6 research outputs found
Bayesian Credibility for GLMs
We revisit the classical credibility results of Jewell and B\"uhlmann to
obtain credibility premiums for a GLM using a modern Bayesian approach. Here
the prior distributions can be chosen without restrictions to be conjugate to
the response distribution. It can even come from out-of-sample information if
the actuary prefers.
Then we use the relative entropy between the "true" and the estimated models
as a loss function, without restricting credibility premiums to be linear. A
numerical illustration on real data shows the feasibility of the approach, now
that computing power is cheap, and simulations software readily available
Computational Bayesian Methods for Insurance Premium Estimation
Bayesian Inference is used to develop a credibility estimator and a
method to compute insurance premium risk loadings. Algorithms to
apply both methods to Generalized Linear Models (GLMs) are provided.
We call our credibility estimator the entropic premium. It is
a Bayesian point estimator that uses the relative entropy as the loss
function. The risk measures Value-at-Risk (VaR) and
Tail-Value-at-Risk (TVaR) are used to determine premium risk
loadings. Our method considers the number of insureds and their
durations as random variables. A distribution to model the duration
of risks is introduced. We call it unifed, it has support on
the interval (0,1), it is an exponential dispersion family and it
can be used as the response distribution of a GLM
Property and Casualty Premiums based on Tweedie Families of Generalized Linear Models
We consider the problem of estimating accurately the pure premium of a property and casualty insurance portfolio when the individual aggregate losses are assumed to follow a compound Poisson distribution with gamma jump sizes. Generalized Linear Models (GLMs) with a Tweedie response distribution are analyzed as a method for this estimation. This approach is compared against the standard practice in the industry of combining estimations obtained separately for the frequency and severity by using GLMs with Poisson and gamma responses, respectively. We show that one important difference between these two methods is the variation of the scale parameter of the compound Poisson-gamma distribution when it is parametrized as an exponential dispersion model. We conclude that both approaches need to be considered during the process of model selection for the pure premium
Generalised linear models for aggregate claims; to Tweedie or not?
The compound Poisson distribution with gamma claim
sizes is a very common model for premium
estimation in Property and Casualty
insurance. Under this distributional assumption,
generalised linear models (GLMs) are used to
estimate the mean claim frequency and severity,
then these estimators are simply multiplied to
estimate the mean aggregate loss.
The Tweedie distribution allows to parametrise the
compound Poisson-gamma (CPG) distribution as a
member of the exponential dispersion family and
then fit a GLM with a CPG distribution for the
response. Thus, with the Tweedie distribution it
is possible to estimate the mean aggregate loss
using GLMs directly, without the need to
previously estimate the mean frequency and
severity separately.
The purpose of this educational note is to explore
the differences between these two estimation
methods, contrasting the advantages and
disadvantages of each
Generalised linear models for aggregate claims; to Tweedie or not?
The compound Poisson distribution with gamma claim
sizes is a very common model for premium
estimation in Property and Casualty
insurance. Under this distributional assumption,
generalised linear models (GLMs) are used to
estimate the mean claim frequency and severity,
then these estimators are simply multiplied to
estimate the mean aggregate loss.
The Tweedie distribution allows to parametrise the
compound Poisson-gamma (CPG) distribution as a
member of the exponential dispersion family and
then fit a GLM with a CPG distribution for the
response. Thus, with the Tweedie distribution it
is possible to estimate the mean aggregate loss
using GLMs directly, without the need to
previously estimate the mean frequency and
severity separately.
The purpose of this educational note is to explore
the differences between these two estimation
methods, contrasting the advantages and
disadvantages of each