Risk measurement involves estimating some functional of the distribution of loss.

Monte Carlo simulation is often used to estimate the mean of a distribution, but

some risk measures, such as value at risk and tail conditional expectation, are

not means of a distribution from which one can sample. This calls for nested

simulation, in which risk factors are sampled at an outer level of simulation,

while the inner level of simulation provides estimates of loss given each

realization of the risk factors. In our examples, the outer level simulates

tomorrow's stock prices, and the inner level estimates the loss of a portfolio of

stock options given tomorrow's stock prices. We present a general method for

providing a confidence interval for the risk measurement given statistical error

at two levels of simulation. This method could require a large computational

budget, so we discuss efficient procedures for providing a confidence interval

and point estimates for tail conditional expectation.