Anatomy of a Multiplicative Rating Plan

What a Multiplicative Rating Plan Describes

A class-rating plan assigns a rate to a risk using characteristics that help distinguish its expected cost. To read such a plan, identify four things: the base rate, the rating variables, the selected levels of those variables, and the relativities attached to those levels. Together, these components indicate a rate for a particular risk.

A rate is quoted for an exposure unit, such as one vehicle-year or each $100 of insured value. The indicated premium for a policy is found by applying that rate to the number of exposure units, subject to any additional terms in the rating plan. Keeping the rate and the premium distinct prevents a common unit error.

Definition: A base rate is the rate for a defined reference risk, stated for a specified coverage, time period, and exposure unit. A rating variable is a risk characteristic used to distinguish rates, such as territory or occupancy. A level is one possible value or category of a rating variable. A level relativity is the rate factor assigned to a level relative to a reference level of that variable.

In a simple multiplicative plan, each variable has a reference level with relativity 1.00. Other levels have relativities above or below 1.00: a relativity of 1.20 represents a factor 20% above the reference level, while 0.90 represents a factor 10% below it. These factors are applied to the base rate under the plan’s multiplicative assumption.

For a risk with $m$ rating variables, write the simple plan as

R=B\prod_{j=1}^{m} r_j

Here $R$ is the indicated rate per exposure unit, $B$ is the base rate, and $r_j$ is the relativity for the risk’s selected level of variable $j$. This formula presumes that the plan uses a multiplicative structure, that the factors refer to the same coverage and rating basis as the base rate, and that any effects represented by the factors are combined without additional interactions. If a plan specifies interactions or a different calculation rule, follow that rule rather than this simplified formula.

How the Parts Fit Together

A rating table is easiest to use in a consistent order. First identify the coverage and exposure unit, then locate the base rate and the relevant reference levels. Next, select the risk’s level for each rating variable and read off its relativity. Finally, apply the plan’s calculation rule and check that the result is still expressed in the stated exposure unit.

Set the rating basis.
Confirm the coverage, time period, and exposure unit. A rate per vehicle-year is not directly interchangeable with a rate per $100 of insured value.

Locate the base rate.
Read the rate for the plan’s reference risk. It is the starting rate, not automatically the rate for every insured.

Select each level.
For each rating variable, match the risk’s characteristic to the applicable category in the plan.

Apply the relativities and check units.
Calculate the indicated rate, then multiply by the number of exposure units if the question asks for an indicated premium.

Why the Relativities Multiply

The multiplication is a modeling assumption about how the plan combines the differences from its reference risk. It can be justified directly from the meaning of a relativity when the factors are assumed to act proportionally and separately.

Suppose $B$ is the rate for the reference risk, and $r_1$ is the proportional rate factor for the first selected level. The rate after applying that factor is $B r_1$. If a second variable has a factor $r_2$ that applies proportionally to that resulting rate, the rate becomes $(B r_1)r_2$. Repeating this reasoning gives

R=((B r_1)r_2)\cdots r_m =B\prod_{j=1}^{m}r_j

Each factor is relative to its own variable’s reference level, so those reference levels have factor 1.00. The derivation does not prove that real risk characteristics always combine this way; it shows why the stated multiplicative plan produces the formula. If combinations of characteristics have additional effects not captured by separate factors, this simple structure may not represent them adequately.

Worked Example: Indicated Rate for an Auto Risk

A hypothetical insurer’s annual plan has a base rate of $600 per vehicle-year for a reference risk: suburban territory and commuting use. Its table assigns a relativity of 1.20 to urban territory and 0.90 to pleasure use. Find the indicated annual rate for one vehicle garaged in an urban territory and used for pleasure.

Method and assumptions. This is a deterministic rate-table calculation using the stated multiplicative plan. The base and relativities are fixed inputs. No interest rate, mortality assumption, or loss distribution and parameters are used: the calculation does not discount cash flows or estimate a probability distribution.

Identify the components. The base rate is $600 per vehicle-year. The rating variables are territory and use. The selected levels are urban territory, with relativity 1.20, and pleasure use, with relativity 0.90.

Calculate the rate. Apply the territory relativity first:

\$600 \times 1.20=\$720 \text{ per vehicle-year}

Then apply the use relativity:

\$720 \times 0.90=\$648 \text{ per vehicle-year}

Thus the indicated annual rate is $648.00 per vehicle-year. For one vehicle-year, the corresponding indicated premium is also $648.00 before any charges or adjustments outside this plan. The result is checked by $600\times1.20\times0.90=648$, with the original per-vehicle-year unit preserved.

Common mistake avoided. Adding the percentage changes would give $600(1+0.20-0.10)=\$660$, which is not the result of this multiplicative plan. The 10% reduction applies to the rate after the territory factor has been applied, not directly as an additive adjustment to the original base rate.

Rate Per Unit and Total Indicated Premium

A rate may need one more step before it becomes a policy premium. If the rate is quoted per exposure unit, multiply it by the number of those units. For property coverage, for example, a rate per $100 of insured value must be multiplied by the insured value expressed in $100 units, not by the dollar value itself.

Worked Example: Property Rate Per $100 of Insured Value

A hypothetical property plan has a base rate of $0.80 per $100 of insured value. For a building’s selected roof level, the relativity is 0.95; for its seasonal occupancy level, the relativity is 1.25. The building is insured for $240,000. Find the indicated rate per $100 and the annual indicated premium.

Method and assumptions. Treat the listed values as fixed entries in a deterministic multiplicative rate plan. There is no interest rate, mortality assumption, or loss distribution and parameters in this calculation; no present-value or probability calculation is being made.

Identify the components. The base rate is $0.80 per $100. The rating variables are roof level and occupancy level. The selected relativities are 0.95 and 1.25.

Find the rate. First apply the roof relativity:

\$0.80 \times 0.95=\$0.76 \text{ per \$100}

Then apply the occupancy relativity:

\$0.76 \times 1.25=\$0.95 \text{ per \$100}

Convert insured value to exposure units. The exposure is

\$240{,}000 \div \$100=2{,}400 \text{ units}

The annual indicated premium is therefore

\$0.95 \times 2{,}400=\$2{,}280.00

The indicated rate is $0.95 per $100, and the annual indicated premium is $2,280.00. As a check, the result has been applied to 2,400 units, which represent $240,000 at $100 per unit.

Common mistake avoided. Multiplying $0.95 by $240,000 would ignore the rate’s unit and overstate the premium by a factor of 100. Convert the insured value to the stated exposure unit before calculating the premium.

Reading a Plan Without Over-Interpreting It

The base rate and relativities describe a rating structure; they do not, by themselves, explain every step involved in setting or collecting a final policy price. For example, the plan may state a minimum premium, a separate charge, or other adjustments. Include such items only when they are specified and relevant to the question. Do not silently fold them into the base rate or the level relativities.

Also keep these concepts separate:

Rating variable: the characteristic used in the plan, such as territory.
Level: the particular category selected for the risk, such as urban.
Relativity: the factor assigned to that level, such as 1.20.
Base rate: the rate for the plan’s reference risk and stated exposure unit.
Indicated rate: the rate resulting from applying the plan’s factors to the base rate.
Indicated premium: the indicated rate applied to the risk’s exposure amount.

A relativity is not itself a dollar rate, and a base rate is not necessarily the rate for the risk being priced. Both are interpreted in relation to the plan’s coverage and reference basis. Before calculating, check that each selected level belongs to the stated variable and that the units remain consistent throughout.

Key takeaway: For a stated simple multiplicative plan, identify the reference base rate, select the risk’s level for each rating variable, apply those levels’ relativities, and preserve the rate’s exposure unit. Convert the rate to an indicated premium only when the exposure amount is known.

Worked Example: Several Vehicles on One Rating Basis

A hypothetical commercial auto plan has a base rate of $1,000 per truck-year for a light truck used within a short local radius. A heavy-truck level has relativity 1.30, and a radius of 26 to 50 miles has relativity 1.10. A small fleet contains four trucks, each with one truck-year of exposure, and all four have the selected heavy-truck and radius levels. Find the indicated rate per truck-year and the total indicated premium.

Method and assumptions. Use the provided deterministic multiplicative table, with identical characteristics and exposure for each truck. No interest rate, mortality assumption, or loss distribution and parameters are required; the plan values are treated as fixed.

Find the rate for one truck-year. The selected rating variables are truck class and operating radius. Apply the heavy-truck relativity:

\$1{,}000 \times 1.30=\$1{,}300 \text{ per truck-year}

Apply the radius relativity:

\$1{,}300 \times 1.10=\$1{,}430 \text{ per truck-year}

Find the fleet premium. Four trucks with one truck-year each represent four truck-years:

\$1{,}430 \times 4=\$5{,}720.00

The indicated rate is $1,430 per truck-year, and the total indicated premium is $5,720.00. As a check, $1{,}000\times1.30\times1.10=1{,}430$, and four identical exposures produce four times that rate.

Common mistake avoided. The rate is not $5,720 per truck-year. That amount is the premium for four truck-years; dividing it by four returns the indicated rate of $1,430 per truck-year.

Common Rating-Plan Traps

When interpreting a short rating schedule, the main errors are usually not difficult arithmetic; they are errors in identifying what a number represents. Check for these issues before settling on a result:

Using the wrong reference basis. Confirm the base rate’s coverage, period, and exposure unit before applying it.
Choosing a factor from the wrong variable. A number in a rating table is not usable until its variable and level are identified.
Replacing the plan’s rule with an additive shortcut. If the stated plan is multiplicative, multiply the relativities as specified; do not add their percentage deviations from 1.
Confusing rate with premium. A rate per exposure unit becomes a premium only after applying the number of exposure units.
Assuming every plan is multiplicative. Use the formula only when that structure is stated or otherwise specified. A different rule or an interaction requires a different calculation.

These checks are useful in routine pricing work as well as in exam-style calculations aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. They help ensure that a correct numerical operation answers the actual rating question.

Check Your Understanding

For each problem, identify the relevant plan components before calculating. Unless stated otherwise, treat the rating schedule as a deterministic multiplicative plan with fixed inputs; no interest, mortality, or loss-distribution assumptions are needed.

A hypothetical auto plan has a base rate of $750 per vehicle-year. A rural territory has relativity 0.88 and delivery use has relativity 1.15. What is the indicated rate for one rural delivery vehicle? (Answer: $759.00 per vehicle-year.)
A hypothetical property plan has a base rate of $1.40 per $100 insured. A selected roof level has relativity 1.10, and a protection level has relativity 0.90. For $180,000 insured value, find the rate per $100 and annual indicated premium. (Answer: $1.386 per $100 and $2,494.80 annually.)
A hypothetical vehicle plan has a base rate of $420 per vehicle-year. A territory factor is 1.25 and a use factor is 0.80. If a policy covers three vehicles for one year each, find the indicated rate per vehicle-year and the total indicated premium. (Answer: $420 per vehicle-year and $1,260.00 total.)
A hypothetical equipment plan has a base rate of $900 per machine-year. The selected age level has relativity 1.12, and the selected operating-class level has relativity 0.95. Calculate the indicated rate for one machine-year and identify the rating variables’ role in the calculation. (Answer: $957.60 per machine-year; age and operating class are the rating variables.)
A hypothetical property schedule quotes a base rate of $0.56 per $100 of insured value. For a selected construction level the relativity is 1.05, and for a selected occupancy level it is 0.92. A building has $125,000 insured value. State the base rate, name the two rating variables, calculate the indicated rate, and calculate the annual premium. (Answer: Base rate $0.56 per $100; variables are construction and occupancy; indicated rate $0.54096 per $100; annual premium $676.20.)

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