Modern Actuarial Risk Theory - R. Kaas, et al., Kluwer, 2001.pdf - Książki(1) - marek33x

Modern Actuarial

Risk Theory

Rob Kaas

University of Amsterdam, The Netherlands

Marc Goovaerts

Catholic University ofLeuven, Belgium and

University of Amsterdam, The Netherlands

Jan Dhaene

Catholic University of Leuven, Belgium and

University of Amsterdam, The Netherlands

and

Michel Denuit

Université Catholique de Louvain, Belgium

KLUWER ACADEMIC PUBLISHERS

NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN:

0-306-47603-7

Print ISBN:

0-7923-7636-6

New York, Boston, Dordrecht, London, Moscow

Dordrecht

No part of this eBook may be reproduced or transmitted in any form or by any means, electronic,

mechanical, recording, or otherwise, without written consent from the Publisher

Created in the United States of America

Visit Kluwer Online at:

http://kluweronline.com

and Kluwer's eBookstore at:

http://ebooks.kluweronline.com

Foreword

Risk Theory has been identified and recognized as an important part of actuarial

education; this is for example documented by the Syllabus of the Society of

Actuaries and by the recommendations of the Groupe Consultatif. Hence it is

desirable to have a diversity of textbooks in this area.

I welcome the arrival of this new text in risk theory, which is original in several

respects. In the language of figure skating or gymnastics, the text has two parts, the

compulsory part and the free-style part. The compulsory part includes Chapters

1–4, which are compatible with official material of the Society of Actuaries.

This feature makes the text also useful to students who prepare themselves for

the actuarial exams. Other chapters are more of a free-style nature, for example

Chapter 10 ( Ordering of Risks, a speciality of the authors). And I would like to

mention Chapter 8 in particular: to my knowledge, this is the first text in risk theory

with an introduction to Generalized Linear Models.

Special pedagogical efforts have been made throughout the book. The clear

language and the numerous exercises are an example for this. Thus the book can

be highly recommended as a textbook.

I congratulate the authors to their text, and I would like to thank them also in the

name of students and teachers that they undertook the effort to translate their text

into English. I am sure that the text will be successfully used in many classrooms.

H.U. Gerber

Lausanne, October 3, 2001

Preface

This book gives a comprehensive survey of non-life insurance mathematics. It was

originally written for use with the actuarial science programs at the Universities

of Amsterdam and Leuven, but its Dutch version has been used at several other

universities, as well as by the Dutch Actuarial Society. It provides a link to the

further theoretical study of actuarial science. The methods presented can not only

be used in non-life insurance, but are also effective in other branches of actuarial

science, as well as, of course, in actuarial practice.

Apart from the standard theory, this text contains methods that are directly rele-

vant for actuarial practice, for instance the rating of automobile insurance policies,

premium principles and IBNR models. Also, the important actuarial statistical

tool of the Generalized Linear Models is presented. These models provide extra

features beyond ordinary linear models and regression which are the statistical

tools of choice for econometricians. Furthermore, a short introduction is given to

credibility theory. Another topic which always has enjoyed the attention of risk

theoreticians is the study of ordering of risks.

The book reflects the state of the art in actuarial risk theory. Quite a lot of the

results presented were published in the actuarial literature only in the last decade

of the previous century.

Models and paradigms studied

An essential element of the models of life insurance is the time aspect. Between

paying premiums and collecting the resulting pension, some decennia generally

vii

viii

PREFACE

elapse. This time-element is less prominently present in non-life insurance math-

ematics. Here, however, the statistical models are generally more involved. The

topics in the first five chapters of this textbook are basic for non-life actuarial

science. The remaining chapters contain short introductions to some other topics

traditionally regarded as non-life actuarial science.

1. The expected utility model

The very existence of insurers can be explained by way of the expected utility

model. In this model, an insured is a risk averse and rational decision maker, who

by virtue of Jensen’s inequality is ready to pay more than the expected value of

his claims just to be in a secure financial position. The mechanism through which

decisions are taken under uncertainty is not by direct comparison of the expected

payoffs of decisions, but rather of the expected utilities associated with these pay-

offs.

2. The individual risk model

In the individual risk model, as well as in the collective risk model that follows

below, the total claims on a portfolio of insurance contracts is the random variable

of interest. We want to compute, for instance, the probability that a certain capital

will be sufficient to pay these claims, or the value-at-risk at level 95% associated

with the portfolio, being the 95% quantile of its cumulative distribution function

(cdf). The total claims is modelled as the sum of all claims on the policies, which

are assumed independent. Such claims cannot always be modelled as purely dis-

crete random variables, nor as purely continuous ones, and we provide a notation

that encompasses both these as special cases. The individual model, though the

most realistic possible, is not always very convenient, because the available data

is used integrally and not in any way condensed. We study other techniques than

convolution to obtain results in this model. Using transforms like the moment

generating function helps in some special cases. Also, we present approximations

based on fitting moments of the distribution. The Central Limit Theorem, which

involves fitting two moments, is not sufficiently accurate in the important right-

hand tail of the distribution. Hence, we also look at two more refined methods

using three moments: the translated gamma approximation and the normal power

approximation.

3. Collective risk models

A model that is often used to approximate the individual model is the collective risk

model. In this model, an insurance portfolio is viewed as a process that produces

claims over time. The sizes of these claims are taken to be independent, identically

distributed random variables, independent also of the number of claims generated.

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