The interaction between variable annuity providers and their customers under a dynamic approach

We consider an insurer offering various single premium variable annuity contracts with a guaranteed lifetime withdrawal benefit (GLWB) rider to policyholders. All the contracts have the same structure, but differ for the contractual parameters or the risk/return profile of the reference fund in which the premium is invested. For comparability, the premium is the same for all contracts, and the contractual parameters are set in such a way that the contracts are fairly priced. To price them, the insurer fixes a risk-neutral probability measure and computes the initial contract value as if the policyholder chose a withdrawal strategy in order to maximize the expected present value of her future cash flows. However, the policyholder can have different preferences; in particular, we assume that she acts in order to maximize the expected discounted utility, under the physical measure, of the cash flows. This leads to a ranking of the various contracts from the policyholder’s perspective and, in turn, for the insurer, since the expected present value, under the insurer’s probability, of the cash flows induced by the actual policyholder’s strategy does not exceed the single premium, thus leaving room for a gain. In the paper, after defining the whole model with related optimization problems of insurer and policyholder, we numerically compare the two orders implied by our approach for different preferences of both parties involved.