Assessing the Risk in Sample Minimum Risk Portfolios

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. While mean variance portfolio theory has been around for nearly 50 years, its use has become widespread mostly during the past decade. This is primarily due to the decline in the cost of acquiring and processing financial market data. The decline in the cost of computing has also made large-scale optimization feasible for most investors, making it possible to work directly with returns on individual securities when constructing efficient portfolios instead of first grouping them into asset classes. The use of a large collection of primitive securities has the advantage of helping construct more efficient portfolios. The disadvantage is that it becomes more difficult to assess the true variance of those portfolios when covariances of asset returns are not known and their estimates are used as inputs to the optimizer. It has been observed in the literature that the in-sample value of the variance of an optimal portfolio constructed using historical return data is an optimistic estimate of the portfolio’s true variance,0 hereafter referred to as the out-of-sample variance. This bias, commonly referred to as in-sample optimism, increases with the number of assets used to construct mean-variance efficient portfolios. In this paper we identify the reason for this in-sample optimism and suggest methods for correcting for it. Since our focus is in assessing the risk of an efficient portfolio and not its mean, we focus on the global minimum variance and minimum benchmark tracking-error variance portfolios constructed using an estimated covariance matrix. We refer such portfolios as sample minimum risk portfolios (SMRPs) in this paper. We demonstrate that the in-sample optimism can be substantial in certain situations. It might be argued that by suitably scaling up the in-sample variance by a factor that is related to the degree-of-freedom of the distribution of the estimated covariance matrix, we may be able to correct for in-sample optimism. When returns over time are drawn from an i.i.d. multivariate Normal distribution, we show that scaling up the in-sample variance by a degrees-of-freedom related factor provides an unbiased estimate of the variance of the true global minimum variance portfolio. Since the out-of-sample variance of the sample minimum risk portfolio is strictly larger on average, this procedure does not adequately correct for the in-sample optimism.