actuar (version 0.1-3)

bstraub: Buhlmann-Straub Credibility Model

Description

bstraub calculates credibility premiums in the B�hlmann-Straub credibility model.

Usage

bstraub(ratios, weights,
        heterogeneity = c("iterative", "unbiased"),
        TOL = 1e-06, echo = FALSE)

Arguments

ratios

matrix of ratios (contracts in lines, years in columns)

weights

matrix of weights corresponding to ratios

heterogeneity

estimator of the between contract heterogeneity parameter used in premium calculation; "iterative" for the Bischel-Straub estimator; "unbiased" for the usual B�hlmann-Straub estimator (see below)

TOL

maximum relative error in the iterative procedure

echo

boolean, whether to echo iterative procedure or not

Value

• A list with the following components:
• premiums
  vector of credibility premiums
• individual
  vector of contract weighted averages
• collective
  collective premium estimator
• weights
  vector of contracts total weights, as used in credibility factors
• s2
  estimator of the within contract heterogeneity parameter
• unbiased
  unbiased estimator of the between contract heterogeneity parameter
• iterative
  iterative estimator of the between contract heterogeneity parameter

Estimation of a

The B�hlmann-Straub unbiaised estimator (heterogeneity = "unbiased") of the between contracts heterogeneity parameter is $$\hat{a} = c \left( \sum_{i = 1}^I w_{i\cdot} (X_{iw} - X_{ww})^2 - (I - 1)\hat{s}^2 \right),$$ where $c = w_{\cdot\cdot}/(w_{\cdot\cdot}^2 - \sum_{i = 1}^I w_{i\cdot}^2)$ and $I$ is the number of contracts. The Bishel-Straub pseudo-estimator (heterogeneity = "iterative") is obtained recursively as the solution of $$\hat{a} = \frac{1}{I - 1} \sum_{i=1}^I z_i (X_{iw} - X_{zw})^2.$$ The fixed point algorithm is used up, with a relative error of TOL stopping criteria.

Details

The credibility premium of contract $i$ is given by $$z_i X_{iw} + (1 - z_i) X_{zw},$$ where $$z_{i} = \frac{w_{i\cdot} \hat{a}}{w_{i\cdot} \hat{a} + \hat{s}^2},$$ $X_{iw}$ is the weighted average of the ratios of contract $i$, $X_{zw}$ is the weighted average of the matrix of ratios using credibility factors and $w_{i\cdot}$ is the total weight of a contract. $\hat{s}^2$ is the estimator of the within contract heterogeneity and $\hat{a}$ is the estimator of the between contract heterogeneity. Missing data are represent by NA in both the matrix of ratios and the matrix of weights. The function can cope with complete lines of NA in case a contract has no experience.

References

Goulet, V. (1998), Principles and Application of Credibility Theory, Journal of Actuarial Practice, Volume 6, ISSN 1064-6647. Goovaerts, M. J. and Kaas, R. and van Heerwaarden, A. E. and Bauwelinckx, T. (1990), Effective actuarial methods, North-Holland.

Examples

data(hachemeister)

## Credibility premiums calculated with the iterative estimator
bstraub(hachemeister$claims, hachemeister$weights)

## Credibility premiums calculated with the unbiased estimator
bstraub(hachemeister$claims, hachemeister$weights, heterogeneity = "unbiased")