Combining Deductibles and Policy Limits

Why the Limit Structure Matters

A deductible makes the insured responsible for an initial part of a covered claim. A limit restricts how much is covered, but the word “limit” can refer to different amounts. It may cap the insurer’s payment, or it may cap the gross loss amount to which coverage applies. Those contract structures can produce different payments for the same claim.

The distinction matters in classification and rating work because the payment amount is the quantity whose expected value is being estimated. A calculation that silently switches between limit definitions can overstate or understate the insurer’s payment. The methods here are aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus.

Definition: Let $X$ be the covered loss amount for one claim, $d$ the per-claim deductible, and $Y^L$ the insurer’s payment per loss. A deductible is applied to each claim separately. The exact payment function depends on what the policy limit caps.

Two Common Limit Structures

First consider a limit $u$ that is the insurer’s maximum payment per claim. The insurer pays the part of the loss above the deductible, subject to a cap of $u$. The deductible is subtracted from the loss, and the limit caps the resulting insurer payment.

Y^L=\min\bigl((X-d)^+,u\bigr), \qquad (z)^+=\max(z,0).

The payment is zero when $X\le d$, increases with the loss between $d$ and $d+u$, and reaches the payment limit once $X\ge d+u$. Thus, the loss amount at which the insurer’s payment becomes capped is the deductible plus the insurer-payment limit.

Alternatively, a policy may state a limit $L$ on the gross loss amount covered before the deductible is applied. The covered loss is first capped at $L$, then the deductible is subtracted, with no payment below zero.

Y^L=\bigl(\min(X,L)-d\bigr)^+.

If $L>d$, the insurer’s maximum payment under this structure is $L-d$, not $L$. If $L\le d$, the deductible is at least as large as the gross-loss cap, so the insurer pays nothing.

Key distinction: An insurer-payment limit $u$ caps payment at $u$, so payment can reach that cap when $X\ge d+u$. A gross-loss limit $L$ caps the loss before the deductible, so payment cannot exceed $L-d$ when $L>d$.

Applying the Payment Functions

A reliable calculation begins by identifying the contractual meaning of the limit. Then apply the appropriate function. For a limit on insurer payment, subtract the deductible and cap the result. For a limit on gross loss, cap the loss and then subtract the deductible. The following steps make the distinction explicit.

Read what the limit caps.
Determine whether the stated amount is the insurer’s maximum payment or the maximum gross loss covered.

Apply the deductible.
For a payment limit, find $(X-d)^+$. For a gross-loss limit, begin with $\min(X,L)$ and subtract $d$, flooring the result at zero.

Check the relevant threshold.
A payment limit $u$ is reached at $X=d+u$. A gross-loss limit $L$, when $L>d$, produces maximum payment $L-d$ once $X\ge L$.

Worked Example: Reading a Limit on Insurer Payment

A hypothetical equipment policy has a per-claim deductible of $2,500 and a limit of $9,000 on the insurer’s payment. A covered loss is $14,500. Calculate the insurer’s payment and identify the loss amount at which the payment limit is reached.

Method. The stated limit caps insurer payment, so use $Y^L=\min((X-d)^+,u)$, with $X=\$14,500$, $d=\$2,500$, and $u=\$9,000$.

First subtract the deductible: $(X-d)^+=\$14,500-\$2,500=\$12,000$. Then apply the payment limit: $Y^L=\min(\$12,000,\$9,000)=\$9,000$.

The payment limit is reached when $X\ge d+u=\$2,500+\$9,000=\$11,500$. Since the loss of $14,500 exceeds $11,500, the insurer pays exactly $9,000.

Check. The payment does not exceed the $9,000 insurer-payment limit. A common mistake is to subtract the deductible from the payment limit as well, reporting $6,500. That would incorrectly treat the stated $9,000 limit as a gross-loss cap; the policy says it caps insurer payment.

Why the Formulas Have These Thresholds

The payment cases follow directly from the order of operations in each contract structure. For an insurer-payment limit, if $X\le d$, the loss does not exceed the deductible and the payment is zero. If $d<X<d+u$, the amount above the deductible is less than the limit, so the payment is $X-d$. If $X\ge d+u$, the amount above the deductible is at least $u$, so the payment is capped at $u$.

Y^L= \begin{cases} 0, & X\le d,\\ X-d, & d<X<d+u,\\ u, & X\ge d+u. \end{cases}

For a gross-loss limit $L>d$, losses up to $d$ still produce no payment. Above $d$ and below $L$, the payment is $X-d$. At and above $L$, only $L$ of gross loss is covered, so the maximum payment is $L-d$. This also shows why a gross-loss cap is equivalent to an insurer-payment cap of $L-d$, provided $L>d$: in either case, payment is the lesser of the amount above the deductible and $L-d$.

\bigl(\min(X,L)-d\bigr)^+ =\min\bigl((X-d)^+,L-d\bigr), \qquad L>d.

The equality follows by considering the same three ranges. When $X\le d$, both sides are zero. When $d<X<L$, both equal $X-d$. When $X\ge L$, both equal $L-d$. This equivalence is useful for checking a result, but it does not make the contract wording interchangeable: the stated limit amount has a different meaning in each structure.

Worked Example: Comparing the Same Number Under Two Limit Definitions

A hypothetical commercial property policy has a $3,000 per-claim deductible and states a $11,000 limit. Calculate the insurer’s payment for a covered loss of $14,000 under (a) a limit on insurer payment and (b) a limit on gross loss.

(a) Limit on insurer payment. Here $u=\$11,000$. The loss above the deductible is $\$14,000-\$3,000=\$11,000$. Therefore, $Y^L=\min(\$11,000,\$11,000)=\$11,000$. The insurer’s maximum payment is reached at $X=d+u=\$3,000+\$11,000=\$14,000$.

(b) Limit on gross loss. Here $L=\$11,000$. First cap the gross loss: $\min(\$14,000,\$11,000)=\$11,000$. Then subtract the deductible: $Y^L=(\$11,000-\$3,000)^+=\$8,000$. The gross-loss cap therefore permits a maximum insurer payment of $8,000.

Check. The two payments differ by $3,000, the deductible. The common mistake is to report $11,000 in both cases because the stated limit is $11,000. That is correct only when $11,000 caps insurer payment; under a gross-loss cap, the deductible reduces the maximum payment.

Expected Payment Under a Severity Model

For a random loss $X$, the expected insurer payment is $E[Y^L]$. With a discrete severity distribution, multiply each possible payment by its probability and add. With a continuous severity distribution, average the payment function over the distribution. In either case, use the payment function matching the contract; the expectation does not resolve ambiguity about the limit definition.

Worked Example: Expected Payment From Discrete Losses

A hypothetical policy has a $3,000 deductible and a limit of $11,000. The limit may be interpreted either as an insurer-payment limit or as a gross-loss limit. The covered loss $X$, in thousands of dollars, has the following distribution. Calculate expected insurer payment under both interpretations.

Loss $X$ ($000)	Probability
0	0.10
2	0.15
5	0.25
9	0.20
14	0.20
20	0.10

The probabilities sum to $0.10+0.15+0.25+0.20+0.20+0.10=1.00$. Under the insurer-payment interpretation, use $Y^L=\min((X-3)^+,11)$, with amounts in thousands of dollars. The payments at losses $0,2,5,9,14,20$ are respectively $0,0,2,6,11,11$.

The expected payment is $0.10(0)+0.15(0)+0.25(2)+0.20(6)+0.20(11)+0.10(11)=0+0+0.50+1.20+2.20+1.10=5.00$ thousand dollars, or $5,000.

Under the gross-loss interpretation, use $Y^L=(\min(X,11)-3)^+$. The payments at the same six losses are $0,0,2,6,8,8$. The expected payment is $0.10(0)+0.15(0)+0.25(2)+0.20(6)+0.20(8)+0.10(8)=0+0+0.50+1.20+1.60+0.80=4.10$ thousand dollars, or $4,100.

Check. The insurer-payment cap allows a maximum payment of $11,000, while the gross-loss cap allows a maximum payment of $8,000. The expected payments, $5,000 and $4,100, are each below their respective maximums. The mistake to avoid is applying the insurer-payment formula to both interpretations without first determining what the $11,000 caps.

Worked Example: Expected Payment Under a Continuous Severity Model

A hypothetical policy covers one loss $X$, where $X$ has a continuous uniform distribution on the interval from $0 to $20,000. Thus its density is $1/20{,}000$ per dollar over that interval. The policy has a $3,000 deductible and a $12,000 limit. Calculate expected payment when (a) the limit caps insurer payment and (b) the limit caps gross loss.

(a) Limit on insurer payment. Use $Y^L=\min((X-\$3,000)^+,\$12,000)$. Payment is zero up to $3,000, increases as $X-\$3,000$ between $3,000 and $15,000, and is $12,000 from $15,000 through $20,000. Therefore,

E[Y^L] =\frac{1}{20{,}000} \left[ \int_{3{,}000}^{15{,}000}(x-3{,}000)\,dx +\int_{15{,}000}^{20{,}000}12{,}000\,dx \right].

The first integral is $\tfrac12(12{,}000)(12{,}000)=72{,}000{,}000$ dollars squared. The second is $(5{,}000)(12{,}000)=60{,}000{,}000$ dollars squared. Their sum is $132{,}000{,}000$, and division by $20{,}000$ gives an expected payment of $6,600.

(b) Limit on gross loss. Use $Y^L=(\min(X,\$12,000)-\$3,000)^+$. Payment increases from $0 to $9,000 over losses from $3,000 to $12,000, then remains $9,000 up to $20,000. Thus,

E[Y^L] =\frac{1}{20{,}000} \left[ \int_{3{,}000}^{12{,}000}(x-3{,}000)\,dx +\int_{12{,}000}^{20{,}000}9{,}000\,dx \right].

The first integral is $\tfrac12(9{,}000)(9{,}000)=40{,}500{,}000$ dollars squared. The second is $(8{,}000)(9{,}000)=72{,}000{,}000$ dollars squared. Their sum is $112{,}500{,}000$; division by $20{,}000$ gives an expected payment of $5,625.

Check. The maximum payments are $12,000 and $9,000, respectively, and both expected payments are below those caps. The calculation uses the given uniform density and integrates payment over the entire loss range; omitting the capped portion above the relevant threshold would understate the expected payment.

Common Pitfalls and Checks

Applying the deductible twice. Under a limit on insurer payment, the payment is $\min((X-d)^+,u)$. Do not subtract $d$ from $u$; $u$ already denotes the maximum insurer payment.
Treating a gross-loss limit as an insurer-payment limit. If the limit caps gross loss at $L$, the maximum payment is $L-d$, provided $L>d$.
Using the wrong threshold. A payment limit $u$ is reached at gross loss $d+u$, whereas a gross-loss cap $L$ is reached at $X=L$.
Ignoring the deductible when $X\le d$. The payment is zero, even if the gross loss is positive.

Before finalizing a claim payment or expected payment, check that payment is nonnegative, does not exceed the maximum allowed by the stated limit structure, and is zero at losses no greater than the deductible. For an expected payment, also confirm that the probabilities sum to one or that the continuous density covers the stated severity range.

Check Your Understanding

For each problem, identify the limit structure before calculating. Amounts are in dollars unless stated otherwise.

A hypothetical policy has a $1,500 deductible and a $5,000 limit on insurer payment. A covered claim is $7,400. Find the payment and the loss amount at which the insurer-payment cap is first reached. (Answer: Payment $5,000; cap reached at a loss of $6,500.)
A hypothetical policy has a $2,000 deductible and an $8,000 gross-loss limit. A covered claim is $10,500. Find the payment and the maximum payment allowed by this structure. (Answer: Payment $6,000; maximum payment $6,000.)
A covered loss is $6,500, the deductible is $2,500, and the limit is $4,000 on insurer payment. Calculate the payment and explain whether the limit binds. (Answer: Payment $4,000; the limit binds because the loss above the deductible is $4,000.)
A hypothetical policy has a $2,000 deductible and a $9,000 limit on insurer payment. Loss amounts, in thousands, are $0,4,10,18$, with probabilities $0.20,0.30,0.30,0.20$, respectively. Calculate the expected payment. (Answer: $4,800.)
A hypothetical policy has a $2,000 deductible and an $8,000 gross-loss limit. The loss $X$ is uniform from $0 to $12,000, with density $1/12{,}000$ over that interval. Calculate expected payment. (Answer: $3,500.)

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