Why Loss Reconciliation Matters
Reinsurance changes who ultimately bears a covered loss; it does not make that loss disappear. For each claim or defined portfolio period, a sound allocation accounts for the covered gross loss as the sum of the reinsurers’ recoveries and the insurer’s retained loss. The key is to calculate each recovery on the loss basis specified by its contract and to apply contracts in the stipulated order.
The starting amount in this tutorial is the covered gross loss: the loss amount eligible for reinsurance after the applicable underlying insurance terms have been applied. The examples assume that each stated amount is covered, that the reinsurers pay valid recoveries in full, and that there are no expenses, reinstatement premiums, exclusions, or other adjustments. In practice, those terms may change the amounts or the accounting.
“Gross,” “ceded,” and “retained” must refer to compatible amounts. For example, a recovery on an insurer’s post-quota-share retention cannot also be treated as though it were calculated on the original gross claim. Doing so would allocate part of the same loss twice. Contract order determines the amount to which each later cover applies.
Recoveries Depend on the Contract Basis
For a quota-share agreement with cession proportion \(q\), the reinsurer’s share of an eligible gross loss \(G\) is \(qG\), and the insurer’s share before other reinsurance is \((1-q)G\). For a surplus-share agreement, the ceded proportion depends on the insured amount and the treaty’s stated retention and capacity. These proportional cessions divide the eligible loss in the stated proportions.
For an excess-of-loss layer with attachment \(A\) and limit \(L\), applied to a specified loss amount \(z\), the recovery is the part of \(z\) above \(A\), subject to a maximum payment of \(L\). The limit here is the layer’s maximum payment, or layer width. A contract may define its own basis \(z\), such as a gross per-occurrence loss or a loss remaining after another cession; that basis must be identified before calculating the recovery.
The inner maximum makes the recovery zero when the loss does not exceed the attachment. The outer minimum caps it at the layer limit. A layer is exhausted when the loss reaches or exceeds \(A+L\).
Worked Example: Quota Share Followed by Per-Occurrence Excess of Loss
A hypothetical insurer has a covered claim of $120,000. A 25% quota-share cover applies first to the gross claim. The insurer’s remaining share is then protected by a per-occurrence layer of $20,000 excess of $30,000. Find the recovery from each contract and the insurer’s final retention. Assume no other terms affect payment.
Method. Allocate the quota-share amount first. Use only the resulting insurer share as the basis for the excess-of-loss layer.
Step 1: Quota-share recovery. The reinsurer receives 25% of the covered gross claim:
The insurer’s share after this cession is:
Step 2: Excess-of-loss recovery. The layer attaches at $30,000 and has a maximum payment of $20,000. Its recovery on the $90,000 remaining share is:
Step 3: Final retention and check. The insurer retains $90,000 minus the excess-of-loss recovery, or $70,000. Total recoveries are $30,000 plus $20,000, or $50,000. Thus:
The amounts reconcile. The common mistake avoided here is applying the excess-of-loss layer to the original $120,000 rather than the post-quota-share amount. That would produce a $20,000 recovery in this case too, but it would use the wrong basis and could give a different answer for other claims or layer terms.
Why the Reconciliation Identity Holds
The identity is a conservation statement: each ceded recovery removes an amount from the loss borne by the insurer, while the remainder is retained. It can be justified directly for a sequence of covers, provided each recovery is an allocation against the same covered loss and is not counted more than once.
Let the initial amount be \(B_0=G\). This is the amount before the reinsurance allocations being analyzed.
If the first contract allocates \(C_1\), the amount remaining is \(B_1=B_0-C_1\).
If the next recovery is \(C_2\), calculated on the amount available under its terms, the new remainder is \(B_2=B_1-C_2\). Continue in the contractually required order.
After \(n\) allocations, \(B_n=G-\sum_{j=1}^{n}C_j\). Therefore \(G=\sum_{j=1}^{n}C_j+B_n\), where \(B_n\) is the final retained amount.
The proof relies on the recoveries being valid allocations against this loss. If two covers overlap, or if a contract pays on a separately defined basis, one cannot simply subtract both calculated amounts without checking the wording and allocation rules. A reconciliation check is necessary, but it does not replace reading the contract.
Worked Example: Surplus Share Followed by Excess of Loss
A hypothetical policy has a sum insured of $500,000 and a covered claim of $150,000. A surplus-share treaty retains the first $100,000 of sum insured and has enough capacity to cover the remaining $400,000. The treaty cedes the same proportion of each eligible loss as the proportion of the sum insured ceded. After that cession, a $10,000 excess-of-loss layer attaches at $20,000 and applies to the insurer’s remaining share. Calculate the cessions and final retention.
Method. First determine the surplus-share proportion from the insured amount. Apply that proportion to the covered claim, then calculate the excess recovery on the insurer’s remaining loss.
Step 1: Surplus-share proportion. The amount ceded from the $500,000 sum insured is $500,000 minus the $100,000 retention, or $400,000. The cession proportion is:
Step 2: Surplus-share loss cession. The ceded loss is 80% of $150,000:
The insurer’s remaining share is:
Step 3: Excess-of-loss recovery. The layer covers the amount above $20,000, up to $10,000. On the $30,000 remaining share:
Step 4: Final retention and check. The insurer retains $30,000 minus $10,000, or $20,000. Total recoveries are $120,000 plus $10,000, or $130,000:
The allocation reconciles. The common mistake avoided is using the $100,000 sum-insured retention as though it were the insurer’s retained share of this $150,000 claim. The claim is allocated in the surplus-share proportion, so the insurer initially retains 20% of the claim, or $30,000.
Reconciling Occurrence Covers with Stop-Loss
A stop-loss cover is applied to an aggregate amount over a stated period, such as the insurer’s annual loss after specified per-occurrence recoveries. If the aggregate loss entering the stop-loss cover is \(T\), and the stop-loss attaches at \(A\) with limit \(L\), its recovery is \(\min(\max(T-A,0),L)\). The amount \(T\) must be calculated from the correct underlying losses; an annual recovery cannot be added to per-occurrence recoveries without first identifying which amounts those occurrence recoveries left for the insurer.
Worked Example: Per-Occurrence Recoveries Followed by Annual Stop-Loss
A hypothetical insurer has three covered claims in one year: $40,000, $80,000, and $150,000. A per-occurrence layer pays $20,000 excess of $30,000 on each claim. After those recoveries, an annual stop-loss cover pays $25,000 excess of $180,000 on the insurer’s aggregate remaining losses. Find each occurrence recovery, the stop-loss recovery, the final retained annual loss, and the reconciliation.
Method. Calculate the per-occurrence payments claim by claim, sum the insurer’s remaining losses, and then apply the annual layer to that aggregate.
Step 1: Per-occurrence recoveries. For the $40,000 claim, the amount above $30,000 is $10,000, below the $20,000 limit, so the recovery is $10,000. For the $80,000 claim, the amount above attachment is $50,000, capped at $20,000. For the $150,000 claim, the amount above attachment is $120,000, also capped at $20,000.
Step 2: Aggregate remaining losses. The insurer’s remaining amounts on the three claims are respectively $40,000 minus $10,000, or $30,000; $80,000 minus $20,000, or $60,000; and $150,000 minus $20,000, or $130,000. Therefore:
Step 3: Stop-loss recovery. The amount above the $180,000 attachment is $40,000. The $25,000 limit caps the recovery:
Step 4: Final retention and check. The insurer retains $220,000 minus $25,000, or $195,000. Total recoveries are $10,000 plus $20,000 plus $20,000 plus $25,000, or $75,000. The annual gross loss is $40,000 plus $80,000 plus $150,000, or $270,000. Hence:
This checks. The common mistake avoided is applying the stop-loss attachment to the $270,000 gross annual loss when its stated basis is the $220,000 remaining after per-occurrence recoveries.
Practical Reconciliation Checks
For a sequence of contracts, write down the amount entering each contract before calculating its recovery. A compact allocation table can make the bases and arithmetic visible:
| Stage | Amount entering stage | Recovery at stage | Amount remaining |
|---|---|---|---|
| Initial covered loss | Gross covered loss \(G\) | — | \(G\) |
| Proportional cession | Amount defined by contract | Contractual share of basis | Basis less cession |
| Excess-of-loss layer | Specified gross or net basis | Amount above attachment, capped at limit | Basis less recovery |
| Aggregate stop-loss | Specified period total | Aggregate amount above attachment, capped at limit | Aggregate less recovery |
A reconciliation is meaningful only when the gross amount, recoveries, and retained amount refer to the same covered losses and time period. If a claim is excluded from one treaty, or a layer is defined on a different basis, record that distinction rather than forcing an apparent equality by mixing amounts. For the stated allocation, the following checks are useful:
- Each recovery is nonnegative and does not exceed its applicable contract limit.
- An excess-of-loss layer pays nothing below attachment and does not pay more than its layer width.
- A recovery calculated on a post-cession amount uses that amount, not the original gross amount.
- The final retained amount is not negative under a direct allocation of recoveries against the covered loss.
- Total recoveries plus final retention equal the covered gross loss on the same basis.
The central exam technique is to treat the contract order as part of the data. Calculate the first allocation, carry forward the appropriate remaining amount, and then apply the next contract to its stated basis. A correct final sum is a valuable check, but it cannot make an incorrectly calculated recovery valid.
Check Your Understanding
For each hypothetical arrangement, use the stated contract order and verify the loss allocation.
- A covered claim is $85,000. A quota-share treaty cedes 20% of the gross claim. Find the ceded amount and the insurer’s retention. (Answer: Ceded $17,000; retained $68,000.)
- A covered claim is $96,000. A per-occurrence layer pays $15,000 excess of $40,000 directly on the gross claim. Find the recovery and retention. (Answer: Recovery $15,000; retained $81,000.)
- A covered claim is $200,000. A 30% quota-share cession applies first; then a $25,000 excess-of-loss layer attaches at $50,000 on the insurer’s post-quota-share share. Find both recoveries and final retention. (Answer: Quota-share recovery $60,000; excess-of-loss recovery $25,000; retained $115,000.)
- Two covered claims are $35,000 and $100,000. A $20,000 per-occurrence layer attaches at $30,000 on each claim. An annual stop-loss cover then pays $15,000 excess of $100,000 on the aggregate remaining loss. Find total recoveries and final annual retention. (Answer: Per-occurrence recoveries $5,000 and $20,000; stop-loss recovery $10,000; total recoveries $35,000; retained $100,000.)
- A policy has a $300,000 sum insured. A surplus-share treaty retains $100,000 and has enough capacity for the balance; it cedes the same proportion of the covered loss as of the sum insured. The covered claim is $90,000. A $12,000 excess-of-loss layer attaches at $18,000 to the post-surplus-share retention. Find total recoveries and final retention. (Answer: Surplus-share recovery $60,000; excess-of-loss recovery $12,000; total recoveries $72,000; retained $18,000.)