Coinsurance With a Policy Limit

Why a Coinsurance Payment Can Have a Cap

Coinsurance makes the insurer responsible for a stated proportion of an eligible loss. A contract may also restrict the insurer’s total payment to a maximum dollar amount. These provisions work together: the coinsurance share determines how quickly the insurer’s payment grows, while the maximum payment stops that growth once the cap is reached.

Here, the contract applies coinsurance to the amount of loss above an ordinary deductible, then caps the insurer’s payment. The threshold is not the deductible plus the cap: because the insurer pays only a proportion of the amount above the deductible, a larger amount of loss may be needed to reach the cap.

Definition: Let $X\ge 0$ be the loss, $d\ge 0$ the ordinary deductible, $c$ the insurer’s coinsurance proportion with $0<c\le 1$, and $u>0$ the maximum insurer payment per loss. The insurer’s payment per loss is $$ Y^L=\min\{c(X-d)_+,u\}, \qquad (X-d)_+=\max\{X-d,0\}. $$ The deductible is applied first, the insurer pays proportion $c$ of the amount above it, and the maximum payment $u$ is then applied.

The parameter $c$ is the insurer’s share, not the policyholder’s share. For example, if the insurer pays 80% of eligible loss, then $c=0.80$. The remaining share of eligible loss is not insured by the coinsurance provision, and losses below the deductible generate no payment. This tutorial uses $u$ for a cap on the insurer’s payment; it is not a limit on the amount of loss considered.

Payment Function and Cap Threshold

For a fixed loss amount $X$, first compare it with the deductible. If $X\le d$, there is no payment. If $X>d$, the uncapped payment is $c(X-d)$. The cap applies when this uncapped amount is at least $u$.

c(X-d)\ge u \quad\Longleftrightarrow\quad X\ge d+\frac{u}{c}.

Define $T=d+u/c$. This is the loss threshold at which the insurer’s payment reaches the maximum. Since $c>0$, dividing by $c$ preserves the inequality. At $X=T$, the uncapped payment is exactly $u$; for any larger loss it would exceed $u$, so the cap holds the payment at $u$.

Y^L= \begin{cases} 0, & 0\le X\le d,\\ c(X-d), & d<X<T,\\ u, & X\ge T, \end{cases} \qquad T=d+\frac{u}{c}.

The cases agree at their boundaries. At $X=d$, the middle expression would be $c(d-d)=0$. At $X=T$, it would be $c(T-d)=c(u/c)=u$. Thus including equality in either adjoining case would give the same payment. The stated intervals simply make the threshold convention explicit.

Apply the deductible.
Find the amount eligible for coinsurance: $(X-d)_+$.

Apply the insurer’s share.
Multiply the eligible amount by $c$ to obtain the uncapped payment.

Apply the payment cap.
Take the smaller of the uncapped payment and $u$.

Locate the threshold if needed.
Solve $c(X-d)=u$, giving $T=d+u/c$.

Why the Threshold Formula Holds

The threshold follows directly from the point at which the uncapped payment reaches the contractual maximum. For losses above the deductible, but before the cap binds, each additional dollar of loss produces $c$ dollars of insurer payment. The insurer needs a payment of $u$, so the eligible loss required is $u/c$, not $u$. Adding the deductible gives the total loss threshold.

\text{eligible loss needed}=\frac{\text{maximum insurer payment}}{\text{insurer share}} =\frac{u}{c}, \qquad \text{total loss threshold}=d+\frac{u}{c}.

This derivation assumes the contract applies the insurer’s proportion to loss above the deductible and caps the resulting insurer payment. If a contract instead specifies a limit on covered loss, the limit acts on a different quantity, so the payment function and threshold may differ. Contract wording determines which interpretation is appropriate.

Worked Example: Calculate Payments Below and Above the Cap

A hypothetical property policy has a $500 deductible, pays 80% of loss above the deductible, and has a maximum insurer payment of $2,400 per loss. Find the loss threshold for the cap and the insurer’s payment for losses of $2,750 and $4,000.

Method and threshold. Here $d=500$, $c=0.80$, and $u=2{,}400$. The loss threshold is $$ T=d+\frac{u}{c} =500+\frac{2{,}400}{0.80} =500+3{,}000 =3{,}500. $$

Loss of $2,750. The amount above the deductible is $2{,}750-500=2{,}250$. The uncapped insurer payment is $0.80(2{,}250)=1{,}800$. Since $1{,}800<2{,}400$, the cap does not reduce the payment. Thus $Y^L=\$1{,}800.00$.

Loss of $4,000. The amount above the deductible is $4{,}000-500=3{,}500$. The uncapped payment is $0.80(3{,}500)=2{,}800$, which exceeds the $2,400 maximum. Thus $Y^L=\$2{,}400.00$.

Check and common mistake avoided. The first loss is below the $3,500 threshold and receives less than the cap; the second is above it and receives exactly the cap. Adding the deductible directly to the cap would give $2,900, an incorrect threshold because it ignores the insurer’s 80% share.

Reading the Contract and Checking Boundary Cases

The payment function is a pointwise rule: for each realized loss, use that loss as $X$ and calculate the payment. If $X$ is modeled as a random severity, the same rule applies to each possible value of $X$. No probability distribution is needed to derive the function or the threshold.

A reliable check is to calculate the uncapped payment at the proposed threshold. If it does not equal $u$, the threshold calculation or the contract interpretation is wrong. Also check the directions of the inequalities: losses below the deductible pay zero, losses between the deductible and threshold have a payment that increases with $X$, and losses at or beyond the threshold pay $u$.

Worked Example: Confirm the Payment at the Threshold

A hypothetical equipment policy has a $1,000 deductible, an insurer share of 60%, and a maximum insurer payment of $3,000. Calculate the cap threshold, then calculate payments for losses of $5,900, $6,000, and $7,000.

Method and threshold. Set the uncapped payment equal to the cap: $$ T=1{,}000+\frac{3{,}000}{0.60} =1{,}000+5{,}000 =6{,}000. $$

Loss of $5,900. The amount above the deductible is $5{,}900-1{,}000=4{,}900$. The uncapped payment is $0.60(4{,}900)=2{,}940$, below the cap, so $Y^L=\$2{,}940.00$.

Loss of $6,000. The amount above the deductible is $6{,}000-1{,}000=5{,}000$. The payment is $0.60(5{,}000)=3{,}000$, exactly the cap, so $Y^L=\$3{,}000.00$.

Loss of $7,000. The amount above the deductible is $7{,}000-1{,}000=6{,}000$. The uncapped payment is $0.60(6{,}000)=3{,}600$, so the capped payment is $Y^L=\$3{,}000.00$.

Check and common mistake avoided. At the threshold, the payment is exactly the maximum; above it, the payment remains fixed. Using $T=d+u=4{,}000$ would be incorrect: at a $4,000 loss the insurer’s uncapped payment is only $0.60(4{,}000-1{,}000)=\$1{,}800$, not $3,000.

A Payment Cap Is Not a Loss Limit

A payment cap sets the largest amount the insurer pays. It does not make the loss itself stop at the threshold, and it does not mean that the insurer pays coinsurance on only the first $u$ dollars above the deductible. Once the insurer payment reaches $u$, any further increase in loss leaves the payment unchanged at $u$.

For comparison, suppose a hypothetical contract instead limited the amount of loss above the deductible eligible for coinsurance to $u$. Under that different wording, the insurer payment would be $c\min\{(X-d)_+,u\}$, whose maximum would be $cu$, not $u$. In the payment-cap contract taught here, the function is $\min\{c(X-d)_+,u\}$, whose maximum is $u$. Confusing these formulas changes both the maximum payment and the loss threshold.

Worked Example: Apply the Insurer Share Before the Payment Cap

A hypothetical repair policy has a $400 deductible, pays 50% of loss above the deductible, and limits the insurer’s payment to $1,800. For a loss of $4,500, find the insurer payment and the threshold at which the cap begins.

Method and threshold. The cap applies to the insurer payment, so $$ T=400+\frac{1{,}800}{0.50} =400+3{,}600 =4{,}000. $$

Payment for the stated loss. The eligible loss is $4{,}500-400=4{,}100$. The uncapped insurer payment is $0.50(4{,}100)=2{,}050$. Since this exceeds $1,800, the insurer pays $Y^L=\$1{,}800.00$.

Check and common mistake avoided. The loss of $4,500 exceeds the $4,000 threshold, so the cap must bind. Capping the eligible loss at $1,800 first and then applying 50% would produce $900, but that calculation treats $1,800 as a limit on eligible loss, contrary to the stated maximum insurer payment.

Common Exam Traps

Using the cap as if it were the eligible-loss limit. A maximum insurer payment $u$ is reached when $c(X-d)=u$, not when $X-d=u$.
Forgetting the deductible. The threshold is the amount needed above the deductible plus the deductible itself: $d+u/c$.
Using the policyholder share in place of $c$. If the insurer pays 70%, use $c=0.70$, even though the policyholder bears the remaining 30% of eligible loss.
Applying the cap before the coinsurance proportion. For a payment cap, the correct order is deductible, insurer proportion, then maximum payment.
Assuming payment keeps increasing after the threshold. Once $X\ge T$, the insurer payment remains $u$, regardless of how much larger the loss becomes.

Key takeaway: For a deductible $d$, insurer coinsurance share $c$, and maximum insurer payment $u$, the payment is $Y^L=\min\{c(X-d)_+,u\}$. The cap begins at loss $T=d+u/c$, where the uncapped insurer payment first equals $u$.

Check Your Understanding

For each hypothetical contract, apply the deductible, insurer share, and payment cap in that order.

A policy has $d=\$300$, $c=0.80$, and $u=\$1{,}600$. Find the cap threshold and payment for $X=\$2{,}100$. (Answer: threshold $\$2{,}300$; payment $\$1{,}440.00$)
A policy has $d=\$500$, $c=0.50$, and $u=\$1{,}250$. Find the threshold and payment when $X=\$3{,}000$. (Answer: threshold $\$3{,}000$; payment $\$1{,}250.00$)
A policy has $d=\$1{,}200$, $c=0.70$, and $u=\$2{,}800$. Find the threshold and payment for $X=\$4{,}000$. (Answer: threshold $\$5{,}200$; payment $\$1{,}960.00$)
A policy has $d=\$600$, $c=0.60$, and $u=\$2{,}400$. Determine whether the cap applies to losses of $\$4{,}500$ and $\$5{,}000$, and give each payment. (Answer: threshold $\$4{,}600$; no cap at $\$4{,}500$, payment $\$2{,}340.00$; cap applies at $\$5{,}000$, payment $\$2{,}400.00$)
A policy has $d=\$800$, $c=0.40$, and $u=\$1{,}600$. For a loss of $\$6{,}000$, find the threshold, insurer payment, and amount of loss remaining with the policyholder after that payment. (Answer: threshold $\$4{,}800$; insurer payment $\$1{,}600.00$; policyholder amount $\$4{,}400.00$)

What You'll Learn