Pricing Insurance portfolio related to the risk of insolvency

In this talk we will discuss some optimization problems in insurance. We consider a portfolio containing heterogeneous risks. The premiums of the policyholders might not cover the amount of the payments which an insurance company pays the policyholders. When setting the premium, this risk has to be taken into consideration. On the other hand the premium that the insured pays has to be fair. This fairness is measured by a function of the difference between the risk and the premium paid. For a given small probability of insolvency, we find the premium for each class, such that the difference function is minimized. Further results are achieved by doing the calculations in terms of utility instead of money. We find that by choosing the appropriate utilities function it is possible to
derive a wide range of premium principles as the optimal solution. Finally, we expand the results to the long-run model by considering a Markov chain in order to calculate the probabilities of insolvency during the years.