Combining A Per-Loss Deductible And Limit

Why the Order of the Terms Matters

A per-loss deductible and a maximum insurer payment both affect what the insurer pays on one loss. The deductible determines the amount the insured bears before the insurer begins paying. The payment limit caps the insurer’s share once that share reaches the stated maximum. To calculate the payment correctly, first identify what the limit caps: a maximum insurer payment is not necessarily the same as a cap on the loss amount to which the deductible applies.

This tutorial considers a single covered loss at a time. It assumes that the loss amount is nonnegative, the deductible is applied to that loss, and the limit is a maximum dollar amount the insurer pays for that loss. There are no coinsurance provisions, exclusions, expenses, or other policy adjustments. Occurrence-based and aggregate limits are outside this rule.

Definition: Let $X$ be the amount of a covered loss, let $d\geq 0$ be the per-loss deductible, and let $u>0$ be the maximum insurer payment per loss. The insurer payment per loss, denoted by $Y^L$, is $$ Y^L=\min\bigl((X-d)_+,u\bigr), $$ where $(X-d)_+=\max(X-d,0)$. The insurer pays nothing up to the deductible, then pays the amount above the deductible until that payment reaches $u$.

The Payment Function and Its Thresholds

There are two loss thresholds. The first is $d$: at or below this loss amount, the deductible absorbs the entire loss. The second is $d+u$: at this loss amount, the amount above the deductible is exactly the maximum insurer payment. Losses above $d+u$ do not increase the insurer’s payment.

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

The middle expression applies only while the amount above the deductible is less than the payment limit. At $X=d+u$, the insurer payment is already $u$, so it belongs in the capped region as well. The amount retained by the insured, assuming no other payment or adjustment, is $X-Y^L$. It includes the deductible and, for sufficiently large losses, the amount above the insurer’s maximum payment.

Worked Example: Locate the Breakpoints and Calculate Payments

A hypothetical property policy has a per-loss deductible of $750 and a maximum insurer payment of $2,500 per loss. For losses of $500, $2,100, $4,000, and $5,200, find the two thresholds, the insurer payment, and the amount retained by the insured for each loss. Assume the limit is explicitly stated as a maximum insurer payment.

Method. Use $Y^L=\min((X-d)_+,u)$, with $d=750$ and $u=2{,}500$. The deductible threshold is $750$, and the cap threshold is $d+u=750+2{,}500=3{,}250$.

For $X=500$, the loss is below the deductible: $(500-750)_+=0$, so $Y^L=\min(0,2{,}500)=0$. The insured retains $500-0=\$500$.

For $X=2{,}100$, the loss exceeds the deductible but is below the cap threshold: $(2{,}100-750)_+=1{,}350$, so $Y^L=\min(1{,}350,2{,}500)=\$1{,}350$. The insured retains $2{,}100-1{,}350=\$750$.

For $X=4{,}000$, the amount above the deductible is $4{,}000-750=3{,}250$, which exceeds the $2,500 maximum payment. Thus $Y^L=\min(3{,}250,2{,}500)=\$2{,}500$, and the insured retains $4{,}000-2{,}500=\$1{,}500$.

For $X=5{,}200$, the amount above the deductible is $5{,}200-750=4{,}450$, also above the maximum. The insurer pays $Y^L=\$2{,}500$, and the insured retains $5{,}200-2{,}500=\$2{,}700$.

Check. The insurer pays zero at a loss below $750, pays the full amount above the deductible at $2,100, and never pays more than $2,500. The two losses above $3,250 therefore have the same insurer payment.

Common mistake avoided. Do not assume that reaching the deductible also means reaching the limit. The cap begins only when $X-d=u$, at $X=d+u=\$3{,}250$.

Why the Formula Has These Three Regions

The payment rule follows directly from the policy terms. Before the deductible is met, the insurer has no payment obligation under this simplified coverage. After the deductible is met, the amount eligible for payment is $X-d$. Since that amount cannot exceed the stated maximum insurer payment $u$, the insurer pays the smaller of $X-d$ and $u$.

Below the deductible.
If $X\leq d$, then $X-d\leq 0$, so $(X-d)_+=0$ and $Y^L=0$.

Above the deductible but below the cap threshold.
If $d<X<d+u$, then $0<X-d<u$. The smaller of $X-d$ and $u$ is $X-d$, so $Y^L=X-d$.

At or above the cap threshold.
If $X\geq d+u$, then $X-d\geq u$. The smaller amount is $u$, so $Y^L=u$.

This also verifies the breakpoints. At $X=d$, the first and middle expressions both give zero. At $X=d+u$, the middle expression would give $(d+u)-d=u$, which agrees with the capped expression. The payment is therefore continuous at both thresholds, even though its rate of increase changes.

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

The second piecewise rule describes the insured’s retained amount under the stated assumptions. For an intermediate loss, the insured bears exactly the deductible. For a loss above $d+u$, the insurer’s payment is capped, so the insured bears both the deductible and the excess above the insurer’s maximum payment.

Worked Example: Apply a Stated Maximum Insurer Payment

A hypothetical equipment policy has a $1,200 per-loss deductible and a $10,000 maximum insurer payment per loss. A covered equipment loss is $18,600. Calculate the insurer payment and the insured’s retained amount, and identify the cap threshold.

Method. Since the limit is stated as a maximum insurer payment, use $Y^L=\min((X-d)_+,u)$, with $X=18{,}600$, $d=1{,}200$, and $u=10{,}000$.

First find the thresholds. The deductible threshold is $1,200. The cap threshold is $d+u=1{,}200+10{,}000=\$11{,}200$.

The loss exceeds $11,200, so the payment is capped. Equivalently, $(18{,}600-1{,}200)_+=17{,}400$, and $Y^L=\min(17{,}400,10{,}000)=\$10{,}000$.

The insured retains $18{,}600-10{,}000=\$8{,}600$. This consists of the $1,200 deductible plus $18{,}600-11{,}200=\$7{,}400$ above the cap threshold; $1{,}200+7{,}400=\$8{,}600$.

Check. The payment is no greater than $10,000, and the insurer payment plus the insured’s retained amount is $10{,}000+8{,}600=\$18{,}600$, the original loss.

Common mistake avoided. A limit described as a maximum insurer payment is applied after the deductible has determined the amount eligible for payment. The cap threshold is $d+u$, not $u$.

Check What the Limit Caps

Policy wording determines which formula applies. In this tutorial, $u$ is the maximum amount the insurer pays for one loss. A different term might instead cap the loss amount to which the deductible is applied. Those are different contract structures, so do not substitute one interpretation for the other.

For example, if a policy limits the covered loss amount to $u$ before applying the deductible, one possible payment rule is $(\min(X,u)-d)_+$. That rule is not the same as $\min((X-d)_+,u)$. With $X=\$7{,}000$, $d=\$1{,}000$, and $u=\$4{,}000$, the maximum-insurer-payment rule gives $\min(6{,}000,4{,}000)=\$4{,}000$. A limit on the loss amount before applying the deductible instead gives $(4{,}000-1{,}000)_+=\$3{,}000$. The wording and definition of the limit must be read before calculating.

Worked Example: Expected Payment Under a Discrete Loss Model

A hypothetical insurer models the amount $X$ of one covered loss using the following discrete distribution. The policy has a $1,000 per-loss deductible and a $4,000 maximum insurer payment per loss. Find the expected insurer payment, the probability of a positive payment, and the probability that the insurer pays the maximum.

Loss $X$	Probability
$0	0.10
$1,000	0.20
$4,000	0.40
$7,000	0.30

Method. Apply $Y^L=\min((X-d)_+,u)$ to each possible loss before taking the probability-weighted average. Here $d=1{,}000$, $u=4{,}000$, and the cap threshold is $d+u=5{,}000$.

At $X=0$, the payment is $\min((0-1{,}000)_+,4{,}000)=0$. At $X=1{,}000$, it is $\min(0,4{,}000)=0$. At $X=4{,}000$, it is $\min(3{,}000,4{,}000)=\$3{,}000$. At $X=7{,}000$, it is $\min(6{,}000,4{,}000)=\$4{,}000$.

The expected insurer payment is the sum of each payment multiplied by its probability:

E[Y^L] =(0)(0.10)+(0)(0.20)+(3{,}000)(0.40)+(4{,}000)(0.30) =0+0+1{,}200+1{,}200 =\$2{,}400.

A positive payment occurs for losses above $1,000, which here are $4,000 and $7,000. Thus $P(Y^L>0)=0.40+0.30=0.70$. The insurer pays the maximum for losses at or above $5,000; among the listed outcomes, only $7,000 qualifies. Therefore $P(Y^L=4{,}000)=0.30$.

Check. The probabilities sum to $0.10+0.20+0.40+0.30=1.00$. The expected payment of $2,400 lies between the minimum possible payment of $0 and maximum of $4,000.

Common mistake avoided. Do not average the losses first and then apply the deductible and limit. The payment function is nonlinear, so calculate the payment at each possible loss and then take its probability-weighted average.

Common Exam Traps

Using $u$ as the cap threshold. When $u$ caps the insurer payment, the loss threshold is $d+u$. The payment reaches its maximum when the amount above the deductible equals $u$.
Subtracting the deductible after imposing a payment cap. This can incorrectly reduce a limit that is defined as the maximum insurer payment. Apply the deductible to the loss first, then cap the resulting insurer payment.
Applying the deductible more than once. The rule here is per loss. Do not subtract it again from the payment after using $(X-d)_+$.
Forgetting the insured’s share above the cap. Once the insurer payment reaches $u$, a larger loss increases the amount retained by the insured, not the insurer payment.
Ignoring boundary cases. At $X=d$, the payment is zero; at $X=d+u$, it is exactly $u$. These values agree with the adjacent regions.

Key takeaway: For a per-loss deductible $d$ and a maximum insurer payment $u$, calculate $Y^L=\min((X-d)_+,u)$. The payment changes at $X=d$ and $X=d+u$. Confirm that the policy’s stated limit really caps the insurer payment before using this formula.

Check Your Understanding

For each problem, treat the limit as a maximum insurer payment per loss unless the wording says otherwise.

A hypothetical policy has a $600 deductible and a $2,400 maximum insurer payment. Find the thresholds and insurer payments for losses of $300, $1,800, $3,200, and $5,000. (Answer: thresholds $600 and $3,000; payments $0, $1,200, $2,400, and $2,400.)
A covered loss is $6,900. The per-loss deductible is $1,500 and the maximum insurer payment is $6,000. Find the insurer payment and the amount retained by the insured. (Answer: insurer payment $5,400; insured retains $1,500.)
A hypothetical policy has an $800 deductible and a $3,200 maximum insurer payment. For a loss of $4,100, determine whether the payment is capped and calculate the insurer payment and retained amount. (Answer: capped; insurer payment $3,200; insured retains $900.)
A loss $X$ has a discrete distribution: $500 with probability 0.20, $1,500 with probability 0.30, $3,500 with probability 0.25, and $6,000 with probability 0.25. The deductible is $500 and the maximum insurer payment is $2,500. Find the expected insurer payment, the probability of a positive payment, and the probability of payment at the maximum. (Answer: expected payment $1,550; probability of a positive payment 0.8000; probability of maximum payment 0.5000.)
A hypothetical claim has loss $6,000, deductible $1,000, and a stated maximum insurer payment of $3,500. Calculate the insurer payment and amount retained, and state the cap threshold. (Answer: insurer payment $3,500; insured retains $2,500; cap threshold $4,500.)

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