Benchmarking of Unconditional VaR and ES Calculation Methods: A Comparative Simulation Analysis with Truncated Stable Distribution

This paper analyzes Value at Risk (VaR) and Expected Shortfall (ES) calculation methods in terms of bias and dispersion against benchmarks computed from a fat-tailed parametric distribution. The daily log returns of the Nikkei-225 stock index are modeled by a truncated stable distribution. The VaR and ES values of the fitted distribution are regarded as benchmarks. The fitted distribution is also used as a sampling distribution; sample returns with different sizes are generated for the simulations of the VaR and ES calculations. Two parametric methods: normal distribution and generalized Pareto distribution and two non-parametric methods: historical simulation and kernel smoothing are selected as the targets of this analysis. A comparison of the simulated VaR, ES, and the ES/VaR ratio with the benchmarks at multiple confidence levels reveals that the normal distribution approximation has a significant downward bias, especially in the ES calculation. The estimates by the other three methods are much closer to the benchmarks on average, although some of them become unstable with smaller sample sizes and/or at higher confidence levels. Specifically, ES tends to be more biased and unstable than VaR at higher confidence levels.