Optimal Retirement Tontines for the 21st Century: With Reference to Mortality Derivatives in 1693
43 Pages Posted: 29 May 2013
Date Written: May 28, 2013
Historical tontines promised enormous rewards to the last few survivors at the expense of those died early. And, while this design appealed to the gambling instinct, it is a suboptimal way to manage and generate retirement income. This is why fair life annuities making constant payments -- where the insurance company is exposed to longevity risk -- induces greater lifetime utility. But, tontines do not have to be structured as a fixed cash-flow shared among a shrinking number of survivors and insurance companies do not actually sell fair life annuities, partially due to aggregate longevity risk.
In this paper we derive the tontine structure that maximizes lifetime utility, but doesn't expose the sponsor to any longevity risk. Technically speaking we solve the Euler Lagrange equation and examine its sensitivity to (i.) the size of the tontine pool, (ii.) individual longevity risk aversion, and (iii.) subjective health status. The optimal tontine varies with the individual's longevity risk aversion and the number of participants. And, the historical (flat, constant) tontine structure is only optimal in the limit as longevity risk aversion goes to infinity.
We then introduce a structure called a natural tontine whose payout declines in exact proportion to the (expected) survival probabilities, which is near-optimal for all values of longevity risk aversion and the size of the tontine pool. We compare the utility of optimal tontines to the utility of loaded life annuities under reasonable demographic and economic conditions and find that the life annuity's advantage over tontines, is minimal. Our contribution is to leverage economic theory to design the next generation of tontine annuities.
We also use our framework to review and analyze the first-ever mortality-derivative issued by the British government, known as King Williams's tontine of 1693. Although it is widely acknowledged that mortality-derivatives were mis-priced in their early years, it is worth noting that both life annuities and tontines co-existed during that period. We shed light on the preferences and beliefs of those who invested in the tontines vs. the annuities and conclude by arguing that tontines should be re-introduced and allowed to co-exist with life annuities. Individuals would likely select a portfolio of tontines and annuities that suit their personal preferences for consumption and longevity risk, as they did over 320 years ago.
Keywords: insurance, retirement, annuities, longevity risk, lifecycle, derivatives, economic history
JEL Classification: G11, C61, D91
Suggested Citation: Suggested Citation