Derive candidate tariff segments from a smooth risk-factor effect

Approximate the smooth effect estimated by risk_factor_gam() with intervals for a continuous risk factor. The resulting boundaries provide a candidate categorical representation that can be inspected before inclusion in a pricing GLM or tariff structure.

Usage

derive_tariff_segments(
  object,
  segmentation_penalty = 0,
  seed = 1,
  max_iterations = 10000,
  population_size = 200,
  complexity = NULL,
  alpha = NULL,
  niterations = NULL,
  ntrees = NULL
)

Arguments

object: A "risk_factor_gam" object returned by risk_factor_gam(). Legacy "riskfactor_gam" and "fitgam" classes are accepted for compatibility.
segmentation_penalty: Non-negative numeric penalty on additional tree splits. Larger values generally favour fewer tariff segments. The default 0 retains the historical behaviour and applies no explicit split penalty; it can therefore produce a relatively detailed candidate segmentation. There is no universal actuarial value: compare candidate penalties and assess the resulting volume and stability by segment.
seed: Single finite whole number used to reproduce the evolutionary search.
max_iterations: Positive integer. Maximum number of evolutionary search iterations. This is an advanced algorithm-control parameter.
population_size: Positive integer. Number of candidate trees maintained during the evolutionary search. This is an advanced algorithm-control parameter.
complexity: Deprecated. Use segmentation_penalty instead.
alpha: Deprecated. Use segmentation_penalty instead.
niterations: Deprecated. Use max_iterations instead.
ntrees: Deprecated. Use population_size instead.

Value

A list of class "tariff_segments" with components:

gam_prediction: Data frame with the fitted GAM curve.
risk_factor: Name of the continuous risk factor.
model_type: Model type: "frequency", "severity", or "pure_premium".
classification_data: Data frame used to derive the segments.
risk_factor_values: Observed risk factor values in portfolio row order.
segment_boundaries: Numeric vector with segment boundaries.
assigned_segments: Factor with the tariff segment assigned to each observed risk factor value.
segment_summary: Data frame with portfolio counts, distinct risk-factor values and the observed response components for each candidate segment. Use summary() as the public interface for this table.
segmentation_penalty: Penalty applied to additional tree splits.

For backward compatibility, the old components prediction, x, model, data, x_obs, splits, class_boundaries, assigned_groups, and tariff_classes are also returned.

Details

Method

An evolutionary regression tree from evtree::evtree() is fitted to the predicted GAM effect over the distinct observed risk-factor values. The tree therefore approximates the estimated univariate curve; it is not fitted directly to individual claim outcomes or portfolio loss. Internal tree split points are translated into interval boundaries. If no internal split is supported by the fitted search, one interval spanning the observed range is returned.

The method follows the data-driven binning approach described by Henckaerts et al. (2018). segmentation_penalty, population_size, max_iterations and seed control the stochastic search rather than an actuarial minimum-volume rule. Reusing the same inputs and seed makes the result reproducible.

Each distinct observed risk-factor value has equal influence when the tree approximates the fitted curve. Exposure, claim count or another actuarial weight is deliberately not applied again in this step. The relevant portfolio information has already influenced the curve through the statistical specification used by risk_factor_gam(), such as the exposure offset in a frequency model, claim-count weights in a severity model or exposure weights in a risk-premium model. Applying a second weight during segmentation would introduce an additional portfolio-distribution choice after the GAM has been estimated.

Exposure and claim count remain available through summary(). They are diagnostics for assessing the support and practical stability of candidate segments, but they do not influence the estimated boundaries.

Actuarial interpretation

The returned segments approximate the shape of the fitted univariate GAM; they are not automatically a final tariff classification. Before use in a multivariate model, the boundaries should be assessed against exposure and claim volume, stability across periods, operational rounding and the interaction with other risk factors. Particular care is required for boundaries in sparsely populated tails.

summary() reports the number of portfolio records, number of distinct risk-factor values and available exposure and claim volume within each proposed segment. These diagnostics support actuarial review but do not constitute an automatic acceptance rule. Minimum-volume requirements and operational rounding should be selected with reference to portfolio size, model purpose and governance standards.

In practical pricing work, the candidate boundaries are often used to form an initial set of relatively broad model groups. The actuary reviews summary() and, where necessary, combines thinly populated segments or increases segmentation_penalty until the groups have sufficient exposure and claim information for stable estimation. The resulting factor can then be included in an unrestricted GLM.

This broad first-stage grouping avoids estimating a separate free GLM coefficient for every fine tariff interval when observations are unevenly distributed over the continuous risk factor. After fitting the GLM, add_smoothing() can use the broad model effect together with the original continuous variable to construct a regularised pattern over finer breaks. These finer breaks may reflect operational or commercial tariff boundaries, while their relativities remain linked through the smoothing specification rather than being estimated independently for every small segment.

The staged approach therefore separates statistical support from final tariff granularity: broad groups provide the information used by the GLM, while smoothing can translate that information into a finer and more regular tariff structure. Smoothing does not create additional observations, so the resulting classes should still be assessed for stability, extrapolation and commercial suitability.

The first and last boundaries equal the observed range used by the GAM. Applying the segmentation to new data outside that range results in an informative error rather than silent extrapolation.

Use autoplot.tariff_segments() to compare the smooth curve and boundaries. Use add_tariff_segments() to apply the resulting boundaries to portfolio data using the original continuous risk factor.

References

Antonio, K. and Valdez, E. A. (2012). Statistical concepts of a priori and a posteriori risk classification in insurance. Advances in Statistical Analysis, 96(2), 187–224. doi:10.1007/s10182-011-0152-7

Grubinger, T., Zeileis, A., and Pfeiffer, K.-P. (2014). evtree: Evolutionary learning of globally optimal classification and regression trees in R. Journal of Statistical Software, 61(1), 1–29. doi:10.18637/jss.v061.i01

Henckaerts, R., Antonio, K., Clijsters, M., & Verbelen, R. (2018). A data driven binning strategy for the construction of insurance tariff classes. Scandinavian Actuarial Journal, 2018(8), 681–705. doi:10.1080/03461238.2018.1429300

Wood, S.N. (2011). Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. JRSS B, 73(1), 3–36. doi:10.1111/j.1467-9868.2010.00749.x

Author

Martin Haringa

Examples

if (FALSE) { # \dontrun{
age_segments <- risk_factor_gam(
  MTPL,
  risk_factor = "age_policyholder",
  claim_count = "nclaims",
  exposure = "exposure"
) |>
  derive_tariff_segments(
    segmentation_penalty = 10,
    seed = 1
  )

autoplot(age_segments, show_observations = TRUE)
summary(age_segments)

MTPL |>
  add_tariff_segments(age_segments, name = "age_policyholder_segment")
} # }