The compound distribution is a model for describing the aggregate claims arised in a group of independent insureds. Let be the number of claims generated by a portfolio of insurance policies in a fixed time period. Suppose is the amount of the first claim, is the amount of the second claim and so on. Then represents the total aggregate claims generated by this portfolio of policies in the given fixed time period. In order to make this model more tractable, we make the following assumptions:

- are independent and identically distributed.
- Each is independent of the number of claims .

The number of claims is associated with the claim frequency in the given portfolio of policies. The common distribution of is denoted by . Note that models the amount of a random claim generated in this portfolio of insurance policies. See these two posts for an introduction to compound distributions (An introduction to compound distributions, Some examples of compound distributions).

When the claim frequency follows a Poisson distribution with a constant parameter , the aggreagte claims is said to have a compound Poisson distribution. After a general discussion of the compound Poisson distribution, we discuss the property that an independent sum of compound Poisson distributions is also a compound Poisson distribution. We also present an example to illustrate basic calculations.

**Compound Poisson – General Properties**

*Distribution Function*

where , is the common distribution function of and is the n-fold convolution of .

*Mean and Variance*

*Moment Generating Function and Cumulant Generating Function*

Note that the moment generating function of the Poisson is . For a compound distribution in general, .

*Skewness*

**Independent Sum of Compound Poisson Distributions**

First, we state the results. Suppose that are independent random variables such that each has a compound Poisson distribution with being the Poisson parameter for the number of claim variable and being the distribution function for the individual claim amount. Then has a compound Poisson distribution with:

- the Poisson parameter:
- the distribution function:

The above result has an insurance interpretation. Suppose we have independent blocks of insurance policies such that the aggregate claims for the block has a compound Poisson distribution. Then is the aggregate claims for the combined block during the fixed policy period and also has a compound Poisson distribution with the parameters stated in the above two bullet points.

To get a further intuitive understanding about the parameters of the combined block, consider as the Poisson number of claims in the block of insurance policies. It is a well known fact in probability theory (see [1]) that the indpendent sum of Poisson variables is also a Poisson random variable. Thus the total number of claims in the combined block is and has a Poisson distribution with parameter .

How do we describe the distribution of an individual claim amount in the combined insurance block? Given a claim from the combined block, since we do not know which of the constituent blocks it is from, this suggests that an individual claim amount is a mixture of the individual claim amount distributions from the blocks with mixing weights . These mixing weights make intuitive sense. If insurance bock has a higher claim frequency , then it is more likely that a randomly selected claim from the combined block comes from block . Of course, this discussion is not a proof. But looking at the insurance model is a helpful way of understanding the independent sum of compound Poisson distributions.

To see why the stated result is true, let be the moment generating function of the individual claim amount in the block of policies. Then the mgf of the aggregate claims is . Consequently, the mgf of the independent sum is:

The mgf of has the form of a compound Poisson distribution where the Poisson parameter is . Note that the component in the exponent is the mgf of the claim amount distribution. Since it is the weighted average of the individual claim amount mgf’s, this indicates that the distribution function of is the mixture of the distribution functions .

**Example**

Suppose that an insurance company acquired two portfolios of insurance policies and combined them into a single block. For each portfolio the aggregate claims variable has a compound Poisson distribution. For one of the portfolios, the Poisson parameter is and the individual claim amount has an exponential distribution with parameter . The corresponding Poisson and exponential parameters for the other portfolio are and , respectively. Discuss the distribution for the aggregate claims of the combined portfolio.

The aggregate claims of the combined portfolio has a compound Poisson distribution with Poisson parameter . The amount of a random claim in the combined portfolio has the following distribution function and density function:

The rest of the discussion mirrors the general discussion earlier in this post.

*Distribution Function*

As in the general case,

where , and is the n-fold convolution of .

*Mean and Variance*

*Moment Generating Function and Cumulant Generating Function*

To obtain the mgf and cgf of the aggregate claims , consider . Note that is the weighted average of the two exponential mgfs of the two portfolios of insurance policies. Thus we have:

*Skewness*

Note that

**Reference**

- Hogg R. V. and Tanis E. A.,
*Probability and Statistical Inference*, Second Edition, Macmillan Publishing Co., New York, 1983.

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Note that the component \displaystyle \sum \limits_{i=0}^{k} \frac{\lambda_i}{\lambda}M_i(t) in the exponent is the mgf of the claim amount distribution. Since it is the weighted average of the individual claim amount mgf’s, this indicates that the distribution function of Y is the mixture of the distribution functions F_i.

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Thank you!