# Other Coverage Modifications

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 $1-F(d)$.