$Y = \alpha \cdot (X - d)_+$ $\text{where } \alpha \in (0,1] = \text{coinsurance percentage (insurer's share)}$ A Coinsurance Percentage ($\alpha$) is the fraction of the covered loss (after any deductible) that the insurer agrees to pay, with the insured retaining the remaining fraction $1 - \alpha$. Coinsurance shares risk between insured and insurer after the deductible is satisfied. When $\alpha = 1$ the insurer covers 100% of the excess; lower values mean the insured co-pays a portion. It scales both the expected payment and variance by $\alpha$ and $\alpha^2$, respectively. > [!example]- Expected Payment with Deductible and Coinsurance {💡 Example} > 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]- Answer > > With $Y = 0.80 \cdot (X - 500)_+$: > > $E[Y] = 0.80 \times 1{,}200 = 960$ > > $\text{Var}(Y) = (0.80)^2 \times 4{,}000{,}000 = 0.64 \times 4{,}000{,}000 = 2{,}560{,}000$