Importance Sampling for Tail Risk in Discretely Rebalanced Portfolios

We develop an importance sampling (IS) algorithm to estimate the lower tail of the distribution of returns for a discretely rebalanced portfolio-one in which portfolio weights are reset at regular intervals. We use a more tractable continuously rebalanced portfolio to design the IS estimator. We analyze a limiting regime based on estimating probabilities farther in the tail while letting the rebalancing frequency increase. We show that the estimator is asymptotically efficient for this sequence of problems; its relative error grows in proportion to the fourth root of the number of rebalancing dates.