Policy Limits, Deductibles, and Reinsurance Recoveries

From Policy Loss to Reinsurance Recovery

A claim can pass through two different contracts before the insurer determines its final cost. The policy first determines how much the insurer owes the policyholder after applying the deductible and policy limit. A reinsurance treaty may then reimburse some of that insurer payment. The two calculations use different contract terms: a policy deductible is not a treaty retention, and a policy limit is not a reinsurance layer limit.

The calculation therefore begins with the loss covered by the policy, not with the treaty. Only after determining the insurer’s gross policy payment should you apply the reinsurance terms, using the treaty’s stated subject loss. This tutorial assumes hypothetical losses and contract wording; actual policy and treaty language governs the order and basis of payment.

Definition: Let $X$ be the amount of a covered loss before the policy deductible. For a policy with deductible $d$ and an insurer-payment limit $u$ applied after the deductible, the insurer’s payment per loss is $$ Y^L=\min\bigl((X-d)^+,u\bigr), \qquad z^+=\max(z,0). $$ Assuming the policyholder is responsible for the deductible and any amount above the limit, the policyholder’s payment is $X-Y^L$. The insurer’s gross policy payment is $Y^L$, before reinsurance.

Apply the Policy Terms Before the Treaty

Under the convention in the definition, the policyholder bears the first $d$ dollars of loss. Once the loss exceeds $d$, the insurer pays the excess, up to a maximum insurer payment of $u$. The insurer’s payment can be expressed piecewise as

Y^L= \begin{cases} 0, & X\le d,\\ X-d, & d<X<d+u,\\ u, & X\ge d+u. \end{cases}

This piecewise form shows why the compact formula works: below the deductible the payment is zero; between the deductible and the exhaustion point $d+u$, it is $X-d$; above that point it stays at $u$. Thus the insurer’s payment is the amount above the deductible, capped at $u$. The expression applies only when $u$ is a cap on the insurer’s payment after the deductible.

Policy wording can instead define $u$ as the maximum covered loss amount before the deductible is subtracted. In that convention, the policy first caps the loss at $u$, then applies the deductible:

Y^L=\bigl(\min(X,u)-d\bigr)^+.

These conventions can give different payments. Always identify what the stated limit caps. Do not substitute one formula for the other merely because both contracts use the word “limit.”

Worked Example: Policy Limit Applied After the Deductible

A hypothetical commercial equipment policy has a $2{,}500$-dollar deductible and a $50{,}000$-dollar limit on the insurer’s payment per loss. A covered loss is $72{,}500$ dollars. The treaty reimburses $30{,}000$ dollars excess of $20{,}000$ dollars on the gross policy payment for this risk. Find the policyholder’s payment, gross policy payment, treaty recovery, and insurer’s net payment.

Method. First apply the policy deductible and payment limit. Then apply the per-risk excess-of-loss layer to the gross policy payment.

Step 1: Calculate the gross policy payment. The loss above the deductible is $72{,}500-2{,}500=70{,}000$ dollars. The policy limit caps the insurer’s payment at $50{,}000$ dollars, so $\min(70{,}000,50{,}000)=50{,}000$ dollars.

Step 2: Calculate the policyholder’s payment. The policyholder is responsible for the portion of the loss not paid by the insurer: $72{,}500-50{,}000=22{,}500$ dollars. This consists of the $2{,}500$-dollar deductible and $20{,}000$ dollars above the policy’s payment limit.

Step 3: Calculate the treaty recovery. The treaty is $30{,}000$ dollars excess of $20{,}000$ dollars. Its maximum recovery is $30{,}000$ dollars. On a $50{,}000$-dollar subject loss, the amount above the attachment is $50{,}000-20{,}000=30{,}000$ dollars. The recovery is therefore $\min(30{,}000,30{,}000)=30{,}000$ dollars.

Step 4: Calculate the insurer’s net payment. Subtract the treaty recovery from the gross policy payment: $50{,}000-30{,}000=20{,}000$ dollars.

Check. The allocation reconciles to the loss: $22{,}500+30{,}000+20{,}000=72{,}500$ dollars. The $22{,}500$ dollars is the policyholder’s responsibility; the treaty does not reduce that amount under the stated terms.

Apply the Treaty to Its Stated Subject Loss

For an excess-of-loss layer, the attachment is the amount of subject loss the insurer retains before the layer begins to pay. The layer limit is the maximum amount the reinsurer pays for that layer. A layer described as $L$ dollars excess of $A$ dollars has limit $L$ and attachment $A$; its maximum recovery is $L$, not an amount that can be inferred from the attachment alone.

Definition: If $S$ is the loss subject to a layer with attachment $A$ and limit $L$, the recovery is $$ R=\min\bigl((S-A)^+,L\bigr). $$ The insurer’s net subject loss is $S-R$. This formula assumes the treaty applies to the stated subject loss on the stated risk or occurrence, and that no other treaty terms alter the layer’s payment.

The formula follows from the layer’s two payment boundaries. If $S\le A$, no loss penetrates the attachment, so recovery is zero. If $A<S<A+L$, the loss within the layer is $S-A$. If $S\ge A+L$, the entire layer is exhausted and recovery cannot exceed $L$. Combining those cases gives $\min((S-A)^+,L)$.

The treaty’s subject loss $S$ must be identified from the wording. A per-risk layer may apply to a single risk’s gross policy payment; a per-occurrence layer may apply to the sum of specified payments for one occurrence. Do not add losses across policies or events unless the treaty defines them as part of the same subject loss. Nor should you apply a treaty attachment to the original loss $X$ when the contract says it applies to gross policy payments.

Worked Example: Several Policy Payments Under One Occurrence Layer

A hypothetical wind event causes covered losses of $18{,}000$, $47{,}000$, and $92{,}000$ dollars on three separate policies. Each policy has a $5{,}000$-dollar deductible and a $40{,}000$-dollar limit on the insurer’s payment per loss, applied after the deductible. A per-occurrence treaty covers $60{,}000$ dollars excess of $25{,}000$ dollars, based on the sum of the three gross policy payments. Calculate the policyholder payments, the treaty recovery, and the insurer’s net payment for the occurrence.

Method. Calculate each policy payment separately, add those payments to obtain the treaty’s occurrence subject loss, and then apply the layer formula.

Step 1: Calculate the first policy payment. The loss above its deductible is $18{,}000-5{,}000=13{,}000$ dollars, below the $40{,}000$-dollar limit. Its gross policy payment is $13{,}000$ dollars, and its policyholder payment is $18{,}000-13{,}000=5{,}000$ dollars.

Step 2: Calculate the second policy payment. The loss above its deductible is $47{,}000-5{,}000=42{,}000$ dollars. The limit reduces the insurer’s payment to $40{,}000$ dollars. The policyholder pays $47{,}000-40{,}000=7{,}000$ dollars.

Step 3: Calculate the third policy payment. The loss above its deductible is $92{,}000-5{,}000=87{,}000$ dollars. The limit again caps the insurer’s payment at $40{,}000$ dollars. The policyholder pays $92{,}000-40{,}000=52{,}000$ dollars.

Step 4: Aggregate the treaty subject loss. The occurrence treaty is based on the three gross policy payments, so $S=13{,}000+40{,}000+40{,}000=93{,}000$ dollars. The total policyholder payments are $5{,}000+7{,}000+52{,}000=64{,}000$ dollars.

Step 5: Calculate recovery and net payment. The layer has a $60{,}000$-dollar limit and a $25{,}000$-dollar attachment. The amount above the attachment is $93{,}000-25{,}000=68{,}000$ dollars, so recovery is $\min(68{,}000,60{,}000)=60{,}000$ dollars. The insurer’s net payment is $93{,}000-60{,}000=33{,}000$ dollars.

Check. Total loss is $18{,}000+47{,}000+92{,}000=157{,}000$ dollars. The allocation is $64{,}000+60{,}000+33{,}000=157{,}000$ dollars. The layer is exhausted because the amount above its attachment exceeds its $60{,}000$-dollar limit.

When the Policy Limit Is Defined Before the Deductible

The limit convention matters most when the loss reaches the policy limit. If $u$ caps the insurer’s payment after the deductible, a loss large enough to exhaust the limit produces an insurer payment of $u$. If instead $u$ caps the amount of loss eligible for coverage before the deductible is subtracted, the maximum insurer payment is generally $u-d$, provided $u>d$. In either case, the reinsurance calculation still uses the gross policy payment if that is the treaty’s stated basis.

Worked Example: Limit on Covered Loss Before the Deductible

A hypothetical policy states that the maximum covered loss amount is $80{,}000$ dollars before applying the $4{,}000$-dollar deductible. A covered loss is $96{,}000$ dollars. A per-risk reinsurance layer is $50{,}000$ dollars excess of $25{,}000$ dollars and applies to the gross policy payment. Find the policyholder payment, gross policy payment, treaty recovery, and insurer’s net payment.

Method. Because the stated policy limit caps covered loss before the deductible, cap $X$ at $u$ first, then subtract $d$. Next apply the treaty to the resulting gross policy payment.

Step 1: Cap the covered loss and apply the deductible. The capped covered loss is $\min(96{,}000,80{,}000)=80{,}000$ dollars. The insurer pays $(80{,}000-4{,}000)^+=76{,}000$ dollars.

Step 2: Calculate the policyholder payment. The policyholder is responsible for $96{,}000-76{,}000=20{,}000$ dollars: the $4{,}000$-dollar deductible plus $16{,}000$ dollars above the covered-loss limit.

Step 3: Calculate the treaty recovery. The amount of the $76{,}000$-dollar gross policy payment above the $25{,}000$-dollar attachment is $51{,}000$ dollars. The layer limit is $50{,}000$ dollars, so recovery is $\min(51{,}000,50{,}000)=50{,}000$ dollars.

Step 4: Calculate the insurer’s net payment. The insurer retains $76{,}000-50{,}000=26{,}000$ dollars after reinsurance.

Check. The total allocation is $20{,}000+50{,}000+26{,}000=96{,}000$ dollars. Applying the after-deductible payment-limit formula here would incorrectly produce a $80{,}000$-dollar gross payment; the policy wording makes the limit a cap on covered loss before the deductible.

Common Exam Trap: Confusing a Layer Limit With Its Attachment

A phrase such as “$40{,}000$ dollars excess of $60{,}000$ dollars” means that the layer attaches at $60{,}000$ dollars and can pay at most $40{,}000$ dollars. It does not mean the layer pays $60{,}000$ dollars, and it cannot produce a recovery greater than $40{,}000$ dollars.

For example, suppose the subject loss is $190{,}000$ dollars and the treaty is $40{,}000$ dollars excess of $60{,}000$ dollars. The incorrect calculation is $190{,}000-60{,}000=130{,}000$ dollars of recovery, which ignores the layer limit. The correct calculation is $\min(190{,}000-60{,}000,40{,}000)=40{,}000$ dollars. The insurer’s net subject loss is $190{,}000-40{,}000=150{,}000$ dollars.

Before calculating, write down both treaty parameters and their roles: attachment $A$, layer limit $L$. Then identify the subject loss $S$ from the contract basis and calculate $\min((S-A)^+,L)$. Finally, check that the recovery is nonnegative, does not exceed $S$, and does not exceed $L$.

Key takeaway: Apply the policy deductible and limit to determine gross insurer payment first. Apply the treaty’s attachment and layer limit to the treaty-defined subject loss second. Keep the policyholder’s responsibility, reinsurance recovery, and insurer’s net payment as separate amounts, then reconcile them to the original loss.

Check Your Understanding

For each problem, identify the stated policy-limit convention and treaty subject loss before calculating. Amounts are in dollars.

A policy has a deductible of $2{,}500$ and a $28{,}000$ limit on the insurer’s payment after the deductible. The covered loss is $34{,}500$. Find the gross policy payment and policyholder payment. (Answer: gross policy payment $28{,}000$; policyholder payment $6{,}500$.)
Gross policy payments from one occurrence are $22{,}000$, $40{,}000$, and $35{,}000$. An occurrence layer is $35{,}000$ excess of $30{,}000$, based on their sum. Find the recovery and insurer’s net occurrence payment. (Answer: recovery $35{,}000$; net payment $62{,}000$.)
A covered loss is $73{,}000$, the deductible is $3{,}000$, and the policy’s insurer-payment limit after the deductible is $60{,}000$. A per-risk treaty is $25{,}000$ excess of $20{,}000$, based on gross policy payment. Find the policyholder payment, recovery, and net insurer payment. (Answer: policyholder payment $13{,}000$; recovery $25{,}000$; net insurer payment $35{,}000$.)
A policy caps covered loss at $75{,}000$ before applying its $4{,}000$ deductible. The covered loss is $91{,}000$. A treaty is $40{,}000$ excess of $30{,}000$, based on gross policy payment. Find the gross policy payment, recovery, and net insurer payment. (Answer: gross policy payment $71{,}000$; recovery $40{,}000$; net insurer payment $31{,}000$.)
Two policies have covered losses of $44{,}000$ and $78{,}000$, each with a $4{,}000$ deductible and a $50{,}000$ insurer-payment limit applied after the deductible. They arise from one occurrence. An occurrence treaty is $45{,}000$ excess of $35{,}000$, based on their total gross policy payments. Find the total gross policy payment, recovery, net insurer payment, and total policyholder payment. (Answer: gross policy payment $90{,}000$; recovery $45{,}000$; net insurer payment $45{,}000$; policyholder payments $32{,}000$.)

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