# Tag Archives: Coverage Modification

## Frequency with Respect to Exposure and Coverage Modifications

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

1. Exposure Modification (not in syllabus) — increasing or decreasing the number of risks or time periods of coverage in the portfolio
2. 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:
1. Poisson:  $\lambda$
2. Negative Binomial:  $r$
(Geometric is a Negative Binomial with $r = 1$)
3. Binomial:  $m$
(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 $r = 2$ and $\beta = 0.5$.  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 $r = 3$ and $\beta = 0.5$  Note that since the mean and variance are $r\beta$ and $r\beta(1+\beta)$ 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 $d$ 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 $X$, then adding a deductible would change the random variable to $(X-d)_+$. Scaling in the presence of a deductible or claim limit also affects the frequency distribution.  The following lists parameters affected by coverage modifications:
1. Poisson:  $\lambda$
2. Negative Binomial:  $\beta$
3. Binomial:  $q$
4. Geometric:  $\beta$
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 $\lambda = 5$.  A deductible is imposed so that only 80% of losses result in payments.  What is the new distribution?
Answer:  It is Poisson with $\lambda = 0.8(5) = 4$.
Example 3:  The frequency of payment $N$ is modeled as a negative binomial with parameters $r = 3$ and $\beta = 0.5$.  Losses $X$ are pareto distributed with parameters $\alpha = 2$ and $\theta = 100$.  The deductible is changed from $d=10$ to $d=15$.  What are the new parameters in the frequency distribution?
Answer:  Firstly, $N$ is the frequency of payment.  So it reflects the current deductible.  If you wanted the distribution of $N$ without deductible, you would divide $\beta$ by $\Pr{(X>10)}$.  Now to find the distribution of $N$ with the deductible of 15, you multiply $\beta$ by $Pr(X>15)$.  To summarize:
$\begin{array}{rll} \beta_{new} &=& \displaystyle \beta_{old}\times\frac{\Pr(X>15)}{\Pr(X>10)} \\ \\ &=& \displaystyle 0.5\times \frac{0.75614367}{0.82644628} \\ \\ &=& 0.45746692 \end{array}$
Coinsurance $\alpha$ is the fraction of losses covered by the policy.  For example, $\alpha = 0.8$ means if a loss is incurred, 80% will be paid by the insurance company.  A claims limit $u$ is the maximum amount that will be paid.  The order in which coinsurance, claims limits, and deductibles is applied to a loss is important and will be specified by the problem.  The expected payment per loss when all three are present in a policy is given by
$E\left[Y\right] = \alpha \left[E\left[X\wedge u\right] - E\left[X \wedge d\right]\right]$
where $Y$ is the payment variable and $X$ is the original loss variable.  The second moment is given by
$E\left[Y^2\right] = \alpha^2\left(E\left[(X\wedge u)^2\right] - E\left[(X \wedge d)^2\right]-2d\left(E\left[X \wedge u\right]-E\left[X \wedge d\right]\right)\right)$
The second moment can be used to find the variance of payment per loss.  If inflation $r$ is present, multiply the second moment by $(1+r)^2$ and divide $u$ and $d$ by $(1+r)$.   For payment per payments, divide the expected values by $P(X>d)$ or $1-F(d)$.