Diagnosing Common Reinsurance Calculation Errors

Why Reinsurance Calculations Go Wrong

Reinsurance calculations often involve simple arithmetic but careful interpretation. A percentage must be applied to the amount specified by the treaty; an excess-of-loss retention and limit play different roles; and an occurrence cover generally responds to the combined covered losses from an occurrence, not to each claim in isolation. A correct-looking calculation can therefore be wrong because it uses the wrong base or unit.

The reliable approach is to identify the covered amount, the treaty’s attachment and cap, and the order in which covers apply before calculating. All examples below describe hypothetical arrangements. Unless stated otherwise, amounts are in dollars, recoveries are limited to covered losses, and there are no exclusions, expenses, reinstatements, or other treaty terms affecting the calculations.

Definition: A ceded loss is the amount recoverable from a reinsurer under the stated treaty. A retained loss is the amount remaining with the insurer after applying the treaty or treaties. For a loss allocation, the basic check is gross loss = ceded loss + retained loss. This equality assumes the calculation includes all relevant recoveries and retained amounts.

First Identify the Treaty and Its Base

A quota-share treaty cedes a stated proportion $q$ of the covered gross amount. If the treaty applies to a gross loss $X$, then the ceded amount is $qX$ and the insurer’s retained amount is $(1-q)X$. The percentage $q$ is the ceded share; it is not the retained share.

An excess-of-loss cover instead responds to loss above a specified retention, subject to a maximum recovery called the limit. For a single covered amount $X$, let $R$ be the retention and $L$ the limit. The amount in excess of the retention is $\max(X-R,0)$, but the reinsurer cannot pay more than $L$.

\text{Excess-of-loss recovery}=\min\bigl(\max(X-R,0),L\bigr).

For an occurrence cover, $X$ in this formula is the total covered loss from that occurrence, not automatically each claim within it. The treaty wording determines what constitutes an occurrence and which losses are covered. In the examples, the occurrence groupings are given explicitly.

Why the Excess-of-Loss Formula Works

The formula follows directly from the attachment and cap. If the covered loss does not exceed the retention, there is no loss above the attachment and the recovery is zero. If the loss exceeds the retention, the amount above it is $X-R$. That excess is payable only up to the limit.

\text{Recovery} = \begin{cases} 0, & X\le R,\\ X-R, & R<X<R+L,\\ L, & X\ge R+L. \end{cases}

These cases are equivalent to $\min(\max(X-R,0),L)$. They also show why retention and limit cannot be used interchangeably: $R$ determines where the cover attaches, while $L$ determines the most the reinsurer can pay. Once the layer is exhausted, additional loss above $R+L$ is not recovered under that layer.

Worked Example: Applying a Quota Share to the Correct Amount

A hypothetical insurer has a 30% quota-share treaty on gross covered losses. Three claims are $48,000, $120,000, and $32,000. Find the ceded and retained loss for each claim and in total. Name the calculation method and identify the common mistake it avoids.

Method. Apply the treaty’s 30% ceded share to each gross loss. The retained share is $1-0.30=0.70$, or 70%.

For the $48,000 claim, the ceded amount is $0.30(48{,}000)=\$14{,}400$, and the retained amount is $0.70(48{,}000)=\$33{,}600$.

For the $120,000 claim, the ceded amount is $0.30(120{,}000)=\$36{,}000$, and the retained amount is $0.70(120{,}000)=\$84{,}000$.

For the $32,000 claim, the ceded amount is $0.30(32{,}000)=\$9{,}600$, and the retained amount is $0.70(32{,}000)=\$22{,}400$.

The total gross loss is $48{,}000+120{,}000+32{,}000=\$200{,}000$. Total ceded loss is $14{,}400+36{,}000+9{,}600=\$60{,}000$. Total retained loss is $33{,}600+84{,}000+22{,}400=\$140{,}000$. The check is $\$60{,}000+\$140{,}000=\$200{,}000$, equal to gross loss.

Common mistake avoided: treating the retained 70% as the ceded share. That would incorrectly report $140,000 ceded and $60,000 retained. The treaty says the reinsurer takes 30%, so the ceded amount must be 30% of gross loss.

Keep Retention and Limit Separate

A convenient way to avoid confusing the two values is to write down the layer’s attachment, maximum recovery, and exhaustion point before applying the formula. For a cover with retention $R$ and limit $L$, the layer attaches at $R$ and is exhausted at $R+L$. The phrase “$L$ excess of $R$” describes a layer with limit $L$ above retention $R$, not a limit equal to $R$.

Worked Example: Distinguishing Retention from Limit

A hypothetical per-risk excess-of-loss treaty covers $250,000 excess of a $100,000 retention on each claim. The covered claims are $80,000, $160,000, and $390,000. Calculate the recovery and retained amount for each claim, then reconcile the totals. Name the method and the mistake it avoids.

Method. Here $R=\$100{,}000$ and $L=\$250{,}000$, so the layer is exhausted at $R+L=\$350{,}000$. For each claim $X$, calculate $\min(\max(X-R,0),L)$, then subtract the recovery from the claim to find the retained amount.

For the $80,000 claim, $X<R$. Recovery is $\min(\max(80{,}000-100{,}000,0),250{,}000)=\$0$. Retained loss is $80{,}000-0=\$80{,}000$.

For the $160,000 claim, the amount above retention is $160{,}000-100{,}000=\$60{,}000$, below the $250,000 limit. Recovery is $60,000, and retained loss is $160{,}000-60{,}000=\$100{,}000$.

For the $390,000 claim, the amount above retention is $390{,}000-100{,}000=\$290{,}000$. The recovery is capped at the $250,000 limit, so retained loss is $390{,}000-250{,}000=\$140{,}000$.

Total gross loss is $80{,}000+160{,}000+390{,}000=\$630{,}000$. Total recovery is $0+60{,}000+250{,}000=\$310{,}000$. Total retained loss is $80{,}000+100{,}000+140{,}000=\$320{,}000$. The reconciliation is $\$310{,}000+\$320{,}000=\$630{,}000$.

Common mistake avoided: treating $250,000 as the retention or as the total claim cap. The retention is $100,000, and the maximum recovery is $250,000. For the $390,000 claim, the insurer retains both the first $100,000 and the $40,000 above the exhausted layer.

Apply Occurrence Cover to the Occurrence Total

A per-risk or per-claim cover uses the covered amount for an individual risk or claim, as specified by the contract. An occurrence cover uses the combined covered loss assigned to an occurrence. Applying an occurrence attachment separately to each claim can miss losses that collectively exceed the retention. Conversely, combining claims from separate occurrences can produce a recovery the treaty does not provide.

Worked Example: Grouping Claims Before an Occurrence Recovery

A hypothetical occurrence excess-of-loss treaty has a $100,000 retention and a $200,000 limit per occurrence. One occurrence produces two covered claims of $70,000 and $80,000. A separate occurrence produces one covered claim of $150,000. Calculate the recovery and retained amount for each occurrence and in total. Name the method and the error it avoids.

Method. Add covered claims within each stated occurrence, then apply the layer formula once to each occurrence total.

For the first occurrence, the total is $70{,}000+80{,}000=\$150{,}000$. Its recovery is $\min(\max(150{,}000-100{,}000,0),200{,}000)=\$50{,}000$. Its retained amount is $150{,}000-50{,}000=\$100{,}000$.

For the separate occurrence, the total is $150,000. Its recovery is also $\min(\max(150{,}000-100{,}000,0),200{,}000)=\$50{,}000$, leaving $100,000 retained.

Across both occurrences, gross loss is $150{,}000+150{,}000=\$300{,}000$, recovery is $50{,}000+50{,}000=\$100{,}000$, and retained loss is $100{,}000+100{,}000=\$200{,}000$. The check is $\$100{,}000+\$200{,}000=\$300{,}000$.

Common mistake avoided: applying the $100,000 retention individually to the first occurrence’s $70,000 and $80,000 claims. That would produce zero recovery for both claims and miss the $50,000 recovery on their $150,000 occurrence total. The separate $150,000 occurrence is calculated on its own; losses from the two occurrences are not combined into one attachment.

Check Treaty Order Before Reusing an Amount

When a hypothetical arrangement includes more than one treaty, the stated order and basis of application matter. A later treaty may apply to gross losses, to losses after a quota-share cession, or to a defined occurrence total. Do not assume the base: use the wording provided. The following example specifies that the excess-of-loss cover applies after the quota-share cession.

Worked Example: Applying Excess of Loss After a Quota Share

A hypothetical insurer first cedes 20% of each gross claim under a quota-share treaty. A per-risk excess-of-loss treaty then applies to the insurer’s remaining amount, with a $100,000 retention and a $200,000 limit. For one gross claim of $250,000, find each recovery and the insurer’s final retained loss. Name the method and the error it avoids.

Method. Follow the stated sequence: calculate quota-share cession first, then apply excess of loss to the post-quota-share amount.

The quota-share recovery is $0.20(250{,}000)=\$50{,}000$. The amount remaining for the insurer before the excess-of-loss treaty is $250{,}000-50{,}000=\$200{,}000$.

The excess above the $100,000 retention is $200{,}000-100{,}000=\$100{,}000$. This is less than the $200,000 limit, so the excess-of-loss recovery is $100,000. Final retained loss is $200{,}000-100{,}000=\$100{,}000$.

Total ceded loss is $50{,}000+100{,}000=\$150{,}000$. The reconciliation is $\$150{,}000+\$100{,}000=\$250{,}000$, equal to the gross claim.

Common mistake avoided: applying excess of loss to the original $250,000 after the quota share has already reduced the insurer’s covered amount. On the gross claim, that layer would appear to recover $150,000; adding the $50,000 quota-share recovery would incorrectly report $200,000 ceded and only $50,000 retained. The stated order requires the excess-of-loss calculation to use $200,000.

A Practical Error-Checking Routine

Before finalizing a reinsurance allocation, label the amount entering each calculation. Then check that the treaty’s type, base, and scope match the calculation. A short sequence helps keep the reasoning transparent.

Identify the covered amount.
Decide whether the treaty applies to a gross claim, a post-cession claim, or the total for a specified occurrence.

Read the treaty terms literally.
For quota share, separate the ceded percentage from the retained percentage. For excess of loss, label retention $R$ and limit $L$ separately.

Apply the cover at the right level.
Calculate claim by claim for a per-risk cover, or aggregate the covered claims within each occurrence for an occurrence cover.

Reconcile the allocation.
Check that total ceded loss plus total retained loss equals the relevant gross covered loss.

The reconciliation is necessary but not sufficient: two incorrect amounts can still add to the gross total. Check the contract interpretation as well as the arithmetic. In particular, verify that a quota-share percentage was applied to the intended base, that the excess layer begins at the stated retention and is capped at the stated limit, and that claims were grouped according to the stated cover.

This calculation discipline is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus. The central habit is simple: name the amount entering the treaty before calculating what leaves it as recovery.

Check Your Understanding

For each hypothetical arrangement, determine the requested recovery or allocation and check the treaty base and scope.

A 25% quota-share treaty applies to two gross claims of $40,000 and $90,000. Find total ceded and retained losses. (Answer: $32,500 ceded; $97,500 retained.)
A per-risk excess-of-loss cover has a $75,000 retention and a $125,000 limit. Three covered claims are $50,000, $140,000, and $260,000. Find total recovery and retained loss. (Answer: $190,000 recovery; $260,000 retained.)
An occurrence cover has a $120,000 retention and a $180,000 limit. One occurrence contains claims of $45,000 and $95,000; a separate occurrence has a $210,000 claim. Find total recovery and retained loss. (Answer: $110,000 recovery; $240,000 retained.)
A 30% quota share is applied first to a $300,000 gross claim. A per-risk excess-of-loss cover then applies to the post-quota-share amount, with an $80,000 retention and a $120,000 limit. Find total ceded and final retained loss. (Answer: $210,000 ceded; $90,000 retained.)
A $100,000-retention, $200,000-limit occurrence cover applies to one occurrence with claims of $65,000 and $85,000. A calculation reports zero recovery by applying the retention to each claim separately. Correct the recovery and retained amount, and state the calculation error. (Answer: $50,000 recovery; $100,000 retained; the claims should be aggregated to a $150,000 occurrence loss before applying the retention.)

What You'll Learn