Per-Risk Excess of Loss Recoveries

Why the Order of Calculation Matters

Per-risk excess-of-loss reinsurance protects an insurer against large payments on individual risks. For each risk, the insurer first determines its payment under the underlying policy. The treaty then reimburses a specified layer of that payment, subject to its retention and limit. These are two separate calculations: the policy deductible and limit determine the amount the insurer owes the policyholder, while the treaty retention and limit determine how much of that payment is ceded.

This distinction is central to aligned with the CAS Exam 5 (Basic Ratemaking and Estimated Claim Liabilities) syllabus. A treaty generally responds to the insurer’s covered payment—not automatically to the policyholder’s gross loss. The precise treaty wording governs what counts as loss, but the calculations here use the standard assumption that the treaty applies to the insurer’s payment after the underlying policy terms.

Definition: Let $X$ be the gross loss on one risk. Suppose the policy has a deductible $d$ and a maximum insurer payment $u$, where $u$ is a cap on payment after the deductible. The insurer’s policy payment per loss is $$ Y^L=\min\bigl((X-d)^+,u\bigr), \qquad (z)^+=\max(z,0). $$ For a per-risk excess-of-loss treaty with retention $R$ and limit $L$, the recovery on that risk is $$ C=\min\bigl((Y^L-R)^+,L\bigr). $$ The insurer’s net payment on the risk is $N=Y^L-C$.

Here $R$ is the treaty’s attachment point measured from zero on the applicable insurer-payment basis. The limit $L$ is the maximum amount the reinsurer pays on one risk; the covered layer extends from $R$ up to $R+L$. If a treaty is instead described by an upper exhaustion point, convert it to a layer limit by subtracting the retention. Always check the stated convention.

The policy formula above assumes that $u$ caps the insurer’s payment after the deductible. Some policies define their limit differently—for example, as a limit on covered loss before the deductible. In that case, the policy payment must be calculated from that wording rather than by using this formula unchanged. The treaty calculation still follows the policy payment once it has been correctly determined.

How the Per-Risk Layer Works

For a single risk, a treaty recovery is zero until the insurer’s policy payment exceeds $R$. Above $R$, the reinsurer pays dollar for dollar until the recovery reaches $L$. Any payment above the layer remains with the insurer. This gives three regions:

Policy payment $Y^L$	Treaty recovery $C$	Insurer net $N$
$Y^L\le R$	0	$Y^L$
$R<Y^L<R+L$	$Y^L-R$	$R$
$Y^L\ge R+L$	$L$	$Y^L-L$

The formula can be justified by measuring how much of the policy payment falls inside the treaty layer. The layer occupies payment amounts above $R$, but no higher than $R+L$. If $Y^L\le R$, none of the payment enters the layer. If $R<Y^L<R+L$, the part in the layer has length $Y^L-R$. If $Y^L\ge R+L$, the entire layer of width $L$ is reached. Combining those cases yields

C= \begin{cases} 0, & Y^L\le R,\\ Y^L-R, & R<Y^L<R+L,\\ L, & Y^L\ge R+L, \end{cases} =\min\bigl((Y^L-R)^+,L\bigr).

Since the treaty recovery is part of the policy payment reimbursed by the reinsurer, the insurer’s net is $N=Y^L-C$. The check $Y^L=C+N$ should hold for every risk.

Worked Calculations

Worked Example: A Policy Payment Capped Before the Treaty Layer

A hypothetical policy has gross loss $X=125$, deductible $d=25$, and maximum insurer payment $u=75$. A per-risk treaty has retention $R=40$ and limit $L=25$. Find the policy payment, treaty recovery, and net payment. Treat the stated loss and terms as deterministic; no probability distribution, mortality assumption, or interest rate is needed.

Step 1: Apply the policy deductible. The loss remaining after the deductible is $$ (X-d)^+=(125-25)^+=100. $$

Step 2: Apply the policy payment cap. $$ Y^L=\min(100,75)=75. $$ The insurer’s payment is $75$, not $100$, because the policy caps payment after the deductible.

Step 3: Apply the treaty retention and limit. The amount above the retention is $75-40=35$, so the recovery before the treaty cap would be $35$. The limit is $25$, giving $$ C=\min\bigl((75-40)^+,25\bigr)=\min(35,25)=25. $$

Step 4: Calculate and check the net payment. $$ N=Y^L-C=75-25=50, \qquad C+N=25+50=75=Y^L. $$ Thus the policy payment is $75$, the ceded recovery is $25$, and the insurer’s net payment is $50$. The treaty limit binds because the amount above the retention exceeds $25$.

Worked Example: Applying One Treaty Separately to Several Risks

A hypothetical insurer has three separate risks with gross losses $55$, $105$, and $180$. Each policy has deductible $d=15$ and maximum insurer payment $u=100$. A per-risk treaty has $R=60$ and $L=40$. Find the total policy payments, total recoveries, and total net payments. Losses are stipulated amounts; no probability distribution, mortality assumption, or interest rate is used.

Step 1: Calculate policy payments risk by risk. For the first risk, $$ Y^L_1=\min((55-15)^+,100)=\min(40,100)=40. $$ For the second risk, $$ Y^L_2=\min((105-15)^+,100)=\min(90,100)=90. $$ For the third risk, $$ Y^L_3=\min((180-15)^+,100)=\min(165,100)=100. $$ Therefore the total policy payment is $40+90+100=230$.

Step 2: Apply the treaty separately to each policy payment. $$ C_1=\min((40-60)^+,40)=0, $$ $$ C_2=\min((90-60)^+,40)=30, $$ $$ C_3=\min((100-60)^+,40)=40. $$ The third recovery reaches, but does not exceed, the treaty limit. Total recovery is $0+30+40=70$.

Step 3: Find net payments and check totals. The net payments are $40-0=40$, $90-30=60$, and $100-40=60$. Their total is $160$, and $$ 230=70+160. $$ Thus total policy payments are $230$, total recoveries are $70$, and total net payments are $160$. The per-risk limit is applied to each risk’s recovery, not once to the sum of all three risks.

Worked Example: A Large Gross Loss That Does Not Exhaust the Treaty

A hypothetical commercial policy has gross loss $X=155$, deductible $d=10$, and maximum insurer payment $u=90$. Its per-risk treaty has retention $R=50$ and limit $L=30$. Find the policy payment, recovery, and net payment. Treat the loss as deterministic; probability, mortality, and interest assumptions are not required.

Step 1: Calculate the policy payment. After the deductible, the loss is $155-10=145$. The policy payment is capped at $90$: $$ Y^L=\min((155-10)^+,90)=\min(145,90)=90. $$

Step 2: Calculate the recovery. The policy payment exceeds the treaty retention by $90-50=40$, but the treaty pays at most $30$: $$ C=\min((90-50)^+,30)=\min(40,30)=30. $$

Step 3: Calculate the net and check. $$ N=90-30=60, \qquad C+N=30+60=90=Y^L. $$ The net payment is $60$, not $50$: once the treaty layer is exhausted, the insurer retains the amount above the layer as well as the retention.

Multiple Risks and Common Errors

A per-risk treaty is applied to each risk separately. If there are $n$ risks, first calculate $Y^L_j$, $C_j$, and $N_j$ for each risk $j$. Then add the corresponding amounts:

C_{\mathrm{total}}=\sum_{j=1}^{n} C_j,\qquad N_{\mathrm{total}}=\sum_{j=1}^{n} N_j,\qquad \sum_{j=1}^{n}Y^L_j=C_{\mathrm{total}}+N_{\mathrm{total}}.

Do not add losses across risks and apply one retention to the combined amount. That would change the treaty from per-risk coverage into a different structure. Similarly, a limit $L$ on each risk does not generally limit the total recovery across all risks to $L$.

A frequent exam trap is applying the treaty to gross loss $X$ before calculating the policy payment. For example, in the first worked example, directly calculating $\min((125-40)^+,25)=25$ happens to produce the same recovery as the correct method. That coincidence can conceal the error. If the treaty were applied to the policy payment, as required here, the input is $Y^L=75$, not gross loss $125$. Where the policy deductible or limit changes how much enters the treaty basis, using $X$ can produce a different and incorrect answer.

Common mistake: Treating the treaty retention as if it were the policy deductible, or treating the treaty limit as if it were the policy limit. The policy deductible and limit determine $Y^L$. Only after $Y^L$ is known should the treaty retention $R$ and per-risk limit $L$ be applied. A useful final check is $0\le C\le L$ and $N=Y^L-C\ge0$.

Another error is to subtract the treaty recovery from gross loss rather than from the insurer’s policy payment. Under the stated assumptions, the insurer does not owe the policyholder the deductible amount, and a policy cap may also leave part of gross loss unpaid. Reinsurance changes the insurer’s financing of its payment; it does not enlarge the policy obligation to the insured.

Check Your Understanding

For each problem, calculate policy payments first and then apply the per-risk treaty. Assume the treaty applies to the insurer’s policy payment and that each stated $u$ is a maximum payment after the deductible.

A hypothetical risk has $X=84$, $d=12$, and $u=60$. Its treaty has $R=45$ and $L=20$. Find $Y^L$, $C$, and $N$. (Answer: $Y^L=60,\ C=15,\ N=45$.)
Three risks have gross losses $44$, $82$, and $130$. Each has $d=10$ and $u=75$; the treaty has $R=50$ and $L=30$. Find total policy payments, total recoveries, and total net payments. (Answer: policy payments $181$, recoveries $47$, net payments $134$.)
A policy has $X=175$, $d=20$, and $u=110$. The treaty has $R=70$ and $L=25$. Find the policy payment, recovery, and net payment, and state whether the treaty layer is exhausted. (Answer: $Y^L=110,\ C=25,\ N=85$; the layer is exhausted.)
Three separate risks have losses $35$, $95$, and $160$. Each policy has $d=5$ and $u=100$. A per-risk treaty has $R=55$ and $L=50$. Find total policy payments, recoveries, and net payments. (Answer: policy payments $220$, recoveries $80$, net payments $140$.)
Two risks have gross losses $150$ and $115$. Each policy has $d=40$ and $u=80$. A per-risk treaty has $R=65$ and $L=25$. Find each risk’s payment, recovery, and net payment, then give the totals. (Answer: policy payments $80$ and $75$; recoveries $15$ and $10$; net payments $65$ and $65$; totals $155$, $25$, and $130$, respectively.)

Policy payment \(Y^L\)	Treaty recovery \(C\)	Insurer net \(N\)
\(Y^L\le R\)	0	\(Y^L\)
\(R<Y^L<R+L\)	\(Y^L-R\)	\(R\)
\(Y^L\ge R+L\)	\(L\)	\(Y^L-L\)

What You'll Learn