Valuation of GLWB variable annuities with long-term care option and accumulation phase

In the present work, we propose an evaluation framework for pricing guaranteed lifetime withdrawal benefit variable annuities equipped with long-term care (LTC) option by considering both an initial accumulation phase and a consequent income phase. Such insurance products allow for additional purchases during the accumulation phase, dynamic withdrawals in the income phase, and full surrender rights throughout the contract. In addition, LTC benefits could be provided in both the accumulation and income phases if the policyholder becomes disabled. The contract value is defined through a stochastic control problem, which is solved using dynamic programming. The validity of the bang-bang condition for the set of discrete withdrawal strategies is proved by backward induction and without assuming convexity or monotonicity of the contract value function. Finally, by exploiting technical bases for LTC insurance in Italy and assuming an exponential Lévy process for the asset price, we present an exhaustive numerical example aiming to analyze the sensitivity of both the contract’s price and the optimal withdrawal strategy to the contractual parameters. We further examine the robustness of the initial contract value and the associated optimal strategy under alternative specifications of the asset dynamics.