Tontines, 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.
Following Donnelly and Young (2017), we present a novel approach of combining annuities and tontines by forming a tontine with a minimum guarantee which can lead to a better risk sharing between policyholders and insurers than annuities, where the insurer carries all the mortality risk, and tontines, where the policyholders carry most of the mortality risk.
We then 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 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.
Further, we look at the effect of subjective mortality beliefs on the perceived attractiveness of annuities and tontines. Given actuarially fair pricing with no subjective mortality beliefs (that is, the insurer's and the policyholder's perceptions coincide), annuities yield a higher lifetime utility than tontines (see also Milevsky and Salisbury (2015)). Staying in an actuarially fair pricing framework, we nd that this result might be reversed if the policyholder's subjective survival probabilities for her peers are lower than the ones used by the insurance company. We prove that, assuming such subjective beliefs, there exists a critical tontine pool size from which on the tontine is always preferred over the annuity.