Adding Competitive And Market Conditions

Why Competitive and Market Conditions Matter

A renewal decision is made in a market, not in isolation. A policyholder may compare the offered renewal premium with prices available from other insurers, and local conditions may affect which alternatives are accessible. If those conditions are omitted, a model can attribute renewal differences to the offered price or other policy characteristics when part of the observed pattern is associated with the competitive environment.

The challenge is to represent that environment using information that is comparable, correctly timed, and aligned with the policyholder’s market. A competitor quote collected after the renewal decision, for example, cannot serve as a valid predictor of that decision. Nor is a state-wide average necessarily a suitable substitute for the market available to a policyholder in one local area.

This tutorial is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. It focuses on building and interpreting competitor-price and market-condition variables for renewal models, not on estimating a causal effect of price.

Definition: Let $Y_i=1$ if policy $i$ renews at the specified renewal opportunity and $Y_i=0$ otherwise. A competitor-price predictor summarizes prices or quotes for alternatives available to policy $i$; a market-condition predictor summarizes a relevant feature of the policy’s market, such as a local price level or competitive pressure. Predictors must be defined using information available at the intended prediction time.

Choosing and Interpreting Price Measures

A raw competitor premium is difficult to interpret without the corresponding renewal offer. A useful relative-price measure compares a competitor quote $C_i$ with the insurer’s renewal premium $O_i$. If a single comparable quote is used, define $z_i=\ln(C_i/O_i)$. With multiple comparable quotes, a specified summary—such as their arithmetic mean—can be used in place of $C_i$.

The ratio should compare like with like: the same coverage basis, comparable limits and deductibles, and prices associated with the same renewal opportunity. Otherwise, a difference in premium may reflect a difference in coverage rather than a difference in competitive price. The logarithm makes the predictor symmetric in percentage terms: doubling the ratio changes $z_i$ by $\ln 2$, while halving it changes $z_i$ by $-\ln 2$.

Model specification: Suppose renewal probability is modeled by logistic regression:

\operatorname{logit}(p_i) = \ln\left(\frac{p_i}{1-p_i}\right) = \beta_0+\beta_1 z_i+\beta_2 m_i, \qquad p_i=P(Y_i=1\mid z_i,m_i).

Here $m_i$ is a defined local-market measure. This specification assumes that, conditional on the included predictors, the log odds are linear in $z_i$ and $m_i$, and that the modeled renewal outcome and observation unit are consistently defined. The model also assumes the predictors used for scoring are available at the scoring time. It does not, by itself, establish that a change in competitor price causes a change in renewal probability.

The sign of $\beta_1$ depends on the definition of the ratio. For $z_i=\ln(C_i/O_i)$, a larger $z_i$ means competitor prices are higher relative to the insurer’s offer. A positive coefficient is therefore consistent with higher renewal odds in that situation, all else equal. Reverse the ratio and the interpretation reverses. State the numerator and denominator before interpreting any coefficient.

Worked Example: Converting Comparable Quotes to a Renewal Probability

For a hypothetical policy, the insurer’s renewal premium is $O=\$1{,}250$. Two comparable competitor quotes obtained for the same renewal opportunity are $\$1{,}020$ and $\$1{,}080$. A fitted illustrative model is $\operatorname{logit}(p)=0.30+1.00\ln(C/O)$. Calculate the quote summary, relative-price predictor, logit, and renewal probability.

Method. Use the arithmetic mean of the two comparable quotes, then apply the stated logit model. This example avoids the common mistake of using the own premium in the numerator or treating a logit value as a probability.

First calculate the competitor quote summary:

C=\frac{\$1{,}020+\$1{,}080}{2} =\frac{\$2{,}100}{2} =\$1{,}050.

The relative price ratio and its logarithm are

\frac{C}{O}=\frac{1{,}050}{1{,}250}=0.84, \qquad z=\ln(0.84)=-0.174353.

Substitute into the model:

\operatorname{logit}(p) =0.30+1.00(-0.174353) =0.125647.

Convert the logit to a probability:

p=\frac{1}{1+\exp(-0.125647)} \approx 0.5314.

Thus the fitted renewal probability is 0.5314, or 53.14%, to four decimal places. As a check, a positive logit must correspond to a probability above 0.5, and 0.125647 does. The result is a model prediction under the specified inputs, not a causal estimate of the effect of changing the quote.

Market Geography, Aggregation, and Timing

A market measure should match the level at which meaningful competitive conditions vary. Depending on the available data and business question, that may be a rating territory, a county, or another defined area. The geographic unit should be assigned consistently to the policy’s insured location and should not be chosen after observing the renewal outcome.

When combining areas, weights matter. An unweighted average gives a small area the same influence as a large one. If the intended measure represents the average condition faced by policies, policy counts or exposure weights may be more appropriate. Other targets may require different weights, but the target must be stated. In particular, aggregating a market-level variable to individual policies does not mean every policyholder in that market experienced the same alternatives.

Worked Example: Aggregating Market Conditions with Exposure Weights

A hypothetical insurer summarizes a local competitive-pressure index in four areas. The index values are $0.10$, $0.20$, $-0.10$, and $0.30$; their corresponding renewal exposures are 20, 30, 30, and 20 units. A policy in the combined market has $C/O=1.10$. Its illustrative model is $\operatorname{logit}(p)=-0.20+0.80\ln(C/O)+1.50m$. Calculate the exposure-weighted market index and predicted renewal probability.

Method. Use the exposure-weighted mean because the intended summary is the average index across renewal exposure. This avoids giving every area equal influence regardless of its size.

The total exposure is $20+30+30+20=100$. The weighted index is

m=\frac{20(0.10)+30(0.20)+30(-0.10)+20(0.30)}{100} =\frac{2+6-3+6}{100} =0.11.

The policy’s relative-price predictor is

z=\ln(1.10)=0.095310.

Then the linear predictor is

\operatorname{logit}(p) =-0.20+0.80(0.095310)+1.50(0.11) =-0.20+0.076248+0.165 =0.041248.

Therefore,

p=\frac{1}{1+\exp(-0.041248)} \approx 0.5103.

The predicted renewal probability is 0.5103 to four decimal places. The arithmetic mean of the four area indices would be $0.125$, not $0.11$; it would answer a different question by weighting the four areas equally. The calculation is internally consistent: the weighted index uses all 100 exposure units, and the resulting positive logit gives a probability just above 0.5.

Timing is equally important. A predictor intended for a renewal decision must be measured as of a defined date, such as the date the renewal offer is generated or the date the customer is expected to decide. A quote obtained only after that date may be a consequence of shopping behavior or the renewal decision itself. Including it can create leakage: the model appears predictive in historical data because it uses information that would not have been available when a real prediction was required.

The observation window should also be consistent. If one policy’s competitor quote reflects an early shopping period and another’s reflects a late period, their values may not represent comparable conditions. Document the quote date, policy effective date, offer date, geography, coverage basis, and aggregation rule. When market data are updated periodically, use the latest value genuinely available by the prediction cutoff, not a later revision.

Missing Quotes and Interpretation Limits

Competitor quotes are often missing for reasons that may matter. A policyholder may not shop, a quote may not be captured, or a particular risk may not be eligible for a competitor’s product. A missing quote is not a zero-dollar quote. Coding it as zero would make the relative-price logarithm undefined and would incorrectly imply an extraordinarily cheap alternative.

Possible treatments include using a market-level benchmark, adding a missingness indicator, fitting a model designed for the available data, or restricting the analysis to a population with sufficiently comparable quote information. None is automatically correct. In particular, a missingness indicator can capture differences associated with quote availability, but it does not remove selection bias or make the missing quote known. Whatever procedure is chosen must be reproducible in validation and at deployment.

Worked Example: A Missing Individual Quote with a Market Benchmark

A hypothetical policy has an own renewal premium of $\$1{,}000$, but no individual competitor quote was recorded by the prediction cutoff. The regional-month benchmark, calculated from other available comparable quotes, is $\$1{,}100$. An illustrative model uses that benchmark as $C$, sets $M=1$ when the individual quote is missing, and is $\operatorname{logit}(p)=-0.10+0.80\ln(C/O)-0.40M$. Calculate the prediction and state an important limitation.

Method. Apply the specified benchmark and missingness indicator together. This example avoids the mistake of replacing the missing quote with zero or silently treating the benchmark as an observed individual quote.

The proxy ratio and log relative price are

\frac{C}{O}=\frac{1{,}100}{1{,}000}=1.10, \qquad \ln(C/O)=\ln(1.10)=0.095310.

Since the individual quote is missing, $M=1$. Thus

\operatorname{logit}(p) =-0.10+0.80(0.095310)-0.40(1) =-0.10+0.076248-0.40 =-0.423752.

The predicted probability is

p=\frac{1}{1+\exp(0.423752)} \approx 0.3956.

The prediction is 0.3956 to four decimal places. The result relies on the stated model and benchmark procedure; it does not mean this policyholder actually received a $\$1{,}100$ quote. The benchmark and missingness treatment must be constructed using information available at prediction time and applied consistently to validation and production data.

Why the Coefficient Is Not a Causal Effect

A logistic coefficient has a precise conditional association interpretation. Holding the other included predictors fixed, increasing $z_i$ by one unit multiplies the modeled renewal odds by $\exp(\beta_1)$. Because $z_i$ is a log price ratio, a proportional change in the ratio has a particularly useful interpretation.

Derivation: For a change in $C/O$ by a factor $r>0$, $z_i$ changes by $\ln r$. Under the stated logistic model, the log odds change by $\beta_1\ln r$. Exponentiating gives an odds multiplier of $\exp(\beta_1\ln r)=r^{\beta_1}$. This is an odds comparison, not a probability difference.

For example, if $\beta_1=0.8$, doubling $C/O$ multiplies modeled odds by $2^{0.8}$, approximately 1.741, conditional on the other predictors. It does not multiply probability by 1.741. The probability change depends on the starting probability: the same odds multiplier produces different probability changes at different baselines.

Nor does the conditional association prove that changing the competitor quote would change renewal behavior by the model’s implied amount. Quote availability, customer shopping, coverage differences, and unmeasured risk or preference characteristics may be related to both the predictor and outcome. The model can support prediction when validated for its intended use, but a causal claim requires an appropriate causal design and assumptions beyond ordinary predictive logistic regression.

A useful construction and review sequence is:

Define the decision and outcome.
Specify the renewal opportunity, prediction date, and whether the outcome is renewal at that opportunity.

Set a comparable market measure.
Fix quote coverage, timing, geography, and any aggregation or weighting rule before calculating predictors.

Handle missingness explicitly.
Record what is missing and apply a stated procedure using only information available by the prediction cutoff.

Interpret at the right level.
Read coefficients as conditional associations on the log-odds scale, and check model performance on data separated in time from model development.

The central practical discipline is to distinguish what the model knows from what the analyst wishes it knew. A precise definition of quote timing, market geography, aggregation, and missingness makes the predictor auditable and the prediction interpretable. Without those definitions, a numerically correct logistic calculation can still answer the wrong question.

Check Your Understanding

Work each problem using the stated model and rounding probabilities to four decimal places.

A hypothetical policy has an own renewal premium of $\$1{,}400$ and a comparable competitor quote of $\$1{,}260$. The model is $\operatorname{logit}(p)=-0.20+1.20\ln(C/O)$. Find the logit and renewal probability. (Answer: $\ln(C/O)=-0.105361$, logit $=-0.3264$, $p=0.4191$.)
Two market areas have index values $0.05$ and $-0.15$, with renewal exposures 40 and 60. The model is $\operatorname{logit}(p)=0.10+2m$. Calculate the exposure-weighted index and probability. (Answer: $m=-0.07$, logit $=-0.04$, $p=0.4900$.)
A policy’s individual quote is missing. The stated regional benchmark is $\$900$, the own renewal premium is $\$1{,}000$, and the model is $\operatorname{logit}(p)=0.20+0.70\ln(C/O)-0.30M$, with $M=1$ for a missing quote. Calculate the prediction using the stated benchmark procedure. (Answer: logit $=-0.1738$, $p=0.4567$.)
A model uses a competitor quote recorded two weeks after the renewal decision, because that is the most complete quote field in the historical file. Identify the principal modeling problem and the correction. (Answer: The predictor is measured after the decision and may contain leakage; define a prediction cutoff and use only comparable information available by that date, or omit the unavailable quote.)
A positive coefficient is fitted for $z=\ln(C/O)$. Explain what happens to modeled odds when $C/O$ is multiplied by $r$, and whether the coefficient alone gives a probability change. (Answer: Odds are multiplied by $r^{\beta_1}$; it does not give a fixed probability change, because that change depends on the starting probability.)

What You'll Learn