Premiums for Excess-of-Loss Covers

Introduction

An excess-of-loss treaty changes the insurer’s financial position in two distinct ways. The insurer pays a reinsurance premium for the cover, and may receive a claim recovery when a covered loss penetrates the treaty layer. These amounts are related to the same protection, but they are not interchangeable: the premium is a price for cover, while a recovery depends on losses that actually occur.

For a basic retained-premium calculation, use the gross premium for the covered business and subtract the stated reinsurance premium. Do not substitute a claim recovery for the reinsurance premium, and do not infer a reinsurance premium from a layer’s attachment and limit alone. The treaty’s pricing terms must supply that premium.

Definition: For a specified period and portfolio, the insurer’s retained premium is the gross premium received or recorded for the covered business, less the reinsurance premium paid or payable for that cover, when no other premium adjustments are included. In symbols, $P_{\rm retained}=P_{\rm gross}-P_{\rm reins}$. This basic identity assumes that the amounts are on the same basis and period, and excludes items such as ceding commissions, taxes, brokerage, and other expenses unless the problem explicitly includes them.

Premium Accounting and Layer Recoveries

Let $X$ be a covered loss. For an excess-of-loss layer attaching at $d$ with a maximum payment (layer limit) $u$, the recovery per loss is

Y^L=\min\bigl((X-d)_+,u\bigr), \qquad (z)_+=\max(z,0).

This formula determines the reinsurer’s payment on a loss under the specified layer. It does not determine the premium charged for the treaty. The attachment and limit define the protection; the reinsurance premium is a separate contractual amount, which may be stated directly or set by pricing terms not supplied by the layer formula.

The premium identity follows from the insurer’s premium account. If the insurer records gross premium $P_{\rm gross}$ and pays reinsurance premium $P_{\rm reins}$, the amount left from premium receipts after that payment is $P_{\rm gross}-P_{\rm reins}$. A claim recovery enters a different account: it offsets claims paid or payable, not the premium paid for cover.

\begin{aligned} \text{retained premium} &= P_{\rm gross}-P_{\rm reins},\\ \text{retained claims} &= \text{gross claims}-\text{reinsurance recoveries}. \end{aligned}

Consequently, if an underwriting result is requested and expenses are ignored, the corresponding calculation is gross premium less reinsurance premium less retained claims. A recovery can improve that result by reducing retained claims, but it does not increase retained premium.

Identify the premium amounts.
Use the gross premium for the specified portfolio and period, and the stated reinsurance premium for the same scope and period.

Calculate retained premium.
Subtract the reinsurance premium from gross premium. Include commissions or other adjustments only if the problem provides them and defines how they affect the amount.

Handle losses separately.
Apply the treaty’s attachment and limit to each covered loss, then total the recoveries if required. Do not put this recovery into the premium subtraction.

Worked Applications

Worked Example: Retained Premium and a Separate Claim Recovery

A hypothetical insurer records annual gross premiums of $1,260,000 for a portfolio. It pays a stated annual premium of $186,000 for a per-risk excess-of-loss treaty with a $250,000 attachment and a $300,000 limit per risk. During the year, one covered risk produces a loss of $430,000. Calculate retained premium and the recovery on that loss. Assume no commissions, taxes, or expenses. No interest rate or mortality assumption is relevant; the premium amounts and claim are stated, so no loss distribution is needed.

Method. First calculate retained premium from the two stated premium amounts. Then apply the layer formula to the loss. The treaty is per risk, so the stated loss is evaluated as one risk’s loss.

Retained premium. The gross premium is $1,260,000 and the reinsurance premium is $186,000:

P_{\rm retained}=\$1{,}260{,}000-\$186{,}000=\$1{,}074{,}000.

Recovery. The amount above the attachment is $430,000 - $250,000 = $180,000. The payment cannot exceed the $300,000 layer limit, so:

Y^L=\min(\$180{,}000,\$300{,}000)=\$180{,}000.

The insurer therefore retains $1,074,000 of premium and receives a $180,000 recovery on this loss. The recovery is not subtracted from the reinsurance premium, nor added to retained premium. As a separate check on the loss allocation, the insurer retains $430,000 - $180,000 = $250,000 of this claim, equal to the attachment.

Worked Example: A Per-Risk Treaty Across Two Losses

A hypothetical insurer writes two risks with gross premiums of $95,000 and $65,000. It pays a stated annual reinsurance premium of $72,000 for a per-risk layer of $50,000 excess of $100,000, meaning a $100,000 attachment and a $50,000 maximum recovery on each risk. The two losses are $138,000 and $42,000. Calculate total gross premium, retained premium, each recovery, and total recovery. Assume no commissions, taxes, or expenses. No interest rate or mortality assumption is relevant; the losses are specified, so no probability distribution is required.

Method. Add the two gross premiums, subtract the treaty premium once, and then apply the per-risk layer separately to each loss.

Premium calculation.

\begin{aligned} P_{\rm gross} &= \$95{,}000+\$65{,}000=\$160{,}000,\\ P_{\rm retained} &= \$160{,}000-\$72{,}000=\$88{,}000. \end{aligned}

First risk’s recovery. The loss exceeds the $100,000 attachment by $138,000 - $100,000 = $38,000. This is below the $50,000 layer limit, so the recovery is $38,000.

Second risk’s recovery. The $42,000 loss is below the $100,000 attachment, so the recovery is $0.

\begin{aligned} Y^L_1&=\min(\$38{,}000,\$50{,}000)=\$38{,}000,\\ Y^L_2&=\min(0,\$50{,}000)=\$0,\\ \text{total recovery}&=\$38{,}000+\$0=\$38{,}000. \end{aligned}

The premium calculation checks against the stated amounts: $160,000 - $72,000 = $88,000. The $38,000 recovery is a separate result of the two losses. In particular, the fact that one loss generates a recovery does not change the stated $72,000 treaty premium.

Worked Example: Expected Recovery Is Not the Reinsurance Premium

A hypothetical insurer has gross annual premiums of $400,000 and pays a stated annual premium of $215,000 for a per-risk layer of $200,000 excess of $500,000. For a randomly selected covered risk, the annual loss $X$ has the following discrete distribution: $P(X=\$200{,}000)=0.5$, $P(X=\$600{,}000)=0.3$, and $P(X=\$1{,}100{,}000)=0.2$. Calculate the expected recovery for that risk and the insurer’s retained premium. Assume no commissions, taxes, or expenses. No interest rate or mortality assumption is relevant. The supplied probabilities and loss amounts fully specify the loss distribution.

Method. Calculate the layer recovery at each possible loss, weight each recovery by its probability, and add the weighted amounts. Separately subtract the stated treaty premium from gross premium.

Recovery at each loss. At $200,000, the loss is below the $500,000 attachment, so the recovery is $0. At $600,000, the amount above attachment is $100,000, below the $200,000 limit, so recovery is $100,000. At $1,100,000, the amount above attachment is $600,000, capped by the $200,000 limit, so recovery is $200,000.

The probabilities sum to $0.5+0.3+0.2=1.0$. Thus the expected recovery is:

\begin{aligned} E[Y^L] &=(0.5)(\$0)+(0.3)(\$100{,}000)+(0.2)(\$200{,}000)\\ &=\$0+\$30{,}000+\$40{,}000\\ &=\$70{,}000. \end{aligned}

The insurer’s retained premium is calculated from the stated annual amounts:

P_{\rm retained}=\$400{,}000-\$215{,}000=\$185{,}000.

The expected recovery of $70,000 is not the reinsurance premium of $215,000. It is an expected claim payment for one randomly selected risk under the supplied distribution; the treaty premium is the stated price for the cover. Any comparison of these values must respect their scope as well: the expected recovery here is per selected risk, whereas the stated treaty premium applies to the insurer’s covered arrangement.

Common Mistakes and Contract Scope

A frequent error is to treat a reinsurance recovery as if it were a premium credit. For example, suppose gross premium is $500,000, reinsurance premium is $80,000, and the insurer receives a $45,000 recovery. Retained premium remains $500,000 - $80,000 = $420,000. The $45,000 recovery belongs in the claim calculation; retained claims are reduced by $45,000. Adding it to retained premium to report $465,000 mixes two different cash-flow categories.

Another error is to calculate a reinsurance premium from the attachment, limit, or recovery amount when the problem has not given a pricing rule. Those contract terms determine the covered claim payment, not its price. If the reinsurance premium is stated, use it. If it is not stated, a retained-premium amount cannot be determined without additional pricing information.

Check the scope before subtracting. Gross and ceded premiums must refer to compatible periods and business. A stated annual treaty premium should not be subtracted from a different period’s gross premium without an explicit instruction to prorate or otherwise adjust. Similarly, a portfolio-level reinsurance premium should not automatically be allocated to individual risks unless an allocation rule is provided.

Key takeaway: Retained premium is a premium-account calculation: gross premium minus the stated reinsurance premium, subject to the problem’s stated adjustments. A layer recovery is a claim-account calculation: apply the attachment and limit to covered losses. Keep the two calculations separate, even when both are reported for the same treaty and period.

Check Your Understanding

For each question, keep the premium calculation separate from any claim recovery. Unless stated otherwise, assume no commissions, taxes, or expenses; no interest rate or mortality assumption is relevant.

A hypothetical insurer records $82,000 of gross premium and pays a stated $17,000 premium for an excess-of-loss treaty. What is its retained premium? (Answer: $65,000)
A per-risk layer is $150,000 excess of $100,000. One covered loss is $310,000, and the stated annual reinsurance premium is $48,000. What is the recovery on this loss, and what information is needed to calculate retained premium? (Answer: Recovery = $150,000; gross premium is also needed to calculate retained premium.)
A portfolio has gross premium of $520,000 and pays $84,000 for a per-occurrence layer of $250,000 excess of $100,000. One covered occurrence produces a loss of $260,000. Find retained premium and the recovery. (Answer: Retained premium = $436,000; recovery = $160,000.)
For one risk, loss $X$ is $0 with probability 0.4, $200,000 with probability 0.35, and $700,000 with probability 0.25. A per-risk layer is $300,000 excess of $100,000. Gross portfolio premium is $900,000 and the stated treaty premium is $125,000. Calculate expected recovery per risk and retained premium. (Answer: Expected recovery = $110,000 per risk; retained premium = $775,000.)
A hypothetical insurer has two separate books with gross premiums of $240,000 and $360,000. It pays stated annual treaty premiums of $30,000 for the first book and $50,000 for the second. A claim recovery of $40,000 occurs on the second book. Find retained premium for each book and in total, and state where the recovery is accounted for. (Answer: First-book retained premium = $210,000; second-book retained premium = $310,000; total retained premium = $520,000; the $40,000 recovery is accounted for against claims, not premium.)

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