MONOGRAPHS ON STATISTICS AND APPLIED PROBABILITY

General Editors

D.R. Cox, D.V. Hinkley, D. Rubin, B.W. Silverman

Stochastic Population Models in Ecology and Epidemiology M.S. Bartlett (1960)

2 Queues D.R. Cox and W.L. Smith (1961)

3 Monte Carlo Methods 1.M. Hammersley and D.C. Handscomb (1964)

4 The Statistical Analysis of Series of Events D.R. Cox and P.A. W. Lewis (1966)

5 Population Genetics W.J. Ewens (1969)

6 Probability, Statistics and Time M.S. Bartlett (1975)

7 Statistical Inference S.D. Silvey (1975)

8 The Analysis of Contingency Tables B.S. Everitt (1977)

9 Multivariate Analysis in Behavioural Research A.E. Maxwell (1977)

10 Stochastic Abundance Models S. Engen (1978)

11 Some Basic Theory for Statistical Inference E.J.G. Pitman (1979)

12 Point Processes D.R. Cox and V. Isham (1980)

13 Identification of Outliers D.M. Hawkins (1980)

14 Optimal Design S.D. Silvey (1980)

15 Finite Mixture Distributions B.S. Everitt and D.J. Hand (1981)

16 Classification A.D. Gordon (1981)

17 Distribution-free Statistical Methods 1.S. Maritz (1981)

18 Residuals and Influence in Regression R.D. Cook and S. Weisberg (1982)

19 Applications of Queueing Theory G.F. Newell (1982)

20 Risk Theory, 3rd edition R.E. Beard, T. Pentikainen and E. Pesonen (1984)

21 Analysis of Survival Data D.R. Cox and D. Oakes (1984)

22 An Introduction to Latent Variable Models B.S. Everitt (1984)

23 Bandit Problems D.A. Berry and B. Fristedt (1985)

24 Stochastic Modelling and Control M.B.A. Davis and R. Vinter (1985)

25 The Statistical Analysis of Compositional Data J. Aitchison (1986)

26 Density Estimation for Statistical and Data Analysis B. W. Silverman (1986)

27 Regression Analysis with Applications G.B. Wetherill (1986)

28 Sequential Methods in Statistics, 3rd edition G.B. Wetherill (1986)

29 Tensor Methods in Statistics P. McCullagh (19~7)

30 Transformation and Weighting in Regression R.J. Carroll and D. Ruppert (1988)

31 Asymptotic Techniques for Use in Statistics O.E. Barndoff-Nielsen and D.R. Cox (1989)

32 Analysis of Binary Data, 2nd edition D.R. Cox and E.J. Snell (1989)

33 Analysis of Infectious Disease Data N.G. Becker (1989)

34 Design and Analysis of Cross-Over Trials B. Jones and M.G. Kenward (1989)

35 Empirical Bayes Methods, 2nd edition J.S. Maritz and T. Lwin (1989)

36 Symmetric Multivariate and Related Distributions K.-T. Fang, S. Kotz and K. Ng (1989)

37 Generalized Linear Models, 2nd edition P.McCullagh and J.A. Neider (1989)

38 Cyclic Designs J.A. John (1987)

39 Analog Estimation Methods in Econometrics C.F. Manski (1988)

40 Subset Selection in Regression A.J. Miller (1990)

41 Analysis of Repeated Measures M. Crowder and D.J. Band (1990)

Risk Theory THE STOCHASTIC BASIS OF INSURANCE

R. E. BEARD O.B.E., F.I.A., F.I.M.A., PROFESSOR

Leicestershire, England

T. PENTIKAINEN PHIL. Dr, PROFESSOR h.c.

Helsinki, Finland

E. PESONEN PHIL. Dr

Helsinki, Finland

THIRD EDITION

CHAPMAN AND HALL

LONDON· NEW YORK· TOKYO· MELBOURNE· MADRAS

UK

USA

JAPAN

Chapman and Hall, II New Fetter Lane, London EC4P 4EE

Chapman and Hall, 29 West 35th Street, New York NYIOOOI

Chapman and Hall Japan, Thomson Publishing Japan, Hirakawacho Nemoto Building, 7F, 1-7-11 Hirakawa-cho, Chiyoda-ku, Tokyo 102

AUSTRALIA Chapman and Hall Australia, Thomas Nelson Australia, 480 La Trobe Street, PO Box 4725, Melbourne 3000

INDIA Chapman and Hall India, R. Sheshadri, 32 Second Main Road, CIT East, Madras 600 035

First edition 1969 Second edition 1977 Third edition 1984 Reprinted 1987, 1990

© 1969, 1977, 1984 R.E. Beard, T. Pentikiiinen, E. Pesonen ISBN 978-94-011-7682-8 ISBN 978-94-011-7680-4 (eBook) DOI 10.1007/978-94-011-7680-4

All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, or stored in any retrieval system of any nature, without the written permission of the copyright holder and the publisher, application for which shall be made to the publisher.

British Library Cataloguing in Publication Data

Beard, R.E. Risk theory: the stochastic basis of insurance.--3rd ed.-{Monographs on statistics and applied probability) I. Risks (Insurance) I. Title II. Pentikiiinen, T. III. Pesonen, E. IV. Series 368 HG8782

Lihrary of Congress Cataloging in Publication Data

Beard, R. E. (Robert Eric) Risk theory. (Monographs on statistics and applied probability) Bibliography: p. Includes indexes. I. Insurance-Mathematics. 2. Risk (Insurance)

3. Stochastic processes. I. Pentikiiinen, Teivo. II. Pesonen, E. (Erkki) III. Title. IV. Series. HG878J.B34 1984 368 83-25180

Contents

Preface IX

Nomenclature XIV

1 Definitions and notation 1 1.1 The purpose of the theory of risk I 1.2 Stochastic processes in general 5 1.3 Positive and negative risk sums 6 1.4 Main problems 7 1.5 On the notation 11 1.6 The moment generating function, the character-

istic function, and the Laplace transform 15

2 Claim number process 18 2.1 Introduction 18 2.2 The Poisson process 19 2.3 Discussion of conditions 20 2.4 Some basic formulae 22 2.5 Numerical values of Poisson probabilities 23 2.6 The additivity of Poisson variables 27 2.7 Time-dependent variation of risk exposure 29 2.8 Formulae concerning the mixed Poisson

distribution 35 2.9 The Polya process 38 2.10 Risk exposure variation inside the portfolio 43

3 Compound Poisson process 47 3.1 The distribution of claim size 47 3.2 Compound distribution of the aggregate claim 50 3.3 Basic characteristics of F 52

vi CONTENTS

3.4 The moment generating function 59 3.5 Estimation of S 60

3.5.1 Individual method 60 3.5.2 Statistical method 62 3.5.3 Problems arising from large claims 65 3.5.4 Analytical methods 67 3.5.5 Exponential distribution 68 3.5.6 Gamma distribution 69 3.5.7 Logarithmic-normal distribution 72 3.5.8 The Pareto distribution 74 3.5.9 The two-parametric Pareto and the quasi-

log-normal distributions 76 3.5.10 The family of Benktander distributions 79 3.5.11 Other types of distribution 83

3.6 The dependence of the S function on reinsurance 84 3.6.1 General aspects 84 3.6.2 Excess of loss reinsurance- 85 3.6.3 Quota share reinsurance 85 3.6.4 Surplus reinsurance 88 3.6.5 Technique using the concept of degree of loss 90

3.7 Decomposition of the portfolio into sections 94 3.8 Recursion formula for F 100 3.9 The normal approximation 104

3.10 Edgeworth series 107

3.11 Normal power approximation 108

3.12 Gamma approximation 121

3.13 Approximations by means offunctions belonging to the Pearson family 123

3.14 Inversion of the characteristic function 124 3.15 Mixed methods 124

4 Applications related to one-year time-span 126 4.1 The basic equation 126 4.2 Evaluation of the fluctuation range of the annual

underwriting profits and losses 130 4.3 Some approximate formulae 138 4.4 Reserve funds 142 4.5 Rules for the greatest retention 145

CONTENTS vii

4.6 The case of several Ms 152 4.7 Excess of loss reinsurance premium 154 4.8 Application to stop loss reinsurance 156 4.9 An application to insurance statistics 159 4.10 Experience rating, credibility theory 162

5 Variance as a measure of stability 171 5.1 Optimum form of reinsurance 171 5.2 Reciprocity of two companies 175 5.3 Equitability of safety loadings: a link to theory of

multiplayer games 179

6 Risk processes with a time-span of several years 183 6.1 Claims 183 6.2 Premium income P(l, t) 198 6.3 Yield of investments 205 6.4 Portfolio divided in sections 211 6.5 Trading result 214 6.6 Distribution of the solvency ratio u 220 6.7 Ruin probability 'I'T(U), truncated convolution 227 6.8 Monte Carlo method 233

6.8.1 Random numbers 233 6.8.2 Direct simulation of the compound

Poisson function 239 6.8.3 A random number generator for the

cycling mixed compound Poisson variable X(t) 241

6.8.4 Simulation of the solvency ratio u(t) 245 6.9 Limits for the finite time ruin probability 'P T 250

7 Applications related to finite time-span T 258 7.1 General features of finite time risk processes 258 7.2 The size of the portfolio 263 7.3 Evaluation of net retention M 265 7.4 Effect of cycles 266 7.5 Effect of the time-span T 266 7.6 Effect of inflation 267 7.7 Dynamic control rules 272 7.8 Solvency profile 278

viii CONTENTS

7.9 Evaluation of the variation range of u( t) 281 7.10 Safety loading 284

8 Risk theory analysis of life insurance 288 8.1 Cohort analysis 288 8.2 Link to classic individual risk theory 292 8.3 Extensions of the cohort approach 295 8.4 General system 300

9 Ruin probability during an inrmite time period 308 9.1 Introduction 308 9.2 The infinite time ruin probability 311 9.3 Discussion of the different methods 315

10 Application of risk theory to business planning 319 10.1 General features of the models 319 10.2 An example ofrisk theory models 322 10.3 Stochastic dynamic programming 330 10.4 Business objectives 336 10.5 Competition models 345

Appendixes 349 A Derivation ofthe Poisson and mixed Poisson processes 349 B Edgeworth expansion 355 C Infinite time ruin probability 357 D Computation of the limits for the finite time ruin

probability according to method of Section 6.9 367 E Random numbers 370 F Solutions to the exercises 373

Bibliography 396

Author index 403

Subject index 405

Preface

The theory of risk already has its traditions. A review of its classical results is contained in Bohlmann (1909). This classical theory was associated with life insurance 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 insurance, to which risk theory has, in fact, its most rewarding applications, was mainly outside the field of interest of the risk theorists. 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 (1909, 1919), which, thanks to H. Cramer (1926), e.O. Segerdahl and other Swedish authors, became generally known as the 'collective theory of risk'. As regards questions of insurance, the problem was essentially the study ofthe progress of the business from a probabilistic point of view. In this form the theory has its applica­tions to non-life insurance as well as to life insurance. This new way of expressing the problem has proved fruitful. In recent years the fundamental assumptions of the theory, and the range of applica­tions, have been significantly enlarged. The advancement of the general theory of stochastic processes and its numerous sub­branches and applications has been reflected in the development of risk theory. The explosive development of computers has made it

x PREFACE

feasible to treat problems which previously could not be handled because of their complicated structure. For example, it is now possible to create models to describe the insurance business as a whole, and the interactions between its sectors, instead oflimiting the consideration to isolated sub-problems like the range of risk fluctuations, reinsurance, safety loadings, reserve funds, and so on.

Today the theory of risk generates an interesting and far-reaching field for research. The development of the theory is still far from complete, as is demonstrated by the many papers which continue to be published on the subject. A defect, as in so many other new and rapidly developing branches of human knowledge, is that the theory has become difficult for practising actuaries to follow. This is regrettable, because a knowledge of this theory deepens actuarial intuition and helps towards an understanding of the insurance business as a process characterized by varying progress and fluctua­tions from year to year. The modern theory of risk can also give an actuary concrete assistance in the form of practical applications. It is true that many problems in this field, for example, problems of a company's solvency, reinsurance requirements, safety loadings in premiums and many others, are such that risk theory alone is incapable of providing a definite solution. This is because in practical work it is often necessary to take into consideration many aspects with which risk theory is not competent to deal. In reinsurance arrangements, for example, attention has to be given to many political aspects of insurance such as reserves, reciprocity and liquid­ity. In spite ofthis, when choosing a form of reinsurance and calculat­ing suitable net retentions and safety loadings, and in business planning in general, risk theory provides effective tools to estimate the fluctuations in the business retained by a company; such fluctuations should obviously always be kept within the limits of the company's resources. Thus the theory of risk can facilitate important considerations of financial interest and be useful in making final decisions.

To disseminate knowledge of the theory of risk it seemed essential to provide an introduction to the theory based upon the elements of probability theory which form part of actuarial study and which provide some of the basic ideas concerning risk theory. Furthermore, there is a need for a summary of the results of the present theory, easily available for practical application.

For this reason, one of the authors, Pentikiiinen, published an

PREFACE xi

elementary textbook of risk theory in the Finnish language in 1955, primarily designed for the use of Finnish actuaries as an introduction to the theory. Many participants at meetings of ASTIN (Actuarial Studies in Non-Life Insurance- a section of the Inter­national Actuarial Association) expressed a wish for a concise book of this kind in English and devoted primarily to practical applica­tions. The authors attempted to produce such a book. The first edition was completed in 1969. The basis of compilation was that the Finnish authors rewrote and brought up-to-date the earlier Finnish textbook and passed it to the English author, who, for his part, worked it into shape, taking into consideration British circum­stances and paying special attention to general actuarial education in English-speaking countries.

Since the publication of the first edition there has been a continued growth of interest in the subject. ASTIN 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 important part played by the theory of risk.

The present and essentially renewed third edition has been worked very much along the same lines, taking into account recent developments of theories and applications. The compilation of the text was mainly made in connection with the university lectures of Pentikainen, and many of the new approaches were developed and tested in a comprehensive research work concerning solvency and the reserve technique (Pentikainen 1982, Rantala 1982).

Since the first edition was published several other textbooks on risk theory have become available, for example Biihlmann (1970), Gerber (1979) and Seal (l966b and 1978). The existence of these excellent presentations of risk theory has very much facilitated the compilation of the third edition of this book. Because the other authors have followed more strictly mathematical lines, it has been appropriate to assume a more pragmatic approach here. The authors hope that all these books will complement each other and give the interested reader a more comprehensive view of the theory than would be possible in any single monograph.

To prevent the book becoming too large and developing beyond

xii PREFACE

the limits of a primary textbook, it has been necessary to limit the subject matter. This has been a very difficult task, due to the very abundant field which the theory and its applications cover today, and from necessity many interesting aspects of the theory have been omitted. Furthermore various alternatives, lines and methods of presentation are possible. Our aim has been for simplicity - the more so because the main purpose of this book is to serve as a first introduction to the theory of risk, since there are several publications dealing with advanced aspects of parts of the theory. On the other hand, the authors have been quite conscious of the danger of over­simplification, which could reduce the theory to 'pseudoscience'; moreover, ignorance of the basic assumptions of the theory could lead to serious mistakes when applying the theory to various actuarial problems. For this reason the basic foundations of the theory have not been omitted. The main lines, the practical one and the theoretical one, are unfortunately not easy to fit together, and in the present state of the theory a firm bridge between the practical problems and exact theory is often not fully developed. Bearing in mind that our main purpose is practical, we have also been obliged to present formulae which are based on approximations without well-mapped confidence limits. We have also attempted a compromise between accuracy and simplicity, transferring some cumbersome considerations to the appendices, which can be omitted at a first reading.

We sincerely hope that this book will prove to be only a first step for each reader in his introduction to the theory, and that sufficient interest will be stimulated to provoke a more extensive and profound investigation. An extensive bibliography has been included to assist the reader in this direction.

The book has been written on the assumption that the reader has a knowledge of the elementary aspects of probability theory. On the other hand, familiarity with the axiomatic theory of probability and stochastic processes is not assumed and the text is compiled accordingly, not covering, for instance, strictly axiomatic existence proofs of the processes concerned, as they are irrelevant to the aims of the book.

In order to render the book more readable, as far as possible, for those with different interests and different degrees of familiarity with probability calculus, the parts of the text that utilize special techniques, or consider details of limited interest only, are marked **

PREFACE xiii

or referred to appendixes. These sections as well as many of the proofs, can be omitted on the first reading and the reader can proceed directly to the final formulae.

Martti Pesonen, PhD, participated in the final collation of the text.

The authors are grateful to the numerous experts in various countries for the generous donation of very helpful criticisms and advice.

The typing of the manuscript for printing, taken from numerous texts and formulae, was undertaken by Brita Aalto, to whom we owe our special thanks for accurate work requiring great patience.

Last but not least, we wish to thank Chapman and Hall Ltd and in particular Mr Richard Stileman and Ms Mary Ann Ommanney for their generous cooperation in the preparation of the book and for the final linguistic checking of the text.

Leicestershire and Helsinki April, 1983

Robert Eric Beard Teivo Pentikiiinen

Erkki Pesonen

We are sorry to announce that our co-author and colleague of long standing Professor Robert E. Beard passed away in November, 1983 without having witnessed the completion of this new edition of Risk Theory.

In grateful recognition for his life-long work in the development and advancement of non-life insurance mathematics we dedicate this book to the memory of Professor Beard.

T.P. E.P.

Nomenclature

Equations in parentheses, items without parentheses. For principles of the notations see Section 1.5. Many of the variables may be both stochastic or deterministic. For brevity this feature is not specified, i.e. bold face letters are not used.

A(v) discounted claims (8.2.7). aj zero-moments of claim size (3.3.1). B gross premiums = P;/(1 - c)

(6.2.8). B(t!, tz) gross premiums from period

[t!, tzJ (6.4.6). bj zero-moment of loss degree

(3.6.16). C loading for expenses of administra­

tion = cB gross or net 6.S(a), (8.1.2), 10.2(1).

c loading factor for expenses 6.2(f), (6.S.4).

c j factor synchronizing inflation (7.6.2).

cl' Cz control caetI. (7.7.1). D(t) dividends 6.S(a), 1O.2(g); time

lag function (6.6.7). d = D/B (10.2.8). d(t) number of deaths (8.1.4). dJ. distribution function E( ) expected value. Eu equilibrium level of u (6.6.9). e base of nat. logarithms = 2.718282. F(k) = F(k ; n) dJ. ofthe claim number

(2.4.1). P(k) mixed dJ. of the claim number

(2.7.4).

F(X) dJ. of aggregate claims (3.2.3). F(X; tj' tz) dJ. of aggregate claims

for period [tj' tzJ (6.1.23). F(x) dJ. of the standardized

aggregate claim (3.9.4). f(X) density of X (3.11.18). f = X/B claims ratio (6.S.4). f(1, t) accumulated claim ratio

(6.6.S). G utility dJ. 1O.4(d). G(uo' u; tl' t z) (6.6.17a). G,(u) dJ. of u(t) (6.6.17b) GNP gross national product G(r) loss degree dJ. (3.6.13). G(x) Edgeworth expansion (3.10.1) g = y/6 3.11(h). H(q) structure d.f. (2.7.1). h.Polya parameter (2.9) or Yz/24

3.1I(h). I(t) investment income 6.S.(a),

1O.2(d). i growth rate specified by suffices ig growth rate 6.1 (c). ijgp = ijrgp (6.S.8a) i j interest rate 6.3(a). ip rate of premium inflation 6.2(a). ix rate of claim inflation 6.1(j). J number of sections or classes 3.7(a).

NOMENCLATURE

j current index, section index 3.7(a). K error factor (4.3.5). k number of claims 2.1(a). L(s) Laplace transform (1.6.9). [(t) cohort size (8.l.l). M net retention 3.6. M(s) moment generating function

(1.6.1). m mean value of one claim = a1

(3.3.2). m.gJ. moment generating function

1.6. m(t) mean value of claim for year t

6A(b). m(Z) measure of tail (3.5.25). N(x) normal dJ., mean 0, st. dev. I

(3.9.3). N/x) NP dJ. (3.11.10). n expected number of claims 2A(a),

(6.1.2), 6A(b). n(l, t) expected number of claims

for period [I, tJ (6.1.24). Ott) run-off error (10.2.7b). P = E(X) risk premium (3.3.7),

(4.l.l). PSL risk premium of stop loss

reinsurance (4.8.1). P X!L risk premium of excess of loss

reinsurance (4.7.1). P A = (1 + A)P safety loaded premium

(6.2.5). P, = P)B (6.5.14). P(t 1 , t2 ) risk premium from period

[t1' tJ (6A.7). p(k; t) claim number probability of

period (0, tJ (2.l.l).

Pk(n) = prob{k = kin} claim number probability (2.2.1), (2A.2).

Pk(n) claim number probability of mixed dJ. (2.7.2).

pit) termination frequency (8.2.3). Q insured sum 3.6A(a); moving

expenses 10.2(f).

q claims number intensity (2.6.5), structure variable 2.7(c).

qi discrete claim size frequency (3.8.2).

q(t) mortality (8.1.1). R insolvency coefficient (9.2.1). R(r) rectangular dJ. (6.8.1). R(t) discounted income (6.6.5). R 1 , R2 , R~, R~ control limits

(7.7.1), (7.7.2). r growth or accumulation factor

specified by suffices, degree of loss (3.6.12).

r 2 = azlm2 risk index (3.3.8). r 3 = a3/m3 (3.3.S). rg = I + ig 6.1(c). rgp = rg X rp (6.5.7). rgx = rg x rx (6. I.l S). rigp = rJrgp (6.5.Sb). rp = 1 + ip (6.2.1), (S.3.6). r(t 1 , t2 ) (6.6.la). /'x = 1 + ix (6.1.10). r(Z) extinction rate (3.5.26). S sum insured S.2. S(Z) dJ. of claim size (3.l.l).

xv

Sk* k:th convolution of S(Z) (3.2.2). T time period, planning horizon. Tz length of a cycle (6.IA). t time tp time lag (6.2.2). V risk reserve, solvency margin (4.1.5), often the initial minimum

capital V(O) = V 0 which satisfies the solvency conditions.

V, ruin barrier (4.1.5), 6.7(a). u solvency ratio = V/P or V/B

(6.5A).

u, = V/B V = var(') variance V(t) premium reserve (8.1.2). v cycle phase (6.1.3); discounting

factor I/ri 8.2. v(y) = v/y) NP transformation

xvi

(3.11.1), (3.I1.17a). W technical reserves 6.5(c). Wk waiting time of the k :th event,

exercise 2.6.2. W(Q) dJ. of sums insured (3.6.14). WT(U) conditioned survival function

(6.7.2). w = WIB (6.5.4), maturity age

(8.1.3). wit) exogenous impulse (6.1.6). X aggregate claims 3.2(a). X(t) aggregate claims assigned to

year t (6.1.1). X(t l , t2 ) aggregate claims assigned

to years t 1, tl + 1, ... , t2

(6.1.1). x = (X - f1x)l(Jx (3.9.1), also (3.3.13). x(l, t) (6.6.15). x, = confidence coefficient defined

by the NP function e = Ny! -xJ Y = P A - X = underwriting profit Yet), YeO, t) profit (8.1.6), (8.1.7a),

(8.3.1), (lO.2.7a). Y standardized normally distributed

variable 3.11 (b). yeO, t) = YeO, t)IB(t) (8.1.7b). Yo discounted random profit (8.2.1). y, confidence coefficient, e = N( - yJ Z size of one claim (3.1.1), credibility

coeff. (4.10.3). Z(t) risk sum (8.2.4). Z cycle variable (6.1.3). zm cycle amplitude (6.1.3). rJ. Pareto parameter (3.5.20) or

41Y~ (3.12.1). rJ.j moment about zero of the claim

number (2.4.3), (1.5.5a). aj moment about zero of the mixed

d.f. (2.8.1). Pj moment about zero of aggregate

claims (3.3.3). r(h) = complete gamma function

NOMENCLATURE

2.9(a). rex, h) incomplete gamma function

(2.9.1). Y skewness 1.5(g). y skewness of mixed d.f. (2.8.6). Y 2 kurtosis 1.5(g). Y2 kurtosis of mixed dJ. (2.8.6). Yx skewness of aggregate claims

(3.3.7). Y 2X kurtosis of aggregate claims

(3.3.7). Yq skewness of the structure function

(2.8.5).

l1f1u 7.9(c). e ruin probability. e(t) noise (6.1.5). e(X) unit step function (1.5.3). '1 auxiliary variable (3.5.18). A safety loading (4.1.4), (6.5.12). Ab gross premium B adjusted safety

loading (6.2.10). Ap risk premium adjusted safety

loading (6.2.9). Atot = A = total safety loading (6.5.12) f1r mean value of the claim ratio f

(6.6.6). l1i = ith central moment (1.5.5b). l1u mean value of the solvency ratio

u (6.6.8). f1x = f1x (1, t) mean value of

aggregate claims X (3.3.7), (6.1.13).

Vj = n/n (3.7.3). 1!j = mx/mx (3.7.8). p risk intensity parameter (2.2.1),

see also (3.3.11), relative mean risk, exercise 5.2.2.

pit) lapse ratio (8.3.2). (J standard deviation. (j standard deviation of mixed dJ.

(2.8.6). (J q standard deviation of the

structure function (2.8.5).

NOMENCLATURE

O"u standard deviation of the solvency ratio u (6.6.10).

O"x standard deviation of aggregate claims X (3.3.7).

O"x (t l , t2) = standard deviation related to X(t l , t2 ) (6.1.19).

qJe (6.9.5). qJee (6.9.7). qJ(u) characteristic function (1.6.8). qJ(t) ruin state probability (6.6.19). 'P iU) ruin probability related to

period (0, T] (6.7.1), (9.2.1). w cycle frequency = 2n/Tz (6.1.4).

Some conventions x+ or ( )+ = X or = the expression

in the parentheses if x or the expression;:" 0, otherwise = 0.

[x] = the integer equal or next smaller than x.

xvii

Au B union of sets A and B = a set consisting of all the elements of A and B

An B intersection of sets = a set consisting of the joint elements of A and B

A :J B B is a subset of A. "Ix valid for all x.

1 1

3x i valid at least for one of the Xi'

* e.g. F * G(X) convolution (\.5.8). ** mark for technically detailed

section or paragraph which can be passed on first reading. f - 1 is the inverse of f.

- I as a subscript relates to the preceding year, e.g. u_ 1 (6.5.2).