Why Model Price-Change Interactions?
A single estimated price response assumes the modeled relationship between renewal price change and renewal odds is the same for every policyholder, after accounting for the other predictors. That can be too restrictive. Tenure, product, distribution channel, or prior claims may be associated with differences in how renewal outcomes vary with a quoted price change. An interaction allows the price-change slope to differ across a specified characteristic.
The purpose is not to assert that a characteristic causes a customer to respond differently. A fitted interaction describes variation in the model’s conditional association, given the included predictors and the data used to estimate it. Its interpretation depends on how variables are coded, which group is the reference, and the scale of the model. These distinctions are essential when translating estimates into modeled retention comparisons.
Logistic Interactions and Their Interpretation
Let \(Y=1\) indicate that a policy renews and \(Y=0\) indicate that it does not. Write \(p=P(Y=1\mid\text{predictors})\). A logistic renewal model uses the logit link, \(\log(p/(1-p))\), for the conditional renewal probability. For a price change \(x\) and tenure \(t\), one useful specification is
Here, \(t_0\) is a chosen reference tenure, so \(z=0\) at that tenure. Suppose \(x\) is measured in percentage points: a five-percentage-point increase is entered as \(x=5\), not \(x=0.05\). The coefficient \(\beta\) is the change in log odds for a one-point increase in price change when \(z=0\). The interaction coefficient \(\delta\) is the change in that log-odds slope for each additional unit of \(z\). Thus the price-change slope at tenure \(t\) is \(\beta+\delta z\), and the modeled odds multiplier for a one-point increase is \(\exp(\beta+\delta z)\).
The probability response is not that slope itself. Since the logistic function is \(p=1/(1+\exp(-\eta))\), where \(\eta\) is the linear predictor, differentiation gives
This derivative is a local probability change per one-point increase in \(x\). It varies with \(p\) as well as with tenure. For a finite price change, calculate the predicted probability at each price rather than treating the local derivative as a constant.
Worked Example: Price Response by Tenure
Worked Example: Price Response by Tenure
In a hypothetical renewal model, \(x\) is the quoted price change in percentage points and \(z=t-5\), where \(t\) is tenure in years. The fitted model is
Calculate the price-change slope, the one-point odds multiplier, and the predicted renewal probability at \(x=5\) for policies with two and eight years of tenure. State the method and identify a common mistake.
Method: substitute each tenure into the interaction, then evaluate the logistic model. At two years, \(z=2-5=-3\). The price slope is
Therefore, at two years of tenure, a one-point increase in price change multiplies modeled renewal odds by \(\exp(-0.05)=0.9512\). At \(x=5\), the linear predictor is
The predicted probability is \(1/(1+\exp(-0.83))=0.6964\), or 69.64%.
At eight years, \(z=8-5=3\). The price slope is \(-0.08+(-0.01)(3)=-0.11\), giving a one-point odds multiplier of \(\exp(-0.11)=0.8958\). At \(x=5\),
Check: The interaction changes the price slope from \(-0.05\) at two years to \(-0.11\) at eight years. The probabilities, 69.64% and 68.35%, are consistent with the two calculated linear predictors. The common mistake is to use \(-0.08\), the slope at the reference tenure of five years, for every tenure. At eight years, that would incorrectly give an odds multiplier of \(\exp(-0.08)=0.9231\), rather than the correct \(\exp(-0.11)=0.8958\).
Interactions with Product, Channel, and Claims
For a categorical characteristic, use an indicator for a non-reference group. If \(B=1\) denotes product B and \(B=0\) denotes reference product A, a price interaction can be written as
For product A, the price slope is \(\beta\); for product B, it is \(\beta+\delta\). The coefficient \(\gamma\) is the difference in log odds between products at \(x=0\), not a general difference that must hold at every price change. The interaction coefficient \(\delta\) is the difference between the two groups’ price slopes. The same coding and interpretation apply to a binary channel or prior-claims indicator.
When the characteristic has more than two categories, choose a reference category and include an indicator for each other category. A price interaction is then included for each non-reference indicator. Each group’s price slope is the main price coefficient plus the interaction coefficient for that group; the reference group’s slope is the main price coefficient. State the reference group explicitly when interpreting results.
Worked Example: Different Price Slopes by Product
A hypothetical model uses price change \(x\) in percentage points and \(B=1\) for product B; product A is the reference. Its fitted equation is
Compare the one-point odds multipliers and predicted renewal probabilities at \(x=5\) for the two products. Also identify the reference-group interpretation and a common mistake.
Method: calculate the group-specific slopes before predicting. For product A, set \(B=0\). Its price slope is \(-0.06\), so the one-point odds multiplier is \(\exp(-0.06)=0.9418\). At \(x=5\), its linear predictor is \(1.00-0.06(5)=0.70\), giving
For product B, set \(B=1\). Its price slope is \(-0.06-0.04=-0.10\), and its one-point odds multiplier is \(\exp(-0.10)=0.9048\). At \(x=5\), its linear predictor is
Check: Product B has a more negative modeled price slope, but its predicted probability at this particular price change is higher because it also has the positive group coefficient \(0.30\). This is not contradictory: the slope and the probability level answer different questions. The common mistake is to treat \(-0.04\), the interaction coefficient, as product B’s full price slope. Its full slope is \(-0.06-0.04=-0.10\).
Combining Interactions for a Policy-Level Comparison
A fitted model can include more than one interaction, such as price change by tenure and price change by prior claims. For a policy with specified values, collect every term containing \(x\) to find its price slope. This is a direct way to check that no interaction contribution has been omitted. A group’s modeled odds multiplier over a finite change from \(x=0\) to \(x=c\), with other predictors fixed, is the exponential of \(c\) times its price slope when the predictor is linear in \(x\) on the log-odds scale.
Worked Example: Combining Tenure and Prior Claims
In a hypothetical renewal model, \(x\) is price change in percentage points, \(z=t-5\) is centered tenure, and \(C=1\) indicates at least one prior claim while \(C=0\) indicates none. The fitted equation is
For five-year policies, compare the predicted probabilities at \(x=0\) and \(x=4\) for the two claims groups. Calculate each group’s price slope and the odds multiplier from \(x=0\) to \(x=4\). Name the method and the common mistake it avoids.
Method: substitute the policy characteristics, collect the price terms, and calculate each probability from its own linear predictor. At five years, \(z=5-5=0\). For policies with no prior claims, \(C=0\), so the price slope is \(-0.07\). The odds multiplier over four points is \(\exp(4(-0.07))=\exp(-0.28)=0.7558\). At \(x=0\), \(\eta=0.90\), giving \(p=1/(1+\exp(-0.90))=0.7109\). At \(x=4\), \(\eta=0.90-0.07(4)=0.62\), giving \(p=1/(1+\exp(-0.62))=0.6502\).
For policies with prior claims, \(C=1\), so the price slope is \(-0.07-0.04=-0.11\). The four-point odds multiplier is \(\exp(4(-0.11))=\exp(-0.44)=0.6440\). At \(x=0\), \(\eta=0.90+0.25=1.15\), giving \(p=1/(1+\exp(-1.15))=0.7595\). At \(x=4\), \(\eta=1.15+4(-0.11)=0.71\), giving \(p=1/(1+\exp(-0.71))=0.6704\).
Check: Each four-point odds multiplier equals the exponential of four times that group’s slope. The claims group starts with higher modeled odds at \(x=0\), but its more negative price slope narrows the probability difference at \(x=4\). The common mistake is to apply only the main price coefficient \(-0.07\) to both groups, omitting the price-by-claims interaction.
Justifying the Odds-Ratio Interpretation
The interaction interpretation follows directly from the model’s odds. For the tenure model, hold \(z\) fixed and increase \(x\) by one. The linear predictor changes by
Because the logit is the logarithm of the odds, the difference in log odds is \(\beta+\delta z\). Exponentiating both sides shows that the odds at \(x+1\), divided by the odds at \(x\), equal \(\exp(\beta+\delta z)\). This is why an interaction changes the group-specific odds multiplier. The probability difference does not generally equal this multiplier or the log-odds slope: converting odds to probability is nonlinear, as the derivative \(p(1-p)(\beta+\delta z)\) shows.
Modeling Choices and Common Traps
An interaction should answer a defined modeling question, not simply provide another way to search for a favorable result. Decide which differences are substantively plausible, code the variables consistently, and interpret estimates on the scale of the fitted model. Centering a continuous variable such as tenure makes the main price coefficient apply at a meaningful reference tenure. It changes the reference point for the main effects, but it does not change the fitted values when the model is otherwise equivalent.
- Confusing a main effect with a group-specific slope. With an interaction, the price coefficient alone applies only at the reference value or reference group.
- Reading an odds multiplier as a probability-point change. An odds multiplier applies to odds. To compare renewal probabilities, calculate them from the model for the specified policies.
- Ignoring variable units. A coefficient for a one-percentage-point price change differs in scale from a coefficient for a one-unit change in a decimal price ratio. State the coding before interpreting coefficients.
- Overstating what the model establishes. An estimated difference in conditional responses is not, by itself, evidence that changing a policyholder characteristic would cause a different response.
- Leaving the reference group unstated. Without the reference coding, a reader cannot tell which group a main effect describes.
The practical value of interaction modeling is that it replaces a single portfolio-wide price slope with explicit, testable group differences. Results can inform comparisons of modeled renewal response, provided the predictions are interpreted at the specified policy characteristics and price changes. This treatment is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus.
Check Your Understanding
For each problem, identify the relevant group-specific price slope before interpreting the prediction or odds multiplier.
- A logistic model is \(\operatorname{logit}(p)=0.80-0.05x+0.02z-0.006xz\), where \(x\) is price change in percentage points and \(z=t-4\) for tenure \(t\) in years. Find the price slope and one-point odds multiplier at \(t=9\). (Answer: slope \(=-0.08\); odds multiplier \(=\exp(-0.08)=0.9231\).)
- A model is \(\operatorname{logit}(p)=0.70-0.04x+0.20G-0.03xG\), where \(G=1\) denotes the non-reference channel. Find the price slopes for both channels and the non-reference channel’s odds multiplier for a three-point price increase. (Answer: reference slope \(=-0.04\); non-reference slope \(=-0.07\); three-point odds multiplier \(=\exp(-0.21)=0.8106\).)
- A model is \(\operatorname{logit}(p)=1.00-0.06x+0.10z-0.02xz\), with centered tenure \(z=t-6\). For \(t=8\) and \(x=3\), find the linear predictor and predicted renewal probability. (Answer: \(z=2\), \(\eta=1.00-0.18+0.20-0.12=0.90\), and \(p=0.7109\).)
- A binary prior-claims model has price slope \(-0.05\) for the no-claims reference group and interaction coefficient \(-0.025\) for the claims group. Find the claims-group slope and its one-point odds multiplier. (Answer: slope \(=-0.075\); odds multiplier \(=\exp(-0.075)=0.9277\).)
- A fitted model has price terms \(-0.08x-0.01xz+0.03xC\), where \(z=t-5\), \(C=1\) indicates a prior claim, and \(x\) is in percentage points. For \(t=7\), \(C=1\), and a four-point price increase, find the price slope and odds multiplier. (Answer: slope \(=-0.08-0.01(2)+0.03=-0.07\); odds multiplier \(=\exp(4(-0.07))=\exp(-0.28)=0.7558\).)