Optimal Longevity Risk Transfer and Investment Strategies

Abstract

Given the rising cost of maintaining defined benefit pensions, there has been a surge of activities in recent years by defined benefit plan sponsors to transfer their pension risk through strategies such as buy-ins and buy-outs. As buy-in and buy-out transaction pipelines grow, insurers actively participating in the buy-in and buy-out markets are exposed to significant longevity risk embedded in pension schemes. In this article, we investigate how to maximize a bulk annuity insurer’s value with reinsurance and/or longevity securities, subject to constraints that control longevity and investment risks as well as an overall risk. We apply duality and the martingale approach to derive an optimal longevity risk transfer strategy. Our results show that longevity risk transfer interacts with an insurer’s investment decision for value maximization. Our analysis also highlights the interdependence of different longevity risk management tools to achieve an overall risk target.

ACKNOWLEDGMENT

We are grateful to the anonymous referee for very helpful comments.

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Notes

1 Assume that the number of retirees at time 0, $N(u,0),$ is so large that, by the law of large numbers, the number of survivors at time $t$ can be described by $$N(u,t)={\sum }_{i=1}^{N(u,0)}{I}_i(u,t)\cong N(u,0)S(u,t),$$ where $I_i(u,t)$ is an indicator function, which is 1 if the retiree aged $u$ at time 0 survives at time t or 0 if he dies between time 0 and $t.$

2 Our annuity setup is flexible enough to fit different types of annuities. In particular, if the rate of return underlying $C(\omega )$ is constant, $k(0,\omega )$ and $\left[1+\rho (u,0,\omega )\right]$ can be scaled to ensure the same payment each year. In this case, this annuity is a single-premium immediate annuity.

3 If an insurer is allowed to lose an unlimited amount of money, it will adopt a strategy similar to the doubling strategy and make an arbitrarily large amount of money.

4 Hakansson (1970) used a discrete-time model to examine the losses due to imperfect portfolio balance arising from infrequent portfolio revision. He obtained a non-closed-form solution for a one-step optimization problem. Rogers (2001) found a series expansion for the continuous-time model when the time interval $\Delta t$ is small. He concluded that the cost of an infrequent portfolio revision is surprisingly small and the loss due to imperfect portfolio balance is typically small.

5 Multiple annuity payments complicate the optimization problem. Each annuity payment can be regarded as a security. For the time horizon after some of those securities mature and disappear, the optimization must anticipate the shuffling of the portfolio because of the disappeared securities. For the case in which all annuity payments exceed the time horizon of the optimization, the above problems do not arise (see Korn and Kraft 2002; and Ekeland and Taflin 2005).

6 For the subclass of $\mathcal{D}$ given by $${\mathcal{D}}^{\prime }\triangleq \left\{\nu \in \mathcal{D};{\mathcal{X}}_{\nu }(y)<\infty ,\forall y\in (0,\infty )\right\}$$ and every $\nu \in {\mathcal{D}}^{\prime },{\mathcal{X}}_{\nu }(·)$ is continuous and strictly decreasing. Thus its inverse ${\mathcal{Y}}_{\nu }(·)$ exists.