Applying Coinsurance With Deductibles And Limits

Why the Order of Coverage Terms Matters

A claim may be subject to a deductible, a coinsurance percentage, and a policy limit. Each term affects the amount ultimately paid, but the order and the amount to which each term applies depend on the contract. In particular, a limit on the insurer’s payment is not necessarily the same as a limit on the loss amount eligible for coinsurance.

This tutorial considers hypothetical per-loss contracts. For each example, the contract wording is stated explicitly before the calculation. The focus is how to translate that wording into a payment rule, not to assume that every policy applies the terms in the same order. The methods are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $X$ be the ground-up loss for one claim, $d$ the per-loss deductible, and $c$ the insurer’s coinsurance percentage, with $0<c\leq 1$. The insured’s coinsurance percentage is $1-c$. The amount remaining after a deductible is $(X-d)_+=\max(X-d,0)$. A limit $u$ may cap the insurer’s payment or, under different wording, cap the loss amount to which coinsurance applies. Those are distinct contract provisions.

Deductible, Coinsurance, and an Insurer-Payment Limit

First consider contract wording that says: subtract the deductible from each loss; the insurer pays percentage $c$ of the remaining amount; and the insurer’s payment is capped at $u$. With this order, the insurer’s payment per loss, denoted by $Y^L$, is

Y^L=\min\left\{c(X-d)_+,u\right\}.

The formula makes each step visible. The positive-part expression prevents a negative payment when the loss does not exceed the deductible. The coinsurance percentage is applied to the post-deductible amount. Finally, the minimum imposes a maximum on the insurer’s calculated payment.

Assume $d\geq 0$, $u\geq 0$, and $c>0$. If $u>0$, the insurer-payment limit starts to bind when the uncapped payment reaches $u$, or when $c(X-d)=u$. Solving for the loss gives the threshold $X=d+u/c$.

Y^L= \begin{cases} 0, & X\leq d,\\ c(X-d), & d<X\leq d+\dfrac{u}{c},\\ u, & X>d+\dfrac{u}{c}. \end{cases}

At the threshold, the uncapped payment equals the limit, so either adjacent expression gives the same payment. If $u=0$, the insurer’s payment is zero for every loss under this formula. The uncapped middle region is absent if the limit is reached immediately after the deductible.

Why the Payment Formula Holds

The formula follows directly from the contract’s sequence. For $X\leq d$, no loss remains after the deductible, so the insurer pays zero. For $X>d$, the post-deductible loss is $X-d$, and the insurer’s share before applying the limit is $c(X-d)$. A payment cap of $u$ means the final payment cannot exceed either that calculated share or $u$; therefore it is the smaller of the two amounts.

This reasoning also explains why the limit cannot be applied casually to a different quantity. If the contract instead caps the post-deductible loss eligible for coinsurance at $u$, the payment rule is

Y^L=c\min\left\{(X-d)_+,u\right\}.

Under that wording, the insurer pays a percentage of the capped eligible loss. The maximum payment is $cu$, not $u$. Under an insurer-payment limit, the maximum payment is $u$. To choose the correct formula, identify exactly what the contract says the limit caps.

Identify the loss basis.
Confirm that the deductible and limit apply per loss, rather than to an occurrence or a policy-period total.

Apply the deductible.
Calculate $(X-d)_+$, the loss remaining after the deductible.

Apply coinsurance.
Multiply the eligible post-deductible amount by the insurer’s percentage $c$.

Apply the stated limit.
Cap the insurer’s payment or the eligible loss, whichever quantity the wording specifies.

Check the allocation.
For a single loss with no other payment, verify that the insurer’s payment plus the insured’s retained loss equals $X$.

Worked Calculations

Worked Example: A Payment Limit That Binds

A hypothetical policy covers a loss of $X=\$5{,}000$ per claim. The contract applies a $\$500$ deductible first, then has the insurer pay 80% of the remaining loss, subject to a maximum insurer payment of $\$3{,}000$. Calculate the insurer’s payment and the amount retained by the insured.

Method. Apply the deductible, calculate the insurer’s coinsurance share, and then cap that payment. This method avoids treating the $\$3{,}000$ as a cap on the post-deductible loss.

The post-deductible loss is

X-d=\$5{,}000-\$500=\$4{,}500.

The uncapped insurer payment is

c(X-d)=0.80(\$4{,}500)=\$3{,}600.

Because $\$3{,}600$ exceeds the insurer-payment limit of $\$3{,}000$, the actual payment is

Y^L=\min(\$3{,}600,\$3{,}000)=\$3{,}000.

The insured retains $\$5{,}000-\$3{,}000=\$2{,}000$. To check the components: the retained amount consists of the $\$500$ deductible, the insured’s 20% coinsurance share of the $\$4{,}500$ post-deductible loss ($\$900$), and the $\$600$ by which the cap reduces the otherwise-calculated insurer payment ($\$3{,}600-\$3{,}000$). Thus $\$500+\$900+\$600=\$2{,}000$, and payment plus retention equals the original $\$5{,}000$ loss.

Worked Example: Two Different Meanings of a Limit

A hypothetical claim has ground-up loss $X=\$10{,}000$, a $\$1{,}000$ deductible, and a 70% insurer coinsurance percentage. Compare two possible wordings, each using a limit of $\$5{,}000$: (A) a maximum insurer payment of $\$5{,}000$; and (B) a maximum post-deductible loss of $\$5{,}000$ eligible for coinsurance.

Method. Calculate the post-deductible loss once, then apply each limit to the quantity specified. The main mistake to avoid is using the same formula for both types of limit.

The post-deductible loss is

(X-d)_+=\$10{,}000-\$1{,}000=\$9{,}000.

Wording A: limit on the insurer’s payment. The uncapped insurer share is $0.70(\$9{,}000)=\$6{,}300$. Therefore,

Y^L=\min(\$6{,}300,\$5{,}000)=\$5{,}000.

The insured retains $\$10{,}000-\$5{,}000=\$5{,}000$. This consists of the $\$1{,}000$ deductible, the insured’s 30% share of the $\$9{,}000$ post-deductible loss ($\$2{,}700$), and the $\$1{,}300$ reduction from the otherwise-calculated insurer payment caused by the cap ($\$6{,}300-\$5{,}000$). These components total $\$5{,}000$.

Wording B: limit on eligible post-deductible loss. Only $\$5{,}000$ of the $\$9{,}000$ post-deductible loss is eligible for coinsurance, so

Y^L=0.70\min(\$9{,}000,\$5{,}000)=\$3{,}500.

The insured retains $\$10{,}000-\$3{,}500=\$6{,}500$. This consists of the $\$1{,}000$ deductible, the 30% share of the $\$5{,}000$ eligible amount ($\$1{,}500$), and $\$4{,}000$ of post-deductible loss beyond the eligible-loss cap ($\$9{,}000-\$5{,}000$). The components total $\$6{,}500$, and the payment plus retention equals $\$10{,}000$.

Worked Example: Finding the Loss Threshold and Calculating Two Payments

A hypothetical policy applies a $\$250$ per-loss deductible, pays 80% of the post-deductible loss, and caps the insurer’s payment at $\$2{,}000$. For losses of $\$1{,}500$ and $\$4{,}000$, find the payment for each loss and the total payment. Assume the limit applies separately to each loss.

Method. Find the threshold at which the uncapped payment reaches the limit, then calculate the two claim payments separately. This avoids accidentally treating a per-loss limit as a combined limit.

The threshold loss is

d+\frac{u}{c} =\$250+\frac{\$2{,}000}{0.80} =\$250+\$2{,}500 =\$2{,}750.

For the $\$1{,}500$ loss, the post-deductible amount is $\$1{,}500-\$250=\$1{,}250$. Its uncapped payment is $0.80(\$1{,}250)=\$1{,}000$, which is below the limit. The payment is therefore $\$1{,}000$, and the insured retains $\$1{,}500-\$1{,}000=\$500$.

For the $\$4{,}000$ loss, the post-deductible amount is $\$4{,}000-\$250=\$3{,}750$. Its uncapped payment is $0.80(\$3{,}750)=\$3{,}000$, which exceeds the limit. The payment is $\$2{,}000$, and the insured retains $\$4{,}000-\$2{,}000=\$2{,}000$.

The total payment is $\$1{,}000+\$2{,}000=\$3{,}000$. The total loss is $\$1{,}500+\$4{,}000=\$5{,}500$, and total retention is $\$500+\$2{,}000=\$2{,}500$. The check is $\$3{,}000+\$2{,}500=\$5{,}500$. The second claim’s retained $\$2{,}000$ can also be reconciled as a $\$250$ deductible, a $\$750$ coinsurance share, and a $\$1{,}000$ reduction in the otherwise-calculated insurer payment due to the cap.

Common Mistakes and Checks

The most reliable way to avoid ordering errors is to write the contract’s sequence in words before substituting numbers. A limit described as a maximum insurer payment belongs outside the coinsurance calculation. A limit on eligible loss belongs inside it. Confusing these positions can materially change the payment, as the second worked example demonstrates.

Applying the percentage to the wrong amount. If the deductible is applied first, multiply $c$ by the post-deductible amount, not by the ground-up loss.
Using the insured’s percentage as the insurer’s percentage. If the insurer pays 80%, then $c=0.80$ and the insured’s share before any limit effect is 20%.
Calling all unpaid post-deductible loss a cap reduction. The cap reduction is the uncapped insurer payment minus the actual capped payment. It is not the entire post-deductible amount less the actual payment.
Applying a per-loss limit to a total across claims. Apply it to each claim when the contract says it is per loss. Combining payments under a policy-period aggregate is a separate coverage feature.
Ignoring the conservation check. For a single loss with no other payment or recovery, the insurer’s payment plus the insured’s retained amount must equal the ground-up loss.

For a payment limit, a useful additional check is that the payment cannot exceed $u$ and cannot exceed $c(X-d)_+$. For a limit on eligible loss, the payment cannot exceed $cu$. These checks follow from the respective formulas and can expose a misread contract term.

Key takeaway: Apply a deductible, coinsurance percentage, and limit only to the quantities specified by the contract. For a deductible-first contract with an insurer-payment limit, use $Y^L=\min\{c(X-d)_+,u\}$. If the limit instead caps the eligible post-deductible loss, use $Y^L=c\min\{(X-d)_+,u\}$.

Check Your Understanding

For each hypothetical contract, identify what the limit caps and calculate the requested amount. Assume all limits are per loss unless stated otherwise.

A loss is $\$2{,}400$. The insurer pays 75% after a $\$400$ deductible, subject to a maximum insurer payment of $\$1{,}500$. Find the payment and retention. (Answer: payment $\$1{,}500$; retention $\$900$.)
A loss is $\$6{,}000$. The insurer pays 80% after a $\$1{,}000$ deductible, subject to a maximum insurer payment of $\$3{,}500$. Find the payment and retention. (Answer: payment $\$3{,}500$; retention $\$2{,}500$.)
A loss is $\$7{,}000$. After a $\$500$ deductible, only $\$3{,}000$ of post-deductible loss is eligible for coinsurance; the insurer pays 60% of that eligible amount. Find the payment and retention. (Answer: payment $\$1{,}800$; retention $\$5{,}200$.)
A contract has a $\$600$ deductible, a 75% insurer coinsurance percentage, and a maximum insurer payment of $\$2{,}700$. Find the loss at which the payment limit first binds, then find the payment and retention for a $\$5{,}000$ loss. (Answer: threshold $\$4{,}200$; payment $\$2{,}700$; retention $\$2{,}300$.)
A loss is $\$12{,}000$, the deductible is $\$2{,}000$, and the insurer pays 80%. Compare a $\$6{,}000$ maximum insurer payment with a $\$6{,}000$ cap on post-deductible loss eligible for coinsurance. Give both payments and their difference. (Answer: insurer-payment limit gives $\$6{,}000$; eligible-loss limit gives $\$4{,}800$; difference $\$1{,}200$.)

What You'll Learn