On the Realized Risk of High-Dimensional Markowitz Portfolios

We study the realized risk of Markowitz portfolio computed using parameters estimated from data and generalizations to similar questions involving the out-of-sample risk in quadratic programs with linear equality constraints. We do so under the assumption that the data is generated according to an elliptical model, which allows us to study models where we have heavy-tails, tail dependence, and leptokurtic marginals for the data. We place ourselves in the setting of high-dimensional inference where the number of assets in the portfolio, p, is large and comparable to the number of samples, n, we use to estimate the parameters. Our approach is based on random matrix theory. We consider both the impact of the estimation of the mean and of the covariance. Our work shows that risk is underestimated in this setting, and further, that in the class of elliptical distributions, the Gaussian case yields the least amount of risk underestimation. The problem is more pronounced for genuinely elliptical distributions and Gaussian computations give an overoptimistic view of the situation. We also propose a robust estimator of realized risk and investigate its performance in simulations.