Statistical analysis of portfolio risk and performance measures: The influence function approach

Risk measures (e.g., volatility, VaR, ETL/CVaR), performance measures (e.g., Sharpe, Sortino, STARR) and optimal portfolios all play an essential role in risk and asset management and are often based on estimated statistics from historical data. Consequently these results are subject to estimation error and can be very sensitive to perturbations of the underlying distribution and adversely influenced by outliers. In addition, there are a bewildering variety of risk and performance measures and there is a need for a convenient way to compare these and provide guidelines for "best" choices. The statistical influence function developed in the robust statistics literature provides an excellent tool to study these issues, in particular to: (a) visualize and display the relative characteristics of the various portfolio risk and performance measures; (b) compute the influence from the outliers and inliers; (c) compute close-form large sample variance of estimators. Our main goal in this study is to introduce the use of influence function in finance and provide a systematic tool to study statistical properties and characteristics of the variety of risk measures and performance measures, as well as the optimal portfolios. In the first chapter, we give a review of relevant literature pertaining to risk measures and performance measures. We also review the three common approaches to rank portfolios and discuss their distinctions and inter-connections. In chapter 2, we give a basic introduction to influence functions and our motivation of research. Chapter 3, 4 and 5 consists of the main body of our computational and analytic results. Chapter 3 and 4 deal with the statistics of stand-alone risk measures and performance measures, respectively, while chapter 5 discusses the mean-variance optimal portfolios from multivariate distribution.