Combining Quota Share and Excess-of-Loss

Why the Loss Basis Matters

A quota-share treaty and an excess-of-loss treaty divide loss in different ways. Quota share assigns a fixed proportion of a specified loss to a reinsurer. Excess of loss pays only the part of a specified loss that falls above an attachment point, subject to a limit. When both treaties apply, the calculation depends on which loss amount each treaty covers. A correct calculation states those bases before finding any payment.

For the calculations in this tutorial, the quota-share treaty cedes a proportion $q$ of the original loss $X$. The excess-of-loss treaty then covers the insurer’s retained share after that quota-share cession. This is a defined structure for the examples, not a conclusion to infer from the names of the treaties. Contract wording must identify the loss basis, and the examples state it explicitly.

Definition: Let $X$ be a nonnegative loss and let $q$, with $0\le q\le1$, be the quota-share proportion ceded to the quota-share reinsurer. The quota-share payment is $qX$, and the loss retained before excess-of-loss protection is $Z=(1-q)X$. If the excess-of-loss treaty covers $Z$, has attachment $d$, and has maximum recovery $u$, its payment is $R=\min((Z-d)_+,u)$, where $(a)_+=\max(a,0)$. The insurer’s net loss is $N=Z-R$. Here $u$ is the maximum amount recoverable under the layer.

The attachment $d$ is the amount of the specified basis that the insurer must retain before the excess-of-loss reinsurer pays. The limit $u$ caps the reinsurer’s payment; it is not the upper endpoint of the loss interval. Thus the layer pays nothing when $Z\le d$, pays $Z-d$ when $d<Z<d+u$, and pays $u$ when $Z\ge d+u$.

R= \begin{cases} 0, & Z\le d,\\ Z-d, & d<Z<d+u,\\ u, & Z\ge d+u. \end{cases}

Allocation and Reconciliation

For a single loss on this stated basis, there are three amounts to track: the quota-share reinsurer’s payment, the excess-of-loss reinsurer’s payment, and the insurer’s net payment. The original loss is the total to be allocated; neither treaty creates an additional loss.

$$ X=qX+R+N $$

This identity follows by substituting $Z=(1-q)X$ and $N=Z-R$:

$$ qX+R+N=qX+R+(Z-R)=qX+Z=qX+(1-q)X=X. $$

The identity is both a useful justification and a practical check. If the allocated amounts do not add back to the original loss, a payment has been omitted, counted twice, or calculated on an unintended basis. Reconciliation alone does not establish that the treaty basis is correct, however: a wrong allocation can still add up correctly.

Identify the loss amount.
Decide whether the calculation concerns one claim, a group of claims, or an annual aggregate. Use the amount specified by the treaty wording.

Calculate the quota-share payment.
For a quota-share proportion $q$ of the original loss $X$, calculate $qX$. The insurer’s pre-excess retained amount is $(1-q)X$.

Apply the excess layer to its stated basis.
For the structure used here, apply $\min(((1-q)X-d)_+,u)$. Do not substitute the gross loss for the retained loss.

Find and check the insurer’s net loss.
Subtract the excess-of-loss recovery from the pre-excess retained amount, then verify that all three shares sum to $X$.

Worked Allocations

Worked Example: A Single Claim Below the Layer Limit

A hypothetical policy produces a claim of $X=\$100{,}000$. A quota-share treaty cedes $40\%$ of the original claim. The excess-of-loss treaty covers the insurer’s quota-share-retained amount, attaches at $d=\$50{,}000$, and has a maximum recovery of $u=\$30{,}000$. Find all three shares. There is no time value or mortality assumption; the claim amount is deterministic.

Method. Calculate the quota-share payment first, then apply the excess layer to the specified retained basis.

The quota-share reinsurer pays $0.40(\$100{,}000)=\$40{,}000$. The insurer retains $1-0.40=0.60$ of the claim before excess-of-loss protection, so the excess treaty’s basis is $Z=0.60(\$100{,}000)=\$60{,}000$.

The amount above the attachment is $\$60{,}000-\$50{,}000=\$10{,}000$. Since this is below the $\$30{,}000$ maximum recovery, the excess-of-loss reinsurer pays $R=\min(\$10{,}000,\$30{,}000)=\$10{,}000$.

The insurer’s net payment is $N=\$60{,}000-\$10{,}000=\$50{,}000$. The shares reconcile: $\$40{,}000+\$10{,}000+\$50{,}000=\$100{,}000$.

Check and common mistake avoided. The excess layer was applied to $\$60{,}000$, not the original $\$100{,}000$. Applying it to the gross claim would produce $\min(\$100{,}000-\$50{,}000,\$30{,}000)=\$30{,}000$, which is not the recovery under the stated retained-loss basis.

Worked Example: Expected Per-Claim Shares Under a Discrete Severity Model

For a hypothetical portfolio, the individual claim severity $X$, in thousands of dollars, is $20$, $50$, or $100$, with probabilities $0.20$, $0.50$, and $0.30$, respectively. A quota-share treaty cedes $25\%$ of each original claim. A per-claim excess-of-loss treaty covers the insurer’s retained portion of each claim, with attachment $d=30$ and maximum recovery $u=25$, both in thousands of dollars. Find the expected quota-share payment, expected excess-of-loss payment, and expected insurer net payment per claim. No interest or mortality assumption is relevant.

Method. Calculate the three allocations at each possible severity, then weight each allocation by its probability. All amounts below are in thousands of dollars.

Claim $X$	Probability	Quota share $0.25X$	Retained basis $0.75X$	Excess recovery	Insurer net
20	0.20	5	15	0	15
50	0.50	12.5	37.5	7.5	30
100	0.30	25	75	25	50

For $X=20$, the retained basis is $15$, below the attachment, so the excess recovery is zero. For $X=50$, the basis is $37.5$, giving a recovery of $\min(37.5-30,25)=7.5$. For $X=100$, the basis is $75$, giving $\min(75-30,25)=25$, at the layer limit.

The expected quota-share payment is $0.20(5)+0.50(12.5)+0.30(25)=1+6.25+7.5=14.75$. The expected excess-of-loss payment is $0.20(0)+0.50(7.5)+0.30(25)=0+3.75+7.5=11.25$. The expected insurer net payment is $0.20(15)+0.50(30)+0.30(50)=3+15+15=33$.

The expected original claim is $0.20(20)+0.50(50)+0.30(100)=4+25+30=59$. The expected shares check: $14.75+11.25+33=59$. Thus the expected total allocated equals the expected loss.

Check and common mistake avoided. The excess layer applies to $0.75X$ for each claim, before taking the expectation. Applying a layer to the expected loss alone would not generally give the expected recovery, because the layer payment is nonlinear.

Worked Example: Annual Aggregate Loss With an Aggregate Excess Layer

A hypothetical portfolio has annual aggregate loss $X$, in thousands of dollars, with $P(X=120)=0.20$, $P(X=180)=0.50$, and $P(X=260)=0.30$. A quota-share treaty cedes $25\%$ of the annual aggregate loss. An annual aggregate excess-of-loss treaty covers the insurer’s annual retained aggregate after that cession, attaches at $d=100$, and has maximum recovery $u=40$, all in thousands. Find the expected annual payments to each reinsurer and the expected insurer net loss. The distribution is discrete as stated; no interest or mortality assumption is relevant.

Method. Treat each possible value as an annual aggregate, calculate the quota-share cession and retained aggregate, and apply the aggregate excess layer to that retained amount. Do not apply this aggregate layer separately to individual claims.

When $X=120$, the quota-share payment is $0.25(120)=30$, and the retained aggregate is $90$. The excess treaty pays $\min((90-100)_+,40)=0$, leaving the insurer with $90$.

When $X=180$, the quota-share payment is $45$, and the retained aggregate is $135$. The excess treaty pays $\min(135-100,40)=35$, leaving the insurer with $135-35=100$.

When $X=260$, the quota-share payment is $65$, and the retained aggregate is $195$. The excess treaty pays $\min(195-100,40)=40$, at its maximum recovery, leaving the insurer with $195-40=155$.

The expected quota-share payment is $0.20(30)+0.50(45)+0.30(65)=6+22.5+19.5=48$. The expected aggregate excess-of-loss payment is $0.20(0)+0.50(35)+0.30(40)=0+17.5+12=29.5$. The expected insurer net loss is $0.20(90)+0.50(100)+0.30(155)=18+50+46.5=114.5$.

The expected annual aggregate loss is $0.20(120)+0.50(180)+0.30(260)=24+90+78=192$. The expected allocation checks: $48+29.5+114.5=192$.

Check and common mistake avoided. The excess treaty in this example is annual aggregate excess of loss, so its input is the annual retained aggregate. Treating $d=100$ as a per-claim attachment would describe a different coverage basis and would not answer the stated question.

Expected Allocations and Limits of the Calculation

When the allocation identity holds for every possible value of $X$, taking expectations gives a corresponding identity for expected payments, provided the expected loss is finite:

$$ E[X]=E[qX]+E[R]+E[N]. $$

For a fixed quota-share proportion, $E[qX]=qE[X]$. The excess-of-loss expectation must generally be calculated using the distribution of its stated basis $Z$: $E[\min((Z-d)_+,u)]$. In general, it is not valid to calculate that expectation by simply applying the layer formula to $E[Z]$. The discrete example showed why: the layer responds differently to outcomes below the attachment, inside the layer, and above its limit.

A claim-level calculation and an annual aggregate calculation can use the same payment formula but have different inputs. For a per-claim treaty, apply the layer to each claim’s specified retained amount. For an annual aggregate treaty, apply it to the specified annual retained total. The contract basis determines which amount enters the formula.

Common Exam Traps

Using gross loss for the layer without checking the stated basis. If the excess treaty covers the quota-share-retained amount, its input is $(1-q)X$, not $X$.
Confusing a recovery limit with an upper loss endpoint. Under the definition here, $u$ is the maximum payment, so the recovery is $\min((Z-d)_+,u)$.
Forgetting the insurer’s retained amount before excess protection. The insurer net is $Z-R$, not $X-R$, when the quota-share payment has already been ceded.
Using the wrong treaty basis. A per-claim layer and an annual aggregate layer do not use the same loss amount, even if their attachment and limit are numerically identical.
Stopping after calculating recoveries. Report the insurer net and all reinsurer payments, then check that the amounts sum to the original covered loss or aggregate.

Key takeaway: Name the loss basis for each treaty before calculating. On the structure used here, quota share cedes $qX$, excess of loss covers the insurer’s retained amount $(1-q)X$, and the insurer pays what remains after the excess recovery. Reconcile the three amounts to $X$.

Check Your Understanding

For each problem, use the loss basis stated in its wording and report the requested allocations.

A hypothetical claim is $\$80{,}000$. Quota share cedes $30\%$ of the original claim, and per-claim excess of loss covers the retained amount with attachment $d=\$40{,}000$ and maximum recovery $u=\$15{,}000$. Find the quota-share payment, excess recovery, and insurer net. (Answer: $\$24{,}000,\ \$15{,}000,\ \$41{,}000$)
A hypothetical claim is $\$50{,}000$. Quota share cedes $20\%$ of the original claim. The excess layer covers the retained amount, attaches at $d=\$30{,}000$, and has maximum recovery $u=\$10{,}000$. Find all three shares. (Answer: quota share $\$10{,}000$; excess recovery $\$10{,}000$; insurer net $\$30{,}000$)
A hypothetical per-claim severity $X$, in thousands of dollars, equals $40$ or $100$, each with probability $0.50$. Quota share cedes $50\%$ of the original claim. Per-claim excess of loss covers the retained amount, attaches at $d=20$, and has maximum recovery $u=30$. Find the expected quota-share payment, expected excess recovery, and expected insurer net per claim. (Answer: $35,\ 15,\ 20$ thousand dollars, respectively)
A hypothetical annual aggregate $X$, in thousands of dollars, equals $100$ with probability $0.40$ and $200$ with probability $0.60$. Quota share cedes $25\%$ of the annual aggregate, and annual aggregate excess of loss covers the retained aggregate with $d=100$ and $u=50$. Find the expected payments to the quota-share reinsurer and excess-of-loss reinsurer, and the expected insurer net. (Answer: $40,\ 30,\ 90$ thousand dollars, respectively)
A hypothetical claim is $\$250{,}000$. Quota share cedes $40\%$ of the original claim. Per-claim excess of loss covers the insurer’s retained amount, attaches at $d=\$100{,}000$, and has maximum recovery $u=\$40{,}000$. Find all three shares and verify the total. (Answer: quota share $\$100{,}000$; excess recovery $\$40{,}000$; insurer net $\$110{,}000$; total $\$250{,}000$)

What You'll Learn