Benefit Limit

Study guide: Exam P-1

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.

$Y = \min\!\bigl(\alpha(X - d)_+,\; u\bigr)$

$\text{where } u = \text{maximum benefit (benefit limit)}$

The formula caps the insurer's payment: the deductible comes off first, coinsurance $\alpha$ applies to the rest, and $u$ caps the result. That matches P-21-05's definition (a benefit limit sets an upper bound on how much the insurer will pay for any loss) and the way SOA's sample questions apply a limit to the benefit itself
A policy can instead cap the covered loss and apply coinsurance after the cap. P-21-05's health policy pays costs up to $5{,}000$ and reimburses $80\%$ of them, so costs of $6{,}000$ are reimbursed $0.80 \times \min(6{,}000,\ 5{,}000) = 4{,}000$ , where the payment-cap formula would give $\min(4{,}800,\ 5{,}000) = 4{,}800$ . The policy wording decides the order (see Payment Random Variable); with no coinsurance the two agree
The benefit limit is the insurer's counterpart to the Deductible: the deductible takes the first part of each loss off the insurer, while the limit caps what it pays on a large one
Under the payment cap, losses above $d + u/\alpha$ result in the insurer paying exactly $u$ and the insured bearing the remainder

Payment with Deductible and Benefit Limit A policy has deductible $d = 100$ and benefit limit $u = 400$ with no coinsurance. If $X = 600$ , what does the insurer pay?

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).