5.5 Interpreting Coefficients & Multiplicative Relativities

From GLM Coefficients to Multiplicative Relativities

The primary objective of property and casualty predictive modeling is to construct an intuitive, statistically sound, and regulatory-compliant rating manual. Under a Generalized Linear Model with a log link, regression coefficients translate directly into multiplicative rating factors (also known as relativities).

Consider a frequency model with $K$ categorical rating variables (e.g., Territory, Driver Age Class, Vehicle Symbol). Using standard dummy coding (reference cell coding), the linear predictor for policy $i$ is:

$\ln(\mu_i) = \beta_0 + \sum_{k=1}^K \sum_{j=1}^{J_k - 1} \beta_{k,j} X_{i,k,j}$

Where:

• $\beta_0$ is the intercept parameter.
• $J_k$ is the number of categorical levels in variable $k$.
• $X_{i,k,j} = 1$ if policy $i$ belongs to level $j$ of variable $k$, and $0$ otherwise.
• One category for each variable is designated as the reference (base) category, for which no dummy variable is created (i.e., its parameter is fixed at $\beta_{k,\text{base}} = 0$).

Exponentiating both sides of the equation recovers the expected frequency on the original scale:

$\mu_i = \exp(\beta_0) \times \prod_{k=1}^K \prod_{j=1}^{J_k - 1} \exp(\beta_{k,j} X_{i,k,j})$

Definitions of Core Rating Components

1. Base Rate (or Base Frequency):

$\text{Base Rate} = \exp(\beta_0)$

The base rate represents the expected frequency (or loss cost) for a hypothetical baseline policyholder whose characteristics fall into the reference category for every rating factor.

1. Reference Category Relativity: For the designated base level of any rating variable, $X = 0$, which yields a relativity of exactly $1.000$:

$\text{Relativity}_{\text{base}} = \exp(0) = 1.000$

1. Rating Factor (Relativity) for Level $j$: For any non-base level $j$, the rating factor $R_{k,j}$ represents the proportional multiplier relative to the base class:

$R_{k,j} = \exp(\hat{\beta}_{k,j})$

• If $\hat{\beta} = 0.250$, the rating factor is $\exp(0.250) = 1.2840$ (a $+28.4%$ surcharge).
• If $\hat{\beta} = -0.150$, the rating factor is $\exp(-0.150) = 0.8607$ (a $-13.93%$ discount).

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Actuarial Best Practices for Base Category Selection

In statistical software, default dummy coding frequently assigns the reference level to the first category alphabetically or numerically (e.g., Territory 01 or Driver Age 16). In actuarial pricing, allowing software defaults to dictate the base category is a serious mistake.

Actuaries adhere to three core principles when selecting the reference level for each rating variable:

1. Maximum Exposure Volume (Highest Credibility)

The reference category should be the level with the largest earned exposure (e.g., adult drivers age 35–49, standard suburban territory, mid-size passenger sedan). Because all other level coefficients represent contrasts against the base level:

$\hat{\beta}_j = \ln\left( \frac{\mu_j}{\mu_{\text{base}}} \right)$

Choosing a base category with massive exposure minimizes the variance of the intercept $\text{Var}(\hat{\beta}_0)$ and minimizes the covariance between contrast coefficients. If a tiny niche category with few claims is chosen as the base (e.g., teenage drivers or exotic sports cars), sampling noise in that reference category propagates instability across every single relativity in the rating algorithm.

2. Risk Homogeneity and Stability

The base category should exhibit consistent, stable loss experience from year to year. Selecting volatile categories (such as coastal hurricane territories or high-performance motorcycles) introduces year-over-year rate turbulence into the base rate.

3. Intuitive Benchmark for Regulators and Consumers

In regulatory filings and consumer marketing, rate manuals are judged on transparency. Regulators expect the base class to represent a standard, average, or low-risk benchmark. Surcharging youthful drivers or high-density urban areas relative to a mature, suburban standard is intuitively understood; discounting all drivers relative to an ultra-high-risk teenage base class confuses regulators and consumers.

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Statistical Significance and Confidence Intervals for Relativities

To determine whether a rating factor differs significantly from the baseline, actuaries evaluate the estimated coefficient's standard error.

The Wald Test

For any coefficient $\hat{\beta}_j$, we test the null hypothesis that the level does not differ from the base category:

$H_0: \beta_j = 0 \iff H_0: \text{Relativity}_j = 1.000$ $H_1: \beta_j \ne 0 \iff H_1: \text{Relativity}_j \ne 1.000$

The Wald statistic $z$ is computed as:

$z = \frac{\hat{\beta}_j}{\text{SE}(\hat{\beta}_j)} \sim \mathcal{N}(0, 1)$

Alternatively, the squared Wald statistic follows a chi-square distribution with 1 degree of freedom: $W = z^2 \sim \chi^2_1$. If $|z| > 1.96$ (or $W > 3.841$), the relativity is statistically significant at the $\alpha = 0.05$ level.

Calculating Confidence Intervals for Relativities

[!IMPORTANT] The Asymmetric Confidence Interval Rule A frequent exam error is constructing confidence intervals directly on the exponentiated relativity scale: $\exp(\hat{\beta}) \pm 1.96 \cdot \text{SE}(\hat{\beta})$. This is mathematically invalid because the distribution of the exponentiated estimator $\hat{R} = \exp(\hat{\beta})$ is skewed (log-normal), not symmetric Gaussian.

The correct procedure calculates the $95%$ confidence limits on the linear predictor scale and exponentiates the lower and upper bounds:

$\text{CI}_{95\%}(\beta_j) = \left[ \hat{\beta}_j - 1.96 \cdot \text{SE}(\hat{\beta}_j), \; \hat{\beta}_j + 1.96 \cdot \text{SE}(\hat{\beta}_j) \right]$

$\text{CI}_{95\%}(R_j) = \left[ \exp\left(\hat{\beta}_j - 1.96 \cdot \text{SE}(\hat{\beta}_j)\right), \; \exp\left(\hat{\beta}_j + 1.96 \cdot \text{SE}(\hat{\beta}_j)\right) \right]$

Worked Numerical Example

An actuary fits a commercial property frequency GLM. For the "Wood Frame" construction class, the model yields:

• Coefficient: $\hat{\beta} = 0.2850$
• Standard Error: $\text{SE}(\hat{\beta}) = 0.0420$

Step 1: Calculate the point relativity:

$R = \exp(0.2850) = 1.3298$

(Wood Frame construction carries an indicated $+32.98%$ frequency surcharge relative to the Fire-Resistive base class).

Step 2: Calculate bounds on the linear scale:

$\text{Lower Bound}_{\beta} = 0.2850 - 1.96(0.0420) = 0.2850 - 0.08232 = 0.20268$ $\text{Upper Bound}_{\beta} = 0.2850 + 1.96(0.0420) = 0.2850 + 0.08232 = 0.36732$

Step 3: Exponentiate to obtain relativity confidence bounds:

$\text{Lower Limit}_R = \exp(0.20268) = 1.2247$ $\text{Upper Limit}_R = \exp(0.36732) = 1.4438$

The $95%$ confidence interval for the relativity is $[1.225, 1.444]$. Because the interval does not contain $1.000$, the surcharge is statistically significant at $p < 0.05$. Notice that the interval is asymmetric around the point estimate ($1.330 - 1.225 = 0.105$ below, while $1.444 - 1.330 = 0.114$ above).

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Assembling the Multiplicative Rating Algorithm

Actuarial ratemaking typically splits pure premium modeling into two separate components:

$\text{Pure Premium} = \mathbb{E}[\text{Loss Cost}] = \mathbb{E}[\text{Frequency}] \times \mathbb{E}[\text{Severity}]$

1. Frequency Model: Poisson or Negative Binomial GLM with log link and $\ln(\text{exposure})$ offset.
2. Severity Model: Gamma GLM with log link and prior weights equal to claim count.

Integrating Frequency and Severity Relativities

Because both models employ the log link, their predictions combine multiplicatively for each rating factor $k$:

$\text{Pure Premium}_i = \left( \text{Base Freq} \times \prod_{k=1}^K R_{\text{Freq}, k} \right) \times \left( \text{Base Sev} \times \prod_{k=1}^K R_{\text{Sev}, k} \right)$

$\text{Pure Premium}_i = (\text{Base Freq} \times \text{Base Sev}) \times \prod_{k=1}^K \left( R_{\text{Freq}, k} \times R_{\text{Sev}, k} \right)$

$\text{Pure Premium}_i = \text{Base Pure Premium} \times \prod_{k=1}^K R_{\text{Combined}, k}$

Where:

• $\text{Base Pure Premium} = \text{Base Freq} \times \text{Base Sev}$
• $R_{\text{Combined}, k} = R_{\text{Freq}, k} \times R_{\text{Sev}, k}$

From Pure Premium to Final Gross Policy Premium

To convert modeled pure premium into final charged premium, actuaries incorporate underwriting expenses, taxes, licenses, and underwriting profit loadings:

$\text{Gross Premium} = \frac{\text{Pure Premium} + \text{Fixed Expense per Exposure}}{1 - \text{Variable Expense Ratio} - \text{Target Underwriting Profit Ratio}}$

Worked Policy Rating Table

Calculation of Policy Pure Premium:

$\text{Pure Premium} = \$360.00 \times 1.375 \times 1.890 \times 1.610 \times 0.765 = \$1,152.88$

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Re-Basing Relativities & Mathematical Premium Invariance

After model estimation, business, marketing, or regulatory requirements may dictate changing the reference base category. For instance, the marketing department may request that the most popular policy deductible ($1,000) serve as the base category (Relativity $= 1.000$), even though the GLM was originally fit with a $500 deductible reference class.

The Re-Basing Transformation Rules

Suppose variable $k$ has estimated relativities $R_1, R_2, \dots, R_J$ relative to Old Base Level 1 ($R_1 = 1.000$). To shift the base to New Base Level $m$ (so that $R'_m = 1.000$):

1. Transform every level relativity by dividing by the old relativity of the new base:

$R'_j = \frac{R_j}{R_m} \quad \forall \quad j \in \{1, \dots, J\}$

1. Transform the Base Rate by multiplying by the old relativity of the new base:

$\text{Base Rate}' = \text{Base Rate} \times R_m$

Mathematical Proof of Premium Invariance

For any policyholder in level $j$, the indicated premium under the re-based rating manual is:

$\text{Premium}'_j = \text{Base Rate}' \times R'_j = (\text{Base Rate} \times R_m) \times \left( \frac{R_j}{R_m} \right)$

$\text{Premium}'_j = \text{Base Rate} \times R_j = \text{Premium}_j$

The term $R_m$ cancels out entirely. The total premium charged to every single policyholder in the book is completely invariant to the choice of base category.

Worked Re-Basing Example

• Original Base Rate: $250.00
• New Re-Based Base Rate: $250.00 \times 0.800 = \mathbf{$200.00}$

Verification of policy premiums:

• Policy with $500 Deductible: Old $= $250 \times 1.000 = $250.00$; New $= $200 \times 1.250 = \mathbf{$250.00}$.
• Policy with $1,000 Deductible: Old $= $250 \times 0.800 = $200.00$; New $= $200 \times 1.000 = \mathbf{$200.00}$.
• Policy with $2,500 Deductible: Old $= $250 \times 0.600 = $150.00$; New $= $200 \times 0.750 = \mathbf{$150.00}$.

Premium invariance holds with absolute mathematical precision across all policy tiers.

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Actuarial Exam Traps & Pitfalls

[!WARNING] Trap 1: Symmetric Exponentiated Confidence Intervals Never calculate confidence intervals as $\exp(\hat{\beta}) \pm 1.96 \cdot \text{SE}(\hat{\beta})$. The delta method or direct exponentiation of linear bounds must be used: $[\exp(\hat{\beta} - 1.96 \cdot \text{SE}), \exp(\hat{\beta} + 1.96 \cdot \text{SE})]$. The resulting confidence interval is always asymmetric around the point relativity.

[!WARNING] Trap 2: Forgetting to Adjust the Base Rate When Re-Basing Dividing rating relativities by a new base factor without adjusting the base rate alters the overall premium collected by the company. The base rate must be multiplied by the same factor used to divide the relativities to preserve portfolio revenue neutrality.

[!WARNING] Trap 3: Credibility Weighting Relativities Directly on the Multiplicative Scale When applying classical credibility $Z$ to complement of credibility relativities $R_0$, actuaries must credibility-weight the relativities carefully. In log-link models, credibility is theoretically applied on the linear scale: $\hat{\beta}{\text{cred}} = Z \hat{\beta} + (1 - Z) \beta_0$, so that $R{\text{cred}} = \exp(\hat{\beta}_{\text{cred}}) = R^Z \cdot R_0^{1-Z}$ (geometric weighting), or linearly weighted as $Z R + (1 - Z) R_0$ with an off-balance correction.