hierCredibility: Hierarchical credibility model of Jewell

Description

Fit a random effects model, without contract-specific risk factors, using the hierarchical credibility model of Jewell.

Usage

hierCredibility(
  Yijkt,
  wijkt,
  sector,
  group,
  data,
  muHat = NULL,
  type = c("additive", "multiplicative"),
  returnData = FALSE
)

Value

An object of type hierCredibility with the following slots:

call: the matched call
type: Whether additive or multiplicative hierarchical credibility model is used.
Variances: The estimated variance components. s2 is the estimated variance of the individual contracts, tausq the estimate of \(Var(V[j])\) and nusq is the estimate of \(Var(V[jk])\).
Means: The estimated averages at the portfolio level (intercept term \(\mu\)), at the first hierarchical level (\(bar(Y)[\%.\% j \%.\% \%.\%]^z\)) and at the second hierarchical level (\(bar(Y)[\%.\% jk \%.\%]\)).
Weights: The weights at the first hierarchical level \(z[j\%.\%]\) and at the second hierarchical level \(w[\%.\%jk\%.\%]\).
Credibility: The credibility weights at the first hierarchical level \(q[j\%.\%]\) and at the second hierarchical level \(z[jk]\).
Premiums: The overall expectation \(widehat(\mu)\), sector expectation \(widehat(V)[j]\) and group expectation \(widehat(V)[jk]\).
Relativity: The estimated random effects \(widehat(U)[j]\) and \(widehat(U)[jk]\) of the sector and group, respectively.
RawResults: Objects of type data.table with all intermediate results.
fitted.values: the fitted mean values, resulting from the model fit.

Arguments

Yijkt: variable name of the response variable (the loss cost within actuarial applications).
wijkt: variable name of the exposure weight.
sector: variable name of the first hierarchical level.
group: variable name of the second hierarchical level that is nested within the first hierarchical level.
data: an object that is coercible by as.data.table, containing the variables in the model.
muHat: estimate for the intercept term. Default is NULL and in this case, the estimator as given in Ohlsson (2005) is used.
type: specifies whether the additive (Dannenburg, 1996) or multiplicative (Ohlsson, 2005) formulation of the hierarchical credibility model is used. Default is additive.
returnData: Logical, indicates whether the data object has to be returned. Default is FALSE.

References

Campo, B.D.C. and Antonio, Katrien (2023). Insurance pricing with hierarchically structured data an illustration with a workers' compensation insurance portfolio. Scandinavian Actuarial Journal, doi: 10.1080/03461238.2022.2161413

Dannenburg, D. R., Kaas, R. and Goovaerts, M. J. (1996). Practical actuarial credibility models. Amsterdam: IAE (Institute of Actuarial Science and Econometrics of the University of Amsterdam).

Jewell, W. S. (1975). The use of collateral data in credibility theory: a hierarchical model. Laxenburg: IIASA.

Ohlsson, E. (2005). Simplified estimation of structure parameters in hierarchical credibility. Presented at the Zurich ASTIN Colloquium.

Examples

Run this code

library(actuar)
library(actuaRE)
data("hachemeister", package = "actuar")
Df = as.data.frame(hachemeister)
X  = as.data.frame(cbind(cohort = c(1, 2, 1, 2, 2), hachemeister))
Df = reshape(X, idvar = "state", varying = list(paste0("ratio.", 1:12),
 paste0("weight.", 1:12)), direction = "long")
fitActuar  = cm(~ cohort + cohort:state, data = X, ratios = ratio.1:ratio.12,
weights = weight.1:weight.12, method = "Ohlsson")
fitActuaRE = hierCredibility(ratio.1, weight.1, cohort, state, Df)
summary(fitActuar)
summary(fitActuaRE)