Efficient Nested Simulation of Tail Risk Measures for Variable Annuities

Abstract

Estimating tail risk measures for portfolios of complex Variable Annuities (VA) is an important enterprise risk management task which usually requires nested simulation. In the nested simulation, the outer simulation stage involves projecting scenarios of key risk factors under the real world measure, while the inner stage is used to value payoffs under guarantees of varying complexity, under a risk neutral measure.

In this thesis we propose and analyze three different two-stage simulation procedures that improve the computation efficiency of nested simulation. All three proposals allocate the inner simulations to the specific outer scenarios that are most likely to generate larger losses. These scenarios are identified using a proxy evaluation in the Stage 1 simulation. The proxy evaluation is used only to rank the outer scenarios, not to estimate the tail risk measure directly. The proxy evaluation can be based on a closed-form calculation which works very efficiently for simpler contracts, or a pilot nested simulation using likelihood ratio estimators which accommodates very complex path-dependent contracts. Then in the Stage 2 simulation we allocate the remaining computational budget to the scenarios identified in Stage 1. Our numerical experiments show that, in the VA context, our proposals are significantly more efficient than a standard Monte Carlo experiment, measured by relative mean squared errors (RMSE), when both are given the same computational budget.