Why a Retrospective Plan Has a Minimum
A retrospective premium responds to the insured’s experience during a defined period. When losses are low, the formula result may be less than the amount the insurer is contractually entitled to collect as a minimum. The minimum premium sets a floor: the amount charged cannot fall below that stated contractual amount.
The key calculation is a comparison. First find the unbounded retrospective premium from the plan formula. Then compare it with the minimum, using amounts stated on the same basis. The minimum changes the calculated premium only when the unbounded result is below the floor.
Calculate the Unbounded Result Before Comparing
For the standard structure used here, assume the contract defines a basic premium \(B\), a loss conversion factor \(LCF\), covered losses \(L\), and a tax multiplier \(T\). Converted losses are \(LCF \times L\), and the unbounded retrospective premium is:
This expression applies when the stated contract uses these components and applies the tax multiplier to their sum. Other contracts may define components or the minimum-premium basis differently; follow the contract rather than assuming this formula or the order of operations is universal. In particular, the examples below specify that the minimum is compared with the final, tax-multiplied amount. The amounts are for the same policy period, and no other premium bounds or adjustments are included.
A minimum may be written directly as a dollar amount, or as a percentage of another specified premium measure. For a minimum stated as a percentage \(q\) of standard premium \(S\), the dollar floor is \(M=qS\), provided that this is how the contract defines the minimum. Establish \(M\) before comparing it with \(R\); do not compare a percentage directly with a dollar amount.
Why the Floor Formula Works
The contract requires the premium to satisfy two conditions: it must equal the formula result when that result is at least the minimum, and it must never be less than the minimum. These conditions determine the amount directly.
The formula result already meets the minimum, so the premium remains \(R\). The floor adds nothing.
Charging \(R\) would violate the minimum condition. The premium must instead be \(M\).
In either case, the premium is the larger of the two amounts: \(P=\max(R,M)\). The increase due solely to the floor is \(P-R=\max(M-R,0)\).
This also explains the equality case. If \(R=M\), both conditions are met by that same amount. The floor is reached, but there is no difference between the formula result and the amount charged.
Applying a Dollar Minimum
Worked Example: Formula Result Above the Minimum
A hypothetical policy has a basic premium of $24,000, covered losses of $30,000, a loss conversion factor of 1.10, and a tax multiplier of 1.08. The contract sets a $50,000 minimum on the final premium after the tax multiplier. Calculate the premium and the increase attributable to the minimum.
Method. Calculate the unbounded result first, then compare it with the contractual floor. This avoids treating the minimum as a replacement for the formula when the formula already exceeds it.
Step 1: Convert losses. The converted losses are \(1.10 \times \$30{,}000=\$33{,}000\).
Step 2: Add the basic premium. Before applying the tax multiplier, the amount is \(\$24{,}000+\$33{,}000=\$57{,}000\).
Step 3: Apply the tax multiplier. The unbounded retrospective premium is \(R=\$57{,}000 \times 1.08=\$61{,}560\).
Step 4: Apply the minimum. Compare \(R=\$61{,}560\) with \(M=\$50{,}000\). Since \(R>M\), the floor does not increase the premium. The premium is \(\max(\$61{,}560,\$50{,}000)=\$61{,}560\), and the floor increase is \(\$61{,}560-\$61{,}560=\$0\).
Check. The result is at least $50,000, as the contract requires. The minimum is not an amount to add to the formula result.
Worked Example: A Binding Dollar Minimum
For a hypothetical policy, the basic premium is $18,000, covered losses are $8,000, the loss conversion factor is 1.25, and the tax multiplier is 1.05. The contract requires at least $36,000 on the final, tax-multiplied premium basis. Find the premium and the amount by which the floor raises it.
Method. Compute the unbounded amount using the plan components, then use the greater of that amount and the minimum.
Step 1: Convert losses. \(1.25 \times \$8{,}000=\$10{,}000\).
Step 2: Add the basic premium. The pre-tax total is \(\$18{,}000+\$10{,}000=\$28{,}000\).
Step 3: Apply the tax multiplier. \(R=\$28{,}000 \times 1.05=\$29{,}400\).
Step 4: Compare with the minimum. Since \(\$29{,}400 < \$36{,}000\), the floor binds. The premium is \(P=\max(\$29{,}400,\$36{,}000)=\$36{,}000\).
Step 5: Quantify the increase. The floor raises the amount by \(\$36{,}000-\$29{,}400=\$6{,}600\).
Check. The charged premium equals the minimum, and the difference between the charged premium and the unbounded result is $6,600.
Minimums Stated as a Percentage
If a contract states a minimum as a percentage of standard premium, convert that percentage to dollars before comparing it with the retrospective result. The standard premium is the reference amount specified by the contract; it is not interchangeable with the basic premium in the retrospective formula.
Worked Example: Converting a Percentage Minimum to Dollars
A hypothetical policy has a basic premium of $16,000, covered losses of $7,500, a loss conversion factor of 1.20, and a tax multiplier of 1.10. Its standard premium is $40,000. The contract defines the minimum as 70% of standard premium and compares it with the final, tax-multiplied retrospective premium. Calculate the premium and the floor increase.
Method. First convert the percentage minimum to dollars. Then calculate the unbounded premium and compare the two dollar amounts.
Step 1: Find the minimum in dollars. \(M=0.70 \times \$40{,}000=\$28{,}000\).
Step 2: Convert losses. \(1.20 \times \$7{,}500=\$9{,}000\).
Step 3: Add the basic premium and converted losses. The pre-tax total is \(\$16{,}000+\$9{,}000=\$25{,}000\).
Step 4: Apply the tax multiplier. \(R=\$25{,}000 \times 1.10=\$27{,}500\).
Step 5: Apply the floor. Since \(\$27{,}500 < \$28{,}000\), the floor binds, and the premium is \(P=\$28{,}000\).
Step 6: Find the increase. \(\$28{,}000-\$27{,}500=\$500\).
Check. The minimum equals 70% of the stated $40,000 standard premium, and the charged premium exceeds the unbounded result by exactly $500.
Common Mistake: Applying the Minimum at the Wrong Stage
A floor only has the meaning given to it by the contract. If the contract defines a minimum on the final, tax-multiplied premium, compare the floor with \(R\) after applying the tax multiplier. Applying the floor to a pre-tax subtotal instead changes the calculation and may produce an incorrect premium.
In the percentage-minimum example, the pre-tax subtotal is $25,000, the multiplier is 1.10, and the final-basis minimum is $28,000. The correct formula result is \(R=\$25{,}000 \times 1.10=\$27{,}500\), so the correct premium is $28,000. A wrong calculation would apply the $28,000 floor to the pre-tax subtotal and then multiply: \(\max(\$25{,}000,\$28{,}000)\times1.10=\$30{,}800\). That exceeds the contractual final-basis minimum by applying the tax multiplier to the floor a second time. The correction is to calculate \(R\) on the specified basis first, then compare \(R\) directly with \(M\).
This correction depends on the contract’s wording. If a contract instead defines the minimum before a particular multiplier, the calculation must follow that stated basis. Always identify what the minimum applies to before doing arithmetic.
Exam and Calculation Checks
- Keep the bases consistent. Convert a percentage minimum into dollars and make sure it is on the same basis as the result being compared.
- Do not add the minimum. The floor is a lower bound, not an additional premium component.
- Separate binding from equality. When \(R<M\), the floor increases the premium. When \(R=M\), it does not increase it. When \(R>M\), it has no effect.
- Report the increase explicitly. The amount added by the floor is \(\max(M-R,0)\), not the full minimum.
These checks keep the minimum’s purpose clear: it changes the result only to the extent needed to prevent the premium from falling below the contractual floor. This treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each problem, calculate the unbounded result first. Unless stated otherwise, the minimum applies to the final amount after the tax multiplier.
- A hypothetical policy has a basic premium of $12,000, converted losses of $4,000, a tax multiplier of 1.10, and a $20,000 minimum. Find the premium and the increase due to the floor. (Answer: unbounded result $17,600; premium $20,000; increase $2,400.)
- A policy has a basic premium of $22,500, covered losses of $5,000, a loss conversion factor of 1.20, and a tax multiplier of 1.04. Its standard premium is $42,000, and its minimum is 75% of standard premium. Find the premium and the floor increase. (Answer: minimum $31,500; unbounded result $29,640; premium $31,500; increase $1,860.)
- A policy has a basic premium of $20,000, converted losses of $10,000, a tax multiplier of 1.10, and a $33,000 minimum. Does the minimum increase the amount due? (Answer: unbounded result $33,000; premium $33,000; increase $0 because the amounts are equal.)
- A policy has a basic premium of $15,000, covered losses of $4,000, a loss conversion factor of 1.50, a tax multiplier of 1.20, and a $24,000 minimum. Find the final premium and state whether the floor binds. (Answer: unbounded result $25,200; premium $25,200; the floor does not bind; increase $0.)
- A policy has a basic premium of $31,000, covered losses of $12,000, a loss conversion factor of 0.75, and a tax multiplier of 1.05. The standard premium is $50,000, and the minimum is 80% of standard premium. Find the unbounded result, the minimum, and the premium. (Answer: unbounded result $42,000; minimum $40,000; premium $42,000; increase $0.)