Monte Carlo Methods for Economic Capital

Economic capital (EC) is a risk measure used by financial firms to specify capital levels to protect (with high probability) against large unforeseen losses. Defined as the difference between an (extreme) quantile and the mean of the loss distribution, the EC is often estimated via Monte Carlo methods. Although simple random sampling (SRS) may be effective in estimating the mean, it can be inefficient for the extreme quantile in the EC. Applying importance sampling (IS) may lead to an efficient quantile estimator but can do poorly for the mean. Measure-specific IS (MSIS) instead uses IS to estimate only the quantile, and the mean is independently handled via SRS. We analyze large-sample properties of EC estimators obtained via SRS only, IS only, MSIS, IS using a defensive mixture, and a double estimator using both SRS and IS to estimate both the quantile and the mean, establishing Bahadur-type representations for the EC estimators and proving they obey central limit theorems. We provide asymptotic theory comparing the estimators when the loss is the sum of a large number of independent and identically distributed random variables.