An optimal stopping problem for equity-linked life insurance policies

Milazzo Alessandro, Università di Torino

Equity-linked policies combine the life insurance protection, provided by the participation in equity performance, together with a number of financial guarantees and options. A minimum rate is guaranteed by the insurer in order to protect the policyholder’s capital against market downturns. Moreover, the policyholder has the right to early terminate the contract (surrender option) and to receive the account value. In general, a penalty, which decreases in time, is applied by the insurer in case of early surrender. These financial guarantees are subsidised by a fee paid by the policyholder, typically in the form of a fixed proportion of the account value. At maturity, the policyholder can choose to obtain the account value or convert it into an annuity stream (variable annuity). These features of equity-linked contracts are liabilities to the issuer and constitute a potential hazard to company solvency. The pricing and risk management of these contracts must take into account a variety of risks, such as mortality risk, market risk and policyholder behaviour risk. This paper proposes an analytical study of equity-linked policies with minimum rate guaranteed embedding a surrender option, in presence of a deterministic force of mortality and of a generic decreasing surrender charge function. This leads to an optimal stopping problem. We set up a free-boundary analysis to study the optimal time to surrender the contract and to investigate the shape of the regions where it is optimal to either wait or to surrender immediately (respectively, the so called continuation and stopping regions). We study the regularity of the value function and of the optimal stopping boundary. We then characterise this boundary as the unique solution to a family of integral equations of Volterra type, which we solve numerically. Examples are illustrated in case of Gompertz-Makeham mortality.

Area: CS20 - Stochastic Modeling in Finance and Insurance I (Tiziano de Angelis)

Keywords: optimal stopping; life insurance; free-boundary;

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