Portfolio size as function of the premium: modelling and optimization

Abstract

An insurance company has a large number N of potential customers characterized by i.i.d. r.v.'s giving the arrival rates of claims. Customers are risk averse, and a customer accepts an offered premium p according to his A-value. The modelling further involves a discount rate d>r of customers, where r is the risk-free interest rate. Based on calculations of the customers' present values of the alternative strategies of insuring and not insuring, the portfolio size is derived, and also the rate of claims from the insured customers is given. Furthermore, the value of p which is optimal for minimizing the ruin probability is derived in a diffusion approximation to the Cramér–Lundberg risk process with an added liability rate L of the company. The solution involves the Lambert W function. Similar discussion is given for extensions involving customers having only partial information on their A and stochastic discount rates.

Keywords:

• adverse selection
• certainty equivalent
• Cramér–Lundberg model
• diffusion approximation
• inverse Gamma distribution
• Lambert W function

2000 Mathematics Subject Classification::

• 91B30
• 60G99
• 91G99

Acknowledgement

Michael Taksar was supported by a grant from the Velux Foundation.

Notes

^1. Manuscript was completed by Søren Asmussen and Bent Jesper Christensen.

^2. [email protected]

^3. Deceased 12 February, 2012.

Additional information

Notes on contributors

Søren Asmussen

1 1. Manuscript was completed by Søren Asmussen and Bent Jesper Christensen.

Bent Jesper Christensen

1 2 2. [email protected]

Michael Taksar

3 3. Deceased 12 February, 2012.