Bounds for Ruin Probabilities and Value at Risk

Financial institutions are concerned about the risk of extreme events, called tail risks, for which the underlying distribution is unknown. Thus, it is difficult to estimate the tail probabilities from actual observations. This paper offers a methodology for calculating confidence intervals for tail risk probabilities by deriving upper and lower semiparametric bounds, given only the first two moments of the distribution, which may be reasonably estimated from actual observations. We apply this methodology to determine confidence intervals for two tail event probabilities. The first, we call ruin probability, occurs when two financial variables simultaneously have extremely low values. The second tail event, which we call value at risk probability, occurs when a portfolio of two financial variables takes a very low value. In all cases we are working with the physical measure or distribution, rather than a pricing or risk neutral measure. That is, we are finding actual or physical ruin and VaR probabilities. In order to numerically obtain the confidence intervals, we use a so-called sum of squares optimization program. To illustrate our results we present relevant numerical experiments.