# Insurance statistics

```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).
Text
D. ZAPPA, Appunti di Statistica Assicurativa - Introduzione ai modelli lineari generalizzati,
EDUCatt, Milan, 2006.
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.
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