Covariance Misspecification in Asset Allocation

Series of returns to broad asset classes often possess histories of unequal length and have been subject to smoothing. Estimates of covariances are generally based on the common, although shorter, series length, and covariances for smoothed series are necessarily biased downward. These characteristics pose serious problems that can generate suboptimal and misleading allocations among asset classes. The article discusses elements of the underlying theory in proposing an informationally efficient covariance estimator. The estimator is then compared with conventional covariance estimates in an empirical application. Covariance estimates are found to be sensitive to both truncated estimates involving shorter series and the effects of smoothing.Problems related to estimation of mean returns are well known, but problems related to covariance estimation are thought to be benign. In applications, however, solutions to quadratic optimization problems often fail because of ill-conditioned covariance matrices. This problem is especially relevant for the search for optimal allocations among broad asset classes whose covariance estimates must first circumvent a number of obstacles—in particular, return series of unequal length and return smoothing.We find that cross-moments estimated on truncated return series (because of different series lengths) are informationally inefficient and that smoothing biases volatilities downward. In both cases, covariance estimates can produce seriously misleading portfolio allocations. Drawing together the theoretical literature on these issues, we propose a procedure for covariance estimation that is informationally rich, more efficient than in traditional methods, and free of the biases associated with naive estimates. Empirically, we compare this improved estimator with the naive estimator in applications to allocations for asset classes typically considered by large institutional investors.We use a framework from the literature that maximizes the informational efficiency in constructing covariance estimates for series of different length. We extend this framework to provide an algorithm for updating covariances sequentially, and we include a formal process that addresses some common types of return smoothing. We also analyze relative portfolio performance in a Monte Carlo experiment that isolated and measured the allocative inefficiency and higher risk common to portfolios based on naive covariance estimates for truncated and smoothed series.Using historical returns to seven asset classes and applying the corrections for truncation and smoothing, we first estimated a corrected covariance matrix, which we contrasted to the corresponding naive covariances. We then addressed allocative inefficiency in the Monte Carlo experiment, in which we used this corrected covariance matrix with a corresponding vector of asset-class returns to generate repeated samples of quarterly returns. We then estimated corrected and naive covariance matrices and mean returns for each sample, and we used these data as arguments in separate mean–variance-optimization problems to solve for the optimal portfolio. Because the “true” returns and covariances that generated these samples were known a priori, these simulated portfolios could be compared with the “full-information” portfolio.On the one hand, we found that for N = 10,000 repetitions, the average quarterly misallocation (measured as deviation in asset-class weights from the true weight vector) for the naive portfolio was 28.66 percent. And a Wilcoxon test easily rejected the null hypothesis that the weight distributions for the full-information and naive estimates are equal.On the other hand, the corrected covariances reduced misallocation by 8.14 percentage points per quarter, and we failed to reject the null hypothesis that the corrected and full-information portfolio weight distributions share the same median. Moreover, the median quarterly naive return underperformed both the full-information and corrected median quarterly returns. The naive portfolio’s tendency to take on extreme positions could be clearly seen in this portfolio’s return volatility (2.51 percent) per quarter relative to the volatility of the full-information portfolio (2.17 percent) and corrected portfolio (2.22 percent).Finally, a long-only constraint exacerbated underperformance of the naive portfolio; it forced approximately 13 percent more extreme (zero-weight) positions than did the full-information portfolio, whereas the corrected covariance portfolio produced only 5 percent more extreme positions.In general, naive covariance estimates generate portfolios with lower returns and higher risks than corrected estimates, and naive covariance estimates underestimate true value at risk.