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Insurance statistics PROF. DIEGO ZAPPA COURSE AIMS Statistics plays a major role both in non-life and - to a lesser extent - life insurance. An example in non-life insurance is the increasingly widespread use of premium customization models based on the policy-holder's characteristics. Whereas in life insurance, an example is the use of models for projecting mortality tables. On this basis, the course presents the main models used by insurance companies to assess the estimated risk of claims and insurance premiums in general, using any available covariate data on policy-holders. After the necessary overview, commentary and explanation of why the well-known classical Gaussian linear models are not of significant interest in the insurance field, the course proceeds with the introduction of inherently non-linear models. Following this, the origin, use and assessment of generalized linear models (GLMs) are explained thoroughly. All the lectures take place in a computer laboratory, and alternate continuously between theory and practical applications to datasets taken from real cases in the insurance world. Students use MS Excel and especially the freeware application R, which is also increasingly used in firms. The R Rcmdr library is extensively explained. This includes - among other things - a module for inferencing GLMs through its glm function. COURSE CONTENT THINGS STUDENTS SHOULD KNOW BEFORE STARTING THE COURSE Before starting the course, students should at least know the elements of mathematics and statistics taught in first degree level courses, have learned how to estimate the parameters of a regression model and be familiar with the main probability models. Basics of linear algebra (concept of vector, matrix, product, determinant and inverse matrix) and - given the significant use of the tool adequate familiarity with a personal computer with Internet browser and MS Excel are useful so students may interpret results more easily. THINGS STUDENTS MAY LEARN DURING THE COURSE The course begins by presenting a summary of the origin, implications and interpretation of classical Gaussian linear models. This is followed by some useful extensions on the use of dummy variables, using different contrast matrices, and a presentation of Box-Cox transformations. After dealing with these subjects, methodological connections with (intrinsically) nonlinear models are illustrated. The above topics are necessary in order to introduce generalized linear models (GLMs). In particular, students are shown how it is possible to overcome many of the limitations of classical modelling by referring to the very general assumption of studying a random variable whose distribution belongs to the exponential family with p explanatory variables. In particular, the course deals with: – the choice of link function and linear predictor; – the weighted least squares algorithm; – inference of GLMs. After these topics, generalized models are discussed: normal, Poisson, gamma, Bernoulli/binomial (logistic), gamma-Poisson (negative binomial) and log-linear. After completing these topics, students should be able to construct models for: – estimating the risk of claims; – estimating the number of claims; – estimating the monetary value of a claim. This is essential in order to calculate customized premiums on the basis of available covariates. The presentation of the study topics is completed by application to datasets drawn from real cases. Time permitting, seminars will also take place, held by people who work in the field, on how GLMs are used by insurance companies in pertinent activities, especially non-life insurance (e.g. for calculating motor-vehicle liability insurance premiums in order to estimate provisions for claims). READING LIST Text D. ZAPPA, Appunti di Statistica Assicurativa - Introduzione ai modelli lineari generalizzati, EDUCatt, Milan, 2006. Complementary reading P. DE JONG-G.Z. HELLER, Generalized linear models for insurance data, University Press, Cambridge, 2008. A. DOBSON, AN INTRODUCTION TO GENERALIZED LINEAR MODELS, Chapman and Hall, 1990. I.B. HOSSACK-J.H. POLLARD - B. ZEHNWIRTH, Introductory statistics with applications in general insurance, Cambridge University Press, 1999, 2nd ed. TEACHING METHOD Lectures in the computer laboratory. Use of software: MS Excel and R. ASSESSMENT METHOD The examination consists of an oral test on a topic chosen by the student from the course syllabus. NOTES Further information can be found on the lecturer's webpage http://www2.unicatt.it/unicattolica/docenti/index.html or on the Faculty notice board. at