Portfolio size as funktion of the premium: modeling and optimization

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Abstract

An insurance company has a large number N of potential customers characterized by i.i.d. r.v.'s A1,…,AN 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.
Original languageEnglish
PublisherT.N. Thiele Centre, Department of Mathematics, Aarhus University
Number of pages17
Publication statusPublished - 2013
SeriesThiele Research Reports
Number01

Keywords

  • Certainty equivalent
  • Cramér-Lundberg model
  • diffusion approximation
  • discounting
  • inverse Gamma distribution
  • Lambert W function
  • present value
  • risk aversion
  • ruin probability

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