A note on the computation of sharp numerical bounds for the distribution of the sum, product or ratio of dependent risks

In this paper, an approximation method for computing numerically the cumulative distribution function of the sum of d random variables is developed. The method leads to numerical bounds for the distribution of the sum of dependent risks. The bounds are fast to compute and converge to the exact value if the joint probability density function exists. They also allow to evaluate sharp numerical bounds on the Value-at-Risk measure. Moreover, the fact that the approximation is deterministic, hence without uncertainty on the resulting values, is an advantage over MC simulation techniques. Applications in actuarial science and finance illustrate the accuracy of the procedure. We also present analogous bounds for the distribution of the product or the ratio of two random variables, which can be useful for actuarial or financial applications.