Stratification Issues in Estimating Value-at-Risk

This paper considers efficient estimation of value-at-risk, which is an important problem in risk management. The value-at-risk is an extreme quantile of the distribution of the loss in portfolio value during a holding period. An effective importance sampling technique is described for this problem. The importance sampling can be further improved by combining it with stratified sampling. In this setting, an effective stratification variable is the likelihood ratio itself. The paper examines issues associated with the allocation of samples to the strata, and compares the effectiveness of the combination of importance sampling and stratified sampling to that of stratified sampling alone.