Per-Occurrence Excess-of-Loss Claim Recoveries

Why the Order of Calculation Matters

One event can produce losses on several insured risks. A storm, for example, might damage multiple buildings, or a single accident might cause claims under several policies. Under per-occurrence excess-of-loss reinsurance, the covered losses attributed to one occurrence are combined first. The retention and limit are then applied to that combined amount.

This order is the defining feature of the cover. It differs from a per-risk contract, where the terms are applied separately to each risk’s loss. Applying an occurrence retention to each claim instead of to the combined occurrence loss can materially misstate the recovery.

Definition: A per-occurrence excess-of-loss cover reimburses the insurer for covered losses from one occurrence that exceed a stated occurrence retention, subject to a maximum recovery for that occurrence. If the covered claim amounts from occurrence $j$ are $L_{j1}, L_{j2}, \ldots, L_{jn_j}$, its occurrence loss is $L_j=\sum_{k=1}^{n_j}L_{jk}$. For retention $R$ and recovery limit $M$, the recovery is $C_j=\min\{\max(L_j-R,0),M\}$, and the insurer’s retained occurrence loss is $I_j=L_j-C_j$.

Here, $R$ is the amount of occurrence loss borne before the reinsurance responds, and $M$ is the most the reinsurer pays for that occurrence. Thus, “$M$ excess of $R$” means a layer attaching above $R$ with width $M$. The maximum occurrence loss that can be covered by this layer is $R+M$; losses above that amount remain with the insurer.

The calculations below assume that all listed claims are covered, that the contract attributes them to the stated occurrence, and that the stated retention and limit apply to the occurrence total. They also assume no coinsurance, exclusions, reinstatements, or other contract terms that change the payment. In practice, the policy wording determines which claims are grouped as one occurrence and how the retention and limit operate.

Calculate the Recovery from the Combined Loss

The recovery has three possible outcomes. If the occurrence loss does not exceed the retention, there is no recovery. If it exceeds the retention but does not exhaust the limit, the reinsurer pays the excess over the retention. If it exceeds the retention plus the limit, the recovery is capped at the limit.

C_j= \begin{cases} 0, & L_j \le R,\\ L_j-R, & R < L_j < R+M,\\ M, & L_j \ge R+M. \end{cases} \qquad I_j=L_j-C_j

Equivalently, the recovery is the smaller of the available excess, $\max(L_j-R,0)$, and the limit $M$. This formulation makes both contract terms visible: the retention controls when payment begins, and the limit caps the payment. The insurer’s retained loss is the occurrence loss minus the recovery; it includes both the amount below the retention and any amount above the exhausted layer.

Group the claims.
Identify the claims attributed to the same occurrence under the contract. Do not combine separate occurrences merely because they happened in the same year.

Find the occurrence loss.
Add the covered claim amounts for that occurrence to obtain $L_j$.

Apply the occurrence terms once.
Calculate the excess above $R$, floor it at zero, and cap it at $M$ to find $C_j$.

Reconcile the allocation.
Calculate $I_j=L_j-C_j$ and check that $I_j+C_j=L_j$.

Why the Formula Holds

The reinsurer cannot pay while the occurrence loss is within the insurer’s retention, so the payment is zero for $L_j\le R$. Once the loss exceeds $R$, the amount above the retention is $L_j-R$. That amount is payable only up to the contract limit. Taking the smaller of the nonnegative excess and $M$ therefore gives exactly the three cases in the piecewise formula.

\text{payment before the cap}=\max(L_j-R,0)

Capping that amount at $M$ gives $C_j=\min\{\max(L_j-R,0),M\}$. Since the reinsurer and insurer together bear the full covered occurrence loss, the insurer’s share is the balance $I_j=L_j-C_j$. This also explains why a recovery cannot exceed either the occurrence loss or the contract limit.

Worked Example: Several Claims, One Occurrence

A hypothetical windstorm produces covered claims of $45,000, $80,000, and $25,000. The insurer has a per-occurrence cover of $150,000 excess of $100,000. Calculate the recovery and the insurer’s retained occurrence loss.

Method. Combine the claims first, then apply the occurrence retention and limit once. This avoids treating an occurrence retention as if it were a separate deductible for each claim.

Step 1: Find the occurrence loss. \[ L=45{,}000+80{,}000+25{,}000=150{,}000. \]

Step 2: Find the excess above the retention. \[ \max(L-R,0)=\max(150{,}000-100{,}000,0)=50{,}000. \]

Step 3: Apply the limit. \[ C=\min(50{,}000,150{,}000)=50{,}000. \]

Step 4: Calculate the insurer’s retained loss and check. \[ I=150{,}000-50{,}000=100{,}000, \qquad I+C=100{,}000+50{,}000=150{,}000=L. \] The recovery is $50,000, and the insurer retains $100,000.

Worked Example: A Recovery That Exhausts the Limit

A hypothetical equipment failure is attributed to one occurrence and generates covered claims of $70,000, $95,000, $110,000, and $65,000. The reinsurance provides $160,000 excess of $120,000 per occurrence. Find the recovery and retained loss.

Method. Aggregate all four claims before testing the attachment and cap. The common error is to test each claim separately against the occurrence retention.

Step 1: Aggregate the claims. \[ L=70{,}000+95{,}000+110{,}000+65{,}000=340{,}000. \]

Step 2: Calculate the excess above the retention. \[ L-R=340{,}000-120{,}000=220{,}000. \]

Step 3: Cap the recovery at the limit. \[ C=\min(220{,}000,160{,}000)=160{,}000. \] The occurrence loss exceeds the layer’s exhaustion point of $R+M=120{,}000+160{,}000=280{,}000$, so the limit is fully used.

Step 4: Find retained loss and reconcile. \[ I=340{,}000-160{,}000=180{,}000, \qquad I+C=180{,}000+160{,}000=340{,}000. \] The reinsurer pays $160,000, and the insurer retains $180,000. The retained amount consists of the $120,000 below the retention and $60,000 above the exhausted layer.

Worked Example: Apply the Terms Separately to Two Occurrences

A hypothetical insurer reports two distinct covered events. Occurrence A has claims of $40,000, $90,000, and $130,000. Occurrence B has claims of $60,000, $50,000, and $30,000. The contract is $120,000 excess of $100,000 per occurrence. Calculate each recovery and the combined totals.

Method. Apply the occurrence terms independently to each event, then add the resulting recoveries and retained losses. Do not pool the two occurrences before applying the terms.

Occurrence A. Its loss is \[ L_A=40{,}000+90{,}000+130{,}000=260{,}000. \] The excess above the retention is $260{,}000-100{,}000=160{,}000$, so \[ C_A=\min(160{,}000,120{,}000)=120{,}000, \qquad I_A=260{,}000-120{,}000=140{,}000. \]

Occurrence B. Its loss is \[ L_B=60{,}000+50{,}000+30{,}000=140{,}000. \] The excess above the retention is $140{,}000-100{,}000=40{,}000$, which is below the limit. Thus, \[ C_B=\min(40{,}000,120{,}000)=40{,}000, \qquad I_B=140{,}000-40{,}000=100{,}000. \]

Combined check. Total loss is $260{,}000+140{,}000=400{,}000$; total recovery is $120{,}000+40{,}000=160{,}000$; and total retained loss is $140{,}000+100{,}000=240{,}000$. The reconciliation $160{,}000+240{,}000=400{,}000$ confirms the allocation.

Occurrence Terms Are Not Per-Risk Terms

The distinction is about the unit to which the retention and limit apply. For a per-occurrence cover, the unit is the combined covered loss arising from one occurrence. For a per-risk cover, the terms apply separately to each risk’s loss. The same claim amounts can therefore produce different recoveries under the two arrangements.

For example, with a $100,000 retention, three claims of $45,000, $80,000, and $25,000 produce a $50,000 recovery under the per-occurrence calculation above. Applying a $100,000 retention separately to each claim would produce no recovery because each individual claim is below $100,000. That calculation uses the wrong unit for an occurrence contract.

Question	Per-occurrence cover	Per-risk cover
What is tested against the retention?	Combined covered loss from one occurrence	Each covered risk loss separately
When is the limit applied?	Once to the recovery for that occurrence	Separately to each risk, according to the contract terms
What must be identified first?	Which claims belong to the same occurrence	Which loss belongs to each insured risk

Common Mistakes and Checks

A useful final check is to verify both the direction and the size of the result. The recovery must be nonnegative, cannot exceed $M$, and cannot exceed the occurrence loss. The retained amount must also be nonnegative, and recovery plus retention must equal the covered occurrence loss.

Common mistake: applying the retention claim by claim. With claims of $45,000, $80,000, and $25,000 and an occurrence retention of $100,000, an incorrect calculation is $\max(45{,}000-100{,}000,0)+\max(80{,}000-100{,}000,0)+\max(25{,}000-100{,}000,0)=0$. This treats the terms as per-risk terms. The correct occurrence calculation combines the claims first: $L=150{,}000$, so the recovery before any limit cap is $150{,}000-100{,}000=50{,}000$.

Common mistake: pooling separate occurrences. A per-occurrence limit is applied separately to each occurrence, not automatically once to all losses in a year. First group claims using the contract’s occurrence definition; then calculate the recovery for each group and sum the results.

Common mistake: forgetting the cap. If $L_j-R$ exceeds $M$, the recovery is $M$, not the full excess. Retained loss then includes the amount above $R+M$, as well as the amount below $R$.

Key takeaway: For each occurrence, add its covered claims, subtract the occurrence retention only from the combined loss, and cap the resulting recovery at the occurrence limit. Then calculate insurer retention as occurrence loss minus recovery and check that the two shares reconcile.

Check Your Understanding

For each problem, use the stated occurrence grouping and calculate the recovery and insurer-retained loss.

A single occurrence produces covered claims of $28,000, $42,000, and $55,000. The cover is $100,000 excess of $80,000 per occurrence. What are the recovery and retained loss? (Answer: Recovery $45,000; retained loss $80,000.)
One occurrence produces claims of $90,000, $45,000, and $75,000. The cover is $80,000 excess of $100,000 per occurrence. Calculate the recovery and retained loss. (Answer: Recovery $80,000; retained loss $130,000.)
Occurrence A produces claims of $30,000, $50,000, and $60,000. A distinct occurrence B produces claims of $20,000 and $40,000. The terms are $100,000 excess of $75,000 per occurrence. Find the recovery and retained loss for each occurrence, then the totals. (Answer: A recovery $65,000 and retained loss $75,000; B recovery $0 and retained loss $60,000; total recovery $65,000 and total retained loss $135,000.)
A covered occurrence has three claims: $80,000, $70,000, and $30,000. The contract is $90,000 excess of $50,000. Determine whether the limit is exhausted, and calculate the recovery and retained loss. (Answer: The limit is exhausted; recovery $90,000 and retained loss $90,000.)
Three claims of $40,000, $70,000, and $60,000 arise from one occurrence. A $150,000 excess-of-loss cover has a $100,000 occurrence retention. Calculate the correct recovery and retained loss, and state the recovery produced by the incorrect method of applying the retention separately to each claim. (Answer: Correct recovery $70,000 and retained loss $100,000; incorrect claim-by-claim recovery $0.)

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