Why the Order of Operations Matters
A claim can pass through two separate contracts before the insurer's ultimate cost is known. The policy first determines how much of the ground-up loss is payable to the policyholder. A reinsurance treaty may then reimburse the insurer for some portion of that covered payment. The treaty attachment must be applied to the loss basis stated in the treaty, not automatically to the original loss.
This distinction matters whenever the policy deductible or limit changes the claim amount. Applying a treaty attachment to ground-up loss when it is defined on covered claim amounts can produce a different recovery. Conversely, applying it to the covered amount when the treaty is expressly defined on ground-up losses is also incorrect. The contract wording establishes the basis.
Here, \(u\) means the maximum indemnity paid by the policy after applying the deductible. If policy wording states a limit on total loss including the deductible, convert it to the corresponding maximum payment before using this representation. The formulas describe a per-claim layer; they do not impose an annual aggregate limit or reinstatement provision.
The sequence for a treaty on covered claim amounts is therefore \(X\rightarrow Y\rightarrow R_C\). First calculate the policy payment. Then compare that payment with the treaty attachment, and cap the excess at the treaty limit. The ground-up alternative \(R_G\) is a separate contract basis, not an intermediate calculation.
Worked Claim Calculations
Worked Example: Applying a Treaty to a Limited Policy Payment
A hypothetical policy has a deductible of $1,500 and a maximum payment of $5,000 per claim after the deductible. The ground-up loss is $8,000. A per-claim treaty on covered claim amounts attaches at $2,000 and has a $2,500 limit. Find the policy payment and reinsurance recovery, and compare them with applying the same treaty terms to ground-up loss.
Method. Calculate the policy payment first, then apply the treaty layer to that payment.
The amount above the deductible is $$ X-d=\$8{,}000-\$1{,}500=\$6{,}500. $$ The payment cap reduces this amount to $$ Y=\min(\$6{,}500,\$5{,}000)=\$5{,}000. $$ The covered-claim excess above the treaty attachment is $$ Y-a=\$5{,}000-\$2{,}000=\$3{,}000. $$ Capping this excess at the treaty limit gives $$ R_C=\min(\$3{,}000,\$2{,}500)=\boxed{\$2{,}500}. $$
For comparison, applying the treaty directly to ground-up loss gives $$ R_G=\min(\$8{,}000-\$2{,}000,\$2{,}500) =\min(\$6{,}000,\$2{,}500) =\boxed{\$2{,}500}. $$ The recoveries happen to be equal because both treaty calculations exceed the layer limit. This equality does not mean that the two bases are interchangeable.
Check. The covered payment is $5,000, the covered-basis recovery is $2,500, and the ground-up-basis recovery is also $2,500. The treaty recovery does not exceed its $2,500 limit or the covered payment.
Common mistake avoided. Subtracting the treaty attachment from the covered payment gives $5,000 − $2,000 = $3,000, not $2,500. That excess must still be capped at the treaty limit; the correct recovery is $2,500.
Worked Example: Expected Recovery for an Exponential Loss
A hypothetical ground-up claim has an exponential severity distribution with mean $5,000, so \(S_X(x)=e^{-x/5000}\) for \(x\geq 0\). A policy has a $1,000 deductible and a maximum payment of $10,000 after the deductible. A treaty on covered claim amounts attaches at $1,500 and has a $2,000 per-claim limit. Find the expected treaty recovery.
Method. Use the layer payment's tail-integral representation. For a nonnegative random variable \(Z\), \(E[Z]=\int_0^\infty P(Z>t)\,dt\). Here the recovery cannot exceed $2,000, so $$ E[R_C]=\int_0^{2000}P(R_C>t)\,dt. $$
For \(0\leq t<\$2{,}000\), the treaty recovery exceeds \(t\) exactly when the covered payment exceeds the attachment plus \(t\): $$ R_C>t \quad\Longleftrightarrow\quad Y>\$1{,}500+t. $$ The threshold \(\$1{,}500+t\) ranges from $1,500 to $3,500, below the policy payment cap of $10,000. Thus the policy cap does not truncate this layer. Since \(Y=\min((X-\$1{,}000)_+,\$10{,}000)\), these thresholds are exceeded exactly when \(X>\$2{,}500+t\). Consequently, $$ \begin{aligned} E[R_C] &=\int_0^{2000}P(X>\$2{,}500+t)\,dt\\ &=\int_0^{2000}e^{-(2500+t)/5000}\,dt\\ &=5000\left(e^{-0.5}-e^{-0.9}\right)\\ &=5000(0.6065306597-0.4065696597)\\ &=\$999.805\text{ approximately}. \end{aligned} $$
Rounded to the nearest cent, the expected recovery is \(\boxed{\$999.81}\). This is an expectation across hypothetical claims, not the recovery on one particular claim.
Check. The result is below the treaty limit of $2,000, as an expected recovery must be for a recovery that never exceeds $2,000. The policy cap is also above the maximum covered payment needed to exhaust the treaty layer, $1,500 + $2,000 = $3,500.
Common mistake avoided. The covered-basis attachment of $1,500 is not the ground-up threshold. The policy deductible shifts the corresponding ground-up threshold to $1,000 + $1,500 = $2,500. Omitting that shift would calculate a different treaty basis.
Worked Example: A Policy Limit Changes the Recovery
A hypothetical portfolio has one claim, with ground-up loss \(X\) taking values $1,000, $2,500, $4,000, or $6,000 with respective probabilities 0.20, 0.30, 0.30, and 0.20. The policy deductible is $1,000 and its maximum payment after the deductible is $3,000. A per-claim treaty attaches at $1,500 and has a $1,000 limit. Find the recovery for each loss and its expected value on the covered-claim basis; then compare with the ground-up basis.
Method. Calculate the covered payment at each possible loss, then apply the treaty to each payment. The probabilities sum to \(0.20+0.30+0.30+0.20=1.00\).
At \(X=\$1{,}000\), the amount above the deductible is zero, so \(Y=\$0\) and \(R_C=\$0\). At \(X=\$2{,}500\), \(Y=\min(\$2{,}500-\$1{,}000,\$3{,}000)=\$1{,}500\), so \(R_C=\$0\). At \(X=\$4{,}000\), \(Y=\min(\$3{,}000,\$3{,}000)=\$3{,}000\), so \(R_C=\min(\$3{,}000-\$1{,}500,\$1{,}000)=\$1{,}000\). At \(X=\$6{,}000\), the payment cap again gives \(Y=\$3{,}000\) and \(R_C=\$1{,}000\).
| Ground-up loss \(X\) | Probability | Covered payment \(Y\) | Covered-basis recovery \(R_C\) | Ground-up recovery \(R_G\) |
|---|---|---|---|---|
| $1,000 | 0.20 | $0 | $0 | $0 |
| $2,500 | 0.30 | $1,500 | $0 | $1,000 |
| $4,000 | 0.30 | $3,000 | $1,000 | $1,000 |
| $6,000 | 0.20 | $3,000 | $1,000 | $1,000 |
The expected covered-basis recovery is $$ E[R_C]=(0.20)(\$0)+(0.30)(\$0)+(0.30)(\$1{,}000)+(0.20)(\$1{,}000)=\$500. $$ For the ground-up treaty, the recoveries are $0, $1,000, $1,000, and $1,000, respectively. Therefore, $$ E[R_G]=(0.20)(\$0)+(0.30)(\$1{,}000)+(0.30)(\$1{,}000)+(0.20)(\$1{,}000)=\$800. $$ The policy deductible and limit reduce the expected recovery by $300 in this example because the treaty is defined on covered payments rather than on ground-up losses.
Check. Each covered-basis recovery is no more than $1,000, and its expected value is $500. The ground-up-basis expected value is $800; the difference follows from the different bases, not from a change in treaty terms.
Why the Layer Formula Works
The layer payment can be understood as accumulating small units of recovery. For a fixed covered payment \(Y=y\), the treaty pays one dollar for each dollar of the interval from \(a\) up to \(a+L\) that lies below \(y\). The length of that interval is \(\min((y-a)_+,L)\). Equivalently, it counts the values of \(t\) between 0 and \(L\) for which \(y>a+t\).
Taking expectations and interchanging the expectation and integral for this bounded, nonnegative quantity gives
This justifies the method used in the exponential example. If \(Y=\min((X-d)_+,u)\), then for thresholds \(a+t<u\), $$ P(Y>a+t)=P(X>d+a+t). $$ If \(a+t\geq u\), that probability is zero because the covered payment cannot exceed \(u\). Thus, when \(u>a\), the expected covered-basis recovery can also be written as
This form assumes the stated payment-cap convention and a nonnegative ground-up loss. It accounts for the deductible by shifting the survival threshold and for the policy limit by shortening the interval over which recovery can occur. It does not apply unchanged if the treaty uses a different loss basis or if the policy and treaty define limits differently.
Exam Checks and Common Traps
A reliable calculation labels the loss basis at every stage: ground-up loss \(X\), covered payment \(Y\), and treaty recovery \(R_C\) or \(R_G\). Check whether the treaty wording refers to covered claim payments or ground-up loss, and establish what “policy limit” means in the stated contract before calculating. A layer can be unaffected by a policy limit if the covered payment cap is high enough to reach the treaty's exhaustion point; otherwise, part of the treaty layer may never be reached.
- Applying the deductible twice. Calculate \(Y\) once. Do not subtract the policy deductible again from a covered payment.
- Ignoring the policy cap. The payment above the deductible may still be reduced by the policy's maximum payment.
- Stopping before applying the treaty limit. First find the excess above attachment, then take the smaller of that excess and \(L\).
- Assuming a shared attachment basis. A treaty on covered payments and a treaty on ground-up losses can produce different recoveries even with identical numerical attachment and limit.
The core discipline is to translate the policy terms into the claim amount actually covered before applying a treaty written on that amount. Only after confirming the treaty's basis should the attachment and limit be used.
Check Your Understanding
For each problem, identify the stated treaty basis before calculating. Monetary answers are in dollars.
- A claim is $5,200. The policy deductible is $700 and its maximum payment after the deductible is $3,000. A covered-claim treaty attaches at $1,800 and has a $1,000 limit. Find the covered payment and recovery, then find the recovery if the treaty instead applies to ground-up loss. (Answer: covered payment $3,000; covered-basis recovery $1,000; ground-up-basis recovery $1,000.)
- A ground-up loss is $2,100. The policy deductible is $900, with a payment cap of $4,000. A treaty on covered payments attaches at $1,500 and has a $1,000 limit. Find both the covered-basis and ground-up-basis recoveries. (Answer: covered payment $1,200; covered-basis recovery $0; ground-up-basis recovery $600.)
- A ground-up loss is exponential with mean $4,000. A policy has a $500 deductible and a $8,000 maximum payment after the deductible. A treaty on covered payments attaches at $1,000 and has a $1,500 limit. Find the expected recovery to the nearest cent. (Answer: \(4000(e^{-0.375}-e^{-0.75})\), approximately $859.69.)
- A loss is $1,500, $3,000, or $5,000 with probabilities 0.25, 0.50, or 0.25, respectively. The policy deductible is $1,000 and the payment cap is $2,500. A treaty on covered payments attaches at $1,000 and has a $1,500 limit. Find the expected recovery on the covered basis and on the ground-up basis. (Answer: covered-basis expected recovery $875; ground-up-basis expected recovery $1,250.)