Improved Analytical Bounds for Some Risk Quantities

Simple analytical lower and upper bounds are obtained for stop-loss premiums and ruin probabilities of compound Poisson risks in case the mean, variance and range of the claim size distribution are known. They are based on stop-loss external distributions and improve the bounds derived earlier from dangerous extremely distributions. The special bounds obtained in case the relative variance of the claim size is unknown, but its maximal value is known, are related to other actuarial results. Keywords: analytical bounds, stop-loss, ruin probability, stochastic orderings, relative variance, atomic distributions