Bounds on Tail Probabilities and Value at Risk Given Moment Information

Abstract

We solve a moment problem to compute the best upper and lower bounds on the expected value E[φ(X)], subject to constraints E[X^i] = μ_i for i = 1, 2,...,n. By setting φ(x)=I_(-\inf,t], the indicator function for the event X ≤ t, we calculate the bounds on Pr(X ≤ t) = E[I_(-\inf,t]]. The bounds can be narrowed if more information about distribution classes is added. Specifically, we show how to find the bounds on a variable with unimodal distribution. In addition, given a set of moments, we present a moment-constrained maximum entropy method that provides “point” estimates on tail probabilities. As robustness check, we investigate how the bounds can be narrowed if more moments are considered. In addition, to check the accuracy and reliability of our numerical bounds, we compare our method with De Schepper and Heijnen (2010) which provides explicit expressions for the bounds.

The semiparametric bounds are useful in risk analysis where there is only incomplete information concerning the random variable X, such as an insurance loss or an asset return. We show how the inversion of these bounds leads to approximations to bounds on Value at Risk (VaR). Besides helping to construct a representative distribution with given moments, the moment-constrained maximum-entropy method can be used to define risk neutral probabilities for asset pricing. To illustrate this idea, we present a numerical example.

Keywords: Moment problem, Semidefinite programming, VaR, Maximum entropy

JEL Classification: C14, C61, G22, G32