WILLIAM RUSSELL PU~ C1 NATIONALE 1\JEDERLANDEN N.V. RESEARCH DEP·\R'nfEL\T P.O. BOX796 3000 AT ROTTERDA.\1 THE :--.'ETHERI.A.'\'DS M.J. GOOVAERTSANDW.J. HOOGSTAD SURVEYS OF ACTUARIAL STUDIES NO.4 HG 80,5 .G"S .1987 PREFACE In the second volume of these Surveys of Actuarial Studies, on "Rate Making", the subJects of "Credibility Theory" and "Large Claims" were considered, but only to a liml. ted extent. This was mentioned in two reviews, and it was suggested that separate volumes might be devoted to these subjects. With regard to "Large Claims", in a review by Mr. B. Ajne (Astin Bulletin, vol. 15, nr 1 (1985), p. 67) it is suggested to further elaborate on the topic of large claims: "The problem of large claims is a nuisance in tariff construction work, at least as soon as personal injury cla1ms or fire cla1ms are present. So, as a practitioner, one could have hoped for a fuller treatment, perhaps including the division of claims into more than two groups (e.g. normal cla1ms, excess claims, superexcess claims) and/or some help from the theory of outlying observations. Maybe one could hope for another volume in the series on this subject?". We have considered this issue, but we have ser1ous doubts whether we will be able to publish a volume on large claims. Up t1ll now the number of articles published on this topic is rather limited, especially as far as the influence of large claims on standard products, such as automobile and home owners insurances, is concerned. In another review, by Mr. E. Straub (Ml.tteilungen der Vere1nigung Schweizerischer Versicherungsmathematiker, 1984, Heft 1, p. 113) the subject of credibility theory was mentioned: " ... es ist kein Kompendium und kann es mit 130 Seiten auch gar nicht se1n (allein iiber Credibility liesse sich mehr schreiben) , . .. ". This part 4 of our ser1es "Surveys oi_ Actuarial Studies" specifically deals with credibility theory, and thus ~ill treat the SubJect into far more depth than was the case l.n volume 2. This volume was written by Will Hoogstad, who works in our company, and Marc Goovaerts, Professor at the unl.versities of Leuven (Belgium) and Amsterdam. We would especially like to thank Mr. Goovaerts for his valuable contribution. We hope this volume will contribute towards a better understanding of credibility theory and thereby wl.ll prov1de a link to further pract1cal applications. April 1987 Research Department Nationale-Nederlanden N.V. G.W. de Wit 3 893638 TABLE OF CONTENTS Preface 3 Table of contents 4 Introduction 7 General Guideline 15 Chapter 1. A mathematical model 19 Appendix 1.1 31 Chapter 2. Exact credibility 33 Chapter 3. The classical model of Blihlmann 37 a. Model and assumptions 37 b. Comments 38 c. Computations 39 d. Remarks 39 e. Numerical example 41 Chapter 4. The BUhlmann-Straub model 43 a. Model and assumptions 43 b. Comments 44 c. Computations 47 d. Remarks 48 e. Alternative estimators 51 f. Numerical example 51 Chapter 5. The Hachemeister regression model 53 a. Model and assumptions 53 b. Comments 53 c. Computations 54 d. Remarks 55 e. Numerical example 57 Chapter 6. The De Vylder non-linear regression model 61 a. Model and assumptions 61 b. Comments 62 c. Computations 63 d. Remarks 64 e. Alternative estimators 65 f. Numerical example 66 4 Chapter 7. The De Vylder semi-linear model 71 a. Model and assumptions 71 b. Conunents 71 c. Computations 73 d. Remarks 74 e. Numerical example 75 Chapter B. The De Vylder optimal semi-linear model 79 a. Model and assumptions 79 b. Conunents 79 c. Computations 80 d. Remarks 82 e. Numerical example 84 Chapter 9. The hierarchical model of Jewell 87 a. Model and assumptions 87 b. Computations 89 c. Numerical example 91 Chapter 10. Special applications of credibility theory 93 10.1 Loss reserving methods by credib1lity 93 a. Model and assumptions 93 b. Conunents 95 c. Computations 96 d. Remarks 98 e. Numerical example 98 10.2. Large claims and credibility theory 99 a. Model and assumptions 99 b. Comments 100 c. Computations 101 d. Remarks 102 e. Numerical example 103 Chapter 11. Credibility for loaded premiums 105 11.1. Credibility for variance loaded premiums 105 11.2. Credibility for Esscher premiums 107 Bibliography 111 5 INTRODUCTION Credibility theory prov1des us with techn1ques to determine insurance premiums for contracts that belong to a more or less heterogeneous portfolio, in case there is limited or irregular claims exper1ence for each contract but ample claims experience for the portfolio. It is the art and sc1ence of using both kinds of experience to adjust the insurance premiums and to improve their accuracy. The general and by now famous credibility formula C = (1 - Z) .B + Z.A originated in the United States during the years before World War I and was suggested in the field of workmen's compensation insurance. The industry-wide premium rate charged for a particular occupational class is represented by B. But an employer having a favourable record w1th this class tries to lower his premium to A, the rate based on his own experience. Because observat1ons of one employer are to a large extend ruled by random fluctuations, Whitney [1918) suggested a balance C between the two extremes A and B. Some 70 years ago he wrote: "The problem of experience rating arises out of the necess1ty, from the standpoint of equity to the individual risk, of stnk1ng a balance between class-experience on the one hand and risk exper1ence on the other". It was felt that the mixing-factor Z should reflect the volume of the employer's experience. The larger this volume, the more credib1lity, by means of a high value of Z, is attached to the desired premium A. Thus it became common parlance to denote Z as "the credibility factor" or simply "the credibility". The theory of credibility 1s concerned w1th the quest1on of how much weight should be g1ven to th1s actual cla1ms experience. Of course, not only downward but also upward sh1fts 1n individual premiums are possible, although the employer's pressure 1n such cases will not be felt strongly. 7 After these early findings credibility theory developed in the direction of what is now called "limited fluctuation credibJ.lity theory". Due to the fact that it was created by North-American actuaries, to some it is also known as "American credibility theory". Although we will not treat this branch in the present survey, we will now briefly outline some of its features. Without mak1.ng reference to the formula above, this theory or1.ginated with a paper by Mowbray [1914] "How extensive a payroll exposure is necessary to give a dependable pure premium?". Also in a workmen's compensation context, he poses the quest1.on of how many insureds, covered by the same contract, are necessary to have a fully credible estimate of A that can serve as a premium for the next year. Or, reformulated, how much (individual) claims experience is needed? We quote his solution to this problem: "A dependable pure premium is one for which the probability is high, that it does not differ from the true pure premium by more than an arbitrary limit". With a relatJ.vely simple mathematical model it is possible, after setting some tolerance-lJ.mits and using the Central Limit Theorem, to compute the number of insureds required for a reliable (credible) estimate of this "true" premium. This number is interpreted as a standard for full credib1.l1.ty. In cases where the number of insureds at least equals this number, it amounts to putting Z 1. However, this solution left as an open question how to act when the number of insureds is too small. Of course, Mowbray himself did not raise that question because his paper predated that of Whitney. This problem, known as partial credibility, led to numerous articles and a number of popular, heuristic formulas for Z. All these formulas assess partial credibility as a value between 0 and 1 and most of them are dependent on the actual and the required number of insureds· Despite all of these practical efforts, the need for a sound mathematical model was felt deeply. For a survey of limited fluctuation credibility theory that includes a bibliography and a mathematical addendum we refer to Longley-Cook [1962]. 8 The theoretical foundation of credibility theory was not established unt1l the 1965 ASTIN Colloquium, where Bi.ihlmann presented his "distribution free" credibility formula (published in [1967)), based on a least squares criterion. This initiated a new branch in the theory, now called "greatest accuracy credibility theory" or simply "European credibility theory". Both Bailey [1950) and Robbins [1955) published results before, but these were not derived in a distribution free context. Bailey [1945) vaguely pointed at this approach. However, his article is hardly understandable due to notational difficulties. This rapidly growing branch of credibility theory forms the scope of this publication. But before focussing on th1s and some of the mathematical background, let us cast some light on the place of credibility within the rate making process. Most of the actuaries working in practice probably agree with the top-down approach in tarification as proposed by H. Bi.ihlmann during an Oberwolfach meeting. He explained how, for an insurance portfolio, the collective matching of liabilities and premium income is the primary concern (this is the top level) while a fair distribution (the down level) of the premium income among the different contracts has to be realized afterwards. This distribution of the total revenue among the d1fferent contracts could e.g. be done by means of premium principles. However, credibility theory provides us with a sound statistical tool for a fair distribution of the premium income among the different contracts in a portfolio. The matching of premiums and liabilities must be the insurer's main goal. Apart from situations where the prem1um is prescribed by the government or ruled by considerations of solidarity, the larger part of the premium payable consists of the (approx1mated) risk prem1um, i.e. the expected future claim amount. In the sequel we will ma1nly restrict ourselves to these risk premiums and shall often denote them by "premiums" only. The treatment of credibility theory for loaded premiums is only briefly dealt with in one of the last chapters. A limited number of specialized papers is ment1oned in the references. 9 For a general actuarial guideline in the rate making process, we refer to the former issue "Rate making" [1983] in this ser1es. Upon read1ng this, it will become clear that credibil1ty is only a part, and even a dispensable part, of the whole process. The actuary's first task is to determine the characteristics of insureds which, in his (subjective) opinion, are essential to distinguish between them, the so-called tariff variables. Unfortunately, not all of these tariff variables will be observable, only some of them have data available and even fewer appear to be of statistical relevance. A special and very important tariff variable is past claims experience, a representative of both observable and non-observable variables. Preferably simultaneously, but often afterwards, the opt1mal tariff classes (i.e. optimal sub-sets of the tariff variables) are to be calculated. These classes imply a structure that consists of so-called cells. Within every cell there are a number (possibly zero) of insureds with identical risk characteristics. Now the question of the determinat1on of the insurance premiums for each of the cells arises. There are two solutions. The first possibility is to specify an additive or multiplicative model, in which the (transformed) claims experience variable should be described, as well as possible, by the chosen tariff structure. Belonging to the same methodological approach is the method of maximum likelihood to estimate the parameters of a pre-specified distribution function for the above ment1oned variable. The other possibihty consists of our credibility approach. The Biihlmann model and 1ts generalizations allow for a distribution free estimation of the 1nsurance premiums as a weighted average of the cell-experience and the portfol1o exper1ence. However, it should be mentioned that these models in general are only suited to deal with the claims experience variable and exactly one other tariff variable. In case more tariff variables are involved, the models should be modified. In theory, combinat1ons of both methods are also possible. To be specif1c, consider the follow1ng example in automobile insurance. 10 On the basis of available and relevant tariff factors, such as the we1ght of the car and age of the driver the heterogeneous portfolio 1s (given the tariff classes) split into groups of insureds which are less heteroge neous. With the aid of one of the f1rst techniques, a premium for each group is calculated. This premium reflects the average claim amount within that group. Nevertheless, not all drivers are equally skillful or careful. Ind1vidual claims experience can tell us more about these hidden risk characte ristics. So, within each group a second selection is possible: base the individual driver's credibility adjusted premium on h1s own and the group's claims experience. Another part of the heterogeneity will be eliminated and the new premium is closer to the true premium. Of course, in pract1ce this procedure 1s too laborious to handle and insurers use a bonus-scale with fixed discounts and surcharges 1n percentages of the group-premium to incorporate the individual (number of) claims experience. Another example can be found in Biihlmann and Straub [ 1970] . We fix our attention to the annual loss ratios of the different kinds of treaties of a reinsurance portfolio observed during a number of years. For a fixed treaty, we are interested in the expected indiv1dual loss ratio. The observed ratio (A in our general formula) is easy to calculate from the data available. Nevertheless, in most cases the data is too scarce and far too irregular to provide for a reliable estimate. Hence credib1lity theory is a useful tool to apply. If the portfolio is large enough, the collective loss ratio (B) can be considered a good estimate for the expected loss ratio over the portfolio. Now credibility theory g1ves for every ind1v1dual treaty a weighing factor Z that reflects the reliability of the individual and group data. Some credibility techniques are able to handle inflated cla1ms figures. An example is Hachemeister's method [1975]. His article also deals with private passenger bodily 1njury insurance. Claim amounts for a few U.S.A.-states are observed for a number of quarters. They show a tendency to increase in time, due to (inter alia) 1nflat1on. We are interested 1n the state-specific inflation factor. It 1s supposed that 1nflation is not the same in all U.S.A.-states, hence, these states form a heterogeneous collective. 11