Per-Claim Excess-of-Loss Reinsurance

What a Per-Claim Excess-of-Loss Treaty Covers

A per-claim excess-of-loss treaty reimburses the insurer for part of an individual claim above an agreed attachment point. The attachment applies separately to each claim, not to the total of claims over a period. This distinction determines both the payment on each claim and the total recovery across a portfolio.

Let $X$ be the gross amount of one claim before reinsurance, and suppose $X\geq 0$. Let $M\geq 0$ be the treaty attachment point, and let $L>0$ be the maximum amount the reinsurer pays on any one claim. The treaty covers the layer from $M$ to $M+L$. A claim below $M$ produces no recovery; a claim above $M+L$ still produces only the maximum recovery $L$.

Definition: The ceded payment on one claim is $C=\min((X-M)_+,L)$, where $(z)_+=\max(z,0)$. The insurer’s retained payment is $R=X-C$. Thus $X=C+R$ for every claim. Here $L$ is the limit on the reinsurer’s payment, so the top of the covered layer is $M+L$.

The payment map is piecewise:

C= \begin{cases} 0, & X\leq M,\\ X-M, & M<X<M+L,\\ L, & X\geq M+L, \end{cases} \qquad R= \begin{cases} X, & X\leq M,\\ M, & M<X<M+L,\\ X-L, & X\geq M+L. \end{cases}

At the boundary $X=M+L$, the ceded payment is already $L$; the boundary convention does not change the amount. The retained payment is not always capped at $M$: once a claim exceeds $M+L$, the insurer retains the excess above the treaty layer as well as the first $M$.

Distributions of Ceded and Retained Payments

A payment distribution can have point masses even when the original claim severity is continuous. In particular, all claims at or below $M$ map to a ceded payment of zero, and all claims at or above $M+L$ map to a ceded payment of $L$. For a general severity distribution, those groups may also include probability at the boundary.

Result: For $0\leq c<L$, the ceded payment distribution satisfies $P(C\leq c)=F_X(M+c)$. Its mass at zero is $P(C=0)=F_X(M)$, and its mass at the full layer is $P(C=L)=P(X\geq M+L)=1-F_X((M+L)-)$. The left limit in the final expression matters if $X$ has an atom at $M+L$.

For $0<c<L$, the event $C\leq c$ is exactly the event $X\leq M+c$: claims up to $M$ have zero ceded payment, and claims between $M$ and $M+c$ have ceded payment at most $c$. This gives the stated distribution formula. Claims at or above $M+L$ all map to $C=L$, which explains the full-layer mass. The retained distribution follows from the same claim-by-claim map $R=X-C$; in general it must account for the flat section $R=M$ when $M<X<M+L$.

If $X$ is continuous with density $f_X$, the ceded payment has an atom of size $F_X(M)$ at zero and an atom of size $S_X(M+L)$ at $L$. Between zero and $L$, its density is $f_X(M+c)$. These statements rely on continuity for the density and the simplified full-layer probability; the CDF and boundary expressions above also accommodate discrete or mixed severities.

Expected Payments From the Layer

For any nonnegative claim severity, the expected ceded payment can be calculated directly from the survival function. The layer has width $L$, so its payment can be written as the area under indicators that the claim reaches successive points in the layer:

C=\int_0^L \mathbf{1}_{\{X>M+y\}}\,dy, \qquad E[C]=\int_0^L P(X>M+y)\,dy =\int_0^L S_X(M+y)\,dy =\int_M^{M+L} S_X(x)\,dx.

The first equality holds claim by claim. If $X\leq M$, the indicator is zero throughout the layer. If $M<X<M+L$, it is one for a length $X-M$ of the integration interval. If $X\geq M+L$, it is one across the full length $L$. Taking expectations and integrating the indicator probabilities gives the formula. Interchanging expectation and integration is valid here because the integrand is nonnegative and bounded by one.

The formula uses the survival function $S_X(x)=P(X>x)$. For discrete or mixed severities, it remains valid as an integral; boundary values at individual points do not affect the integral. Once $E[C]$ is known, the expected retained payment follows from $R=X-C$:

$$ E[R]=E[X]-E[C]. $$

This subtraction is often the simplest way to find the retained mean. It is valid when the expectations are finite, and follows from the claim-by-claim identity $R=X-C$. It does not say that the retained severity distribution is obtained by simply truncating $X$ at $M$; claims above $M+L$ leave additional loss with the insurer.

Worked Example: Exponential Severity and Expected Portfolio Recovery

A hypothetical insurer models each gross claim as exponential with mean $2,000, so $S_X(x)=e^{-x/2000}$ for $x\geq 0$. A per-claim treaty has attachment $M=\$1,000$ and limit $L=\$3,000$. The insurer expects 12 claims, and claim severities are independent of the claim count. Find the zero-payment and full-layer probabilities, the expected ceded and retained payments per claim, and the expected aggregate ceded and retained amounts.

Method. Use the per-claim payment map and integrate the severity survival function over the treaty layer. The top of this layer is $M+L=\$4,000$.

The probability of no ceded payment is $\displaystyle P(C=0)=P(X\leq 1{,}000)=1-e^{-1000/2000}=1-e^{-0.5}=0.3935$, to four decimal places. The probability of a full-layer payment is $\displaystyle P(C=3{,}000)=P(X\geq 4{,}000)=e^{-4000/2000}=e^{-2}=0.1353$, also to four decimal places. Continuity means equality at either boundary has probability zero.

The expected ceded payment per claim is $\displaystyle E[C]=\int_{1000}^{4000}e^{-x/2000}\,dx =2000(e^{-0.5}-e^{-2})$. Using $e^{-0.5}=0.6065307$ and $e^{-2}=0.1353353$, this is $\$2{,}000(0.4711954)=\$942.39$, rounded to the nearest cent.

The gross mean is $E[X]=\$2{,}000.00$. Therefore the expected retained payment per claim is $\$2{,}000.00-\$942.39=\$1{,}057.61$.

Let $N$ be the number of claims, with $E[N]=12$, independent of claim severities. By conditioning on $N$, the expected aggregate payment of either type is the expected count times its expected payment per claim. Thus expected aggregate ceded payment is $12(\$942.3908)=\$11{,}308.69$, and expected aggregate retained payment is $12(\$1{,}057.6092)=\$12{,}691.31$. Their sum, $\$24{,}000.00$, equals $12$ times the gross mean and checks the calculation.

Common mistake avoided: the top of the layer is $\$4{,}000$, not $\$3{,}000$. The stated $\$3{,}000$ is the limit on the reinsurer’s payment, in addition to the $\$1{,}000$ attachment.

Worked Example: Discrete Severity With Boundary and Payment Atoms

A hypothetical portfolio has claim severity $X$ with the following distribution. A per-claim treaty has attachment $M=\$1{,}000$ and limit $L=\$2{,}000$. Find the ceded and retained payment distributions and their means.

Gross claim $X$	Probability
$500	0.20
$1,500	0.30
$2,500	0.30
$4,500	0.20

Step 1: Map each claim. The treaty’s layer runs from $\$1{,}000$ to $\$3{,}000$, with a maximum ceded payment of $\$2{,}000$.

For $X=\$500$, $C=\$0$ and $R=\$500$. For $X=\$1{,}500$, $C=\$1{,}500-\$1{,}000=\$500$ and $R=\$1{,}000$. For $X=\$2{,}500$, $C=\$2{,}500-\$1{,}000=\$1{,}500$ and $R=\$1{,}000$. For $X=\$4{,}500$, the uncapped layer amount would be $\$3{,}500$, so $C=\$2{,}000$ after applying the limit and $R=\$2{,}500$.

Step 2: Collect equal payments and calculate means. The ceded payment distribution has values $\$0,\$500,\$1{,}500,\$2{,}000$ with probabilities $0.20,0.30,0.30,0.20$, respectively. Its mean is $\displaystyle E[C]=0(0.20)+500(0.30)+1{,}500(0.30)+2{,}000(0.20)=\$1{,}000.00$.

The retained payment distribution has values $\$500,\$1{,}000,\$2{,}500$ with probabilities $0.20,0.60,0.20$, respectively. Its mean is $\displaystyle E[R]=500(0.20)+1{,}000(0.60)+2{,}500(0.20)=\$1{,}200.00$. The gross mean is $500(0.20)+1{,}500(0.30)+2{,}500(0.30)+4{,}500(0.20)=\$2{,}200.00$, and $\$1{,}000.00+\$1{,}200.00=\$2{,}200.00$, as required.

Common mistake avoided: the mass at the full layer is the probability of $X\geq \$3{,}000$, which here is $0.20$, not the probability of $X>\$3{,}000$. An atom exactly at the top of the layer also receives the full payment.

Worked Example: Apply the Treaty Separately to Every Claim

A hypothetical insurer has five gross claims of $\$700$, $\$1{,}000$, $\$1{,}800$, $\$2{,}500$, and $\$4{,}200$. Its per-claim treaty has attachment $M=\$1{,}000$ and limit $L=\$1{,}500$. Find each recovery and retained amount, then check the portfolio totals.

Method. Apply $C=\min((X-\$1{,}000)_+,\$1{,}500)$ to each claim before summing. The layer ends at $\$2{,}500$.

For the $\$700$ claim, $C=\$0$ and $R=\$700$. For the $\$1{,}000$ claim, $C=\$0$ and $R=\$1{,}000$. For the $\$1{,}800$ claim, $C=\$800$ and $R=\$1{,}000$. For the $\$2{,}500$ claim, $C=\min(\$1{,}500,\$1{,}500)=\$1{,}500$ and $R=\$1{,}000$. For the $\$4{,}200$ claim, the uncapped layer amount is $\$3{,}200$, so $C=\$1{,}500$ and $R=\$2{,}700$.

Total gross claims are $\$700+\$1{,}000+\$1{,}800+\$2{,}500+\$4{,}200=\$10{,}200$. Total ceded recovery is $\$0+\$0+\$800+\$1{,}500+\$1{,}500=\$3{,}800$. Total retained claims are $\$700+\$1{,}000+\$1{,}000+\$1{,}000+\$2{,}700=\$6{,}400$. The check is $\$3{,}800+\$6{,}400=\$10{,}200$.

Common mistake avoided: applying an attachment once to the $\$10{,}200$ portfolio total would describe a different contract. This treaty applies its attachment and limit to each of the five claims separately.

From Per-Claim Payments to Aggregate Amounts

If there are $N$ claims with severities $X_1,\ldots,X_N$, the aggregate ceded and retained amounts are the sums of the corresponding per-claim payments:

C_{\mathrm{agg}}=\sum_{j=1}^{N}\min((X_j-M)_+,L), \qquad R_{\mathrm{agg}}=\sum_{j=1}^{N}\left[X_j-\min((X_j-M)_+,L)\right].

For a fixed number $n$ of claims with a common severity distribution, linearity of expectation gives $E[C_{\mathrm{agg}}]=nE[C]$, whether or not the severities are independent. If $N$ is random and independent of identically distributed severities, conditioning on $N$ gives $E[C_{\mathrm{agg}}]=E[N]E[C]$, provided the expectations are finite. The same reasoning gives $E[R_{\mathrm{agg}}]=E[N]E[R]$. These mean formulas do not determine the aggregate distribution; they summarize expected amounts only.

Common Errors to Check

Confusing the limit with the top of the layer. A limit $L$ on ceded payment and attachment $M$ give a top of $M+L$.
Applying the treaty to the wrong quantity. A per-claim treaty maps each $X_j$ separately; it does not apply one attachment to the annual total.
Capping the insurer’s retained payment at $M$. That is true only within the layer. Above $M+L$, the insurer retains $X-L$, which can exceed $M$.
Ignoring payment atoms. Claims below the attachment accumulate at zero ceded payment, and claims reaching the top of the layer accumulate at the full limit.
Multiplying by the claim count too early. First find the expected payment per claim under the treaty. Multiply by a fixed count, or by $E[N]$ under the stated independent random-count assumptions.

Key takeaway: A per-claim excess-of-loss treaty is a transformation of each gross severity. Define the layer’s attachment and limit precisely, calculate ceded and retained amounts claim by claim, and only then sum or take expectations across claims.

Check Your Understanding

For each problem, identify the per-claim layer carefully before calculating.

A hypothetical exponential claim severity has mean $1,000. A treaty has attachment $500 and limit $1,000. Find the expected ceded payment per claim and the probabilities of zero and full-layer payments. (Answer: expected ceded payment $383.40; zero-payment probability 0.3935; full-layer probability 0.2231.)
A claim is $400 with probability 0.25, $1,000 with probability 0.25, $2,400 with probability 0.30, and $5,000 with probability 0.20. Under attachment $1,000 and limit $2,000, find the expected ceded and retained payments. (Answer: expected ceded payment $820.00; expected retained payment $1,250.00.)
A treaty has attachment $1,000 and limit $1,500. For gross claims of $900, $1,600, and $3,100, find total gross claims, total ceded recovery, and total retained claims. (Answer: gross $5,600; ceded $2,100; retained $3,500.)
A portfolio has a fixed 8 claims. Each claim has expected gross severity $900 and expected ceded payment $250 under its per-claim treaty. Find the expected aggregate gross, ceded, and retained amounts. (Answer: gross $7,200; ceded $2,000; retained $5,200.)
A hypothetical exponential severity has mean $2,500. A treaty has attachment $500 and limit $1,000. Find the probability that the ceded payment is zero and the probability it equals the full limit. (Answer: zero-payment probability 0.1813; full-layer probability 0.5488.)

What You'll Learn