Pricing workflow building blocks
In insurancerating: Actuarial Tools for Insurance Pricing Models

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)

insurancerating provides building blocks for common actuarial pricing tasks in GLM-based tariff analysis. The package does not prescribe a single pricing method. Instead, it supports practical steps that often appear in insurance pricing work: portfolio analysis, model interpretation, tariff refinement and model validation.

This vignette gives a compact overview of those building blocks and how they can be combined.

library(insurancerating)

1. Start with portfolio experience

A pricing analysis often starts by checking how the observed portfolio behaves by risk factor. This is useful before modelling, but also later when reviewing whether fitted relativities are plausible.

factor_analysis() summarises exposure, claim frequency, average severity, risk premium and related metrics by one or more risk factors.

fa <- factor_analysis(
  MTPL,
  risk_factors = "zip",
  claim_count = "nclaims",
  claim_amount = "amount",
  exposure = "exposure"
)

head(fa)

The output helps answer practical questions such as:

where exposure is concentrated
whether observed differences are credible or noisy
whether a segment is driven by a small number of claims
which risk factors may need closer modelling or refinement

For numeric variables with long or skewed tails, outlier_histogram() can help inspect extreme observations before fitting severity models or constructing tariff segments.

outlier_histogram(
  MTPL2,
  x = "premium",
  upper = 100,
  density = FALSE
)

2. Assess large losses

Large claims can dominate severity and risk premium analysis. In capped severity workflows, it is often useful to assess a cap first, decompose the historical claim amounts, and then decide how the excess burden should be allocated. A low threshold increases pricing responsiveness but introduces volatility. A high threshold improves stability but may understate structural differences between segments.

portfolio <- data.frame(
  policy_id = 1:10,
  sector = rep(c("Industry", "Retail"), each = 5),
  claim_count = c(0, 1, 1, 1, 1, 0, 1, 1, 1, 1),
  claim_amount = c(
    0, 25000, 120000, 50000, 175000,
    0, 40000, 90000, 150000, 300000
  ),
  policy_years = rep(1, 10)
)

thresholds <- assess_excess_threshold(
  portfolio,
  claim_amount = "claim_amount",
  thresholds = c(25000, 50000, 100000, 150000),
  exposure = "policy_years",
  group = "sector",
  claim_count = "claim_count"
)

if (requireNamespace("gt", quietly = TRUE)) {
  as_gt(thresholds)
} else {
  thresholds
}

After choosing a threshold, redistribute_excess_loss() caps the large losses and allocates the excess burden. The same calculation supports two modelling workflows:

| Output | Typical redistribution weight | Interpretation | Typical use | |---|---|---|---| | "redistributed_claim" | Claim count or expected claim count | Excess loss redistributed across claims | One severity model | | "excess_loading" | Earned exposure | Excess risk premium per exposure unit | Separate component of total risk premium |

The total excess loss is preserved in both workflows.

adjusted <- redistribute_excess_loss(
  portfolio,
  claim_amount = "claim_amount",
  threshold = 100000,
  claim_count = "claim_count",
  risk_factor = "sector",
  redistribution_method = "partial",
  output = "redistributed_claim"
)

Portfolio redistribution gives every claim the same excess amount per claim. Risk-factor redistribution uses only the experience of each group. Partial redistribution balances portfolio stability, group responsiveness and the credibility of observed excess experience.

severity_data <- adjusted[adjusted$claim_count > 0, ]
severity_model <- glm(
  claim_amount_adjusted_average ~ sector,
  weights = claim_count,
  family = Gamma(link = "log"),
  data = severity_data
)

The adjusted average claim amount already contains the redistributed cost of large losses. A separate excess-premium component must therefore not be added again. This is a concise workflow and generally works well when modelled groups contain sufficient claims and sparse factor levels have been combined.

The amount allocated to an individual row is not an observed claim amount for that row. In a sparse sector, a severity model can therefore interpret allocated portfolio excess as evidence about that sector. Inspect claim volume, combine sparse levels using an actuarially meaningful hierarchy and validate coefficient stability over time. A minimum such as 20 claims can be a useful starting point, but is illustrative and must be selected for the portfolio at hand.

When retained severity and allocated excess should remain conceptually separate, use earned exposure as the redistribution weight:

loading_result <- redistribute_excess_loss(
  portfolio,
  claim_amount = "claim_amount",
  threshold = 100000,
  claim_count = "claim_count",
  redistribution_weight = "policy_years",
  risk_factor = "sector",
  redistribution_method = "partial",
  output = "excess_loading"
)

frequency_model <- glm(
  claim_count ~ sector + offset(log(policy_years)),
  family = poisson(link = "log"),
  data = loading_result
)
retained_severity_model <- glm(
  claim_amount_capped ~ sector,
  weights = claim_count,
  family = Gamma(link = "log"),
  data = loading_result[loading_result$claim_count > 0, ]
)

loading_result$predicted_frequency <-
  predict(frequency_model, type = "response") / loading_result$policy_years
loading_result$predicted_retained_severity <- predict(
  retained_severity_model,
  newdata = loading_result,
  type = "response"
)
loading_result$predicted_retained_risk_premium <-
  loading_result$predicted_frequency *
  loading_result$predicted_retained_severity
loading_result$predicted_total_risk_premium <-
  loading_result$predicted_retained_risk_premium +
  loading_result$excess_loading

Here excess_loading is an amount per policy year. Claim volume can still determine partial credibility while policy years determine both the allocation shares and the unit of the loading. This approach avoids treating allocated portfolio excess as an observed individual claim severity.

3. Translate continuous factors into tariff segments

Many tariffs use grouped versions of continuous variables such as age, vehicle age or insured value. risk_factor_gam() can be used to inspect the fitted shape of a continuous risk factor. derive_tariff_segments() can then derive candidate segment boundaries from that pattern.

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

age_segments <- derive_tariff_segments(age_gam)
age_segments

The derived segments can be added back to the portfolio with add_tariff_segments().

portfolio <- MTPL |>
  add_tariff_segments(age_segments, name = "age_policyholder_segment")

head(portfolio[, c("age_policyholder", "age_policyholder_segment")])

These functions are intended to support actuarial judgement, not replace it. Candidate segment boundaries should still be reviewed for credibility, stability and practical usability.

4. Fit and interpret a GLM

GLMs are widely used in insurance pricing because they provide an interpretable multiplicative structure. After fitting a model, rating_table() expresses the coefficients in tariff-table form.

portfolio$zip <- as.factor(portfolio$zip)

freq_model <- glm(
  nclaims ~ zip + age_policyholder_segment + offset(log(exposure)),
  family = poisson(),
  data = portfolio
)

rt <- rating_table(
  freq_model,
  model_data = portfolio,
  exposure = "exposure"
)

head(rt$df)

Observed portfolio experience can be attached to the rating table with add_portfolio_experience(). When data is supplied, the function calculates the observed experience for the rating-table risk factors automatically. This makes the comparison between model relativities and observed experience explicit.

rt |>
  add_portfolio_experience(
    data = portfolio,
    claim_count = "nclaims",
    exposure = "exposure"
  ) |>
  autoplot(risk_factors = "zip", metric = "frequency")

5. Refine tariff effects when needed

Raw model output may be statistically valid but still unsuitable for direct tariff use. Sparse levels, noisy estimates or non-monotonic adjacent effects can make a tariff hard to explain or maintain.

The refinement workflow makes these adjustments explicit:

refined_model <- prepare_refinement(freq_model) |>
  add_smoothing(
    model_variable = "age_policyholder_segment",
    source_variable = "age_policyholder",
    weights = "exposure"
  ) |>
  add_restriction(restrictions) |>
  refit()

Common refinement tasks include:

smoothing adjacent tariff levels
fixing selected coefficients to actuarial or commercial assumptions
applying sublevel relativities within a broader GLM factor level
refitting the model while preserving the intended tariff structure

These tools are most useful when the statistical model already captures the main risk structure and the remaining work is tariff refinement.

6. Validate model behaviour

Pricing models should be checked before their output is used in a tariff. insurancerating contains helpers for several common checks:

check_overdispersion() for Poisson frequency models
check_residuals() for simulation-based residual diagnostics using DHARMa
bootstrap_performance() for predictive stability with metrics such as RMSE
rating_grid() to inspect observed rating-grid combinations

For example:

check_overdispersion(freq_model)

check_residuals(freq_model) |>
  autoplot()

Validation does not make a tariff decision by itself. It gives evidence about model fit, stability and areas that may need further review.

Typical workflow

One possible workflow is:

Inspect the portfolio with factor_analysis() and outlier_histogram().
Assess large-loss thresholds with assess_excess_threshold() where capped severity or excess-loss loadings are relevant.
Use redistribute_excess_loss() either to create one redistributed severity response or to calculate a separate excess loading per exposure unit.
Analyse continuous risk factors with risk_factor_gam().
Create candidate tariff segments with derive_tariff_segments().
Fit GLMs for frequency, severity or pure premium.
Interpret coefficients with rating_table().
Compare fitted relativities with observed experience using add_portfolio_experience().
Apply refinement where needed with prepare_refinement(), add_smoothing(), add_restriction() or add_relativities().
Validate the resulting model with the model-performance helpers.