Pricing principles

Insurance pricing is not only a statistical modelling exercise. It is the process of translating observed experience into a tariff that is:

statistically sound
commercially viable
stable over time
interpretable and explainable

This vignette outlines the key concepts that underpin pricing workflows in insurancerating.

Exposure

Exposure measures the amount of risk observed in the portfolio, typically as time under coverage.

In practice, exposure equals policy-years:

one year –> exposure = 1
six months –> exposure = 0.5

In motor insurance, this is often expressed as vehicle-years.

more exposure –> more credible observations
less exposure –> more volatile outcomes

Key principle
Pricing metrics are expressed per unit of exposure.

Frequency, severity, and risk premium

Insurance losses are typically decomposed into two components:

frequency: number of claims per unit of exposure
severity: average claim size

From these, the risk premium is derived:

risk premium = expected loss per unit of exposure

This can be written as:

risk premium = frequency × severity

or equivalently:

risk premium = total loss / exposure

The risk premium is also referred to as:

pure premium
burning cost

It represents the expected cost of claims, excluding expenses and margins.

Why this decomposition matters

Separating frequency and severity is useful because:

they are driven by different risk factors
they often require different model assumptions
they behave differently across segments

Typical modelling choices:

frequency → Poisson GLM
severity → Gamma GLM
risk premium → derived or modelled directly

In practice, both approaches are used:

separate frequency/severity models
or a direct burning cost model

From analysis to tariff

Pricing is not just about estimating expected losses. The process typically consists of four steps:

Exploration
Analyse risk factors and identify patterns in the data
Estimation
Fit statistical models (typically GLMs)
Refinement
Adjust coefficients to ensure:
- stability
- monotonicity
- commercial acceptability
Translation
Convert model output into a tariff structure

Key principle
The refinement step is where actuarial judgement plays a key role.

The role of factor analysis

Before fitting models, it is essential to understand the data. factor_analysis() provides a structured way to analyse:

frequency
severity
risk premium
exposure

Example:


library(insurancerating)

fa <- factor_analysis(
  MTPL,
  x = "zip",
  nclaims = "nclaims",
  exposure = "exposure",
  severity = "amount"
)

head(fa)
#> # A tibble: 4 × 7
#>   zip      amount nclaims exposure frequency average_severity risk_premium
#>   <fct>     <dbl>   <int>    <dbl>     <dbl>            <dbl>        <dbl>
#> 1 1     116178669    1593   11081.     0.144           72931.       10485.
#> 2 2      59751985    1008    7783.     0.130           59278.        7678.
#> 3 3      58988962    1038    7588.     0.137           56829.        7774.
#> 4 0        821510      29     207.     0.140           28328.        3972.

This helps to answer questions such as:

Are differences between segments credible?
Are there segments with low exposure?
Are patterns stable or driven by noise?

From model to tariff

GLMs are widely used in insurance pricing because they provide:

interpretable coefficients
multiplicative structure
compatibility with tariff construction

However, raw model output is rarely used directly.

Typical issues include:

non-monotonic patterns
volatility in low-exposure segments
overly granular differences

This is why refinement is essential.

Refinement includes:

smoothing coefficients
imposing monotonic trends
applying business constraints
incorporating expert judgement

The goal is not to improve statistical fit, but to create a tariff that is:

stable
explainable
commercially usable

In insurancerating, this is done through:


prepare_refinement(model) |>
  add_smoothing(...) |>
  add_restriction(...) |>
  refit()

Balancing model fit and usability

A key principle in pricing is:

The best statistical model is not always the best tariff.

Trade-offs include:

accuracy vs stability
granularity vs interpretability
statistical fit vs commercial constraints

For example:

a highly flexible model may overfit noise
a perfectly smooth tariff may ignore real risk differences

The role of the actuary is to balance these aspects.

Summary

Insurance pricing combines:

data analysis
statistical modelling
business judgement

Key concepts include:

exposure as the measure of risk volume
frequency and severity as building blocks of losses
risk premium as the core pricing metric

The goal is not only to model risk, but to translate it into a tariff that works in practice.

Next steps

For a practical introduction, see:

Getting started

For coefficient refinement:

Refinement workflow

Actuarial pricing philosophy

Insurance pricing is often presented as a modelling exercise. In practice, it is primarily a process of portfolio steering.

Models estimate expected losses. Tariffs determine which risks enter and remain in the portfolio.

Pricing as portfolio steering

A pricing model does not only describe risk — it influences it.

Higher premiums discourage certain risks
Lower premiums attract others

As a result, pricing decisions directly affect:

portfolio composition
future claims experience
overall profitability

This means pricing should always be considered in a forward-looking context.

Risk differentiation as a core principle

A central objective of pricing is risk differentiation:

higher-risk segments → higher premiums
lower-risk segments → lower premiums

Well-calibrated differentiation improves:

portfolio quality
predictability of results
alignment between price and risk

Poor differentiation leads to:

adverse selection
cross-subsidisation
unstable performance

Pure statistical output is rarely suitable for direct use in tariffs.

This is because:

data can be sparse in certain segments
models can capture noise instead of signal
coefficients may fluctuate across adjacent levels

Refinement introduces structure:

smoothing reduces volatility
monotonicity enforces logical consistency
restrictions incorporate business rules

The goal is not to “improve the model”, but to ensure:

the tariff behaves in a predictable and explainable way.

Stability over time

A good tariff is not only accurate today, but also stable over time. Large fluctuations between renewals can lead to:

poor customer experience
operational complexity
unintended portfolio shifts

This requires:

controlled updates
gradual changes
monitoring of portfolio impact

The role of expert judgement

Insurance pricing cannot be fully automated. Expert judgement is required to:

interpret model output
decide on appropriate smoothing
apply constraints based on business context
balance competing objectives

This is particularly important when:

exposure is low
historical data is not representative
external factors influence risk

Balancing objectives

Pricing involves multiple, often competing objectives:

statistical accuracy
commercial competitiveness
interpretability
operational simplicity

No single model optimises all dimensions. The role of the pricing framework is to make these trade-offs:

explicit
consistent
reproducible

Pricing as a controlled process

A structured workflow helps ensure that pricing decisions are:

transparent
auditable
consistent across portfolios

In insurancerating, this is reflected in:

separation of analysis, modelling, and refinement
explicit transformation steps
reproducible outputs

Final perspective

Insurance pricing is not about finding the “best model”. It is about constructing a tariff that:

reflects underlying risk
behaves predictably
supports portfolio objectives

Statistical models are a tool in that process — not the end goal.