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 modernfi nance and insurance: it has been recommended, although with several warnings, as a measure of risk and the basis for capital requirement determination both by the guidelines of international committees (such as Basel 2 and 3, Solvency 2 etc.) 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 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).