Impact of multimodality of distributions on VaR and ES calculation

Unimodal probability distribution - its probability density having only one peak, like Gaussian distribution or Student-t distribution - has been widely used for Value-at-Risk (VaR) computation by both regulators and risk managers. However, a histogram of financial data may have more than one modes. In this case, first, a unimodal distribution may not provide a good fit on the data due to the number of modes, which may lead to bias for risk measure computation; second, these modes in the tails contain important information for trading and risk management purposes. For capturing information in the modes, in this paper, we discuss how to use a multimodal distribution - its density having more than one peaks - from Cobb's distribution family (Cobb et al., 1983), to compute the VaR and the Expected Shortfall (ES) for the data having multimodal histogram. In general, the cumulative distribution function of a multimodal distribution belonging to Cobb's distribution family does not have a closed-form expression. It means that we cannot compute the VaR and the ES, or generate random numbers using classical inversion approaches. Thus we use Monte-Carlo techniques to compute the VaR and the ES for Cobb's distribution; first, based on rejection sampling algorithm, we develop a way to simulate random numbers directly from a multimodal probability density belonging to Cobb's family; second, we compute the VaR and the ES by using their empirical estimators and these simulated random numbers. In the end, with these techniques, we compute the VaR and the ES for one data associated with market risk and another data associated with operational risk.