Assessing the Risk in Sample Minimum Risk Portfolios

Abstract

We show that the in-sample estimate of the variance of a global minimum risk portfolio constructed using an estimated covariance matrix of returns will on average be strictly smaller than its true variance. Scaling the in-sample estimate upward by a standard degrees-of-freedom related factor or using the Bayes covariance matrix estimator can be inadequate; the correction is likely to be twice as large as the standard correction when returns are i.i.d. multivariate Normal. We develop a Jackknife-type estimator of the optimal portfolio's variance that is valid when returns are i.i.d.; and a variation that may be better when returns exhibit volatility persistence.

We empirically demonstrate the need to correct for in-sample optimism by considering an optimal portfolio of 200 stocks that has the lowest tracking error when the S&P500 is the benchmark and three years of daily return data are used for estimating covariances. When the optimal portfolio is constructed using the sample covariance matrix, the standard deviation of the tracking error is 1.46 percent whereas its in-sample estimate is 0.94 percent. Standard degrees of freedom correction gives an estimate of 1.10 percent; our correction, 1.24 percent; and the weighted Jackknife, 1.36 percent.

Keywords: Minimum risk portfolios, in-sample optimism

JEL Classification: G10, G11