بهینه سازی پرتفوی : تحمیل محدودیت های راست

We reassess the recent finding that no established portfolio strategy outperforms the naively diversified portfolio, 1/N, by developing a constrained minimum-variance portfolio strategy on a shrinkage theory based framework. Our results show that our constrained minimum-variance portfolio yields significantly lower 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 average Sharpe 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 consistently and significantly. At the same time, our developed portfolio strategy achieves a comparatively low turnover and exhibits no excessive short interest.

Since the foundation of modern portfolio theory by Markowitz (1952), the development of new portfolio strategies has become a horserace-like challenge among researchers. The sobering finding that theoretically optimal, utility maximizing portfolios perform poorly out-of-sample1 can be attributed to the error prone estimation of expected returns, leading to unbalanced optimization results. This result directed researchers’ attention to the minimum-variance portfolio, the only portfolio on the efficient frontier that simply requires the variance–covariance matrix as input parameter for the optimization. For instance, Merton, 1980, Jorion, 1985 and Nelson, 1992 remark that variance–covariance estimates are relatively stable over time and can, hence, be predicted more reliably than expected returns. Nevertheless, DeMiguel et al. (2009b) have argued that no single portfolio strategy from the existing portfolio selection literature outperforms the naively diversified portfolio, 1/N, consistently in terms of out-of-sample Sharpe ratio. Similar to Fletcher (2009), we evaluate in this paper a broader range of minimum-variance portfolios to challenge the findings of DeMiguel et al. (2009b). Additionally, we develop a constrained minimum-variance portfolio strategy that outperforms 1/N in terms of a lower out-of-sample variance and a higher Sharpe ratio while, at the same time, yielding a turnover and short interest that do not hamper the practical implementation of this portfolio strategy. We propose a minimum-variance portfolio strategy with flexible upper and lower portfolio weight constraints. Incorporating these constraints into the portfolio optimization process trades off the reduction of sampling error and loss of sample information (Jagannathan and Ma, 2003). On the one hand, weight constraints ensure that portfolio weights are not too heavily driven by the sampling error inherent in parameter estimates based on historical data, which often leads to highly concentrated portfolios.2 On the other hand, portfolio weight constraints cause a misspecification of the optimization problem because the resulting portfolio weights are less driven by potentially useful sample information (Green and Hollifield, 1992). Consequently, incorporating portfolio weight constraints into the portfolio optimization problem is promising if the input parameters are error prone. We calibrate portfolio weight constraints such that the desired reduction of sampling error and the concomitant loss of sample information is traded-off. Using this shrinkage theory based framework, we introduce portfolio weight constraints which depend on the error inherent in the empirical variance–covariance matrix estimate. In particular, we impose the set of lower and upper portfolio weight constraints that minimizes the sum of the mean squared errors (MSE) of the covariance matrix entries. The latter serves as a loss function, quantifying the trade-off between the reduction of sampling error and loss of sample information. Our empirical results show that our constrained minimum-variance portfolio with lower and upper portfolio weight constraints achieves substantial out-of-sample variance reductions in comparison to various minimum-variance portfolios. We observe that the variance of our portfolio strategy achieves the lowest variances among all twelve considered portfolio strategies. In terms of risk adjusted performance, we observe that our portfolio strategy generates a 32.5% higher Sharpe ratio than 1/N. This Sharpe ratio increase is statistically significant on five out of six datasets. Further, we observe that our portfolio strategy achieves on average a higher Sharpe ratio than every other benchmark strategy. This finding 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 solitary lower or lower and upper weight constraints results in an equally effective risk reduction. The risk adjusted performance of both portfolios is on average equally good. However, we find that the constrained minimum-variance portfolio with lower and upper portfolio weight constraints achieves a less volatile Sharpe ratio over the various datasets than the constrained minimum-variance portfolio with solitary lower portfolio weight constraints. This is reflected in the statistical significance of the outperformance over 1/N. While imposing lower and upper weight constraints yields on five out of six datasets a significantly higher Sharpe ratio than 1/N, imposing solitary lower weight constraints yields only on two datasets a significantly higher Sharpe ratio. Further, we observe that the constrained minimum-variance portfolio with lower and upper portfolio weight constraints yields a lower turnover and short interest than the constrained minimum-variance portfolio with solitary lower weight constraints. Hence, we find that imposing lower and upper portfolio weight constraints is beneficial with respect to the resulting out-of-sample performance and the practical implementation of the portfolio strategy. The impact of our ex ante calibrated weight constraints varies with respect to the size of the investment universe. While the imposed constraints are loose for small portfolios, they are comparatively tight for larger investment universes. Specifically, we observe for portfolios comprising 30 or more assets, that the ex ante calibrated lower portfolio weight constraints are close to zero, i.e. a short-sale constraint. While the lower portfolio weight constraints of our minimum-variance strategies with solitary lower, respectively lower and upper weight constraints are similar across all investment universes, we find that the tightness of the additional upper constraint is particularly pronounced for larger universes. Our paper contributes to three lines of literature. First, we add to the prevalent discussion whether optimized portfolios represent a preferable investment vehicle over 1/N by amending the empirical evidence of DeMiguel et al., 2009b and Fletcher, 2009. Contrary to recent contributions to this ongoing discussion by Pflug and Pichler, 2012 and Kritzman et al., 2010 that assess the conditions rendering 1/N an optimal strategy, i.e. the reasons for the unsatisfying performance of optimal portfolios, we develop a portfolio strategy that outperforms 1/N consistently and significantly. Thus, our paper relates to Chevrier and McCulloch, 2008, DeMiguel et al., 2009a and Tu and Zhou, 2011, who claim to develop portfolio strategies achieving consistently higher Sharpe ratios than 1/N. However, neither Chevrier and McCulloch (2008) nor Tu and Zhou (2011) provide statistical inference for their results. Second, we extend previous work on portfolio optimization in presence of constraints. Alexander and Baptista, 2006 and Alexander et al., 2007 evaluate the imposition of drawdown constraints, while DeMiguel et al., 2009a and Gotoh and Takeda, 2011 assess the impact of norm constraints on the portfolio optimization process. 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 aforementioned work on weight constraints is concerned with arbitrarily chosen or ex post determined upper and/or lower constraints, we postulate a framework to determine these constraints ex ante. Our new approach should thus perform better out-of-sample given that flexible ex ante constraints are better able to suit the data at hand. Third, our framework represents a new approach to the estimation of the variance–covariance matrix for portfolio optimization. Similar to the Ledoit and Wolf, 2003, Ledoit and Wolf, 2004a and Ledoit and Wolf, 2004b shrinkage strategies, our approach imposes a data-dependent structure on the variance–covariance matrix. Our approach, however, requires fewer assumptions 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 portfolio performance. The remainder of the paper is organized as follows. Section 2 outlines our methodology and data, while Section 3 contains the empirical results. Section 4 reports the robustness checks, Section 5 concludes.

In this paper, we reassess the finding by DeMiguel et al. (2009b) that no existing portfolio strategy outperforms 1/N in terms of Sharpe ratio out-of-sample. Evaluating a broader range of minimum-variance portfolios than DeMiguel et al. (2009b), we find that the Ledoit and Wolf (2004a) strategy and the 1-norm constrained minimum-variance portfolio calibrated to minimize the out-of-sample variance, suggested by DeMiguel et al. (2009a), deliver empirically higher Sharpe ratios than 1/N. Yet, we find that the Sharpe ratio increases of the aforementioned strategies are only on one, respectively three out of the six considered datasets significant. Hence, our results corroborate the findings by DeMiguel et al. (2009b) that no established portfolio strategy outperforms 1/N significantly. Contrary to the established portfolio strategies, our constrained minimum-variance portfolio strategy developed in this paper achieves the goal of generating statistically significantly higher Sharpe ratios than 1/N. Our portfolio strategy builds on shrinkage estimation theory, trading-off the reduction of sampling error and loss of sample information by imposing a data dependent structure on the empirical variance–covariance matrix estimate. The advantage of our approach over the established shrinkage approaches to estimating the variance–covariance matrix is that we neither require any distributional assumptions such as iid returns, nor do we require the identification of a shrinkage target, which has a significant impact on the out-of-sample portfolio performance as our empirical results suggest. The findings document that our constrained minimum-variance portfolio 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 model based minimum-variance portfolio. Compared to the Ledoit and Wolf, 2003 and Ledoit and Wolf, 2004a shrinkage strategies and norm constrained minimum-variance portfolios suggested by DeMiguel et al. (2009a), we observe a similar out-of sample variance. In terms of risk adjusted performance, we find that our portfolio strategy outperforms the value weighted portfolio and 1/N consistently in terms of Sharpe ratio. On average, we observe that our portfolio strategy achieves across the six considered datasets a 32.5% higher Sharpe ratio than 1/N. We find that the Sharpe ratio increase is significant on five out of six datasets. The outperformance of our constrained minimum-variance portfolio is not plagued by excessive turnovers or short interest. We rather observe that our portfolio strategy achieves a comparatively low turnover and short interest of 16.7% and 91.0%, respectively. This is in the range of the short interest and turnover of the established Min-1F, Ledoit and Wolf, 2003, Ledoit and Wolf, 2004a and Ledoit and Wolf, 2004b and norm constrained strategies suggested by DeMiguel et al. (2009a). Hence, our portfolio strategy represents a practically implementable approach to outperforming 1/N consistently.