Getting Started

Factor analysis

A pricing analysis commonly starts with a descriptive review of the portfolio.

Before fitting a model, it is useful to assess:

• how experience differs across factor levels
• whether observed differences are supported by sufficient exposure and claim volume
• whether the observed pattern is plausible

This is done with factor_analysis().

Basic factor analysis

We start by analysing a single risk factor.

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

fa
#>   zip    amount nclaims   exposure frequency average_severity risk_premium
#> 1   1 116178669    1593 11080.6274 0.1437644         72930.74    10484.846
#> 2   2  59751985    1008  7782.6301 0.1295192         59277.76     7677.608
#> 3   3  58988962    1038  7587.5644 0.1368028         56829.44     7774.427
#> 4   0    821510      29   206.8438 0.1402024         28327.93     3971.644

The output provides commonly used portfolio metrics such as:

• frequency = claims / exposure
• average severity = loss / claims
• risk premium = loss / exposure

Loss ratio and average premium can also be calculated when an earned-premium column is supplied. MTPL does not contain that quantity, so they are not used in this example.

Visualising factor behaviour

autoplot(fa, metrics = c("exposure", "frequency", "risk_premium"))

This provides a direct view of:

• the distribution of exposure
• the variation in claim frequency
• the variation in risk premium

These are descriptive, univariate results. They show the observed experience and volume by level, but do not control for correlations with other risk factors.

Why continuous variables are treated separately

Continuous variables can be modelled directly. In many traditional insurance tariffs, they are instead translated into a limited number of segments to improve stability, interpretation and implementation. This example uses the following sequence:

1. analysed as continuous variables
2. translated into tariff segments
3. used in a GLM as categorical rating factors

Grouping makes the resulting tariff effect discrete and directly implementable, but introduces a segmentation choice that should be reviewed.

Analysing the shape with a GAM

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

autoplot(age_freq, show_observations = TRUE)

This step is used to inspect:

• non-linear patterns
• local volatility
• areas with low exposure
• plausible breakpoints for tariff segments

Deriving tariff segments

age_segments <- derive_tariff_segments(age_freq)
autoplot(age_segments)

summary(age_segments)
#>   segment portfolio_records risk_factor_values   exposure claim_count
#> 1 [18,25]              1543                  8 1331.17534         348
#> 2 (25,32]              4254                  7 3648.72055         653
#> 3 (32,39]              4919                  7 4247.34795         615
#> 4 (39,51]              8366                 12 7421.35890        1009
#> 5 (51,58]              3594                  7 3245.45479         372
#> 6 (58,65]              3058                  7 2790.83288         272
#> 7 (65,84]              4181                 19 3900.75890         394
#> 8 (84,95]                85                 10   72.01644           5
#>    frequency
#> 1 0.26142311
#> 2 0.17896684
#> 3 0.14479624
#> 4 0.13595893
#> 5 0.11462184
#> 6 0.09746194
#> 7 0.10100599
#> 8 0.06942859

This derives candidate segment boundaries from the fitted continuous effect. Exposure and claim volume have already informed the GAM estimate. They are reported by summary() for review, but are not applied again when the fitted curve is divided into intervals. The evolutionary tree approximates the GAM curve rather than the individual claim outcomes. Each distinct risk-factor value on the fitted curve has equal influence on this approximation; exposure is not applied as a second weight.

The resulting segments should be reviewed against exposure, observed experience, stability and practical tariff requirements before they are used in a model.

Adding tariff segments to the data

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

set_reference_level() sets the reference level to the level with the highest exposure. This changes the coefficient parameterisation, not the fitted values. A high-exposure level is often a useful baseline because its relativity is supported by a substantial part of the portfolio.

Why GLMs are used

Generalized linear models are widely used in insurance pricing. They can:

• accommodate non-normal response distributions
• produce interpretable multiplicative effects
• be translated into tariff relativities

A common decomposition is:

• frequency –> Poisson GLM
• severity –> Gamma GLM

Frequency model

mod_freq <- glm(
  nclaims ~ age_cat,
  offset = log(exposure),
  family = poisson(),
  data = dat
)

The response is the observed claim count for each policy-period. Because log(exposure) is included as an offset, predict(mod_freq, type = "response") returns the expected number of claims for that record’s exposure. Dividing that prediction by exposure gives claim frequency per exposure-year.

Severity model

severity_data <- dat |>
  filter(nclaims > 0, amount > 0) |>
  mutate(average_claim_amount = amount / nclaims)

mod_sev <- glm(
  average_claim_amount ~ age_cat,
  weights = nclaims,
  family = Gamma(link = "log"),
  data = severity_data
)

amount is the total claim cost recorded for a portfolio row. Dividing by nclaims gives average claim severity. Claim count is then used as the model weight because a row containing several claims represents more severity observations than a row containing one claim. Frequency and severity are modelled separately because they describe different parts of the loss process.

Fitting a tariff representation

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

This second GLM is not statistically required after fitting frequency and severity. Here it is used as a tariff-construction step: it approximates the combined technical risk with a compact multiplicative structure based on age_cat and zip. Exposure weights give more influence to tariff cells that represent a larger part of the portfolio.

The response is a fitted technical target rather than an observed claim outcome. The model is therefore used to obtain an interpretable and implementable tariff representation, not as a replacement for the underlying frequency and severity analyses.

Rating table

rt <- rating_table(burn_unrestricted) |>
  add_portfolio_experience(
    data = premium_df,
    claim_count = "nclaims",
    exposure = "exposure",
    claim_amount = "amount",
    metric = "risk_premium"
  )
rt
#>    risk_factor       level est_burn_unrestricted exposure
#> 1  (Intercept) (Intercept)          8201.0782374       NA
#> 2      age_cat     [18,25]             1.9848879     1331
#> 3      age_cat     (25,32]             2.0914417     3649
#> 4      age_cat     (32,39]             1.0312559     4247
#> 5      age_cat     (39,51]             1.0000000     7421
#> 6      age_cat     (51,58]             0.5826148     3245
#> 7      age_cat     (58,65]             0.6059124     2791
#> 8      age_cat     (65,84]             0.7803242     3901
#> 9      age_cat     (84,95]             0.6199937       72
#> 10         zip           0             1.0000000      207
#> 11         zip           1             1.0000000    11081
#> 12         zip           2             1.0000000     7783
#> 13         zip           3             1.0000000     7588

rating_table() expresses fitted model effects as tariff relativities for the original factor levels, including the reference level. The attached portfolio experience makes it possible to compare those modelled relativities with the observed risk-premium pattern. Exposure remains important when judging whether an apparent difference is sufficiently credible.

Visualising coefficients

autoplot(rt, metric = "risk_premium")

This plot can be used to assess:

• the relative size of coefficients
• the structure across levels
• the exposure behind each level
• whether additional refinement may be needed

At this stage, the relevant questions are:

• are coefficients sufficiently stable?
• do they follow the expected pattern?
• are some levels driven by limited exposure?

Overdispersion

check_overdispersion(mod_freq)
#> Dispersion ratio =     1.187
#> Pearson's Chi-squared = 35590.309
#> p-value =   < 0.001
#> Overdispersion detected.

For a Poisson frequency model, a dispersion ratio materially above one can indicate remaining heterogeneity or an unsuitable variance assumption. The result should be considered together with residual and factor-level checks.

Model performance

model_performance(mod_freq)
#> # Comparison of Model Performance Indices
#> 
#>  Model   |   AIC    |    BIC    | RMSE  
#> ---------+----------+-----------+------ 
#> mod_freq | 22949.04 | 23015.512 | 0.362

This reports AIC, BIC and response-scale RMSE for the expected claim-count model. These measures are most meaningful when models use the same response, records, weights and offsets.

Bootstrap performance

bp <- bootstrap_performance(
  mod_freq,
  dat,
  n_resamples = 50,
  sample_fraction = 0.8,
  sampling = "bootstrap",
  show_progress = FALSE
)
autoplot(bp)

This refits the model on repeated bootstrap samples and evaluates RMSE on out-of-bag records. The resulting distribution describes sensitivity to the sampled portfolio records; it is not a prediction interval for future claims.

A single fit statistic is not sufficient. In pricing practice, the numerical results should be reviewed together with exposure, observed experience, residual diagnostics and stability of effects over time. The dedicated model validation vignette covers these checks in more detail.