On Valuing Equity-Linked Insurance and Reinsurance Contracts∗

Insurance companies are increasingly facing losses that have heavy exposure to capital market risks through the issuance of equity-linked insurance policies. In this paper, we determine the continuous premium rate that an insurer charges via the principle of equivalent utility. Using exponential utility, we obtain the resulting premium rate in terms of a risk-neutral expectation. We also consider the related problem of pricing double-trigger reinsurance contracts, paying a function of the risky asset and losses, once the insurer has fixed her premium rate. We solve the Hamilton-Jacobi-Bellman equation arising in the indifference pricing problem and show that the price satisfies a PDE with a non-linear shift term. Although a closed form solution is not, generally, attainable, we obtain analytical results in some special cases. Finally, we recast the pricing PDE as a linear stochastic control problem and provide an explicit finite-difference scheme for solving the PDE numerically.