Outliers, influence functions, and robust portfolio optimization

Portfolio optimization and calculation of the efficient frontier requires estimation of the covariance matrix and mean vector for the portfolio returns as a fundamental step. It is almost universal that portfolio optimization is carried out based on the classical covariance matrix and vector sample mean estimates, which are maximum-likelihood estimates when the returns have a joint Gaussian distribution. However, financial returns are often strongly non-Gaussian in character, and exhibit multivariate outliers. Thus the Gaussian maximum likelihood rationale is not very convincing. Furthermore, it is well-known in the statistical literature that the classical covariance matrix estimate and sample means can be very greatly influenced by a small fraction of outliers. See for example, Gnanadesikan, and Kettenring (1972), Devlin, Gnanadesikan and Kettenring (1975, 1981), and Hampel et al. (1986). Thus, optimal portfolio weights and the efficient frontier can be greatly influenced by a small fraction of outliers in the returns. This fact appears to have been largely overlooked in the finance literature on portfolio optimization