Importance sampling for the simulation of reinsurance losses

Abstract

Importance sampling is a well developed method in statistics. Given a random variable $X$, the problem of estimating its expected value $\mu$ is addressed. The standard approach is to use the sample mean as an estimator $\bar x$. In importance sampling, a suitable variable $L$ is introduced such that the random variable $X/L$ has an estimator with a smaller variance than that of $\bar x$. As a result, a smaller sample size can lead to the same estimation accuracy. In the simulation of reinsurance financial terms for catastrophe loss, choosing a general variable $L$ is difficult: Even before the application of financial terms, the loss distribution is often not modelled by a closed-form distribution. After that, a wide range of financial terms can be applied that makes the final distribution unpredictable. However, it is evident that the heavy tail of the resulting net loss distribution makes the use of importance sampling desirable. We propose an importance sampling technique using a power function transformation on the cumulative distribution function. The benefit of this technique is that no prior knowledge of the loss distribution is required. It is a new technique that has not been documented in the literature. The transformation depends on the choice of the exponent $k$. For a specific example we investigate desirable values of $k$.