A Sequential Importance Sampling for Estimating Multi-Period Tail Risk

Plain or crude Monte Carlo simulation (CMC) is commonly applied for estimating multi-period tail risk measures such as value-at-risk (VaR) and expected shortfall (ES). After fitting a volatility model to the past history of returns and estimating the conditional distribution of innovations, one can simulate the return process following the fitted volatility model with the estimated conditional distribution of innovations. Repeated generation of the return processes with the desired length gives a sufficient number of simulated multi-period returns. Then, the multi-period VaR and ES are directly estimated from the empirical distribution of them. CMC is easily applicable. However, it needs to generate a huge number of multi-period returns for the accurate estimation of a tail risk measure, especially when the confidence level of the measure is close to 1. To overcome this shortcoming, we propose a sequential importance sampling, which is a modification of CMC. In the proposed method. The sampling distribution of innovations is chosen differently from the estimated conditional distribution of innovations so that the simulated multi-period losses are more severe than in the case of CMC. In other words, the simulated losses over the VaR that is wanted to estimate are common in the proposed method, which reduces very much the estimation error of ES, and requires the less simulated samples. We propose how to find the near optimal sampling distribution. The multi-period VaR and ES are estimated from the weighted empirical distribution of the simulated multi-period returns. We propose how to compute the weight of a simulated multi-period return. An empirical study is given to backtest the estimated VaRs and ESs by the proposed method, and to compare the performance of the proposed sequential importance sampling with CMC.