arXiv2013
Historical tontines promised enormous rewards to the last survivors at the expense of those who died early. While this design appealed to the gambling instinct, it is a suboptimal way to manage longevity risk during retirement. This is why fair life annuities making constant payments -- where the insurance company is exposed to the longevity risk -- induces greater lifetime utility. However, tontines do not have to be designed using a winner-take-all approach 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. We examine its sensitivity to the size of the tontine pool; individual longevity risk aversion; and subjective health status. The optimal tontine varies with the individual's longevity risk aversion $γ$ and the number of participants $n$, which is problematic for product design. That said, we introduce a structure called a natural tontine whose payout declines in exact proportion to the (expected) survival probabilities, which is near-optimal for all $γ$ and $n$. 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. We also review and analyze the first-ever mortality-derivative issued by the British government, known as King Williams's tontine of 1693. We shed light on the preferences and beliefs of those who invested in the tontines vs. the annuities and argue 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.