Building An Implementable Rating Factor Table

From Model Output to a Rating Table

A fitted model can produce a prediction for each risk, but an implemented rating plan usually needs a practical table: a base rate and a set of factors that a rating system can apply consistently. The conversion is not just a formatting exercise. Reference levels must be defined, interactions must be represented, and rounding must be checked against both individual predictions and the intended portfolio total.

This tutorial focuses on that conversion. It is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. Assume a selected model and its estimated coefficients are already available; the task here is to express those estimates in a usable table and verify the table’s arithmetic.

Definition: A rating factor table expresses a modeled rate as a base rate multiplied by factors associated with the risk’s rating characteristics. A reference level is the selected level of a rating variable whose factor is set to 1.00. The base rate is the modeled rate for a risk at the reference level of every modeled variable, provided the model has no interactions that change that reference prediction.

For a log-link model with categorical predictors and no interactions, the fitted rate for risk $i$ can be written as

\widehat{r}_i = \exp(\widehat{\beta}_0) \prod_{j=1}^{p}\exp(\widehat{\beta}_{j,\ell_{ij}}).

Here $\widehat{r}_i$ is the predicted rate per exposure unit, $\widehat{\beta}_0$ is the estimated intercept, and $\widehat{\beta}_{j,\ell_{ij}}$ is the coefficient for the level of variable $j$ applying to risk $i$. In a reference-coded model, the coefficient for each reference level is zero, so its factor is $\exp(0)=1$. The base rate is $\exp(\widehat{\beta}_0)$, when all predictors are at their reference levels. This expression assumes the fitted model uses a log link and that the included predictors are represented by the factors shown; an omitted interaction or other modeled term cannot be recovered merely by multiplying main-effect factors.

Reference Levels, Factors, and Interactions

A reference level is a normalization choice, not a claim that the reference risk is average or risk-free. Its factor is 1.00 because the other levels are expressed relative to it. Changing the reference level changes the displayed coefficients and factors, but it does not change the model’s predictions if the intercept and all factors are transformed consistently.

For a coefficient $\widehat{\beta}$ on a log-rate scale, the corresponding multiplicative factor is $\exp(\widehat{\beta})$. For example, a coefficient of $\ln(1.25)$ corresponds to a factor of 1.25: holding other modeled terms fixed, the predicted rate is 25% higher than at the reference level. Factors multiply; they are not added to the base rate.

If the selected model includes an interaction, a table containing only the separate main-effect factors generally will not reproduce the model. An interaction factor applies to a combination of levels. It can be shown as a separate modifier or incorporated into a joint table, but it must be applied exactly once. A joint table is often easier to use when the interacting variables have a manageable number of levels.

Worked Example: Setting Reference Levels and Calculating a Rate

A hypothetical property insurer selects a log-link model with an intercept of $\ln(480)$, a territory coefficient of $\ln(1.15)$ for Territory C, and a vehicle coefficient of $\ln(1.25)$ for vehicle class Utility. Territory A and vehicle class Sedan are the reference levels. There is no interaction. Find the base rate and the predicted rate for a Territory C Utility risk. Verify the prediction by both the factor table and the log-link expression.

Method and reference levels. The base rate is the predicted rate at Territory A and Sedan. Both reference factors equal 1.00, so the base rate is $\exp(\ln(480))=\$480.00$ per exposure unit.

Convert coefficients to factors. The Territory C factor is $\exp(\ln(1.15))=1.15$. The Utility factor is $\exp(\ln(1.25))=1.25$.

Apply the table. The Territory C Utility predicted rate is $480 \times 1.15 \times 1.25$. First, $480 \times 1.15=552$; then $552 \times 1.25=\$690.00$ per exposure unit.

Check using the model expression. The linear predictor is $\ln(480)+\ln(1.15)+\ln(1.25)=\ln(480 \times 1.15 \times 1.25)=\ln(690)$. Exponentiating gives $\$690.00$, matching the factor-table result. The common mistake avoided here is adding the percentage changes to the base rate: the factors multiply under the log-link model.

Why the Multiplication Works

The factor representation follows directly from the log link. Suppose a no-interaction model has a reference-coded linear predictor $\eta_i=\widehat{\beta}_0+\sum_j\widehat{\beta}_{j,\ell_{ij}}$. Its fitted rate is $\widehat{r}_i=\exp(\eta_i)$. The exponential of a sum is the product of exponentials, so

\widehat{r}_i = \exp\left(\widehat{\beta}_0+\sum_j\widehat{\beta}_{j,\ell_{ij}}\right) = \exp(\widehat{\beta}_0) \prod_j\exp(\widehat{\beta}_{j,\ell_{ij}}).

Thus the intercept becomes the base rate and each coefficient becomes a multiplicative factor. With an interaction coefficient $\widehat{\gamma}_{ab}$ applying to a particular pair of levels $a,b$, the fitted rate also includes $\exp(\widehat{\gamma}_{ab})$. If that modifier is omitted, the resulting table describes a different model and generally gives different predictions.

Worked Example: Preserving an Interaction in the Table

A hypothetical insurer models annual rates with a base rate of $\$600$, a Territory B factor of 1.20, a Sport vehicle factor of 1.50, and an interaction factor of 0.90 applying only to the Territory B and Sport combination. Territory A and Standard vehicle are reference levels. Calculate the rate for all four combinations and the total predicted premium for exposures of 10, 20, 30, and 40 units in the order A-Standard, A-Sport, B-Standard, and B-Sport. Check whether main-effect factors alone would reproduce the B-Sport prediction.

Method. Multiply the base rate by applicable main-effect factors and, for Territory B Sport only, by the interaction factor.

A-Standard. Both variables are at reference levels, so the rate is $600 \times 1.00 \times 1.00=\$600.00$.

A-Sport. The rate is $600 \times 1.00 \times 1.50=\$900.00$.

B-Standard. The rate is $600 \times 1.20 \times 1.00=\$720.00$.

B-Sport. Apply both main effects and the interaction: $600 \times 1.20=720$, $720 \times 1.50=1{,}080$, and $1{,}080 \times 0.90=\$972.00$.

Check the portfolio total. The four predicted premiums are $10 \times 600=\$6{,}000$, $20 \times 900=\$18{,}000$, $30 \times 720=\$21{,}600$, and $40 \times 972=\$38{,}880$. Their sum is $\$84{,}480$.

Check the interaction. Using only main effects would predict $\$1{,}080$ for B-Sport, not $\$972$. The interaction reduces that cell’s prediction by $1-0.90=10\%$. The common mistake avoided is omitting the joint effect because each variable already has its own factor.

Rounding and Portfolio Calibration

A table may require factors to be displayed to a fixed number of decimal places. Rounding changes the values used in production, so the rounded table can produce different predictions from the fitted model. The impact should be measured on the actual portfolio exposure distribution, not inferred from the number of digits alone: a small factor change in a high-exposure segment can matter more to total premium than a larger change in a small segment.

Let $e_i$ be the exposure for risk or cell $i$, $\widehat{r}_i$ the intended model rate, and $\widetilde{r}_i$ the rate from the rounded table. Their corresponding portfolio totals are

T=\sum_i e_i\widehat{r}_i, \qquad \widetilde{T}=\sum_i e_i\widetilde{r}_i.

If the objective is to restore the intended total while retaining the rounded table’s relative factors, multiply all rounded-table rates by a common calibration scalar $c=T/\widetilde{T}$. This works because the calibrated total is $c\widetilde{T}=(T/\widetilde{T})\widetilde{T}=T$. It assumes the same exposure basis and portfolio mix are used for both totals, and that a common scalar is permitted. It does not restore every original cell prediction: it matches the aggregate total while retaining the rounded table’s changed relativities.

Worked Example: Calibrating Rounded Factors to the Model Total

A hypothetical insurer has a base rate of $\$500$, a Territory B factor of 1.237, and a Vehicle Y factor of 1.416. Territory A and Vehicle X have factors of 1.00. The deployed table rounds Territory B to 1.24 and Vehicle Y to 1.42. The portfolio has exposures of 100 A-X, 50 A-Y, 80 B-X, and 20 B-Y units. Calculate the intended and rounded-table totals, then find a common calibration scalar that makes the rounded table reproduce the intended total.

Method. Calculate each cell rate and multiply it by the cell exposure. Use unrounded factors for the intended total and rounded factors for the deployed-table total.

Intended A-X and A-Y. The A-X rate is $\$500$, giving $100 \times 500=\$50{,}000.00$. The A-Y rate is $500 \times 1.416=\$708$, giving $50 \times 708=\$35{,}400.00$.

Intended B-X and B-Y. The B-X rate is $500 \times 1.237=\$618.50$, giving $80 \times 618.50=\$49{,}480.00$. For B-Y, the combined factor is $1.237 \times 1.416=1.751592$; the rate is $500 \times 1.751592=\$875.796$, and the premium is $20 \times 875.796=\$17{,}515.92$.

Intended total. Adding the four cell premiums gives $50{,}000+35{,}400+49{,}480+17{,}515.92=\$152{,}395.92$.

Rounded-table total. The A-X rate remains $\$500$, giving $\$50{,}000.00$. The A-Y rate is $500 \times 1.42=\$710$, giving $\$35{,}500.00$. The B-X rate is $500 \times 1.24=\$620$, giving $\$49{,}600.00$. The B-Y rate is $500 \times 1.24 \times 1.42=\$880.40$, giving $\$17{,}608.00$. The rounded-table total is $\$152{,}708.00$.

Calibrate. The required scalar is $c=152{,}395.92/152{,}708.00\approx0.997956$. Multiplying the rounded-table total by this scalar gives approximately $\$152{,}395.92$. If implemented through a common multiplier on the base rate, the calibrated base rate before any further currency rounding is $500 \times 0.997956\approx\$498.98$. The common mistake avoided is calibrating with the rounded factors’ simple average; the correct scalar uses the exposure-weighted portfolio totals.

Deployment Checks and Practical Rules

Before release, verify that the table represents the selected model and that the production calculation uses the same definitions, exposure units, and applicable factors. A concise audit sequence is:

Fix the references.
State the reference level for each variable and set its factor to 1.00. Confirm that the base rate corresponds to the all-reference risk under the model’s parameterization.

Translate every modeled term.
Exponentiate log-link coefficients to obtain factors. Include interaction terms, or encode them in joint table entries, rather than silently dropping them.

Apply the deployment precision.
Round according to the specified table rules, then recalculate rates using the rounded values. Do not check only the printed factors.

Reconcile predictions and totals.
Compare selected cell rates with the model predictions, calculate the exposure-weighted portfolio total, and apply an allowed common calibration scalar if needed.

A clean total does not prove that every table entry is correct. For example, an omitted interaction might be partly offset by rounding changes elsewhere, leaving the aggregate close to its target while mispricing a particular combination. Therefore, inspect representative cells, including reference risks, high-impact combinations, and every interaction combination, as well as the total. Also confirm that factors are applied once, in the intended order, and to the correct rate basis.

Do not add relativities. Under the stated multiplicative model, a rate is the base rate times the applicable factors.
Do not treat a reference level as average. Its factor of 1.00 is a normalization convention.
Do not omit interactions. A main-effect-only table will not reproduce a model with material interaction terms.
Do not confuse matching totals with matching predictions. A common scalar can restore the aggregate premium but cannot undo cell-level changes caused by rounding.
Do not calibrate on a different portfolio basis without a reason. The exposures used to test the table should match the intended calibration basis.

Key takeaway: A deployable factor table must preserve the model’s reference structure and every included term, then be checked using the actual rounded values. Verify both risk-level predictions and the exposure-weighted premium total; an aggregate calibration alone is not a substitute for checking the cells.

Check Your Understanding

For each problem, assume a multiplicative log-link rating table unless an interaction is specified. Work through the factor arithmetic and the stated exposure basis.

A selected model has base rate $\$480$, factor 1.15 for Territory C, and factor 1.25 for Utility vehicles, with no interaction. What is the Territory C Utility rate? (Answer: $\$690.00$ per exposure unit)
A table has base rate $\$250$, Territory A and Basic vehicle factors of 1.00, Territory B factor 1.30, and Plus vehicle factor 1.20. Exposures are 40 A-Basic, 20 A-Plus, 30 B-Basic, and 10 B-Plus. What is the predicted portfolio premium total? (Answer: $\$29{,}650.00$)
A two-level class table has factors 1.00 and 1.40, with exposures of 100 and 50 units, respectively. The target premium total is $\$96{,}000$. What base rate reproduces that total before rounding? (Answer: $\$564.71$ per exposure unit, rounded to cents)
A model portfolio total is $\$152{,}395.92$, while the rounded table produces $\$152{,}708.00$ on the same exposures. What common scalar calibrates the rounded table to the model total, to six decimal places? (Answer: $0.997956$)
A hypothetical model has a $\$600$ base rate, Territory B factor 1.20, Sport vehicle factor 1.50, and an interaction factor of 0.90 for B-Sport only. What is the B-Sport rate, and what would a main-effect-only table incorrectly predict? (Answer: $\$972.00$; main-effect-only prediction $\$1{,}080.00$)

What You'll Learn