Risk Theory: The Stochastic Basis of Insurance

206 Pages·1977·6.02 MB·English

by  Robert Eric Beard O.B.E., F.I.A., F.I.M.A., Teivo Pentikäinen Phil. Dr., Erkki Pesonen Phil. Dr. (auth.)

MONOGRAPHS
ON APPLIED PROBABILITY AND STATISTICS
General Editor8: M. S. BARTLETT, F.R.S.
and D. R. COX, F.R.S.
RISK THEORY
The Stochastic Basis
of Insurance
Risk Theory
THE STOCHASTIC BASIS
OF INSURANCE
R. E. BEARD, O.B.E., F.I.A., F.I.M.A.
London, England
T. PENTIKAINEN, Phil. Dr.
Helsinki, Finland
E. PESONEN, Phil. Dr.
Helsinki, Finland
SECOND EDITION
LONDON
CHAPMAN AND HALL
A Halsted Press book
John Wiley & Sons, New York
First published 1969
by Methuen & 00. Ltd
Second edition 1977
published by Ohapman and Hall Ltd
11 New Fetter Lane,
London E04P 4EE
© 1969, 1977 R. E. Beard,
T. Pentikiiinen, E. Pesonen
ISBN-13: 978-94-009-5783-1
All rights reserved. No part of
this book may be reprinted, or reproduced
or utilized in any form or by any electronic,
mechanical or other means, now known or hereafter
invented, including photocopying and recording,
or in any information storage or retrieval
system, without permission in writing
from the publisher
Distributed in the U.S.A. by Halsted Press,
a Division of John Wiley & Sons, Inc., New York
Library of Congress Cataloging in Publication Data
Beard, Robert Eric.
Risk theory.
(Monographs on applied probability a.nd statistics)
Bibliography: p.
Includes indexes.
1. Insurance - Mathematics. 2. Risk (Insurance)
I. Pentikainen, Teivo, joint author. II. Pesonen,
Erkki, joint author. III. Title.
HG8781.B34 1977 368'.001 '519 77-1637
ISBN-13: 978-94-009-5783-1 e-ISBN-13: 978-94-009-5781-7
DOl: 10.1007/978-94-009-5781-7
Softcover reprint of the hardcover 1st edition 1977
Contents
Preface to Second Edition page xi
Preface to First Edition. xiii
I. Definitions and Notations
1.1. The Purpose of the Theory of Risk I
1.2. Random Processes in General 3
1.3. Positive and Negative Risk Sums 3
1.4. Main Problems 5
2. Process with Constant Size of One Claim
2.1. Introduction 7
2.2. The Poisson Process 8
2.3. Discussion of Assumptions 8
2.4. Numerical Calculations 10
2.5. Application I 12
2.6. Application 2 15
3. Generalized Poisson Distribution
3.1. The Distribution Function of the Size of a Claim 18
3.2. Generalized Poisson Function 21
3.3. The Mean and Standard Deviation of F(x) 22
3.4. Characteristic Function 24
3.5. Estimation of S(z)
3.5.1. Individual Method 25
3.5.2. Statistical Method 27
3.5.3. Problems Arising from Large Olaims 29
3.5.4. The Dependence of the S-Function on Reinsurance 30
3.5.5. Analytical Methods 33
3.5.6. Exponential Function 34
3.5.7. A Generalization of the Exponential Type 36
3.5.8. Other Types of Distribution 37
3.6. Decomposition of S(z) 37
vii
CONTENTS
4. Normal Approximation and Edgeworth Series for F(x)
4.1. The Normal Approximation 41
4.2. Edgeworth Series 42
4.3. Normal Power Expansion 43
4.4. The Accuracy of the Normal Approximation 47
5. Applications of the Normal Approximation
5.1. The Basic Equation 52
5.2. Net Retention 54
5.3. Reserve Funds 58
5.4. Statutory Basis of Reserve Funds 63
5.5. The Rule of Greatest Retention 64
5.6. The Case of Several M's 65
5.7. An Application to Insurance Statistics 68
5.8. Experience Rating, Credibility Theory 69
6. The Esscher Approximation
6.1. Introduction 76
6.2. The Accuracy of the Esscher Formula 79
6.3. Some Hints for Numerical Computations 82
6.4. Examples of Numerical Applications 84
7. Monte Carlo Method
7.1. Random Numbers 91
7.2. Simulation of Generalized Poisson Function 94
7.3. Discussion on the Accuracy and a Modification 95
8. Other Methods of Calculating the Generalized Poisson
Function
8.1. Inversion of the Characteristic Function 98
8.2. A Modification of the Esscher Method 99
8.3. Step Function Approximation of S(z) 99
8.4. Exponent Polynomials 100
8.5. Mixed Methods 100
8.6. Statistical Method 101
9. Variance as a Measure of Stability
9.1. Optimum Form of Reinsurance 103
9.2. Reciprocity of Two Companies 106
viii
CONTENTS
10. Varying Basic Probabilities
10.1. Introduction 110
10.2. Compound Poisson Process 114
10.3. Direct Numerical Computation of the Compound
Poisson Function 121
10.4. The Polya Process 124
10.5. Application to Stop Loss Reinsurance 129
n.
The Ruin Probability During a Finite Time Period
11.1. The Ruin Function in Finite Time Periods 132
11.2. Calculation of lJ'N(U) by a Monte Carlo Method 134
12. The Ruin Probability During an Infinite Time Period
12.1. Introduction 137
12.2. Ruin Probability 138
12.3. Applications 147
! 2.4. Some Approximation Formulae 151
12.5. Discussion on the Different Methods 156
13. Application of Risk Theory to Business Planning 160
Appendix A. Derivation of the Poisson Process and Compound
Poisson Processes 167
Appendix B. The Edgeworth Expansion 173
Solutions to the Exercises 175
Bibliography 182
Author Index 187
Subject Index 188
IX
Preface to Second Edition
Since the publication of Risk Theory in 1969, there has been a
continued growth of interest in the subject. ASTIN, the section of
the International Actuarial Association concerned with the subject,
now has well over a thousand members and there are few actuarial
societies which do not include some aspects of risk theory in their
education and training. A number of Universities and technical
institutions now have courses of study and, on the application side,
the growth in the concept of risk management, namely the technique
of total financial management planning, has emphasized the im
portant part played by the theory of risk.
We have taken this opportunity to correct a number of misprints
which have come to light. Fortunately developments have not
invalidated the text as an elementary introduction to the subject,
but the opportunity has also been taken to rewrite Chapter 13 to
reflect the current direction of development in applications. Were
a new book being written, some changes of emphasis would be
appropriate and we have indicated these in the Preface, together
with additional references to avoid modification of the list and
alterations to the basic text.
In Chapter 5.4, dealing with the statutory requirements for
excess reserves above the usual provisions for unexpired risks and
outstanding claims, reasons were advanced that as regards fluctu
ations in risk this margin should be fixed in proportion to the square
root of the size of the business. The European Economic Community
have now issued a non-life directive setting out the rules for calcu
lation of the minimum solvency margin so that member countries
will be modifying their own legislation to conform to the provisions
of the directive.
The discussion on Credibility Theory in Chapter 5.8 was largely
concerned with the basic principles originally developed in the
U.S.A. for premium rating purposes. Over the past few years there
has been a considerable research in this subject but the extent to
Xl
PREFACE
whioh the developments are appropriate in an elementary text book
is open to doubt. Fortunately the proceedings of the conference
arranged by the Society of Actuaries Research Committee in
September 1974 provide an effective review of the ourrent position
(Credibility, Theory and Applications, Ed. P. M. Kahn, Academic
Press, 1975).
It is doubtful if any practical use is now made of the Esscher
approximation and the N-P method is much more convenient and
of adequate accuracy in most practical work. Thus the first half of
Chapter 6 is now largely of historical interest.
Chapter 11 dealing with ruin probability during a finite time
interval does not give an adequate view of the current importanoe
of this topic but the position is fluid because of the considerable
effort being expended in the search for practical methods of calcu
lation. Formulae are, in general, complicated and involve extensive
computer based quadratures or simulation techniques. The paper
by Seal in the Scandinavian Actuarial Journal (The Numerical
Calculation of U(w,t) the Probability of Non-ruin in an Interval
(O,t) 1974) gives a recent treatment and a fairly complete list of
relevant references.
In many countries studies are currently in progress in the develop
ment of models for business planning where the basic operations
involve a stochastic process. Not only are insurance companies
interested but in many commercial and industrial firms the needs
are significant so that a very large field exists for applications.
Chapter 13 has been recast to provide a natural starting point for
developments. This is largely based on recent work by Pentikiiinen,
and further references are included in that chapter.
As regards other publications, reference should be made to the
important text books by Biihlmann (Mathematical Methods iIi Risk
Theory, Springer, 1970) and by Seal (Stochastic Theory of a Risk
Business, Wiley, 1969) which provide more advanced treatment of
some of the topics in Risk Theory. Numerous papers have been
presented at the various colloquia organized by ASTIN and pub
lished in the ASTIN Bulletin. Further papers will also be found in
SAJ (Scandinavian Actuarial Journal) and in the Transactions of
the Casualty Actuarial Society.
Robert Eric Beard
London and Helsinki Teivo Pentikiiinen
August 1976 Erkki Pesonen
xii
Preface to First Edition
The theory of risk already has its traditions. A review of its classical
results is contained in Bohlmann's paper published in the transactions
of the International Congress of Actuaries, Vienna, 1909. This classical
theory was associated with life assurance mathematics and dealt
mainly with deviations, which were expected to be produced by
random fluctuations in individual policies. According to this theory,
these deviations are discounted to some initial instant; the square
root of the sum of the squares of the capital values calculated in this
way then gives a measure for the stability of the portfolio. A theory
constituted in this manner is not, however, very appropriate for
practical purposes. The fact is that it does not give an answer to such
questions as, for example, within what limits a company's probable
gain or loss will lie during different periods. Further, non-life
assurance, to which risk theory has, in fact, its most rewarding
applications, was mainly outside the risk theorists' interest. Thus
it is quite understandable that this theory did not receive very
much attention and that its applications to practical problems of
insurance activity remained rather unimportant.
A new phase of development began following the studies of Filip
Lundberg, which, thanks to H. Cramer, C. O. Segerdahl, and other
Swedish authors, has become generally known as the 'collective
theory of risk'. As regards questions of insurance the problem was
essentially the study of the progress of the business from a pro
babilistic point of view. In this form the theory has its applications
to non-life insurance as well as to life assurance. This new way of
expressing the problem has proved fruitful and the development of
the theory has since been continued by several other authors. In
recent years the fundamental assumptions of the theory, and thus
the range of its applications, have been significantly enlarged by the
use of more general probability models, which allow, for example,
for certain types of fluctuation in the basic probabilities.
xiii