Boosting insights in insurance tariff plans with tree-based machine learning methods

1 Introduction

• The introduction describes insurance machine-learning research beyond GLMs as limited, with prior work addressing particular tasks such as claim frequency, loss cost, or fraud detection.Tree-based methods are also noted for discovering interaction effects that may inform insurers.Source: §1 · pp. 2–3
• Insurance pricing must be transparent, fair, and avoid extreme risk personalization while balancing customer segmentation with risk pooling.The introduction links poor reflection of portfolio heterogeneity to good risks lapsing and adverse selection.Source: §1 · p. 2
• Decision trees partition policyholders into observable risk groups with constant predictions, while ensembles can improve predictive performance at the cost of transparency.The paper uses interpretation tools to examine ensemble models and their decision processes.Source: §1 · p. 2
• Claim data combine excess-zero, exposure-varying counts for frequency with scarce, potentially long-tailed observations for severity, challenging standard machine-learning methods.The introduction notes that binary classification of claim frequency loses exposure information and information about policyholders with multiple claims.Source: §1 · p. 3
• The study develops full tariffs from both claim frequency and severity, using elaborate cross-validation and categorical, continuous, and spatial risk factors.It compares tree-based pricing with GLMs on statistical performance, interpretation, and business implications.Source: §1 · p. 3

2 State-of-the-art insurance pricing models

• Claim frequency is claims per unit of exposure and severity is average cost per claim; the technical price assumes independence between them.Alternatives that allow dependence are noted in the literature.Source: §2 · p. 3
• For frequency, the GLM can use a Poisson count response and an exposure offset; severity uses positive-claim observations and weights cases by claim count.The severity response may follow a right-skewed distribution such as gamma or log-normal.Source: §2 · p. 4
• The benchmark strategy fits smooth GAM effects, then bins continuous risk factors with decision trees and clusters spatial effects into postcode groups for a categorical GLM.The resulting interpretable GLM serves as the study’s benchmark pricing model.Source: §2 · p. 4, Figure 1 · p. 4

3.1 Algorithmic essentials

• Regression trees use recursive binary splits to partition the predictor space, with each leaf predicting a constant value; ensembles can improve accuracy by reducing a single tree’s variance.Random forests average independently grown trees while decorrelating them through random feature selection; boosting iteratively fits trees to pseudo-residuals, with shrinkage controlling update size.Source: §3.1 · pp. 5–7

3.2 Loss functions for insurance data

• Poisson deviance accommodates claim counts and exposure, while gamma deviance is the preferred loss for right-skewed, long-tailed claim severity.For frequency trees, leaf predictions use claims divided by exposure; severity predictions use total losses divided by claim counts.Source: §3.2 · p. 9

3.3 Tuning strategy

• The grid search selects tuning parameters, while hyperparameters set implementation constraints such as a minimum node size of 1% and 75% subsampling for ensembles.The shrinkage parameter is set to 0.01; a Poisson-rate prior adjustment is used only for frequency trees and forests to prevent zero-rate estimates in claim-free nodes.Source: §3.3 · p. 10
• The six-fold scheme stratifies data by claim frequency and severity, tunes parameters by five-fold cross-validation within each training split, and evaluates models on rotating holdout folds.Multiple test folds provide multiple performance measures, enable stability checks, and avoid dependence on one chosen test set.Source: §3.3 · p. 11, Figure 3 · p. 12

3.4 Interpretability matters: opening the black box

• Variable importance aggregates loss improvements at a feature’s splits, while partial dependence averages predictions across other features and individual conditional expectations can reveal heterogeneous effects.Interactions can make opposing individual effects cancel in a partial dependence plot, falsely suggesting no effect.Source: §3.4 · pp. 12–13

4.1 Quick scan of the MTPL data

• The 163,212-policyholder Belgian MTPL portfolio is zero-heavy, has mostly small but some very large claims, and includes varying exposure periods.These distributions motivate Poisson and gamma deviance rather than squared-error loss for frequency and severity, respectively.Source: §4.1 · p. 14, Figure 4 · p. 14

4.2 Optimal tuning parameters

• Frequency trees have 20–38 splits, while severity trees have only one or two, reflecting the smaller severity sample and harder prediction task.Across folds, random-forest tree counts are unstable, whereas boosting tree counts are more stable; five of six severity boosters use stumps, making them additive.Source: §4.2 · p. 15, Table 3 · p. 15

4.3 Model interpretation

• Frequency variable importance is similar across methods, especially for bonus-malus scale and age, whereas severity rankings diverge; boosting’s leading severity factors match GLM selections.Figure 5 compares regression trees, random forests, and gradient boosting across folds and by frequency versus severity.Source: §4.3 · p. 16, Figure 5 · p. 16
• Boosting captures smoother age and spatial risk effects than a single tree; Brussels remains the highest-risk area, and tree-based age effects flatten at the boundaries.For age, boosting predicts frequencies from 0.12 to 0.20, compared with 0.12 to 0.165 for the regression tree; the flat boundaries reduce extrapolation risk.Source: §4.3 · pp. 17–18, Figure 6 · p. 17, Figure 7 · p. 18
• Bonus-malus risk generally rises with scale, but some regression-tree individual effects decrease; the analysis does not enforce monotonicity constraints.Actuaries could address this with post-hoc smoothing or an implementation allowing constraints a priori.Source: §4.3 · pp. 18–19, Figure 8 · p. 19

4.4 Hunting for interaction effects

• For the frequency booster, longitude and latitude form the strongest reported interaction, while grouped plots expose changing effects across five equally sized groups.The H-statistic measures interaction strength but not its behavior; grouped partial-dependence curves are compared to inspect how effects change across groups.Source: §4.4 · pp. 19–20, Table 4 · p. 20, Figure 9 · p. 20
• The grouped plots show that vehicle-power effects are steepest for young policyholders and slightly higher for gasoline cars, while power risk also rises with bonus-malus level.For vehicle coverage, adding material-damage cover reduces frequency risk less for vehicles aged zero to three than for other age ranges.Source: §4.4 · pp. 20–21, Figure 9 · p. 20

4.5 Statistical out-of-sample performance

• The gradient boosting machine consistently has the lowest out-of-sample Poisson deviance for frequency, while severity methods show no clear winner across folds.The GAM’s severity deviance peaks when smooth effects must extrapolate beyond the training range; tree-based models’ flat edge regions avoid that issue.Source: §4.5 · pp. 21–22, Figure 10 · p. 22

5 Model lift: from analytic to managerial insights

• The analysis uses out-of-sample predictions for each policyholder; portfolio premiums replicate total losses on average, and separate folds yield the same model ranking.Table 5 presents predicted premium and observed-loss totals at portfolio and fold levels.Source: §5 · p. 22, Table 5 · p. 23

5.1 Tools to measure model lift

• Loss-ratio lift compares exposure-balanced bins: an upward trend indicates that the competing tariff better aligns risk, while a technically accurate benchmark should have ratios near 100%.The double lift chart directly compares two tariff structures.Source: §5.1 · p. 23
• The Gini index summarizes tariff differentiation: a larger value for the competing tariff is associated with a more profitable portfolio, and average profit equals half the index.The ordered Lorenz curve compares losses and premiums ordered by competing-to-benchmark premium relativities.Source: §5.1 · p. 24

5.2 Adverse selection and profits

• Gradient boosting minimizes the maximum Gini index across benchmarks, indicating the lowest adverse-selection risk; against the GLM, it implies around 3.3% average insurer profit.The profit is averaged over possible retention strategies based on relativities.Source: §5.2 · pp. 25–26, Table 6 · p. 26

5.3 Solidarity and risk differentiation

• Gradient boosting and GLM premiums have similar distributions: annual medians are €155 and €156, while half of relative differences fall within [−12%, +12%].Their similar overall premiums indicate a similar balance between differentiation and solidarity.Source: §5.3 · p. 26
• At around age 75, the gradient boosting tariff gives an average 30% discount over the GLM; its relative premiums also favor low-powered cars, while the GLM favors high-powered cars.Figure 13 compares relative premium differences by policyholder age and car power.Source: §5.3 · pp. 26–27, Figure 13 · p. 27

• The study develops complete real-life tariff plans with tree-based methods, using Poisson and gamma deviance, elaborate cross-validation, business metrics, and interpretability analysis.Source: §6 · p. 27
• The authors provide documented R functions for training, prediction, evaluation, cross-validation, interpretation, and economic-lift assessment, and extend rpart with additional deviance options.The extensions enable regression trees with gamma deviance and random forests with Poisson and gamma deviance.Source: §6 · p. 28
• The gradient boosting machine is consistently selected as best by out-of-sample performance and model-lift criteria.Regulatory issues such as filing requirements may prevent its direct deployment; an internal model could identify variables and interactions for a GLM, though building a competitive GBM-inspired GLM is left for future work.Source: §6 · p. 28

A List of variables in the MTPL data

• Table 7 describes the available variables in the MTPL data.Source: Table 7 · p. 31

B Search grid for the tuning parameters

• Table 8 gives the tuning-parameter search grid; γ is used only for frequency models with Poisson deviance.Source: Table 8 · p. 32

C Random forests for claim frequency data

• Across six frequency random forests with 100 to 5000 trees, age effects remain very similar, indicating that model structure does not change drastically once enough trees are included.Source: C Random forests for claim frequency… · p. 32, Figure 14 · p. 32
• Figure 14’s left and right panels show partial-dependence effects for policyholder age and municipality of residence, respectively.Source: Figure 14 · p. 32, C Random forests for claim frequency… · p. 32

D Filing requirements for pricing models

• Regulations may require insurers to file rating models on paper.The displayed frequency and severity models were trained with D3 held out.Source: D Filing requirements for pricing models · p. 33
• Sections D.1 and D.2 show GLMs and regression trees, while five other GLMs and regression trees are omitted for lack of space.Source: D Filing requirements for pricing models · p. 33
• Ensembles can be filed by printing individual trees, but the resulting pages are impractical and not very insightful for regulators.Source: D Filing requirements for pricing models · p. 33

D.1 GLM

• Table 9 presents the frequency GLM specification trained with D3 held out as the test set.Source: Table 9 · p. 33
• Both GLM specifications use data with D3 kept as the hold-out test set.Source: Table 9 · p. 33, Table 10 · p. 33
• Table 10 presents the severity GLM specification trained with D3 held out as the test set.Source: Table 10 · p. 33

D.2 Regression tree

• Figure 15 presents regression trees for claim frequency and severity.Source: Figure 15 · p. 34
• The frequency tree is shown large in the top-left, while the severity tree is small in the bottom-right.Source: Figure 15 · p. 34
• Both regression trees were trained with D3 kept as the hold-out test set.Source: Figure 15 · p. 34