From a Rating Table to One Risk’s Rate
A multiplicative rating plan turns a base rate and a risk’s characteristics into a rate for that risk. In practice, the calculation has two distinct parts: first select the correct entry for each characteristic, then multiply those entries by the base rate. A factor read from the wrong row or column can produce a plausible-looking but incorrect result, so the lookup is as important as the arithmetic.
This tutorial focuses on applying a supplied rating plan to one risk. It does not estimate or revise the factors. Treat the base rate and factor values as given, and assume that the risk’s information is sufficient to place it into one level for each listed factor. Unless stated otherwise, all rates and premiums below are hypothetical, annual, and before taxes, fees, or other adjustments.
The Calculation Rule
Let \(B\) be the base rate, and let \(r_1,r_2,\ldots,r_k\) be the relativities selected for the risk’s \(k\) rating characteristics. The indicated rate is
A relativity is a multiplier, not a dollar amount. For example, a relativity of \(1.12\) applies a 12% increase to the rate to which it is applied; a relativity of \(0.90\) applies a 10% decrease. In a multiplicative plan, the factors compound: each one scales the result of the preceding calculation.
The multiplication order does not change the final result, but a reliable working method is to record the base rate, list every factor selected from the table, multiply the relativities, and then apply their product to the base rate. Keep the base rate’s unit visible throughout. If it is quoted per $1,000 of insured value, the calculated rate is still per $1,000 until it is multiplied by the risk’s exposure measured in those units.
Why the Factors Are Multiplied
The product follows from how the factors are defined. Each selected relativity scales the rate by its own amount. Applying \(r_1\) to the base rate gives \(B r_1\). Applying \(r_2\) to that adjusted rate gives \((B r_1)r_2\). Continuing in this way for \(k\) factors gives
For a simple numerical justification, suppose one factor raises a rate by 20%, so its multiplier is \(1.20\), and another raises it by 10%, so its multiplier is \(1.10\). The two successive adjustments give \(1.20\times1.10=1.32\), a combined increase of 32%, not 30%. The second 10% increase applies to the amount already increased by the first factor.
This justification depends on the plan’s factors being multiplicative. If a problem specifies an additive adjustment, a fixed dollar charge, a cap, or some other rule, follow that stated rule instead; do not automatically treat it as another relativity.
Worked Applications
Worked Example: Selecting Three Factors for a Vehicle
A hypothetical personal auto plan has a base rate of $500 per vehicle-year. The table gives these relativities:
| Rating characteristic | Level | Relativity |
|---|---|---|
| Territory | North | 1.12 |
| Driver class | Class 2 | 0.90 |
| Vehicle use | Commuting | 1.05 |
A risk is in the North territory, has a Class 2 driver, and uses the vehicle for commuting. Find its annual indicated rate per vehicle. No interest rate, mortality assumption, or probability distribution is used: this is deterministic rating-plan arithmetic, and the listed rate and factors are the complete inputs.
Step 1: Select the table entries. The risk’s territory selects \(1.12\), its driver class selects \(0.90\), and its vehicle use selects \(1.05\). Each listed characteristic contributes one factor.
Step 2: Multiply the relativities. \(1.12\times0.90=1.008\), and \(1.008\times1.05=1.0584\). Thus the combined multiplier is \(1.0584\).
Step 3: Apply the multiplier to the base rate. \(500\times1.0584=\$529.20\) per vehicle-year. The indicated annual rate is $529.20 per vehicle.
Check. The combined multiplier is above 1, so the result should exceed the $500 base rate. The calculated rate does. No rounding beyond cents was needed.
Method and mistake avoided. The method is a factor-by-factor lookup followed by multiplication. Adding the percentage changes instead would give \(500(1+0.12-0.10+0.05)=\$535.00\), which is not the rate under this multiplicative plan.
Worked Example: Keeping the Exposure Unit Straight
A hypothetical commercial property plan uses a base rate of $720 per $1,000 of insured value. Its table gives a construction relativity of \(0.85\) for masonry, a protection relativity of \(1.10\) for the specified protection class, and an occupancy relativity of \(1.20\) for a workshop. A masonry workshop in that protection class has $350,000 of insured value. Find its indicated rate per $1,000 and its annual premium. No interest rate, mortality assumption, or loss distribution is needed; the calculation uses only the supplied rating plan and insured value.
Step 1: Identify the three selected factors. The applicable values are construction \(0.85\), protection \(1.10\), and occupancy \(1.20\). The base rate is not a flat $720 per property; its stated unit is $1,000 of insured value.
Step 2: Find the combined multiplier. \(0.85\times1.10=0.935\), and \(0.935\times1.20=1.122\).
Step 3: Calculate the indicated rate. \(720\times1.122=\$807.84\). The indicated rate is $807.84 per $1,000 of insured value.
Step 4: Convert the insured value to exposure units. \(\$350{,}000\div\$1{,}000=350\) units. The annual premium is \(807.84\times350=\$282{,}744.00\).
Check. The combined multiplier exceeds 1, so the rate per $1,000 is higher than the $720 base rate. Multiplying that rate by 350 exposure units gives a premium larger than the rate, as expected for this insured value. Money is reported to the cent.
Method and mistake avoided. Calculate the rate per stated exposure unit first, then multiply by the number of exposure units. Treating $807.84 as the total premium, or multiplying it by 350,000 rather than 350, would ignore the base rate’s “per $1,000” unit.
Worked Example: Including a Reference-Level Factor
A hypothetical equipment policy has a base rate of $640 per policy-year. Its rating table assigns these factors: equipment age, 1.08 for newer and 1.15 for older; territory, 0.92 for Zone A and 1.00 for Zone B; and annual mileage band, 0.95 for low mileage and 1.10 for high mileage. A risk covers older equipment in Zone A with low annual mileage. Find its indicated annual rate. No interest rate, mortality assumption, or probability distribution is involved.
Step 1: Match each risk characteristic to its table entry. Older equipment selects \(1.15\), Zone A selects \(0.92\), and low mileage selects \(0.95\). The value \(1.00\) belongs to Zone B, so it does not apply to this risk.
Step 2: Multiply the selected factors. \(1.15\times0.92=1.058\), and \(1.058\times0.95=1.0051\).
Step 3: Apply the combined multiplier. \(640\times1.0051=\$643.264\), which rounds to $643.26 per policy-year to two decimals.
Check. The combined multiplier \(1.0051\) is just above 1, so the indicated rate should be just above $640. The calculated rate is $3.264 above the base before rounding.
Method and mistake avoided. The method includes exactly the table entry for each characteristic that describes the risk. Do not select the more familiar or default-looking \(1.00\) entry just because it leaves a rate unchanged; Zone A’s actual factor is \(0.92\).
Checks That Prevent Rate Errors
Before finalizing a one-risk calculation, check both the lookup and the units. A multiplication can be arithmetically correct while using the wrong factor level, and a correct indicated rate can still be reported as the wrong premium if its exposure unit is lost.
- Confirm the base rate. Use the base rate stated for the relevant coverage and exposure unit, not a rate from another table or coverage.
- Match every characteristic to its level. Read the risk description carefully and select the corresponding row or column. Do not substitute a nearby level.
- Use each applicable factor once. Do not omit a characteristic included in the plan or apply the same selected factor twice.
- Multiply, do not add, relativities. In a multiplicative plan, a factor of \(0.90\) means multiply by \(0.90\); it is not a subtraction of 0.90 dollars.
- Keep rate and premium distinct. A rate is quoted per exposure unit. Multiply it by the risk’s number of exposure units only when asked for the premium.
- Round at the end. Carry the product through the calculation before rounding the final money amount to cents, unless the instructions specify another precision.
A rate below the base rate is not automatically an error: a combined multiplier below 1 produces a lower rate. Likewise, one factor above 1 does not guarantee that the final rate exceeds the base, because other selected factors may reduce it. The combined product, rather than any single factor viewed alone, determines the direction and size of the adjustment.
Check Your Understanding
For each hypothetical problem, show the selected factors and keep the stated exposure unit in view. Answers are rounded to cents where a money amount is requested.
- A plan has a base rate of $460 per unit-year. A risk selects territory \(1.10\), class \(0.95\), and use \(1.04\). What is its indicated rate per unit-year? (Answer: $499.93 per unit-year)
- A property plan has a base rate of $900 per $1,000 of insured value. A risk selects construction \(0.80\), protection \(1.25\), and occupancy \(1.06\), and has $240,000 of insured value. Find the rate per $1,000 and the annual premium. (Answer: $954.00 per $1,000; $228,960.00 annual premium)
- A hypothetical equipment plan has a base rate of $375 per policy-year. For one risk, the selected factors are age \(0.88\), territory \(1.15\), and usage \(1.20\). Find the annual indicated rate. (Answer: $455.40 per policy-year)
- A plan’s base rate is $1,250 per exposure unit. A risk selects factors \(0.94\), \(1.08\), and \(1.15\). Calculate the combined multiplier and indicated rate. (Answer: combined multiplier 1.16748; $1,459.35 per exposure unit)
- A plan has a base rate of $540 per unit-year. A risk selects factors \(1.12\), \(0.90\), and \(1.06\); a separate characteristic is at its listed reference level with factor \(1.00\). Find the indicated rate and explain whether omitting the \(1.00\) factor changes the result. (Answer: $576.98 per unit-year; omitting the factor of 1.00 does not change the result)