Coefficient Interpretation Across Model Structures

Why Model Structure Changes Coefficient Interpretation

A rating-variable coefficient does not have one universal interpretation. Its meaning depends on what the model predicts and on the link function. A coefficient in a frequency model changes expected claim count or rate; a coefficient in a severity model changes expected dollars per claim; and a coefficient in a pure-premium model changes expected loss dollars per exposure unit directly. A rating factor can affect more than one component, so interpreting only one coefficient may not describe the total change in predicted loss cost.

This tutorial compares these interpretations using hypothetical insurance portfolios. The examples concern expected losses before expenses, profit provisions, taxes or investment income. No interest or mortality assumption is needed: the modeled losses are not discounted and the exposures are not life-contingent.

Definition: A frequency-severity model separately models expected claim frequency and expected claim severity. The expected pure premium, or loss cost per exposure unit, is the product of those two means when the expected aggregate loss factors into expected claim count times a common expected severity. A direct pure-premium model models expected aggregate loss per exposure unit directly.

Log Links and Multiplicative Effects

For a model with a log link, the conditional mean is the exponential of the linear predictor. For a rating variable $x$ with coefficient $\beta$, increasing $x$ by $\Delta x$ changes the linear predictor by $\beta\Delta x$. Therefore the new conditional mean is the old mean multiplied by $\exp(\beta\Delta x)$. The coefficient itself is a change in log mean; it is not usually the percentage change in the mean.

\frac{\text{new conditional mean}}{\text{old conditional mean}} =\exp(\beta\Delta x), \qquad \text{percentage change} =100\left[\exp(\beta\Delta x)-1\right]\%.

For a one-unit increase, the multiplicative relativity is $\exp(\beta)$. For example, a coefficient of $0.10$ implies a relativity of $\exp(0.10)$, not a 10% increase exactly. The exact percentage increase is $100[\exp(0.10)-1]\%$, approximately 10.52%. For a categorical rating variable coded as an indicator, $\exp(\beta)$ compares the indicated category with its reference category, holding the other modeled variables fixed.

Frequency and Severity Effects on Pure Premium

Let $N$ be the number of claims for one exposure unit, and let $X_j$ be the severity of claim $j$. Suppose, conditional on the rating variables, claim count has mean $\lambda$, and each claim has the same conditional expected severity $m$. Also assume that claim severity does not depend on the number of claims, conditional on those rating variables. Then the aggregate loss $L=\sum_{j=1}^{N}X_j$ has expected value $\lambda m$. This follows by conditioning on the claim count: given $N=n$, the expected sum is $nm$, so taking expectations gives $E[L]=E[N]m=\lambda m$.

If the frequency model and severity model both use log links, let $\beta_f$ and $\beta_s$ be the coefficients of the same rating variable in those models. For a change of $\Delta x$, frequency is multiplied by $\exp(\beta_f\Delta x)$, and severity by $\exp(\beta_s\Delta x)$. Their product, the pure premium, is consequently multiplied by $\exp[(\beta_f+\beta_s)\Delta x]$. This addition applies to coefficients on the log scale; it does not mean that the separate percentage changes should simply be added.

\lambda(x)=\exp(\eta_f(x)), \qquad m(x)=\exp(\eta_s(x)), \qquad P(x)=\lambda(x)m(x).

\frac{P(x+\Delta x)}{P(x)} =\exp(\beta_f\Delta x)\exp(\beta_s\Delta x) =\exp\!\left[(\beta_f+\beta_s)\Delta x\right].

Worked Example: Combining Frequency and Severity Relativities

A hypothetical insurer models annual claim count per exposure unit as Poisson with mean $\lambda(x)=0.08\exp(0.20x)$. Conditional claim severities are independent Gamma random variables with shape $5$ and scale $500\exp(-0.05x)$ dollars, so their mean is $m(x)=2{,}500\exp(-0.05x)$ dollars. There is no discounting or mortality. Compare the pure premium at $x=0$ and $x=1$, and identify the combined relativity.

Method and assumptions. The expected claim count at a rating value is the Poisson mean. The mean of a Gamma distribution with shape $k$ and scale $\theta$ is $k\theta$, so the stated severity mean is $5[500\exp(-0.05x)]=2{,}500\exp(-0.05x)$. Conditional severity does not depend on claim count, so expected aggregate loss per exposure unit is frequency times mean severity.

At $x=0$. Frequency is $0.08\exp(0)=0.08$ claims per exposure unit. Mean severity is $2{,}500\exp(0)=\$2{,}500$ per claim. Thus the pure premium is $0.08(2{,}500)=\$200.00$ per exposure unit.

At $x=1$. The frequency factor is $\exp(0.20)=1.2214028$, giving frequency $0.08(1.2214028)=0.0977122$. The severity factor is $\exp(-0.05)=0.9512294$, giving mean severity $2{,}500(0.9512294)=\$2{,}378.0736$. Multiplying the unrounded component means gives $0.0977122(2{,}378.0736)$, approximately $\$232.3668$.

Combined check. The coefficients sum to $0.20-0.05=0.15$, so the combined relativity is $\exp(0.15)=1.1618342$. Applying it to the reference pure premium gives $200(1.1618342)=\$232.3668$, agreeing with the component calculation. Rounded to the nearest cent, the new pure premium is \$232.37 per exposure unit, an increase of about 16.18%.

The common mistake avoided here is adding the separate percentage changes as if they were exact. The correct calculation multiplies the two relativities, equivalently adding their log-scale coefficients.

Direct Pure-Premium Models and Exposure Offsets

A direct pure-premium model predicts expected loss dollars per exposure unit without separately modeling claim count and claim severity. With a log link, a coefficient for a rating variable directly describes the multiplicative effect on that expected loss cost. If the response is total aggregate loss for an observation with exposure $e$, an exposure offset $\log(e)$ makes its modeled mean $e$ times the pure-premium mean. The offset adjusts the expected total for the amount of exposure; it is not an estimated rating-variable coefficient.

For a concrete mean specification, suppose total loss $Y_e$ follows a Tweedie model with conditional mean $e\mu$, variance $2e\mu^{1.5}$, and $e>0$. Here $\mu$ is the expected pure premium per exposure unit. A log-link model can be written $\log E[Y_e]=\log(e)+\eta$. The coefficient interpretation for $\mu$ follows from the mean equation; the variance specification does not change that interpretation.

Worked Example: Applying a Direct Pure-Premium Coefficient

A hypothetical portfolio uses the direct model just specified, with $\mu=480\exp(0.035x-0.12z)$ dollars per exposure unit. The reference risk has $x=0$ and $z=0$. A policy has $x=2$, $z=1$, and exposure $e=0.60$. Find its expected pure premium and expected total loss. Treat the Tweedie variance parameters as $p=1.5$ and $\phi=2$; no interest or mortality adjustment applies.

Method. The modeled mean per exposure unit is $\mu$. First calculate the rating-variable contribution to the linear predictor, then apply the log link. Finally multiply by exposure to obtain expected total loss.

The reference pure premium is $\$480$. For this policy, the predictor change is $0.035(2)-0.12(1)=0.07-0.12=-0.05$. Therefore the relativity is $\exp(-0.05)=0.9512294$, and the pure premium is $480(0.9512294)=\$456.5901$, or \$456.59 per exposure unit to the nearest cent.

Expected total loss is exposure times pure premium: $0.60(456.5901)=\$273.9541$, or \$273.95 to the nearest cent. As a check, the reference expected total at exposure $0.60$ is $0.60(480)=\$288$, and applying the same relativity gives $288(0.9512294)=\$273.9541$.

The common mistake avoided is treating the offset as part of the risk relativity or applying it twice. It scales expected total loss, but the rating-variable factor $0.9512294$ applies to the per-exposure-unit pure premium.

When Frequency Uses a Logit Link

Not every frequency model uses a log link. If the modeled response is whether a policy has at least one claim during a period, a logistic model may be used. Its coefficient changes the log odds of a claim, not the log probability. If the baseline probability is $p_0$ and a rating-variable change multiplies the odds by $r$, the new probability is $rp_0/(1-p_0+rp_0)$. With a fixed expected severity $m$, the pure premium for this one-claim-or-no-claim setting is probability times severity.

Worked Example: Converting an Odds Effect to a Loss Cost

In a hypothetical product, a policy either has zero claims or one claim in the year. At the reference rating value the claim probability is $0.10$. A logistic frequency model assigns a coefficient $\ln(2)$ to a one-unit increase in $x$, so the odds double. Conditional on a claim, severity is Gamma with shape $4$ and scale $\$450$, giving mean severity $\$1{,}800$. Find the new annual expected loss cost. There is no discounting or mortality.

Method and assumptions. The occurrence count is Bernoulli with the modeled claim probability. The conditional claim severity has the stated Gamma distribution and is independent of occurrence given $x$. Its mean is $4(450)=\$1{,}800$.

At the reference value, the odds are $0.10/(1-0.10)=1/9$. The coefficient $\ln(2)$ doubles these odds, giving new odds $2/9$. Converting odds back to probability gives $p=(2/9)/(1+2/9)=2/11=0.181818\ldots$. The expected loss cost is $p(1{,}800)=(2/11)(1{,}800)=\$327.2727\ldots$, or \$327.27 to the nearest cent.

At the reference value, the expected loss cost is $0.10(1{,}800)=\$180$. The new cost is not twice the reference cost: doubling the odds does not double the probability. The common mistake avoided is applying the odds ratio directly as a probability relativity.

Common Interpretation Traps

Before converting a coefficient to a premium or loss cost, identify the modeled mean and the link. A coefficient is interpreted in relation to that mean, not automatically in relation to the final premium charged. The following checks help prevent common errors:

Do not read a log-link coefficient as an exact percentage. Use $\exp(\beta\Delta x)$ for the relativity and subtract one to obtain the percentage change.
Do not use a frequency coefficient as a pure-premium coefficient. In a frequency-severity structure, include both component effects when both change with the rating variable.
Do not add percentage changes across components. Multiply the frequency and severity factors, or add their coefficients on the log scale.
Do not interpret a logit coefficient as a probability change. Convert baseline probability to odds, apply the odds multiplier, and convert back.
Do not confuse an offset with a rating effect. An exposure offset scales expected aggregate loss, while a pure-premium relativity changes expected loss per exposure unit.

The model structure matters because the same underlying risk difference may be represented in different places. A variable can raise expected frequency while reducing average severity, or a direct model can represent their combined effect on pure premium. The final comparison should therefore be made on the predicted loss-cost scale relevant to the rating decision, not by comparing raw coefficients from different model structures.

Key takeaway: Under log links, exponentiate the coefficient change to obtain a mean relativity. In a frequency-severity model, combine the two component relativities to get the pure-premium effect. In a direct pure-premium model, the coefficient already applies to the loss-cost mean. For a logit frequency model, convert through odds and probability before calculating expected loss cost.

Check Your Understanding

For each question, calculate the requested effect before rounding. Monetary answers are rounded to the nearest cent.

A Poisson frequency model has mean $0.04\exp(0.08x)$ claims per exposure unit. What is the predicted frequency when $x$ rises from 0 to 3? (Answer: $0.05085$ claims per exposure unit.)
A Gamma severity model has shape $3$, scale $\$300$ at the reference class, and a log-link class coefficient of $0.12$. What is mean severity in a class two units higher? (Answer: $\$1{,}144.12$ per claim.)
A frequency model and a severity model have log-link coefficients $0.15$ and $-0.04$, respectively, for the same variable. If the reference pure premium is $\$250$, what is the pure premium after a two-unit increase? (Answer: $\$311.52$ per exposure unit.)
A logistic claim-occurrence model has baseline probability $0.20$, and a one-unit rating change multiplies the odds by $1.5$. Conditional mean severity is $\$2{,}400$. What is the new expected loss cost in a zero-or-one-claim period? (Answer: $\$654.55$ per exposure unit.)
A direct log-link pure-premium model predicts $\$620$ per exposure unit at the reference value. A one-unit increase has coefficient $-0.10$, and the policy has exposure $0.75$. What is its expected total loss? (Answer: $\$420.75$.)

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