Workshop: Calculate A Claim Under A Multi-Term Policy

Introduction

A claim can pass through several policy terms before the insurer’s payment is determined. A deductible may leave an initial portion of the loss with the policyholder; coinsurance may divide the remaining amount; and a per-loss limit may cap the insurer’s payment. The calculation is reliable only when the terms are applied in the order specified by the contract.

This tutorial considers a hypothetical single covered loss and a contract that applies the deductible first, calculates the insurer’s coinsurance share of the amount above the deductible second, and then applies a limit to the insurer’s payment. No interest rate, mortality assumption, or probability distribution is needed: each example is a deterministic claim calculation. The methods are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Definition: Let $X=x$ be the amount of a single covered loss, $d$ the per-loss deductible, $c$ the insurer’s coinsurance percentage, and $u$ the maximum insurer payment per loss. Assume $d\geq 0$, $u\geq 0$, and $0<c\leq 1$. The contract considered here applies $d$ first, makes the insurer responsible for share $c$ of the amount above $d$, and caps the insurer’s payment at $u$. The insurer payment per loss is denoted by $Y^L$.

The Payment Formula and Contract Order

The amount remaining after the deductible is $(x-d)_+=\max(x-d,0)$. Before the limit is applied, the insurer’s share is $c(x-d)_+$. The limit caps that payment, not the loss itself. Thus the payment is the smaller of the uncapped insurer amount and $u$.

Y^L=\min\bigl(c(x-d)_+,\,u\bigr).

When the contract says the coinsurance percentage is the policyholder’s share rather than the insurer’s, first convert it: if the policyholder pays proportion $q$ of the post-deductible amount, then the insurer’s share is $c=1-q$. Using the wrong share reverses the allocation.

The insurer’s payment is not the only amount of interest. For a covered loss with no other payments or adjustments, the policyholder’s total out-of-pocket amount is $x-Y^L$. Below the deductible this is the whole loss. Above the deductible and before the limit binds, it includes the deductible and the policyholder’s coinsurance share. After the insurer payment reaches $u$, the policyholder is also responsible for any additional loss.

Important contract condition: The formula above assumes that $u$ limits the insurer’s payment after coinsurance. Some contracts instead define a limit on covered loss before coinsurance, or specify another sequence. Those terms can produce a different result, so state the assumed order before calculating.

Why the Breakpoint Has This Form

A breakpoint is the loss amount at which the uncapped insurer share first reaches the per-loss limit. The cap binds when $c(x-d)_+\geq u$. For $x>d$, solve the equality $c(x-d)=u$: divide by $c$, then add $d$. This gives the loss amount at which the payment reaches $u$.

c(x-d)=u \quad\Longrightarrow\quad x=d+\frac{u}{c}.

This breakpoint applies when $c>0$, as assumed here. It separates the linear part of the payment from the capped part. It does not mean that every loss below the breakpoint produces a positive payment: losses at or below the deductible still produce zero payment.

Result: For $d\geq 0$, $u\geq 0$, and $0<c\leq 1$, the single-loss payment is

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

For $d<x<d+u/c$, the insurer payment increases linearly with the loss. For $0\leq x\leq d$, it is zero. For $x\geq d+u/c$, it is $u$. At the breakpoint itself, $c(x-d)=u$, so either the uncapped expression or the capped expression gives the same payment. If $u=0$, the limit makes the payment zero for every loss; the compact minimum formula handles this case directly.

Read the terms.
Identify the deductible, whether the stated coinsurance share belongs to the insurer or policyholder, and what amount the limit caps.

Apply the deductible.
Calculate $(x-d)_+$. If $x\leq d$, the insurer payment is zero under this contract.

Calculate the insurer share.
Multiply the post-deductible amount by $c$, the insurer’s share.

Apply the per-loss limit.
Compare the uncapped amount with $u$. The insurer payment is the smaller value.

Reconcile the loss.
Subtract the insurer payment from $x$ to obtain the policyholder’s remaining share, if requested.

Worked Calculations

Worked Example: The Limit Binds After Coinsurance

A hypothetical policy has a per-loss deductible of $d=\$500$, an insurer coinsurance share of $c=0.80$, and a limit of $u=\$4{,}000$ on the insurer’s payment. A covered loss is $x=\$7{,}000$. Calculate the insurer payment and the policyholder’s amount. The method is to apply the terms in contract order; no interest, mortality, or probability model is assumed.

Step 1: Apply the deductible. The amount above the deductible is $\$7{,}000-\$500=\$6{,}500$.

Step 2: Calculate the uncapped insurer share. This is $0.80(\$6{,}500)=\$5{,}200$.

Step 3: Apply the limit. The insurer pays the smaller of $\$5{,}200$ and $\$4{,}000$, so $Y^L=\$4{,}000$. As a check, the breakpoint is $d+u/c=\$500+\$4{,}000/0.80=\$5{,}500$. Since $\$7{,}000\geq \$5{,}500$, the limit should bind.

Step 4: Reconcile. The policyholder’s amount is $\$7{,}000-\$4{,}000=\$3{,}000$. This consists of the $\$500$ deductible, $20\%$ of the first $\$5{,}000$ above the deductible ($\$1{,}000$), and the $\$1{,}500$ loss above the $\$5{,}500$ breakpoint: $\$500+\$1{,}000+\$1{,}500=\$3{,}000$. The reconciliation agrees.

Common mistake avoided: Stopping at the uncapped amount of $\$5{,}200$ ignores the per-loss payment limit.

Worked Example: A Loss Below the Limit Breakpoint

A hypothetical contract has $d=\$1{,}200$, an insurer share of $c=0.75$, and a per-loss insurer payment limit of $u=\$6{,}000$. A covered loss is $\$7{,}000$. Find the insurer payment and the policyholder’s amount. The calculation is deterministic; no interest, mortality, or loss distribution is assumed.

Step 1: Find the amount above the deductible. $\$7{,}000-\$1{,}200=\$5{,}800$.

Step 2: Calculate the uncapped insurer share. $0.75(\$5{,}800)=\$4{,}350$.

Step 3: Check the limit. The uncapped payment, $\$4{,}350$, is less than $u=\$6{,}000$, so the insurer pays $\$4{,}350$. Equivalently, the breakpoint is $\$1{,}200+\$6{,}000/0.75=\$9{,}200$, and the loss of $\$7{,}000$ is below it but above the deductible.

Step 4: Calculate the policyholder’s amount. $\$7{,}000-\$4{,}350=\$2{,}650$. The same amount is the $\$1{,}200$ deductible plus $25\%$ of the $\$5{,}800$ post-deductible loss: $\$1{,}200+\$1{,}450=\$2{,}650$. The two calculations agree.

Common mistake avoided: Do not assume that the limit always reduces the payment. Here it does not bind because the insurer’s calculated share is below the limit.

Worked Example: Separate the Deductible, Coinsurance, and Limit Effects

A hypothetical policy applies a deductible of $d=\$2{,}500$, an insurer share of $c=0.60$, and a limit of $u=\$3{,}000$ on the insurer’s payment. The covered loss is $\$8{,}500$. Calculate both parties’ amounts, checking the loss at which the limit begins to bind.

Step 1: Apply the deductible. The post-deductible amount is $\$8{,}500-\$2{,}500=\$6{,}000$.

Step 2: Calculate the uncapped insurer share. $0.60(\$6{,}000)=\$3{,}600$.

Step 3: Apply the limit and check the breakpoint. The insurer pays $\min(\$3{,}600,\$3{,}000)=\$3{,}000$. The breakpoint is $d+u/c=\$2{,}500+\$3{,}000/0.60=\$7{,}500$. Because $\$8{,}500$ exceeds $\$7{,}500$, the limit binds.

Step 4: Find the policyholder’s amount. The remainder is $\$8{,}500-\$3{,}000=\$5{,}500$. Check the components: the deductible is $\$2{,}500$; the policyholder’s $40\%$ share of the first $\$5{,}000$ above the deductible is $\$2{,}000$; and the loss above the breakpoint is $\$8{,}500-\$7{,}500=\$1{,}000$. Their sum is $\$5{,}500$.

Common mistake avoided: After the limit binds, the insurer does not continue to pay $60\%$ of additional loss. The payment remains at $\$3{,}000$.

Worked Example: Infer the Loss From an Uncapped Payment

A hypothetical policy has deductible $d=\$900$, insurer share $c=0.70$, and per-loss insurer payment limit $u=\$3{,}500$. The insurer’s payment on one covered loss is $\$2{,}100$. Find the loss amount, assuming the payment is below the limit and the stated contract order applies. No probability model is assumed.

Step 1: Use the uncapped payment relation. Since the payment is below the limit, it equals $c(x-d)$. Thus the amount above the deductible is $\$2{,}100/0.70=\$3{,}000$.

Step 2: Add the deductible. The loss is $x=\$900+\$3{,}000=\$3{,}900$.

Step 3: Verify the assumption. The breakpoint is $\$900+\$3{,}500/0.70=\$5{,}900$. Since $\$3{,}900<\$5{,}900$ and exceeds the deductible, its uncapped payment is $0.70(\$3{,}900-\$900)=\$2{,}100$, as given. The inferred loss is consistent.

Common mistake avoided: Dividing by the insurer percentage gives the post-deductible amount, not the total loss; add the deductible afterward.

Common Traps and Checks

A multi-term claim calculation is short, but a small change in interpretation can change the result. In particular, “coinsurance of 20%” is incomplete unless the contract identifies which party bears that share. This tutorial uses $c$ for the insurer’s share. If the policyholder bears $20\%$, then $c=0.80$.

A second trap is applying the limit to the wrong amount. For example, suppose $d=\$1{,}000$, $c=0.80$, $u=\$3{,}000$, and $x=\$6{,}000$, with the limit applying to the insurer payment. The post-deductible amount is $\$5{,}000$, and the uncapped insurer share is $0.80(\$5{,}000)=\$4{,}000$. The correct payment is $\min(\$4{,}000,\$3{,}000)=\$3{,}000$. A wrong calculation caps the post-deductible loss first and then applies coinsurance: $0.80\min(\$5{,}000,\$3{,}000)=\$2{,}400$. That calculation treats $u$ as a limit on covered loss, which is not the contract assumed here.

A final check is to verify that the insurer’s payment never exceeds either the uncapped amount or the limit: $Y^L\leq c(x-d)_+$ and $Y^L\leq u$. Also verify that the insurer payment plus the policyholder remainder equals the covered loss when there are no other adjustments. These checks catch arithmetic errors, but they do not replace reading the contract order.

Key takeaway: For the stated contract, apply the deductible, multiply the amount above it by the insurer’s share, and cap that result at the per-loss insurer limit. The payment is zero through the deductible, increases linearly between the deductible and $d+u/c$, and remains at $u$ thereafter.

Check Your Understanding

For each hypothetical single-loss policy, use the stated order: deductible, insurer coinsurance share, then a limit on insurer payment. Unless a question asks otherwise, find the insurer payment.

A policy has $d=\$400$, $c=0.80$, and $u=\$2{,}400$. A covered loss is $\$2{,}400$. Calculate the insurer payment. (Answer: $\$1{,}600$)
A policy has $d=\$1{,}500$, $c=0.60$, and $u=\$3{,}000$. A covered loss is $\$8{,}000$. Find the insurer payment and the policyholder’s amount. (Answer: insurer $\$3{,}000$; policyholder $\$5{,}000$)
A policy has $d=\$750$, $c=0.90$, and $u=\$2{,}700$. A covered loss is $\$3{,}000$. Find the insurer payment and check whether the limit binds. (Answer: $\$2{,}025$; the limit does not bind)
A policy has $d=\$1{,}200$, $c=0.75$, and $u=\$5{,}000$. The insurer pays $\$3{,}300$ on a single loss, with the limit not binding. Infer the loss amount and verify that the assumption is consistent. (Answer: $\$5{,}600$; the breakpoint is approximately $\$7{,}866.67$, so the limit does not bind)
A policy has $d=\$2{,}000$, $c=0.80$, and $u=\$4{,}000$. A loss is exactly $\$7{,}000$. Find the payment and explain whether the loss is at, below, or above the breakpoint. (Answer: $\$4{,}000$; the breakpoint is $\$7{,}000$, so the payment is exactly at the limit)

What You'll Learn