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We first study the properties of solutions of quadratic programs with linearequality constraints whose parameters are estimated from data in thehigh-dimensional setting where p, the number of variables in the problem, is ofthe same order of magnitude as n, the number of observations used to estimatethe parameters. The Markowitz problem in Finance is a subcase of our study.Assuming normality and independence of the observations we relate the efficientfrontier computed empirically to the "true" efficient frontier. Ourcomputations show that there is a separation of the errors induced byestimating the mean of the observations and estimating the covariance matrix.In particular, the price paid for estimating the covariance matrix is anunderestimation of the variance by a factor roughly equal to 1-p/n. Thereforethe risk of the optimal population solution is underestimated when we estimateit by solving a similar quadratic program with estimated parameters. We alsocharacterize the statistical behavior of linear functionals of the empiricaloptimal vector and show that they are biased estimators of the correspondingpopulation quantities.

High-dimensionality effects in the Markowitz problem and other quadratic programs with linear constraints: Risk underestimation