Approximations of Ruin Probability by Tri-atomic or Tri-exponential Claims

We find the necessary and sufficient conditions fot fitting five given moments by a triatomic distributions. We consider three examples drawn from fire (large spread), individual life (medium spread) and group life (small spread) insurance data, fit them with triatomics, and compute the ruin probabilities using well knnown formulas for discrete and for combination of exponentials claims amounts. We then compare our approximations with the exact values that appeared in the literature. In the fire (large spread) example, we found the triatomic to have 4.8% average relative error of 0.03%. In the small spread example, tre triatomic has maximum relative error of 0.14% and average relative error of 0.02%.