Worked Chain: Quota Share Followed by Per-Claim Excess Cover

Why the Order of Reinsurance Matters

A reinsurance chain divides an original claim among several parties. In this tutorial, a quota-share treaty first cedes a fixed proportion of each claim. A per-claim excess-of-loss treaty then covers part of the amount remaining after that quota share. The order matters: the excess attachment is applied to the post-quota amount, not automatically to the original claim.

The calculations are made claim by claim before being aggregated. For each claim, identify the quota-share payment, the excess-layer payment, and the insurer’s final retention. Their sum must equal the original loss. This payment-by-payment check is useful both for finding expected costs and for detecting a misapplied attachment or limit.

Definition: Let $X$ be the original loss on one claim, let $q$ be the quota-share proportion ceded, and let $M$ and $L$ be the attachment and limit of a per-claim excess layer. Assume $0\le q\le 1$, $M\ge 0$, and $L\ge 0$. The quota-share reinsurer pays $Q=qX$, leaving $Z=(1-q)X$ after the quota share. The excess reinsurer pays $C=\min\{(Z-M)_+,L\}$, where $(w)_+=\max(w,0)$. The insurer retains $R=Z-C$. Thus $X=Q+C+R$.

The excess layer pays nothing while $Z\le M$, pays the amount above $M$ while that amount is below $L$, and pays its full limit once $Z\ge M+L$. The insurer retains the portion of $Z$ below the attachment and, if $Z$ exceeds the top of the layer, the portion above the layer. Unless the contract defines the attachment differently, applying it to $Z$ is essential to this chain.

Post-quota amount $Z$	Excess payment $C$	Insurer retention $R$
$Z\le M$	0	$Z$
$M<Z<M+L$	$Z-M$	$M$
$Z\ge M+L$	$L$	$Z-L$

From Claim Payments to Expected Aggregate Payments

For a portfolio with $N$ claims, let $X_j$ be the original loss on claim $j$. Apply the contract to each loss separately: $Q_j=qX_j$, $Z_j=(1-q)X_j$, $C_j=\min\{(Z_j-M)_+,L\}$, and $R_j=Z_j-C_j$. The aggregate amounts are the sums of the corresponding claim payments.

If $N$ is independent of identically distributed severities, with finite means, the expected aggregate payment for any one layer is the expected claim count multiplied by that layer’s expected payment per claim. This result applies to each layer separately; it does not require calculating the full distribution of aggregate loss.

E\!\left[\sum_{j=1}^{N} Q_j\right]=E[N]E[Q],\qquad E\!\left[\sum_{j=1}^{N} C_j\right]=E[N]E[C],\qquad E\!\left[\sum_{j=1}^{N} R_j\right]=E[N]E[R].

The independence assumption is between the claim count and the severities. If claim severities depend on the number of claims, the product formula need not hold; the expected aggregate payment must then be calculated using the relevant conditional severity model. The layer definitions themselves still apply claim by claim.

Derivation: Conditional on $N=n$, the aggregate excess payment is $\sum_{j=1}^{n}C_j$. By linearity of expectation and identical claim distributions, $E[\sum_{j=1}^{n}C_j\mid N=n]=nE[C]$. Taking expectations over $N$ gives $E[\sum_{j=1}^{N}C_j]=E[N]E[C]$. The same argument applies to $Q_j$ and $R_j$. Also, on each claim $Q_j+C_j+R_j=X_j$, so the expected aggregate payments across all three parties must sum to the expected original aggregate loss.

Worked Payment Chains

In each example, the losses and all reinsurance payments are measured at the same valuation date. No expenses, premiums, reinstatements, or time value of money are included. The excess contract is applied to the amount remaining after the quota share.

Worked Example: Discrete Severity and a Fixed Claim Count

A hypothetical portfolio has exactly two claims. Each claim has severity $X$ equal to $2,000, $4,000, or $8,000, with probabilities 0.25, 0.50, and 0.25, respectively. A 25% quota share is followed by a per-claim excess layer with attachment $M=$$3,000 and limit $L=$$1,000 on the post-quota amount. Find the expected aggregate payment to each party.

Method and contract parameters. The claim count is fixed at $N=2$. The quota share is $q=0.25$, so the post-quota fraction is $1-q=0.75$. For each severity, calculate $Q=0.25X$, $Z=0.75X$, $C=\min\{(Z-3000)_+,1000\}$, and $R=Z-C$.

For $X=$$2,000, the quota-share payment is $Q=$$500 and the post-quota amount is $Z=$$1,500. Since $Z\le 3000$, the excess payment is $C=$$0 and the insurer retains $R=$$1,500.

For $X=$$4,000, $Q=$$1,000 and $Z=$$3,000. The amount is exactly at the attachment, so $C=$$0 and $R=$$3,000.

For $X=$$8,000, $Q=$$2,000 and $Z=$$6,000. The amount above the attachment is $3,000, but the layer limit is $1,000, so $C=$$1,000 and $R=$$5,000.

The expected original loss per claim is

$$ E[X]=0.25(2000)+0.50(4000)+0.25(8000) =500+2000+2000=4500. $$

The expected quota-share payment per claim is $E[Q]=0.25(4500)=$$1,125. The expected excess payment per claim is $E[C]=0.25(0)+0.50(0)+0.25(1000)=$$250. The expected insurer retention per claim is $E[R]=0.25(1500)+0.50(3000)+0.25(5000)=375+1500+1250=$$3,125.

Because there are exactly two claims, multiply each per-claim expectation by two. The expected aggregate payments are $2,250 to the quota-share reinsurer, $500 to the excess reinsurer, and $6,250 retained by the insurer. The expected original aggregate loss is $2(4500)=$$9,000, and $2,250 + $500 + $6,250 = $9,000, as required.

Common mistake avoided. Applying the $3,000 attachment to the gross severities would give an excess payment on the $4,000 claim. That is not the specified contract: its post-quota amount is exactly $3,000, so the excess payment is zero. Always determine which amount the attachment acts on before calculating the layer.

Worked Example: Uniform Severity and a Capped Layer

A hypothetical portfolio has exactly two claims, each with severity $X\sim\operatorname{Uniform}(0,8000)$. A 25% quota share is followed by a per-claim excess layer attaching at $3,000 and limited to $1,000 on the post-quota amount. Calculate the expected aggregate excess payment and the expected aggregate insurer retention.

Method and parameters. Here $E[X]=(0+8000)/2=$$4,000 and $1-q=0.75$, so $Z=0.75X$. The layer starts paying when $0.75X=3000$, or $X=3000/0.75=$$4,000. It reaches its $1,000 limit when $0.75X=3000+1000=4000$, or $X=4000/0.75=$$5,333.33 repeating.

As $X$ increases from $4,000 to $5,333.33 repeating, the excess payment rises from zero to $1,000. For larger losses it stays at $1,000. The area under the payment function over the severity interval is a triangle plus a rectangle. Using the exact threshold $16000/3$, the triangle has base $16000/3-4000=4000/3$ and height $1,000; the rectangle has width $8000-16000/3=8000/3$ and height $1,000.

E[C]=\frac{\frac12(4000/3)(1000)+(8000/3)(1000)}{8000} =\frac{2{,}000{,}000/3+8{,}000{,}000/3}{8000} =\frac{10{,}000{,}000/3}{8000} =416.6667\ldots

Thus the expected excess payment per claim is $416.67 to the nearest cent, and the expected aggregate excess payment for two claims is $2(416.6667\ldots)=$$833.33 to the nearest cent.

The expected quota-share payment per claim is $0.25(4000)=$$1,000. The expected post-quota amount per claim is $0.75(4000)=$$3,000. Therefore the expected insurer retention per claim is $3000-416.6667\ldots=$$2,583.33 to the nearest cent, and the expected aggregate insurer retention is $5,166.67 to the nearest cent. The expected aggregate original loss is $8,000; the unrounded layer expectations sum to $2,000 + $833.3333… + $5,166.6667… = $8,000.

Common mistake avoided. The attachment threshold in terms of the original loss is $4,000, not $3,000, because only 75% of each original claim remains after the quota share. Likewise, the layer limit is reached at an original loss of $5,333.33 repeating, not at $4,000.

Worked Example: Binomial Frequency and Discrete Severity

A hypothetical book has $N\sim\operatorname{Binomial}(3,0.4)$ claims. Claim count is independent of severity, and each claim is $1,000, $3,000, or $5,000 with probabilities 0.20, 0.50, and 0.30. A 40% quota share is followed by a per-claim excess layer attaching at $1,500 and limited to $900 on the post-quota amount. Find the expected aggregate payments to the quota-share reinsurer, the excess reinsurer, and the insurer.

Method and assumptions. The expected claim count is $E[N]=3(0.4)=1.2$. The post-quota fraction is $1-q=0.60$. Calculate the three per-claim payments first, then multiply their expectations by 1.2.

For a $1,000 claim, $Q=0.40(1000)=$$400 and $Z=0.60(1000)=$$600. Since $Z\le 1500$, $C=$$0 and $R=$$600.

For a $3,000 claim, $Q=$$1,200 and $Z=$$1,800. The excess amount is $1800-1500=$$300, which is less than the $900 limit. Thus $C=$$300 and $R=1800-300=$$1,500.

For a $5,000 claim, $Q=$$2,000 and $Z=$$3,000. The excess above attachment is $1,500, so the layer pays its full $900 limit. The insurer retains $R=3000-900=$$2,100.

The expected original severity is $0.20(1000)+0.50(3000)+0.30(5000)=200+1500+1500=$$3,200. The expected quota-share payment per claim is $0.20(400)+0.50(1200)+0.30(2000)=80+600+600=$$1,280. The expected excess payment per claim is $0.20(0)+0.50(300)+0.30(900)=0+150+270=$$420. The expected insurer retention per claim is $0.20(600)+0.50(1500)+0.30(2100)=120+750+630=$$1,500.

Multiplying by $E[N]=1.2$, the expected aggregate payments are $1,536 to the quota-share reinsurer, $504 to the excess reinsurer, and $1,800 retained by the insurer. The expected aggregate original loss is $1.2(3200)=$$3,840. The three expected payments sum to $1,536 + $504 + $1,800 = $3,840.

Common mistake avoided. The binomial mean is 1.2, not three: three is the maximum possible count. Also, the excess payment on the $5,000 claim is capped at $900, even though the amount above attachment is $1,500.

Checks and Exam Traps

A reliable solution keeps the contract sequence visible. First calculate the quota-share amount and the post-quota amount. Next apply the per-claim attachment and limit to that post-quota amount. Finally calculate the insurer’s retention as the post-quota amount minus the excess payment. This avoids accidentally treating the two ceded payments as if they were both calculated directly from the gross claim.

Establish the specified basis.
Identify whether the attachment applies to the original loss or, as here, to the post-quota amount. Record the quota proportion, attachment, and limit.

Calculate payments by claim.
For each severity, find $Q=qX$, $Z=(1-q)X$, $C=\min\{(Z-M)_+,L\}$, and $R=Z-C$.

Aggregate expectations.
Find each expected payment per claim. If count and severity are independent, multiply each by $E[N]$.

Reconcile the totals.
Check $Q+C+R=X$ for each claim type and check that expected aggregate payments sum to expected aggregate original loss.

This method produces expected layer costs but does not by itself determine the distribution of aggregate payments or a risk measure for them. Those questions require the appropriate aggregate-loss calculation. For the expected-cost calculation here, the key is correctly defining the per-claim payment functions and applying them in the stated order.

Key takeaway: In a quota-share-then-excess chain, the excess layer is applied to $(1-q)X$ when the contract specifies a post-quota attachment. Calculate and reconcile all three payments claim by claim before extending them to the portfolio.

Check Your Understanding

For each problem, assume the excess attachment is applied to the post-quota amount. Unless stated otherwise, claim count is independent of severity. Round monetary answers to the nearest cent where needed.

A portfolio has exactly three claims, each of severity $5,000. A 20% quota share is followed by an excess layer attaching at $3,000 with a $1,500 limit. Find the aggregate quota-share payment, excess payment, and insurer retention. (Answer: quota share, $3,000; excess cover, $3,000; insurer retention, $9,000.)
The claim count is Poisson with mean 2, and every claim is $4,000. A 25% quota share is followed by an excess layer attaching at $2,500 with a $500 limit. Find the three expected aggregate payments. (Answer: quota share, $2,000; excess cover, $1,000; insurer retention, $5,000.)
There are exactly four claims. Each severity is $2,000 or $6,000 with probability 0.50 each. A 50% quota share is followed by an excess layer attaching at $2,000 with a $1,000 limit. Find the expected aggregate payments to each party. (Answer: quota share, $8,000; excess cover, $2,000; insurer retention, $6,000.)
The claim count is $\operatorname{Binomial}(5,0.2)$. Severity is $1,000 or $4,000 with probability 0.50 each. A 25% quota share is followed by an excess layer attaching at $2,000 with a $1,000 limit. Find the expected aggregate payments to each party. (Answer: quota share, $625; excess cover, $500; insurer retention, $1,375.)
There are exactly three claims, each with severity uniformly distributed from $0 to $10,000. A 20% quota share is followed by an excess layer attaching at $4,000 with a $1,000 limit. Find the expected aggregate excess payment and expected aggregate insurer retention. (Answer: excess cover, $1,312.50; insurer retention, $10,687.50.)

Post-quota amount \(Z\)	Excess payment \(C\)	Insurer retention \(R\)
\(Z\le M\)	0	\(Z\)
\(M<Z<M+L\)	\(Z-M\)	\(M\)
\(Z\ge M+L\)	\(L\)	\(Z-L\)

What You'll Learn