Computing the distribution of the sum of dependent random variables via overlapping hypercubes

The original motivation of this work comes from a classic problem in finance and insurance: that of computing the Value-at-Risk (VaR) of a portfolio of dependent risky positions, i.e., the quantile at a certain level of confidence of the loss distribution. In fact, it is difficult to overestimate the importance of the concept of VaR in modern finance and insurance. It has been recommended, although with several warnings, as a measure of risk and the basis for capital requirement determination by both the guidelines of international committees (such as Basel 2 and 3 and Solvency 2) and the internal models adopted by major banks and insurance companies. However, the actual computation of the VaR of a portfolio constituted by several dependent risky assets is often a hard practical and theoretical task. To this purpose, here we prove the convergence of a geometric algorithm (alternative to Monte Carlo and quasi-Monte Carlo methods) for computing the Value-at-Risk of a portfolio of any dimension, i.e., the distribution of the sum of its components, which can exhibit any dependence structure. Moreover, although the original motivation is financial, our result has a relevant measure-theoretical meaning. What we prove, in fact, is that the H-measure of a d-dimensional simplex (for any \(d\ge 2\) and any absolutely continuous with respect to Lebesgue measure H) can be approximated by convergent algebraic sums of H-measures of hypercubes (obtained through a self-similar construction).