An Upper Confidence Bound Approach to Estimating Coherent Risk Measures

Abstract

Coherent risk measures have received increasing attention in recent years among both researchers and practitioners. The problem of estimating a coherent risk measure can be cast as estimating the maximum expected loss taken under a set of probability measures. In this paper, we consider the set of probability measures is finite, and study the estimation of a coherent risk measure via an upper confidence bound (UCB) approach, where samples of the portfolio loss are simulated sequentially from one of the probability measures. We study in depth the so-called Grand Average estimator, and establish statistical guarantees, including its strong consistency, asymptotic normality, and asymptotic mean squared error. We also construct asymptotically valid confidence intervals.