Introduction to Reinsurance
An introduction to modelling insurance claims in the presence of reinsurance
In an earlier article, we explored modelling losses in the context of a general insurer selling motor insurance¹. In that article we implicitly assumed that the insurer would settle each claim in full, but that is unlikely to be the case in practice. In practice, the general insurer would likely make use of reinsurance. Suppose we again have a general insurer selling motor insurance. In this context, the insurer is exposed to the risk of policyholders being involved in accidents and the insurer having to pay claims — i.e. cover the cost to repair or replace the vehicle. The insurer will often choose to transfer some of this risk to another insurer, specifically, a reinsurer. The reinsurer would enter an arrangement with the insurer under which for each claim the insurer receives, part of the claim would be paid by the reinsurer instead. The insurer would therefore need to understand the distribution of their claims both before and after accounting for reinsurance.
Reinsurance arrangements can be segmented into two broad categories: proportional reinsurance, and non-proportional reinsurance. In proportional reinsurance, the reinsurer commits to pay a certain percentage of individual claims. For simplicity, we will consider a proportional reinsurance arrangement under which the reinsurer will pay a fixed proportion of every claim received by the insurer. For example, the reinsurer may commit to pay 60% of each claim the insurer receives. In this instance, if the insurer receives a claim amounting to £50,000, the insurer would pay £20,000 (or 40%) and the reinsurer would pay £30,000 (or 60%). The bar graph below illustrates how 10 claims would be shared by an insurer and reinsurer under a proportional reinsurance structure.
For the purposes of modelling, dealing with proportional reinsurance is simple. If the underlying claims distribution is known then the distribution of the claims net of reinsurance can be found by scaling the random variable by the retention percentage. Similarly, the distribution of the reinsurer’s share of claims can be found by scaling the random variable by the complement of the retention percentage (i.e. 1 — retention percentage). The presence of proportional reinsurance, as shown by the calculations below, reduces both the expectation and variance of the claims paid by the insurer.
In a non-proportional reinsurance arrangement, the reinsurer will pay a limited amount of the excess of claims above a certain threshold. For simplicity, we will consider a non-proportional reinsurance arrangement under which the reinsurer will pay, without limit, the excess of each individual claim above a certain threshold. For example, the reinsurer may commit to pay the excess of all individuals claims above £20,000. In this instance, if the insurer receives a claim amounting to £50,000, the insurer would pay £20,000 and the remaining £30,000 would be paid by the reinsurer. Any claim less than or equal to £20,000 would be paid in full by the insurer. The bar graph below illustrates how 10 claims would be shared by an insurer and reinsurer under an excess of loss reinsurance structure.
Modelling claim amounts under excess of loss reinsurance is more complex than under proportional reinsurance, but still straight forward. The key lies in the definition of the random variable representing the amount of the claim paid by the insurer.
Using the above definition, the moments of the distributions of the net claim and reinsurer’s share of claims can be found by integrating as per normal. The presence of excess of loss reinsurance, as shown by the calculations below, also reduces both the expectation and variance of the claims paid by the insurer since this amount is capped at the retention level.
Another option for the insurer to reduce both the expectation and variance of their claims paid is to introduce a policy excess or deductible. In the presence of a policy excess, the insurer will only pay claims in excess of a certain limit. All claims that amount to less than (or are equal to) the limit will be paid entirely by the policyholder themselves. As illustrated below, this setup is equivalent to the policyholder being an “insurer” with an excess of loss reinsurance arrangement in place. The bar graph below illustrates how 10 claims would be shared by a policyholder and insurer under a policy excess structure.
The repository with the R code to reproduce the graphs and calculations can be found here: https://github.com/ValentineChisango/A212-CS2
[1] My article on Loss Distributions can be found here: https://vmchisango.medium.com/introduction-to-loss-distributions-dbc3eacff971
