Abstract
AbstractTontines, retirement products constructed in such a way that the longevity risk is shared in a pool of policyholders, have recently gained vast attention from researchers and practitioners. Typically, these products are cheaper than annuities, but do not provide stable payments to policyholders. This raises the question whether, from the policyholders' viewpoint, the advantages of annuities and tontines can be combined to form a retirement plan which is cheaper than an annuity, but provides a less volatile retirement income than a tontine. In this article, we analyze and compare three approaches of combining annuities and tontines in an expected utility framework: the previously introduced “tonuity”, a product very similar to the tonuity which we call “antine” and a portfolio consisting of an annuity and a tontine. We show that the payoffs of a tonuity and an antine can be replicated by a portfolio consisting of an annuity and a tontine. Consequently, policyholders achieve higher expected utility levels when choosing the portfolio over the novel retirement products tonuity and antine. Further, we derive conditions on the premium loadings of annuities and tontines indicating when the optimal portfolio is investing a positive amount in both annuity and tontine, and when the optimal portfolio turns out to be a pure annuity or a pure tontine.
Highlights
Annuities providelong payment streams to the policyholder and constitute a possible way to build protection against the increasing threat of the individual’s incapability to keep her living standards at older ages
Our main theoretical result shows that, from the policyholder’s point of view, a portfolio consisting of an annuity and a tontine can outperform any tonuity and antine
Note that the payoffs of any tonuity and antine can be replicated by a portfolio consisting of an annuity and a tontine by choosing the payoffs of the annuity and the tontine appropriately
Summary
Annuities provide (life-)long payment streams to the policyholder and constitute a possible way to build protection against the increasing threat of the individual’s incapability to keep her living standards at older ages. In the portfolio, the optimal payoff of the tontine coincides roughly with optimal tontine designs discussed in the literature (cf Milevsky and Salisbury, 2015; Chen et al, 2019), the corresponding annuity payoff structure deviates substantially from this literature as it first increases and decreases rather strongly, leading to a bell-shaped curve. At rather advanced retirement ages, the tontine payments are relatively high due to the few surviving policyholders, which leads to a decrease in the annuity payoff Based on these optimal income streams, we can implicitly determine the fractions of wealth initially invested in the annuity and the tontine, respectively. The newly proposed antine seems not to be a desirable product from the policyholder’s perspective and is frequently outperformed by the tonuity This is probably due to the design of the antine which leaves policyholders with volatile payments in the advanced retirement ages and is still rather expensive compared to tontines.
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