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...

We prove the convergence of a deterministic algorithm to compute the distribution function of the sum of d >1 dependent random variables, with given joint distribution, via the approximation of the probability measure of a d-dimensional symplex by overlapping hypercubes.

Necessary and sufficient conditions for uniform geometric convergence in the relative supremum norm of the Metropolis-Hastings simulation algorithm with a general generating function are established. An explicit expression for the convergence rate is given.