The use of the triangular approximation for some complicated risk measurement calculations

We introduce the triangular approximation to the normal distribution in order to extract closed- and semi-closed-form solutions that are useful in risk measurement calculations. In risk measurement models there is usually a normal distribution together with some other distributions in a portfolio of risks. Exceedance probability or value-at-risk (VaR) calculations for these portfolios require simulations. However, with the use of the triangular approximation to the normal density we can have closed-form solutions for risk measurements using actuarial models that include not only insurance risk, such as gamma- and Pareto-distributed losses, but also financial risk. We also approximate the collective risk model under lognormally distributed severities and estimate its VaR. We evaluate the accuracy of the analytic solutions versus the Monte Carlo estimates and, according to our results, the analytic solutions requiring much less computational time are quite good.