Order of Treaty Application

Why Treaty Order Matters

When a claim is subject to more than one reinsurance treaty, the recovery under each treaty depends on the loss amount to which that treaty applies. If an excess-of-loss treaty applies to a claim after a quota-share cession, its subject loss is smaller than the original claim. If quota share instead applies to the amount remaining after an excess-of-loss recovery, the proportional cession is calculated on a different base. Unless the contract specifies the bases and order, the calculation is ambiguous.

The calculations here concern hypothetical treaties applied per claim. They do not describe annual aggregate protection. Treat each claim separately and add the results only when the treaties apply claim by claim and the problem asks for portfolio totals.

Definition: Let $x$ be one gross claim, $q$ the fraction ceded under quota share, and $r=1-q$ the insurer's retained fraction. A quota-share cession on subject loss $z$ is $qz$, leaving $rz$. For an excess-of-loss treaty with attachment $a$ and limit $L$, the recovery on subject loss $z$ is $$ T(z)=\min\bigl((z-a)_+,L\bigr), \qquad (w)_+=\max(w,0). $$ The insurer retains the subject loss less the recovery. The attachment is the amount the insurer bears before the layer responds, and the limit is the maximum recovery.

State the Basis Before Calculating

A useful way to specify order is to state, for each treaty, the loss amount to which it applies. In a quota-share-then-excess-of-loss sequence, quota share first cedes $qx$, leaving $rx$; the excess-of-loss recovery is $T(rx)$. In an excess-of-loss-then-quota-share sequence, the first recovery is $T(x)$; quota share then applies to the remaining $x-T(x)$.

\begin{aligned} C_{Q\to X}(x)&=qx+T(rx),& R_{Q\to X}(x)&=rx-T(rx),\\ C_{X\to Q}(x)&=T(x)+q\bigl(x-T(x)\bigr),& R_{X\to Q}(x)&=r\bigl(x-T(x)\bigr). \end{aligned}

Here $C$ is total ceded recovery across both treaties and $R$ is the insurer's final retained claim. Each ceded amount is calculated on the stated subject loss; it is not automatically a share of the original gross claim. The notation $Q\to X$ or $X\to Q$ describes the calculation sequence, not a universal contractual convention. Actual treaty wording determines the subject loss and priority.

Worked Example: One Claim Under Two Unscaled Sequences

A hypothetical insurer has a gross claim of $x=\$200$. It cedes 40% under quota share, so $q=0.40$ and $r=0.60$. A per-claim excess-of-loss treaty pays up to $L=\$100$ above attachment $a=\$50$. Compare the two sequences, using the same dollar attachment and limit on the subject loss at each step.

Method: quota share first. The quota-share recovery is $0.40(\$200)=\$80$. The remaining subject loss is $0.60(\$200)=\$120$. The excess-of-loss recovery is $$ T(\$120)=\min(\$120-\$50,\$100)=\min(\$70,\$100)=\$70. $$ Total ceded recovery is $\$80+\$70=\$150$. The insurer retains $\$120-\$70=\$50$.

Method: excess of loss first. The first recovery is $$ T(\$200)=\min(\$200-\$50,\$100)=\min(\$150,\$100)=\$100. $$ The amount remaining is $\$200-\$100=\$100$. Quota share then cedes $0.40(\$100)=\$40$, leaving the insurer with $0.60(\$100)=\$60$. Total ceded recovery is $\$100+\$40=\$140$.

Check and trap avoided. In the first sequence, $\$150+\$50=\$200$; in the second, $\$140+\$60=\$200$. The results differ because the excess-of-loss treaty sees $\$120$ in the first sequence but $\$200$ in the second. Applying the quota-share percentage to the original claim in the second sequence would incorrectly calculate a $\$80$ quota-share recovery; its stated base is the $\$100$ remaining claim.

Portfolio Calculations Follow the Same Basis

For per-claim treaties, apply the specified sequence to each claim and then sum the recoveries and retained amounts. Do not first combine claims into an annual total unless the treaty is explicitly written on an aggregate basis. A portfolio total can conceal claim-level differences, so a clear calculation table is often helpful.

Worked Example: Compare Sequences Across Three Claims

A hypothetical insurer has three claims of $\$30$, $\$100$, and $\$180$. Quota share cedes $q=0.30$, leaving $r=0.70$. The per-claim excess-of-loss treaty has attachment $a=\$40$ and limit $L=\$60$. In each sequence, apply the same attachment and limit to the loss amount reaching the excess-of-loss treaty.

Quota share first. For the $\$30$ claim, quota share cedes $0.30(\$30)=\$9$, leaving $\$21$; the layer pays $T(\$21)=\$0$. For the $\$100$ claim, quota share cedes $\$30$, leaving $\$70$; the layer pays $\min(\$70-\$40,\$60)=\$30$, so total cession is $\$60$ and retention is $\$40$. For the $\$180$ claim, quota share cedes $\$54$, leaving $\$126$; the layer pays $\min(\$126-\$40,\$60)=\$60$, so total cession is $\$114$ and retention is $\$66$.

The quota-share recoveries total $\$9+\$30+\$54=\$93$, and the excess-of-loss recoveries total $\$0+\$30+\$60=\$90$. Total ceded recovery is $\$183$; total retained loss is $\$21+\$40+\$66=\$127$. The gross claims total $\$30+\$100+\$180=\$310$, and $\$183+\$127=\$310$.

Excess of loss first. On the three gross claims, layer recoveries are $T(\$30)=\$0$, $T(\$100)=\$60$, and $T(\$180)=\$60$. The remaining losses are $\$30,\$40,\$120$. Quota share then recovers $\$9,\$12,\$36$, respectively. The total recoveries by claim are therefore $\$9,\$72,\$96$; insurer retentions are $\$21,\$28,\$84$.

Total excess-of-loss recovery is $\$120$, total quota-share recovery is $\$57$, and total ceded recovery is $\$177$. Total retention is $\$21+\$28+\$84=\$133$. The check is $\$177+\$133=\$310$, the portfolio's gross loss total.

Check and trap avoided. The two sequences produce different portfolio recoveries: $\$183$ versus $\$177$. The layer's limit is applied to each claim separately, not once to the portfolio total. Applying the $\$60$ limit just once to the sum of claims would change the treaty basis.

When Scaling the Layer Removes the Difference

A change of order does not have to change the insurer's final retention. If quota share leaves a fraction $r$ of each loss, scaling both the excess-of-loss attachment and limit by $r$ makes the layer on the post-quota-share loss equal to $r$ times the gross-loss layer. This is a particular, explicitly defined arrangement—not a general property of sequential treaties.

\begin{aligned} T_r(rx) &=\min\bigl((rx-ra)_+,rL\bigr)\\ &=r\min\bigl((x-a)_+,L\bigr)\\ &=rT(x), \qquad r>0. \end{aligned}

The second equality follows because multiplication by a positive $r$ preserves the order of the nonnegative amounts being compared: $(rx-ra)_+=r(x-a)_+$, and $\min(ru,rv)=r\min(u,v)$. Therefore, when the quota-share-first layer has attachment $ra$ and limit $rL$, its total cession is $qx+rT(x)$. In the reverse sequence, excess of loss first cedes $T(x)$, then quota share cedes $q(x-T(x))$. That total is also $qx+rT(x)$. Thus the insurer retains the same amount under these stated scaled terms.

Worked Example: Verify Equivalence With Scaled Terms

A hypothetical gross claim is $\$150$. Quota share cedes $q=0.30$, leaving $r=0.70$. On gross loss, the excess-of-loss terms are attachment $a=\$80$ and limit $L=\$40$. Compare excess of loss first with quota share afterward against quota share first with the layer terms scaled to the post-quota-share basis.

Excess of loss first. The gross-loss layer pays $$ T(\$150)=\min(\$150-\$80,\$40)=\$40. $$ The remaining claim is $\$150-\$40=\$110$. Quota share cedes $0.30(\$110)=\$33$, leaving retention $0.70(\$110)=\$77$. Total cession is $\$40+\$33=\$73$.

Quota share first with scaled layer terms. Quota share cedes $0.30(\$150)=\$45$, leaving $0.70(\$150)=\$105$. The scaled attachment is $0.70(\$80)=\$56$, and the scaled limit is $0.70(\$40)=\$28$. The layer pays $$ \min(\$105-\$56,\$28)=\min(\$49,\$28)=\$28. $$ The insurer retains $\$105-\$28=\$77$, and total cession is $\$45+\$28=\$73$.

Check and trap avoided. Both sequences satisfy $\$73+\$77=\$150$ and produce the same retention. If the quota-share-first calculation used the unscaled $\$80$ attachment and $\$40$ limit, it would instead pay $\min(\$105-\$80,\$40)=\$25$, yielding retention $\$80$. That is a different treaty basis, not a contradiction of the scaled-term result.

Interpret the Contract, Not Just the Formula

The order calculation is only meaningful after identifying what each treaty covers. For every treaty, record its subject loss, attachment, limit, and priority. In particular, determine whether the excess-of-loss layer covers the gross claim or the amount remaining after quota share, and whether quota share applies to gross loss or to the post-layer amount. If the wording instead defines separate recoveries on the same gross loss, the allocation is not the sequential structure calculated here.

A useful consistency check is that total cessions plus final retention equal the gross claim, provided the specified recoveries are allocated without overlap. A mismatch often indicates that a recovery was calculated on the wrong base, a quota-share percentage was applied twice, or a layer payment was not limited correctly. For a portfolio of per-claim treaties, verify the identity claim by claim before summing.

Key takeaway: There is no universal order rule for quota share and excess of loss. Define each treaty's subject loss first. With unscaled attachment and limit, reversing the sequence can change recoveries; appropriately scaled terms can preserve the insurer's final retention.

Check Your Understanding

For each hypothetical per-claim arrangement, use the stated sequence and basis. Amounts are in dollars.

A $\$90$ claim is subject to 20% quota share and then excess of loss with attachment $\$50$ and limit $\$30$. Find total ceded recovery and insurer retention. (Answer: Ceded $\$40$; retained $\$50$.)
For a $\$160$ claim, excess of loss with attachment $\$60$ and limit $\$70$ applies first, followed by 25% quota share on the amount remaining. Find each recovery and the final retention. (Answer: Excess-of-loss recovery $\$70$, quota-share recovery $\$22.50$, total ceded $\$92.50$, retained $\$67.50$.)
A portfolio has claims of $\$100$ and $\$200$. Compare total ceded recovery and retention when 50% quota share precedes per-claim excess of loss with attachment $\$30$ and limit $\$50$, versus when the same unscaled layer applies first and quota share applies to the remainder. (Answer: Quota share first: ceded $\$220$, retained $\$80$. Excess of loss first: ceded $\$200$, retained $\$100$.)
A $\$150$ claim has 30% quota share and gross-basis excess-of-loss terms of attachment $\$80$ and limit $\$40$. If quota share applies first, scale both layer terms by the 70% retained fraction. Find the layer recovery, total cession, and retention; then state whether these match excess of loss first followed by quota share. (Answer: Scaled layer recovery $\$28$, total cession $\$73$, retention $\$77$; these match the reverse sequence.)
A $\$75$ claim is subject first to 20% quota share and then to an unscaled excess-of-loss layer with attachment $\$40$ and limit $\$50$. Calculate total cession and retention. Then calculate them for the reverse sequence, with quota share applying to the post-layer amount. (Answer: Quota share first: ceded $\$35$, retained $\$40$. Excess of loss first: ceded $\$43$, retained $\$32$.)

What You'll Learn