Erwin Straub Non-Life Insurance Mathematics With 12 Figures Springer-Verlag Berlin Heidelberg GmbH 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 978-3-642-05741-0 Library ofCongress Cataloging-in-Publication Data Straub, Erwin Non-life insurance mathematics/Erwin Straub. p. cm. ISBN 978-3-642-05741-0 ISBN 978-3-662-03364-7 (eBook) DOI 10.1007/978-3-662-03364-7 1.lnsurance-Mathematics. I. Title HG8781.S75 1988 368'.01-dc19 88-11959 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 provi sions 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 1s t edition 1988 Typesetting: Macmillan (lndia) Limited, Bangalore; 214113140-543210 Printed on acid-free paper 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 experi ence and on the other for practitioners knowing mathematics from so me distance only, but it is definitely too elementary and incomplete for a thorough bred 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 Table of Contents Chapter 1. Problems Chapter 2. Tools ......................................... . 7 2.1. The Model ........................................ . 7 2.2. Distributions for K and X ............................. 16 2.3. Moments ......................................... . 21 2.4. The Total Claims Cost Z ............................ . 29 2.5. Cramer's Inequality 36 2.6. Dependent Variables 42 Chapter 3. Premiums ....................................... 52 3.1. Pragmatic Principles ................... . . . . . . . . . . . . .. 52 3.2. Theoretical Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54 3.3. Experience Rating ................................... 59 Chapter 4. Reinsurance ..................................... 68 Chapter 5. Retentions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76 Chapter 6. Statistics ........................................ 89 Chapter 7. Reserves 102 Chapter 8. Solutions 116 8.1. Negative Binomial 116 8.2. Exact Credibility 119 8.3. Closing the Circle 126 References 131 Subject Index 133 Chapter 1. Problems Actuaries are the people who deal with all kinds of mathematical and statistical problems in insurance - that's why we speak of actuarial problems. With the more recent application of actuarial methods also to Property and Liability insurance, it has become customary to distinguish between Life and Non-Life actuarial sciences. Although such a distinction does not always make sense-in Health insurance, for instance, the two domains overlap widely - the present book deals exclusively with so-called Non-Life problems and their possible solutions. Insurance companies accept risks, i.e. potential claims, from their clients, the insureds, against a certain price called a premium. If a risk or a portfolio of risks is too large for a company, it will pass on parts of it to one or several other companies, its reinsurers, whereby that part which finally remains with the first company is called its retention. That is very briefty how the insurance industry operates. If a risk is very large, some of the reinsurers in their turn may have to pass on parts of their acceptances to yet other companies, so that the original risk or portfolio is in the end covered by an entire network of insurance and reinsurance arrangements between a number of companies, each of them carrying the retention deemed appropriate. On page 2, there is a picture of such a network (Fig. 1). In order to get quickly an overview of the landscape of Non-Life actuarial problems arising in practice, let a layman very naively ask a question like, "What is an appropriate retention for a given company and how do you determine it?" To this the expert's answer will be, "Hmm-well, it depends ... you see ..." . It certainly depends on many things. In the first place, a wealthy company can clearly afford to retain more for its own account than a poor one, for it has more capital to stake. Secondly, it depends whether management is at all willing to take risks; a conservative manager will display little risk willingness, contrary to a coutageous or even foolhardy entrepreneur. Furthermore, the underwriter will have to make up his mind on the premium: Is the risk or are the policy conditions likely to produce a profit if there are no unusual claims? (Whatever that means!) And, last but not least, a very close look must of course be taken at the risk itself, for there are better balanced risks and/or portfolios on the one hand and highly unbalanced ones on the other. After some pondering on these indeed very sketchy explanations regarding the problem of how to fix retentions, our naive layman-even if only vaguely- 2 I. Problems Their retro cessionaires 35.7% • A retrocessionaire "or secc)Od order" Figure 1 understands by instinct that basically things must be rat her simple and obvious, in that the retention of each company in the above network can or must be all the higher if the three items "capital", "risk willingness" and "profit margin" are high. It should be all the lower, however, the more unbalanced the risk iso Thus he can conclude that . capital x risk willingness x profit margin retentIon = --==--------:---:--=---:-----'=-------=- un balancedness must be the magie formula, and he can furthermore immediately see that, depending on which four of the five items appearing in this formula are given, the problem of calculating the fifth one represents a fundamental actuarial task. So, for instance, as the calculation of premiums is connected with the term "profit margin" and the above can be rewritten as . _ retention x unbalancedness fi pro t margm - . I . k ·11· caplta x ns WI mgness we have here an indication of how to assess a premium. I. Problems 3 "So, if it is all that easy, why do we need actuaries at all?" we may ask ourselves desperately. No need to worry, however, because the above rule of thumb has just been written down by mere instinct and without any intelligent proof whatsoever. It is precisely one of the objectives ofthis book to show that a proper risk-theoretical approach to insurance phenomena leads straight to this formula if we allow for maximum possible simplifications in the mathematical model. Also, the. instinctive formula is purely qualitative for it does not tell us, for instance, how to express risk willingness numerically. Finally, why should the mutual relationship between the five items in the formula be a multiplicative one and not of some other type? Be that as it may, the formula is handy, even correct, under certain assump tions, as we shall see later on. Within this first chapter, however, we shall only use it to broadly describe die five main types of problems we meet in practice, namely (i) Rating, i.e. premium calculation characterised by bringing our instinctive formula into the shape of . _ retention x unbalancedness fi pro t margm - . I . k ·11· caplta x ns Wl mgness telling us among other things that in some way or other the loading contained in a premium ought to be dependent on the degree of unbalancedness of the risk or portfolio under consideration. (ii) Assessing reserves in the sense of contingency reserves, risk capital or catastrophe funds through . I _ retention x unbalancedness caplta - fi . . k ·11· pro t margm x ns Wl mgness (iii) Underwriting limits. How to fix retentions? Weil, somehow in accordance with . capital x profit margin x risk willingness retentton = --='----=---_:_~-=-_:_-----==-­ unbalancedness if we follow basic common sense and provided there are no other prevailing criteria induced, for example, by external solvency prescriptions. (iv) Risk behaviour in general, i.e. a company's entrepreneurial attitude towards risk indicated by . k ·11· retention x unbalancedness ns Wl mgness = ---:-.- ---::----::----:--- capltal x profit margin This may serve, for example, to measure how conservative or not a company is or ought to be or to compare the risk behaviour of two or several companies. 4 I. Problems (v) The maximum tolerable risk load a company may be able to bear on its gross account, given its rate level, retention and financial strength as weIl as its general behaviour towards risk, abbreviated by the formula capital x profitability x risk willingness un b a Ia nee d ness = . retentIOn This theoretically tolerated unbalancedness may be set against the actual fluctuations of business results observed in the past. The above may be viewed as a sketch of five global problems dealing with items that concern the company or account as a wh oie, namely the general rate level (i), the overall reserves (ii), the absolute retention (iii), the general risk willingness or aversion (iv) and the potential for more or less extreme fluctuations of the company's total result (v). Parallel to such "holistic" problems, there are corresponding questions of how to differentiate for individual risk categories or sub-portfolios, how to distribute or how to graduate certain overall quantities at lower levels of the company structure. Here is a short indication of wh at may be called relative problems as opposed to the above-mentioned holistic or absolute ones: (i) Experience rating. There is this basic dilemma in every practical rating situation: On the one hand, according to the principle of fair premiums, each risk ought to finance its own claims in the long run (say asymptotically), while on the other hand it is the very idea of insurance that a given portfolio forms a collective of risks "with equal rights", each of them paying the same pure risk premium equal to the portfolio's average yearly loss costs. Now these two principles are in perfect harmony in the case of an ideally homogeneous risk collective, but what if, as always in reallife, there are different risk categories or even different individual risks (as in a Motor Liability portfolio, for example)? How reliable is the individual's claims experience compared to the portfolio average? How should premium rates be varied by risk category? How should the degree of heterogeneity of a portfolio be measured? - Such questions are an swered by the so-called credibility theory, one ofthe oldest branches ofNon-Life mathematics. (ii) Allocating reserves. Onee a given part ofthe company's total equity is ear marked as risk capital, Le. as an overall shock-absorber against fluctuations of the yearly overall result, one may need to break it down into individuallines of business. Furthermore, it is not apriori evident that the sum of per line risk capitals must equal the overall risk capital. They may either overlap or be supplemented by reserves for common catastrophes. (iii) Relative retentions. Quite obviously, it would not be optimal to have the same amount for the company's own account on each risk, but how should the Concluding Remarks and Exercises 5 retention by risk class be graduated reasonably? What appropriate statistical tools are there to answer this question? (iv) Risk behaviour. There may be different degrees of risk acceptance or aversion as we move from one sub-portfolio to another. (v) Unbalancedness. The overall unbalancedness is generated by the tluctua tions of claims costs of individual lines andjor risk classes and here again "the sum must not necessarily be equal to the total". There are furthermore different types of tluctuations or rather different sources: Random tluctuations due to the occurrencejnon-occurrence of (exceptionally large) claims, cydical market tluc tuations and last but not least the financial risk of a company due to changes of the value of the total of its investments. With the above .attempt to sketch five overall and five relative Non-Life actuarial tasks, the whole field is of course not completely covered - it should nevertheless give a good first survey. Concluding Remarks and Exercises Insurance is a business of risks (not the only one, however) but then what is a risk? - It is the danger of losing something, a loss potential, the loss may be human lives, health or wealth. Observe that the loss depends on the viewpoint, on the level so to speak: If a house bums down, for instance, nothing is lost from nature's point of view. There is only a transformation - a transformation from wood into ash and co al and smoke, from walls into debris and dust, from living into dead matter, from potential energy into heat and light and so on, by a simple chemical reaction called oxydation or fire, to put it more plainly. Nothing is lost whatsoever from an overall point of view. If, however, the house that burned down was my house, then it may mean a tremendous loss to me - unless it was weil insured andjor I am a wise philosopher. Exercise J Make up a list of at least twenty risks an average family is confronted with. Every person, every company and institution, every country, mankind as a wh oie, each life, as a matter of fact everything is exposed to a number of specific risks, has to live with these risks and has to apply his specific kind of risk management usually by fight or tlight, i.e. so me type of protection against or avoidance of risk.