The effect of policyholders’ rationality on unit-linked life insurance contracts with surrender guarantees

Abstract

We study the valuation of unit-linked life insurance contracts with surrender guarantees. Instead of solving an optimal stopping problem, we propose a more realistic approach accounting for policyholders’ rationality in exercising their surrender option. The valuation is conducted at the portfolio level by assuming that the surrender rate of the representative agent is bounded from below and from above. The lower bound corresponds to purely exogenous surrender, and the difference between the upper bound and the lower bound represents the rationality of the policyholders. The valuation problem is formulated by a PDE approach and solved with the finite difference method. We show that the rationality of the policyholders has a significant effect on average contract value and hence on the fair contract design. We also present the separating boundary between purely exogenous surrender and endogenous surrender. This provides implications on the predicted surrender activity of the policyholders.

Keywords:

• Unit-linked life insurance contracts
• Surrender guarantee
• Limited rationality
• Fair contract analysis

JEL Classification:

• G13
• G22
• C65

Acknowledgments

We thank Anna Rita Bacinello, An Chen, Christian Hilpert, Klaus Sandmann, Birgit Schnorrenberg, Filip Uzelac, two anonymous referees, participants of the 15th International Congress on Insurance: Mathematics and Economics, 2011 China International Conference on Insurance and Risk Management and the CEQURA Conference 2012 for helpful comments and discussions. Jing Li acknowledges the financial support of Bonn Graduate School of Economics.

Notes

The pre-death/surrender value refers to the contract value when the contract has not been terminated either due to the premature death or surrender of the policyholder.

 Kuo et al. (2003) empirically investigate a mix of different life insurance contracts. For these contracts, the interest rate can be seen as endogenous risk factor and is found to affect the surrender behaviour of the policyholders. The unemployment rate can be interpreted as an exogenous risk factor that is also effecting the surrender behaviour of the policyholders. In this paper, the equity takes the role of the endogenous risk factor and exogenous risk factors are assumed to be diversifiable.

For those competent policyholders who are able to exercise their surrender option optimally, less premiums are charged than those are needed to support the contracts. It is the irrational policyholders who have born the extra costs.

We assume that a and are regular enough to allow for a unique strong solution of (2), see, e.g. Section 5.2 of Karatzas and Shreve (1991) for details.

The specification is admissible since is assumed to be independent of W (and hence of S) and moreover, S has continuous paths almost surely.

We order the events generating a payoff with decreasing priority as follows: death, surrender and survival. This order has no impact on the results since both random times and admit an intensity and hence .

If surrender option is exercised at , it will lead to a lower contract value, which could even lie below the value of a pure European-style contract. We will demonstrate this issue in Section 4 with numerical examples.

For examples of penalty functions refer to Palmer (2006).

Following Stanton (1995) we can also incorporate a proportional cost applicable in case of surrender by replacing the surrender guarantee by , with , for , where is the deterministic proportional cost, with .

Alternatively, the volatility surface could be found within the local volatility model. However, this would not change the results qualitatively, and hence, we take the constant volatility model for simplicity.