August 28, 2008 · 4:56 am

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:

**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:

Filed under Coverage Modifications, Deductibles, Frequency Models, Limits

Tagged as Binomial, Coverage Modification, Deductibles, Exposure Modification, Frequency, Frequency Distribution, Geometric, Limits, Negative Binomial, Pareto, Poisson

August 24, 2008 · 11:22 pm

August 24, 2008 · 5:56 am

Frequency models count the number of times an event occurs.

- The number of customers to arrive each hour.
- The number of coins lucky Tom finds on his way home from school.
- How many scientists a Tyrannosaur eats on a certain day.
- Etc.

This is in contrast to a severity model which measures the magnitude of an event.

- How much a customer spends.
- The value of a coin that lucky Tom finds.
- The number of calories each scientist provides.
- Etc.

The following distributions are used to model event frequency. For notation,

means

.

##

Poisson:

Properties: