Empirical likelihood for value-at-risk and expected shortfall

When estimating risk measures, whether from historical data or by Monte Carlo simulation, it is helpful to have confidence intervals that provide information about statistical uncertainty. We provide asymptotically valid confidence intervals and confidence regions involving value-at-risk (VaR), conditional tail expectation and expected shortfall (conditional VaR), based on three different methodologies. One is an extension of previous work based on robust statistics, the second is a straightforward application of bootstrapping, and we derive the third using empirical likelihood. We then evaluate the small-sample coverage of the confidence intervals and regions in simulation experiments using financial examples. We find that the coverage probabilities are approximately nominal for large sample sizes, but are noticeably low when sample sizes are too small (roughly, less than 500 here). The new empirical likelihood method provides the highest coverage at moderate sample sizes in these experiments. We want to measure the risk of a given portfolio that has random profits at the end of a predetermined investment period. We can sample from the distribution of the portfolio’s profits using Monte Carlo simulation based on a stochastic model of financial markets. Our focus will be on estimating risk measures for our portfolio based on simulated profits and providing information in the form of confidence intervals and regions about the statistical uncertainty of these estimates. We address only this Monte Carlo sampling error in estimating risk, not the model risk that includes errors introduced by using an incorrect model of financial markets and statistical error in estimating the model’s parameters from data. We will emphasize moderate Monte Carlo sample sizes, which are appropriate when