Collective Risk Models

In the collective risk model, we are trying to model a compound distribution of the total claim amount. The total claim amount is the sum of individual claims . The individual claims are independent and identically distributed (i.i.d.) random variables. The distribution of the individual claims is called the claim size distribution. The distribution of the total claim amount is called the compound distribution. It can be given as the from

where is the number of claims and are the individual claims.

The number of claims is a random variable that follows a specific case distribution. If follows a Poisson distribution, the compound distribution is called a compound Poisson distribution. If follows a binomial distribution, the compound distribution is called a compound binomial distribution, etc.

In the real world of insurance, the compound distributions can model various scenarios. For example, the compound Poisson distribution can model the total claim amount in a car insurance policy,

where the number of accidents follows a Poisson distribution.

The compound binomial distribution can model the total claim amount in a health insurance policy,

where the number of doctor visits follows a binomial distribution.

Compound distribution properties

When we model the compound distribution, we are wondering about the pattern of this distribution, especially the first three moments, like mean, variance and skewness.

Mean (First moment)

The first moment of the compound distribution should utilise the double expectation theory. Suppose we are modelling a compound distribution with . The expected value of can be given by

Variance (Second moment)

In the fist step, we need to introduce the conditional variance for the mixture distribution if the variable is given by the .

Conditional variance formula

To prove the above forluma, we use the following and in a more general way.

Recall the variance formula

If we have the following condition, we have the variable given , the conditional variance will be

If we take the expectation, we can have:

On the other hand, we can derive the variance for the conditional expectation of given , following the variance formula:

If we combine the above two equation, we can get the following equation:

Variance for mixture distribution

Assume we have a random variable and also follows a random distribution, we can apply the above result to the based on the mixture distribution criteria.

Moment generating function

Based on the standard definition of first moment and second moment, we can easily derive the mean and variance for the mixture distribution. However, we need to investigate the moment generating function to find the higher moment for the mixture distribution.

Based on the above relationship, we can also derive the general MGF for all mixed distributions.

Given the vairables , the moment generating function is:

You can go the the Moment generating function section to check more details

Summarise the above equations, we can have the following three relationships:

• Mean:
• Variance:
• MGF:

Special case with

Assume we have a random variable , which has a constant result for all claims and another random variable following an unknown distribution with mean, , and variance . Hence, we can get

variance

and moment generating function

If we apply the above results for mixture distribution , we can have

• Mean:

• Variance:

• MGF:

Special case with Poisson distribution

Suppose we have a random variable S follows a compound Poisson distribution with . According to the Poisson properties, we can have

Then, we can easily derive the mean, variance and moment generating function for the compound Poisson distribution.

• Mean:

• Variance:

• MGF:

Skewness (Third moment)

In Probability and Statistics, we have introduced the skewness for a random variable as the third moment of the distribution. The skewness is defined as

which equals to the ratio between the expected value of difference between value and expectation and the standard deviation.

The easiest way to calculate the third moment is to utilise cumulant moment generating function , which has been defined as

Then we can have:

The proof has been skipped. It will be added later.

Third moment for compound Poisson distribution

Suppose we keep using the previous compound Poisson distribution . Then, the CGF is

The third central moment of is

According to the above equation, we can have