In Defense of Portfolio Optimization: What If We Can Forecast?

Abstract

We challenge the academic consensus that estimation error makes mean–variance portfolio strategies inferior to passive equal-weighted approaches. We demonstrate analytically, via simulation, and empirically that investors endowed with modest forecasting ability benefit substantially from a mean–variance approach. An investor with some forecasting ability improves expected utility by increasing the number of assets considered. We frame our study realistically using budget constraints, transaction costs, and out-of-sample testing for a wide range of investments. We derive practical decision rules to choose between passive and mean–variance optimization and generate results consistent with much financial market practice and the original Markowitz formulation.

Disclosure: The authors report no conflicts of interest.

Editor’s Note:

This article was externally reviewed using our double-blind peer-review process. When the article was accepted for publication, the authors thanked the reviewers in their acknowledgments. Nick Baltas was one of the reviewers for this article.

Submitted 8 June 2018

Accepted 20 March 2019 by Stephen J. Brown.

This article is referred to by:

“In Defense of Portfolio Optimization: What If We Can Forecast?”: A Comment

Acknowledgment

The authors would like to thank Professor Stephen Brown, Nick Baltas, Steven Thorley, the editorial team at the Financial Analysts Journal, and one anonymous reviewer for their very helpful comments and suggestions. We would also like to acknowledge the contributions of Professor Mark Kritzman, Dr. Mark Thompson, and participants at seminars in Cambridge, Oxford, and Sydney in shaping the paper.

Notes

¹ See, for example, Amenc, Goltz, Le Sourd, and Martellini (2008).

² Markowitz (1959) and Levy and Markowitz (1979) showed that quadratic approximation provides a reasonable and robust working assumption for a broad range of utility functions and return distributions. But in neither the 1959 nor the 1979 research is the assertion made that normality or quadratic utility holds, nor is either one a requirement for the model. See also Kritzman and Markowitz (2017). An alternative approach broadly within this framework would be to have regime-dependent risk aversion (see Chow, Jacquier, Kritzman, and Lowry 1999).

³ Note also that there are accepted Bayesian approaches to deal with parameter uncertainty.

⁴ Certainty equivalent is the amount of payoff that an agent would have to receive to be indifferent between that payoff and a given gamble.

⁵ Kritzman (2006) also demonstrated that when the assets are close substitutes, substantial misallocation induced by estimation error results in only relatively small changes in portfolio ex ante distributions and, hence, only a limited reduction in expected utility.

⁶ Significant contributions include Solnik (1993), Ulf and Maurer (2006), Campbell and Shiller (1988), Fama and French (1988), Lintner (1975), Fama and Schwert (1977), Campbell (1987), Poterba and Summers (1988), and Lo and MacKinlay (1988).

⁷ Among authors using a myopic agent (one that optimizes one-period-ahead utility) approach are Fleming, Kirby, and Ostdiek (2001, 2003); Jagannathan and Ma (2003); DeMiguel et al. (2009); and Kirby and Ostdiek (2012).

⁸ DeMiguel et al. (2009) divided the unconstrained mean–variance weights by their sum to ensure that the weights satisfied the budget constraint. Our results show that this seemingly innocuous approach has some unusual effects. We, instead, incorporated the budget constraint directly into the optimization problem by using the result given by Ingersoll (1987).

⁹ The online supplemental material is available at www.tandfonline.com/doi/suppl/10.1080/0015198X.2019.1600958.

¹⁰ We should sound a note of caution: These results hold when alpha, beta, and gamma are held constant. Adding assets to the investment universe will change the return and covariance matrixes, however, and marginal assets may become more prone to forecast errors.

¹¹ Formally,

¹² $I{C}^{*}\equiv \inf \{IC:L_{mv}(w^{*},\widehat{w})=L_{1/N}(w^{*},w^{1/N})\},$

¹³ where $L_y\left(w^{*},\widehat{w}\right)$is the expected utility loss of using weights $\widehat{w}$ instead of the optimal weight strategy. Here, we set the expected utility loss of strategy y using weights $\widehat{w}$ instead of optimal weights $\widehat{w}*$ equal for the different portfolios. We could equally well define IC* in terms of expected utility.

¹⁴ Provided $V_{1/N}{}^{}$is positive.

¹⁵ See Appendix B and Appendix C in the online supplemental material, available at www.tandfonline.com/doi/suppl/10.1080/0015198X.2019.1600958, for more details and the full proof, respectively.

¹⁶ Kenneth French’s data library is available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

¹⁷ As an illustration, the correlation between the “soda” and “gold” industries in the 48-industry classification is just 0.06; their parent sectors in the 5-industry classification have a correlation of 0.88.

¹⁸ M² converts the Sharpe ratio to a measure of the risk-adjusted return premium to the market, as proposed by Modigliani and Modigliani (1997). Mathematically, $M^2=S{R}_p{\sigma }_m-{\mu }_m,$ where $S{R}_p$ is the portfolio Sharpe ratio, ${\mu }_m$ is the mean market return, and ${\sigma }_m$ is the market standard deviation. We set ${\mu }_m=0.73\%$ and ${\sigma }_m=4.27\%$ as in Kane et al. (2010).

¹⁹ DeMiguel et al. (2009) forced the asset weights to sum to unity by dividing the unconstrained mean–variance weights by the absolute value of the sum of the weights, as follows: $w=\frac{{\displaystyle {\sum }_{}^{*-1}{\mu }^{*}}}{\left[\left|i^{\prime }{\displaystyle {\sum }_{}^{*-1}{\mu }^{*}}\right|\right]}.$

²⁰ We acknowledge that superior estimates of risk can be attained by using factor models, such as the Fama and French (1993) three-factor model (rarely used by practitioners), macroeconomic risk models, or statistical factor models. Fundamental factor models, such as those developed by Barra and Axioma, are routinely used by practitioners to reduce estimation error. The results that we present based on a one-factor model (if they support the mean–variance approach) should, therefore, be conservative. We note also that our models involve calculating a matrix inverse and that the work of Fan, Fan, and Lv (2008) implies that, relative to a k-factor model, we have incurred large losses in accuracy in this context. Rather than choosing a k-factor model among many possibilities, we chose the simplest and most parsimonious model to demonstrate the impact of forecasting ability. A reviewer has pointed out, however, that our single-factor specification may be helping improve performance of the mean–variance approach by reducing estimation error in the covariance matrix; we acknowledge this possibility.

²¹ Full data sources and prior literature supporting the inclusion of these variables are available in the appendixes in the online material (available at www.tandfonline.com/doi/suppl/10.1080/0015198X.2019.1600958).

²² We acknowledge that improved tests of Sharpe ratios could be accomplished by using bootstrapping procedures, as in Ledoit and Wolf (2008), but these procedures involve some implementation costs.