrained strategiessuggested by DeMiguel et al. (2009a). Hence, our portfolio strategyrepresents a practically implementable approach to outperforming1/N consistently.

Acknowledgements

We thank an anonymous referee, Michael Chernov, LorenzoGarlappi, Rasa Karapandza, Raman Uppal, Pedro Santa Clara, Mi-chael Wolf, and participants at the 8th Cologne Colloquium on As-set Management, 48th Southwestern Finance Conference, 12thConference of the Swiss Society for Financial Market Research,59th Midwest Finance Conference, seminar participants at EBSBusiness School, as well as DWS Investments for their helpful com-ments. All remaining errors are our own.

References

Alexander, G.J., Baptista, A.M., 2006. Portfolio selection with a drawdownconstraint. Journal of Banking and Finance 30, 3171–3189.

Alexander, G.J., Baptista, A.M., Yan, S., 2007. Mean-variance portfolio selection withat-risk constraints and discrete distributions. Journal of Banking and Finance 31,3761–3781.

Basak, G.K., Jagannathan, R., Ma, T., 2009. Jackknife estimator for tracking errorvariance of optimal portfolios. Management Science 55, 990–1002.

Black, F., Litterman, R., 1992. Global portfolio optimization. Financial AnalystsJournal 48, 28–43.

Chan, L.K.C., Karceski, J., Lakonishok, J., 1999. On portfolio optimization: forecastingcovariances and choosing the risk model. Review of Financial Studies 12, 937–974.

Chevrier, T., McCulloch, R.E., 2008. Using economic theory to build optimalportfolios. Working paper, Booth School of Business.

Chopra, V.K., 1993. Improving optimization. Journal of Investing 8, 51–59.Chopra, V.K., Ziemba, W.T., 1993. The effects of errors in means, variances, and

covariances on optimal portfolio choice. Journal of Portfolio Management 4, 6–11.

1242 P. Behr et al. / Journal of Banking & Finance 37 (2013) 1232–1242

DeMiguel, V., Garlappi, L., Nogales, F.J., Uppal, R., 2009a. A generalized approach toportfolio optimization: improving performance by constraining portfolionorms. Management Science 55, 798–812.

DeMiguel, V., Garlappi, L., Uppal, R., 2009b. Optimal versus naive diversification:how inefficient is the 1/N portfolio strategy? Review of Financial Studies 22,1915–1953.

Fletcher, J., 2009. Risk reduction and mean-variance analysis: an empiricalinvestigation. Journal of Business Finance and Accounting 36, 951–971.

Frost, P.A., Savarino, J.E., 1986. An empirical Bayes approach to efficient portfolioselection. Journal of Financial and Quantitative Analysis 21, 293–305.

Frost, P.A., Savarino, J.E., 1988. For better performance: constrain portfolio weights.Journal of Portfolio Management 15, 29–34.

Gotoh, J., Takeda, A., 2011. On the role of norm constraints in portfolio selection.Computational Management Science 5, 1–31.

Grauer, R.R., Shen, F.C., 2000. Do constraints improve portfolio performance?Journal of Banking and Finance 24, 1253–1274.

Green, R., Hollifield, B., 1992. When will mean-variance efficient portfolios be welldiversified? Journal of Finance 47, 1785–1809.

Jagannathan, R., Ma, T., 2003. Risk reduction in large portfolios: why imposing thewrong constraints helps. Journal of Finance 58, 1651–1684.

Jorion, P., 1985. International portfolio diversification with estimation risk. Journalof Business 58, 259–278.

Jorion, P., 1986. Bayes–Stein estimation for portfolio analysis. Journal of Financialand Quantitative Analysis 21, 279–292.

Kritzman, M., Page, S., Turkington, D., 2010. In defense of optimization: the fallacyof 1/N. Financial Analyst Journal 66, 31–39.

Ledoit, O., Wolf, M., 2003. Improved estimation of the covariance matrix of stockreturns with an application to portfolio selection. Journal of Empirical Finance10, 603–621.

Ledoit, O., Wolf, M., 2004a. A well-conditioned estimator for large-dimensionalcovariance matrices. Journal of Multivariate Analysis 88, 365–411.

Ledoit, O., Wolf, M., 2004b. Honey, I shrunk the sample covariance matrix. Journal ofPortfolio Management 30, 110–119.

Ledoit, O., Wolf, M., 2008. Robust performance hypothesis testing with the Sharperatio. Journal of Empirical Finance 15, 850–859.

Ledoit, O., Wolf, M., 2011. Robust performance hypothesis testing with the variance.Wilmott Magazine (September), 86–89.

Marcucci, J., Quagliariello, M., 2009. Asymmetric effects of the business cycle onbank credit risk. Journal of Banking and Finance 33, 1624–1635.

Markowitz, H., 1952. Portfolio selection. Journal of Finance 7, 77–91.Merton, R.C., 1980. On estimating the expected return on the market: an

exploratory investigation. Journal of Financial Economics 8, 323–361.Michaud, R.O., 1989. The Markowitz optimization enigma: is optimized optimal?

Financial Analysts Journal 45, 31–42.Nelson, D., 1992. Filtering and forecasting with misspecified ARCH models: getting

the right variance with the wrong model. Journal of Econometrics 52, 61–90.Pflug, G.C., Pichler, D.W.A., 2012. The 1/N investment strategy is optimal under high

model ambiguity. Journal of Banking and Finance 36, 410–417.Politis, D., Romano, J., 1994. The stationary bootstrap. Journal of the American

Statistical Association 89, 1303–1313.Politis, D.N., Romano, J.P., 1992. A circular block-resampling procedure for

stationary data. In: LePage, R., Billard, L. (Eds.), Exploring the Limits ofBootstrap. John Wiley, New York.

Sharpe, W.F., 1963. A simplified model for portfolio analysis. Management Science9, 277–293.

Tu, J., Zhou, G., 2011. Markowitz meets Talmud: a combination of sophisticated andnaive diversification strategies. Journal of Financial Economics 99, 204–215.