Optimal retirement income tontines

Abstract

Tontines were once a popular type of mortality-linked investment pool. They promised enormous rewards to the last survivors at the expense of those died early. And, while this design appealed to the gambling instinct, it is a suboptimal way to generate retirement income. Indeed, actuarially-fair life annuities making constant payments -where the insurance company is exposed to longevity risk -induce greater lifetime utility. However, tontines do not have to be structured the historical way, i.e. with a constant cash flow shared amongst a shrinking group of survivors. Moreover, insurance companies do not sell actuarially-fair life annuities, in part due to aggregate longevity risk.We derive the tontine structure that maximizes lifetime utility. Technically speaking we solve the Euler-Lagrange equation and examine its sensitivity to (i.) the size of the tontine pool n, and (ii.) individual longevity risk aversion γ. We examine how the optimal tontine varies with γ and n, and prove some qualitative theorems about the optimal payout. Interestingly, Lorenzo de Tonti's original structure is optimal in the limit as longevity risk aversion γ → ∞. We define the natural tontine as the function for which the payout declines in exact proportion to the survival probabilities, which we show is near-optimal for all γ and n. We conclude by comparing 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 the optimal tontine is minimal.In sum, this paper's contribution is to (i.) rekindle a discussion about a retirement income product that has been long neglected, and (ii.) leverage economic theory as well as tools from mathematical finance to design the next generation of tontine annuities.

Equitable Retirement Income Tontines: Mixing Cohorts Without Discriminating

“…the capital they have invested. In Lorenzo Tonti's original scheme, as well as the structure proposed in Milevsky and Salisbury (2015), all investors (in the same pooling class) were assumed to be of the same age and paid the same price. When smaller groups were segmented into age bands they lost the benefit of large numbers.…”

“…This brings us to the tontine structure introduced in Milevsky and Salisbury (2015), in which a predetermined dollar amount is shared among survivors at every t. Let d(t) be the rate at which funds are paid out per initial dollar invested, a.k.a. the tontine payout function.…”

“…Milevsky and Salisbury (2015) find this optimum in some generality. The following summarizes the conclusion in the case of CRRA utility U (c) = (2015)).…”

“…We will work with γ = 1 so U (c) = log c, and the above formula simplifies considerably. By results of Milevsky and Salisbury (2015), the optimald(t) is the tontine that is natural for the age-x i cohort, in other words,d(t) =…”

The Modern Tontine: An Innovative Instrument for Longevity Risk Management in an Aging Society

“…However, the products introduced had only moderate success due to low market acceptance and the lack of a perfect hedge. Milevsky and Salisbury (2015) derive an optimal tontine design by accounting for sensitivity of both the tontine size and the longevity risk aversion for each tontine member. By doing so, they raise the question whether an optimally designed tontine with low implications regarding capital requirements for the sponsor will gain more attention in times of risk-based capital standards and conclude that, due to higher volatility of the payments, the tontine provides a lower utility than a traditional life annuity.…”