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
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:
- Negative Binomial:
(Geometric is a Negative Binomial with )
(Only valid if the new value remains an integer)
: You own a portfolio of 10 risks. You model the frequency of claims with a negative binomial having parameters
. The number of risks in your portfolio increases to 15. What are the parameters for the new distribution?
: The frequency distribution now has parameters
Note that since the mean and variance are
respectively, the new mean and variance are multiplied by 1.5.
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:
: 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?
: It is Poisson with
: The frequency of payment
is modeled as a negative binomial with parameters
are pareto distributed with parameters
. The deductible is changed from
. What are the new parameters in the frequency distribution?
is the frequency of payment. So it reflects the current deductible. If you wanted the distribution of
without deductible, you would divide
. Now to find the distribution of
with the deductible of 15, you multiply
. To summarize:
If a policy pays a reward to the participants when losses are below a certain level, this is a particular type of problem which Weishaus calls a “bonus” problem. The bonus, dividend, or refund amount is expressed as a maximum between 0 and the refunded amount. For example, a 15% refund is paid on the difference between the $100 premium and the loss . No refund is paid if losses exceed $100. The refund amount can be expressed as
The key to finding the expected refund is knowing how to manipulate the max function and rewrite it as a min. We can rewrite as
So the expected value is given by
Before I begin, please note: I hated this chapter. If there are any errors please let me know asap!
A deductible is an amount that is subtracted from an insurance claim. If you have a $500 deductible on your car insurance, your insurance company will only pay damages incurred beyond $500. We are interested in the following random variables: and .
- Payment per Loss:
- Limited Payment per Loss:
We may also be interested in the payment per loss, given payment is incurred (payment per payment
Since actuaries like to make things more complicated than they really are, we have special names for this expected value.
It is denoted by and is called mean excess loss
in P&C insurance and
is called mean residual life
in life insurance.
Weishaus simplifies the notation by using the P&C notation without the random variable subscript. I’ll use the same.