The computation of the worst conditional expectation

Recent advances in risk theory identify risk as a measure related to the tail of a probability distribution function, since it represents the "worst" outcomes of the random variable. Measures like Value-at-Risk, Conditional Value-at-Risk, Expected Shortfall and so on have become familiar operational tools for many financial applications. In this paper, one of these measures, the Worst Conditional Expectation with threshold alpha of a discrete random variable Z, shortly WCE[Z], is considered. The main properties of WCE are established, proving that it belongs to the class of the coherent risk measures, of which the Conditional Value-at-Risk is the most popular one. We remark the differences between WCE and CVaR, proving that they are the same measure for continuous random variable, but that for discrete random variables CVaR is an approximation of the “real” worst outcome of the variable. Then, it has been found that computing the WCE of a discrete random variable can be formulated as a fractional integer programming problem with a single linear constraint, but its complexity is NP-hard, therefore it must be solved by implicit enumeration. Due to its similarity with the knapsack problem, it has been found that a good upper bound and a sharp data structure allow the implementation of a branch\&bound that is able to solve realistic size problems in less than one hundredth of a second.