insurancerating (version 0.8.2)

add_shrinkage: Shrink categorical tariff relativities towards a common level

Description

Reduce differences between the relativities of one categorical risk factor before the refined GLM is fitted. add_shrinkage() combines each current relativity with a central level on the logarithmic scale. Extreme relativities move further in absolute terms, while their ordering is retained.

Usage

add_shrinkage(model, model_variable, credibility = 0.9, weights = NULL)

Value

A rating_refinement object containing an ordered shrinkage step. The returned object stores the original and adjusted relativities, level weights, inferred weight source and normalization information. The GLM is fitted only when refit() is called.

Arguments

model

A rating_refinement object created with prepare_refinement(). Shrinkage is applied to the current relativities at this point in the ordered refinement workflow.

model_variable

Character string naming the categorical risk factor to shrink. This may also identify a tariff factor created by an earlier add_relativities() step. After add_restriction(), use the first column of its restriction table; the second column is resolved internally.

credibility

Numeric scalar between 0 and 1. This is the weight given to the current risk-factor relativity. The remaining weight is assigned to the common centre. The default 0.9 retains 90 percent of the current logarithmic effect.

weights

NULL, "equal", or a character string naming a numeric, non-negative column in the refinement data. NULL derives the weighting basis from explicit model weights or a simple exposure offset. "equal" gives every level equal weight.

Author

Martin Haringa

Details

Shrinkage can be used when the direction of a fitted risk-factor pattern is credible, but the difference between its highest and lowest relativities is considered too large for the available experience or the intended tariff. It is a structured actuarial adjustment rather than a new statistical fit.

For level \(i\), the unnormalised adjusted relativity is

$$ \tilde{r}_i = \exp\{Z \log(r_i) + (1-Z)\log(c)\}, $$

where \(r_i\) is the current relativity, \(Z\) is credibility, and \(c\) is the weighted geometric centre. A credibility of 1 leaves the relativities unchanged. A credibility of 0 removes the differences between levels.

The adjusted relativities are subsequently rescaled so that their weighted arithmetic mean equals the weighted arithmetic mean before shrinkage. With portfolio weights such as exposure or claim count, this prevents shrinkage itself from changing the weighted level of the risk factor. The final GLM refit may still change the intercept or other fitted quantities; use audit_refinement() to assess that combined portfolio effect.

Weight selection

weights = NULL first uses explicit GLM weights when these were supplied during model fitting. Otherwise, a single column in an offset of the form log(column) is used. This commonly selects claim count for a weighted severity GLM and exposure for a frequency or risk-premium GLM. If neither source is unambiguous, the function asks for an explicit choice.

Set weights to a column name to control the basis directly. For example, exposure is generally appropriate for frequency or risk-premium relativities, while claim count is generally appropriate for severity relativities. Set weights = "equal" to give every risk-factor level the same weight. In that case the equal-level mean is preserved, which does not necessarily preserve the level of the observed portfolio.

Interpretation

credibility is a user-supplied refinement parameter. It should not be interpreted as an automatically estimated Buhlmann or Buhlmann-Straub credibility factor. Its value should be supported by portfolio stability, validation over time and the intended degree of tariff differentiation. The selected value and weighting basis are retained in the refinement specification and shown by summary().

Following a restriction

When shrinkage follows add_restriction(), model_variable remains the first column of the restriction table: the categorical risk factor whose levels are being adjusted. The second column is a numeric implementation column containing the fixed relativities used during refit(); it is not a separate categorical risk factor. add_shrinkage() resolves that internal column automatically from the stored restriction metadata.

See Also

prepare_refinement(), add_smoothing(), add_restriction(), add_rebasing(), add_relativities(), refit(), audit_refinement()

Examples

portfolio <- data.frame(
  claims = c(1, 2, 1, 3, 2, 4, 1, 5),
  exposure = c(1, 1, 1, 1, 2, 1, 1, 1),
  sector = factor(rep(c("Industry", "Office", "Retail", "Transport"), 2))
)

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

refinement <- prepare_refinement(model, data = portfolio) |>
  add_shrinkage(
    model_variable = "sector",
    credibility = 0.9,
    weights = "exposure"
  )

summary(refinement)
refined_model <- refit(refinement)
rating_table(refined_model)

# Use equal level weights explicitly when portfolio weighting is not wanted.
equal_level_refinement <- prepare_refinement(model, data = portfolio) |>
  add_shrinkage(
    model_variable = "sector",
    credibility = 0.8,
    weights = "equal"
  )