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The European Journal of Finance

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Multi-asset portfolio optimization and out-of-sample performance: an evaluation ofBlack–Litterman, mean-variance, and naïvediversification approaches

Wolfgang Bessler, Heiko Opfer & Dominik Wolff

To cite this article: Wolfgang Bessler, Heiko Opfer & Dominik Wolff (2017) Multi-asset portfoliooptimization and out-of-sample performance: an evaluation of Black–Litterman, mean-variance, and naïve diversification approaches, The European Journal of Finance, 23:1, 1-30,DOI: 10.1080/1351847X.2014.953699

To link to this article: http://dx.doi.org/10.1080/1351847X.2014.953699

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The European Journal of Finance, 2017Vol. 23, No. 1, 1–30, http://dx.doi.org/10.1080/1351847X.2014.953699

Multi-asset portfolio optimization and out-of-sample performance: an evaluationof Black–Litterman, mean-variance, and naïve diversification approaches

Wolfgang Besslera∗, Heiko Opferb and Dominik Wolffa

aCenter for Finance and Banking, Justus-Liebig-University Giessen, Giessen, Germany; bDeka Investment GmbH,Frankfurt, Germany

(Received 16 November 2012; final version received 7 August 2014)

The Black–Litterman model aims to enhance asset allocation decisions by overcoming the problems ofmean-variance portfolio optimization. We propose a sample-based version of the Black–Litterman modeland implement it on a multi-asset portfolio consisting of global stocks, bonds, and commodity indices,covering the period from January 1993 to December 2011. We test its out-of-sample performance relativeto other asset allocation models and find that Black–Litterman optimized portfolios significantly outper-form naïve-diversified portfolios (1/N rule and strategic weights), and consistently perform better thanmean-variance, Bayes–Stein, and minimum-variance strategies in terms of out-of-sample Sharpe ratios,even after controlling for different levels of risk aversion, investment constraints, and transaction costs.The BL model generates portfolios with lower risk, less extreme asset allocations, and higher diversifica-tion across asset classes. Sensitivity analyses indicate that these advantages are due to more stable mixedreturn estimates that incorporate the reliability of return predictions, smaller estimation errors, and lowerturnover.

Keywords: portfolio optimization; Black–Litterman; mean-variance; minimum-variance; Bayes–Stein;naïve diversification; 1/N ; Markowitz

JEL Classification: C61; G11

1. Introduction

The traditional mean-variance (MV) portfolio optimization (Markowitz 1952) has played aprominent role in modern investment theory and has been widely discussed and tested in theliterature. In theory, it provides the investor with the optimal asset allocation if the portfoliovariance and return are the only relevant parameters and future asset returns and the covariancematrix are given. However, when applied in an asset management setting, estimation errors ininput parameters (Jobson and Korkie 1981a; Michaud 1989), corner solutions (Broadie 1993),and high transaction costs, resulting from extreme portfolio reallocations (Best and Grauer 1991),often result in a poor out-of-sample portfolio performance.

Asset managers frequently try to cope with these shortcomings by implementing constraintson the portfolio weights and turnover. In the literature, several variations and extensions of MVare discussed that attempt to overcome its limitations. These suggestions range from imposingportfolio constraints (Frost and Savarino 1988; Jagannathan and Ma 2003; Behr, Guettler, andMiebs 2013) to the use of factor models (Chan, Karceski, and Lakonishok 1999), and Bayesianmethods for estimating the MV input parameters (Jorion 1985, 1986; Pastor 2000; Pastor and

∗Corresponding author. Email: wolfgang.bessler@wirtschaft.uni-giessen.de

© 2014 Taylor & Francis

2 W. Bessler et al.

Stambaugh 2000). In a recent study, DeMiguel, Garlappi, and Uppal (2009) analyze whetherMV strategies and various extensions suggested in the literature are able to outperform a naïve-diversified 1/N portfolio across a range of different stock data sets. They report that none ofthe different MV strategies consistently outperforms a naïve equally weighted portfolio (1/N) inan out-of-sample setting. However, Kirby and Ostdiek (2012) question these results, suggestingthat the results are mainly due to their research design and their data sets. They show that byfocusing on the tangential portfolio, DeMiguel, Garlappi, and Uppal (2009) place the MV modelat an inherent disadvantage in terms of turnover and estimation risk. Nevertheless, Kirby andOstdiek (2012) also find that the MV approach hardly outperforms the 1/N strategy at statisticallysignificant levels when transaction costs are included.

In contrast to the professional asset management industry, the academic literature has paid lit-tle attention to the performance of the Black and Litterman (1992) model, which was derivedas an alternative approach to portfolio optimization more than 20 years ago. Since then theBlack–Litterman (BL) model has experienced an increasing attention among quantitative port-folio managers (Satchell and Scowcroft 2000; Jones, Lim, and Zangari 2007). In the literature,only a few authors analyze the rationale of the BL model, provide examples for applying themethodology, or propose extensions of the model (Lee 2000; Satchell and Scowcroft 2000; Dro-betz 2001; Herold 2005; Idzorek 2005; Meucci 2006; Chiarawongse et al. 2012; Bessler andWolff 2013). So far, there exists only little empirical evidence for the out-of-sample performanceof the BL model.

Our study contributes to the literature by empirically testing the BL model performance out-of-sample. The main objectives are first to analyze whether the BL model is able to alleviatethe typical shortcomings of MV optimization, and second to evaluate its performance relativeto alternative asset allocation strategies in an out-of-sample setting with realistic investmentconstraints. We propose a sample-based approach of the BL model that allows us to computeits out-of-sample performance based on historic data sets. For the period from January 1993 toDecember 2011, we compute the out-of-sample performance of BL, MV, and Bayes–Stein (BS),as one of the most prominent extensions of MV, minimum variance (MinVar), as well as for twonaïve diversification approaches and evaluate the performance of each strategy. Furthermore, wepropose variations of the BL model by incorporating different reference portfolios and finallyanalyze expansionary and recessionary sub-periods separately.

Relative to other studies on asset allocation strategies such as DeMiguel, Garlappi, and Uppal(2009), Kirby and Ostdiek (2012), Behr, Guettler, and Miebs (2013), Murtazashvili and Voz-lyublennaia (2013), our study extends the literature with respect to the following three aspects.First, and most importantly, we include the BL model in the analysis. Second, we analyze all opti-mization strategies for three different investor types with maximum desired portfolio volatilitiesof 5%, 10% and 15%, while earlier studies usually focus only on the tangential portfolio. Third,we implement portfolio optimization strategies based on multi-asset portfolios. Earlier studiescited above focus solely on stock portfolios. Our empirical analysis covers the asset classesdeveloped and emerging stocks, government and corporate bonds, as well as commodities. Inparticular, we expect that the additional asset classes, government bonds and commodities, offerhigh diversification potentials, due to their historically low correlations with stocks. Given thatgovernment bonds usually gain in value during stock market downturns, asset allocation modelsthat shift wealth from stocks to bonds during these periods should benefit relative to naïve strate-gies such as the 1/N approach. Therefore, it is important to analyze how different asset allocationstrategies performed in a multi-asset context over the last two decades.

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Our empirical results offer new insights into several dimensions. We find that for a multi-assetuniverse all portfolio optimization strategies outperform relative to naïve equally weighted (1/N)or strategically weighted portfolios. Yet the outperformance is insignificant for most MV and BSportfolios, which is consistent with the results of Kirby and Ostdiek (2012) and DeMiguel, Gar-lappi, and Uppal (2009). However, the BL portfolios exhibit significantly higher Sharpe ratiosthan naïve strategies and consistently outperform MV. Additionally, BL portfolios have lowerportfolio turnover and offer more diversification across asset classes relative to MV portfolios.Our sub-period analysis suggests that the BL model outperforms MV and naïve-diversified port-folios particularly in recessionary periods. Sensitivity analyses indicate that the superiority of theBL model results from more stable mixed return estimates that include the reliability of returnestimates (‘views’) and lead to a lower portfolio turnover. A variety of sensitivity analyses indi-cates that our results are robust to different estimation windows, optimization constraints, andto restrictions of the optimized portfolio weights. Our results also hold for different levels oftransaction costs and when transaction costs are included in the optimization function as well asfor larger and different sets of asset universes.

The remainder of this study is organized as follows. In Section 2, we discuss the asset alloca-tion models employed, the related literature as well as the employed performance measures. Weprovide the data and some descriptive statistics in Section 3 and present and discuss our empiri-cal results in Section 4. Section 5 includes a description and discussion of the robustness checks.Section 6 concludes.

2. Methodology

We analyze the out-of-sample performance of different optimization techniques and naïve diver-sification rules. The optimization models employed include three alternative implementationsof the Black–Litterman (BL) model, the mean-variance (MV) and minimum-variance (MinVar)portfolios, as well as the Bayes–Stein (BS) approach as one of the most prominent extensions ofMV. The naïve diversification rules analyzed are equally weighted portfolios (1/N) and strate-gically weighted portfolios (st.w.). Table 1 summarizes all asset allocation models used in thisstudy. For the period from January 1993 to December 2011, we calculate monthly optimizedportfolios at the first trading day of each month, based on the different optimization approaches.

In line with DeMiguel, Garlappi, and Uppal (2009) and Daskalaki and Skiadopoulos (2011),we employ a rolling sample approach. This means that at any point in time t (month), weuse data available up to and including time t (k observations) to compute optimized portfolioweights. These weights are then used to compute the out-of-sample realized portfolio return overthe period [t, t + 1]. We repeat this process by moving the sample period one month forwardand computing the optimized weights for the next month. Rolling estimation windows offer theadvantage of being more responsive to structural breaks than expanding estimation windows. Toensure the robustness of our results, we analyze different estimation window lengths. All opti-mization models include realistic investment constraints. First, we include a budget restrictionaccording to Equation (1) which ensures that portfolio weights sum to one:

N∑i=1

ωi = 1, (1)

where ωi is the portfolio weight of asset i and N is the number of assets in the portfolio. Second,we exclude short selling since many institutional investors are restricted to long positions only

4 W. Bessler et al.

Table 1. Overview of asset allocation models included.

Number Model Abbreviation

Naïve asset allocation rules

1 1/N with rebalancing (benchmark strategy) 1/N2 Naïve diversification (benchmark weights) with rebalancing st.w.

Portfolio optimization models

3 Sample-based mean-variance MV4 Bayes–Stein BS5 Minimum-variance MinVar6 Black–Litterman with strategic weights as reference portfolio BL-st.w.7 Black–Litterman with 1/N as reference portfolio BL-1/N8 Black–Litterman with minimum variance as reference portfolio BL-MinVar

Notes: This table lists the various asset allocation models that we include in this study. The last columnof the table gives the abbreviation used to refer to the strategy in the tables in which the performance ofthe various strategies is compared.

(Equation (2)):

ωi ≥ 0, with i = 1, . . . , N . (2)

Third, we limit the maximum portfolio volatility in order to distinguish between differentinvestor types in terms of their maximum desired portfolio risk (Equation (3)):

√ω′

∑ω ≤ σP, (3)

where ω is the vector of portfolio weights, � is the covariance matrix of the asset returns, andσP is a predefined portfolio volatility constraint. The volatility constraint represents an uppervolatility bound, rather than a target volatility for the optimized portfolio. Such a volatility boundprovides the advantage that the asset allocation model may shift large fractions of wealth to low-risk government bonds during stock market downturns, thereby preventing losses, whereas fora specific target volatility (e.g. 10% p.a.) substantial fractions of wealth have to be invested instocks even if stock return estimates are negative. In the remainder of this section, we brieflydescribe the methodology of the various asset allocation strategies implemented in this study.

2.1 Naïve diversification rules

We compute two naïve-diversified portfolios that serve as benchmark portfolios for the optimiza-tion models. First, we calculate a 1/N strategy that invests equally in the N included assets. Thisnaïve diversified 1/N portfolio is a popular investment strategy for private investors (Benartziand Thaler 2001). In a recent study, Plyakha, Uppal, and Vilkov (2012) report that equal-weighted (1/N) stock portfolios with monthly rebalancing achieve higher total mean returns,four factor alphas (Fama and French 1993; Carhart 1997), and Sharpe ratios compared to bothvalue-weighted and price-weighted portfolios.1

As a second naïve diversification strategy, we compute strategically weighted portfolios (st.w.)in which each asset obtains a strategic weight that is constant over time. We account for threedifferent investor clienteles – a ‘conservative’, a ‘moderate’, and an ‘aggressive’ one – and setdifferent strategic weights for bonds, commodities, and stocks depending on the investor type. To

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Table 2. Strategically weighted portfolios.

Benchmark portfolio weights

Investor typeBonds

(%)Stocks

(%)Commodities

(%)

Historic volatilityof benchmark

portfolio (% p.a.)

Optimizationconstraint: max.

portfolio volatility(% p.a.)

Conservative 80 15 5 4.58 5.00Moderate 45 40 15 6.66 10.00Aggressive 10 65 25 9.72 15.00

Notes: This table provides the strategic weights for the three analyzed investor types: conservative, moderate, and aggres-sive, which are used to compute naïve-diversified portfolios and implied return estimates. Within the asset class stocks,emerging market stocks obtain a strategic weight of 25%, while developed market stocks obtain a strategic weight of75%. Accordingly, in the asset class bonds, high-yield corporate bonds have a strategic weight of 25% and governmentbonds 75%. We assume that the three investor types prefer a maximum expected portfolio volatility of 5%, 10% and15%, respectively, which is compatible with the historic volatilities of the benchmark portfolios.

determine strategic weights we rely on the results of earlier studies and on discussions with prac-titioners. Analyzing multi-asset portfolios including US stocks, bonds, and commodities for theperiod from 1974 to 1997, Anson (1999) suggests that a moderate investor should allocate about15% to commodities. Usually the risk of commodities is similar to equities (Bodie and Rosansky1980; Gorton and Rouwenhorst 2006) and substantially larger than for bonds. Therefore, we setthe strategic allocation to commodities and stocks lower for conservative investors and higherfor aggressive investors. For the conservative, moderate, and aggressive investor clienteles, thestrategic weights for commodities are 5%, 15%, and 25% and for stocks 15%, 40%, and 65%,respectively. Accordingly, the strategic weights for bonds are 80% for the conservative investor,which might be, for instance, reasonable for pension funds, 45% for the moderate investor, and10% for the aggressive investor type. Table 2 summarizes the strategic weights for the differentinvestor types. The maximum desired portfolio volatilities are set to 5%, 10%, and 15% for theconservative, moderate, and aggressive investor clienteles, respectively, which is in line withthe historic volatilities of the strategically weighted portfolios for these investor types beforethe evaluation period from 1988 to 1992. We employ these volatility bounds as constraints forthe optimization strategies. As for the optimized portfolios, both naïve-diversified portfolios arerebalanced at every first trading day of each month. Rebalancing, however, only maintains thenaïve 1/N or strategic weight for each asset. As robustness check, we also compute equally andstrategically weighted buy-and-hold benchmark portfolios that do not include any rebalancing.Our findings also hold for the buy-and-hold benchmark portfolios, and the performance differsonly marginally between the portfolios with and without rebalancing. To be parsimonious, weonly report the results with rebalancing.2

2.2 Minimum-variance portfolio

The minimum-variance (MinVar) strategy selects portfolio weights that minimize the varianceof the portfolio returns. The minimization problem is

minω

ω′ ∑ω. (4)

To implement this policy we rely on the sample covariance matrix � using rolling estima-tion windows as described above. The advantage of the minimum-variance approach is that it

6 W. Bessler et al.

does not require any return estimates, which are usually subject to large estimation errors. Theeffects of estimation errors in the covariance matrix are about 10 times smaller than the effectsof estimation errors in returns (Chopra and Ziemba 1993). In line with all other optimizationstrategies, we include realistic investment constraints, specifically a budget restriction accordingto Equation (1) and disallow short selling according to Equation (2).

2.3 Mean-variance (MV) optimized portfolio

In the MV approach (Markowitz 1952), the investor optimizes a trade-off between risk andreturn. The mean-variance optimization problem is

maxω

U = ω′ μ − δ

2ω′ ∑ ω, (5)

where U is the investor’s utility, μ is the vector of expected return estimates, and δ is the coef-ficient of risk aversion. The Markowitz optimization framework assumes normally distributedreturns or mean-variance preferences, focusing only on the mean and variance of returns andignoring higher moments. Although this seems to be a critical assumption, Landsman and Nešle-hová (2008) show that it is sufficient that returns are elliptically symmetrically distributed sothat all investor preferences are equivalent to mean-variance preferences. Therefore, it is rea-sonable to rely on the mean-variance framework for portfolio optimization even if asset returnsare non-normal, but are symmetric. To implement the mean-variance strategy, we employ thesample mean μ and the sample covariance matrix � as described above. We include a budgetrestriction according to Equation (1), disallow short selling according to Equation (2), and limitthe maximum allowed portfolio volatility (Equation (3)). The risk aversion coefficient is initiallyset to 2 and variations are analyzed as robustness check.

2.4 Bayes–Stein shrinkage portfolio

The Bayes–Stein (BS) approach is one of the most prominent extensions of MV and thereforeincluded in our analysis. The BS approach, proposed by Jorion (1985, 1986), is based on theidea of shrinkage estimation by Stein (1955) and James and Stein (1961) and attempts to reduceestimation error of the input parameters of MV by employing a Bayesian approach for estimatingreturns and the covariance matrix. The optimization procedure is the same as in the MV approachpresented in Equation (5). The intuition of BS is to reduce estimation errors by shrinking thesample mean μ toward the expected return of the minimum-variance portfolio μmin with

μmin = w′min μ =

�1′�−1

�1′�−1⇀

1μ, (6)

where⇀

1 is a vector of ones and � is the covariance matrix. BS shrinks the sample estimates byusing return estimates of the form:

μBS = (1 − γ )μ + γ μmin �1 with γ = N + 2

(N + 2) + K(μ − μmin�1)′�−1(μ − μmin�1). (7)

The BS approach estimates the covariance matrix as � = ((K − 1)/(K − N − 2))�, whereμmin is the expected return of the minimum-variance portfolio, � is the usual unbiased samplecovariance matrix, N is the number of assets included, and K is the sample size (Jorion 1986). In

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addition to shrinking the return estimates, the Bayes–Stein approach also adjusts the covariancematrix that we implement as described in Jorion (1986). We employ the same optimization pro-cedure, optimization constraints, and estimation windows as for the MV approach and analyzevariations of all input parameters as a robustness check.

2.5 Black–Litterman portfolio

The BL model is an alternative approach for dealing with estimation errors in return estimates.It combines two sources of information: ‘neutral’ (‘implied’) returns and ‘subjective’ returnestimates. The latter are also referred to as ‘views’. The advantage of the BL approach is thatinvestors can either provide return estimates for each asset or stay neutral for some assets theyfeel less comfortable in making return forecasts. Moreover, the reliability of each return esti-mate can be incorporated allowing investors to distinguish between qualified estimates and pureguesses.

The basic idea is that investors should only depart from the reference portfolio, which mightbe the market or any other benchmark portfolio, if they are able to provide reliable estimates offuture returns. The BL model uses implied returns as a prior that we compute based on the assetweights of the reference portfolio. The implied excess returns used in the original Black andLitterman (1992) approach are derived using reverse optimization, assuming that the observablemarket or benchmark weights ω∗ are the result of a mean-variance optimization. Implied returnsused in the BL model are computed as

∏e= δ

∑ω∗, (8)

where∏

e is the vector of implied asset excess returns, � is the covariance matrix, δ is theinvestor’s risk aversion coefficient, and ω∗ is the vector of observable market or benchmarkweights.

The BL framework combines the vector of implied returns∏

with the investor’s ‘views’expressed in the vector Q, incorporating the reliability of each ‘view’ quantified in a matrix �. Toderive the combined return estimates, the original Black and Litterman (1992) paper referencesTheil’s mixed estimation model (1971). Figure 1 illustrates the procedure of the BL approach.

The combined return estimates are written as

μBL = [(τ�)−1 + P′�−1P]−1[(τ�)−1 + P′�−1Q] (9)

in which P is a binary matrix containing the information for which asset a subjective returnestimate exists, and τ is a factor that measures the reliability of implied return estimates. Thecombined return estimate is a matrix-weighted average of implied returns and ‘views’ withrespect to the correlation structure (Lee 2000). The weighting factors are the uncertainty mea-sures of implied returns τ and subjective return estimates �, which we discuss in the followingsection.

In the BL approach, the posterior covariance matrix is (Satchell and Scowcroft 2000)∑

BL=

∑+[(τ�)−1 + P′�−1P]−1. (10)

After computing combined return estimates and the posterior covariance matrix, a traditionalrisk-return optimization is conducted maximizing the investor’s utility function presented inEquation (5).

8 W. Bessler et al.

Figure 1. The procedure of the Black–Litterman approach (Idzorek 2005).

2.5.1 Implied returns and the parameter τ

The reference portfolio used to compute implied returns might have a strong influence on theout-of-sample performance of the BL model. Therefore, we employ three alternative referenceportfolios: the 1/N portfolio (BL-1/N), the strategically weighted portfolio (BL-st.w.) describedin Section 2.1, and the minimum-variance portfolio (BL-MinVar) described in Section 2.2. Theminimum-variance portfolio might be particularly reasonable for conservative investors, as theymight prefer to invest in the lowest risk portfolio if the reliability of their return estimates is low.However, for some portfolio managers it might be advantageous to stick to a certain benchmarkif return estimates are uncertain.

The parameter τ controls how distinctly the optimized portfolio may depart from the referenceportfolio. It reflects the uncertainty of implied returns and can be chosen based on a desired track-ing error. Allaj (2013) provides a discussion of the parameter τ . For very small values (τ → 0)the combined returns converge to implied returns and the BL optimized portfolio converges tothe reference portfolio. For large values (τ →∞) the combined returns converge to the ‘views’and the BL optimized portfolio converges to the MV portfolio in which the ‘views’ are the under-lying return estimates. In the literature, the values used for τ typically range between 0.025 and0.300 (Black and Litterman 1992; He and Litterman 1999; Drobetz 2001; Idzorek 2005). We startwith setting the parameter τ at a level of 0.100 and analyze variations in a sensitivity analysis.

2.5.2 ‘Subjective’ return estimates and their reliabilityAs described above, the BL model combines implied returns and subjective return estimates.The literature on the BL model does not provide a clear answer how to derive subjective returnestimates and the reliability of these estimates. Several studies simply assume exogenously given

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estimates (He and Litterman 1999; Lee 2000; Drobetz 2001; Idzorek 2005) and suggest confi-dence intervals of the return estimates as a measure of uncertainty (Black and Litterman 1992).Since the portfolio performance critically depends on the exogenously assumed estimates, theseapproaches are hardly capable to evaluate the performance of the BL model in comparisonto MV and naïve-diversified benchmark portfolios. Moreover, mutually estimating returns andtheir respective confidence intervals might be a challenging task for analysts and might hindera successful implementation of the BL model. Therefore, we employ sample means as subjec-tive return estimates based on rolling estimation windows as in the other optimization models(MV and BS). However, in contrast to MV and BS, the BL model additionally incorporates thereliability of these estimates that are expressed in the matrix �.

For a portfolio of N assets, � is a N × N diagonal matrix with the reliability measures forthe assets on its diagonal. A low reliability in all ‘views’ results in optimized portfolio weightsclose to the reference portfolio. Meucci (2010) proposes to assume simply an overall level ofconfidence in the ‘views’ that is constant over time by setting � as

� =(

P

(1

c�

)P′

). (11)

In this approach, it is assumed that the reliability of ‘views’ is proportional to the reliabilityof implied returns with the proportionality factor 1/c. As a result no additional and time-varyinginformation on the reliability of ‘views’ is included. We suggest that the out-of-sample perfor-mance of the BL model is superior if reasonable and time-varying information on the reliabilityof ‘views’ is included. Therefore, we measure the reliability of each ‘view’ i based on historicforecast errors εi. For this, we employ the same rolling estimation windows as for return esti-mates and estimate the uncertainty of return estimates as the variance of historic forecast errors.We analyze different estimation window lengths as robustness check. The idea is that in uncer-tain market environments when the last months return estimates differ strongly from the realizedreturns, the reliability for the subsequent return estimate is likely to be lower, resulting in portfo-lios closer to the reference portfolio. In contrast, in stable market conditions when the last monthsreturn forecasts are close to the realized returns, the return estimate for the next period should bemore reliable and the optimized portfolio may depart more from the reference portfolio.

We analyze the contribution of this historic reliability measure by comparing the out-of-sampleportfolio performance of our approach with the assumption of an overall time-invariant level ofconfidence in the ‘views’ as in Meucci (2010) by substituting � according to Equation (11).

2.6 Performance measures

We calculate several performance measures to evaluate the optimized portfolios. First, we com-pute the moments of the net portfolio returns (after transaction costs) for each optimizationstrategy i. Further, we determine the out-of-sample net Sharpe ratio as the fraction of the out-of-sample mean net excess return (mean return after transaction costs less risk-free rate) dividedby the standard deviation of out-of-sample net returns.

SRi = RNet,i − Rf

σNet,i. (12)

Using Sharpe ratios as performance measure is often criticized, because asset return distri-butions are usually non-normal. However, Meyer (1987) shows that the general LS condition

10 W. Bessler et al.

(location and scale) of returns is sufficient to rely on the μ-σ framework. Therefore, relyingon Sharpe ratios is reasonable even for asymmetric return distributions and fat tails as long asthe LS condition holds. We use the two-sample statistic to test if the difference in Sharpe ratiosof two portfolios is significant (Opdyke 2007). In contrast to earlier Sharpe ratio tests (Jobsonand Korkie 1981b; Lo 2002), this test is appropriate under very general conditions – stationaryand ergodic returns. Most important for our analysis, the test permits autocorrelated and non-normal distributed returns and allows for a likely high correlation between the portfolio returnsof different strategies.3

As alternative performance measure, we compute the Omega ratio (Shadwick and Keating2002) also referred to as gain-loss ratio. It measures the average gains to average losses in whichwe define gains as returns above the risk-free rate and losses as returns below the risk-free rate.Hence, investments with a larger Omega measure are preferable. Formally, the omega measure is

Omegai = (1/T)∑T

t=1 max(0, rt,i − rf)

(1/T)∑T

t=1 max(0, rf − rt,i), (13)

where ri,t is the return of asset i at time t and rf is the risk-free rate. The advantage of the Omegameasure is that it does not require any assumption on the distribution of returns.

As alternative risk measures besides volatility, we compute the maximum drawdown (MDD)as proposed by Grossman and Zhou (1993). The MDD reflects the maximum accumulated lossthat an investor may suffer during the entire investment period if she buys the portfolio at a highprice and subsequently sells at the lowest price. As the Omega measure, it does not require anyassumption on the return distribution. We compute the percentage maximum drawdown (MDD)of strategy i as

MDDi = Maxi,t∗∈(0,T)

[Maxi,t∈(0,t∗)

(Pi,t − Pi,t∗

Pi,t

)], (14)

where Pi,t is the price of portfolio i at time t, when the portfolio is bought, and Pi,t∗ is the priceof portfolio i at time t*, when the portfolio is sold.

Furthermore, we compute the portfolio turnover, in line with Daskalaki and Skiadopoulos(2011) and DeMiguel, Garlappi, and Uppal (2009), which quantifies the extent of trading requiredto implement a certain strategy. The portfolio turnover PTi of strategy i is the average absolutechange of the portfolio weights ω over the T rebalancing points in time and across the N assets:

PTi = 1

T

T∑t=1

N∑j=1

(|ωi,j,t+1 − ωi,j,t+|) (15)

in which ωi,j,t is the weight of asset j at time t under strategy i; ωi,j,t+ is the portfolio weight beforerebalancing at t + 1; and ωi,j,t+1 is the desired portfolio weight at t + 1, after rebalancing. ωi,j,t

is usually different from ωi,j,t+ due to changes in asset prices between t and t + 1.We account for trading costs by assuming proportional transaction costs of 30 basis points of

the transaction volume in the base case and compute all performance measures after transactioncosts. As a robustness check, we analyze the impact of different transaction costs.

3. Data

To construct multi-asset portfolios we include global stocks, bonds, and commodity indices inthe investment universe. We use the MSCI World and MSCI Emerging Markets stock indices

The European Journal of Finance 11

to cover both developed and emerging markets. Emerging markets usually provide higher stockreturns than developed markets because they have a higher exposure to additional risk factorssuch as illiquidity or institutional and political conditions (Iqbal, Brooks, and Galagedera 2010).Chiou, Lee, and Chang (2009) find that international diversification is beneficial for US investors,reducing portfolio volatility, and improving risk-adjusted returns.

Government bond returns typically have low or even negative correlations with stock returnsand often prices of government bonds increase during stock market downturns. To ensure theirrole as a low-risk investment we employ US government bonds, thereby eliminating default andcurrency risks. To represent US government bond investments we rely on the Bank of Amer-ica/Merill Lynch US Government Bond Index (all maturities). In addition, we include the Bankof America/Merill Lynch US High Yield 100 Bond Index to add exposure to corporate defaultrisk. This index is expected to provide higher returns and higher risks compared to governmentbonds, but lower returns and lower risks compared to stock indices.

The S&P GSCI Light Energy Index represents a diversified commodity investment, providinginvestors with an exposure to a wide range of commodity price changes. While the main deter-minants of the S&P GSCI Index are energy prices, the S&P GSCI Light Energy Index offersmore balance across different commodity classes. At the end of our sample period, it reflectsthe price developments on the future markets for energy (37.4%), agricultural products (31.2%),and livestock (10%), as well as industry (14%) and precious metals (7.4%). Commodities shouldhave low correlations with the traditional asset classes such as stocks and bonds, because theirprices depend on different risk factors such as weather, geographical conditions, and supplyconstraints. Moreover, as several studies document a positive correlation between commod-ity returns and future inflation, investing in commodities is often viewed as a hedge againstinflation (Bodie and Rosansky 1980; Erb and Harvey 2006; Gorton and Rouwenhorst 2006).Several studies also find that by including commodities, the efficient frontier of stock-bond port-folios improved (for instance, Satyanarayan and Varangis 1996; Abanomey and Mathur 1999;Anson 1999; Jensen, Johnson, and Mercer 2000). More recent evidence suggests that diversifi-cation benefits of commodities are regime-dependent (Cheung and Miu 2010), but that portfoliobenefits of commodities in out-of-sample optimized portfolios are ambiguous (Daskalaki andSkiadopoulos 2011; You and Daigler 2013). However, given that investors still perceive com-modities as an important asset class in portfolio optimization, we include commodities in ouranalysis.

We obtain monthly total return index data for the period from January 1988 to December2011 from Thomson Reuters Datastream. All data are denominated in US dollar. In line withDeMiguel, Garlappi, and Uppal (2009), we use the three months US T-Bill rate as the risk-freerate. Because we need several years of historical data to set up the first optimized portfolio theevaluation period ranges from January 1993 to December 2011. Table 3 provides descriptivestatistics of the monthly asset returns for the full evaluation period.

The table shows similar annualized mean returns for stock and bond indices ranging from6.16% to 8.00% p.a. For the entire period, the average return of the commodity index is slightlylower than the average risk-free rate of 3.12%, resulting in a negative Sharpe ratio. The USGovernment Bond Index generates the highest Sharpe ratio of 0.655. The maximum drawdowns(MDD) of the assets reveal that the maximum loss an investor could have suffered during theevaluation period by investing in stocks was between 55.16% and 63.04% of the invested cap-ital. This figure was 59.95% for commodities, 27.21% for the US High Yield Bond Index, and5.29% for the US Government Bond Index. The Jarque–Bera statistic is significant for all asset

12 W. Bessler et al.

Table 3. Descriptive statistics of asset returns (January 1993–December 2011).

MSCI World MSCI EM US Gov. BondsUS High Yield

BondsS&P GSCI

Light Energy

Mean return p.a. (%) 6.71 8.00 6.16 7.31 2.66SD p.a. (%) 16.61 25.40 4.64 8.44 16.58Skewness − 0.90 − 0.93 − 0.04 − 1.47 − 1.29Kurtosis 5.34 5.93 4.26 10.44 8.45Sharpe ratio 0.22 0.19 0.66 0.50 − 0.03MDD (%) 55.16 63.04 5.29 27.21 59.95JB 82.47* 114.61* 15.10* 608.24* 339.20*Observations 228 228 228 228 228

Notes: This table provides sample moments, Sharpe ratios, maximum drawdown, and Jarque–Bera statisticsof the five asset classes considered in the empirical analysis. The period covers the months from January 1993to December 2011. ‘Mean return p.a.’ denotes annualized time-series mean of monthly returns, while ‘SDp.a.’ denotes the associated annualized standard deviation. ‘Skewness’ and ‘Kurtosis’ represent the third andfourth moment of the return distribution. ‘Sharpe ratio’ shows the annualized Sharpe ratios of the respectiveasset classes using the average 1993–2011 risk-free interest rate of 3.12% per year. MDD shows the maximumdrawdown of the respective asset class during the period from January 1993 to December 2011 and ‘JB’ is theJarque–Bera statistic for testing normality of returns.*Statistically significant at the 1% level.

classes. Hence, the assumption of normal distributed returns has to be rejected. However, as men-tioned above, Meyer (1987) shows that the general LS condition (location and scale) of returnsis sufficient to apply the mean-variance framework.

Table 4 presents evidence on potential diversification effects in terms of pair-wise correlationcoefficients. Over the entire period the diversification benefits when investing only in stocksare limited. The correlation between the MSCI World and MSCI Emerging Markets is highlysignificant and larger than 0.8, indicating a strong co-movement of developed and emergingstock markets. While the US-High-Yield-Bond Index and commodities offer a slightly betterdiversification effect with correlation coefficients ranging between 0.35 and 0.65, the highestdiversification potential during our sample period is provided by investing in the US GovernmentBond Index, which is reflected in negative correlation coefficients and positive Sharpe ratios.Consequently, we expect to find significant portfolio benefits by applying the BL, BS, and MVframeworks on a multi-asset portfolio, including bonds and commodities, rather than on a stock-only portfolio.

4. Empirical results

4.1 Results for the full sample

Table 5 summarizes the out-of-sample performance of the different asset allocation strategies forthe three investor types ‘conservative’, ‘moderate’, and ‘aggressive’ for the base case over thefull evaluation period from January 1993 to December 2011. In the base case, we use 36 monthsrolling estimation windows for the covariance matrix and 12 months for return estimates. Weuse shorter estimation windows for return estimates as we expect the correlation structure andvariances to be more stable. In robustness checks, we provide the results for different estimationwindows (5.2). The strategically weighted portfolio (st.w.) is computed according to the assetweights in Table 2. Within the asset class ‘stocks’, the strategic weights are specified to be 25%for emerging markets and 75% for developed markets. Accordingly, for the asset class ‘bonds’,

The European Journal of Finance 13

Table 4. Correlation matrix of asset classes (January 1993–December 2011).

MSCI World MSCI EM US Gov. BondsUS High Yield

BondsS&P GSCI

Light Energy

MSCI World 1MSCI Emerging Markets 0.811* 1US Gov. Bond Index − 0.182* − 0.221* 1US High Yield Bond Index 0.652* 0.619* − 0.069 1S&P GSCI Light Energy 0.470* 0.463* − 0.128** 0.353* 1

Notes: This table provides the correlation matrix for the asset classes considered in the analysis over the period fromJanuary 1993 to December 2011.*Statistically significant at the 1% level.**Statistically significant at the 5% level.

the strategic weights are set to 75% for government bonds and to 25% for high-yield corpo-rate bonds. In the equal-weighted (1/N) portfolio, all five asset classes obtain an equal portfolioweight of 20%.

The performance evaluation in Table 5 reveals that the three BL approaches, with differentreference portfolios, yield larger Sharpe ratios and Omega measures than MV, BS, MinVar, andboth naïve-diversified portfolios for all investor types. The Sharpe ratios of the BL portfolios aresignificantly larger (5% level) than that of the naïve-diversified 1/N strategy. In contrast, the MVstrategy does not result in a significant outperformance relative to the 1/N strategy.

Our results suggest that the impact of the reference portfolio on the performance of the BLapproach is rather low. For all analyzed reference portfolios, the BL optimization leads to aconsistently superior performance relative to MV, BS, and both naïve-diversified portfolios. Thisresult holds for all investor types. We also observe that for the conservative investor type, the BL-MinVar approach performs slightly better than the other BL approaches. For aggressive investors,however, the BL model based on strategic weights as reference portfolio performs marginallybetter.

Our risk measures volatility and maximum drawdown (MDD) indicate a consistently lowerrisk for all BL optimized portfolios in comparison to MV and BS, independent of the investortype. At first, it seems surprising that the ex post realized volatilities differ from the ex antedetermined volatility constraints of 5%, 10%, and 15%, but the explanation is rather straightfor-ward. On the one hand, the sample volatility estimates include estimation errors that may lead toa discrepancy between ex ante estimated volatility and ex post realized volatility. On the otherhand, the optimization framework with volatility constraint as described in Equation (5) doesnot necessarily favor the asset allocation with the largest possible volatility that is close to thevolatility constraint. The optimized portfolio is rather determined by the trade-off between riskand return. For instance, in recessionary periods with negative expected stock returns, the opti-mization framework is likely to allocate a large fraction of the portfolio to low-risk governmentbonds, independent of the volatility constraint. Therefore, the average volatility over the entireevaluation period is below the volatility constraint for all optimization models.

Table 5 also presents the average yearly portfolio turnover as an indicator for the magnitudeof trading volume and transaction costs required to implement a certain strategy. However, allperformance measures already include transaction costs. The results reveal that for all investortypes the BL approach requires a lower portfolio turnover and, therefore, has lower transactioncosts relative to the MV approach. ‘The BL-st.w.’ approach, for instance, yields an average yearly

14W

.Bessler

etal.

Table 5. Empirical results for the full sample (evaluation period 1993–2011).

BL-st.w. BL-1/N BL-MinVar BS MV Min Var st.w. 1/N st.w. B&H 1/N B&H

Investor type: conservativeNet mean return p.a. (%) 7.81 7.79 7.86 7.31 7.29 6.09 6.29 6.08 6.51 6.25Volatility p.a. (%) 5.16 5.22 5.13 5.27 6.08 4.17 4.86 11.08 5.07 11.08Net Sharpe ratio 0.91†† 0.89†† 0.92†† 0.79†† 0.68 0.71 0.65 0.27 0.67 0.28Net Omega measure 1.96 1.93 1.98 1.78 1.69 1.70 1.66 1.24 1.67 1.25Net MDD (%) 5.99 6.07 6.17 7.30 9.41 7.45 13.99 40.59 14.19 40.50VaR99% (%) 2.77 3.04 2.87 3.33 3.92 2.74 3.16 9.72 9.70 12.33Avrg. number of assets 3.32 3.79 3.20 2.91 2.69 3.71 5.00 5.00 5.00 5.00Avrg. turnover p.a. 2.13 2.06 2.11 1.96 2.67 0.77 0.18 0.29 0.00 0.00

Investor type: moderateNet mean return p.a. (%) 9.58 9.51 9.28 8.27 8.21 – 6.03 6.08 6.39 6.25Volatility p.a. (%) 8.65 8.87 8.05 9.40 10.37 – 9.16 11.08 9.52 11.08Net Sharpe ratio 0.75∗∗##†† 0.72†† 0.76†† 0.55 0.49 – 0.32 0.27 0.34 0.28Net Omega measure 1.77 1.73 1.81 1.48 1.46 – 1.27 1.24 1.31 1.25Net MDD (%) 9.46 11.02 9.21 14.35 16.18 – 35.10 40.59 35.30 40.50VaR99% (%) 6.36 6.91 5.52 6.87 7.53 – 7.65 9.72 − 9.70 − 12.33Avrg. number of assets 3.26 3.48 2.86 2.33 2.16 – 5.00 5.00 5.00 5.00Avrg. turnover p.a. 3.32 3.29 3.28 3.34 4.44 – 0.27 0.29 0.00 0.00

Investor type: aggressiveNet mean return p.a. (%) 11.72##†† 11.32##†† 10.37##†† 9.46 10.35 – 5.81 6.08 6.27 6.25Volatility p.a. (%) 11.68 11.30 9.85 12.38 13.62 – 14.27 11.08 14.74 11.08Net Sharpe ratio 0.74#†† 0.73##†† 0.74##†† 0.51 0.53 – 0.19 0.27 0.21 0.28Net Omega measure 1.79 1.78 1.82 1.51 1.53 – 1.15 1.24 1.18 1.25Net MDD (%) 14.39 11.67 12.10 22.23 24.59 – 50.99 40.59 52.04 40.50VaR99% (%) 9.43 7.86 6.73 10.26 10.77 – 12.38 9.72 − 9.70 12.33Avrg. number of assets 3.12 3.21 2.63 1.86 1.72 – 5.00 5.00 5.00 5.00Avrg. turnover p.a. 3.62 3.83 3.89 4.76 4.61 – 0.25 0.29 0.00 0.00

Notes: This table reports the portfolio performance measures for the full sample from 1993 to 2011 in the base case. In all optimized portfolios, the maximum expectedvolatility is constrained to 5%, 10%, and 15% p.a. for the conservative, moderate, and aggressive investor type, respectively. All portfolios are rebalanced at the first tradingday of every month. */**, #/##, †/†† indicate a significantly higher Sharpe ratio compared to mean-variance, the strategically weighted benchmark, and the 1/N benchmark atthe significance levels of 1% and 5%, respectively.

The European Journal of Finance 15

turnover equal to 3.32 times the portfolio volume for the moderate investor, while the yearlyturnover for MV equals 4.44 times the portfolio volume. The explanation for this result is thatthe combined return estimates used in the BL approach are more stable over time than the samplemeans used in MV, which is indicated by a lower variance of return estimates (unreported).

Additionally, we find that the average number of assets in the optimized portfolio, as an indi-cator for the magnitude of diversification across asset classes, is higher for BL than for MV andBS portfolios. Consequently, BL portfolios offer more diversification across asset classes andhave less extreme allocations. A possible explanation is that the combined return estimates usedin the BL approach are less heterogeneous. Figure 2 presents the optimized portfolio weights forBL and MV optimization for the three investor types during the 1993–2011 period. In line withthe turnover and diversification measures, the figure reveals less extreme portfolio reallocationsand a stronger benchmark orientation of the BL optimized portfolios relative to MV.

Our findings for the multi-asset data set suggest a slight outperformance of the sample-basedMV approach relative to naïve-diversified strategies. In line with Kirby and Ostdiek (2012), thelevel of outperformance (after transaction costs) of MV is insignificant. However, our findingsdiffer from that of DeMiguel, Garlappi, and Uppal (2009) and Murtazashvili and Vozlyublennaia(2013) who conclude that none of the variations of MV is able to outperform a naïve 1/Nstrategy. One explanation is the difference in the employed data set. While both earlier stud-ies analyze stock-only portfolios, we additionally include government bonds, corporate bonds,and commodities that should result in broader diversification and might enhance the portfoliooptimization benefits. Particularly, by including government bonds, asset allocation models mayoutperform naïve strategies if they actively shift wealth from stocks to bonds during stock mar-ket downturns and vice versa. In fact, Figure 2 shows that for both, BL and MV, and across allinvestor types, the optimized portfolios are shifted completely to government bonds during thestock market downturns between 2000 and 2003 (end of the new economy period) and between2008 and 2009 (the financial crisis).

However, while the multi-asset approach might explain why MV performs superior in ouranalysis relative to the DeMiguel, Garlappi, and Uppal (2009) study, additional analyses arerequired to explain the reasons for the performance differences between BL, BS, and MVapproaches. To obtain additional insights, we analyze the return estimates used in the opti-mization approaches. We find that for all assets the forecast error of estimated to subsequentlyrealized returns, measured as mean absolute error (MAE), is consistently lower for mixed returnestimates employed in the BL approach than for return estimates employed in the BS and MVapproaches. Additionally, we compute the coefficients of determination (R2) as the squared cor-relation between forecasted and subsequently realized returns and use it as an indicator for theex post performance. In line with the results for the MAE, we find that the coefficients of deter-mination of forecasted returns to subsequently realized returns are larger for BL mixed returnestimates than for BS shrinked return estimates and for sample mean returns employed in MV.The analyses of forecast errors are available from the authors upon request.

4.2 Performance of optimized portfolios in different market environments

To offer additional explanations for the performance of the asset allocation strategies in differentmarket environments, we divide the full evaluation period (1993–2011) into several sub-periods.Expansionary and recessionary sub-periods are determined on an ex ante basis from monetarypolicy and stock market signals. We combine both approaches to reduce the number of sub-periods as well as the probability of incorrect signals (Bessler, Holler, and Kurmann 2012). The

16 W. Bessler et al.

Figure 2. BL and MV optimized portfolio weights (base case).Notes: This figure reports the optimized portfolio weights for the full sample from 1993 to 2011 in the basecase. The optimized portfolios are constraint to a maximum expected volatility of 5%, 10%, and 15% forthe conservative, moderate, and aggressive investor type, respectively. All portfolios are rebalanced at thefirst trading day of every month.

monetary cycle is the first change of the short-term interest rate by the central bank that runscounter to the previous trend (Jensen and Mercer 2003). The stock market signal is determinedas the intersection of the 24-months moving average of the MSCI World with the actual indexfrom either below (expansionary state) or above (recessionary state), indicating a change in thebusiness cycle. For the transition from one state to another, it is required that both instruments,the monetary policy and the stock market, provide a consistent signal. In Figure 3, we presentthe definition of sub-periods derived from the joint monetary policy and stock market signals,where shaded areas denote recessionary periods.

The first sub-period ranges from January 1993 to January 2001 and includes a number ofevents such as the Asian crisis, the Russian default, and the buildup of the new economy bubble.

The European Journal of Finance 17

Figure 3. Definition of sub-periods.Notes: The figure shows the definition of individual sub-periods conditional on monetary policy signals aswell as stock market signals. Shaded areas denote down markets/recessionary periods.

This period consists of 97 months and can be characterized as ‘expanding’ with increasing valuesin developed stock markets and relatively high interest rates with T-Bills yielding, on average,4.77% p.a. The second sub-period between February 2001 and June 2004 covers the end of thenew economy period and the subsequent rebound of international stock markets. It includes 41months and is characterized as ‘recessionary’ with bearish stock markets and an average risk-free rate of 1.85% p.a. The third sub-period ranges from July 2004 to February 2008 covering 44months, and containing bullish stock markets and high interest rates. The average risk-free ratein this period is 3.67% p.a. The final sub-period from March 2008 to December 2011 includes46 months and incorporates the financial crisis that led to substantial declines in the values ofequities and alternative asset classes, such as commodities and hedge funds. The average risk-free rate in the fourth period was only 0.37% p.a.

The performance of the out-of-sample optimized portfolios for the four sub-periods is sum-marized in Table 6 for the moderate investor.4 We find that BL optimized portfolios outperformBS and MV in all four sub-periods and perform superior than naïve-diversified portfolios inthree of four sub-periods. For both recessionary sub-periods, we observe substantially higherSharpe ratios for the BL strategies relative to MV. Naturally, due to the shorter time series of sub-periods, the power of the significance test is lower relative to the full period so that differencesin Sharpe ratios are not significant at the 5% level anymore. In comparison to Bayes–Stein andboth naïve-diversified portfolios, all BL approaches achieve a substantially higher performancein both recessionary periods.

In contrast, for both expansionary periods we find a relatively smaller outperformance of BLin comparison to MV and BS. In the third sub-sample, which covers the bullish stock mar-kets between July 2004 and February 2008, the naïve-diversified portfolios outperform all otheroptimization strategies (BL, BS, MV, and MinVar).

For all sub-periods, the BL optimized portfolios are less risky than BS, MV, and both naïve-diversified portfolios. This is indicated by a lower maximum drawdown and is consistent with

18 W. Bessler et al.

Table 6. Analysis of sub-periods.

BL-st.w. BL-1/N BL-MinVar BS MV MinVar st.w. 1/N

January 1993–January 2001Net mean return p.a. (%) 10.25 10.02 9.97 10.93 10.41 6.78 7.86 6.74Volatility p.a. (%) 9.16 9.51 8.07 10.34 11.08 3.92 6.90 8.30Net Sharpe ratio 0.60 0.55 0.65 0.60 0.51 0.51 0.45 0.24Net MDD (%) 9.46 9.03 9.21 11.42 12.93 5.72 15.32 19.41Avrg. number of assets 2.95 3.21 2.70 2.35 2.09 3.70 5.00 5.00Avrg. turnover p.a. 4.16 4.03 4.04 2.95 5.48 0.83 0.22 0.26Avrg. risk rate 4.77% 4.77% 4.77% 4.77% 4.77% 4.77% 4.77% 4.77%Obs. 97 97 97 97 97 97 97 97

February 2001–June 2004Net mean return p.a. (%) 7.77 7.26 8.14 5.25 5.01 5.97 3.71 5.19Volatility p.a. (%) 6.40 6.54 6.02 7.60 8.05 4.84 8.32 10.00Net Sharpe ratio 0.92 0.83 1.04** 0.45 0.39 0.85 0.22 0.33Net MDD (%) 6.52 7.08 4.88 9.43 10.67 3.40 14.19 17.37Avrg. number of assets 2.76 3.10 2.32 2.10 1.90 4.56 5.00 5.00Avrg. turnover p.a. 2.67 3.12 1.95 4.03 4.42 0.62 0.30 0.31Avrg. risk-free rate 1.85% 1.85% 1.85% 1.85% 1.85% 1.85% 1.85% 1.85%Obs. 41 41 41 41 41 41 41 41

July 2004–February 2008Net mean return p.a. (%) 13.81 14.05 13.36 10.71 11.97 6.60 10.51 12.44Volatility p.a. (%) 9.41 9.54 9.38 9.42 11.35 2.80 5.08 6.66Net Sharpe ratio 1.08 1.09 1.03 0.75 0.73 1.05 1.35 1.32Net MDD (%) 6.63 6.69 7.25 8.98 10.48 6.65 34.14 39.57Avrg. number of assets 3.84 3.84 3.59 2.66 2.59 3.95 5.00 5.00Avrg. turnover p.a. 3.45 3.24 4.08 4.24 4.20 0.97 0.22 0.25Avrg. risk-free rate 3.67% 3.67% 3.67% 3.67% 3.67% 3.67% 3.67% 3.67%Obs. 44 44 44 44 44 44 44 44

March 2008–December 2011Net mean return p.a. (%) 5.84 6.21 5.06 3.17 3.00 4.31 0.14 − 0.43Volatility p.a. (%) 8.63 8.73 8.28 8.79 9.75 5.08 15.04 18.10Net Sharpe ratio 0.63 0.67 0.57 0.32 0.27 0.78 − 0.01 − 0.04Net MDD (%) 9.34 11.02 6.90 14.35 16.18 7.45 35.10 40.59Avrg. number of assets 3.81 4.06 3.00 2.21 2.13 2.77 5.00 5.00Avrg. turnover p.a. 2.02 1.97 2.10 2.10 2.57 0.59 0.39 0.38Avrg. risk-free rate 0.37% 0.37% 0.37% 0.37% 0.37% 0.37% 0.37% 0.37%Obs. 47 47 47 47 47 47 47 47

Notes: This table reports portfolio performance measures for the four sub-periods from 1993 to 2011 for the moderateinvestor type. In all optimized portfolios, the maximum expected volatility is constrained to 10% p.a. All portfoliosare rebalanced at the first trading day of every month. */** Indicate a significantly higher Sharpe ratio compared tomean-variance at the significance levels of 1% and 5%, respectively.

the analysis of the full sample. Furthermore, the analysis reveals consistently lower portfolioturnovers and, hence, lower transaction costs for BL compared to MV for all sub-periods. Addi-tionally, for all sub-periods BL portfolios are more diversified across asset classes than MV andBS portfolios as indicated by a higher average number of assets in the optimized portfolios. Over-all, our results suggest that the BL model outperforms MV, BS, and naïve-diversified portfoliosparticularly in recessionary periods. The explanation is that the BL model reacts more quickly tochanges in the economic cycle and adjusts the asset allocation more rapidly.

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5. Robustness checks

We perform a variety of robustness checks and sensitivity analyses to confirm the robustness ofour results. We vary the constraints on optimized portfolio weights and relax the short sellingconstraint (Section 5.1). We analyze variations in all input parameters (Section 5.3) and estima-tion windows (Section 5.2) and modify the level of transaction costs and, alternatively, includetransaction costs in the optimization function (Section 5.4). Finally, we alter the investment uni-verse and employ different sets of assets (Section 5.5). We primarily report the results for theBL-st.w. strategy because the results for the other reference portfolios used in the BL approach(st.w., 1/N, MinVar) are qualitatively similar. The same applies for the three investor types in thatwe focus on the moderate investor type, as the results for the others are similar. All unreportedresults are available upon request.

5.1 Alternative optimization constraints

5.1.1 Restrictions on optimized portfolio weightsIt is possible that the portfolio composition of the benchmark differs substantially from the allo-cation of the optimized portfolios and consequently the naïve-diversified benchmark portfoliois inappropriate. Therefore, we repeat the optimizations relative to the strategically weightedbenchmark and restrict the optimized portfolio weight for each asset so that the maximum abso-lute deviation from the benchmark weight reported in Table 2 does not exceed a certain threshold.This is an approach frequently used by practitioners to cope with the shortcomings of MV opti-mization. Table 7 presents the results for a maximum absolute deviation from the benchmarkweight of 15% points. The results confirm our findings for the base case. This holds for theout-of-sample Sharpe ratios as well as for the portfolio risk (MDD), the portfolio diversification,and the portfolio turnover. Intuitively, restricting asset weights lowers the portfolio turnover andincreases the average number of assets in the portfolios for all strategies.5

5.1.2 Short selling constraintTo investigate whether our findings are sensitive to the short selling constraint, we allow for shortpositions in the optimization. The new results (unreported) confirm that the BL approach yieldssuperior portfolio performance (Sharpe ratios and Omega measures) with lower risk (volatility,MDD, value-at-risk) compared to MV and BS approaches. When short selling is feasible theperformance of all optimization strategies improves despite higher turnover and transaction costs.

5.1.3 Volatility constraintNext, we vary the portfolio volatility restriction, while keeping the strategic weights of stocks,bonds, and commodities constant at the level of 40%, 45%, and 15% (moderate portfolio accord-ing to Table 2), respectively. Panel A of Table 8 shows that our results are robust for volatilityconstraints between 5% and 20%. This holds for the out-of-sample Sharpe ratios, the portfoliorisk (MDD), the portfolio diversification, and the portfolio turnover.

5.1.4 Variations of the risk aversion parameterThe results for variations of the risk aversion coefficient between 0.5 and 10.0 we report in panelB of Table 8. Again, we find that our results are robust to different risk aversion coefficients. Thisholds for the out-of-sample Sharpe ratios, the portfolio risk (MDD), the portfolio diversification,

20 W. Bessler et al.

Table 7. Restriction of optimized portfolio weights.

BL-st.w. BL-1/N BL-MinVar BS MV MinVar st.w. 1/N

ConservativeNet Sharpe ratio 0.95† 0.86† 0.91† 0.77†† 0.73†† 0.74†† 0.65 0.27Net Omega measure 2.02 1.88 1.97 1.77 1.71 1.73 1.66 1.24Net MDD (%) 7.86 8.46 6.49 9.76 10.83 7.67 13.99 40.59Avrg. number of assets 4.42 4.20 3.91 3.48 3.51 3.69 5.00 5.00Avrg. turnover p.a. 1.11 1.00 0.99 1.36 1.53 0.51 0.18 0.29ModerateNet Sharpe ratio 0.64##† 0.65##† 0.62##† 0.52†† 0.49 – 0.32 0.27Net Omega measure 1.60 1.62 1.62 1.48 1.44 – 1.27 1.24Net MDD (%) 21.84 20.46 19.33 25.02 25.99 – 35.10 40.59Avrg. number of assets 4.45 4.48 4.31 3.91 3.88 – 5.00 5.00Avrg. turnover p.a. 1.44 1.32 1.42 1.80 1.87 – 0.27 0.29AggressiveNet Sharpe ratio 0.40## 0.41## 0.44∗∗## 0.25 0.31 – 0.19 0.27Net Omega measure 1.36 1.37 1.41 1.22 1.27 – 1.15 1.24Net MDD (%) 40.37 39.87 38.61 43.31 43.33 – 50.99 40.59Avrg. number of assets 4.21 4.36 4.52 4.28 4.15 – 5.00 5.00Avrg. turnover p.a. 1.49 1.44 1.46 1.89 1.64 – 0.25 0.29

Notes: This table reports the portfolio performance measures for the full sample from 1993 to 2011. In all optimizedportfolios, the asset weights are restricted to a maximum absolute deviation of 15% from the respective benchmarkweight reported in Table 2. The maximum expected volatility is constrained to 5%, 10%, and 15% p.a. for theconservative, moderate, and aggressive investor type, respectively. All portfolios are rebalanced at the first tradingday of each month. */**, #/##, †/†† indicate a significantly higher Sharpe ratio compared to mean-variance, thestrategically weighted benchmark, and the 1/N benchmark at the significance levels of 1% and 5%, respectively.

as well as the portfolio turnover. The BL model performs best for risk aversion coefficientsbetween 2 and 4, which is in line with the assumed moderate investor type.

5.2 Alternative estimation windows

Panels C and D of Table 8 present the results for different estimation windows for the covariancematrix and the return estimates. In line with the base case results, we find consistently highernet Sharpe ratios for the BL approach in comparison to MV and both naïve-diversified port-folios for all analyzed estimation windows. The results are insignificant for much shorter andlonger estimation windows. The portfolio turnover increases dramatically for very short estima-tion windows (one to six months), resulting in substantial transaction costs and a relatively lowerperformance compared to the naïve-diversified portfolios. For long estimation windows ( ≥ 36months for returns), the responsiveness to structural breaks such as stock market downturns isreduced, resulting in lower out-of-sample Sharpe ratios for all optimization approaches. An anal-ysis of the autocorrelation functions of the asset returns confirms that only the last months returnsare significantly correlated with current returns, while returns with a lag longer than 12 monthshardly provide any explanatory power.

We identify optimal estimation windows between 36 and 48 months for the covariance matrixand around 12 months for the return estimates for the BL approach. The insignificant results ofthe BL model for very long and very short return estimation windows highlight the importanceof accurate and responsive return estimates. However, further research is required to analyze theperformance of the BL approach for alternative return estimates.

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Table 8. Variation of optimization constraints, risk aversion coefficients, and estimation windows.

Maximumvolatility p.a.

5.00% 7.50% 10% 15% 20% Benchmark

BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV st.w. 1/N

Panel A: Variation of optimization constraint: maximum allowed portfolio volatility

Net Sharpe ratio 0.92##† 0.79##†† 0.68 0.80∗∗##†† 0.63 0.54 0.75∗∗##†† 0.55 0.49 0.75∗∗##†† 0.51 0.53 0.74##†† 0.50 0.49 0.32 0.27

Net MDD (%) 5.88 9.30 9.41 7.58 9.82 11.54 9.46 14.35 16.18 11.18 22.23 24.59 16.41 27.90 27.98 35.10 40.59

Avrg. number ofassets

3.64 2.91 2.69 3.45 2.59 2.41 3.26 2.33 2.16 3.00 1.86 1.71 2.76 1.52 1.35 5.00 5.00

Avrg. turnoverp.a.

2.00 1.96 2.67 2.80 2.78 3.79 3.32 3.34 4.44 3.91 3.80 4.61 4.07 3.61 4.36 0.27 0.29

Panel B: Variation of risk aversion coefficient

0.5 1 2 4 8 Benchmark

Delta BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV st.w. 1/N

Net Sharpe ratio 0.61 0.54 0.47 0.68 0.54 0.48 0.75∗∗##†† 0.55 0.49 0.78∗∗##†† 0.53 0.50 0.69##†† 0.58 0.48 0.32 0.27

Net MDD 11.62% 14.35% 16.18% 10.35% 14.35% 16.18% 9.46% 14.35% 16.18% 9.94% 14.35% 16.18% 18.59% 10.93% 15.81% 35.10% 40.59%

Avrg. number ofassets

2.79 2.31 2.17 2.95 2.31 2.16 3.26 2.33 2.16 3.93 2.35 2.19 4.53 2.42 2.25 5.00 5.00

Avrg. turnoverp.a.

3.72 3.48 4.48 3.62 3.38 4.49 3.32 3.34 4.44 2.74 3.32 4.44 2.04 3.36 4.35 0.27 0.29

Panel C: Variation of estimation window for covariance matrix

Estimationwindownumber ofmonths

12 24 36 48 60 Benchmark

BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV st.w. 1/N

Net Sharpe ratio 0.63 0.66 0.56 0.73∗∗##†† 0.58 0.48 0.75∗∗##†† 0.55 0.49 0.75∗∗##†† 0.42 0.46 0.73†† 0.44 0.47 0.32 0.27

Net MDD 16.13% 10.18% 18.28% 14.58% 14.04% 15.73% 9.46% 14.35% 16.18% 8.97% 17.55% 16.99% 8.76% 16.82% 18.26% 35.10% 40.59%

Avrg. number ofassets

3.04 2.37 1.90 3.14 2.16 2.09 3.26 2.33 2.16 3.38 2.38 2.24 3.41 2.50 2.29 5.00 5.00

Avrg. turnoverp.a.

4.34 3.29 4.63 3.56 2.86 4.33 3.32 3.34 4.44 3.48 3.51 4.60 3.27 3.31 4.59 0.27 0.29

(Continued).

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Table 8. Continued.

5.00% 7.50% 10% 15% 20% Benchmark

Maximumvolatility p.a. BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV st.w. 1/N

Panel D: Variation of estimation window for historic return estimatesEstimation

windownumber of

months

1 6 12 18 36 Benchmark

BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV BL-st.w. BS MV st.w. 1/N

Net Sharpe ratio 0.45 0.26 0.23 0.57 0.53 0.32 0.75∗∗##†† 0.55 0.49 0.56 0.43 0.33 0.48 0.44 0.34 0.32 0.27

Net MDD 14.48% 19.59% 22.69% 15.65% 15.69% 24.93% 9.46% 14.35% 16.18% 17.45% 19.44% 20.58% 28.95% 22.70% 30.87% 35.10% 40.59%

Avrg. number ofassets

2.97 2.10 2.01 3.10 2.26 2.15 3.26 2.33 2.16 3.38 2.24 2.13 3.69 2.17 2.03 5.00 5.00

Avrg. turnoverp.a.

12.30 14.44 15.29 4.76 5.68 6.41 3.32 3.34 4.44 2.92 2.45 4.07 1.81 2.64 2.80 0.27 0.29

Notes: The table shows sensitivity analysis for different optimization constraints, risk aversion coefficients, and estimation windows for the moderate investor type and the fullsample from 1993 to 2011. ∗/∗∗, #/##, †/†† indicate a significantly higher Sharpe ratio compared to mean-variance, the strategically weighted benchmark, and the 1/N benchmarkat the significance levels of 1% and 5%, respectively.

The European Journal of Finance 23

Relative to BS, the BL approach performs consistently superior with one exception: for a 12-months estimation window for the covariance matrix, the BS approach achieves a slightly higherSharpe ratio than BL. However, further analysis suggests that this result is due to inflating thecovariance matrix in the BS model, which is equivalent to reducing the volatility constraint to6.74%.6 With a volatility constraint of 6.74%, the BL and MV approaches yield a Sharpe ratio of0.83 and 0.56 (unreported), respectively. Hence, when comparing equally constrained portfolios,the BL approach outperforms BS and MV, which confirms our base case results.

5.3 Varying BL model parameters τ and �

We present the results for variations of the BL parameters τ and � in Table 9. The MV, BS,and naïve-diversified portfolios are insensitive to alternative values of τ and � and, hence, areconstant in this analysis. First, we vary the parameter τ in the BL model that captures the confi-dence in implied returns. For values of τ close to zero, the unconstrained optimized BL portfolioconverges to the benchmark portfolio. In contrast, for very large values of τ , the optimized BLportfolio converges to the MV optimized portfolio. Panel A of Table 9 shows that for the com-monly used values for τ between 0.025 and 1.00 (Black and Litterman 1992; He and Litterman1999; Drobetz 2001; Idzorek 2005), our results are robust. This holds for the out-of-sampleSharpe ratios as well as for the portfolio risk (MDD), the portfolio diversification, and the port-folio turnover. Furthermore, we observe that the BL portfolio’s deviation from the benchmarkdeclines with lower values of τ , which results in declining tracking errors. This is consistent withthe interpretation of τ as an uncertainty measure of implied returns and confirms its function tocontrol the desired deviation from the reference portfolio.

Second, we vary the estimation window for �, which quantifies the reliability of ‘views’ basedon the historic estimation errors of return forecasts. Panel B of Table 9 confirms that our resultsare robust for estimation windows of � between 6 and 36 months. Third, we assume an overalltime-invariant level of confidence in the ‘views’ (Meucci 2010) by substituting � accordingto Equation (11). In the approach no additional time-varying information on the reliability of‘views’ is included. Panel C of Table 9 presents the results for different overall time-invariantlevels of confidence, measured by the scalar c. We find that for all confidence levels (scalars cbetween 0.1 and 100), the outperformance of the BL model relative to MV vanishes. Therefore, itseems that including time-varying information on the reliability of return estimates is an essentialfactor for the superior performance of the BL model relative to MV. This leads, on the one hand,to an investment close to the reference portfolio when market conditions are uncertain and, onthe other hand, to larger deviations from this reference portfolio when markets are more stableand return forecasts are more reliable.

5.4 The impact of transaction costs

To analyze the impact of transaction costs, we first vary the variable transaction costs between0 and 50 basis points and then include variable transaction costs in the optimization function.The net Sharpe ratio measures for different levels of transaction costs are provided in Table 10.We find that for all analyzed levels of transaction costs, the BL portfolios outperform all otherstrategies and perform consistently better than MV and both naïve-diversified portfolios.

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Table 9. Variation of BL model parameters.

Panel A: Variation of level of confidence of implied returns (τ )

BS MV BL-st.w.: varying parameter τ st.w.

Parameter τ n/a (τ → ∞) 1 0.3 0.15 0.1 0.05 0.025 (τ → 0)

Net Sharpe ratio 0.55 0.49 0.67 0.67 0.71##†† 0.75∗∗##†† 0.77∗∗##†† 0.51 0.32Net MDD (%) 14.35 16.18 10.79 10.01 9.58 9.46 10.34 15.74 35.10Avrg. number of assets 2.33 2.16 2.72 2.86 3.06 3.26 3.87 2.34 5.00Avrg. turnover p.a. 3.34 4.44 4.01 3.88 3.56 3.32 2.79 4.45 0.27Tracking error (%) 2.17 2.50 2.34 2.20 2.12 2.03 1.68 2.40 –

Panel B: Variation of estimation window for uncertainty measure of views (�)

Estimation window size (months)BS MV BL-st.w varying estimation window for � st.w

for estimating the uncertainty of views n/a (� → 0) 3 6 12 18 24 36 (� → ∞)

Net Sharpe ratio 0.55 0.49 0.52 0.68 0.75∗∗##†† 0.73†† 0.76##†† 0.74†† 0.32Net MDD (%) 14.35 16.18 13.01 13.66 9.46 10.53 10.64 11.08 35.10Avrg. number of assets 2.33 2.16 2.89 3.15 3.26 3.22 3.26 3.30 5.00Avrg. turnover p.a. 3.34 4.44 6.28 4.28 3.32 3.10 2.98 2.98 0.27Tracking error (%) 2.17 2.50 2.17 2.18 2.03 2.07 2.09 2.07 –

Panel C: Variation of assumed overall level of confidence in views (c)

BS MV BL-st.w varying overall confidence level c for views st.w

Parameter c n/a (τ → ∞) 0.1 2 5 10 20 100 (τ → 0)

Net Sharpe ratio 0.55 0.49 0.46 0.52 0.51 0.52 0.51 0.49 0.32Net MDD (%) 14.35 16.18 29.13 15.68 15.60 15.74 15.88 16.09 35.10Avrg. number of assets 2.33 2.16 4.86 2.67 2.44 2.33 2.24 2.17 5.00Avrg. turnover p.a. 3.34 4.44 1.14 4.07 4.38 4.44 4.47 4.44 0.27Tracking error (%) 2.17 2.50 0.51 2.02 2.23 2.33 2.37 2.42 –

Notes: The table shows sensitivity analyses for setting different model parameters in the BL model for the moderate investor type and the full sample from 1993 to 2011. Thestrategically weighted benchmark is the reference portfolio for BL and is used to compute tracking errors. */**, #/##, †/†† indicate a significantly higher Sharpe ratio comparedto mean-variance, the strategically weighted benchmark, and the 1/N benchmark at the significance levels of 1% and 5%, respectively.

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Table 10. The influence of transaction costs.

Variable transactioncosts in basis points BL-st.w. BL-1/N BL-MinVar BS MV st.w. 1/N MinVar

5 0.85##†† 0.82##†† 0.87##†† 0.64 0.61 0.32 0.27 0.76##††

10 0.83##†† 0.80##†† 0.85##†† 0.62 0.58 0.32 0.27 0.75#††

20 0.79##†† 0.76##†† 0.81##†† 0.59 0.54 0.32 0.27 0.73#††

30 0.75∗∗##†† 0.72†† 0.76†† 0.55 0.49 0.32 0.27 0.71#††

40 0.71∗∗†† 0.68∗∗†† 0.72†† 0.51 0.45 0.31 0.26 0.69#†

50 0.67∗∗ 0.64** 0.68 0.47 0.40 0.31 0.26 0.67#†

Notes: The table shows a sensitivity analysis for different levels of transaction costs for the moderate investor typeand the full sample from 1993 to 2011. */**, #/##, †/†† indicate a significantly higher Sharpe ratio compared to mean-variance, the strategically weighted benchmark, and the 1/N benchmark at the significance levels of 1% and 5%,respectively.

Alternatively, we include variable transaction costs in the optimization function. The opti-mization problem with transaction costs is

maxω

U = ω′μ − � ′ϕ − δ

2ω′ ∑ω, (16)

where � is a vector that contains the changes in portfolio weights required to rebalance theportfolio at the monthly rebalancing dates, and ϕ is the vector of variable transaction costs ofthe assets. The results in Table 11 confirm our findings for the base case. This holds for theout-of-sample Sharpe ratios as well as for the portfolio risk (MDD), the portfolio diversification,and the portfolio turnover. As expected, including transaction costs in the optimization functionsubstantially reduces portfolio turnover for all optimization strategies and it slightly enhancesperformance after transaction costs for most strategies.

5.5 Alternative asset universes

To check whether our results are robust to different sets of assets, we repeat our analysisusing different and larger asset universes. First, we include two alternative multi-asset portfo-lios containing a larger number of stock indices. Second, we employ stock-only portfolios as inDeMiguel, Garlappi, and Uppal (2009) to investigate the benefits of additional asset classessuch as bonds and commodities for our asset allocation models relative to naïve-diversifiedportfolios. The multi-asset portfolios include a larger number of stock indices denominated inUSD and Euro, reflecting USD and Euro investors. The USD multi-asset portfolio contains eightMSCI stock indices (France, Germany, UK, Canada, USA, Italy, Japan, and Switzerland), com-plemented by US government bonds (Datastream Government Bond Index) and commodities(S&P GSCI). The evaluation period is 1993–2011. The Euro multi-asset universe includes Euro-pean government bonds, European corporate high-yield bonds, as well as four currency-hedgedregional stock indices (MSCI Europe, MSCI North America, MSCI Pacific, and MSCI EmergingMarkets) and the Euro-currency-hedged S&P GSCI commodity index. Based on the availabilityof data, the evaluation period is 1999–2011.

The first stock-only portfolio is based on international country indices (International Stocks)and includes eight MSCI stock indices (France, Germany, the UK, Canada, the USA, Italy, Japan,and Switzerland) as in DeMiguel, Garlappi, and Uppal (2009). The second stock-only portfolio

26 W. Bessler et al.

Table 11. Considering transaction costs in the optimization function.

BL-st.w. BL-1/N BL-MinVar BS MV MinVar st.w. 1/N

Conservative

Net Sharpe ratio 0.96† 0.86†† 0.92†† 0.82†† 0.73 0.64 0.65 0.27Net Omega measure 2.03 1.90 1.99 1.85 1.72 1.61 1.66 1.24Net MDD (%) 6.50 6.69 6.09 7.35 7.80 8.35 13.99 40.59Avrg. number of assets 4.39 4.28 3.73 2.86 2.72 3.81 5.00 5.00Avrg. turnover p.a. 0.73 0.82 0.82 1.58 2.16 0.21 0.18 0.29

ModerateNet Sharpe ratio 0.76##† 0.83##†† 0.85##†† 0.50 0.50 – 0.32 0.27Net Omega measure 1.78 1.87 1.92 1.47 1.47 – 1.27 1.24Net MDD (%) 13.89 8.45 7.56 13.75 16.18 – 35.10 40.59Avrg. number of assets 4.12 3.85 3.42 2.39 2.21 – 5.00 5.00Avrg. turnover p.a. 1.27 1.34 1.34 3.25 3.81 – 0.27 0.29

AggressiveNet Sharpe ratio 0.72#†† 0.73##†† 0.77##†† 0.52 0.53 – 0.19 0.27Net Omega measure 1.77 1.78 1.86 1.52 1.53 – 1.15 1.24Net MDD (%) 17.98 14.72 11.00 22.23 24.59 – 50.99 40.59Avrg. number of assets 3.28 3.30 2.91 1.97 1.75 – 5.00 5.00Avrg. turnover p.a. 1.71 1.77 1.87 3.67 4.10 – 0.25 0.29

Notes: This table reports the portfolio performance measures for the full sample from 1993 to 2011 when consideringtransaction costs in the optimization function according to Equation (16). The variable transaction costs are set to 30 bpfor all assets. The maximum expected volatility is constrained to 5%, 10%, and 15% p.a. for the conservative, moder-ate, and aggressive investor type, respectively. */**, #/##, †/†† indicate a significantly higher Sharpe ratio compared tomean-variance, the strategically weighted benchmark, and the 1/N benchmark at the significance levels of 1% and 5%,respectively.

includes industry indices for US stocks (US-industries) such as Oil and Gas, Basic Materi-als, Industrials, Consumer Goods, Health Care, Telecom, Utilities, Financials, Technology, andConsumer Services for the 1993–2011 period (Datastream indices).

Panel A of Table 12 presents the performance measures (Sharpe ratios) for the two alter-native multi-asset universes for a moderate investor. The conclusions for the conservative andaggressive investors are similar. The results confirm the base case in that the BL portfolios areconsistently superior relative to the MV, BS, and naïve approaches. The MV and BS portfoliosalso perform slightly better than naïve-diversified portfolios. However, the outperformance ofBS and MV is lower in magnitude and insignificant.

Panel B of Table 12 contains the results for stock-only asset universes.7 Because the risk ofstock-only portfolios is substantially larger than for mixed portfolios, we compute the optimiza-tion models for the aggressive investor with volatility threshold of 15% p.a. As in our basecase results, the BL model performs consistently better than MV and BS. However, similarto DeMiguel, Garlappi, and Uppal (2009) and Murtazashvili and Vozlyublennaia (2013), wefind that for stock-only portfolios none of the optimization models is able to outperform naïve-diversified portfolios such as the 1/N strategy. The likely explanation for this finding is thatin stock-only portfolios the diversification effects and the benefits of reallocating the portfolioover time are lower and are outweighed by estimation errors (as discussed by Murtazashviliand Vozlyublennaia 2013) and transaction costs. We conclude that in order to outperform naïve-diversified portfolios, optimization models require more precise return estimates than provided

The European Journal of Finance 27

Table 12. Alternative asset universes.

Assetuniverse Period

Number ofassets s/b/c BL-st.w. BL-1/N

BL-MinVar BS MV st.w. 1/N

Panel A: Multi-asset universesBase case 1993–2011 2/2/1 0.75∗∗##†† 0.72†† 0.76†† 0.55 0.49 0.32 0.27Multi-asset USD 1993–2011 8/1/1 0.78** 0.77∗∗## 0.74†† 0.68 0.51 0.43 0.28Multi-asset EUR 1999–2011 4/2/1 0.83∗∗##†† 0.76†† 0.81##†† 0.46 0.45 0.27 0.18

Panel B: Stock-only universesInt-stocks 1993–2011 8/0/0 / 0.19 0.15 0.06 0.01 – 0.28US-industries 1993–2011 10/0/0 / 0.34 0.38 0.23 0.29 – 0.36

Notes: This table reports the portfolio performance (Sharpe ratios) for different asset universes. The column number ofassets shows the number of assets considered in the asset universe, where s is the number of stock indices, b is the numberof bond indices, and c is the number of commodity indices. */**, #/##, †/†† indicate a significantly higher Sharpe ratiocompared to mean-variance, the strategically weighted benchmark, and the 1/N benchmark at the significance levels of1% and 5%, respectively.

by sample moments. Alternatively, quantitative investors need an asset universe that includesnot only stocks, but also a broader set of assets, providing larger diversification effects and largerbenefits from dynamically reallocating the portfolio, thereby enabling optimization strategies toprovide superior results relative to naïve diversification strategies.

6. Conclusion

We implement a sample-based version of the BL model and analyze its out-of-sample per-formance relative to BS, MV, minimum-variance, and naïve-diversified portfolios based onmulti-asset rolling sample optimizations for the period from January 1993 to December 2011.To ensure the comparability of all optimization approaches, we use the same sample moments inall approaches. Our empirical results contribute to the literature in several dimensions. We findthat the BL model generates a consistently higher out-of-sample performance (Sharpe ratios andOmega measures) relative to MV, BS, and minimum-variance optimized portfolios. In compar-ison to naïve diversification strategies with equal or strategic portfolio weights, the BL modelsignificantly performs better in almost all analyzed cases even after transaction costs.

Furthermore, BL portfolios are less risky as indicated by lower volatility and ‘maximum draw-down’ (MDD) measures, and are more diversified across asset classes, including a larger numberof assets compared to MV and BS optimized portfolios. Sensitivity analyses suggest that the out-of-sample outperformance of the BL model compared to MV is due to incorporating additionalinformation on the reliability of return estimates, resulting in more stable and more accuratereturn estimates and consequently in a lower portfolio turnover.

We separate the full sample (1993–2011) into four sub-periods based on the monetary cycleand stock market signals and find that the BL model outperforms MV and BS in all four sub-periods, but the outperformance is larger in magnitude during recessionary periods. In line withthe results for the full sample period, we observe the additional benefits of the BL strategy relativeto MV in all sub-periods, such as lower risk (maximum drawdown and volatility), lower portfolioturnover, and broader portfolio diversification.

The robustness checks confirm that our results are insensitive to restricting optimized portfolioweights, to short selling restrictions, to all input parameter variations, and to different levels of

28 W. Bessler et al.

transaction costs. This holds even when transaction costs are included in the optimization func-tion. Finally, our results hold for larger and different sets of assets. For stock-only portfolios,the BL approaches consistently perform better than MV and BS optimization. However, wheninvesting solely in stocks, we find that none of the optimization models is able to outperformconsistently naïve-diversified portfolios such as the 1/N strategy. For these stock portfolios, thediversification effects are lower, and the estimation errors and transaction costs seem to out-weigh the benefits of reallocating the portfolio over time. Hence, to outperform naïve-diversifiedportfolios, optimization models require more precise return estimates than provided by samplemoments or an asset universe that includes not only stocks, but also a broader set of assetsoffering larger diversification effects and higher benefits of reallocating the portfolio over time.Both enable optimization strategies to demonstrate their strengths vis-à-vis naïve diversificationstrategies.

Overall, we provide empirical evidence that the BL model consistently performs superior rel-ative to MV and BS, and significantly outperforms naïve strategies for all analyzed data sets,sub-periods, and robustness checks. Consequently, quantitative portfolio managers may be ableto improve their asset allocation decisions when employing the BL approach rather than the MVor BS framework.

Acknowledgements

We are grateful to Chris Adcock (editor) and two anonymous referees for helpful advice and comments that significantlyimproved the quality of this research. The authors also thank Lawrence Kryzanowski, Paul Söderlind, Michael Frömmel,as well as participants at the European Financial Management Symposium 2012, the 19th Annual Meeting of the GermanFinance Association 2012, the International Annual Conference of the German OR Society 2012, and the Verein fuerSocialpolitik Annual Congress 2012 for helpful comments and suggestions.

Notes

1. Value-weighted portfolios weight constituents proportional to their relative market weights and price-weightedportfolios allocate the fraction of each constituent proportional to its actual market price.

2. Moreover, benchmark portfolios with rebalancing are more common in institutional asset management and we expectthe performance of the portfolios with rebalancing to be less sensitive to the evaluation period compared to the buy-and-hold portfolio because the portfolio composition of the buy-and-hold portfolio varies over time based on therelative performance of the assets during the evaluation period.

3. See Opdyke (2007) for a detailed description of the Sharpe ratio test.4. The results for the other investor types are qualitatively similar so that we only report the performance measures

for the moderate investor. We use the same continuous sample to compute optimized portfolios and divide theresulting return time series into four sub-samples, thereby avoiding the problems of rolling sample estimations on adiscontinuous sample.

5. It seems surprising that restricting asset weights results in higher Sharpe ratios for conservative investors, but toan inferior performance for moderate and aggressive investors. The explanation is that the restriction of the assetweights for the aggressive (moderate) investor results in a maximum portfolio weight of government bonds below22.5% (50%). Therefore, in periods of stock market downturns, the asset allocation algorithm cannot fully reallocatethe portfolio into the safe asset, thereby loosing performance relative to the unrestricted case.

6. In the BS approach the sample covariance matrix is inflated by the factor (T − 1)/(T − N − 2), where T is thesample size and N is the number of assets. While for larger observation windows the covariance matrix inflationin the BS approach plays only a minor role, it gets pronounced for a shorter estimation window of only 12 monthsas the inflation factor increases to 2.2 (T = 12; N = 5). In this case, the expected portfolio volatility increases bythe factor

√2.2 = 1.48. Because the volatility constraint requires the expected portfolio volatility to be below 10%,

inflating the covariance matrix by a factor of 2.2 is equal to setting the volatility constraint to a level of 6.74%(10%/

√2.2) instead of 10%.

The European Journal of Finance 29

7. For the stock-only case, we do not compute strategically weighted portfolios as strategic weights are only used todetermine the fraction of stocks relative to bonds and commodities and we do not have any reason to set differentstrategic weights for different industries or countries on an ex ante basis.

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