Risk Estimation in the Case of Limited Insurance Liability

摘要

There are models, used for the insurance risk estimation.There are two directions of analysis called "Classical risk theory" and "Modem risk theory" in the practice.The modem risk theory includes additional conditions, typical for the insurance company business like taxes, different internal costs and many others.Unfortunately these peculiarities are usually not available for the "outside world". That is why, it is very difficult to do analysis with such details. On the other hand, the classical risk theory is focused on the analytical models of stochastic processes which open a wide field for mathematical application. As a fundamental part of insurance risk theory, the model of Cramer-Lundberg is based on the balance between claims costs of the insurer and the premium payments from the side of the insured persons. The model also includes information about the retention and the initial capital necessary to meet the expected claims costs. The expected claims process is a compound stochastic process, which is usually modeled by continuous distributions. The approach often used for reducing the insurance risk is by using franchise value or just declaring a limit value for the insurer's liability. Including such restriction in the models the claims cost distribution is continuous no more. This involves considering of appropriate approximations for the mixed discrete-continuous distributions of the claims cost. Also all estimations of the level of risk like the retention and the necessary free reserves are affected by the choice of approximate distribution. The influence of different transformations of the random variables, which describe the claims cost in the risk models is considered in the current work. The classical risk model of Cramer-Lundberg for one year fixed period of time was used for estimations of the retention and the free reserves. The experiment we provide is based on empirical distribution for simulating transformed random variables. Fourier approximati