Reinsurance Examples

The following section will show examples about the common distributions we discussed in the previous sections.

Exponential distribution

1. loss function:
2. survival function:
3. density function:
4. expectation:
5. expectation of the minimum of the claim amount and the excess:

Pareto distribution

1. loss function:
2. survival function:
3. density function:
4. expectation:
5. expectation of the minimum of the claim amount and the excess:

if , then

if , then

Gamma distribution

1. loss function:

where is the gamma function, is the shape parameter, and is the rate parameter.

2. survival function:
3. density function:
4. expectation:

Lognormal distribution

1. loss function:
2. survival function:
3. density function:
4. expectation:
5. expectation of the minimum of the claim amount and the excess:

Weibull distribution

1. loss function:
2. survival function:
3. density function:
4. expectation:
5. expectation of the minimum of the claim amount and the excess:

Real world Examples

Case 1 (Exponential distribution)

Let’s take the exponential distribution as an example. The exponential distribution has the following parameters: . The limit of the claim amount is . The expectation of the claim amount is . The expectation of the minimum of the claim amount and the limit is

Case 2 (Pareto distribution)

Suppose the claim amount follows a Pareto distribution with the following parameters: , . The limit of the claim amount is .

The expectation of the claim amount is . The expectation of the minimum of the claim amount and the limit is