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**

If there is an exposure modification, you would adjust the frequency distribution by scaling the appropriate parameter to reflect the change in exposure. The following list provides the appropriate parameter to adjust for each distribution:

- 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**

Coverage modifications shift, censor, or scale the individual risks, usually in the presence of a deductible or claim limit, and they change the conditions that trigger a payment. For example, adding a deductible is considered a coverage modification and this changes the condition for payment because any losses below the deductible do not qualify for payment. If the risk is represented by random variable , then adding a deductible would change the random variable to . Scaling in the presence of a deductible or claim limit also affects the frequency distribution. The following lists parameters affected by coverage modifications:

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

These parameters are scaled by the probability that a payment occurs.

**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: