LAUSR

Recent studies in Lee and Prekopa (Oper Res Lett 45:19–24, 2017) and Lee (Oper Res Lett 45:1204–1220, 2017) showed that a union of partially ordered orthants in $$R^n$$ can be decomposed only into the largest and the second largest chains. This allows us to calculate the probability of the union of such events in a recursive manner. If the vertices of such orthants designate p-level efficient points, i.e., the multivariate quantile or the multivariate value-at-risk (MVaR) in $$R^n$$ , then the number of them, say N, is typically very large, which makes it almost impossible to calculate the multivariate conditional value-at-risk (MCVaR) introduced by Prekopa (Ann Oper Res 193(1):49–69, 2012). This is because it takes $$O(2^N)$$ in case of N MVaRs in $$R^n$$ to find the exact value of MCVaR. In this paper, upon the basis of ideas in Lee and Prekopa (Oper Res Lett 45:19–24, 2017) and Lee (Oper Res Lett 45:1204–1220, 2017), together with proper adjustments, we study efficient methods for the calculation of the MCVaR without resorting to an approximation. In fact, the proposed methods not only have polynomial time complexity but also computes the exact value of MCVaR. We also discuss additional benefits MCVaR has to offer over its univariate counter part, the conditional value-at-risk, by providing numerical results. Numerical examples are presented with computing time in both cases of given population and sample data sets.