Why Audit the Formula, Not Just the Final Premium?
A multiplicative rating formula can look plausible and still produce an incorrect premium. A factor may be assigned to the wrong characteristic, a reference level may be treated as a missing value, or a monthly base rate may be compared with an annual table. An audit therefore follows each risk characteristic from the rating description, through the formula, to the calculated table entry.
The task here is not to estimate a rating structure. It is to check whether a proposed structure has been applied consistently. Assume the formula is intended to multiply a base rate by one factor for each included characteristic, with the factors and base rate expressed on compatible terms.
For a formula audit, record the factors by their meaning, not merely by their numerical values. Each factor should correspond to the stated characteristic and level. A factor of 1.00 is not evidence that the characteristic was forgotten: it may be the stated reference level. Conversely, a factor different from 1.00 must not be applied to a risk that does not have that factor’s level.
A Practical Audit Sequence
Begin with the written rating description and the rating table, then check the formula in a fixed order. This avoids relying on a plausible-looking total as proof that the calculation is right.
Write down every characteristic the formula is intended to rate, such as territory, vehicle class, or use.
Confirm that each factor represents the correct characteristic and that each risk receives the factor for its own level.
Verify that the designated reference level has factor 1.00 and that omitted reference factors are understood as one, not as zero or missing.
Confirm that the base rate and calculated rate use the same exposure basis and time period, and that factors are applied to the intended base.
Calculate selected table entries directly from the formula, compare them with the published entries, and investigate any difference before rounding.
The audit assumes the proposed formula is actually multiplicative as written. If the quoted amount also includes a minimum premium, fixed fee, tax, cap, or adjustment not shown in the formula, the formula alone will not reproduce the final invoice. Such additions must be identified and checked separately; they should not be silently folded into a factor.
Worked Example: Reconstructing a Two-Factor Rating Table
A hypothetical insurer proposes a base annual rate of $480. The territory factors are Urban 1.20, Suburban 1.00, and Rural 0.90. Vehicle-class factors are A 1.25, B 0.80, and C 1.00. Audit the formula and reconstruct the annual rates for all nine combinations.
Method. Map one territory factor and one class factor to each risk, multiply both by the same annual base rate, and check the reference levels. Suburban and class C are references because their factors equal 1.00.
For class A in Urban territory: $$ 480\times1.20\times1.25=576\times1.25=\$720.00. $$ For class A in Suburban territory: $$ 480\times1.00\times1.25=\$600.00. $$ For class A in Rural territory: $$ 480\times0.90\times1.25=432\times1.25=\$540.00. $$
For class B, the three entries are: $$ 480\times1.20\times0.80=\$460.80,\qquad 480\times1.00\times0.80=\$384.00,\qquad 480\times0.90\times0.80=\$345.60. $$
For class C, the three entries are: $$ 480\times1.20\times1.00=\$576.00,\qquad 480\times1.00\times1.00=\$480.00,\qquad 480\times0.90\times1.00=\$432.00. $$
| Vehicle class | Urban | Suburban | Rural |
|---|---|---|---|
| A | $720.00 | $600.00 | $540.00 |
| B | $460.80 | $384.00 | $345.60 |
| C | $576.00 | $480.00 | $432.00 |
Check. The class C, Suburban entry equals the base rate because both factors are 1.00. The Urban-to-Suburban ratio within any class is 1.20, and the class A-to-class C ratio within any territory is 1.25. These comparisons verify that the factors have been applied to the intended dimensions. The common error avoided is treating the reference factor as missing and omitting a nonreference factor instead.
Why Reference Factors and Ratios Help
A reference factor of 1.00 leaves the base rate unchanged for that characteristic. This also makes it useful as a check on relative rates. For two risks that differ in only one characteristic, all shared factors cancel when their rates are divided.
The cancellation holds when the two risks have the same base and the same levels for every characteristic except \(c_j\), and the formula has no additional adjustments that distinguish them. It explains why a within-column or within-row ratio in a rating table is a useful audit. If the ratio does not match the factor ratio, either the table, the formula, or the mapping needs further review.
Worked Example: Checking a Quoted Amount Against Its Time Unit
A hypothetical policy has a base rate of $48 per vehicle-month. A six-month policy has the “Rideshare” use factor 1.50 and the Rural territory factor 0.90. The rating table gives $388.80 for the six-month policy. Determine whether the quoted amount is consistent with the formula, and identify the corresponding annualized amount.
Method. First calculate the adjusted rate per vehicle-month. Then multiply by six months. This keeps the base-rate unit visible throughout the calculation.
The adjusted monthly rate is: $$ \$48\times1.50\times0.90 =\$72\times0.90 =\$64.80\text{ per vehicle-month}. $$
For six months, the calculated amount is: $$ \$64.80\times6=\$388.80. $$
The table’s $388.80 therefore agrees with the formula for a six-month term. For comparison, the corresponding amount for 12 months, if the same rate and factors apply throughout, is: $$ \$64.80\times12=\$777.60. $$
Check. Half of $777.60 is $388.80, as expected for a six-month term at a constant monthly rate. The common mistake avoided is comparing the six-month amount directly with an annual amount, or multiplying by six when the base rate already states a six-month period.
Reconcile Totals by Auditing Each Risk
A total premium can agree with a proposed calculation even when individual risks have been assigned the wrong factors. Audit a sample of risks separately before summing their rates. The table is particularly useful here because it records both the factor assigned to each characteristic and the rate resulting from that assignment.
Worked Example: Catching a Factor-Mapping Error Hidden by a Matching Total
A hypothetical rating formula uses a base annual rate of $360. Vehicle-class factors are A 1.25 and B 1.00; territory factors are Urban 1.00 and Rural 0.90. A portfolio contains one class A Urban vehicle and one class B Rural vehicle. Audit the rate for each vehicle and the portfolio total.
Method. Assign each vehicle its own class and territory factors, multiply those factors by the base, and only then add the resulting rates.
For the class A Urban vehicle: $$ \$360\times1.25\times1.00=\$450.00. $$ For the class B Rural vehicle: $$ \$360\times1.00\times0.90=\$324.00. $$ The correct total is: $$ \$450.00+\$324.00=\$774.00. $$
The table entries confirm the assignments: class A Urban is $450.00, class A Rural is $405.00, class B Urban is $360.00, and class B Rural is $324.00. The two actual risks use the first and fourth entries, respectively.
Common mistake and correction. Applying both nonreference factors to both vehicles would produce $$ \$360\times1.25\times0.90+\$360\times1.25\times0.90 =\$405.00+\$405.00 =\$810.00, $$ but that assigns the wrong factors to each vehicle. The expression $$ \$360\times1.25+\$360\times0.90=\$450.00+\$324.00=\$774.00 $$ gives the correct total only by coincidence; it does not establish correct mapping. In this case the numerical terms happen to equal the two correct rates, but the expression does not explicitly apply each vehicle’s class and territory factors. The valid audit is the vehicle-by-vehicle calculation above.
Check. The total from the two correctly assigned table entries is $774.00. A matching aggregate alone is not enough: the formula must also produce the right rate for each risk.
What a Complete Formula Audit Should Establish
A useful audit records more than the final dollar amount. It should make the chain of calculation reproducible: identify the base and its unit, name the level of every characteristic, show the corresponding factor, and calculate the rate without premature rounding. Where a rating table is available, selected combinations should be recalculated directly and compared with their table entries.
- One characteristic, one intended factor. Do not use a territory factor as if it were a class factor merely because both are multipliers.
- Reference means one. A reference-level factor does not change the rate; it is not a zero multiplier.
- Units must reconcile. A monthly rate, annual rate, per-vehicle rate, and per-exposure rate are not interchangeable without the appropriate conversion.
- Check risk-level results before totals. A sum may conceal a mapping error, a duplicated factor, or an accidental numerical match.
- Separate formula effects from other charges. If a table includes a fee or minimum not represented in the formula, identify and reconcile that item separately.
These checks are useful whenever a rating formula is translated into a table, a calculation worksheet, or a quoted premium. They make errors visible before the final amount is treated as evidence that the underlying assignments are correct. This treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each problem, check the factor assignments and units before evaluating the product.
- A hypothetical annual base rate is $400. Terrain factors are Flat 1.00, Hill 1.15, and Coast 0.95; vehicle-class factors are Light 0.80, Standard 1.00, and Heavy 1.30. What is the annual rate for a Coast, Standard vehicle? (Answer: $380.00)
- A hypothetical base rate is $48 per vehicle-month. A three-month policy has a Use factor of 1.25 and a Territory factor of 0.90. What is the policy amount? (Answer: $162.00)
- A hypothetical annual base rate is $560. Class X has factor 1.00 and Class Y has factor 1.20; Zone A has factor 0.85 and Zone B has factor 1.00; an alarm discount factor is 0.90. Calculate the rate for a Class Y, Zone A risk with an alarm. (Answer: $514.08)
- A hypothetical base rate is $250. Risk 1 is class X with factor 1.40 and West territory with factor 0.90. Risk 2 is class Y with factor 0.80 and East territory with factor 1.10. What is the total rate for the two risks? (Answer: $535.00)
- A hypothetical annual base rate is $900. Standard class and Metro territory are the reference levels, each with factor 1.00. Preferred class has factor 0.75 and Rural territory has factor 0.80. What is the annual rate for a Preferred, Rural risk? (Answer: $540.00)