Description

Fit the following credibility models: Bühlmann, Bühlmann-Straub, hierarchical, regression (Hachemeister) or linear Bayes.

Usage

Arguments

Details

cm is the unified front end for credibility models fitting. The function supports hierarchical models with any number of levels (with Bühlmann and Bühlmann-Straub models as special cases) and the regression model of Hachemeister. Usage of cm is similar to lm for these cases. cm can also fit linear Bayes models, in which case usage is much simplified; see the section on linear Bayes below.

When not "bayes", the formula argument symbolically describes the structure of the portfolio in the form ~ terms. Each term is an interaction between risk factors contributing to the total variance of the portfolio data. Terms are separated by + operators and interactions within each term by :. For a portfolio divided first into sectors, then units and finally contracts, formula would be ~ sector + sector:unit + sector:unit:contract, where sector, unit and contract are column names in data. In general, the formula should be of the form ~ a + a:b + a:b:c + a:b:c:d + ....

If argument regformula is not NULL, the regression model of Hachemeister is fit to the data. The response is usually time. By default, the intercept of the model is located at time origin. If argument adj.intercept is TRUE, the intercept is moved to the (collective) barycenter of time, by orthogonalization of the design matrix. Note that the regression coefficients may be difficult to interpret in this case.

Arguments ratios, weights and subset are used like arguments select, select and subset, respectively, of function subset.

Data does not have to be sorted by level. Nodes with no data (complete lines of NA except for the portfolio structure) are allowed, with the restriction mentioned above.

Value

Function cm computes the structure parameters estimators of the model specified in formula. The value returned is an object of class cm.

An object of class "cm" is a list with at least the following components:

Regression fits have in addition the following components:

The method of predict for objects of class "cm" computes the credibility premiums for the nodes of every level included in argument levels (all by default). Result is a list the same length as levels or the number of levels in formula, or an atomic vector for one-level models.

Hierarchical models

The credibility premium at one level is a convex combination between the linearly sufficient statistic of a node and the credibility premium of the level above. (For the first level, the complement of credibility is given to the collective premium.) The linearly sufficient statistic of a node is the credibility weighted average of the data of the node, except at the last level, where natural weights are used. The credibility factor of node i is equal to

w[i]/(w[i] + a/b),

where w[i] is the weight of the node used in the linearly sufficient statistic, a is the average within node variance and b is the average between node variance.

Regression models

The credibility premium of node i is equal to

y' ba[i],

where y is a matrix created from newdata and ba[i] is the vector of credibility adjusted regression coefficients of node i. The latter is given by

ba[i] = Z[i] b[i] + (I - Z[i]) m,

where b[i] is the vector of regression coefficients based on data of node i only, m is the vector of collective regression coefficients, Z[i] is the credibility matrix and I is the identity matrix. The credibility matrix of node i is equal to

A^(-1) (A + s2 S[i]),

where S[i] is the unscaled regression covariance matrix of the node, s2 is the average within node variance and A is the within node covariance matrix.

If the intercept is positioned at the barycenter of time, matrices S[i] and A (and hence Z[i]) are diagonal. This amounts to use Bühlmann-Straub models for each regression coefficient.

Argument newdata provides the “future” value of the regressors for prediction purposes. It should be given as specified in predict.lm.

Variance components estimation

For hierarchical models, two sets of estimators of the variance components (other than the within node variance) are available: unbiased estimators and iterative estimators.

Unbiased estimators are based on sums of squares of the form

B[i] = sum(j; w[ij] (X[ij] - Xb[i])^2 - (J - 1) a)

and constants of the form

c[i] = w[i] - sum(j; w[ij]^2)/w[i],

where X[ij] is the linearly sufficient statistic of level (ij); Xb[i] is the weighted average of the latter using weights w[ij]; w[i] = sum(j; w[ij]); J is the effective number of nodes at level (ij); a is the within variance of this level. Weights w[ij] are the natural weights at the lowest level, the sum of the natural weights the next level and the sum of the credibility factors for all upper levels.

The Bühlmann-Gisler estimators (method = "Buhlmann-Gisler") are given by

b = mean(max(B[i]/c[i], 0)),

that is the average of the per node variance estimators truncated at 0.

The Ohlsson estimators (method = "Ohlsson") are given by

b = sum(i; B[i]) / sum(i; c[i]),

that is the weighted average of the per node variance estimators without any truncation. Note that negative estimates will be truncated to zero for credibility factor calculations.

In the Bühlmann-Straub model, these estimators are equivalent.

Iterative estimators method = "iterative" are pseudo-estimators of the form

b = sum(i; w[i] * (X[i] - Xb)^2)/d,

where X[i] is the linearly sufficient statistic of one level, Xb is the linearly sufficient statistic of the level above and d is the effective number of nodes at one level minus the effective number of nodes of the level above. The Ohlsson estimators are used as starting values.

For regression models, with the intercept at time origin, only iterative estimators are available. If method is different from "iterative", a warning is issued. With the intercept at the barycenter of time, the choice of estimators is the same as in the Bühlmann-Straub model.

Linear Bayes

When formula is "bayes", the function computes pure Bayesian premiums for the following combinations of distributions where they are linear credibility premiums:

• X|Θ ~ Poisson(Θ) and Θ ~ Gamma(α, λ);

• X|Θ ~ Exponential(Θ) and Θ ~ Gamma(α, λ);

• X|Θ ~ Gamma(τ, Θ) and Θ ~ Gamma(α, λ);

• X|Θ ~ Normal(Θ, σ_2^2) and Θ ~ Normal(μ, σ_1^2);

• X|Θ ~ Bernoulli(Θ) and Θ ~ Beta(a, b);

• X|Θ ~ Binomial(ν, Θ) and Θ ~ Beta(a, b);

• X|Θ = θ ~ Geometric(θ) and Θ ~ Beta(a, b).

• X|Θ ~ Negative Binomial(r, Θ) and Θ ~ Beta(a, b).

The following combination is also supported: X|Θ ~ Single Parameter Pareto(Θ) and Θ ~ Gamma(α, λ). In this case, the Bayesian estimator not of the risk premium, but rather of parameter θ is linear with a “credibility” factor that is not restricted to (0, 1).

Argument likelihood identifies the distribution of X|Θ = θ as one of "poisson", "exponential", "gamma", "normal", "bernoulli", "binomial", "geometric", "negative binomial" or "pareto".

The parameters of the distributions of X|Θ = θ (when needed) and Θ are set in ... using the argument names (and default values) of dgamma, dnorm, dbeta, dbinom, dnbinom or dpareto1, as appropriate. For the Gamma/Gamma case, use shape.lik for the shape parameter τ of the Gamma likelihood. For the Normal/Normal case, use sd.lik for the standard error σ_2 of the Normal likelihood.

Data for the linear Bayes case may be a matrix or data frame as usual; an atomic vector to fit the model to a single contract; missing or NULL to fit the prior model. Arguments ratios, weights and subset are ignored.

Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca, Xavier Milhaud, Tommy Ouellet, Louis-Philippe Pouliot

References

Bühlmann, H. and Gisler, A. (2005), A Course in Credibility Theory and its Applications, Springer.

Belhadj, H., Goulet, V. and Ouellet, T. (2009), On parameter estimation in hierarchical credibility, Astin Bulletin 39.

Goulet, V. (1998), Principles and application of credibility theory, Journal of Actuarial Practice 6, ISSN 1064-6647.

Goovaerts, M. J. and Hoogstad, W. J. (1987), Credibility Theory, Surveys of Actuarial Studies, No. 4, Nationale-Nederlanden N.V.

See Also

subset, formula, lm, predict.lm.

Examples