Coinsurance Percentage

Study guides: Exam P-1, Exam 5

A Coinsurance Percentage means different things on the SOA and CAS exams, and the two are not interchangeable:

Exam P ( $\alpha$ ): the fraction of the covered loss (after any deductible) that the insurer agrees to pay, with the insured retaining the remaining fraction $1 - \alpha$
Exam 5 ( $c$ ): the required insurance-to-value percentage in a property coinsurance clause. The insured must carry insurance of at least $cV$ , e.g. $80\%$ of the property's value $V$ , or the payment on a covered loss is reduced in proportion to the underinsurance. It is not the insurer's share of each loss

In the Exam P sense:

$Y = \alpha \cdot (X - d)_+$

$\text{where } \alpha \in (0,1] = \text{coinsurance percentage (insurer's share)}$

When $\alpha = 1$ the insurer covers 100% of the excess; lower values mean the insured co-pays a portion
It scales the expected payment by $\alpha$ and the variance by $\alpha^2$

In the Exam 5 sense (Werner & Modlin):

$a = \min\!\left(\frac{F}{cV},\ 1.0\right)$

$I = L \times \frac{F}{cV}, \quad \text{where } I \le F \text{ and } I \le L$

$F$ is the face value of the policy, $V$ the value of the property, $L$ the loss after deductible, $a$ the apportionment ratio and $I$ the indemnity. A home worth $\$500{,}000$ insured for $\$300{,}000$ under an $80\%$ requirement has $cV = \$400{,}000$ and $a = 0.75$ , so a $\$200{,}000$ loss is paid $\$150{,}000$

Expected Payment with Deductible and Coinsurance Ground-up losses $X$ have $E[(X-500)_+] = 1{,}200$ and $\text{Var}((X-500)_+) = 4{,}000{,}000$ . The insurer applies coinsurance $\alpha = 0.80$ . Find $E[Y]$ and $\text{Var}(Y)$ .

Answer With $Y = 0.80 \cdot (X - 500)_+$ : $E[Y] = 0.80 \times 1{,}200 = 960$ $\text{Var}(Y) = (0.80)^2 \times 4{,}000{,}000 = 0.64 \times 4{,}000{,}000 = 2{,}560{,}000$