Non-Life Insurance Mathematics

Transcription

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2 Erwin Straub Non-Life Insurance Mathematics With 12 Figures Springer-Verlag Berlin Heidelberg GmbH

3 Professor Dr. Erwin Straub Swiss Reinsurance Company Mythenquai 50/60 P.O.Box CH-8022 Zürich, Switzerland The illustration that appears on the front cover is described in detail on page 48. Mathematics Subject Classification (1980): 62P05 ISBN Library ofcongress Cataloging-in-Publication Data Straub, Erwin Non-life insurance mathematics/erwin Straub. p. cm. ISBN ISBN (ebook) DOI / lnsurance-Mathematics. I. Title HG8781.S '.01-dc CIP This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concemed, specifically those of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions ofthe German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act ofthe German Copyright Law. Springer-Verlag Berlin Heidelberg 1988 Originally published by Springer-Verlag Berlin Heidelberg New York in 1988 Softcover reprint of the hardcover 1 st edition 1988 Typesetting: Macmillan (lndia) Limited, Bangalore; Printed on acid-free paper

4 To the Reader The present book is dedicated to the late w. Leimbacher. It is based on lecture notes which were written between 1975 and 1985 when I taught Non-Life Mathematics at the University of Berne. It is meant to be a textbook on the one hand for university students with limited practical experience and on the other for practitioners knowing mathematics from so me distance only, but it is definitely too elementary and incomplete for a thoroughbred modern Non-Life Actuary. The main purpose is to provide insight into some pertinent practical problems and their possible theoretical solutions, for "there is nothing more practical than a good theory". I do not remember who said this so I cannot give a reference - by the way, references are only fragmentary; I have given only the ones I was most impressed by, the large remaining part of the contents was taken from different places and put together without mentioning the source. The whole thing is admittedly incomplete, imperfect and sometimes perhaps even inaccurate, both with regard to contents and deduction of results. There are three reasons for this: 1) to stir up the reader's mind and emotions to stimulate new work, 2) because all perfection, completeness and correctness is deadly boring, 3) because 1) and 2) are such handy excuses. I would like to express my thanks to the many who supported me in writing this, yet I confine myself to mentioning only a few: Swiss Re, who generously granted me money and time, Hans Bühlmann for being my exemplary teacher for many years, Adette Harnisch and Nicola Chappuis who typed all the twenty-seven versions fast and skillfully and last but not least my students in Berne, Belgrade and Strasbourg. Zürich, August 1988 Erwin Straub

5 Table of Contents Chapter 1. Problems Chapter 2. Tools The Model Distributions for K and X Moments The Total Claims Cost Z Cramer's Inequality Dependent Variables 42 Chapter 3. Premiums Pragmatic Principles Theoretical Principles Experience Rating Chapter 4. Reinsurance Chapter 5. Retentions Chapter 6. Statistics Chapter 7. Reserves Chapter 8. Solutions 8.1. Negative Binomial 8.2. Exact Credibility 8.3. Closing the Circle References Subject Index