TABLE OF CONTENTS In this chapter we focus on the distribution function of the total claim amount S for the portfolio of an insurer. We intend to determine not only the expected value and the variance of the insurer's random capital, but also the probability that the amounts paid exceed a fixed threshold. A model for the total claim amount S is also needed to be able to apply the theory of the previous chapter. To determine the value-at-risk at, say, the 99.9% level, we need also good approximations for the inverse of the cdf, especially in the far tail. In this chapter we deal with models which still recognize the individual, usually different, policies. As is done often in non-life insurance mathematics, the 'time' aspect will be ignored. This aspect is nevertheless important in disability and long term care insurance. For this reason, these types of insurance are sometimes counted as life insurances.
In the insurance practice, risks usually can't be modelled by purely discrete random variables, nor by purely continuous random variables. For instance, in liability insurance a whole range of positive amounts can be paid out, each of them with a very small probability. There are two exceptions: the probability of having no claim, i.e., claim size 0, is quite large, and the probability of a claim size which equals the maximum sum insured, i.e., a loss exceeding that threshold, is also not negligible. For the expected value of such mixed random variables, we use the...
