Retained Loss Under a Layered Reinsurance Program

Tracing Loss Through a Reinsurance Program

A layered reinsurance program cannot be calculated reliably by looking at each contract in isolation. The result depends on which loss base each contract covers, the order in which recoveries apply, and whether a layer still has capacity when a later loss occurs. The cedent’s retained loss is the amount remaining after all recoveries that are available under the stated terms.

The calculations below assume all stated losses are covered, all amounts are in the same currency and on the same basis, and there are no exclusions, expenses, recoveries from other sources, or reinstatement premiums. In practice, the contract wording determines such matters as the covered loss basis and reinstatement terms. The calculation must follow those terms rather than assume a preferred order.

Definition: For a covered gross loss $G$, the cedent’s final retained loss is $G$ less the reinsurance recoveries that apply to that loss, after respecting the contracts’ order, attachment points, limits, and remaining capacity. A proportional contract cedes a stated share of its covered loss base. An excess of loss layer with attachment $a$ and limit $L$ pays the part of the covered loss above $a$, up to $L$, subject to any remaining aggregate capacity.

Set the Loss Base and Order First

A quota share with ceded proportion $q$ cedes $qG$ and leaves $(1-q)G$ with the cedent before other covers apply. If an excess of loss (XOL) contract inures after that quota share, the XOL calculation uses the post-quota-share loss, not the original gross loss. For a layer attaching at $a$ with per-occurrence limit $L$, its recovery before considering any annual capacity restriction is

R_{\mathrm{XOL}}=\min\bigl((X-a)^+,L\bigr), \qquad (z)^+=\max(z,0),

where $X$ is the loss base specified for that layer. If the contract has an aggregate limit or reinstatements, the actual recovery may be smaller than this per-occurrence amount because capacity may be partly or fully used already.

In an inuring sequence, a recovery from an earlier contract reduces the loss base passed to a later contract when the wording specifies that order. For example, if a per-risk XOL contract inures to a catastrophe XOL contract, first calculate the per-risk recoveries; then apply the catastrophe layer to the remaining event loss. Do not apply both contracts to the same unreduced loss unless their terms expressly provide otherwise.

Key takeaway: Write down the loss base for each contract before calculating recoveries. A clear sequence might be gross loss → proportional cession → per-risk XOL recovery → event-level XOL recovery → final retained loss.

Worked Examples

Worked Example: Quota Share Followed by a Per-Occurrence Layer

A hypothetical insurer has a covered gross loss of $1,200,000. A quota share cedes 25% of the covered loss. An XOL contract inures after the quota share and covers $300,000 in excess of $200,000 per occurrence. There is enough annual capacity to pay the full per-occurrence recovery. Calculate the cedent’s final retained loss.

Method. Apply the quota share first, then apply the XOL layer to the cedent’s post-quota-share loss.

Step 1: Calculate the quota share recovery. The ceded proportion is 25%, so

R_{\mathrm{QS}}=0.25(1{,}200{,}000)=300{,}000.

The loss remaining after the quota share is

$$ X=1{,}200{,}000-300{,}000=900{,}000. $$

Step 2: Calculate the XOL recovery. The loss above the $200,000 attachment is

$$ 900{,}000-200{,}000=700{,}000. $$

The layer limit is $300,000, so its recovery is

R_{\mathrm{XOL}}=\min(700{,}000,300{,}000)=300{,}000.

Step 3: Reconcile the final retained loss.

$$ 1{,}200{,}000-300{,}000-300{,}000=600{,}000. $$

The cedent retains $600,000. As a check, the post-quota-share loss was $900,000, of which the XOL contract paid $300,000, leaving $600,000.

Common mistake avoided: Applying the XOL layer to the original $1,200,000 would give a nominal recovery of $300,000 here as well, but that numerical coincidence does not make it correct. The covered XOL base is $900,000 because the quota share inures first; with different values, using gross loss could change the recovery.

Worked Example: Two Occurrences and One Full Reinstatement

A hypothetical cedent has two covered gross losses during a contract year: $600,000 and $1,000,000. A 20% quota share inures before an XOL contract covering $250,000 in excess of $150,000 per occurrence. The XOL contract has an initial annual capacity of $250,000 and one full reinstatement of $250,000. Assume the reinstated capacity is available for the second occurrence, there is no other annual aggregate limit, and no reinstatement premium is included. Calculate total recoveries and total retained loss.

Method. For each occurrence, calculate the quota share recovery and the XOL loss base. Then track the XOL’s remaining annual capacity in occurrence order.

Occurrence 1. The quota share recovery is

$$ 0.20(600{,}000)=120{,}000. $$

The post-quota-share loss is $600{,}000-120{,}000=480{,}000$. The amount above the XOL attachment is $480{,}000-150{,}000=330{,}000$, so the per-occurrence recovery is $\min(330{,}000,250{,}000)=250{,}000$. This exhausts the initial $250,000 annual capacity.

Occurrence 2. The quota share recovery is

$$ 0.20(1{,}000{,}000)=200{,}000. $$

The post-quota-share loss is $1{,}000{,}000-200{,}000=800{,}000$. The amount above the attachment is $800{,}000-150{,}000=650{,}000$, and the per-occurrence layer amount is $\min(650{,}000,250{,}000)=250{,}000$. The reinstated capacity makes this full $250,000 available.

Total quota share recoveries are $120{,}000+200{,}000=320{,}000$. Total XOL recoveries are $250{,}000+250{,}000=500{,}000$. The initial limit plus one full reinstatement provides total nominal capacity of $250{,}000+250{,}000=500{,}000$, so the calculated recoveries use exactly the available amount.

\text{Total retained}=1{,}600{,}000-320{,}000-500{,}000=780{,}000.

The cedent’s total retained loss is $780,000. As a check, the post-quota-share losses total $480{,}000+800{,}000=1{,}280{,}000$, and subtracting the $500,000 XOL recovery also gives $780,000.

Common mistake avoided: A reinstatement restores a specified amount of capacity; it does not erase the first recovery or create unlimited capacity. Here the maximum total XOL recovery is $500,000, not $250,000 per occurrence without limit and not an amount exceeding the initial limit plus the reinstatement.

Worked Example: Per-Risk Cover Inuring to an Event Layer

A hypothetical insurer incurs three covered claims from one event: $400,000, $700,000, and $1,000,000. A 10% quota share applies first. Next, a per-risk XOL contract covers $100,000 in excess of $200,000 per claim, with enough capacity for all three recoveries. A catastrophe XOL contract then covers $500,000 in excess of $400,000 of the event loss remaining after the per-risk recoveries. Its available capacity is at least $500,000. Calculate the event’s final retained loss.

Method. Process each claim through the quota share and per-risk layer. Aggregate the remaining claims for the event layer, then reconcile all recoveries against gross event loss.

Step 1: Apply the 10% quota share. The quota share recoveries on the three claims are $0.10(400{,}000)=40{,}000$, $0.10(700{,}000)=70{,}000$, and $0.10(1{,}000{,}000)=100{,}000$. They total

$$ 40{,}000+70{,}000+100{,}000=210{,}000. $$

The post-quota-share claims are respectively $360,000, $630,000, and $900,000.

Step 2: Apply the per-risk layer. For the $360,000 claim, the amount above $200,000 is $160,000; the layer pays $\min(160{,}000,100{,}000)=100{,}000$. For the $630,000 claim, the amount above attachment is $430,000, so the recovery is $100,000. For the $900,000 claim, the amount above attachment is $700,000, so the recovery is also $100,000. Total per-risk recoveries are

$$ 100{,}000+100{,}000+100{,}000=300{,}000. $$

The losses remaining after both earlier contracts are $260,000, $530,000, and $800,000. Their event total is $260{,}000+530{,}000+800{,}000=1{,}590{,}000$.

Step 3: Apply the catastrophe layer. The event loss above the $400,000 attachment is $1{,}590{,}000-400{,}000=1{,}190{,}000$. The layer limit caps the recovery at

R_{\mathrm{Cat}}=\min(1{,}190{,}000,500{,}000)=500{,}000.

Step 4: Reconcile. Gross event loss is $400{,}000+700{,}000+1{,}000{,}000=2{,}100{,}000$. Therefore,

\text{Final retained}=2{,}100{,}000-210{,}000-300{,}000-500{,}000=1{,}090{,}000.

The cedent retains $1,090,000. As a check, the loss after the quota share and per-risk recoveries is $1,590,000; subtracting the $500,000 catastrophe recovery leaves the same $1,090,000.

Common mistake avoided: Applying the catastrophe layer to the $1,890,000 remaining immediately after the quota share would ignore the stated inuring order. The catastrophe layer instead sees $1,590,000 after the per-risk recoveries.

Why the Reconciliation Works

The retained-loss identity follows from conservation of the covered loss across the cedent and reinsurers. For a single loss, if each recovery $R_j$ is an amount paid to the cedent for that loss, the amount left with the cedent is the gross loss less the sum of those payments:

G=\text{retained loss}+\sum_j R_j \quad\Longrightarrow\quad \text{retained loss}=G-\sum_j R_j.

The identity is an accounting reconciliation, not a substitute for contract calculations. Each $R_j$ must first be calculated on the correct covered loss base, with its attachment, limit, and remaining capacity applied. If two recoveries are calculated on overlapping loss without accounting for inuring order, the identity can still balance arithmetically while the recoveries themselves are wrong.

List the contract sequence.
Record which cover applies first and whether each later contract sees gross loss or loss net of earlier recoveries.

Calculate each loss base.
Apply proportional shares or earlier recoveries before calculating the next contract when the stated order requires it.

Apply attachment and limits.
Calculate the layer amount for each covered claim or event, then cap it at the per-occurrence limit and available aggregate capacity.

Track reinstatements.
Add only the capacity provided by the stated reinstatement terms, and keep a running total of capacity used.

Reconcile to gross loss.
Subtract all contract recoveries from gross loss and, where possible, check against the loss remaining at the end of the sequence.

Common Traps in Layered Programs

Using the wrong base. A layer that inures after a quota share generally applies to the loss net of that cession if the contract specifies that basis. Do not assume every layer responds to gross loss.
Confusing per-occurrence limits with annual capacity. A per-occurrence limit caps a recovery for one occurrence. Aggregate limits and reinstatements determine how much capacity remains across the contract period.
Forgetting earlier recoveries. When one contract inures to another, reduce the later contract’s loss base according to the stated order.
Giving a reinstatement more capacity than specified. A full reinstatement adds the stated restored limit; a partial reinstatement or separate aggregate cap may produce a different total.
Subtracting the same recovery twice. Track each recovery once, and distinguish the cedent’s loss before a layer from the recovery paid by that layer.

The practical value of a layer-by-layer calculation is that it makes both the result and its assumptions auditable. This approach is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Check Your Understanding

For each hypothetical program, follow the stated order and calculate the cedent’s final retained loss.

A gross loss is $800,000. A 30% quota share inures before an XOL layer of $250,000 in excess of $150,000, with sufficient capacity. What is the final retained loss? (Answer: $310,000)
Two gross losses are $400,000 and $900,000. A 25% quota share applies first. An XOL layer covers $200,000 in excess of $100,000 per occurrence, with an initial annual limit of $200,000 and one full reinstatement of $200,000 available for the second loss. What is the total retained loss? (Answer: $575,000)
Two claims from one event are $200,000 and $500,000. A 20% quota share applies first; then a per-risk layer covers $100,000 in excess of $100,000 per claim, with sufficient capacity; then an event layer covers $250,000 in excess of $200,000 of the remaining event loss. What is the final retained event loss? (Answer: $200,000)
Three gross occurrences are $300,000, $700,000, and $500,000. A 40% quota share inures before an XOL layer of $150,000 in excess of $100,000 per occurrence. The XOL has an initial annual limit of $150,000 and one full reinstatement, and recoveries are applied in the order listed. What is total retained loss? (Answer: $600,000)
Two claims from one event are $600,000 and $900,000. A one-third quota share applies first. A per-risk layer of $200,000 in excess of $100,000 per claim then inures to an event layer of $300,000 in excess of $250,000. Both layers have enough capacity. What is the final retained event loss? (Answer: $300,000)

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