Why the Price-Change Encoding Matters
A renewal price change can be expressed in dollars, as a percentage of the expiring premium, or as the logarithm of the ratio of renewal to expiring premium. These measures describe related—but not identical—features of a price offer. For example, a $120 increase is 20% of a $600 expiring premium but only 10% of a $1,200 expiring premium. The predictor chosen for a renewal model therefore determines which offers are treated as comparable.
For a binary renewal outcome, logistic regression models the conditional probability of renewal. In the examples below, \(Y=1\) means that the policy renews and \(Y=0\) means that it does not. The model assumes a Bernoulli outcome conditional on the predictors and a linear predictor for the log odds. Other relevant predictors are held fixed unless stated otherwise.
The percentage change is sometimes recorded as a decimal, such as \(r=0.08\), and sometimes in percentage points, such as \(x=100r=8\). A model must use the same scale as the one used to define its coefficient. The log ratio is dimensionless and is defined only for positive premiums. These predictors should be constructed from premiums that are comparable: a change in limits, deductible, coverage, or premium basis can make a raw premium difference misleading as a measure of price change.
Three Common Encodings
The three encodings answer different questions. A dollar change measures the amount added to or subtracted from the bill. A signed percentage change measures the increase relative to the expiring premium. A log ratio measures the multiplicative change on a logarithmic scale. Their coefficients consequently have different units and interpretations.
| Predictor | Definition | Coefficient interpretation in a logit model |
|---|---|---|
| Dollar change | \(\Delta=P_1-P_0\) | Change in log odds per dollar of premium change |
| Percentage points | \(x=100(P_1-P_0)/P_0\) | Change in log odds per percentage point |
| Log ratio | \(g=\ln(P_1/P_0)\) | Change in log odds per one-unit increase in log ratio |
For any one of these predictors \(z\), a single-predictor logistic model (with other predictors fixed) has the form
The coefficient \(\beta\) describes an association with renewal on the log-odds scale, not a fixed change in renewal probability. The probability effect depends on the starting value of the linear predictor. A negative price-change coefficient means that larger increases are associated with lower renewal odds, conditional on the other model terms; it does not by itself establish a causal effect.
Worked Example: Percentage-Point Encoding
A hypothetical insurer models renewal with \(z=x=100(P_1-P_0)/P_0\), measured in percentage points. The fitted model is \(\operatorname{logit}(p)=0.90-0.02x\). Calculate the predicted renewal probability for a policy moving from $750 to $810, and compare it with a policy moving from $1,500 to $1,620, with all other predictors held fixed.
Method. Calculate the signed percentage change using the expiring premium as the denominator, then use \(x\) in percentage-point units.
For the first policy, \(\Delta=810-750=\$60\), so \(r=60/750=0.08\) and \(x=100(0.08)=8\). The linear predictor is \(\eta=0.90-0.02(8)=0.74\). Thus
For the second policy, \(\Delta=1{,}620-1{,}500=\$120\), so \(r=120/1{,}500=0.08\) and again \(x=8\). Its linear predictor is also \(0.74\), giving the same predicted renewal probability, \(0.6770\). This model treats the two 8% increases alike, despite the different dollar changes. The displayed probabilities are rounded to four decimals.
Check and trap avoided. Both offers have the same ratio \(P_1/P_0=1.08\), confirming the same percentage change. The coefficient is per percentage point, so entering \(0.08\) instead of \(8\) would use the wrong scale.
Dollar Changes and Log Ratios Across Starting Premiums
A dollar-change model assigns the same predictor value to equal dollar increases, whatever the expiring premium. That can be suitable when the absolute amount of the increase is the intended measure, but it does not treat equal percentage increases alike. Conversely, a percentage-change model gives the same predictor value to equal percentage increases even when the dollar amounts differ.
A log-ratio model also gives the same predictor value to equal proportional changes, while expressing that change on a logarithmic scale. If \(P_1/P_0=1.20\), then \(g=\ln(1.20)\), not \(\ln(1.20)\) times some conversion to percentage points. Keep the ratio itself distinct from the percentage-point change.
Worked Example: Absolute-Dollar Encoding
For a hypothetical renewal portfolio, a fitted model is \(\operatorname{logit}(p)=0.80-0.0015\Delta\), where \(\Delta\) is measured in dollars. Compare policies whose premiums change from $800 to $920 and from $1,600 to $1,720. Calculate both percentage increases and both model predictions.
Method. Use the dollar change directly in the stated model, while separately calculating each relative increase to show what the dollar encoding does and does not capture.
For the first policy, \(\Delta=920-800=\$120\). Its percentage increase is \(120/800=0.15=15\%\). The linear predictor is \(\eta=0.80-0.0015(120)=0.80-0.18=0.62\), so
For the second policy, \(\Delta=1{,}720-1{,}600=\$120\), but its percentage increase is \(120/1{,}600=0.075=7.5\%\). The linear predictor remains \(0.80-0.0015(120)=0.62\), so the predicted probability is again \(0.6502\).
Check and trap avoided. The dollar changes are equal, so these predictions agree under this model; the percentage changes are not equal. Replacing the dollar predictor with a percentage predictor would be a different model, not a mere change in notation. Probabilities are rounded to four decimals.
Worked Example: Log-Ratio Encoding
A hypothetical model uses \(\operatorname{logit}(p)=0.60-0.80g\), where \(g=\ln(P_1/P_0)\). Compare a change from $1,000 to $1,200 with changes from $2,000 to $2,400 and from $2,000 to $2,200. Calculate each log ratio and predicted probability.
Method. Divide the renewal premium by the expiring premium before taking the natural logarithm. This preserves the direction of change and makes equal proportional changes produce equal values of \(g\).
For $1,000 to $1,200, the ratio is \(1{,}200/1{,}000=1.20\), so \(g=\ln(1.20)=0.1823216\). The linear predictor is \(0.60-0.80(0.1823216)=0.4541427\), and
For $2,000 to $2,400, the ratio is \(2{,}400/2{,}000=1.20\), again giving \(g=0.1823216\), \(\eta=0.4541427\), and \(p=0.6116\). These two offers have different dollar increases but the same 20% increase.
For $2,000 to $2,200, the ratio is \(2{,}200/2{,}000=1.10\), so \(g=\ln(1.10)=0.0953102\). Then \(\eta=0.60-0.80(0.0953102)=0.5237518\), giving \(p=1/[1+\exp(-0.5237518)]\approx0.6281\). This offer has a $200 increase, just as the first offer does, but its 10% change produces a different prediction under this log-ratio model. Probabilities are rounded to four decimals.
Check and trap avoided. The first ratio is \(1.20\), so its log is approximately \(0.1823\). In particular, a ratio of \(1.20\) must not be confused with a ratio of \(2.00\), whose natural logarithm is \(0.6931\).
Why the Log-Ratio Has a Useful Interpretation
The log-ratio encoding has a direct multiplicative interpretation on the odds scale. Let the renewal odds be \(O=p/(1-p)\). In a logistic model with \(g=\ln(P_1/P_0)\), hold all other model terms fixed and multiply the renewal premium by a factor \(q>0\). The new ratio is \(qP_1/P_0\), so the new log-ratio is \(g+\ln(q)\). Therefore,
This derivation shows why the log-ratio coefficient is convenient when the modeled effect is intended to respond consistently to proportional changes. For example, multiplying the premium ratio by \(1.10\) changes the odds by a factor of \(1.10^\beta\), conditional on the other terms. It does not imply that probability changes by the same percentage: probability is obtained from the odds through \(p=O/(1+O)\).
A percentage-point model has a related interpretation, but its coefficient is per one percentage point, not per one-unit log ratio. For a change from \(r_1\) to \(r_2\), the odds ratio is \(\exp[\beta\,100(r_2-r_1)]\) if the predictor is \(x=100r\). Always state the scale before interpreting a coefficient.
Model Specification and Practical Checks
The price-change predictor alone may not capture systematic differences associated with the expiring premium. If such differences matter, a model can include a separate starting-premium term, for example \(\gamma\ln(P_0/P_{\mathrm{ref}})\), where \(P_{\mathrm{ref}}>0\) is a fixed reference premium. This additive adjustment allows the model to account for starting-premium differences while retaining a common price-change coefficient. Whether that adjustment is appropriate is an empirical and modeling question; it should not be assumed that any one encoding removes all confounding.
Before fitting or scoring a model, verify that the premium fields refer to the same policy and renewal offer, use the correct expiring premium as the denominator, and follow the exact predictor scale used in estimation. Check that a decrease remains negative and that the log-ratio is computed as renewal divided by expiring premium, not the reverse. If the model is intended to describe an association, avoid describing its coefficient as a causal impact without a design that supports causal interpretation.
Check Your Understanding
For each problem, treat the stated logistic model and its price-change scale as given. Round probabilities to four decimals.
- A model is \(\operatorname{logit}(p)=0.70-0.03x\), where \(x\) is the signed percentage change in percentage points. A premium rises from $900 to $954. Find \(x\), the linear predictor, and the renewal probability. (Answer: \(x=6\), \(\eta=0.52\), \(p=0.6271\).)
- A dollar-change model is \(\operatorname{logit}(p)=1.00-0.002\Delta\), with \(\Delta\) in dollars. For a change from $500 to $575, calculate the percentage increase and predicted probability. (Answer: \(15\%\), \(\eta=0.85\), \(p=0.7006\).)
- A log-ratio model is \(\operatorname{logit}(p)=0.40-1.10g\). For a change from $1,000 to $1,100, calculate \(g\), the linear predictor, and the predicted probability. (Answer: \(g=\ln(1.10)=0.0953\), \(\eta=0.2952\), \(p=0.5733\).)
- Under \(\operatorname{logit}(p)=0.60-0.80g\), compare the predicted probabilities for $600 to $720 and $1,200 to $1,320. State why the predictions differ despite equal dollar increases. (Answer: \(g_1=0.1823\), \(p_1=0.6116\); \(g_2=0.0953\), \(p_2=0.6281\); the relative changes are 20% and 10%.)
- A percentage-point model has coefficient \(-0.025\) per point. A policy's baseline renewal probability, before a price-change contribution, is \(0.7500\). Calculate the probability after a 12% increase, holding all other terms fixed. (Answer: odds multiplier \(e^{-0.025(12)}=e^{-0.30}=0.7408\); updated odds \(3(0.7408)=2.2225\); probability \(0.6897\).)