Why a Claim-by-Claim Ledger Matters
A large deductible arrangement does not usually change the amount of a covered loss recorded for a claim. It changes how that loss is divided between the insured and the insurer. A policy-year ledger makes that division visible: for each claim, it records the covered gross loss, the amount reimbursed by the insured, and the insurer's net loss after reimbursement.
The unit of calculation is essential. With a per-claim deductible, the deductible is applied once to the covered loss for each claim. It is not applied once per payment, invoice, or installment. Once every claim has been processed, the ledger can be totaled for the policy year and checked for consistency.
This definition assumes that \(X_j\) is already the covered amount to which the deductible applies. It also assumes that the deductible is fixed per claim, all listed amounts are on the same valuation basis, and there are no reimbursement caps, coinsurance provisions, or other policy adjustments. Those features must be included explicitly when the contract specifies them. Do not apply this per-claim formula to an annual aggregate deductible: that structure uses a different allocation across the policy year.
A Reliable Ledger Method
For each claim, first determine the full covered gross loss. Then apply the contract's reimbursement rule to that claim total. The insurer net loss is the remaining covered amount after reimbursement. A ledger should preserve the claim-level calculation before presenting policy-year totals.
If a claim has multiple covered payments, add them to obtain its gross loss \(X_j\). Do not treat separate payments as separate claims unless the contract defines them as such.
For an uncapped fixed per-claim deductible, use \(R_j=\min(X_j,d)\). The insured cannot reimburse more than the claim's covered gross loss.
Use \(N_j=X_j-R_j\). This gives zero insurer net loss when the gross loss does not exceed the deductible, and the amount above the deductible otherwise.
Sum each column, then verify that total gross loss equals total reimbursement plus total insurer net loss.
The reconciliation is both a mathematical identity and a practical audit check. It follows directly from defining \(N_j=X_j-R_j\), so \(R_j+N_j=R_j+(X_j-R_j)=X_j\) for every claim. Summing those claim-level equalities produces the policy-year equality. If the totals fail to reconcile, investigate the claim grouping, the reimbursement rule, and the arithmetic before relying on the ledger.
Worked Examples
Worked Example: A Basic Policy-Year Ledger
A hypothetical contractor has four covered claims during a policy year. Their gross losses are $12,500, $34,000, $7,800, and $61,250. The policy has a $15,000 deductible per claim, with no reimbursement cap. Prepare the claim-by-claim ledger and reconcile the totals.
Method. Apply \(R_j=\min(X_j,15{,}000)\) separately to each claim, then calculate \(N_j=X_j-R_j\).
For claim 1, \(R_1=\min(12{,}500,15{,}000)=12{,}500\), so \(N_1=12{,}500-12{,}500=0\).
For claim 2, \(R_2=\min(34{,}000,15{,}000)=15{,}000\), so \(N_2=34{,}000-15{,}000=19{,}000\).
For claim 3, \(R_3=\min(7{,}800,15{,}000)=7{,}800\), so \(N_3=7{,}800-7{,}800=0\).
For claim 4, \(R_4=\min(61{,}250,15{,}000)=15{,}000\), so \(N_4=61{,}250-15{,}000=46{,}250\).
| Claim | Gross loss | Insured reimbursement | Insurer net loss |
|---|---|---|---|
| 1 | $12,500 | $12,500 | $0 |
| 2 | $34,000 | $15,000 | $19,000 |
| 3 | $7,800 | $7,800 | $0 |
| 4 | $61,250 | $15,000 | $46,250 |
| Total | $115,550 | $50,300 | $65,250 |
The gross total is \(12{,}500+34{,}000+7{,}800+61{,}250=115{,}550\). Reimbursements total \(12{,}500+15{,}000+7{,}800+15{,}000=50{,}300\), and insurer net losses total \(0+19{,}000+0+46{,}250=65{,}250\). The check is \(50{,}300+65{,}250=115{,}550\), equal to the gross total.
Common mistake avoided: Subtracting the deductible from every claim, including claims below $15,000, would produce negative insurer losses. The \(\max(X_j-d,0)\) form prevents that error.
Worked Example: Multiple Installments for One Claim
A hypothetical equipment claim is paid in two covered installments of $11,000 and $17,000. Two other claims in the same policy year total $15,000 and $43,000. The deductible is $20,000 per claim, with no reimbursement cap. Construct the ledger.
Method. Combine installments belonging to the same claim before applying its deductible. Claim A has gross loss \(11{,}000+17{,}000=28{,}000\). Claims B and C have gross losses of $15,000 and $43,000.
For claim A, \(R_A=\min(28{,}000,20{,}000)=20{,}000\), and \(N_A=28{,}000-20{,}000=8{,}000\).
For claim B, \(R_B=\min(15{,}000,20{,}000)=15{,}000\), and \(N_B=15{,}000-15{,}000=0\).
For claim C, \(R_C=\min(43{,}000,20{,}000)=20{,}000\), and \(N_C=43{,}000-20{,}000=23{,}000\).
| Claim | Covered payments | Gross loss | Reimbursement | Insurer net loss |
|---|---|---|---|---|
| A | $11,000 + $17,000 | $28,000 | $20,000 | $8,000 |
| B | $15,000 | $15,000 | $15,000 | $0 |
| C | $43,000 | $43,000 | $20,000 | $23,000 |
| Total | $86,000 | $55,000 | $31,000 |
The gross total is \(28{,}000+15{,}000+43{,}000=86{,}000\). Total reimbursement is \(20{,}000+15{,}000+20{,}000=55{,}000\), and total insurer net loss is \(8{,}000+0+23{,}000=31{,}000\). The reconciliation check gives \(55{,}000+31{,}000=86{,}000\).
Common mistake avoided: Applying the $20,000 deductible separately to claim A’s two installments would incorrectly produce $28,000 of reimbursement on a $28,000 claim: \(\min(11{,}000,20{,}000)+\min(17{,}000,20{,}000)=11{,}000+17{,}000=28{,}000\). The contract has one deductible per claim, so the correct reimbursement is $20,000 and the insurer net loss is $8,000.
Worked Example: Adding a Per-Claim Reimbursement Cap
A hypothetical policy has a $30,000 deductible per claim and limits the insured's reimbursement to $22,000 per claim. The four covered gross losses for the year are $18,000, $30,000, $47,000, and $80,000. Calculate the ledger. Assume the cap applies to the insured's reimbursement, not to the gross covered loss.
Method. The deductible determines the reimbursement before the cap, while the cap limits the reimbursement actually collected. Thus \(R_j=\min(X_j,d,22{,}000)\), and \(N_j=X_j-R_j\).
For the $18,000 claim, \(R_1=\min(18{,}000,30{,}000,22{,}000)=18{,}000\), so \(N_1=0\).
For the $30,000 claim, the deductible amount is $30,000 but the cap limits reimbursement to $22,000. Thus \(R_2=22{,}000\) and \(N_2=30{,}000-22{,}000=8{,}000\).
For the $47,000 claim, \(R_3=\min(47{,}000,30{,}000,22{,}000)=22{,}000\), so \(N_3=47{,}000-22{,}000=25{,}000\).
For the $80,000 claim, \(R_4=22{,}000\), so \(N_4=80{,}000-22{,}000=58{,}000\).
| Claim | Gross loss | Reimbursement | Insurer net loss |
|---|---|---|---|
| 1 | $18,000 | $18,000 | $0 |
| 2 | $30,000 | $22,000 | $8,000 |
| 3 | $47,000 | $22,000 | $25,000 |
| 4 | $80,000 | $22,000 | $58,000 |
| Total | $175,000 | $84,000 | $91,000 |
The totals reconcile: \(84{,}000+91{,}000=175{,}000\). The cap matters for claims 2 through 4 because their deductible amounts exceed $22,000. For claim 1, reimbursement cannot exceed the $18,000 gross loss.
Common mistake avoided: Treating the deductible as the amount the insurer must always recover would assign $30,000 reimbursement to claim 2. The stated cap controls the reimbursement, leaving an $8,000 insurer net loss on that claim.
Reading and Auditing the Ledger
A complete ledger should distinguish the amount of covered loss from the amount each party ultimately bears. Gross loss remains the starting point; reimbursement is not a reduction in the gross loss entry. For an uncapped deductible, the insurer's net loss is zero when \(X_j\le d\), and equals \(X_j-d\) when \(X_j>d\). At exactly \(X_j=d\), reimbursement equals the full claim and insurer net loss is zero.
When the contract has a reimbursement cap \(c\), and no other adjustments apply, the appropriate claim-level rule is \(R_j=\min(X_j,d,c)\). The insurer net loss is still \(X_j-R_j\). This form makes two boundaries explicit: reimbursement cannot exceed the covered gross loss, and it cannot exceed either the deductible amount or the contractual cap. Always identify what the cap limits; a limit on reimbursement is not automatically a limit on the covered loss.
Payment timing does not change the claim-level allocation if the contract specifies a deductible per claim and the ledger is measuring the final covered amount for the policy year. Installment dates may matter for cash-flow or valuation analyses, but they do not create additional deductibles. If a payment is later adjusted, use the final covered claim amount consistently rather than mixing an earlier gross estimate with a later reimbursement.
Check Your Understanding
For each problem, calculate the requested claim-level or policy-year amounts and use the reconciliation identity as a check.
- A hypothetical policy has a $12,000 per-claim deductible and no reimbursement cap. Three covered claims are $6,200, $14,500, and $38,000. Find total gross loss, total reimbursement, and total insurer net loss. (Answer: gross $58,700; reimbursement $30,200; insurer net loss $28,500.)
- A policy has a $20,000 deductible per claim. Claim A has installments of $8,000 and $13,000; claim B has installments of $4,000, $5,000, and $9,000; claim C is a single $31,000 payment. Find the policy-year totals. (Answer: gross $70,000; reimbursement $58,000; insurer net loss $12,000.)
- A hypothetical contract has a $16,000 per-claim deductible and a $9,000 cap on insured reimbursement per claim. Gross claims are $7,000, $16,000, $24,000, and $40,000. Find total reimbursement and total insurer net loss. (Answer: reimbursement $34,000; insurer net loss $53,000.)
- A policy has a $25,000 per-claim deductible and no cap. Its claims are $25,000, $25,010, $49,990, $50,000, and $103,000. Find total gross loss and total insurer net loss. (Answer: gross $253,000; insurer net loss $128,000.)
- A hypothetical policy has an $18,000 per-claim deductible and no cap. Claim A has payments of $7,000 and $15,000; claim B has payments of $10,000 and $6,000; claim C has payments of $12,000, $20,000, and $14,000. Find each claim's reimbursement and net loss, then give the totals. (Answer: A reimbursement $18,000 and net loss $4,000; B reimbursement $16,000 and net loss $0; C reimbursement $18,000 and net loss $28,000; totals are gross $84,000, reimbursement $52,000, and insurer net loss $32,000.)