An Efficient Method for Mitigating Longevity Value-at-Risk

Abstract

Many of the existing index-based longevity hedging strategies focus on the reduction in variance. However, solvency capital requirements are typically based on the τ-year-ahead Value-at-Risk, with τ = 1 under Solvency II. Optimizing a longevity hedge using variance minimization is particularly inadequate when the cost of hedging is nonzero and mortality improvements are driven by a skewed and/or heavy-tailed distribution. In this article, we contribute a method to formulate a value hedge that aims to minimize the Value-at-Risk of the hedged position over a horizon of τ years. The proposed method works with all stochastic mortality models that can be formulated in a state-space form, even when a non normal distributional assumption is made. We further develop a technique to expedite the evaluation of a value longevity hedge. By utilizing the generic assumption that the innovations in the stochastic processes for the period and cohort effects are not serially correlated, the proposed technique spares us from the need for nested simulations that are generally required when evaluating a value hedge.

ACKNOWLEDGMENTS

An earlier version of this article was presented at the 13th International Longevity Risk and Capital Markets Solutions Conference. The authors are grateful to David Blake, Jeff Mulholland, and other participants at the conference for their insightful comments.

Discussions on this article can be submitted until October 1, 2020. The authors reserve the right to reply to any discussion. Please see the Instructions for Authors found online at http://www.tandfonline.com/uaaj for submission instructions.

Notes

1 In the literature (including the original work of Renshaw and Haberman (2003)), ${\eta }_t^{(k)}$ and ${\eta }_{t-x_a}^{(g)}$ are typically assumed to be independent. Our state-space formulation permits the user to specify the dependence between ${\eta }_t^{(k)}$ and ${\eta }_{t-x_a}^{(g)},$ if he or she finds such dependence significant.

2 We assume that the floating leg is the underlying death probability instead of the observed (crude) death probability. This assumption is consistent with the fact that the LifeMetrics Index (on which q-forwards in today’s market can be written) is developed using smoothed death probabilities instead of crude death probabilities.

3 Skew-t distributions are often applied to financial data with skewness and excess kurtosis (see, e.g., Theodossiou, 1998).

Additional information

Funding

This work is supported by research grants from the Global Risk Institute, the Natural Sciences and Engineering Research Council of Canada (Discovery Grant RGPIN-356050-2013), and the Society of Actuaries Center of Actuarial Excellence Program.