A new Bayesian method for estimation of value at risk and conditional value at risk

Abstract

Value at Risk (VaR) and Conditional Value at Risk (CVaR) have become the most popular measures of market risk in Financial and Insurance fields. However, the estimation of both risk measures is challenging, because it requires the knowledge of the tail of the distribution. Therefore, Extreme Value Theory initially seemed to be one of the best tools for this kind of problems, because using peaks-over-threshold method, we can assume the tail data approximately follow a Generalized Pareto distribution (GPD). The main objection to its use is that it only employs observations over the threshold, which are usually scarce. With the aim of improving the inference process, we propose a new Bayesian method that computes estimates built with all the information available. Informative prior Bayesian (IPB) method employs the existing relations between the parameters of the loss distribution and the parameters of the GPD that models the tail data to define informative priors in order to perform Metropolis–Hastings algorithm. We show how to apply IPB when the distribution of the observations is Exponential, stable or Gamma, to make inference and predictions. .Afterwards, we perform a thorough simulation study to compare the accuracy and precision of the estimates computed by IPB and the most employed methods to estimate VaR and CVaR. Results show that IPB provides the most accurate, precise and least biased estimates, especially when there are very few tail data. Finally, data from two real examples are analysed to show the practical application of the method.