Vincent Goulet

Credibility for Severity Revisited

Abstract It is basic actuarial knowledge that the pure premium of an insurance contract can be written as the product of the expected claim number and the expected claim amount. Actuaries use credibility theory to incorporate the contract’s individual experience into this calculation in a statistically optimal way. For many years, however, the use of credibility was limited to the frequency component. Starting with the paper by Hewitt (1971), there have been various suggestions as to how credibility theory also can be applied to the severity component of the pure premium. The latest such suggestion, Frees (2003), revived the interest in the problem. In this paper, we review four different formulas incorporating frequency and severity into credibility calculations. We then compare by simulation which one is most accurate at predicting a contract’s next-year outcome. It is found that the classical formula of Bühlmann (1967) is as good as the other ones in many cases. Alternatives, however, may offer easier analysis of the separate effects of frequency and severity on the premium. We also show that all the formulas reviewed in this paper stem from the same minimization problem, and we present a general, integrated, solution. At the same time, we complete Gerber (1972) by providing a proof to the main result of this paper and by stating required additional assumptions.

Simulation of Compound Hierarchical Models in R

Abstract Hierarchical probability models are widely used in insurance applications for data classified in a tree-like structure and in Bayesian inference. We propose an R function to simulate data from compound models in which both the frequency component and the severity component can have a hierarchical structure. The model description method is based solely on R expressions, and it allows for models with any number of levels and nodes per level, as well as with very general conditional probability structures. The function is part of the R package actuar.

On Parameter Estimation in Hierarchical Credibility

One can find in the literature three main sets of estimators for the variance components in the hierarchical credibility model. This paper presents these estimators in a unified notation, studies some of their properties important for numerical evaluation and compares their relative performance by simulation. The paper also demonstrates how function cm of the R package actuar can be used to fit hierarchical models to insurance data.

A generalized crossed classification credibility model

Abstract In the crossed classification credibility (CCC) model of Dannenburg [Crossed classification credibility models. In: Transactions of the 25th International Congress of Actuaries, Vol. 4, 1995, pp. 1–35], every contract in an insurance portfolio is assumed to be affected by the same number of risk factors. One can, however, imagine situations where this may be a restriction to the model. We thus propose a generalization of the original model where the number of risk factors can vary per contract. This is done by introducing a new category for each risk factor representing the fact of not being affected by the risk factor. The main effect on formulae is that credibility factors are replaced by some other function when used as weights.

Four-Bar Linkage Synthesis Using Non-convex Optimization

We show how four-bar linkages can be designed using non-convex optimization techniques. Our generative design software takes as input a curve that needs to be reproduced by a four-bar linkage and outputs the best assembly that approximates this curve. We model the problem using quadratic constraints and show how redundant constraints help to solve the problem. We also provide an algorithm that samples the curve based on its characteristics. Experiments show that our software is faster and more precise than existing systems. The current work is part of a larger generative design initiative at Autodesk Research.

A note on optimal parameter estimation under zero-excess assumptions

Abstract In the Buhlmann-Straub credibility model, optimal estimators (in the sense of minimum variance) for the variance parameters were proposed by De Vylder and Goovaerts [Insurance: Mathematics and Economics 11 (1992) 167–171] under zero excess assumptions for the involved random variables. For some results, supplementary restrictive conditions were needed. Using the variance components approach to the model, we show that one of these supplementary conditions can be dropped.

Computation of the Aggregate Claim Amount Distribution Using R and Actuar

actuar is a package providing additional Actuarial Science functionality to the R statistical system. This paper presents the features of the package targeted at risk theory calculations. Risk theory refers to a body of techniques to model and measure the risk associated with a portfolio of insurance contracts. The main quantity of interest for the actuary is the distribution of total claims over a fixed period of time, modeled using the classical collective model of risk theory.