An Importance Sampling Method for Least-Squares Monte Carlo in Risk Measure Estimation

Calculating risk measures is challenging due to the complexity of the loss random variable, which depends on multiple state variables over a risk horizon. A common simplification uses a quadratic approximation of the loss random variable to construct an empirical loss distribution. However, this approach may fail to capture extreme events over longer horizons. A more robust method involves representing the loss as a finite linear combination of higher-degree polynomial basis functions. This raises two main challenges: estimating coefficients for each basis function and managing the potentially large number of simulations needed for tail risk estimation. This paper introduces an efficient algorithm that combines Least-Squares Monte Carlo with Importance Sampling to approximate the loss distribution and estimate extreme loss probabilities. We analyze each method individually, then assess their combined performance in delivering consistent risk measures.