# Portfolio size as function of the premium: modeling and optimization

By Søren Asmussen, Bent Jesper Christensen and Michael Taksar
Thiele Research Reports
No. 01, January 2013
Abstract:

An insurance company has a large number $N$ of potential customers characterized by i.i.d. r.v.'s $A_{1},\ldots ,A_{N}$ giving the arrival rates of claims. Customers are risk averse, and a customer accepts an offered premium $p$ according to his $A$-value. The modeling 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 $n(p)$ is derived, and also the rate of claims from the insured customers is given. Further, 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: Certainty equivalent, Cramér-Lundberg model, diffusion approximation, discounting, inverse Gamma distribution, Lambert W function, present value, risk aversion, ruin probability

Format available: PDF (506.4 kb)