Risk sharing in equity-linked insurance products: Stackelberg equilibrium between an insurer and a reinsurer

TL;DR

This paper models the strategic interaction between an insurer and reinsurer in equity-linked insurance products as a Stackelberg game, deriving equilibrium strategies and analyzing the impact of reinsurance premiums on expected utilities.

Contribution

It introduces a Stackelberg game framework for insurer-reinsurer interactions in equity-linked products and explicitly computes equilibrium strategies for power utility functions.

Findings

Reinsurer chooses the largest reinsurance premium allowing maximal reinsurance purchase by insurer.

Equilibrium strategies are explicitly derived for power utility functions.

Reinsurers should consider charging lower premiums than the equilibrium to influence reinsurance uptake.

Abstract

We study the optimal investment-reinsurance problem in the context of equity-linked insurance products. Such products often have a capital guarantee, which can motivate insurers to purchase reinsurance. Since a reinsurance contract implies an interaction between the insurer and the reinsurer, we model the optimization problem as a Stackelberg game. The reinsurer is the leader in the game and maximizes its expected utility by selecting its optimal investment strategy and a safety loading in the reinsurance contract it offers to the insurer. The reinsurer can assess how the insurer will rationally react on each action of the reinsurer. The insurance company is the follower and maximizes its expected utility by choosing its investment strategy and the amount of reinsurance the company purchases at the price offered by the reinsurer. In this game, we derive the Stackelberg equilibrium for general utility functions. For power utility functions, we calculate the equilibrium explicitly and find that the reinsurer selects the largest reinsurance premium such that the insurer may still buy the maximal amount of reinsurance. Since in the equilibrium the insurer is indifferent in the amount of reinsurance, in practice, the reinsurer should consider charging a smaller reinsurance premium than the equilibrium one. Therefore, we propose several criteria for choosing such a discount rate and investigate its wealth-equivalent impact on the expected utility of each party.