How a Per-Claim Deductible Allocates One Loss
A per-claim deductible assigns the first portion of a covered claim to the insured. For a single claim, the calculation is simple, but its interpretation matters: the insured cannot be assigned more than the covered loss, and the insurer’s net loss cannot be negative. These boundary conditions are especially important when a claim is smaller than the deductible.
This tutorial considers the economic allocation of one covered claim only. It does not determine whether a loss is covered, combine separate claims, or apply a policy limit. Treat the amount \(X\) as the covered loss amount that is subject to the deductible. The deductible is \(d\), a fixed dollar amount per claim. The insured’s deductible reimbursement means the amount of that loss allocated to the insured; the insurer’s net loss is the portion remaining after that allocation.
The deductible applies separately to the claim being calculated. If \(X<d\), the insured bears the entire covered loss and the insurer’s net loss is zero. If \(X=d\), the insured bears the entire loss, again leaving zero net loss to the insurer. If \(X>d\), the insured bears exactly \(d\), while the insurer bears the amount above \(d\).
The allocation also satisfies a useful reconciliation:
This is an allocation of the covered loss, not a statement about the timing of cash flows. For example, a policy may specify how claim payments and later reimbursement are handled, but those payment mechanics do not change the two amounts in this single-claim allocation. To calculate the allocation, first identify the covered loss amount and deductible on the same dollar basis, then calculate each share and verify the reconciliation.
Why the Allocation Formula Works
The definition follows directly from the deductible’s meaning: the insured is responsible for the loss up to \(d\), but cannot be responsible for more than the loss that actually occurred. The insurer receives no allocation of the first \(d\) dollars and bears only any excess.
Consider the two exhaustive cases. If \(X\leq d\), then \(\min(X,d)=X\) and \(\max(X-d,0)=0\). Therefore \(R+N=X+0=X\). If \(X>d\), then \(\min(X,d)=d\) and \(\max(X-d,0)=X-d\). Therefore \(R+N=d+(X-d)=X\). In both cases the insured and insurer shares sum exactly to the covered loss. The \(\max\) operation is necessary in the first case because \(X-d\) would otherwise be negative, even though a negative insurer loss is not an allocation of the covered claim.
Use the stated covered loss \(X\) subject to the deductible, not an amount that still needs a coverage determination.
Convert both to dollars or both to thousands of dollars before subtracting or comparing them.
Use \(R=\min(X,d)\) and \(N=\max(X-d,0)\).
Confirm \(R+N=X\), \(0\leq R\leq X\), and \(N\geq 0\).
Worked Single-Claim Allocations
Worked Example: Claim Above the Deductible
A hypothetical commercial property policy has a per-claim deductible of $7,500. A single covered water-damage claim has a covered loss of $23,200. Find the insured’s deductible responsibility and the insurer’s net loss.
Model and assumptions. The stated $23,200 is the covered loss amount subject to the deductible, with no additional limit or other adjustment in this calculation. The deductible is \(d=\$7,500\). Interest and mortality assumptions are not applicable. No stochastic loss distribution is assumed; the claim amount is stipulated.
Step 1: Compare the loss with the deductible. Since \(\$23,200>\$7,500\), the insured’s responsibility reaches the full deductible.
Step 2: Calculate the insurer’s net loss.
Step 3: Check. The two shares total \(\$7,500+\$15,700=\$23,200\), which equals the covered loss. The method is the per-claim allocation formula. The mistake it avoids is assigning the entire loss to the insured simply because the insured has a deductible; above the deductible, the insurer bears the excess.
Worked Example: Claim Below the Deductible
A hypothetical equipment policy has a per-claim deductible of $4,800. A single covered claim has a covered loss of $3,650. Determine the insured’s responsibility and the insurer’s net loss.
Model and assumptions. The entire $3,650 is a covered amount subject to the deductible. The deductible is \(d=\$4,800\). Interest and mortality assumptions are not applicable. No stochastic loss distribution is assumed; the claim amount is stipulated.
Step 1: Compare the amounts. The covered loss is below the deductible: \(\$3,650<\$4,800\). The insured can be allocated only the loss that occurred.
Step 2: Apply the zero floor to the insurer’s share.
Step 3: Check. The allocation is \(\$3,650+\$0=\$3,650\). The method is still the per-claim allocation formula, including its minimum and maximum operations. The mistake it avoids is reporting \(-\$1,150\) as the insurer’s loss: that subtraction is an intermediate value, not a valid net loss.
Worked Example: Converting Thousands of Dollars
A hypothetical liability claim is reported as a covered loss of 68.4 thousand dollars. The per-claim deductible is $25,000. Calculate the insured’s responsibility and the insurer’s net loss, reporting both in dollars.
Model and assumptions. The reported 68.4 is in thousands of dollars, and the full amount is covered and subject to the deductible. The deductible is \(d=\$25,000\). Interest and mortality assumptions are not applicable. No stochastic loss distribution is assumed; the claim amount is stipulated.
Step 1: Convert the loss to dollars.
Step 2: Compare the amounts and calculate the insured’s share. Since \(\$68,400>\$25,000\), the insured bears the full deductible:
Step 3: Calculate the insurer’s share.
Step 4: Check. The shares sum to \(\$25,000+\$43,400=\$68,400\). The method is to convert units first and then apply the deductible formula. The mistake it avoids is subtracting $25,000 from 68.4 as though both figures were in dollars; that would mix units and produce a meaningless result.
Boundary Cases and Common Exam Traps
The most reliable way to handle a per-claim deductible is to keep the insured share and insurer share conceptually separate. The insured’s share is capped by the loss itself; the insurer’s share is floored at zero. These conditions handle both small claims and claims exceeding the deductible without requiring a separate special rule.
| Relationship | Insured responsibility \(R\) | Insurer net loss \(N\) | Interpretation |
|---|---|---|---|
| \(X<d\) | \(X\) | 0 | The insured bears the whole covered loss |
| \(X=d\) | \(d\) | 0 | The loss exactly exhausts the deductible |
| \(X>d\) | \(d\) | \(X-d\) | The insurer bears the amount above the deductible |
A frequent error is to calculate the insurer’s share as \(X-d\) in every case. For a loss of $3,650 and a deductible of $4,800, that gives \(-\$1,150\). The correction is \(N=\max(X-d,0)=\$0\), while the insured’s responsibility is \(R=\min(X,d)=\$3,650\). A second error is to set the insured’s responsibility equal to \(d\) even when \(X<d\); the correction is to use the minimum, so that the insured is never assigned more than the covered loss.
A third trap is applying the deductible to an amount other than the \(X\) defined for the question. Read carefully to determine which covered amount is being allocated. Do not silently change that amount or add a separate adjustment. For this single-claim calculation, once \(X\) and \(d\) are specified on the same basis, the allocation is fully determined.
Check Your Understanding
For each hypothetical single covered claim, calculate the insured’s deductible responsibility and the insurer’s net loss. Treat the stated covered loss as the amount subject to the deductible; interest, mortality, and a stochastic loss distribution are not applicable.
- A covered claim is $18,400 and the per-claim deductible is $5,000. What are the two allocated amounts? (Answer: insured $5,000; insurer $13,400.)
- A covered claim is $3,275 and the deductible is $4,000. What are the two allocated amounts, and what does the reconciliation show? (Answer: insured $3,275; insurer $0; the shares sum to $3,275.)
- A covered claim is $42,600 and the deductible is $12,500. Calculate both shares and verify their total. (Answer: insured $12,500; insurer $30,100; total $42,600.)
- A claim is stated as 9.75 thousand dollars, and the deductible is $8,000. Convert the loss to dollars before allocating it. (Answer: insured $8,000; insurer $1,750.)
- A single covered loss is $26,300. Compare the insured and insurer shares under a $10,000 deductible and a $30,000 deductible. For each option, verify that the shares sum to the loss. (Answer: with $10,000 deductible, insured $10,000 and insurer $16,300; with $30,000 deductible, insured $26,300 and insurer $0; each pair sums to $26,300.)