Longevity Greeks: What Do Insurers and Capital Market Investors Need to Know?

Abstract

Recently, it has been argued that capital markets may share some of the overwhelming longevity risk exposures borne by the pension and life insurance industries. The transfer of risk can be accomplished by trading standardized derivatives such as q-forwards that are linked to published mortality indexes. To strategize such trades, one may utilize longevity Greeks, which are analogous to equity Greeks that have been used extensively in managing stock price risk. In this article, we first derive three important longevity Greeks—delta, gamma, and vega—on the basis of an extended version of the Lee-Carter model that incorporates stochastic volatility. We then study the properties of each longevity Greek and estimate the levels of effectiveness that different longevity Greek hedges can possibly achieve. The results reveal several interesting facts; for example, in a delta–vega hedge formed by q-forwards, the choice of reference ages does not materially affect hedge effectiveness, but the choice of times to maturity does. These facts may aid insurers to better formulate their hedge portfolios and issuers of mortality-linked securities to determine what security structures are more likely to attract liquidity.

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Notes

1 According to Theorem 2 in Section 5.3, the third moment of ${\kappa }_{t^f}|{\mathcal{F}}_0$ about its mean is zero. It follows that the distribution of $Y_{x^f,t^f-1}(1)={\alpha }_{x^f}+{\beta }_{x^f}{\kappa }_{t^f}$ given ${\mathcal{F}}_0$ is symmetric.

2 All stationary ARCH(1) models (in which a = 0 and b < 1) meet this condition. However, admittedly, not all GARCH(1, 1) models satisfy this condition, even if they are stationary with $a+b<1.$

3 When considering delta–gamma hedges with $t_1^f=5$ and $t_2^f=15,$ these two reference ages result in the highest level of hedge effectiveness. Other reference ages may also be used in this analysis, provided that one of them is less than or equal to 86 and the other is greater than 86.

4 This metric is adopted from the (non-age-specific) explanation ratio considered by N. Li and Lee (2005).

Additional information

Funding

This work is supported by the Society of Actuaries Center of Actuarial Excellence Programme and the Natural Sciences and Engineering Research Council of Canada.