Add sublevel relativities to a refinement workflow

Divide one or more levels of an existing GLM risk factor into more detailed tariff levels using supplied relativities. This can be appropriate when the GLM is estimated on a coarser factor for statistical stability, while a documented actuarial segmentation is required within sufficiently homogeneous model levels.

Usage

add_relativities(
  model,
  model_variable,
  split_variable,
  relativities,
  exposure,
  normalize = TRUE,
  output_variable = paste0(model_variable, "_refined")
)

Arguments

model: Object of class rating_refinement, created with prepare_refinement(). A fitted GLM, including a model returned by refit(), is not accepted directly; retain and modify the corresponding refinement specification instead.
model_variable: Character string. Existing variable in the GLM, or a restricted version created by an earlier add_restriction() step. Levels of the underlying model variable can be split into more detailed tariff segments. When an earlier restriction exists, its coefficients are used automatically.
split_variable: Character string. More granular portfolio variable that defines the detailed groups inside model_variable.
relativities: Named list of data frames, usually created with relativities() and split_level().
exposure: Character string. Exposure column used for weighting and, when requested, normalisation.
normalize: Logical. If TRUE, normalise the supplied relativities by exposure within each split model level.
output_variable: Character string naming the resulting hybrid tariff factor. The default appends _refined to model_variable. A more application-specific name, such as sbi_tariff_segment, can make the intended tariff use clearer in model output and reporting. The name must not overwrite an existing column in the refinement data.

Value

A rating_refinement object containing the stored relativity specification. The pricing GLM is not fitted again until refit() is called.

Details

add_relativities() stores a relativity step on a rating_refinement object. It does not alter the fitted GLM immediately. The split is evaluated in the recorded step order and applied when refit() is called.

model_variable is the variable already used in the GLM. split_variable is the more detailed variable in the portfolio data that will be used to split one or more levels of model_variable. The relativities argument should be a named list describing those splits, usually built with relativities() and split_level(). output_variable names the resulting hybrid tariff factor: levels included in relativities are represented by their detailed split_variable level, while all other levels retain their model_variable level.

Levels of model_variable that are not included in relativities retain their existing model coefficient. In rating_table(), exposure for these retained levels is aggregated from model_variable, while exposure for the newly split levels is aggregated from split_variable within the specified parent model level. Omitting a model level from relativities therefore means that the level remains unsplit; it is not treated as an incomplete specification.

add_relativities() validates the supplied sublevel names against the observed values of split_variable before storing the refinement step. A misspelled or incorrectly spaced category or sublevel therefore produces an immediate error, with a suggestion when a closely matching observed value is available. It also verifies that each sublevel occurs within its specified parent category of model_variable.

When normalize = TRUE, the supplied relativities are normalised using exposure so that their exposure-weighted mean equals one within the split model level. They then redistribute the existing model coefficient across the sublevels without changing its exposure-weighted average. With normalize = FALSE, the supplied relativities are applied directly.

Step order and restrictions

If model_variable was restricted in an earlier add_restriction() step, the restricted coefficients are automatically used as the basis for the derived relativities. The user can continue to supply the original model variable; no additional argument is needed. Supplying the restricted variable explicitly gives the same coefficient basis and does not apply the restriction a second time. Refinement steps are order-dependent, so a restriction added after add_relativities() does not affect an earlier relativity step. Once the restricted coefficients have been used to derive the final split, rating_table() reports output_variable as the tariff factor and does not also show the intermediate restricted variable.

Conversely, add_restriction() can be called after add_relativities() to adjust selected levels of the derived output_variable. The output variable is then recognised as an existing refinement factor; users do not need to set allow_new_risk_factors = TRUE. Levels omitted from the restriction table are fixed at the relativities calculated by this step.

Appropriate use

add_relativities() is intended for refinement within an already reasonably homogeneous GLM segment. It redistributes an existing coefficient across sublevels using exposure-weighted relativities, while preserving the overall level of the original coefficient when normalisation is used. Appropriate applications include mild residual heterogeneity, monotonic tariff differentiation and expert-based segmentation within a stable risk group where the original GLM coefficient remains broadly representative.

Limitations

The method is not a substitute for creating a separate risk segment when the original GLM coefficient is itself distorted. For example, suppose a broad industry segment contains many relatively stable businesses, but a few chemical companies drive most of the losses while representing little exposure. The fitted industry coefficient may then be dominated by the chemical companies' experience. Applying exposure-weighted relativities inside that segment may barely reduce the coefficient for the large exposure group, because the original coefficient is already pulled upward by the outlier subgroup.

In that situation it is often better to create a separate GLM factor level, derive a separate tariff segment, or apply explicit segmentation or acceptation rules, instead of relying only on add_relativities().

Author

Martin Haringa

Examples

portfolio <- data.frame(
  claims = c(1, 2, 1, 3, 2, 4),
  exposure = rep(1, 6),
  construction = factor(c("residential", "commercial", "residential",
                          "commercial", "residential", "commercial")),
  construction_detail = factor(c("flat", "shop", "house",
                                 "office", "flat", "shop"))
)

model <- glm(
  claims ~ construction + offset(log(exposure)),
  family = poisson(),
  data = portfolio
)

relativities <- relativities(
  split_level(
    "residential",
    new_levels = c("flat", "house"),
    relativities = c(0.95, 1.05)
  ),
  split_level(
    "commercial",
    new_levels = c("shop", "office"),
    relativities = c(1.10, 0.90)
  )
)

refined <- prepare_refinement(model, data = portfolio) |>
  add_relativities(
    model_variable = "construction",
    split_variable = "construction_detail",
    output_variable = "construction_tariff_segment",
    relativities = relativities,
    exposure = "exposure"
  )

# A subsequent restriction can revise one derived level. The remaining
# tariff-segment levels are fixed at the relativities calculated above.
refined <- refined |>
  add_restriction(data.frame(
    construction_tariff_segment = "flat",
    construction_tariff_segment_restricted = 1.00
  ))

refined_model <- refit(refined)
rating_table(refined_model, exposure = FALSE)
#>                              risk_factor       level est_refined_model
#> 1                            (Intercept) (Intercept)         2.3746130
#> 2 construction_tariff_segment_restricted        shop         1.0645161
#> 3 construction_tariff_segment_restricted        flat         1.0000000
#> 4 construction_tariff_segment_restricted      office         0.8709677
#> 5 construction_tariff_segment_restricted       house         0.4745763