Markov chain Monte Carlo for risk measures

In this paper, we consider random sums with heavy-tailed increments. By the term random sum, we mean a sum of random variables where the number of summands is also random. Our interest is to construct an efficient method to calculate tail-based risk measures such as quantiles and conditional expectation (expected shortfalls). When assuming extreme quantiles and heavy-tailed increments, using standard Monte Carlo method can be inefficient. In previous works, there exists an efficient method to sample rare-events (tail-events) using a Markov chain Monte Carlo (MCMC) with a given threshold. We apply the sampling method to estimate statistics based on tail-information, with a given rare-event probability. The performance is compared with other methods by some numerical results in the case increments follow Pareto distributions. We also show numerical results with Weibull, and Log-Normal distributions. Our proposed method is shown to be efficient especially in cases of extreme tails.