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

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