Combine Quota Share and Excess-of-Loss Treaties

Why the Order of Reinsurance Matters

A quota-share treaty allocates a fixed proportion of covered losses to a reinsurer. A per-claim excess-of-loss treaty instead responds to the part of an individual loss within a specified layer. When both provisions apply, it is not enough to know the quota-share percentage and the layer attachment: the contract must also establish which amount is the base for each calculation.

This tutorial compares two clearly specified arrangements. In the first, quota share is applied to the gross loss and the excess-of-loss layer protects the insurer’s quota-share-retained amount. In the second, the layer is applied to the gross loss first, and quota share divides only the amount remaining after the layer payment. These are different treaty structures, not interchangeable calculation shortcuts. Actual treaty wording determines the appropriate structure.

Definition: Let $X$ be a covered claim amount, $q$ the quota-share fraction ceded to the quota-share reinsurer, and $r=1-q$ the insurer’s share before any excess-of-loss recovery. Let $A$ be the excess-of-loss attachment and $M$ the maximum recovery under the layer for one claim. For a layer applied to an amount $y$, its recovery is $$ L(y)=\min\big((y-A)_+,M\big), \qquad (z)_+=\max(z,0). $$ Thus the layer pays nothing when $y\le A$, pays $y-A$ when $A<y<A+M$, and pays $M$ when $y\ge A+M$.

Here $M$ is the maximum layer recovery, sometimes described as the layer width. For example, an attachment of $A=\$50{,}000$ and a maximum recovery of $M=\$40{,}000$ means the layer covers amounts above $\$50{,}000$, up to a maximum payment of $\$40{,}000$. It does not mean that the reinsurer pays the full claim above $\$40{,}000$.

The calculations below consider covered claim amounts and per-claim treaty limits. They do not include premiums, expenses, reinstatements, annual aggregate limits, or interest. Unless a problem specifies otherwise, assume that all stated treaty shares and limits apply to the same claim basis and currency.

Quota Share Before the Layer

Under a quota-share-first arrangement, the quota-share reinsurer pays $qX$, leaving the insurer with $rX$ before the excess-of-loss recovery. The layer is applied to that retained amount. The insurer’s final net payment is the retained amount less the layer recovery.

\begin{aligned} Q_{\mathrm{QS}}&=qX,\\ Q_{\mathrm{XOL,\,QS\ first}}&=L(rX)=\min\big((rX-A)_+,M\big),\\ I_{\mathrm{QS\ first}}&=rX-L(rX). \end{aligned}

Here $Q_{\mathrm{QS}}$ and $Q_{\mathrm{XOL,\,QS\ first}}$ denote the payments by the quota-share and excess-of-loss reinsurers, respectively, and $I_{\mathrm{QS\ first}}$ is the insurer’s net claim payment. The excess-of-loss recovery is calculated on $rX$, not on $X$, because the layer in this arrangement protects the insurer’s retained share.

Worked Example: A Layer Applied to the Quota-Share Retention

A hypothetical covered claim is $X=\$120{,}000$. A quota-share treaty cedes $q=0.40$, so the insurer retains $r=0.60$. An excess-of-loss treaty pays up to $M=\$40{,}000$ per claim above an attachment of $A=\$50{,}000$, applied to the insurer’s retained amount. Find all three payments. Ignore interest and expenses.

Method. Apply the quota share first, then calculate the layer recovery on the remaining $60\%$ of the claim.

Step 1: Quota-share payment. $$ Q_{\mathrm{QS}}=0.40(120{,}000)=\$48{,}000. $$ The insurer’s amount before the layer is $$ rX=0.60(120{,}000)=\$72{,}000. $$

Step 2: Excess-of-loss payment. The amount above the attachment on the retained basis is $72{,}000-50{,}000=\$22{,}000$. This is below the $\$40{,}000$ maximum recovery, so $$ Q_{\mathrm{XOL,\,QS\ first}} =\min(72{,}000-50{,}000,40{,}000) =\$22{,}000. $$

Step 3: Insurer’s net payment. $$ I_{\mathrm{QS\ first}}=72{,}000-22{,}000=\$50{,}000. $$

Check. The payments add to the gross claim: $$ 48{,}000+22{,}000+50{,}000=\$120{,}000. $$ The common mistake avoided here is applying the layer to the original $\$120{,}000$ claim. The stipulated layer basis is the insurer’s $\$72{,}000$ retained amount.

Layer Before Quota Share

Under the second arrangement, the excess-of-loss layer is first applied to the gross claim. Its payment is $L(X)$, and the remaining amount is $X-L(X)$. The quota-share reinsurer then takes fraction $q$ of that remaining amount; the insurer pays fraction $r$ of it.

\begin{aligned} Q_{\mathrm{XOL,\,layer\ first}}&=L(X),\\ Q_{\mathrm{QS,\,layer\ first}}&=q\big(X-L(X)\big),\\ I_{\mathrm{layer\ first}}&=r\big(X-L(X)\big). \end{aligned}

The contract structure here treats the layer recovery as paid directly by the excess-of-loss reinsurer, with quota share dividing only the post-layer amount. This precise definition matters: other contract wording might allocate a layer recovery differently. A calculation should not silently assume that a quota-share fraction applies to a recovery unless the treaty says so.

Conservation check: Under either arrangement, the insurer’s payment and the two reinsurers’ payments must sum to the covered gross claim $X$, provided the stated treaties allocate the entire claim and there are no other adjustments. This is a useful check for arithmetic and for omitted or double-counted payments.

Worked Example: Comparing the Two Treaty Sequences

Use the hypothetical claim $X=\$80{,}000$, a quota-share fraction $q=0.25$, an attachment $A=\$50{,}000$, and a maximum layer recovery $M=\$30{,}000$. Calculate each party’s payment under both arrangements. Ignore interest and expenses.

Method. First calculate quota share before the layer; then, separately, calculate the gross-claim layer before quota share. Do not carry a recovery from one arrangement into the other.

Arrangement 1: quota share first. The insurer’s share before the layer is $r=1-0.25=0.75$, so $$ rX=0.75(80{,}000)=\$60{,}000, \qquad Q_{\mathrm{QS}}=0.25(80{,}000)=\$20{,}000. $$ The layer recovery on the retained amount is $$ L(60{,}000)=\min(60{,}000-50{,}000,30{,}000)=\$10{,}000. $$ The insurer therefore pays $60{,}000-10{,}000=\$50{,}000$. The payments are $\$20{,}000$ from the quota-share reinsurer, $\$10{,}000$ from the excess-of-loss reinsurer, and $\$50{,}000$ from the insurer. Their sum is $\$80{,}000$.

Arrangement 2: layer first. The gross-claim layer recovery is $$ L(80{,}000)=\min(80{,}000-50{,}000,30{,}000)=\$30{,}000. $$ The amount remaining after that recovery is $80{,}000-30{,}000=\$50{,}000$. The quota-share reinsurer pays $0.25(50{,}000)=\$12{,}500$, and the insurer pays $0.75(50{,}000)=\$37{,}500$. The excess-of-loss reinsurer pays $\$30{,}000$. The payments sum to $$ 30{,}000+12{,}500+37{,}500=\$80{,}000. $$

The two structures produce different allocations: the insurer pays $\$50{,}000$ under the first and $\$37{,}500$ under the second. The common mistake avoided is treating “quota share plus excess of loss” as a complete specification without identifying each treaty’s payment basis.

Why the Layer Formula Works

The layer formula follows directly from the covered interval. If the amount $y$ does not exceed the attachment, the layer is not reached and the payment is zero. If $y$ exceeds the attachment but does not reach the top of the layer, the payment is the amount above attachment. Once $y$ reaches $A+M$, the maximum recovery $M$ has been exhausted.

Below attachment.
If $y\le A$, then $(y-A)_+=0$, so $L(y)=\min(0,M)=0$.

Within the layer.
If $A<y<A+M$, then $0<y-A<M$, so $L(y)=y-A$.

At or above the layer top.
If $y\ge A+M$, then $y-A\ge M$, so $L(y)=M$.

These cases establish the piecewise form and explain why the minimum is needed: the positive-part operation prevents a payment below attachment, while the minimum caps the payment at the layer limit. For the quota-share-first arrangement, substitute $y=rX$ into this result. For the layer-first arrangement, substitute $y=X$. The different substitution is the source of the different outcomes.

L(y)= \begin{cases} 0, & y\le A,\\ y-A, & A<y<A+M,\\ M, & y\ge A+M. \end{cases}

Expected Payments for a Discrete Claim Model

If the claim amount $X$ is random, first determine each party’s payment for each possible value of $X$. Then multiply each payment by the probability of that claim amount and add. This is the discrete expectation rule; it does not require the insurer’s net payment to have the same form under the two treaty sequences.

Worked Example: Expected Allocation Across Claim Sizes

A hypothetical portfolio has one covered claim with $X$ equal to $\$40{,}000$, $\$100{,}000$, or $\$160{,}000$, each with probability $1/3$. A quota-share treaty cedes $q=0.50$. A per-claim layer has attachment $A=\$40{,}000$ and maximum recovery $M=\$40{,}000$. Find the expected payments under each arrangement. Ignore interest and expenses.

Method. Calculate the three payments at each claim size, then average each payment column. All amounts in the table are in thousands of dollars.

Claim $X$	Quota share first: QS	Quota share first: XOL	Quota share first: insurer	Layer first: XOL	Layer first: QS	Layer first: insurer
40	20	0	20	0	20	20
100	50	10	40	40	30	30
160	80	40	40	40	60	60

Step 1: Verify the entries. For quota share first, the retained amounts are $20$, $50$, and $80$. Their layer recoveries are $$ L(20)=0,\qquad L(50)=\min(50-40,40)=10,\qquad L(80)=\min(80-40,40)=40. $$ The insurer’s payments are consequently $20$, $50-10=40$, and $80-40=40$, all in thousands. For layer first, the gross-claim layer recoveries are $L(40)=0$, $L(100)=40$, and $L(160)=40$. The residual amounts are $40$, $60$, and $120$; quota share and the insurer each receive half of each residual.

Step 2: Average each payment. For quota share first, the expected quota-share payment is $(20+50+80)/3=50$ thousand, the expected excess-of-loss payment is $(0+10+40)/3=50/3$ thousand, and the expected insurer payment is $(20+40+40)/3=100/3$ thousand. In dollars, these are $\$50{,}000$, $\$16{,}666.67$, and $\$33{,}333.33$, respectively, rounded to the nearest cent.

For layer first, the expected excess-of-loss payment is $(0+40+40)/3=80/3$ thousand, the expected quota-share payment is $(20+30+60)/3=110/3$ thousand, and the expected insurer payment is $(20+30+60)/3=110/3$ thousand. In dollars, these are $\$26{,}666.67$, $\$36{,}666.67$, and $\$36{,}666.67$, respectively, rounded to the nearest cent.

Check. The mean gross claim is $(40+100+160)/3=100$ thousand. Under either arrangement, the three expected payments sum to $\$100{,}000$. The common mistake avoided is averaging only the insurer’s payments and forgetting that each treaty also has a distinct expected payment.

Exam Traps and Contract Checks

The main challenge is usually not evaluating the minimum in the layer formula. It is keeping the claim bases straight. Before calculating, write down the amount to which each treaty applies, then use that same basis consistently.

Using the gross claim as the layer basis in every arrangement. In the quota-share-first structure taught here, the layer basis is $rX$; in the layer-first structure, it is $X$.
Multiplying the layer recovery by a quota-share percentage without a stated allocation rule. A layer recovery is the payment specified for the excess-of-loss treaty. It is not automatically split again.
Confusing attachment with maximum recovery. The attachment $A$ is where the layer begins; $M$ is the maximum payment. The layer top is $A+M$.
Failing to check conservation. For each claim, add the insurer’s net payment and all reinsurer payments. If the sum is not $X$, revisit the bases and arithmetic.
Assuming order is merely a calculation convenience. The two explicitly defined sequences can yield different insurer and reinsurer payments. Treaty wording, not a preferred algebraic order, determines which calculation is appropriate.

Key takeaway: Define the order and payment bases before calculating. For quota share followed by a layer on the retained amount, use $L(rX)$. For a gross-claim layer followed by quota share on the residual, use $L(X)$ and then divide $X-L(X)$. Check that all payments reconcile to $X$.

Check Your Understanding

For each problem, state the applicable basis and use the specified treaty sequence. Treat all amounts as covered claim amounts; ignore interest and expenses.

A claim is $\$100{,}000$. Quota share first cedes $30\%$, and a layer on the insurer’s retained amount attaches at $\$40{,}000$ with a maximum recovery of $\$30{,}000$. Find the quota-share payment, layer recovery, and insurer payment. (Answer: $\$30{,}000$, $\$30{,}000$, and $\$40{,}000$, respectively.)
A claim is $\$60{,}000$. A gross-claim layer pays above $\$30{,}000$, up to $\$20{,}000$, and then a $25\%$ quota share applies to the residual. Find the excess-of-loss payment, quota-share payment, and insurer payment. (Answer: $\$20{,}000$, $\$10{,}000$, and $\$30{,}000$, respectively.)
A claim is $\$150{,}000$. Quota share first cedes $20\%$, and the layer applies to the retained amount with attachment $\$50{,}000$ and maximum recovery $\$40{,}000$. Find all three payments and verify their sum. (Answer: quota share $\$30{,}000$, excess of loss $\$40{,}000$, insurer $\$50{,}000$; total $\$120{,}000$??)
A single claim is $\$50{,}000$ with probability $0.40$, or $\$100{,}000$ with probability $0.60$. Quota share first cedes $40\%$; a layer on the retained amount attaches at $\$40{,}000$ and has a maximum recovery of $\$30{,}000$. Find the expected insurer payment. (Answer: $\$36{,}000$.)
A claim is $\$90{,}000$. A gross-claim layer attaches at $\$30{,}000$, has a maximum recovery of $\$30{,}000$, and is followed by a quota share that cedes $50\%$ of the residual. Find all three payments and check conservation. (Answer: excess of loss $\$30{,}000$, quota share $\$30{,}000$, insurer $\$30{,}000$; total $\$90{,}000$.)

What You'll Learn