Before revision:  We develop nested Monte Carlo estimators and construct (asymptotically valid) confidence intervals for risk measures of mean response. We further study the associated budget allocation problem for more efficient nested simulation of the estimators, and propose a novel method to solve the problem .
After revision:  We develop nested Monte Carlo simulation approach to estimate them. Then we develop asymptotical properties such as consistency and asymptotic normality for the proposed nested risk estimators. Finally we study the associated budget allocation problem for more efficient nested simulation of the estimators, and propose a novel method to solve the problem .