premium_change: Interpret the premium effect of a smoothing curve

Description

Translate an effective smoothing curve in a refinement specification into concrete modelled-premium comparisons. By default, each selected value is compared with twice that value. Supplying increment instead compares each value with a fixed increment above it.

Format an object returned by premium_change(). One refinement state is shown as a three-column table. Multiple states are shown side by side; when exactly two are selected, their difference is added in percentage points.

Usage

premium_change(
  x,
  variable = NULL,
  at = NULL,
  change = "double",
  increment = NULL,
  steps = "current",
  basis = c("curve", "segments"),
  ...
)
# S3 method for premium_change
as_gt(x, locale = "en-US", decimals = 1, title = NULL, subtitle = NULL, ...)

Value

A tibble with class premium_change in long format, containing the variable, refinement state, starting and comparison values, evaluated relativities, and premium change as a decimal.

A gt_tbl object.

Arguments

x: For premium_change(), a rating_refinement object containing at least one smoothing step. For as_gt(), an object returned by premium_change().
variable: Optional character string identifying the smoothed model variable or its continuous source variable. This may be omitted when the refinement contains exactly one smoothing lineage.
at: Optional numeric vector of starting values. Each starting and comparison value must lie inside the supported smoothing range of every selected refinement state. Doubling retains the existing requirement that starting values are positive. If NULL, approximately six representative values are selected automatically.
change: Character comparison mode. "double" (default) compares \(x\) with \(2x\). When increment is supplied, omit change; fixed- increment mode is then selected automatically.
increment: Optional positive finite numeric increase in the units of the source variable. When supplied, compares \(x\) with \(x + increment\). It cannot be combined with an explicitly supplied change instruction.
steps: Refinement states to evaluate. Use "current" for the latest state, "all" for every state from the selected smoothing onwards, or a numeric vector such as c(1, 6) for stored refinement positions.
basis: Character string determining the interpretation basis. "curve", the default, evaluates the continuous effective smoothing at the exact values. "segments" compares the effective relativities of the tariff intervals containing those values.
...: Additional arguments are not accepted.
locale: Character string passed to gt for numeric formatting.
decimals: Non-negative integer. Number of decimal places for changes.
title: Optional table title. The source-variable name is used by default.
subtitle: Optional table subtitle.

Details

premium_change() is an interpretation helper for smoothing created with add_smoothing() and subsequently modified with edit_smoothing(). It is not a smoothing method and does not change the refinement specification.

For a multiplicative relativity curve \(R(x)\), doubling reports \(R(2x) / R(x) - 1\). Fixed-increment mode reports \(R(x+h) / R(x) - 1\), where \(h\) is increment. If total modelled premium can be written as \(P(x,z)=C(z)R(x)\), all other multiplicative model effects \(C(z)\) cancel in this ratio. No particular policy profile is therefore required for the interpretation.

The effective curve is reconstructed from the stored refinement history. Consequently, steps = "current" reflects all smoothing edits recorded up to the current state. Numeric step identifiers refer to positions in the complete refinement sequence. If another type of refinement occurs after a smoothing step, the previously effective smoothing is carried forward.

With the default basis = "curve", evaluation uses the continuous effective smoothing line retained by the refinement system. It therefore describes the shape and steepness of the estimated or edited curve at exactly \(x\) and the corresponding comparison value; it does not use neighbouring tariff-segment relativities.

With basis = "segments", both values are assigned to the effective tariff intervals created by the smoothing. Their current segment relativities are compared. This describes the premium effect of the implementable segmented tariff. The result can be zero when both values fall in the same segment and can change discretely when the comparison crosses a segment boundary.

Values are never extrapolated. When at = NULL, six representative starting values are selected from the common range for which both the starting and comparison values are supported in every selected refinement state.

Multiplying an entire curve by a common rebasing constant does not alter the result because that constant cancels in the relativity ratio.

Examples

Run this code

age <- rep(seq(20, 70, by = 5), each = 5)
portfolio <- data.frame(
  claims = rep(c(0, 1, 0, 2, 1), length(age) / 5),
  exposure = 1,
  age = age
)
portfolio$age_band <- cut(
  portfolio$age,
  breaks = c(15, 30, 45, 60, 75),
  include.lowest = TRUE
)
model <- glm(
  claims ~ age_band + offset(log(exposure)),
  family = poisson(),
  data = portfolio
)
refinement <- prepare_refinement(model, data = portfolio) |>
  add_smoothing(
    model_variable = "age_band",
    source_variable = "age",
    breaks = seq(15, 75, by = 5),
    smoothing = "poly",
    degree = 2,
    weights = "exposure"
  )
premium_change(refinement, at = c(20, 25, 30))
premium_change(refinement, at = c(20, 25, 30), increment = 5)
premium_change(refinement, at = c(20, 25, 30), basis = "segments")

edited <- refinement |>
  edit_smoothing(
    model_variable = "age_band",
    from = 20,
    to = 60,
    adjustment = 1.05,
    transition = "linear"
  )
premium_change(edited, at = c(20, 25, 30), steps = c(1, 2))