Efficient Nested Simulation for Conditional Tail Expectation of Variable Annuities

Abstract

Monte Carlo simulations—in particular, nested Monte Carlo simulations—are commonly used in variable annuity (VA) risk modeling. However, the computational burden associated with nested simulations is substantial. We propose an Importance-Allocated Nested Simulation (IANS) method to reduce the computational burden, using a two-stage process. The first stage uses a low-cost analytic proxy to identify the tail scenarios most likely to contribute to the Conditional Tail Expectation risk measure. In the second stage we allocate the entire inner simulation computational budget to the scenarios identified in the first stage. Our numerical experiments show that, in the VA context, IANS can be up to 30 times more efficient than a standard Monte Carlo experiment, measured by relative mean squared errors, when both are given the same computational budget.

ACKNOWLEDGMENTS

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada, funding reference number 203271 (Hardy) and 03755 (Feng). This work was also supported by the Society of Actuaries through a Center of Actuarial Excellence Research Grant and through the Hickman Scholarship held by Ou Dang. Ou Dang also receives support from the Ontario Graduate Scholarship.

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Notes

1 The α-CTE, which was introduced in Wirch and Hardy (1999), is identical to the TailVaR${}_{\alpha },$ which is based on (but not identical to) a measure introduced in Artzner et al. (1999). Both terms are used to mean the expected value of a loss, conditional on the loss falling in the upper $(1-\alpha )$ part of its distribution.