Payment

Study guides: Exam FAM, Exam RET-101

The payment $Y$ (the insurer's payment) is the amount an insurer actually pays a policyholder after all policy adjustments — deductible, coinsurance, and benefit limit — are applied to the raw loss $X$ .

$Y = \min\!\big(c\,(X - d)^+,\ u\big)$

$X$ is the ground-up loss, $d$ the deductible, $c$ the coinsurance factor, and $u$ the benefit limit (maximum payment).
$(X - d)^+ = \max(X - d,\ 0)$ applies the deductible; multiplying by $c$ applies coinsurance; the outer $\min$ caps the payment at $u$ .
This is the payment random variable whose expectation drives premium and reserve calculations.

Expected Payment with a Deductible A loss $X$ is uniformly distributed on $[0, 1000]$ with a $ $200$ deductible and no other adjustments. Find $E[Y]$ .
Answer Here $Y = (X - 200)^+$ : zero for $X \le 200$ , and $X - 200$ for $X > 200$ .
 $\begin{align*} E[Y] &= \int_{200}^{1000} (x - 200)\,\frac{1}{1000}\,dx \\ &= \frac{1}{1000} \cdot \frac{(800)^2}{2} \\ &= \frac{640{,}000}{2{,}000} = 320 \end{align*}$ 
The insurer expects to pay $ $320$ per loss after the deductible.