Overview
The factorized model splits a model of total claim size into two separate and independent models, a model for the total number of claims, and then a model for the claim size.
{% \mathbb{S} = \mathbb{E}[X] \mathbb{E}[N] %}
Number of Claims
The standard distribution used to model a random number of events is the Poisson Distribution. The distribution is fit to the data by fitting the {% \lambda %} parameter to the data, typically by setting it equal to the average number of claims.
Once {% \lambda %} is fit, this will determing the standard deviation of the model. (That is the variance of the distribution is equal to {% \lambda %}).
If the variance of the sample data is markedly different from the average, then the model needs to be adjusted to accomodate this dispersion. One way to do this is to assume a compound model, that is, assume that {% \lambda %} itself is a random variable, with some distribution. (a simple model would be that {% \lambda %} is drawn from a uniform distribution over some specified range). Once the value of {% \lambda %} is fixed, the value of {% N %} can then be fixed.
The effect of the Poisson mixture model is to alter the variance of the distribution. That is we have
{% \mathbb{E}[N] = \mathbb{E}[\lambda] %}
and
{% \mathbb{V}[N] = \mathbb{V}[\lambda] + \mathbb{E}[\lambda] %}
see Bluhm section 2.2.1
Sum of Number of Claims Models
When modeling the number of claims using the Poisson distribution, one often has to add multiple models together. For example, a model built to forecast the number of claims from a group of 100 insured customers, will be inappropriate if the customer base grows to 200. That is, the analyst needs to understand how to add Poisson models together.
The sum of two Poisson distributed
{% L_1 + L_2 \sim Pois(\lambda_1 + \lambda_2) %}
for {% L_1 \sim Pois(\lambda_1) %} and {% L_2 \sim Pois(\lambda_2) %}
see Bluhm section 2.2
In the limiting case, one could build a Poisson model for a customer base of one individual, and then scale the model up to the number of individuals in the customer base.
As an additional consideration, the anlayst must decide if the Poisson model will encompass multiple types of claims, or if each type of claim will be modeled separately. When the claim types are aggregated into a single model, the claim size model (see below) then needs to account for the different types of claims.
Size of the Claim
The distribution of the size of claim, will in general be dependent on the type of insurance. The standard method of modeling the claim size is to select a distribution whose characteristics match those of the assumed claim. (such as, distribution is positive for only positive numbers, and possibly bounded above, etc.....)