Refinement building blocks

Introduction

In many pricing analyses, model estimation is followed by a translation step.

A fitted GLM may capture the structure of the portfolio well, while some fitted effects still need to be reviewed before they are used in a tariff.

Common reasons include:

irregular local variation
lack of monotonicity
externally imposed tariff structures
expert judgement not directly represented in the model
implementation constraints in policy administration systems

For this reason, actuarial pricing work often distinguishes between:

model estimation
tariff refinement
final refit of the pricing structure

insurancerating provides a staged refinement interface:

fit an unrestricted model
initialise a refinement object with prepare_refinement()
add one or more refinement steps
inspect these steps before refit
call refit() to obtain the final fitted model

This separation can make tariff adjustments easier to understand, reproduce, and audit.

When refinement can help

Refinement can help when the estimated model output is useful, but the fitted coefficient pattern needs additional structure before it is used in a tariff.

Typical use cases include:

smoothing a rating factor derived from a continuous variable
imposing monotonicity
restricting coefficients to a predefined relativity structure
introducing expert-based relativities within existing model levels
simplifying the final tariff for practical implementation

In many workflows, refinement is applied to the model that represents the final pricing signal, such as a premium or pure-premium model. In other cases, it may also be useful for selected frequency or severity effects. The relevant question is whether the adjusted coefficient pattern is intended to support the tariff structure that will be reviewed or implemented.

Example setup

The example below starts from one common premium modelling setup:

analyse a continuous variable with a GAM
convert it to tariff segments
fit frequency and severity models
combine both into a premium proxy
fit an unrestricted premium model


library(insurancerating)
library(dplyr)

age_policyholder_frequency <- risk_factor_gam(
  data = MTPL,
  claim_count = "nclaims",
  risk_factor = "age_policyholder",
  exposure = "exposure"
)

age_segments_freq <- derive_tariff_segments(age_policyholder_frequency)

dat <- MTPL |>
  add_tariff_segments(age_segments_freq, name = "age_policyholder_freq_cat") |>
  mutate(across(where(is.character), as.factor)) |>
  mutate(across(where(is.factor), ~ set_reference_level(., exposure)))

freq <- glm(
  nclaims ~ bm + age_policyholder_freq_cat,
  offset = log(exposure),
  family = poisson(),
  data = dat
)

sev <- glm(
  amount ~ zip,
  weights = nclaims,
  family = Gamma(link = "log"),
  data = dat |> filter(amount > 0)
)

premium_df <- dat |>
  add_prediction(freq, sev) |>
  mutate(premium = pred_nclaims_freq * pred_amount_sev)

burn_unrestricted <- glm(
  premium ~ zip + bm + age_policyholder_freq_cat,
  weights = exposure,
  family = Gamma(link = "log"),
  data = premium_df
)

Before refinement, inspect the unrestricted coefficient structure:


rating_table(burn_unrestricted)
#>          level               risk_factor est_burn_unrestricted exposure
#> 1  (Intercept)               (Intercept)          1.228041e+04       NA
#> 2            0                       zip          3.737317e-01      207
#> 3            1                       zip          1.000000e+00    11081
#> 4            2                       zip          7.574226e-01     7783
#> 5            3                       zip          7.325129e-01     7588
#> 6      [18,25] age_policyholder_freq_cat          1.895596e+00     1331
#> 7      (25,32] age_policyholder_freq_cat          1.301496e+00     3649
#> 8      (32,39] age_policyholder_freq_cat          1.053848e+00     4247
#> 9      (39,51] age_policyholder_freq_cat          1.000000e+00     7421
#> 10     (51,58] age_policyholder_freq_cat          8.491823e-01     3245
#> 11     (58,65] age_policyholder_freq_cat          7.258652e-01     2791
#> 12     (65,84] age_policyholder_freq_cat          7.584714e-01     3901
#> 13     (84,95] age_policyholder_freq_cat          5.131699e-01       72
#> 14          bm                        bm          9.980551e-01       NA

rating_table(burn_unrestricted) |>
  autoplot()

At this stage, the coefficients reflect the unrestricted model fit. This output is often informative by itself. If the pattern is too irregular, too granular or difficult to explain, a refinement step can be added explicitly.

The refinement object

Refinement begins with:


ref <- prepare_refinement(burn_unrestricted)
ref
#> <rating_refinement>
#> Base model: glm, lm
#> Steps: 0

A rating_refinement object stores:

the fitted base model
the underlying model data
the refinement steps added through the refinement interface

At this point, the model itself has not been refitted. The refinement object represents a proposed tariff adjustment structure, not yet the final fitted result.

This distinction is useful because refinement steps can be inspected before they are incorporated into the final model.

Smoothing

Purpose

Smoothing can be used when a rating factor derived from a continuous variable contains local variation that is hard to justify in a tariff.

For example, a coefficient pattern such as:

age 30–34 lower
age 34–38 higher
age 38–42 lower again

may be statistically possible, but difficult to explain or maintain. Smoothing adds a more stable structure to the rating factor.

Adding smoothing


ref <- ref |>
  add_smoothing(
    model_variable = "age_policyholder_freq_cat",
    source_variable = "age_policyholder",
    breaks = seq(18, 95, 5),
    weights = "exposure"
  )

The key arguments are:

model_variable: the grouped variable present in the GLM
source_variable: the original continuous portfolio variable
breaks: the preferred commercial cut points
smoothing: the smoothing specification
weights: optional weighting, typically exposure

Inspecting smoothing before refit


print(ref)
#> <rating_refinement>
#> Base model: glm, lm
#> Steps: 1
#> 1. smoothing [age_policyholder_freq_cat]
autoplot(ref, variable = "age_policyholder_freq_cat")

This plot belongs to the pre-refit stage. It shows:

the original fitted coefficients
the proposed smoothed structure

The purpose is to inspect the refinement step itself, before it is incorporated into the final fitted model.

Choosing a smoothing method

Typical smoothing choices are:

"spline": polynomial-style smoothing
"gam": flexible smooth curve
"mpi": monotone increasing
"mpd": monotone decreasing

The appropriate choice depends on the pricing context.

For example:

age may justify a flexible smooth
insured value or power may require a monotonic relationship
low-exposure tails may benefit from exposure weighting

Restrictions

Purpose

Restrictions can be used when coefficients need to follow a predefined structure.

Typical examples include:

bonus-malus systems
governance-approved relativities
externally mandated tariff structures
implementation constraints in policy systems

Restrictions differ from smoothing:

smoothing reshapes the fitted pattern
restriction imposes user-defined coefficients

Adding restrictions


zip_df <- data.frame(
  zip = c(0, 1, 2, 3),
  zip_adj = c(0.8, 0.9, 1.0, 1.2)
)

ref <- ref |>
  add_restriction(restrictions = zip_df)

The restriction table must contain exactly two columns:

the original factor levels
the adjusted coefficients

Inspecting restrictions before refit


autoplot(ref, variable = "zip")

This shows the proposed restricted structure relative to the original fitted model.

Expert-based relativities

Purpose

In some cases, the fitted model uses a broad factor level, while portfolio or business knowledge suggests that more granular differentiation may be useful.

For example, a model may estimate one coefficient for “construction”, while pricing practice distinguishes between:

residential construction
commercial construction
civil engineering

This can be relevant when subgroup exposure is too limited to estimate stable coefficients directly.

Adding relativities


relativities_activity <- relativities(
  split_level(
    "construction",
    c("residential_construction", "commercial_construction"),
    c(1.00, 1.15)
  )
)

ref <- ref |>
  add_relativities(
    model_variable = "business_activity",
    split_variable = "business_activity_split",
    relativities = relativities_activity,
    exposure = "exposure",
    normalize = TRUE
  )

If normalize = TRUE, the relativities are scaled so that their exposure-weighted average remains equal to 1 within the original level.

This preserves the original model signal while introducing finer structure.

Refit

Why refit is required

Refinement steps alter part of the model structure. Once these changes are applied, the remaining coefficients may also adjust.

For that reason, the sequence does not end with add_smoothing() or add_restriction(). The final step is:


burn_refined <- refit(ref)

This refits the model while incorporating the documented refinement steps.

Inspecting the final fitted result

After refit, use rating_table():


rating_table(burn_refined)
#>          level             risk_factor est_burn_refined exposure
#> 1  (Intercept)             (Intercept)     1.070848e+04       NA
#> 2            0                 zip_adj     8.000000e-01      207
#> 3            1                 zip_adj     9.000000e-01    11081
#> 4            2                 zip_adj     1.000000e+00     7783
#> 5            3                 zip_adj     1.200000e+00     7586
#> 6      [18,23] age_policyholder_smooth     1.985214e+00      586
#> 7      (23,28] age_policyholder_smooth     1.524645e+00     2204
#> 8      (28,33] age_policyholder_smooth     1.193787e+00     2790
#> 9      (33,38] age_policyholder_smooth     1.053848e+00     3021
#> 10     (38,43] age_policyholder_smooth     1.020715e+00     3089
#> 11     (43,48] age_policyholder_smooth     9.959711e-01     3041
#> 12     (48,53] age_policyholder_smooth     9.290435e-01     2978
#> 13     (53,58] age_policyholder_smooth     8.282178e-01     2186
#> 14     (58,63] age_policyholder_smooth     7.382691e-01     1974
#> 15     (63,68] age_policyholder_smooth     7.024490e-01     1973
#> 16     (68,73] age_policyholder_smooth     7.265697e-01     1558
#> 17     (73,78] age_policyholder_smooth     7.629301e-01      907
#> 18     (78,83] age_policyholder_smooth     7.318230e-01      246
#> 19     (83,88] age_policyholder_smooth     5.983689e-01       93
#> 20     (88,93] age_policyholder_smooth     5.224158e-01       11
#> 21          bm                      bm     9.977213e-01       NA

At this point, the output no longer represents a proposed refinement plan. It represents the fitted coefficient structure after refinement.

The distinction is:

before refit() –> inspect the refinement plan
after refit() –> inspect the fitted tariff structure

If smoothing, restrictions, and relativities have been applied, they are now embedded in the fitted model output.

Visualising the final structure


rating_table(burn_refined) |>
  autoplot()

Model data and rating grids

After refit, model structure can be extracted with extract_model_data():


md <- extract_model_data(burn_refined)
head(md)
#>   age_policyholder age_policyholder_freq_cat_smooth age_policyholder_smooth
#> 1               18                         1.985214                 [18,23]
#> 2               18                         1.985214                 [18,23]
#> 3               18                         1.985214                 [18,23]
#> 4               18                         1.985214                 [18,23]
#> 5               19                         1.985214                 [18,23]
#> 6               19                         1.985214                 [18,23]
#>   nclaims   exposure amount power bm zip age_policyholder_freq_cat
#> 1       1 1.00000000 261777    40  3   3                   [18,25]
#> 2       0 0.09589041      0    68  5   2                   [18,25]
#> 3       0 0.18630137      0    37  3   2                   [18,25]
#> 4       0 0.18904110      0    33  1   2                   [18,25]
#> 5       0 1.00000000      0    47  6   3                   [18,25]
#> 6       1 0.06849315   6642    68  1   3                   [18,25]
#>   pred_nclaims_freq pred_amount_sev   premium zip_adj
#> 1        0.26210773        68671.20 17999.251     1.2
#> 2        0.02502713        70854.51  1773.285     1.0
#> 3        0.04883103        70854.51  3459.898     1.0
#> 4        0.04975996        70854.51  3525.718     1.0
#> 5        0.26044368        68671.20 17884.979     1.2
#> 6        0.01802897        68671.20  1238.071     1.2

Observed model-point combinations can be obtained with rating_grid():


grid <- rating_grid(burn_refined)
head(grid)
#>   age_policyholder_smooth zip bm count   exposure zip_adj
#> 1                 (23,28]   1  1   414 342.578082     0.9
#> 2                 (23,28]   1  4    26  22.315068     0.9
#> 3                 (23,28]   0  4     1   1.000000     0.8
#> 4                 (23,28]   0  1     7   5.268493     0.8
#> 5                 (23,28]   1  7    33  28.334247     0.9
#> 6                 (23,28]   2 13     5   4.041096     1.0
#>   age_policyholder_freq_cat_smooth
#> 1                         1.524645
#> 2                         1.524645
#> 3                         1.524645
#> 4                         1.524645
#> 5                         1.524645
#> 6                         1.524645

This is typically used for:

tariff review
portfolio summaries
compact prediction input
implementation support

Complete example

One possible refinement sequence is:


zip_df <- data.frame(
  zip = c(0, 1, 2, 3),
  zip_adj = c(0.8, 0.9, 1.0, 1.2)
)

burn_refined <- prepare_refinement(burn_unrestricted) |>
  add_smoothing(
    model_variable = "age_policyholder_freq_cat",
    source_variable = "age_policyholder",
    breaks = seq(18, 95, 5),
    weights = "exposure"
  ) |>
  add_restriction(zip_df) |>
  refit()

rating_table(burn_refined)
#>          level             risk_factor est_burn_refined exposure
#> 1  (Intercept)             (Intercept)     1.070848e+04       NA
#> 2            0                 zip_adj     8.000000e-01      207
#> 3            1                 zip_adj     9.000000e-01    11081
#> 4            2                 zip_adj     1.000000e+00     7783
#> 5            3                 zip_adj     1.200000e+00     7586
#> 6      [18,23] age_policyholder_smooth     1.985214e+00      586
#> 7      (23,28] age_policyholder_smooth     1.524645e+00     2204
#> 8      (28,33] age_policyholder_smooth     1.193787e+00     2790
#> 9      (33,38] age_policyholder_smooth     1.053848e+00     3021
#> 10     (38,43] age_policyholder_smooth     1.020715e+00     3089
#> 11     (43,48] age_policyholder_smooth     9.959711e-01     3041
#> 12     (48,53] age_policyholder_smooth     9.290435e-01     2978
#> 13     (53,58] age_policyholder_smooth     8.282178e-01     2186
#> 14     (58,63] age_policyholder_smooth     7.382691e-01     1974
#> 15     (63,68] age_policyholder_smooth     7.024490e-01     1973
#> 16     (68,73] age_policyholder_smooth     7.265697e-01     1558
#> 17     (73,78] age_policyholder_smooth     7.629301e-01      907
#> 18     (78,83] age_policyholder_smooth     7.318230e-01      246
#> 19     (83,88] age_policyholder_smooth     5.983689e-01       93
#> 20     (88,93] age_policyholder_smooth     5.224158e-01       11
#> 21          bm                      bm     9.977213e-01       NA

rating_table(burn_refined) |>
  autoplot()

Legacy interface

Legacy entry points remain available:


burn_refined_old <- burn_unrestricted |>
  smooth_coef(
    x_cut = "age_policyholder_freq_man",
    x_org = "age_policyholder",
    breaks = seq(18, 95, 5)
  ) |>
  restrict_coef(zip_df) |>
  refit_glm()

These are primarily maintained for backward compatibility.

For new code, the recommended interface is:


prepare_refinement() |> add_*() |> refit()

This keeps the sequence of tariff adjustments explicit.

Refinement building blocks

Introduction

When refinement can help

Example setup

The refinement object

Smoothing

Purpose

Adding smoothing

Inspecting smoothing before refit

Choosing a smoothing method

Restrictions

Purpose

Adding restrictions

Inspecting restrictions before refit

Expert-based relativities

Purpose

Adding relativities

Refit

Why refit is required

Inspecting the final fitted result

Visualising the final structure

Model data and rating grids

Complete example

Legacy interface

Summary

Next steps