Equitable Retirement Income Tontines: Mixing Cohorts Without Discriminating

Abstract

There is growing interest in the design of pension annuities that insure against idiosyncratic longevity risk while pooling and sharing systematic risk. This is partially motivated by the desire to reduce capital and reserve requirements while retaining the value of mortality credits; see for example, Piggott et al. (2005) or Donnelly et al. (2014). In this paper, we generalize the natural retirement income tontine introduced by Milevsky and Salisbury (2015) by combining heterogeneous cohorts into one pool. We engineer this scheme by allocating tontine shares at either a premium or a discount to par based on both the age of the investor and the amount they invest. For example, a 55-year old allocating $10, 000 to the tontine might be told to pay $200 per share and receive 50 shares, while a 75-year old allocating $8, 000 might pay $40 per share and receive 200 shares. They would all be mixed together into the same tontine pool and each tontine share would have equal income rights. The current paper addresses existence and uniqueness issues and discusses the conditions under which this scheme can be constructed equitably -which is distinct from fairly -even though it isn't optimal for any cohort. As such, this also gives us the opportunity to compare and contrast various pooling schemes that have been proposed in the literature and to differentiate between arrangements that are socially equitable, vs. actuarially fair vs. economically optimal.

Modern tontine with bequest: Innovation in pooled annuity products

“…Different rules of calculating the longevity credits are proposed in Donnelly et al (2014); Sabin (2010); Stamos (2008). Bräutigam et al (2017) compare the tontine of Donnelly et al (2014) against that of Milevsky and Salisbury (2016), but are unable to fit the two structures into a single, over-arching structure. Similarly, Donnelly (2015) compares the scheme of Donnelly et al (2014) against the group selfannuitization scheme of Piggott et al (2005), and finds that the financial outcomes due to pooling longevity risk are similar if there are enough people in each scheme.…”

On the Optimal Combination of Annuities and Tontines

“…Nevertheless, dealing with heterogeneity between the individuals in our context surely opens up an interesting perspective for future research (cf. Milevsky and Salisbury, 2016). By choosing the pool size n large enough, it is possible for the insurer to diversify the unsystematic mortality risk.…”

Pricing participating longevity-linked life annuities: a Bayesian Model Ensemble approach

“…Mortality-linked funds (MLF), which bears some similarities to a PAF but in which the mortality credits being paid are deterministic and not stochastic [ 31 ], and annuity overlay funds (AOF) which depart from a PAF in that the fund pools wealth in an investment fund, longevity risk is shared between participants and payout benefits are determined so that the contract is actuarially fair at any instant [ 30 , 32 ], belong also to this category. This category includes modern tontines [ 35 , 48 , 49 ], a self-hedged annuity pool with flexible contributions, heterogenous populations, a customizable annuity plan and additional features like individual unit-linked allocation. As in a standard annuity, there are no benefits paid upon death, and the mortality credits of deceased members are to be allocated among the survival peers.…”