Max–min Optimization Problem for Variable Annuities Pricing

Abstract

In this paper, we study the valuation of variable annuities for an insurer. We concentrate on two types of these contracts, namely guaranteed minimum death benefits and guaranteed minimum living benefits that allow the insured to withdraw money from the associated account. Here, the price of variable annuities corresponds to a fee, fixed at the beginning of the contract, that is continuously taken from the associated account. We use a utility indifference approach to determine the indifference fee rate. We focus on the worst case for the insurer, assuming that the insured makes the withdrawals that minimize the expected utility of the insurer. To compute this indifference fee rate, we link the utility maximization in the worst case for the insurer to a sequence of maximization and minimization problems that can be computed recursively. This allows to provide an optimal investment strategy for the insurer when the insured follows the worst withdrawal strategy and to compute the indifference fee. We finally explain how to approximate these quantities via the previous results and give numerical illustrations of parameter sensitivity.

Insurance–finance arbitrage

“…This approach can be generalized to the quadratic hedging error. Finally, indifference pricing leads to a nonlinear pricing rule and we refer to Blanchet-Scalliet et al (2015), Chevalier et al (2016) for details and further literature. Nonlinear methods to analyze insurance contracts often use the axiomatic approach to risk measures on 𝐿 ∞ with the Fatou property and thus admitting robust presentations, see Tsanakas and Desli (2005); Barigou et al (2019); Engsner et al (2020) while this can also be treated on more general spaces, as for example, in Kaina and Rüschendorf (2009), Cheridito and Li (2009).…”

“…Finally, indifference pricing leads to a nonlinear pricing rule and we refer to Blanchet‐Scalliet et al. (2015), Chevalier et al. (2016) for details and further literature.…”

No Arbitrage in Insurance and the QP-Rule

“…Fourth, indifference pricing leads to a non-linear pricing rule and we refer to Blanchet-Scalliet et al (2015), Chevalier et al (2016) for details and further literature. Finally, there are valuation methodologies utilizing risk measures, often based on an axiomatic view, see Tsanakas and Desli (2005), Pelsser and Stadje (2014), or on hedging, see Chen et al (2020).…”