Calculation of convolution and application to Peer-to-Peer insurance

The conditional mean risk sharing is proved to be a fair and easy to understand way to share the losses between individuals in the group. More precisely, if the conditional expectation of loss that one individual brings to the group is a non-decreasing function of total losses; the risk sharing mechanism is Pareto-optimal. The thesis applies the conditional mean risk sharing into the context of Peer-to-Peer insurance, by providing numerical illustrations with assumption that the total losses follow compound Poisson sum. The explicit expression of conditional mean risk sharing for compound Poisson sum is facilitated by size-biased transform of different types of random variables. With the support of Panjer recursion formula and convolution method, different scenarios are tested regarding assumptions of the compound Poisson sums, such as their claim frequencies, claim severities (Log-normal distribution or Pareto distribution), and the number of participants in the group. When applying conditional expectation of losses instead of the loss itself, in the case of identically and independently distributed claim severities, increasing in number of participants in the group will reduce the variability of losses while the expected contribution remains unchanged.