Varying risk exposure in auto insurance: a weighted tweedie framework for experience rating an cancellation penalties

Abstract: This paper proposes a new family of Tweedie-based ratemaking models that explicitly account for mid-term policy cancellations. Using an automobile insurance dataset from a Canadian insurer, we document a marked difference in claims experience between policyholders who maintain their coverage until maturity and those who cancel their policies mid-term. Building on the classical Tweedie framework, we introduce flexible weighting functions and a premium penalty structure that depend on the level of exposure, allowing for a more realistic representation of the earned premium when coverage is interrupted before the end of the policy period. We compare several weighting structures within the Tweedie framework and examine their theoretical properties, as well as their empirical performance using deviance-based model comparison criteria, an area-between-curves criterion derived from concentration and Lorenz curves, and Murphy diagrams grounded in Bregman dominance. To operationalize the proposed models, monotonicity and non-negativity constraints are imposed on the penalty function, ensuring consistency with actuarial principles. Finally, using real-world data, we show that this approach provides both a strategic and competitive advantage: it allows the insurer to indirectly compensate for large losses through a cancellation surcharge, while preserving actuarial coherence and statistical consistency.

• The paper introduces spline-based exposure functions and three weighting schemes that relax proportional exposure assumptions, reducing test deviance by about 10% versus traditional ratio ratemaking.
• Flexible models show that mid-term cancelers generate higher claim frequency and severity, while cancellation penalties can reallocate roughly 11% of premiums and support lower annual prices.
• The framework improves predictive pricing and enables BMS-specific penalties, but constrained estimation sacrifices consistency guarantees and does not model policyholder responses to revised cancellation charges.

Motivation and contribution

Mid-term cancellations are a common feature of North American auto insurance, yet standard ratemaking treats exposure $t$ as a purely mechanical scaling factor, imposing the proportional mean structure $\mu = t\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$. This paper by Boucher, Coulibaly, and Trufin relaxes that assumption. Building on the Tweedie framework for aggregate loss costs, the authors introduce a flexible exposure function $\gamma(t)$ in the mean, $$\mu = \gamma(t)\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$$, examine which weight function $w = \omega(t)$ is coherent with this structure, and derive an explicit cancellation penalty $\rho(t) = \pi^{\text{Flex}}(\gamma(t) - t)$ that can be embedded in a transparent, contractually disclosed pricing scheme. The work extends the offset-versus-ratio comparison of Boucher and Coulibaly (2026) by relaxing the proportionality assumption itself, and connects the resulting penalty structure to experience rating via the Bonus–Malus Scale (BMS).

Empirical motivation

The empirical analysis uses a 13-year Ontario portfolio from a major Canadian insurer, with over two million vehicle-year records; the first six years build the claims history used to compute BMS levels, and the last seven years form the analysis sample split 75/25 into training and test sets. Contracts are classified by truncation pattern, and the study restricts attention to $\mathcal{XX}$ (full-year exposure, about 34.7%) and $\mathcal{XO}$ (mid-term cancellation, about 65.3%) contracts. The empirical findings are striking: contracts canceled mid-term exhibit substantially higher claim frequency per unit exposure than full-year contracts, and their average cost per claim is also higher. The average loss cost as a function of exposure is clearly not proportional to $t$, directly contradicting the traditional specification. This pattern is consistent across all policy years and persists after conditioning on BMS level, which is positively associated with annualized loss cost and with the propensity to cancel. The authors note that mid-term cancellations are frequently triggered by major losses (e.g., total theft), which creates the strategic possibility of recovering part of large-claim costs through a cancellation surcharge.

Theoretical results

The theoretical core concerns consistency under possible misspecification, using White's quasi-maximum likelihood theory. Assuming the true mean is $$\mu^{\text{True}} = \delta(t)\exp\{\mathbf{x}^\top\boldsymbol{\beta}^{\text{True}}\}$$, the authors show that traditional Tweedie estimators (offset, $\mu = t\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$0, or ratio, $\mu = t\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$1) converge to $\mu = t\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$2 if and only if $\mu = t\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$3. Under the flexible specification $\mu = t\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$4, consistency holds if and only if $\mu = t\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$5 — and since $\mu = t\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$6 is estimated by splines within a GAM, it can approximate an arbitrary $\mu = t\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$7. Notably, consistency does not depend on the choice of weight, dispersion, or variance parameters, so the weight $\mu = t\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$8 can be selected on other grounds.

Three weighting schemes are proposed: the Constant-Weight Model (CWM, $\mu = t\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$9, generalizing the offset approach), the Gamma-Weight Model (GWM, $\gamma(t)$0, generalizing the ratio approach so that weight enters only the Poisson component of the compound Poisson–Gamma representation), and the Exposure-Weighted Model (EWM, $\gamma(t)$1 with flexible mean). The GWM requires an iterative fixed-point algorithm alternating between the spline estimate of $\gamma(t)$2 and the weights; convergence is reported to be very rapid, stabilizing after two or three iterations.

Model comparison

Three evaluation criteria are used: normalized Tweedie deviance on training and test sets, the Area between empirical concentration and Lorenz curves (preferred to the signed ABC because the curves intersect), and Murphy diagrams based on Bregman dominance. With $\gamma(t)$3 fixed from prior calibration, the results are consistent across criteria:

Criterion	Traditional ratio	CWM	GWM	EWM
Deviance (train)	114.64	102.57	102.60	102.56
Deviance (test)	114.77	102.77	102.80	102.76
Area	≈0.13	—	smallest (marginally)	—

The flexible models reduce test deviance by roughly 10% relative to the traditional ratio approach, with negligible train–test gaps, indicating no overfitting. EWM attains the lowest deviances; GWM the smallest Area. A diagnostic on exposure ordering shows why: under the traditional specification, short-exposure contracts are underpriced, whereas the flexible models assign the lowest estimated premiums predominantly to contracts running to maturity.

Constrained penalties and ratemaking

The unconstrained $\gamma(t)$4 is not operationally usable — it can exceed 1 at low exposure, implying penalties exceeding the full annual premium. The authors therefore impose two constraints: (C1) $\gamma(t)$5 increasing in $\gamma(t)$6, and (C2) non-negative penalty $\gamma(t)$7, estimated via constrained splines. A crucial caveat is acknowledged plainly: once constraints are imposed, the quasi-maximum likelihood consistency result no longer applies; the constrained estimator trades statistical optimality for implementability. Murphy diagrams confirm that the unconstrained EWM Bregman-dominates the constrained model at all thresholds $\gamma(t)$8, as theory predicts.

Because even the constrained penalty implies paying nearly 75% of the annual premium after a few weeks of coverage, a smoothing parameter $\gamma(t)$9 is introduced via $$\mu = \gamma(t)\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$$0, interpolating between full penalty ($$\mu = \gamma(t)\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$$1) and pro rata refund ($$\mu = \gamma(t)\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$$2). Predictive scores improve monotonically in $$\mu = \gamma(t)\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$$3 (test Area falls from 0.1237 at $$\mu = \gamma(t)\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$$4 to 0.0306 at $$\mu = \gamma(t)\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$$5; test deviance from 114.77 to 108.12), so the full penalty is also statistically preferred. Embedding $$\mu = \gamma(t)\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$$6 as an offset shows that stronger penalties attenuate the estimated BMS effect, and coefficient paths are not uniformly monotonic in $$\mu = \gamma(t)\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$$7.

Premium reallocation and competitive implications

Approximately 11% of total collected premium is attributable to the penalty component (10.89% training, 10.80% test), and the model recovers roughly 15% more premium from mid-term cancelers than the traditional specification. Because the annual premium is the dominant factor in policyholder choice while penalty structures are not, shifting revenue into cancellation penalties permits lower headline premiums: the full-penalty model yields full-year premium ratios consistently below one relative to the traditional model, with discounts up to 25% for some profiles. The authors frame this as a strategic and competitive advantage, while noting that modified penalties will alter cancellation behavior — an analogue of Lemaire's bonus-hunger effect — which the framework does not model endogenously.

Group-specific penalties

A final extension lets the spline vary by BMS group: $$\mu = \gamma(t)\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$$8, $$\mu = \gamma(t)\,\exp\{\mathbf{x}^\top\boldsymbol{\beta}\}$$9. Searching over all eight possible two-group splits of the BMS scale, the best partition separates levels 95–99 (at least one claim-free year) from levels 100–104 (new or recently claiming policyholders). The smooth functions are statistically different over most of the exposure range, and the group structure suggests imposing stronger penalties on newly insured policyholders, who are also administratively costlier. Applying constraints (C1)–(C2) to the Group 2 curve would charge the full annual premium after only one quarter of coverage — extreme, though the authors note parallels with front-loaded subscription products. Further subdivision of groups yielded no significant differences.

Limitations and open questions

Several limitations are explicit. The dataset is a non-random sample from a single insurer and province; covariates are anonymized and illustrative rather than a full commercial tariff. The BMS level is treated as exogenous despite being endogenous to the claims and pricing process. Vehicle substitutions are not identifiable from the data and are excluded by design. Consistency is lost under the constrained estimation required for implementability, and the behavioral response of cancellation probabilities to changed penalties — which would feed back into the empirical $w = \omega(t)$0 — is acknowledged but not modeled. The cumulative premium curves still deviate from total claim costs under the constraints, leaving open how to capture a larger share of premium from cancelers while respecting regulatory disclosure. Finally, generalizing $w = \omega(t)$1 beyond a single BMS split to richer covariate-dependent penalty surfaces remains formalized only as future work.

Conclusion

The paper demonstrates that the proportionality between exposure and expected loss cost, embedded in standard offset and ratio Tweedie ratemaking, is empirically rejected and statistically consequential: flexible, spline-based exposure functions with coherent weighting schemes reduce out-of-sample deviance by roughly 10% and improve local balance, while an explicit, contractually disclosed cancellation penalty reallocates about 11% of premium and enables materially lower annual premiums. The framework reconciles actuarial coherence, regulatory transparency, and insurer competitiveness, at the cost of consistency guarantees once constraints are imposed and without modeling the behavioral feedback of penalties on cancellation decisions.