Excess of Loss Reinsurance

Sometimes, the reinsurance company will set the excess level with the baseline insurance company. In the excess of loss agreement with the insurer, the reinsurer will handle the excess of each claim over an agreed excess level , which means that the baseline insurance companies should handle the payment if the claim is lower than the excess level. We can write:

represents the claim amount, represents the payment amount of the baseline insurance company, and represents the payment amount of the reinsurance company. The relationship between , , and can be described as follows:

It can also be described as follows:

When we discuss the total claim, we could seperate the calculation into two parts: baseline insurance company and reinsurance company.

Baseline Insurance Company

According to the above expression, the expected payment per claim for the insurer is reduced from to

We can solve the above equation by the following steps:

Letting , we can solve the above equation with the following steps:

In addition, we can also apply another method to find a solution

Let , , then , , then we have:

Substitute the above, we have:

Estimation

when an excess of loss contract has been agreed, the baseline insurance company is only interested in for any loss , and the total claims can be collected for the insurer as a censored sample of losses of the following form:

where is the number of the losses beyond the excess level, is the number of claims that the amount is lower than the excess level (). In trying to get the appropriate loss distribution, the insurer can find the maximised joint likelihood function given by

where is the density function and is the survival function of the underlying density distribution .

For example, we assume that the claim follows an exponential distribution . The joint likelihood function can be written as

Taking logarithm,

Then, we take partial deriviative for . We can have

Reinsurance Company

The reinsurance company will pay the remaining amount of the claim after the baseline insurance company has paid the claim amount up to the limit (). The expectation of the reinsurance company can be calculated as follows:

From the second step, if let , then , , then we have:

It can also be described as follows:

All above expressions are the same. The expectation of the reinsurance company is the expectation of the total claim amount minus the expectation of the baseline insurance company.