Comparing Deductible Options Across Claim Scenarios

Why Compare Deductibles on the Same Claims?

A deductible option cannot be evaluated from its stated amount alone. A fixed deductible can shift more loss to the insured on frequent small claims, while a percentage share can produce a larger insured payment on a sufficiently large claim. An annual aggregate deductible responds to the policy year's total loss, not to each claim separately. Comparing options against the same set of claims makes these differences visible and helps prevent changes in claim mix from being mistaken for changes in contract design.

The comparisons here concern ultimate loss allocation for hypothetical covered claims. Assume no policy limit, coinsurance outside the deductible structure, loss adjustment expense, tax, premium, reimbursement cap, or time value of money unless stated. The insurer initially pays each covered claim in full and then receives any deductible reimbursement from the insured. All dollar amounts are shown to the cent where needed.

Definition: For claim $j$, let $X_j$ be its covered gross loss, $R_j$ the insured's deductible reimbursement to the insurer, and $I_j$ the insurer's ultimate net loss. The insured's retained loss in these comparisons is $R_j$. For each claim, the allocation must reconcile: $$ X_j=R_j+I_j. $$

Under a fixed per-claim deductible $d$, the insured reimburses the lesser of the claim and the deductible. Under a percentage deductible, the insured pays a specified proportion of each claim. An annual aggregate deductible instead applies once to the covered losses accumulated during a policy year. These are distinct structures; a stated dollar amount is not enough to identify how it applies.

Core Formulas and Reconciliation

For a fixed per-claim deductible $d$, the insurer's net loss on claim $j$ is the amount of that claim above the deductible. For a percentage deductible with insured share $\alpha$, where $0\leq\alpha\leq1$, the insured reimburses proportion $\alpha$ of every covered claim.

R_j=\min(X_j,d), \qquad I_j=X_j-R_j=\max(X_j-d,0)

R_j=\alpha X_j, \qquad I_j=(1-\alpha)X_j

The fixed-deductible formulas follow by considering the two possible claim sizes. If $X_j\leq d$, then $R_j=X_j$ and $I_j=0$. If $X_j>d$, then $R_j=d$ and $I_j=X_j-d$. In either case, the insured reimbursement plus insurer net loss equals the gross claim. This also explains why the insurer's net loss is not $X_j+d$: the reimbursement is a recovery from the insured, not an additional claim payment.

For an annual aggregate deductible $A$, let $S_y=\sum_j X_{yj}$ be the total covered loss in policy year $y$. If the insured bears the first $A$ of annual loss and the deductible resets each year, total annual reimbursement is $\min(S_y,A)$, and annual insurer net loss is the excess, if any.

R_y=\min(S_y,A), \qquad I_y=\max(S_y-A,0), \qquad S_y=R_y+I_y

The annual formulas allocate the total for a year. If a claim-level ledger is required, the aggregate amount can be assigned in the order specified by the contract; below, claims are processed chronologically. The aggregate total is unaffected by that bookkeeping choice, but the reimbursement assigned to an individual claim can depend on it.

Worked Comparisons

Worked Example: Three Fixed Deductibles on the Same Claims

A hypothetical insured has five covered claims of $2,400, $7,500, $13,200, $28,000, and $61,000. Compare fixed per-claim deductibles of $5,000, $12,000, and $25,000. Find total insured reimbursement and total insurer net loss under each option.

Method and assumptions. Apply $R_j=\min(X_j,d)$ separately to every claim. There are no limits or other adjustments. First, the gross loss total is $2,400 + $7,500 + $13,200 + $28,000 + $61,000 = $112,100.

$5,000 deductible. The reimbursements are $\min(2,400,5,000)=2,400$, $\min(7,500,5,000)=5,000$, $\min(13,200,5,000)=5,000$, $\min(28,000,5,000)=5,000$, and $\min(61,000,5,000)=5,000$. Total reimbursement is $2,400 + $5,000 + $5,000 + $5,000 + $5,000 = $22,400. Total insurer net loss is $112,100 − $22,400 = $89,700.

$12,000 deductible. Reimbursements are $2,400, $7,500, $12,000, $12,000, and $12,000, totaling $45,900. Insurer net loss is $112,100 − $45,900 = $66,200.

$25,000 deductible. Reimbursements are $2,400, $7,500, $13,200, $25,000, and $25,000, totaling $73,100. Insurer net loss is $112,100 − $73,100 = $39,000.

Check and common mistake avoided. For each option, reimbursement plus insurer net loss equals the same $112,100 gross loss. Applying the deductible to the annual total just once would not calculate a per-claim deductible: the contract applies the deductible separately to each claim.

Worked Example: Fixed Versus Percentage Reimbursement

A hypothetical equipment policy has covered claims of $4,000, $12,000, $24,000, and $60,000. Compare a fixed $10,000 per-claim deductible with a percentage structure under which the insured reimburses 20% of each claim. Find each option's total reimbursement and insurer net loss.

Method and assumptions. Use the fixed-deductible formula for the first option and $R_j=0.20X_j$ for the second. The gross loss total is $4,000 + $12,000 + $24,000 + $60,000 = $100,000.

Fixed deductible. The claim reimbursements are $\min(4,000,10,000)=4,000$, $\min(12,000,10,000)=10,000$, $\min(24,000,10,000)=10,000$, and $\min(60,000,10,000)=10,000$. Total reimbursement is $34,000, so insurer net loss is $100,000 − $34,000 = $66,000.

20% percentage structure. The reimbursements are $0.20(4,000)=800$, $0.20(12,000)=2,400$, $0.20(24,000)=4,800$, and $0.20(60,000)=12,000$. Total reimbursement is $20,000, so insurer net loss is $100,000 − $20,000 = $80,000.

Comparison and check. The fixed option shifts $14,000 more of this claim set to the insured: $34,000 − $20,000 = $14,000. The $60,000 claim is a useful check on the comparison: its fixed reimbursement is $10,000, while its percentage reimbursement is $12,000. Across all claims, each option's reimbursement plus insurer net loss is $100,000. Do not apply 20% only to the amount above $10,000; that would describe a different contract structure.

Worked Example: Annual Aggregate Deductible Versus Per-Claim Deductible

A hypothetical insured has the following covered losses, processed in chronological order: in Year 1, $6,000, $9,000, and $18,000; in Year 2, $4,000 and $22,000. Compare a $10,000 per-claim deductible with an annual aggregate deductible of $15,000 that resets at the start of each year. Under the aggregate option, allocate reimbursement chronologically within each year.

Method and assumptions. For the per-claim option, apply $\min(X_j,10,000)$ to each loss. For the aggregate option, total the losses within each year and apply $\min(S_y,15,000)$. The gross losses are $6,000 + $9,000 + $18,000 = $33,000 in Year 1 and $4,000 + $22,000 = $26,000 in Year 2, for $59,000 overall.

Per-claim option. Year 1 reimbursements are $6,000, $9,000, and $10,000, totaling $25,000. Year 2 reimbursements are $4,000 and $10,000, totaling $14,000. Overall reimbursement is $25,000 + $14,000 = $39,000; insurer net loss is $59,000 − $39,000 = $20,000.

Annual aggregate option. In Year 1, the annual reimbursement is $\min(33,000,15,000)=15,000$. Chronologically, $6,000 is assigned to the first claim, $9,000 to the second, and $0 to the third. The insurer therefore bears $0, $0, and $18,000 on those claims, totaling $18,000. In Year 2, reimbursement is $\min(26,000,15,000)=15,000$. Chronologically, $4,000 is assigned to the first claim and the remaining $11,000 to the second; the insurer bears $0 and $22,000 − $11,000 = $11,000. Year 2 insurer net loss is $11,000.

The aggregate option therefore has total reimbursement of $15,000 + $15,000 = $30,000 and total insurer net loss of $18,000 + $11,000 = $29,000. The check is $30,000 + $29,000 = $59,000. The per-claim option shifts $9,000 more to the insured for this set of claims.

Common mistake avoided. Applying $15,000 separately to every claim would not be an annual aggregate deductible. The $15,000 annual amount is exhausted once in each policy year, so the year-by-year totals—not each individual loss—determine the aggregate reimbursement.

How to Make a Reliable Comparison

A comparison is easiest to audit when each option is applied to the same gross losses and the calculation keeps insured reimbursement separate from insurer net loss. The following sequence works for fixed, percentage, and aggregate structures:

State the contract structure.
Identify whether the deductible applies per claim, as a percentage of each claim, or to an annual aggregate. Record any reset, cap, or claim-order rule that affects the calculation.

Hold the gross claims constant.
List each covered loss and, for an annual aggregate, group the claims by policy year before calculating.

Calculate insured reimbursement.
Apply the formula for the stated structure. Do not substitute a similar-looking deductible formula from a different structure.

Reconcile the allocation.
Subtract total reimbursement from total gross loss to obtain insurer net loss, then verify that the two amounts sum back to gross loss.

The result describes how the specified claims are divided; it does not by itself determine which option is preferable overall. A different mix of claim sizes can change the comparison, and terms such as reimbursement caps, limits, and expenses can change the applicable calculations. For the options compared here, the practical lesson is to identify exactly what unit the deductible applies to before multiplying, summing, or comparing dollar amounts. These scenario-based calculations are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.

Key takeaway: Use identical gross claims for every option, apply the deductible at its stated level, and verify $ \text{gross loss}=\text{insured reimbursement}+\text{insurer net loss} $. A per-claim amount, a percentage share, and an annual aggregate amount do not operate interchangeably.

Check Your Understanding

For each problem, compare the stated structure using the same loss set and check that insured reimbursement and insurer net loss reconcile to gross loss.

A fixed $6,000 per-claim deductible applies to claims of $2,000, $9,000, and $17,000. Find total reimbursement and insurer net loss. (Answer: reimbursement $14,000; insurer net loss $14,000.)
A percentage deductible requires the insured to reimburse 30% of claims of $5,000, $20,000, and $35,000. Find total reimbursement and insurer net loss. (Answer: reimbursement $18,000; insurer net loss $42,000.)
For claims of $8,000, $30,000, and $60,000, compare a fixed $12,000 per-claim deductible with a 25% percentage deductible. Which option produces more total insured reimbursement, and by how much? (Answer: fixed reimbursement $32,000; percentage reimbursement $24,500; the fixed option is higher by $7,500.)
An annual aggregate deductible of $18,000 resets each year. Year 1 losses are $7,000, $13,000, and $20,000; Year 2 losses are $5,000, $9,000, and $30,000. Find total reimbursement and insurer net loss across both years. (Answer: reimbursement $36,000; insurer net loss $48,000.)
Compare a fixed $9,000 per-claim deductible with a 15% percentage deductible on a single claim of $40,000 and then on a single claim of $80,000. State which option reimburses more for each claim and the difference. (Answer: on $40,000, fixed is higher by $3,000; on $80,000, percentage is higher by $3,000.)

What You'll Learn