What a Coinsurance Payment Percentage Means
A policy may state that the insurer pays a specified percentage of an otherwise covered loss. The percentage describes how the eligible loss is divided; it is not, by itself, a deductible or a limit. The first task is therefore to identify the amount to which the percentage applies. Only then should the insurer’s share be calculated.
For this tutorial, assume a simple coinsurance arrangement: there is no deductible, limit, or other payment adjustment, and the stated percentage applies directly to the otherwise covered loss. An otherwise covered loss is the portion of a claim that qualifies for coverage before applying this percentage. If a claim also contains an excluded item, the excluded amount is not part of that covered-loss base.
A percentage must be interpreted according to whose share it describes. If the policy says “the insurer pays 70%,” then \(c=0.70\), and the insured’s share of the otherwise covered loss is \(30\%\). The insured’s share is not an additional percentage to apply to the insurer’s payment; it is the complement of the insurer’s share.
Calculate the Covered-Loss Base First
The percentage applies to the otherwise covered loss, not necessarily to every dollar shown in a claim description. Some amounts may be excluded by the hypothetical policy terms. To keep the allocation clear, distinguish three quantities:
- Ground-up loss: the full loss amount before determining which items qualify for coverage.
- Otherwise covered loss: the portion eligible for coverage before coinsurance is applied.
- Insurer and insured shares: the division of that otherwise covered loss under the stated percentage.
When the entire ground-up loss is otherwise covered, the two amounts are the same. When part is excluded, first remove the excluded amount from the ground-up loss. Then apply the coinsurance percentage to the remainder. The excluded amount remains outside the coinsurance split. It is typically part of the insured’s total cost, but it is not part of the insured’s coinsurance share of the covered loss.
Worked Example: Separate an Excluded Item Before Applying Coinsurance
A hypothetical repair claim totals $24,600. The policy excludes $1,600 of the reported work, while the remaining work is otherwise covered. The policy states that the insurer pays 70% of the otherwise covered loss. Find the covered-loss base, the insurer payment, the insured’s share of the covered loss, and the insured’s total out-of-pocket amount including the excluded work.
Method. Determine the otherwise covered loss first, apply the insurer’s percentage to that base, and then add the excluded amount only when calculating total out-of-pocket cost.
Step 1: Find the otherwise covered loss. Subtract the excluded work from the ground-up claim:
Step 2: Calculate the insurer’s share. The insurer pays 70%, so \(c=0.70\):
Step 3: Calculate the insured’s share of the covered loss. The complementary percentage is \(1-0.70=0.30\):
Step 4: Include the excluded work in total out-of-pocket cost. The insured bears the $6,900 covered-loss share and the $1,600 excluded amount:
Check. The covered-loss shares add to the covered base: \(\$16{,}100+\$6{,}900=\$23{,}000\). Adding the excluded $1,600 gives the original $24,600 claim total, split between the insurer’s $16,100 and the insured’s $8,500.
Common mistake avoided: Applying 70% to the entire $24,600 would incorrectly include an excluded item in the coinsurance base. The percentage applies to $23,000, not $24,600.
Why the Two Shares Are Complements
The share calculation follows from dividing the covered loss into two parts. If the insurer pays the fraction \(c\), the fraction left for the insured must be \(1-c\), because the two fractions together must account for the entire covered loss. Multiplying each fraction by \(L\) gives the dollar shares.
This identity is both a justification and a useful check. It also explains why the insurer’s percentage and the insured’s percentage should not be added as though they were independent charges. They are complementary shares of the same covered-loss base.
The rule is limited to the simple arrangement defined here. If a policy also applies a deductible or a limit, the payment may depend on how those terms interact with coinsurance. Do not assume the simple expression \(cL\) resolves those additional provisions; identify the specified order and terms before calculating.
Worked Example: Calculate Shares Across Several Claims
A hypothetical policy pays 65% of each otherwise covered claim, with no other payment adjustments. During a reporting period, three separate claims have otherwise covered losses of $4,800, $12,000, and $23,500. Find the insurer’s payment and the insured’s covered-loss share for each claim, then find both totals for the period.
Method. Apply the same percentage to each claim’s covered-loss base. Then add the claim-level payments and shares. The total insurer payment can also be checked by applying 65% to the sum of the covered losses.
For the first claim, the insurer pays \(0.65(\$4{,}800)=\$3{,}120\). The insured’s share is \(0.35(\$4{,}800)=\$1{,}680\). The check is \(\$3{,}120+\$1{,}680=\$4{,}800\).
For the second claim, the insurer pays \(0.65(\$12{,}000)=\$7{,}800\). The insured’s share is \(0.35(\$12{,}000)=\$4{,}200\). The check is \(\$7{,}800+\$4{,}200=\$12{,}000\).
For the third claim, the insurer pays \(0.65(\$23{,}500)=\$15{,}275\). The insured’s share is \(0.35(\$23{,}500)=\$8{,}225\). The check is \(\$15{,}275+\$8{,}225=\$23{,}500\).
Step 1: Add the covered losses.
Step 2: Add the insurer payments.
Step 3: Add the insured shares.
Check. The totals reconcile: \(\$26{,}195+\$14{,}105=\$40{,}300\). Applying 65% directly to the combined covered loss also gives \(0.65(\$40{,}300)=\$26{,}195\).
Common mistake avoided: The 65% applies to each claim’s covered amount, not just to one claim or to an average claim. In this simple model, summing claim payments and applying the percentage to the sum agree because multiplication by a constant distributes over addition.
Recovering a Percentage and Checking Its Meaning
Sometimes the payment is known and the coinsurance percentage must be inferred. Divide the insurer’s payment by the otherwise covered loss. This produces the fraction of the covered loss paid by the insurer. The insured’s share is then the complement, and the two dollar amounts should reconcile to the covered loss.
Worked Example: Infer the Insurer’s Percentage From a Payment
A hypothetical policy has no deductible, limit, or other payment adjustment. An otherwise covered equipment loss is $27,500, and the insurer pays $19,250. Determine the insurer’s payment percentage and the insured’s share in dollars. Check that the inferred percentage reproduces the payment.
Method. Divide the insurer payment by the covered loss to find \(c\), then use \(1-c\) to find the insured’s share.
Step 1: Find the insurer’s fraction.
Thus, the insurer pays 70% of the otherwise covered loss.
Step 2: Find the insured’s fraction and dollar share. The insured’s fraction is \(1-0.70=0.30\), so the dollar share is:
Step 3: Check the payment and allocation. The inferred insurer percentage reproduces the stated payment, and the shares sum to the covered loss:
Common mistake avoided: The amount paid is not the percentage. The insurer paid $19,250, but its percentage is the ratio \(\$19{,}250/\$27{,}500=70\%\), not $19,250 or 30%.
Common Interpretation Traps
The most reliable way to avoid errors is to label both the percentage and its base before multiplying. In particular, check that the policy’s percentage is the insurer’s share, rather than assuming that any stated percentage names the insurer. Also check whether the stated amount is the ground-up loss or the otherwise covered loss.
- Using the wrong share. If the insurer pays 60%, the insured bears 40% of the covered loss. Applying 40% as the insurer payment reverses the allocation.
- Using the wrong base. Apply the percentage to the otherwise covered loss. Do not include an excluded amount in that base.
- Confusing the insured’s covered-loss share with total out-of-pocket cost. Excluded amounts may add to what the insured pays, but they are not part of the coinsurance split.
- Failing to reconcile. For the covered-loss base, insurer payment plus insured share must equal the base. A mismatch signals an arithmetic or interpretation error.
Coinsurance payment percentages matter because they allocate a claim between the insurer and insured in a direct, measurable way. Correct calculation depends less on complicated arithmetic than on reading the policy’s stated share and identifying exactly which dollars qualify for the percentage. These distinctions are aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.
Check Your Understanding
For each problem, identify the applicable covered-loss base before calculating. Assume no deductible, limit, or other payment adjustment unless the problem states otherwise.
- A hypothetical policy pays 72% of an otherwise covered loss of $8,400. Find the insurer payment and the insured’s share of the covered loss. (Answer: insurer $6,048; insured $2,352.)
- A reported loss is $13,200, including $2,200 of excluded work. The insurer pays 85% of the otherwise covered loss. Find the insurer payment and the insured’s total out-of-pocket amount, including the excluded work. (Answer: insurer $9,350; insured total out-of-pocket $3,850.)
- Three separate otherwise covered claims are $2,750, $6,400, and $9,350. A hypothetical policy pays 60% of each. Find the total insurer payment and total insured covered-loss share. (Answer: insurer $11,100; insured $7,400.)
- An insurer pays $14,560, equal to 80% of an otherwise covered loss. Find the covered loss and the insured’s share. (Answer: covered loss $18,200; insured share $3,640.)
- For an otherwise covered loss of $31,600, compare two hypothetical policies: one pays 75% and the other pays 90%. Find each insurer payment and the difference between those payments. (Answer: $23,700 and $28,440; the 90% policy pays $4,740 more.)