Combining Quota Share and Excess-of-Loss Covers

Why the Order of the Covers Matters

When an insurer has both quota-share and excess-of-loss reinsurance, each treaty responds to a different calculation basis. Quota share divides a stated proportion of the gross loss between the insurer and the quota-share reinsurer. In the arrangement considered here, per-risk excess-of-loss protection then applies to the loss still retained by the insurer after that quota-share cession. The sequence is therefore essential: calculate the quota-share allocation first, then calculate excess-of-loss recovery on the resulting retained loss.

This tutorial considers hypothetical contracts in which one gross covered claim is associated with each risk, the quota-share percentage applies uniformly to every claim, and the excess-of-loss cover is applied separately to each risk’s post-quota-share loss. There are no aggregate caps, reinstatements, exclusions, or other treaty terms affecting the examples. If a contract specifies a different order or basis, its wording governs.

Definition: Let $X$ be a risk’s gross covered claim, and let $\alpha$ be the quota-share proportion ceded to the quota-share reinsurer, where $0 \leq \alpha \leq 1$. The insurer’s retained proportion is $r=1-\alpha$. Let $A$ be the per-risk excess-of-loss attachment and $L$ its limit, interpreted here as the maximum recovery above the attachment. The quota-share recovery is $\alpha X$, the loss entering the excess-of-loss layer is $rX$, and the excess-of-loss recovery is $\min(\max(rX-A,0),L)$.

Calculate Each Treaty on Its Own Basis

For each risk, begin with the gross claim. The quota-share reinsurer pays its agreed percentage of that claim, leaving the insurer with the remainder. The per-risk excess-of-loss reinsurer then pays only the part of that remainder above the attachment, subject to the layer limit. The insurer retains what remains after both recoveries.

\begin{aligned} \text{Quota-share recovery} &= \alpha X,\\ \text{Post-quota-share loss} &= (1-\alpha)X=rX,\\ \text{Excess-of-loss recovery} &= \min\bigl(\max(rX-A,0),L\bigr),\\ \text{Insurer's final retention} &= rX-\min\bigl(\max(rX-A,0),L\bigr). \end{aligned}

The expression $\max(rX-A,0)$ ensures that the excess-of-loss treaty pays nothing when the retained loss does not exceed its attachment. The outer minimum caps the recovery at $L$. Thus, a layer described as “$7,000 excess of $7,000” would have an attachment of $7,000 and a limit of $7,000: it could pay up to $7,000, not the full loss above the attachment without a cap.

Why the Excess-of-Loss Formula Holds

The formula follows by checking the three possible positions of the post-quota-share loss $rX$ relative to the layer. If $rX\leq A$, the loss has not reached the attachment, so recovery is zero. If $A<rX<A+L$, the loss enters the layer but does not exhaust it, so recovery is $rX-A$. If $rX\geq A+L$, the layer is exhausted and recovery is $L$.

\text{Recovery}= \begin{cases} 0, & rX\leq A,\\ rX-A, & A<rX<A+L,\\ L, & rX\geq A+L. \end{cases}

These three cases are exactly represented by $\min(\max(rX-A,0),L)$: first remove the loss below the attachment, then prevent the resulting amount from exceeding the layer limit. This also gives useful gross-claim thresholds. The layer begins to respond when $rX>A$, or equivalently $X>A/r$ for $r>0$. It is exhausted when $rX\geq A+L$, or $X\geq(A+L)/r$. If $r=0$, the insurer retains no loss after quota share, so this excess-of-loss basis produces no recovery.

Record the treaty terms.
Identify the ceded quota-share percentage, the excess-of-loss attachment, and the excess-of-loss limit. Confirm that the layer applies to post-quota-share retained losses.

Calculate the quota-share allocation.
For each gross claim $X$, calculate $\alpha X$ ceded and $rX=(1-\alpha)X$ retained before excess-of-loss.

Apply the per-risk layer.
Use the post-quota-share retained loss in $\min(\max(rX-A,0),L)$. Apply the attachment and limit separately to each risk under the assumptions here.

Reconcile the allocation.
Add the quota-share recovery, excess-of-loss recovery, and insurer’s final retention. Their sum should equal the gross claim, risk by risk and in total.

Worked Allocations

Worked Example: Correctly Applying the Layer to Retained Claims

A hypothetical insurer has two claims of $15,000 and $35,000. It cedes 40% of each claim under quota share. A per-risk excess-of-loss treaty then attaches at $7,000 on the retained claim and has a limit of $10,000 per claim. Find the total quota-share recovery, total excess-of-loss recovery, and total insurer retention. Treat the $10,000 as the maximum layer payment above the attachment.

Method. Calculate the 40% quota-share cession first. Apply the layer to the remaining 60%, not to the gross claims.

For the $15,000 claim, the quota-share recovery is $0.40(15{,}000)=\$6{,}000$. The post-quota-share loss is $0.60(15{,}000)=\$9{,}000$. The amount above the $7,000 attachment is $\$9{,}000-\$7{,}000=\$2{,}000$, below the $10,000 limit. The excess-of-loss recovery is therefore $2,000, and the insurer retains $\$9{,}000-\$2{,}000=\$7{,}000$.

For the $35,000 claim, the quota-share recovery is $0.40(35{,}000)=\$14{,}000$. The post-quota-share loss is $0.60(35{,}000)=\$21{,}000$. The amount above attachment is $\$21{,}000-\$7{,}000=\$14{,}000$, which is capped at $10,000. The excess-of-loss recovery is $10,000, and the insurer retains $\$21{,}000-\$10{,}000=\$11{,}000$.

The total quota-share recovery is $\$6{,}000+\$14{,}000=\$20{,}000$. The total excess-of-loss recovery is $\$2{,}000+\$10{,}000=\$12{,}000$. The total insurer retention is $\$7{,}000+\$11{,}000=\$18{,}000$.

Check. Gross claims total $\$15{,}000+\$35{,}000=\$50{,}000$, and $\$20{,}000+\$12{,}000+\$18{,}000=\$50{,}000$. The allocation reconciles. A common mistake is to calculate the layer on the gross claims: that uses the wrong basis and overstates the excess-of-loss recovery.

Worked Example: Several Risks and a Capped Layer

A hypothetical portfolio has three separate covered claims: $10,000, $25,000, and $40,000. A quota-share treaty cedes 30%. The insurer then buys per-risk excess-of-loss protection attaching at $8,000, with a $12,000 limit for each risk. Calculate all three allocations by claim and the portfolio totals.

Method. The insurer retains $1-0.30=0.70$ of each gross claim before excess-of-loss. The layer recovery for each risk is the smaller of $12,000 and the positive amount by which its post-quota-share loss exceeds $8,000.

For the $10,000 claim, quota share pays $0.30(10{,}000)=\$3{,}000$, leaving $\$7{,}000$. This is below the attachment, so excess-of-loss pays $0 and the insurer retains $7,000.

For the $25,000 claim, quota share pays $0.30(25{,}000)=\$7{,}500$, leaving $\$17{,}500$. The amount above attachment is $\$17{,}500-\$8{,}000=\$9{,}500$, below the limit. Excess-of-loss pays $9,500, leaving the insurer with $\$17{,}500-\$9{,}500=\$8{,}000$.

For the $40,000 claim, quota share pays $0.30(40{,}000)=\$12{,}000$, leaving $\$28{,}000$. The amount above attachment is $20,000, but the layer is limited to $12,000. Excess-of-loss pays $12,000, and the insurer retains $\$28{,}000-\$12{,}000=\$16{,}000$.

The portfolio’s quota-share recovery is $\$3{,}000+\$7{,}500+\$12{,}000=\$22{,}500$. Its excess-of-loss recovery is $\$0+\$9{,}500+\$12{,}000=\$21{,}500$. The insurer retains $\$7{,}000+\$8{,}000+\$16{,}000=\$31{,}000$.

Check. Gross claims total $75,000, and $\$22{,}500+\$21{,}500+\$31{,}000=\$75{,}000$. Applying the attachment separately matters: the first risk’s unused layer capacity cannot be transferred to another risk under this per-risk calculation.

Worked Example: Finding the Trigger and Exhaustion Points

A hypothetical insurer cedes 40% of each risk’s claim under quota share, then retains the remaining 60% subject to a per-risk excess-of-loss layer attaching at $10,000 with a $15,000 limit. Four gross claims are $8,000, $20,000, $40,000, and $100,000. Find each recovery and retention, and identify the gross-claim amounts at which the layer begins to respond and becomes exhausted.

Method. The post-quota-share proportion is $r=0.60$. The layer begins to respond when $0.60X>10{,}000$, so the gross threshold is $X>10{,}000/0.60=\$16{,}666.67$ (approximately). It is exhausted when $0.60X\geq10{,}000+15{,}000$, so the exhaustion threshold is $X\geq25{,}000/0.60=\$41{,}666.67$ (approximately).

For the $8,000 claim, quota share pays $0.40(8{,}000)=\$3{,}200$, leaving $4,800. The layer pays $0, so final retention is $4,800.

For the $20,000 claim, quota share pays $8,000 and leaves $12,000. The layer pays $\$12{,}000-\$10{,}000=\$2{,}000$, leaving a final retention of $10,000.

For the $40,000 claim, quota share pays $16,000 and leaves $24,000. The amount above attachment is $\$24{,}000-\$10{,}000=\$14{,}000$, below the $15,000 limit. The layer pays $14,000, leaving $10,000.

For the $100,000 claim, quota share pays $40,000 and leaves $60,000. The amount above attachment is $50,000, so the layer pays its full $15,000 limit. Final retention is $\$60{,}000-\$15{,}000=\$45{,}000$.

Total quota-share recovery is $\$3{,}200+\$8{,}000+\$16{,}000+\$40{,}000=\$67{,}200$. Total excess-of-loss recovery is $\$0+\$2{,}000+\$14{,}000+\$15{,}000=\$31{,}000$. Total insurer retention is $\$4{,}800+\$10{,}000+\$10{,}000+\$45{,}000=\$69{,}800$.

Check. Gross claims total $168,000, and $\$67{,}200+\$31{,}000+\$69{,}800=\$168{,}000$. The $40,000 claim has entered the layer but has not exhausted it; the $100,000 claim exceeds the exhaustion threshold and is capped.

Common Calculation Traps

Using gross loss as the excess-of-loss basis. If the contract attaches after quota share, apply the layer to $rX$, not $X$. A gross claim may be large while its post-quota-share retained amount is below the attachment.
Subtracting the attachment from every claim. The layer pays nothing when $rX\leq A$; use the positive-part operation rather than allowing a negative recovery.
Ignoring the limit. Once the amount above attachment exceeds $L$, recovery remains $L$, not the full amount above attachment.
Applying a per-risk limit to the portfolio total. Calculate each risk’s recovery separately, then add the recoveries. A per-risk layer is not a single portfolio-wide layer.
Forgetting the reconciliation. For every risk, quota-share recovery plus excess-of-loss recovery plus final insurer retention must equal the gross covered claim under the stated assumptions.

This sequence is useful whenever a problem specifies that excess-of-loss protection covers a net or retained amount after proportional reinsurance. It is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus, where careful interpretation of treaty order, attachment, and limit is central to allocating losses correctly.

Check Your Understanding

For each hypothetical arrangement, calculate the requested recoveries and retention using the stated sequence: quota share first, then per-risk excess-of-loss on the post-quota-share loss.

A $10,000 claim and a $25,000 claim are subject to 20% quota share, followed by a per-risk layer attaching at $12,000 with an $8,000 limit. Find the total quota-share recovery, excess-of-loss recovery, and insurer retention. (Answer: $7,000; $8,000; $20,000.)
Two claims are $20,000 and $50,000. The insurer cedes 35% under quota share, then buys per-risk excess-of-loss attaching at $10,000 with an $18,000 limit. Find the total allocation to each party. (Answer: Quota-share recovery $24,500; excess-of-loss recovery $21,000; insurer retention $24,500.)
Three separate claims are $8,000, $18,000, and $40,000. Quota share cedes 50%; excess-of-loss attaches at $6,000 with a $9,000 limit per risk. Find the total recoveries and retention, and check the gross-loss reconciliation. (Answer: Quota-share recovery $33,000; excess-of-loss recovery $12,000; insurer retention $21,000; gross total $66,000.)
Claims of $16,000, $32,000, and $80,000 are subject to 25% quota share and then per-risk excess-of-loss attaching at $18,000 with a $24,000 limit. Identify which risks trigger the layer and find the total allocation. (Answer: The $32,000 and $80,000 claims trigger; quota-share recovery $32,000; excess-of-loss recovery $30,000; insurer retention $66,000.)
A single $40,000 claim is subject to 40% quota share, followed by per-risk excess-of-loss attaching at $9,000 with a $12,000 limit. Find the allocation and the gross-claim threshold at which the layer begins to respond and becomes exhausted. (Answer: Quota-share recovery $16,000; excess-of-loss recovery $12,000; insurer retention $12,000; trigger threshold approximately $15,000 and exhaustion threshold $35,000.)

What You'll Learn