“In Defense of Portfolio Optimization: What If We Can Forecast?”: Author Response

This article responds to:

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

We thank Richard O. Michaud, David N. Esch, and Robert O. Michaud for their interest in our paper. In their letter, they suggest that our results are inconsistent with the canonical literature because we ignore model risk and covariance matrix estimation error, and they claim that were we to include realistic levels of both, much higher levels of IC than those we model would be needed to outperform 1/N investment. We respectfully disagree with their conclusion.

With respect to model risk, it is virtually a truism that 1/N would be preferred if assumed model risk were high enough; this raises the question of what sort of model risk we might expect, for example, if we assume mean–variance optimization. The answer, going as far back as Samuelson’s seminal 1970 paper, is that mean–variance optimization is surprisingly robust for return distributions, those he termed “compact distributions.” Many other studies (some cited in Samuelson’s work) make a similar point. Of course, there are certain asset classes and investment situations for which mean–variance optimization would be inappropriate; we do not advocate it as a universal panacea. Nonetheless, we are confident that our results are robust to many standard investment problems.

The letter states that we ignore “the bulk of estimation error—model error and covariance matrix estimation.” We address both points throughout the paper. In the section “Mean–Variance Performance: A Simulation Approach,” we relax the assumption that the covariance matrix is known and show that, where investors can forecast and the opportunity set is large, mean–variance optimization outperforms 1/N even when the covariance matrix needs to be estimated.We then acknowledge that our simulation assumptions might “introduce a bias that favors the mean–variance approach” and so go on to relax those assumptions in our empirical setting. In the section “Out-of-Sample Empirical Evaluation,” where the data-generating process is not the CAPM and the covariance matrix must be estimated, we demonstrate that mean–variance optimization generates higher Sharpe ratios than 1/N in 20 out of 21 high-dimension problems, consistent with our analytic and simulation results.

Michaud, Esch, and Michaud’s letter questions why we found it “necessary to impose inequality constraints on the optimizer.” We note that their own portfolio resampling heuristic is reliant on the long-only constraint to produce results that differ from the standard mean–variance technique (see Scherer 2002). In our case, we apply a long-only constraint because it is applicable to the vast majority of practitioners. For investors with forecasting ability, the long-only constraint hampers their ability to implement their views (see Clarke, de Silva, and Thorley 2002), so in practice applying this constraint is a conservative assumption.

In their letter, Michaud, Esch, and Michaud make reference to the simulation studies of Jobson and Korkie (1981) and their own more recent simulation study (Michaud, Esch, and Michaud forthcoming), which suggest that 1/N beats mean–variance optimization. Such studies typically use a comparatively small amount of data to compute covariance, leading to ill-conditioned covariance matrices driving poor performance. Michaud, Esch, and Michaud (forthcoming) use monthly data (1994–2013) to estimate the covariance matrices of portfolios with some 500 stocks: Such a small sample equates to a single data point per term in the covariance matrix.¹ It is for this reason that dimension reduction techniques and high-frequency data are used in real life. Our results, which demonstrate the clear benefits of mean–variance optimization, reconcile a paradox. If mean–variance optimization is as flawed as Michaud, Esch, and Michaud imply in their letter, why is this technique so pervasive among educated and incentivised practitioners?

Notes

¹ The hypothetical practitioner would be estimating some 125,250 terms $\left(n^2+n\right)/2$ from just 120,000 stock returns.