From a Rating Plan to a Calculation
A rating indication becomes a usable premium only when its components are translated into an explicit calculation. The rating algorithm specifies the sequence of operations; factor tables provide the values to use for the rated risk; and rules define any additional conditional operations. An unclear table lookup or an unstated rule sequence can produce a premium that differs from the intended indication even when the underlying rates are correct.
This tutorial focuses on configuring that calculation. Assume the required rating characteristics have already been assigned to the risk. For a basic multiplicative plan, the premium begins with a base rate and is adjusted by selected relativities. Additional rules may then apply, provided the plan states their conditions and where they enter the sequence.
For a plan with \(k\) multiplicative factors, a simple calculation is
where \(B\) is the base premium for the stated exposure basis, \(r_j\) is the selected value of factor \(j\), and \(P_{\text{ind}}\) is the indicated premium before any separately specified charges or rules. If exposure is not already included in \(B\), the plan must specify how it enters the calculation. For example, a base rate of dollars per vehicle-year must be multiplied by the number of vehicle-years being rated.
A factor is usually interpreted relative to a reference class. A factor of 1.20 increases the premium to 120% of the amount before that factor; a factor of 0.85 reduces it to 85%. These interpretations depend on the factor table’s defined base and do not, by themselves, establish that the table is appropriate for every rating plan.
Tables, Lookups, and Calculation Order
A factor table must specify the valid inputs, the output factor, and how to handle combinations that do not appear in the table. A table may use one characteristic or multiple dimensions. For a two-dimensional table, the configured lookup must identify the intended row and column; a lookup that uses the right values but reverses their roles can still return the wrong factor.
| Configuration item | Question the plan must answer |
|---|---|
| Base rate | What coverage and exposure unit does it represent? |
| Factor table | Which characteristic or table dimensions select the factor? |
| Factor meaning | Is the value a multiplicative relativity, or a different type of adjustment? |
| Rule condition | Which risks qualify, and what information determines eligibility? |
| Rule sequence | Which specified subtotal does the operation act on? |
| Unmatched input | What happens if a required value is missing or outside the defined table? |
For multiplicative factors alone, their order does not change the mathematical result. Since real-number multiplication is commutative and associative,
The displayed order still matters operationally: it makes the calculation traceable and shows which table supplies each value. Order matters mathematically when a later operation acts on a subtotal that may or may not include an earlier charge, discount, or adjustment. Such rules must specify their calculation base rather than relying on an assumed sequence.
Worked Example: Applying Three Multiplicative Factors
A hypothetical insurer configures an annual liability premium using a base rate of $480 per vehicle-year. A vehicle-year is the exposure basis, so one vehicle-year is being rated. The selected territory factor is 1.25, the usage factor is 0.90, and the vehicle-class factor is 1.10. Calculate the indicated premium and verify the result.
Method. Apply the base rate and multiply by each selected table value. There are no additional charges or conditional rules in this example.
Start with the base rate: $480. Apply the territory factor: $480 × 1.25 = $600. Apply the usage factor: $600 × 0.90 = $540. Apply the vehicle-class factor: $540 × 1.10 = $594.
The indicated annual premium is $594.00. Check by combining the factors first: 1.25 × 0.90 = 1.125, and 1.125 × 1.10 = 1.2375. Then $480 × 1.2375 = $594. The sequential and combined calculations agree. The common mistake is to add the factor deviations, which would give an incorrect total relativity of 1 + 0.25 − 0.10 + 0.10 = 1.25 rather than multiplying the factors.
A short derivation explains why multiplicative relativities are applied this way. If each factor is defined as the ratio of the premium for a class to the premium at the reference level of that characteristic, then applying one factor changes the current reference premium by that ratio. Applying a second factor multiplies the already-adjusted premium by its own ratio. For example, if the first factor is \(r_1\) and the second is \(r_2\), the adjusted amount is \((B r_1)r_2 = B(r_1r_2)\). This product follows from the ratio definitions, provided the rating plan specifies factors on a compatible multiplicative basis. It does not justify treating an additive charge or a conditional rule as another relativity.
Worked Examples With Table Selection and Rules
Worked Example: Selecting Entries From Two Tables
A hypothetical annual property plan has a base rate of $720 per insured unit. The territory table gives a factor of 1.15 for Territory North. The construction table gives a factor of 0.85 for masonry construction. The rated risk is in Territory North and has masonry construction. Calculate the indicated premium.
Method. Select the entry from each table using the stated characteristic, then apply both values to the base rate. No other rule is specified.
The territory lookup returns 1.15, so $720 × 1.15 = $828. The construction lookup returns 0.85, so $828 × 0.85 = $703.80. Equivalently, the combined factor is 1.15 × 0.85 = 0.9775, and $720 × 0.9775 = $703.80.
The indicated premium is $703.80. The result is below the base rate because the combined relativity, 0.9775, is below 1. The common mistake is to select the right numerical values but apply the construction factor as a dollar reduction; a factor of 0.85 means 85% of the current premium, not a subtraction of $0.85.
Worked Example: Applying a Policy-Level Discount to a Defined Subtotal
A hypothetical commercial plan rates three identical pieces of equipment. The base annual premium is $390 per piece. Each piece has a territory factor of 1.25 and a protection factor of 0.80. The plan grants a 6% fleet discount when at least three pieces are insured; the discount applies to the aggregate premium after the two piece-level factors and before any separate charges. There are no separate charges. Calculate the policy indicated premium.
Method. First rate each piece using its own base premium and factors. Then aggregate the piece premiums and test the stated fleet condition.
For one piece, $390 × 1.25 = $487.50. Applying the protection factor gives $487.50 × 0.80 = $390.00. For three pieces, the pre-discount aggregate is 3 × $390.00 = $1,170.00.
The policy has three insured pieces, which meets the condition of at least three. The discount is 0.06 × $1,170.00 = $70.20. The indicated policy premium is $1,170.00 − $70.20 = $1,099.80.
Check by applying the discount factor to the aggregate: $1,170.00 × 0.94 = $1,099.80. The common mistake is to apply the discount to the base rate before the piece-level factors without checking the rule’s stated base. Here the final arithmetic happens to agree because the same 6% is applied uniformly to all three piece premiums, but the configured sequence and eligibility test still need to be explicit.
Worked Example: Sequencing a Percentage Surcharge and a Fixed Charge
A hypothetical annual specialty coverage has a base premium of $400. Its selected factors are 1.20 and 0.90. A rule then applies an 8% surcharge to the premium after those factors; a separate $24 policy charge is added after the surcharge. Calculate the indicated amount.
Method. Follow the rule’s explicitly defined bases: calculate the factored premium, apply the percentage surcharge to that amount, and then add the fixed charge.
Apply the first factor: $400 × 1.20 = $480. Apply the second: $480 × 0.90 = $432. The surcharge is 0.08 × $432 = $34.56, so the amount after the surcharge is $432 + $34.56 = $466.56. Add the $24 charge: $466.56 + $24 = $490.56.
The result checks as $432 × 1.08 + $24 = $466.56 + $24 = $490.56. A common sequencing error is to apply the 8% surcharge to the amount including the $24 charge. That would produce $1.08 × ($432 + $24) = $492.48, which is incorrect under the stated rule because the charge is added after the surcharge.
Configuration Checks and Common Traps
A well-configured algorithm should be reproducible from its inputs and definitions. The purpose of a calculation trace is not merely to reach a total; it is to make clear which table entries and rule conditions produced that total. This is useful when reviewing whether an indicated rate has been translated faithfully into a production calculation.
State the coverage basis, exposure unit, and base amount. Do not multiply by exposure again if it is already incorporated in the base premium.
Write the applicable characteristic values and the selected table entries. Confirm that all required dimensions are used and that the chosen values are defined as multiplicative factors.
For each condition, identify whether it applies and exactly which subtotal it changes. Keep fixed charges, percentage changes, and multiplicative factors conceptually distinct.
Recalculate using a combined product for the multiplicative factors, then compare it with the sequential calculation. Verify each rule against its stated base and condition.
Several errors are especially easy to make:
- Adding relativities instead of multiplying them. A factor of 1.10 followed by 0.90 gives a combined factor of 0.99, not 1.00.
- Confusing a factor with a percentage-point adjustment. A 10% reduction is a multiplier of 0.90; it is not a subtraction of 0.10 dollars.
- Using an unstated table default. A missing or unmatched rating value needs a defined treatment. Silently selecting a nearby class or a neutral factor is not a defensible lookup rule.
- Applying a rule to the wrong subtotal. A percentage applied before a fixed charge generally gives a different result from the same percentage applied after it. The calculation specification must settle the order.
- Testing a condition at the wrong level. A rule may apply per exposure, per item, or at the policy level. The plan must define both the unit of eligibility and the unit to which the adjustment applies.
Clear configuration supports calculation review and controlled implementation of a rating plan, aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. It also helps distinguish a genuine difference in the intended indication from an error in selecting a table entry or sequencing a rule.
Check Your Understanding
For each problem, follow the stated table lookups and rule sequence.
- A hypothetical annual plan has a base premium of $560 and selected factors of 1.10, 0.95, and 1.20. Calculate the indicated premium. (Answer: $702.24)
- A hypothetical equipment plan has a base premium of $390 per unit, with factors 1.25 and 0.80 per unit. A 6% discount applies to the aggregate premium if at least four units are insured. Four units are covered, and there are no other charges. Calculate the policy premium. (Answer: $1,466.40)
- A hypothetical coverage has a base premium of $625, factors 0.88 and 1.16, a 7% surcharge applied after the factors, and an $18 charge added after the surcharge. Calculate the indicated amount. (Answer: $700.66)
- A factor table gives a territory factor of 1.05 for Territory West and a deductible factor of 0.90 for the selected deductible. The base premium is $840, and no other rules apply. Calculate the indicated premium and the combined factor. (Answer: combined factor 0.945; premium $793.80)
- A hypothetical policy covers two identical units. Each has a $500 base premium and factors 1.10 and 0.90. A 4% policy discount applies if the aggregate premium after unit-level factors exceeds $900; a $30 charge is added after the discount. Calculate the policy amount. (Answer: $980.40)