Optimal per claim deductibility in insurance with the possibility of risky investments

In this paper we study the problem of optimal level of deductibility in an insurance contract when there is a deductible on each claim and the number of claims is Poisson distributed. Using new results in the theory of risk aversion and in actuarial risk theory, we prove that known results in the case when the deductible is on total claim amount carry over to this more ‘realistic’ model. We also extend the model to allow for investment in a risky asset, analysing the interplay between buying insurance and investing risky.