Policy Limits and Insurer Payment Functions

Why a Policy Limit Changes the Payment

A policy limit places a maximum on the insurer’s payment for a covered claim. If the covered loss is below that maximum, the insurer can pay the full loss. If the covered loss is larger, the insurer pays only up to the limit, and the insured retains the excess. For ratemaking, it is important to describe this payment rule precisely before considering how limits affect average loss costs.

This tutorial focuses on one per-claim limit and no deductible. It does not include coinsurance, exclusions, expenses, or an aggregate cap. Accordingly, $X$ denotes the nonnegative covered ground-up loss for a single claim, measured on the same dollar basis as the limit. The limit $u$ is the maximum insurer payment for that claim.

Definition: For a covered loss $X$ subject to a per-claim limit $u$, with no deductible or other payment adjustment, the insurer payment per loss is $Y^L=\min(X,u)$. For a realized loss $x$, the payment is $y^L(x)=\min(x,u)$. The insured’s retained covered loss is $x-y^L(x)=\max(x-u,0)$.

The limit is a cap on the insurer’s payment, not a reduction applied to every loss. When $x$ is below the limit, there is no limit-related reduction. When $x$ exceeds the limit, the difference $x-u$ is the amount above the cap. It is retained by the insured under the assumptions used here.

The Payment Function in Cases

The minimum expression is compact, but a piecewise definition makes the contract’s operation explicit. At the boundary $x=u$, the full loss equals the maximum payment, so the insurer pays $u$. The two cases meet at that point without a discontinuity.

y^L(x)= \begin{cases} x, & 0\leq x\leq u,\\ u, & x>u. \end{cases}

The corresponding retained amount is zero up to the limit and increases dollar-for-dollar above it:

x-y^L(x)= \begin{cases} 0, & 0\leq x\leq u,\\ x-u, & x>u. \end{cases}

These formulas describe one claim at a time. A per-claim limit is applied separately to each covered claim; it is not automatically a maximum on all payments over a policy period. Whether a contract also contains an aggregate limit or other terms would require a separate specification.

Identify the covered loss.
Use the covered amount $x$ that is subject to the limit, not an amount that includes items outside the stated coverage.

Identify the applicable per-claim limit.
Express the limit $u$ in the same units and on the same dollar basis as the covered loss.

Cap the payment.
Calculate $y^L(x)=\min(x,u)$, then, if needed, calculate the retained amount as $x-y^L(x)$.

Why the Minimum Formula Holds

The formula follows directly from the two requirements of the contract: the insurer does not pay more than the covered loss, and the insurer does not pay more than the per-claim limit. If $x\leq u$, the covered loss is the smaller amount, so the insurer pays $x$. If $x>u$, the limit is the smaller amount, so the insurer pays $u$. In either case, the payment is the smaller of $x$ and $u$.

\begin{aligned} x\leq u &\implies \min(x,u)=x,\\ x>u &\implies \min(x,u)=u. \end{aligned}

The retained-loss formula follows by subtraction. When $x\leq u$, the insurer pays $x$, leaving $x-x=0$. When $x>u$, the insurer pays $u$, leaving $x-u$. Combining the cases gives $\max(x-u,0)$. This remainder is the loss above the limit; it is not the insurer payment.

Worked Example: Payments Below, At, and Above the Limit

A hypothetical policy has a per-claim limit of $u=\$80{,}000$, no deductible, and no other payment adjustment. Three separate covered claims have realized losses of $\$63{,}000$, $\$80{,}000$, and $\$104{,}000$. Calculate the insurer payment and retained amount for each claim. These are stipulated losses, not observations from a probability distribution; no interest or mortality assumption is involved.

Method. Apply $y^L(x)=\min(x,u)$ to each claim separately, then subtract the payment from the covered loss to find the retained amount.

For the $\$63{,}000$ loss, $\min(\$63{,}000,\$80{,}000)=\$63{,}000$. The insurer pays $\$63{,}000$, and the insured retains $\$63{,}000-\$63{,}000=\$0$.

For the $\$80{,}000$ loss, $\min(\$80{,}000,\$80{,}000)=\$80{,}000$. The insurer pays $\$80{,}000$, and the insured retains $\$80{,}000-\$80{,}000=\$0$. A loss exactly at the limit is not reduced below its full amount.

For the $\$104{,}000$ loss, $\min(\$104{,}000,\$80{,}000)=\$80{,}000$. The insurer pays $\$80{,}000$, and the insured retains $\$104{,}000-\$80{,}000=\$24{,}000$.

Check. For each claim, payment plus retained amount equals the covered loss: $\$63{,}000+\$0=\$63{,}000$, $\$80{,}000+\$0=\$80{,}000$, and $\$80{,}000+\$24{,}000=\$104{,}000$. The method avoids treating the excess above the limit as the insurer’s payment.

Applying a Per-Claim Limit to Multiple Claims

When a policy has several claims, calculate the payment for each claim using that claim’s own covered loss and the stated per-claim limit. Then add the individual payments if a total across the claims is requested. Do not first add all losses and apply the per-claim limit once: that would treat the contract as though it had one limit for the whole group.

Worked Example: Separate Limits for Claims in One Policy Period

A hypothetical small-business policy has a $\$45{,}000$ limit for each claim, with no deductible, coinsurance, or aggregate limit. During a policy period, four covered claims have losses of $\$18{,}400$, $\$52{,}700$, $\$44{,}600$, and $\$90{,}000$. Find each insurer payment, the total insurer payment, and the total retained loss. The losses are given amounts; no distribution, interest rate, or mortality assumption is used.

Method. For each claim, use $\min(x,\$45{,}000)$. The cap applies to each of the four claims independently.

The payments are $\min(\$18{,}400,\$45{,}000)=\$18{,}400$, $\min(\$52{,}700,\$45{,}000)=\$45{,}000$, $\min(\$44{,}600,\$45{,}000)=\$44{,}600$, and $\min(\$90{,}000,\$45{,}000)=\$45{,}000$.

Adding the four payments gives $\$18{,}400+\$45{,}000+\$44{,}600+\$45{,}000=\$153{,}000$. The retained amounts are $\$0$, $\$7{,}700$, $\$0$, and $\$45{,}000$, respectively, for a total retained loss of $\$52{,}700$.

Check. The total covered loss is $\$18{,}400+\$52{,}700+\$44{,}600+\$90{,}000=\$205{,}700$, and $\$153{,}000+\$52{,}700=\$205{,}700$. Applying one $\$45{,}000$ cap to the combined loss would be incorrect because the contract states a per-claim limit, not one limit for the policy period.

Reading a Payment When the Loss Is Not Known

Sometimes a problem gives the payment rather than the covered loss. The payment function can then restrict what the loss might have been. If the payment is strictly below the limit, the loss must equal that payment: a loss above the limit would have produced a payment equal to the limit. If the payment equals the limit, however, the exact loss cannot be inferred from the payment alone. The loss may equal the limit or exceed it.

y^L(x)<u \implies x=y^L(x), \qquad y^L(x)=u \implies x\geq u.

Worked Example: Interpreting Capped Payments

A hypothetical equipment policy has a $\$110{,}000$ per-claim limit, no deductible, and no other payment adjustment. An insurer payment record shows $\$84{,}000$ for one covered claim and $\$110{,}000$ for another. For the second claim, a separate covered-loss statement gives a gross amount of $\$146{,}000$. Determine what can be concluded about each claim’s loss and retained amount. Amounts are stipulated at the same valuation date; no discounting, distribution, or mortality assumption is used.

Method. Compare each recorded payment with the limit. A payment below the limit identifies the covered loss under this contract; a payment at the limit identifies only a lower bound unless additional loss information is available.

For the first claim, the payment is $\$84{,}000<\$110{,}000$. Therefore its covered loss must be $\$84{,}000$, and its retained amount is $\$84{,}000-\$84{,}000=\$0$.

For the second claim, the payment equals the limit, so the payment record alone establishes only that the covered loss is at least $\$110{,}000$. The separate covered-loss statement supplies the exact amount, $\$146{,}000$. The insurer pays $\min(\$146{,}000,\$110{,}000)=\$110{,}000$, leaving $\$146{,}000-\$110{,}000=\$36{,}000$ retained.

Check. The second claim’s payment plus retained amount is $\$110{,}000+\$36{,}000=\$146{,}000$. The method avoids claiming that a payment equal to the limit proves the loss was exactly the limit.

Common Mistake: Reversing Payment and Retention

A frequent error is to subtract the limit from a loss above the limit and call the result the insurer’s payment. For a $\$104{,}000$ covered loss and an $\$80{,}000$ per-claim limit, that incorrect calculation is $\$104{,}000-\$80{,}000=\$24{,}000$. It calculates the insured’s retained amount, not the insurer’s payment.

The correction is to apply the payment function: $y^L(\$104{,}000)=\min(\$104{,}000,\$80{,}000)=\$80{,}000$. The retained amount is then $\$104{,}000-\$80{,}000=\$24{,}000$. As a check, the two amounts sum to the covered loss. This distinction is especially useful when a question asks separately for the insurer’s payment, the insured’s responsibility, or the payment function itself.

Key takeaway: With no deductible or other adjustment, a per-claim limit $u$ gives the payment $Y^L=\min(X,u)$. The insurer pays the full covered loss up to the limit and no more than the limit above it; the retained amount is $\max(X-u,0)$. This treatment is aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Check Your Understanding

For each problem, assume the stated amounts are covered losses and that there is no deductible or other payment adjustment. No probability distribution or discounting is needed.

A covered loss is $\$38{,}200$ under a $\$42{,}500$ per-claim limit. Find the insurer payment and retained amount. (Answer: insurer payment $\$38{,}200$; retained amount $\$0$.)
A covered loss is exactly $\$60{,}000$, and the per-claim limit is $\$60{,}000$. Find the insurer payment and retained amount. (Answer: insurer payment $\$60{,}000$; retained amount $\$0$.)
A hypothetical policy has a $\$55{,}000$ per-claim limit and no aggregate limit. Three separate claims are $\$24{,}500$, $\$81{,}000$, and $\$59{,}000$. Find the total insurer payment and total retained loss. (Answer: total insurer payment $\$134{,}500$; total retained loss $\$30{,}000$.)
Under a $\$125{,}000$ per-claim limit, an insurer’s recorded payment is $\$117{,}000$. What is the covered loss, and what is the retained amount? (Answer: covered loss $\$117{,}000$; retained amount $\$0$.)
A policy has a per-claim limit of $\$50{,}000$. Three covered losses are $\$31{,}000$, $\$88{,}000$, and $\$140{,}000$. Find each payment and the total retained loss. (Answer: payments $\$31{,}000$, $\$50{,}000$, and $\$50{,}000$; total retained loss $\$128{,}000$.)

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