Choosing Reference Levels for Rating Factors

Why a Rating Factor Needs a Reference Level

A categorical rating variable divides risks into levels, such as vehicle use, building construction, or territory. A multiplicative rating plan assigns a factor to each level. To interpret those factors, one level must anchor the comparisons: its relativity is set to 1.00, and the other levels are expressed relative to it.

Choosing the anchor is an interpretive decision. A common, clearly defined level often makes a useful reference because comparisons against it are easy to explain. The reference does not have to be the least risky level, the most risky level, or the portfolio average. A different valid reference changes the numerical comparisons but does not, by itself, change the underlying risk differences.

Definition: For a categorical rating variable with levels $l$, a reference level $r$ is the selected level whose relativity is defined to be 1.00. If $q_l$ represents the underlying positive multiplicative factor for level $l$, its relativity against reference level $r$ is $q_l/q_r$. Thus the reference level has relativity $q_r/q_r=1.00$.

Each categorical rating variable has its own reference level. For example, a plan with a vehicle-use factor and a territory factor chooses one reference level for vehicle use and another for territory. The combination of those selected levels forms the reference risk. A reference level is not the same thing as a reference risk: the former belongs to one variable; the latter is a combination of levels across variables.

The anchor also fixes the scale used to describe the levels of a factor. In a multiplicative model, the reference level’s factor is 1.00, so it leaves the base amount unchanged for that variable. A level with relativity 1.25 is 25% above that reference level, while a level with relativity 0.80 is 20% below it, with other factors held constant. These comparisons do not say that the level is 25% above or below every other level.

How to Choose an Anchor

A sound choice is usually a level that is easy to identify, well represented in the intended rating population, and useful for explaining comparisons. A common level can make results more familiar and reduce the chance that the anchor describes only a small or unusual group. If a business or regulatory audience routinely thinks in terms of a particular class, that class may also be a useful reference.

These are practical guidelines, not mathematical requirements that force a unique answer. Any valid level can serve as the reference when its factor is positive and the model is otherwise defined appropriately. The choice should be stated explicitly so readers know what a relativity of 1.00 means. A 1.00 is an anchor, not a claim that the level has average risk or no risk.

Suppose, for illustration, a multiplicative plan has a base amount $B$ and one factor for each of $J$ categorical variables. The level $l_j$ of variable $j$ has a positive factor $q_{j,l_j}$. If $r_j$ is the chosen reference level for variable $j$, define the relativity $R_{j,l_j}$ by dividing that level’s factor by the reference factor.

R_{j,l_j}=\frac{q_{j,l_j}}{q_{j,r_j}} \qquad\text{and}\qquad R_{j,r_j}=1.00

The reference-risk amount is the amount for the combination of all selected reference levels. When the factors combine multiplicatively, it is $B\prod_{j=1}^{J}q_{j,r_j}$. Replacing each level factor with its relativity gives the same amount for every risk:

B\prod_{j=1}^{J}q_{j,l_j} = \left(B\prod_{j=1}^{J}q_{j,r_j}\right) \prod_{j=1}^{J}R_{j,l_j}

This equality explains why the reference level acts as an anchor: the reference factors are absorbed into the reference-risk amount, while each relativity records a comparison with its own variable’s anchor.

Deriving the Relativity and Choosing the Reference

The ratio formula follows directly from the requirement that a reference level have relativity 1.00. For one variable, let $C$ be a common amount before applying its level factor. The amount for level $l$ is $Cq_l$, and the amount for reference level $r$ is $Cq_r$. Their ratio is:

\frac{Cq_l}{Cq_r} = \frac{q_l}{q_r}

The common amount $C$ cancels. Therefore, the level’s comparison with the reference is $q_l/q_r$, not $q_r/q_l$. Setting $l=r$ gives $q_r/q_r=1$, as required. This argument assumes positive factors and a multiplicative relationship; it does not apply unchanged to an additive rating structure.

List the levels for each variable.
Keep the factors for distinct categorical variables separate; each variable will have its own anchor.

Select a useful reference level.
Prefer a clearly defined and, where practical, well-represented level that makes comparisons understandable.

Set its relativity to 1.00.
For each other level, divide its factor by the factor for the selected reference level of that same variable.

Interpret the comparison in context.
State that the relativity compares the level with its reference, holding other rating factors fixed.

Worked Example: Choosing a Reference for One Rating Variable

A hypothetical insurer is reviewing a vehicle-use factor with three levels. The modeled expected annual loss costs supplied for this exercise are $360 for pleasure use, $450 for commuting, and $585 for business use. The company wants its reference to be a familiar, central category; commuting is selected. Find the relativities and verify the implied costs. No interest or mortality assumption is involved, and no probability distribution is assumed; the expected loss costs are supplied directly.

Step 1: Identify the anchor. Commuting is the reference level, so its relativity must be 1.00. Its expected loss cost is $450.

Step 2: Compare each level with commuting. Pleasure use has relativity $360/450=0.80$. Commuting has relativity $450/450=1.00$. Business use has relativity $585/450=1.30$.

Step 3: Check the interpretation. Relative to commuting, the modeled cost for pleasure use is 20% lower, and the modeled cost for business use is 30% higher.

Step 4: Verify the costs. The commuting reference amount remains $450. Applying the other relativities gives $450(0.80)=\$360$ and $450(1.30)=\$585$, matching both supplied costs. The method is direct comparison to the selected anchor; the common mistake it avoids is dividing the reference cost by each level cost.

Worked Example: Selecting a Separate Anchor for Each Variable

A hypothetical property rating plan has two categorical variables. Its supplied multiplicative factors are construction age: new $0.90$, established $1.00$, and older $1.40$; and site setting: urban $1.20$, suburban $1.00$, and rural $0.80$. The modeled amount for an established suburban reference risk is $500. Select the stated central levels as references, identify their relativities, and calculate the amount for an older rural risk. Assume factors combine multiplicatively. No interest or mortality assumption or probability distribution is needed; the factors and reference-risk amount are given.

Step 1: Select one reference per variable. Established is the construction-age reference, and suburban is the site-setting reference. Each has relativity 1.00 within its own variable.

Step 2: Express the requested levels relative to their respective anchors. Older construction has relativity $1.40/1.00=1.40$. Rural setting has relativity $0.80/1.00=0.80$. The two ratios use different variables’ own references; they are not divided by one another.

Step 3: Apply both factors to the reference-risk amount. The older rural amount is $500(1.40)(0.80)=500(1.12)=\$560$.

Step 4: Check the result. The older factor raises the reference amount by 40%, while the rural factor reduces the amount relative to suburban. Their combined factor is 1.12, so the result is 12% above $500. The method avoids treating “reference” as one overall category shared by both variables.

Worked Example: Choosing a Common Level and Checking the Ratio Direction

A hypothetical equipment policy has a categorical equipment-size factor. During a model review, the supplied expected annual loss costs and policy counts are: compact, $240 and 700 policies; standard, $300 and 250 policies; and heavy, $420 and 50 policies. Choose a practical reference, calculate all three relativities, and verify the heavy-level comparison. No interest or mortality assumption or probability distribution is used; the loss costs are supplied values.

Step 1: Choose the anchor. Compact is the most represented level, with 700 of 1,000 policies, and is clearly defined. It is a practical reference. This does not imply that compact has average loss cost.

Step 2: Set the reference to 1.00. Compact has relativity $240/240=1.00$.

Step 3: Calculate the other comparisons. Standard has relativity $300/240=1.25$. Heavy has relativity $420/240=1.75$.

Step 4: Check in dollars. Against the $240 compact reference, standard gives $240(1.25)=\$300$, and heavy gives $240(1.75)=\$420$. The counts supported the practical choice of anchor; they are not part of the ratio calculation.

Method and mistake avoided. Divide each level’s supplied amount by the compact reference amount. Reversing the ratio for standard would give $240/300=0.80$, incorrectly implying standard is below compact. The correct comparison is $300/240=1.25$, because standard is 25% above compact.

Common Interpretive Traps

Assuming the reference must be the lowest-risk level. Any suitable level can be the anchor. A reference of 1.00 means “equal to the selected reference,” not “lowest risk” or “average risk.”
Using one anchor for several variables. Choose a reference separately for each categorical variable. The reference risk is the combination of those selections.
Reversing the ratio. The level being compared goes in the numerator; its variable’s reference goes in the denominator.
Reading a relativity as an absolute amount. A relativity compares levels within one factor. It is not a dollar premium or loss cost by itself.
Claiming more than the comparison supports. “1.30” means 30% above that factor’s reference, holding other factors fixed in the multiplicative plan; it is not a comparison with all risks in the portfolio.

Reference selection is useful because it makes a rating factor interpretable and gives the factor a defined scale. For a plan to be understood and reviewed consistently, state the reference level for every categorical variable and verify that its relativity is exactly 1.00. The choice should make the comparisons clear, while remaining distinct from any claim about portfolio-average risk.

Check Your Understanding

For each problem, identify the reference level first, then compare the other levels with that anchor.

A hypothetical policy factor has supplied modeled costs of $320 for local delivery, $400 for regional delivery, and $520 for long-haul delivery. If regional delivery is the reference, calculate each relativity and verify the long-haul amount. (Answer: local 0.80, regional 1.00, long-haul 1.30; $400 × 1.30 = $520.)
A rating plan has a building-age factor of $0.85$ for newer buildings, $1.00$ for maintained buildings, and $1.25$ for older buildings. Maintained is the reference. What are the newer and older relativities, and what does the older relativity mean? (Answer: newer 0.85, older 1.25; older is 25% above maintained, holding other factors fixed.)
A hypothetical class has supplied costs of $150 for basic service and $210 for extended service. Extended service is selected as the reference. Calculate both relativities and explain why the basic relativity is greater than 1.00 or less than 1.00. (Answer: basic $150/210=0.7143$, extended 1.00; basic is below the selected reference.)
A plan’s reference-risk amount is $600. For the first variable, the selected reference is standard equipment and the requested level has relativity 1.20. For the second variable, the selected reference is suburban and the requested level has relativity 0.75. Assuming multiplicative combination, calculate the amount for the requested combination. (Answer: $600(1.20)(0.75)=\$540$.)
A hypothetical factor has costs of $270 for small units, $360 for medium units, and $450 for large units. Medium is the reference. Calculate all relativities, then identify the incorrect result if an analyst reports small as $360/270$. (Answer: small 0.75, medium 1.00, large 1.25; $360/270$ is the reversed ratio, and the correct small relativity is $270/360=0.75$.)

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