Why Premium Deductions Change the Required Quote
An excess placement can have a quoted premium that differs materially from the amount ultimately retained by the reinsurer. Brokerage, premium taxes, and a ceding commission may each take a share of the premium, but they do not necessarily use the same calculation base. To price consistently, first identify which amounts are deducted from the reinsurer’s premium and which are separate pass-through collections.
The formulas below use a deliberately explicit convention. Let \(P\) be the subject premium before premium taxes. Brokerage and ceding commission are paid out of \(P\); premium tax is calculated as a percentage of \(P\), collected in addition to \(P\), and remitted to the taxing authority. Thus, tax increases the invoice but is not retained by the reinsurer. Actual contracts and tax rules can use other bases, so the stated convention must not be assumed without checking the placement terms.
If brokerage and commission are each percentages of subject premium, write their rates as \(b\) and \(c\). Then \(B=bP\), \(C=cP\), and the retained fraction is \(1-b-c\). With tax rate \(t\) applied to subject premium and billed separately, the premium flow is:
A tax charge is not automatically a premium deduction in the reinsurer’s retained-premium calculation. It is a deduction only if the contract’s cash-flow terms make it one; under the convention above, the tax is collected on top and passed through. Similarly, “commission of 10%” is incomplete unless the contract says what amount the 10% applies to.
Grossing Up a Required Retained Premium
For a target retained amount \(R\), when both brokerage and commission are percentages of \(P\), solve \(R=P(1-b-c)\) for \(P\). The premium tax then determines the invoice, not the subject premium needed to achieve \(R\).
This is a linear gross-up. It applies when the deduction rates are fixed, both deductions use the same subject-premium base, and the tax is assessed on subject premium and collected separately. If any of these conditions changes, the equation must change too.
Worked Example: Grossing Up for Brokerage, Commission, and Tax
A hypothetical excess reinsurer must retain \(\$23{,}400\). Brokerage is 8.0% of subject premium, ceding commission is 7.4% of subject premium, and a 4.0% premium tax is billed in addition to subject premium. Find the subject premium, each deduction, the tax-inclusive invoice, and the amount retained. Round final dollar amounts to cents.
Method. First gross up for the deductions that reduce retained premium. The brokerage and commission rates share the subject-premium base, so the retained fraction is \(1-0.080-0.074=0.846\).
Brokerage, calculated on the unrounded \(P\), is \(0.080(\$27{,}659.574468\ldots)=\$2{,}212.765957\ldots\), or \(\$2{,}212.77\). Ceding commission is \(0.074(\$27{,}659.574468\ldots)=\$2{,}046.808511\ldots\), or \(\$2{,}046.81\). The tax is \(0.040(\$27{,}659.574468\ldots)=\$1{,}106.382979\ldots\), or \(\$1{,}106.38\).
The tax-inclusive invoice, calculated before rounding the subject premium, is \((\$23{,}400/0.846)(1.04)=\$28{,}765.957447\ldots\), or \(\$28{,}765.96\). The reinsurer’s retained amount is \(\$27{,}659.574468\ldots-\$2{,}212.765957\ldots-\$2{,}046.808511\ldots=\$23{,}400\), as required.
Check and common mistake. The retained fraction is \(0.846\), and \(0.846P=\$23{,}400\). Do not multiply the displayed, rounded subject premium \(\$27{,}659.57\) by 1.04 and report \(\$28{,}765.96\): that multiplication gives \(\$28{,}765.9528\), which rounds to \(\$28{,}765.95\). The correct invoice of \(\$28{,}765.96\) comes from using the unrounded subject premium in the calculation.
Why the Gross-Up Formula Holds
The formula follows directly from the cash-flow definition. Suppose the reinsurer receives subject premium \(P\), then pays brokerage equal to \(bP\) and commission equal to \(cP\). Because both deductions are calculated on the same base, the remaining premium is:
For a specified target \(R\), divide both sides by the retained fraction, provided \(1-b-c>0\), to obtain \(P=R/(1-b-c)\). If \(1-b-c\) is zero or negative, no finite positive subject premium can produce a positive retained amount under this formula. The tax rate does not appear in this gross-up because the assumed tax is added to the invoice and remitted, rather than deducted from \(P\).
Worked Example: Separating the Invoice from the Retained Amount
A hypothetical treaty has a subject premium of \(\$50{,}000\). Brokerage is 5% of subject premium, ceding commission is 12% of subject premium, and tax is 3% of subject premium, billed separately. Calculate the invoice, each payment or deduction, and the reinsurer-retained premium. Round final dollar amounts to cents.
Method. Calculate each item using its stated base. Since the subject premium is already given, no gross-up is needed.
Brokerage is \(0.05(\$50{,}000)=\$2{,}500\). Ceding commission is \(0.12(\$50{,}000)=\$6{,}000\). Tax is \(0.03(\$50{,}000)=\$1{,}500\). The invoice is \(\$50{,}000+\$1{,}500=\$51{,}500\).
The reinsurer retains the subject premium after brokerage and commission: \(\$50{,}000-\$2{,}500-\$6{,}000=\$41{,}500\). The tax is remitted separately and does not reduce that amount under the stated convention.
Check and common mistake. The deductions from subject premium total \(\$8{,}500\), leaving \(\$41{,}500\). A common error is to subtract the \(\$1{,}500\) tax again from the reinsurer’s retained amount. That would treat the tax as both a separately billed pass-through and a deduction from premium.
When Commission Uses a Different Base
Some arrangements define commission as a percentage of premium remaining after brokerage, rather than as a percentage of the original subject premium. Let brokerage be \(bP\), and let commission be a fraction \(c\) of \(P-B\). Then \(C=c(P-bP)=cP(1-b)\), and retained premium is:
For a target retained amount, the corresponding gross-up is \(P=R/[(1-b)(1-c)]\). This is not generally the same as \(R/(1-b-c)\). In fact, \(1-b-c\) would apply the commission rate to the original subject premium, not to the amount left after brokerage.
Worked Example: Commission on Premium Net of Brokerage
A hypothetical reinsurer must retain \(\$42{,}300\). Brokerage is 6% of subject premium. Ceding commission is 10% of premium after brokerage has been deducted. Premium tax is 2.5% of subject premium and is billed separately. Find the subject premium, brokerage, commission, tax, invoice, and retained amount. Round final dollar amounts to cents.
Method. The commission base is \(P-B=0.94P\). Therefore, the retained fraction is \(0.94(1-0.10)=0.846\), and the required subject premium is:
Brokerage is \(0.06(\$50{,}000)=\$3{,}000\). Premium after brokerage is \(\$50{,}000-\$3{,}000=\$47{,}000\). Commission is \(0.10(\$47{,}000)=\$4{,}700\). The retained amount is \(\$47{,}000-\$4{,}700=\$42{,}300\).
Tax is \(0.025(\$50{,}000)=\$1{,}250\), so the invoice is \(\$50{,}000+\$1{,}250=\$51{,}250\). The cash-flow check is \(\$51{,}250-\$1{,}250-\$3{,}000-\$4{,}700=\$42{,}300\).
Check and common mistake. The target is met exactly. Using \(1-0.06-0.10=0.84\) as the retained fraction would incorrectly calculate commission on all of \(P\). The correct fraction is \(0.94(0.90)=0.846\) because the contract applies commission after brokerage.
Contract Bases, Rounding, and Practical Interpretation
There is no universal rule that brokerage, commission, and tax must use the same base. A placement may specify tax on a tax-inclusive amount, commission on premium net of brokerage, or another contractual basis. The actuary should translate the wording into equations before doing arithmetic. For example, if a tax is charged on the tax-inclusive invoice itself, the equation is not \(T=tP\); the tax base must be defined explicitly and solved consistently with the invoice convention.
Likewise, a sliding-scale commission, a minimum brokerage, or a commission contingent on results is not necessarily a fixed percentage deduction. A simple linear gross-up applies only within a range where the relevant rates and bases remain fixed. For a piecewise commission schedule, identify the applicable range, solve using that range’s formula, and then verify that the resulting premium actually falls in the range assumed.
Keep precision through the calculation and round final monetary amounts to cents. Individually rounded components may occasionally differ by a cent from a reconciliation made using unrounded amounts. If a displayed rounded premium is multiplied again, the result can differ from the correctly rounded value based on the unrounded premium, as the first example demonstrates. For exam-style work and professional pricing, show the unrounded calculation or enough precision to make the rounding path clear.
Distinguish subject premium from the tax-inclusive invoice and from the amount the reinsurer ultimately retains.
For every percentage, state whether it applies to subject premium, premium after brokerage, or another defined amount.
Include only amounts that reduce reinsurer-retained premium; keep separately billed pass-through taxes distinct.
Gross up if the retained amount is the target, calculate all amounts from unrounded values, and check the cash-flow identity.
This separation is useful whenever a quoted price must be translated into the amount available to support the reinsurer’s obligations. It also prevents a subtle but consequential error: treating every charge on an invoice as though it were a deduction from subject premium. Correct treatment depends on the contract and applicable tax rules, not on the label alone. The calculations in this tutorial are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each hypothetical placement, use the stated convention and calculate from unrounded amounts before rounding final dollar figures to cents.
- A subject premium is \(\$80{,}000\). Brokerage is 4% and commission is 6%, both of subject premium; tax is 5% of subject premium and billed separately. Find the invoice and reinsurer-retained premium. (Answer: invoice \(\$84{,}000.00\); retained premium \(\$72{,}000.00\))
- A reinsurer must retain \(\$30{,}000\). Brokerage is 7% and commission is 8%, both of subject premium; tax is 4% of subject premium and billed separately. Find the subject premium and invoice. (Answer: subject premium \(\$35{,}294.12\); invoice \(\$36{,}705.88\))
- A treaty has subject premium \(\$120{,}000\), brokerage of 3% of subject premium, commission of 15% of premium after brokerage, and tax of 2.5% of subject premium, billed separately. Find the invoice and retained premium. (Answer: invoice \(\$123{,}000.00\); retained premium \(\$98{,}940.00\))
- A reinsurer must retain \(\$50{,}000\). Brokerage is 5% of subject premium, commission is 5% of premium after brokerage, and tax is 6% of subject premium, billed separately. Find the subject premium and tax-inclusive invoice. (Answer: subject premium \(\$55{,}401.66\); invoice \(\$58{,}725.76\))
- For a subject premium of \(\$100{,}000\), compare two hypothetical contracts, each with brokerage of 4% of subject premium and a 3% tax on subject premium billed separately. In Contract A, commission is 10% of subject premium; in Contract B, it is 10% of premium after brokerage. Find each retained amount and the difference. (Answer: Contract A retains \(\$86{,}000.00\); Contract B retains \(\$86{,}400.00\); Contract B retains \(\$400.00\) more)