Hierarchical credibility model of Jewell

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
)

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.

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.

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.http://www.actuaries.org/ASTIN/Colloquia/Zurich/Ohlsson.pdf

Examples

library(actuar)
#> 
#> Attaching package: 'actuar'
#> The following objects are masked from 'package:statmod':
#> 
#>     dinvgauss, pinvgauss, qinvgauss, rinvgauss
#> The following objects are masked from 'package:stats':
#> 
#>     sd, var
#> The following object is masked from 'package:grDevices':
#> 
#>     cm
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)
#> Call:
#> cm(formula = ~cohort + cohort:state, data = X, ratios = ratio.1:ratio.12, 
#>     weights = weight.1:weight.12, method = "Ohlsson")
#> 
#> Structure Parameters Estimators
#> 
#>   Collective premium: 1745.055 
#> 
#>   Between cohort variance: 88476.11 
#>   Within cohort/Between state variance: 11628.45
#>   Within state variance: 139120026 
#> 
#> Detailed premiums
#> 
#>   Level: cohort 
#>     cohort Indiv. mean Weight   Cred. factor Cred. premium
#>     1      1965.436    1.427755 0.9157058    1946.859     
#>     2      1527.011    1.633248 0.9255216    1543.250     
#> 
#>   Level: state 
#>     cohort state Indiv. mean Weight Cred. factor Cred. premium
#>     1      1     2060.921    100155 0.8932938    2048.750     
#>     2      2     1511.224     19895 0.6244749    1523.251     
#>     1      3     1805.843     13735 0.5344614    1871.491     
#>     2      4     1352.976      4152 0.2576359    1494.229     
#>     2      5     1599.829     36110 0.7511373    1585.748     
#> 
summary(fitActuaRE)
#> Call:
#> hierCredibility(Yijkt = ratio.1, wijkt = weight.1, sector = cohort, 
#>     group = state, data = Df)
#> 
#> 
#> Additive hierarchical credibility model
#> 
#> Estimated variance parameters:
#>   Individual contracts: 139120026 
#>   Var(V[jk]): 11628.45 
#>   Var(V[j]): 88476.11 
#> Unique number of categories of cohort: 2
#> Unique number of categories of state: 5 
#> 
#> Estimates at the cohort level:
#> 
#>    cohort       zj Yjz_BarTilde        qj       Vj        Uj
#> 1:      1 1.427755     1965.436 0.9157058 1946.859  201.8044
#> 2:      2 1.633248     1527.011 0.9255216 1543.250 -201.8044
#> 
#> Estimates at the state level:
#> 
#>    cohort state    wjk Yjk_BarTilde       zjk      Vjk       Ujk
#> 1:      1     1 100155     2060.921 0.8932938 2048.750 101.89107
#> 2:      1     3  13735     1805.843 0.5344614 1871.491 -75.36785
#> 3:      2     2  19895     1511.224 0.6244749 1523.251 -19.99963
#> 4:      2     4   4152     1352.976 0.2576359 1494.229 -49.02155
#> 5:      2     5  36110     1599.829 0.7511373 1585.748  42.49796

Usage

Arguments

Value

References

See also

Examples