In a portfolio of risks, there are two types of modifications which can influence the frequency distribution of payments.

*Exposure Modification (not in syllabus**)*— increasing or decreasing the number of risks or time periods of coverage in the portfolio*Coverage Modification*— applying limits, deductible or adjusting for inflation in each individual risk

**EXPOSURE MODIFICATION**

- Poisson:
- Negative Binomial:

(Geometric is a Negative Binomial with ) - Binomial:

(Only valid if the new value remains an integer)

**Example 1**: You own a portfolio of 10 risks. You model the frequency of claims with a negative binomial having parameters and . The number of risks in your portfolio increases to 15. What are the parameters for the new distribution?

**Answer**: The frequency distribution now has parameters and Note that since the mean and variance are and respectively, the new mean and variance are multiplied by 1.5.

**COVERAGE MODIFICATION**

- Poisson:
- Negative Binomial:
- Binomial:
- Geometric:

**Example 2**: The frequency of loss is modeled as a Poisson distribution with parameter . A deductible is imposed so that only 80% of losses result in payments. What is the new distribution?

**Answer**: It is Poisson with .

**Example 3**: The frequency of payment is modeled as a negative binomial with parameters and . Losses are pareto distributed with parameters and . The deductible is changed from to . What are the new parameters in the frequency distribution?

**Answer**: Firstly, is the frequency of payment. So it reflects the current deductible. If you wanted the distribution of without deductible, you would divide by . Now to find the distribution of with the deductible of 15, you multiply by . To summarize: