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

This paper introduces a weighted Tweedie framework for auto insurance ratemaking that explicitly models mid-term policy cancellations through flexible weighting functions and penalty structures, demonstrating via Canadian data that this approach improves statistical consistency and allows insurers to strategically offset large losses with cancellation surcharges.

The Gist

The Big Picture: The "Gym Membership" Problem

Imagine you buy a year-long gym membership. You pay $1,200 upfront.

• Scenario A: You go to the gym every day for a year. You get your money's worth.
• Scenario B: You go for two weeks, get injured, and quit. You ask for a refund. The gym says, "Okay, we'll give you back $1,000 (keeping only $200 for the two weeks you used)."

The Problem: In the real world (specifically car insurance), this "fair refund" logic often fails. The paper argues that people who cancel their car insurance early (like the injured gym-goer) are often riskier than people who stay. They might cancel because they just got into a huge accident, stole a car, or are switching to a cheaper insurer because they know they are a bad driver.

If the insurance company just gives them a simple "pro-rata" refund (money back based on time used), they lose money. They are essentially paying a penalty to the risky drivers for leaving early.

The Solution: A "Flexible Penalty" System

The authors propose a new way to calculate insurance prices using a fancy statistical tool called a Tweedie Model. Think of this model as a smart, shape-shifting ruler that measures risk differently depending on how long you stay.

Here is the breakdown of their ideas:

1. The "Shape-Shifting Ruler" (Flexible Exposure)

• Old Way (The Straight Ruler): The old math assumed risk is a straight line. If you drive for 6 months, you pay exactly 50% of the yearly premium. It's simple, but it's wrong.
• New Way (The Curved Ruler): The authors found that risk isn't a straight line. The first few months of a policy might be riskier (or cheaper) than the middle months. They use a flexible curve (called a spline) that bends and twists to match the actual data. It's like a tailor measuring a customer with a flexible tape measure instead of a rigid stick, getting a perfect fit for the "risk shape."

2. The "Weighted Scale" (Weighting Functions)

In statistics, some data points are more important than others.

• The Analogy: Imagine a scale weighing apples. If you have a bag of 100 tiny apples and one giant pumpkin, the pumpkin should weigh more on the scale.
• The Paper's Twist: They found that short-term policies (people who cancel early) need to be "weighed" differently. Some of their new models give more "weight" to the people who cancel early, acknowledging that their behavior tells the insurer something important about the risk.

3. The "Cancellation Fee" (The Penalty)

This is the most strategic part. The paper suggests that when you cancel early, you shouldn't just get a refund for the time you didn't use. You should pay a penalty.

• How it works: If you cancel after 3 months, the math says you actually "used" more than just 3 months of risk. Maybe the risk was front-loaded, or maybe your cancellation itself signals a problem.
• The Result: The insurer charges you a bit extra for canceling early.
• The Benefit: This extra money helps cover the cost of the big claims that often trigger the cancellation. It also means that good drivers who stay for the full year get a discount. It's like a loyalty reward: "If you stay with us all year, your annual rate goes down because we aren't charging the 'cancellation tax' on you."

4. The "Secret Sauce" (BMS Levels)

The paper also looks at a "Bonus-Malus System" (BMS), which is basically a driving score.

• Low Score (Good Driver): You rarely crash.
• High Score (Bad Driver): You crash often.
• The Discovery: They found that "Bad Drivers" (High Score) are much more likely to cancel their policies early (often after a crash).
• The Fix: They propose that the "Cancellation Penalty" shouldn't be the same for everyone. A "Bad Driver" who cancels early should pay a steeper penalty than a "Good Driver." It's like a gym charging a higher fee to cancel if you've been skipping workouts and only showing up when you need to quit.

Why Does This Matter? (The "So What?")

1. Fairness: It stops "Good Drivers" from subsidizing "Bad Drivers" who leave early.
2. Profitability: Insurance companies lose less money because they capture more of the cost from risky people who leave.
3. Competition: Because the company saves money on the risky cancellations, they can lower the base price for everyone who stays the full year. This makes the insurance company more attractive to good drivers.

Summary in One Sentence

The paper proposes a smarter way to price car insurance where the "refund" for canceling early isn't just a simple math calculation, but a flexible penalty that accounts for the fact that people who leave early are often riskier, allowing the insurance company to charge them more and reward the people who stay with lower rates.

Technical Summary

1. Problem Statement

Traditional auto insurance pricing models typically assume a linear proportionality between risk exposure ($t$, the fraction of the year covered) and the expected loss cost (premium). The standard formulation is $\mu = t \times \exp(x^\top \beta)$. This implies that if a policyholder cancels mid-term, they receive a pro-rata refund, and the insurer charges exactly proportional to the time covered.

However, empirical evidence suggests this assumption is flawed:

• Non-Linear Risk: Policyholders who cancel mid-term (often due to major claims or switching insurers) exhibit significantly higher claim frequencies and severities than those who maintain coverage for the full term.
• Strategic Cancellations: Cancellations are not random; they are often triggered by adverse events (e.g., a total loss requiring a new policy) or by policyholders seeking lower premiums elsewhere.
• Financial Imbalance: The linear model fails to capture the true cost of short-duration policies, leading to cross-subsidization where low-risk, full-year policyholders subsidize high-risk, short-term cancellations.
• Data Limitations: Standard datasets often fail to distinguish between "true" cancellations and vehicle substitutions, complicating the modeling of exposure.

The paper addresses the need for a pricing framework that explicitly models mid-term cancellations and allows for non-linear relationships between exposure and expected loss, enabling insurers to impose strategic penalties on early cancellations to recover costs and improve risk selection.

2. Methodology

The authors propose a Weighted Tweedie Framework that generalizes the standard Generalized Linear Model (GLM) and Generalized Additive Model (GAM) approaches.

A. The Flexible Mean Structure

Instead of the traditional linear exposure term, the authors propose a flexible mean function:
$$\mu = \gamma(t) \exp(x^\top \beta)$$
Where:

• $\gamma(t)$ is a smooth, non-linear function of exposure $t$ (estimated using splines via GAMs).
• $x$ represents covariates (e.g., Bonus-Malus Scale, vehicle type).
• This allows the relationship between exposure and loss to deviate from strict proportionality.

B. Weighting Schemes

The paper investigates three specific weighting structures ($w$) within the Tweedie distribution variance formula ($Var[Y] = \phi w \mu^p$):

1. Constant-Weight Model (CWM): $w = 1$. Generalizes the "offset" approach; all observations have equal weight regardless of duration.
2. Gamma-Weight Model (GWM): $w = (\gamma(t)/\gamma(1))^{p-1}$. Generalizes the "ratio" approach; weight depends on the estimated exposure function.
3. Exposure-Weighted Model (EWM): $w = t^{p-1}$. Retains the traditional ratio weighting but applies it to the flexible mean structure.

C. Penalty Structure and Constraints

To operationalize the model, the authors define a cancellation penalty $\rho(t)$:
$$\rho(t) = \pi_{Flex}(t) - t\pi_{Flex} = \pi_{Flex}(\gamma(t) - t)$$
To ensure actuarial coherence and regulatory compliance, two constraints are imposed on $\gamma(t)$:

• (C1) Monotonicity: $\gamma(t)$ must be increasing (refund decreases as time passes).
• (C2) Non-negativity: $\rho(t) \geq 0$ (the premium charged for duration $t$ must not be less than the pro-rata benchmark).

The authors propose a smoothing parameter $a$ to blend the optimal (constrained) penalty with the pro-rata benchmark:
$$\gamma_{adj}(t, a) = a \cdot \gamma_{con}(t) + (1-a) \cdot t$$
This allows insurers to tune the severity of the penalty.

D. Model Selection Criteria

The authors compare models using:

• Deviance: Standard statistical fit measure.
• ABC (Area Between Curves): Derived from Concentration and Lorenz curves to assess local balance.
• Murphy Diagrams: Based on Bregman dominance to compare estimators across the entire class of loss functions.

3. Key Contributions

1. Generalization of Exposure Modeling: The paper moves beyond the rigid proportionality assumption ($\mu \propto t$) to a flexible function $\gamma(t)$, acknowledging that risk accumulation is not linear over time, especially for cancellations.
2. Theoretical Consistency: The authors prove that under the flexible specification, the parameter estimator is consistent even if the true mean is not proportional to exposure, whereas traditional estimators are only consistent if the true mean is strictly proportional.
3. Strategic Penalty Design: The framework provides a mathematical mechanism to design cancellation penalties that:
   • Indirectly recover costs from major losses (by penalizing the cancellation that often follows a total loss).
   • Lower the annual premium for full-year policyholders (by shifting cost recovery to cancellations).
4. Group-Specific Penalties: The paper extends the model to allow $\gamma(t)$ to vary by risk profile (specifically the Bonus-Malus Scale), showing that high-risk drivers should face steeper cancellation penalties than low-risk drivers.

4. Results

Using a dataset of over 2 million auto insurance contracts from a Canadian insurer:

• Empirical Evidence: Policyholders who cancel mid-term (XO contracts) have significantly higher claim frequencies and severities than those who stay (XX contracts). The relationship between exposure and loss cost is non-linear.
• Model Performance:
  • All flexible models (CWM, GWM, EWM) outperformed the traditional ratio approach in terms of deviance on both training and test sets.
  • The Exposure-Weighted Model (EWM) generally performed best, achieving the lowest deviance and best local balance (lowest Area Between Curves).
  • The flexible models showed no signs of overfitting (test deviance was comparable to training deviance).
• Penalty Impact:
  • Imposing the constraints (C1, C2) and using the optimal penalty ($a=100\%$) resulted in a model that recovered approximately 15% more premium from mid-term cancellers compared to the traditional model.
  • This allowed the insurer to reduce the annual premium for full-year policyholders by up to 25% for certain risk profiles, creating a competitive advantage.
• Group-Specific Findings: When splitting by Bonus-Malus Scale (BMS), the model revealed that new policyholders (BMS 100-104) require a much steeper penalty curve (front-loaded fees) compared to experienced, claim-free drivers (BMS 95-99).

5. Significance

• Actuarial Coherence: The framework resolves the tension between statistical accuracy (modeling non-linear risk) and actuarial principles (ensuring premiums cover expected costs).
• Competitive Strategy: It offers a strategic lever for insurers. By charging higher penalties for mid-term cancellations (which are often correlated with high-risk behavior or major claims), insurers can lower the headline annual premium for loyal, full-year customers, making them more competitive in the market.
• Regulatory Compliance: The approach respects transparency requirements by defining penalties upfront in the contract, rather than relying on opaque post-hoc adjustments.
• Behavioral Insight: The model captures the "behavioral" aspect of insurance, where the decision to cancel is endogenous to the risk profile and claim history, allowing for more precise risk segmentation.

In conclusion, the paper demonstrates that moving from a linear, proportional exposure model to a flexible, weighted Tweedie framework with structured cancellation penalties significantly improves risk classification, portfolio profitability, and market competitiveness.