Innovative combo product design embedding variable annuity and long-term care insurance contracts

This paper presents a novel combo insurance product design consisting of a variable annuity contract embedded with guaranteed minimum income benefit and long-term care insurance riders. This combo product provides enhanced benefits when the policyholder is functionally disabled. The policyholder and provider's joint optimal decision is formulated as a Nash equilibrium of a two-stage non-zero sum game. The provider aims to offer the optimal insurance product that minimises solvency capital requirement (SCR) per unit premium under Solvency II in the first stage. The policyholder aims to purchase the optimal amount of insurance that maximises lifetime utility in the second stage. The Hamiltonian Monte Carlo (HMC) simulation technique is utilised for numerically valuing the combo product whose underlying fund is proportionally invested in multiple asset classes. Due to the natural hedging effect between longevity and disability risks and the option payoff structure, the combo product is a win-win solution for providers and policyholders compared with an LTC annuity or an LTC insurance and a variable annuity with guaranteed minimum income benefit. From the policyholder's perspective, we quantify the extent to which the combo product costs less premium and the policyholder gains more lifetime utility. From the provider's perspective, we show that the combo product requires less SCR per initial unit premium. Product features including the elimination period and the maximum benefit period, are examined, and we show that they can effectively reduce the product premium. We perform fee sensitivity tests on model parameters to reveal insights regarding risk management from the provider's perspective.