The simple analytics of a pooled annuity fund

Journal of Risk and Insurance, Sept, 2005 by John Piggott, Emiliano A. Valdez, Bettina Detzel

ABSTRACT

This article provides a formal analysis of payout adjustments from a longevity risk-pooling fund, an arrangement we refer to as group self-annuitization (GSA). The distinguishing risk diffusion characteristic of GSAs in the family of longevity insurance instruments is that the annuitants bear their systematic risk, but the pool shares idiosyncratic risk. This obviates the need for an insurance company, although such instruments could be sold through a corporate insurer. We begin by deriving the payout adjustment for a single entry group with a single annuity factor and constant expectations. We then show that under weak requirements a unique solution to payout paths exists when multiple cohorts combine into a single pool. This relies on the harmonic mean of the ratio of realized to expected survivorship rates across cohorts. The case of evolving expectations is also analyzed. In all cases, we demonstrate that the periodic-benefit payment in a pooled annuity fund is determined based on the previous payment adjusted for any deviations in mortality and interest from expectations. GSA may have considerable appeal in countries which have adopted national defined contribution schemes and/or in which the life insurance industry is noncompetitive or poorly developed.

INTRODUCTION AND MOTIVATION

From a theoretical perspective, annuitization is a natural mechanism for insuring against longevity risk, especially at retirement. Risk averse individuals value annuities highly (Mitchell, 2002). However, voluntary annuity markets remain thin, and there is evidence that risk sharing through transfers is limited even within families (Hayashi, Altonji, and Kotlikoff, 1996). Annuity demand remains low despite tax concessions, perhaps because annuity loadings are often penal, especially in small economies such as Australia's (see, e.g., Doyle, Mitchell, and Piggott, 2001), or in countries where the financial sector is not well developed. Supply also appears reluctant, perhaps because of an industry perception that systematic risk, in the form of breakthrough life-prolonging technical innovation, may bankrupt an insurance company with a large life annuity portfolio. This has led to a situation where almost no voluntary longevity risk-spreading takes place in the private market, in spite of its clear welfare-enhancing effects (Kotlikoff and Spivak, 1981; Kingston and Piggott, 1999).

Milevsky and Robinson (2001) develop the mathematical analysis that can be made by individuals at retirement who face the choice between voluntary annuitization and self-annuitization. Self-annuitization provides greater liquidity than voluntary annuitization; however, it does so at the cost of possibly outliving resources. Albrecht and Maurer (2002) evaluate that risk by calculating a personal probability of consumption shortfall and show that it is substantial, particularly for high-entry ages.

A possible response is to separate the systematic from the idiosyncratic risk. Groups could be formed to pool idiosyncratic risk within a clear framework with specified legal rights and obligations, but payouts could be conditioned to the mortality experience of the group. This concept of group self-annuitization (GSA) was mooted by Wadsworth, Findlater, and Boardman (2001) and Martineau (2001). A group self-annuity plan will allow retirees to pool together and form a fund that can provide for protection against longevity. With the right implementation, GSA can provide a less expensive form of insurance against the risk of longevity.

This is not the only contribution in recent times to address the problem of systematic longevity risk. An alternative policy strategy that would allow annuity issuers to immunize their systematic longevity risk is for the government to issue "survivor bonds," in which payouts are linked to evolving mortality in a manner analogous to "indexed bonds" whose payouts are linked to evolving price inflation (Blake and Burrows, 2001). Thus far, no such bonds have been issued.

In the United States, annuities where payments reflect evolving mortality have for some time been issued by the teachers insurance and annuity association (TIAA) through its companion organization college retirement equities fund (CREF). Mortality experience of its participants receiving lifetime income is tracked and this historical experience is used as a guide in the annual adjustment to the mortality participation factor. TIAA-CREF reports that historically, the impact of annual mortality adjustments has been relatively modest. (1) Brown et al. (2001) provide additional commentary on the TIAA-CREF product. Weil and Fischer (1974) indicate that in general, the initial benefit payment calculation is based on a conservative interest rate and mortality assumptions. Most of the adjustments have reflected investment experience and very little on errors resulting from mortality projections. However, there appears to be no formal treatment of how benefit adjustments for these types of products should be calculated, even with annuities issued by TIAA-CREF, or what conditions these adjustments should satisfy.