$Y = \min\!\bigl(\alpha(X - d)_+,\; u\bigr)$ $\text{where } u = \text{maximum benefit (benefit limit)}$ A Benefit Limit ($u$) is the maximum amount an insurer will pay on a single claim, capping the insurer's payment regardless of how large the underlying loss $X$ is. The benefit limit is the insurer's counterpart to the deductible: the deductible removes small losses, while the limit removes large ones. Losses above $d + u/\alpha$ (for coinsurance $\alpha$) result in the insurer paying exactly $u$ and the insured bearing the remainder. > [!example]- Payment with Deductible and Benefit Limit {💡 Example} > A policy has deductible $d = 100$ and benefit limit $u = 400$ with no coinsurance. If $X = 600$, what does the insurer pay? > > > [!answer]- Answer > > The uncapped payment would be $X - d = 600 - 100 = 500$. Applying the benefit limit: > > $Y = \min(500, 400) = 400$ > > The insurer pays \$400; the insured absorbs the first \$100 (deductible) and the last \$100 (excess above limit).