Insurer Payment Functions for Ordinary Deductibles

Introduction

A deductible changes which part of a claim the insurer pays; it does not change the amount of the ground-up loss. To calculate the payment correctly, first identify the loss, then subtract the ordinary deductible, and finally apply any limit on the insurer’s payment. Keeping those steps distinct avoids a common error: applying a limit to the ground-up loss when the contract instead limits the insurer’s payment.

This tutorial defines the insurer payment function for one loss under an ordinary deductible. It focuses on writing and evaluating the function, including the case where the contract also caps the payment. It does not require a probability model: each calculation below treats the stated loss amount as a stipulated claim.

Definition: Let $X\geq 0$ be the ground-up amount of one loss, let $d\geq 0$ be an ordinary per-claim deductible, and let $u\geq 0$ be a limit on the insurer’s payment for that claim. The insurer payment per loss is $$ Y^L=\min\bigl((X-d)^+,u\bigr), \qquad (X-d)^+=\max(X-d,0). $$ The insured retains the part of the ground-up loss not paid by the insurer. This definition assumes that the deductible applies once to the claim and that $u$ caps the insurer’s payment, rather than the ground-up loss.

Payment by Loss Range

The positive-part expression $(X-d)^+$ means that the insurer does not pay when the loss is at or below the deductible. Once the loss exceeds $d$, the amount above the deductible is potentially payable. The payment limit $u$ caps that amount. Therefore, the payment function can also be written piecewise:

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

When $u>0$, the payment starts increasing just above $d$ and reaches its cap at $d+u$. At exactly $X=d$, payment is zero. At exactly $X=d+u$, payment is $u$; the same capped amount applies to larger losses. If $u=0$, payment is zero for every loss.

The phrase ordinary deductible matters at the threshold: payment does not begin until the loss exceeds the deductible. The function is continuous at $d$, since both the zero-payment portion and the increasing portion give payment zero there. A deductible is not itself the insurer’s payment limit; it is the amount of each claim that is not paid by the insurer.

Worked Example: Applying a Deductible and a Payment Cap

A hypothetical equipment policy has a per-claim deductible of $d=\$750$ and a maximum insurer payment of $u=\$4{,}000$. A covered ground-up claim is $X=\$6{,}150$. Find the insurer payment and the amount retained by the insured.

Model and assumptions. Use the ordinary per-claim payment function $Y^L=\min((X-d)^+,u)$. The loss amount is stipulated, so no severity distribution or mortality assumption is needed. There is no interest or discounting; all amounts are evaluated at the same claim settlement basis.

Step 1: Apply the deductible. The loss above the deductible is $$ X-d=\$6{,}150-\$750=\$5{,}400. $$ Since this is positive, $(X-d)^+=\$5{,}400$.

Step 2: Apply the insurer payment limit. $$ Y^L=\min(\$5{,}400,\$4{,}000)=\$4{,}000. $$

Step 3: Reconcile the loss. The insured retains $$ X-Y^L=\$6{,}150-\$4{,}000=\$2{,}150. $$ This retained amount consists of the $\$750$ deductible plus $\$1{,}400$ above the insurer’s payment cap. The check is $\$4{,}000+\$2{,}150=\$6{,}150$, equal to the ground-up loss.

Method and mistake avoided. The method is to calculate the excess above the deductible before taking the minimum with the payment cap. Subtracting the cap from the ground-up loss, or treating the cap as the amount paid before applying the deductible, would not implement this contract.

Why the Minimum Formula Holds

The formula follows directly from the contract’s two restrictions on insurer payment. First, the ordinary deductible prevents payment of the first $d$ dollars of a claim, so no more than $(X-d)^+$ is payable. Second, the payment limit prevents the insurer from paying more than $u$. A payment subject to both restrictions must be no greater than either amount. Under the stated contract, the insurer pays the full excess above the deductible until that excess reaches $u$, and pays $u$ thereafter. Thus the payment is the lesser of the two amounts.

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

These cases also verify the piecewise representation. The threshold for reaching the insurer payment cap is $X=d+u$, not $X=u$: the loss must first include the deductible amount before $u$ dollars are available for insurer payment.

When the Contract Limit Is Stated on Ground-Up Loss

A contract may instead state a maximum covered ground-up loss $U$. That is a different description of the limit from a maximum insurer payment $u$. If $U>d$, the most the insurer can pay is the covered ground-up amount above the deductible, $U-d$. The payment function for a ground-up limit is therefore

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

If $U\leq d$, then $(U-d)^+=0$, so the formula gives no insurer payment. For $U>d$, the formula also caps payment at $U-d$ when $X$ exceeds $U$. This conversion is useful only when the limit is in fact stated on ground-up loss; it should not be applied to a limit already stated as the insurer’s maximum payment.

Worked Example: Converting a Ground-Up Limit

A hypothetical property contract has a per-claim deductible of $d=\$1{,}500$ and a maximum covered ground-up loss of $U=\$10{,}000$. Calculate insurer payments for losses of $\$900$, $\$6{,}000$, and $\$12{,}400$.

Model and assumptions. The limit $U$ applies to ground-up loss, not directly to insurer payment. The maximum payment is $U-d=\$10{,}000-\$1{,}500=\$8{,}500$. Each loss is a stipulated amount; no severity distribution, mortality assumption, interest, or discounting is involved.

Loss of $\$900$. Since $X\leq d$, the loss does not exceed the deductible: $$ Y^L=\min\bigl((\$900-\$1{,}500)^+,\$8{,}500\bigr) =\min(\$0,\$8{,}500)=\$0. $$

Loss of $\$6{,}000$. The amount above the deductible is $\$6{,}000-\$1{,}500=\$4{,}500$, below the $\$8{,}500$ payment cap: $$ Y^L=\min(\$4{,}500,\$8{,}500)=\$4{,}500. $$

Loss of $\$12{,}400$. The amount above the deductible is $\$12{,}400-\$1{,}500=\$10{,}900$, which exceeds the payment cap: $$ Y^L=\min(\$10{,}900,\$8{,}500)=\$8{,}500. $$ The insured retains $\$12{,}400-\$8{,}500=\$3{,}900$: the $\$1{,}500$ deductible plus $\$2{,}400$ above the ground-up limit. The reconciliation $\$8{,}500+\$3{,}900=\$12{,}400$ checks the result.

Method and mistake avoided. Convert the ground-up limit to a maximum insurer payment by subtracting the deductible. Using $\$10{,}000$ as the payment cap would overstate the insurer’s payment, because the contract’s $\$10{,}000$ limit includes the deductible portion.

Comparing Payment Functions

When comparing contracts, evaluate each contract using its own deductible and its own payment limit. A contract with a smaller deductible does not necessarily produce a larger payment for every possible loss if its payment cap is also smaller. The breakpoints $d$ and $d+u$ make that comparison manageable: below $d$, payment is zero; between the breakpoints, payment increases dollar for dollar; at and above $d+u$, payment is capped.

Worked Example: Comparing Two Ordinary Deductible Contracts

A hypothetical contractor compares two options for a covered claim. Contract A has $d_A=\$1{,}200$ and payment limit $u_A=\$4{,}800$. Contract B has $d_B=\$2{,}000$ and payment limit $u_B=\$6{,}000$. Compare their payments at a ground-up loss of $\$7{,}400$, and identify where the payment functions reach their caps.

Model and assumptions. For each contract, $Y^L=\min((X-d)^+,u)$. The loss is deterministic, with no severity distribution, mortality assumption, interest, or discounting.

Contract A. Its cap threshold is $d_A+u_A=\$1{,}200+\$4{,}800=\$6{,}000$. Since $\$7{,}400\geq\$6{,}000$, payment is capped: $$ Y_A^L=\min(\$7{,}400-\$1{,}200,\$4{,}800) =\min(\$6{,}200,\$4{,}800)=\$4{,}800. $$

Contract B. Its cap threshold is $d_B+u_B=\$2{,}000+\$6{,}000=\$8{,}000$. Since $\$2{,}000<\$7{,}400<\$8{,}000$, payment is above the deductible but below the cap: $$ Y_B^L=\min(\$7{,}400-\$2{,}000,\$6{,}000) =\min(\$5{,}400,\$6{,}000)=\$5{,}400. $$

At this loss, Contract B pays $\$600$ more, even though its deductible is higher, because its payment limit is also higher. Contract A reaches its cap at a ground-up loss of $\$6{,}000$; Contract B reaches its cap at $\$8{,}000$. The method avoids judging the contracts by deductible alone.

Common Mistakes and Checks

Applying the deductible after the limit. For a payment limit $u$, calculate $(X-d)^+$ first, then take the minimum with $u$.
Confusing limit bases. A cap on insurer payment is $u$; a ground-up limit $U$ generally corresponds to a maximum payment of $U-d$, when $U>d$.
Paying at or below the deductible. Under an ordinary deductible, $X=d$ still gives zero payment. Payment starts only when $X>d$.
Using $X=u$ as the cap threshold. For a payment limit $u$, the payment reaches its cap when $X=d+u$.
Forgetting to reconcile. For one covered loss with no other contract features, insurer payment plus the insured-retained amount must equal $X$.

Key takeaway: Under an ordinary per-claim deductible and a limit on insurer payment, $Y^L=\min((X-d)^+,u)$. Establish whether the limit is on insurer payment or on ground-up loss before substituting values.

Check Your Understanding

For each problem, use the stated contract terms and check the payment against the loss amount where appropriate.

A hypothetical appliance policy has $d=\$900$, a maximum insurer payment of $u=\$3{,}500$, and a claim of $\$5{,}200$. Find the insurer payment and the insured-retained amount. (Answer: insurer payment $\$3{,}500$; insured retains $\$1{,}700$.)
A hypothetical workshop policy has deductible $d=\$1{,}600$ and ground-up limit $U=\$9{,}000$. Find insurer payments for losses of $\$1{,}200$, $\$6{,}500$, and $\$11{,}000$. (Answer: $\$0$, $\$4{,}900$, and $\$7{,}400$, respectively.)
A contract has $d=\$500$ and payment limit $u=\$2{,}500$. For a loss of $\$3{,}000$, state the payment and whether the cap has been reached. (Answer: $\$2{,}500$; yes, the loss equals $d+u=\$3{,}000$.)
Contract A has $d=\$1{,}000$, $u=\$4{,}000$; Contract B has $d=\$1{,}800$, $u=\$5{,}500$. For a ground-up loss of $\$6{,}200$, which pays more, and by how much? (Answer: Contract A pays $\$4{,}000$, Contract B pays $\$4{,}400$; Contract B pays $\$400$ more.)
A contract has $d=\$1{,}250$ and payment limit $u=\$4{,}750$. A claim produces an insurer payment of $\$3{,}300$, with neither deductible nor limit disputed. Find the ground-up loss. (Answer: $\$4{,}550$, since the payment is below the cap and $X-d=\$3{,}300$.)

What You'll Learn