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Jean Piaget Cognitive Theory
ORN ERSTON E CREDIBILITY THEORY: THE CORN ACTUARIAL SCIENCE
James C. C. Hickman* and Linda ind a Hea Heacox cox †
ABSTRACT The Nort h American ociety of Actuarieson Actuarieson its golden annivers anniversary ary American Actuarial Actuari al Journal is honoring t he Society in 1999 by publishing publishing a se series ries of articles articles on the contributions of actuar actuarie ies s to the devel development opment of ideas. In this issue, the second of the series, we explore the development of the credibility idea and the rela relations tionships hips of this development development to dee deep p iss issues ues in the methodology of science cience.. We begin wit h a short short ess essay by James C. C. Hickman Hickman and continue wit h comments by three actuarie actuaries s who wh o w ere deeply involved invol ved with wi th the th e develop development ment of credibilit credib ilit y: Robert Robert A. Bailey Bailey,, Hans Hans Bu¨ hlm ann, and Charles C. Hewitt.
1 . I NTRODUCTION
most most import import ant and enduring enduring contri contri buti buti on t o actuarial science.’’ cience.’’ Rode odermund rmund also also quotes Thomas Thomas Ca Carl rl son, a pr ominent act act uary, uary, as sa saying of Art hur Bail Bail ey, ey, the pioneer in using the Bayesian approach to credibili t y, ‘‘ [ he is] is] proba probably the most most profound profound contr contr ibut or t o casua casualt lt y actuarial actuarial theory the Unit ed States States has has produced.’’ produced.’’ Allen Ma May yerson erson ( 1964) publi shed hed a modern Bay Bayesian esian approac approach h t o credibil it y theory. theory. In hi s discuss cussion of Ma May yerson’s rson’s pape paper, r, Hew Hewit t (1965) said, ‘‘European actuaries have outstripped us in the classical ‘theory of risk.’ Professor Mayerson has distilled t he ess essence ence of Ameri Ameri can can achiev achieve ement in t he area of credibil it y and t he Bayes Bayesian ian approac approach.’’ h.’’ Thes These words about bout credibili credibili t y do not contain contain t he mode moderate adjec adjec-t ives ives expe expected cted in actuari al prose. prose. What What accounts accounts for t his ebulli ence? nce? Reviews iews of the current status status of the theory and prac prac-t ice of credibili credibili ty are avail able in Herzo Herzog g (1994), Klugman, Panjer, and Willmot (1998) and Venter (1990). In this art icle we will prov provide some some historical historical backbackground ground t hat hat will be helpful helpful in apprec ppreciating iating t he interviews with some of the pioneers in actuarial thought on credibili t y and and att empt to justi justi fy t he e extrav xtrava agant gant praise.
The ti t le of t his int roductor y es essay and and t he as associated interviews is lifted from Longley-Cook’s 1962 monograph graph on credibili ty t heory heory.. I t is one of of ma many ny str str ong statements tatements about about the centralit y of credibili ty t heory heory to actuarial thought. In connection with marking the ﬁft ieth annive nniversa rsary of the Society ociety of Actuaries ctuaries,, it seems appropri ppropri ate to rev review some of the hist hist ory of thi s pervasive idea. The conse conservat rvat ive nat nat ure of actuari es has often been been the foun founda dation tion of the public public perce perception ption of the them. m. Moorhea Moorhead d (1989, p. 391) quotes Ma Matt t hew Rode odermund rmund on this perception: ‘‘The image of actuaries depicted on stage and screen, in books and comedy routines, has bee been t hat hat of st st aid, humor less less, unimaginati ve creacreat ures, ures, their minds immerse immersed in numbers, numbers, lacking, lacking, it has been said ad extremum, the personality to be an accounta ccountant.’’ nt.’’ Give Given t his propens propensit it y t o a lack lack of i ma maggination, the extrava xtravagant gant praise praise hea heaped ped on credibili credibili t y theory is all the more surprising. The same Matthew Rodermund (1990) who articulated the conve convent ional cons conse ervat rvat ive image of act act uaries used used exube exuberant rant l anguage nguage t o declare, declare, ‘‘ It is t he concep conceptt of credibili credibili ty that has has bee been the cas casualt ualt y act act uari uari es’
2 . A BUSINES USI NESS S NECESSI TY In the years immediately immediately before before World Wa War I, the rapid rapid industr iali zat zat ion of t he Unit ed St St at es and Canad Canada a wa was dri ving many ada adaptat ptations. ions. Thes These adaptat adaptations ions incl ude uded d new provisions provisions for family economic economic securi ecuri t y. For For example, mple, t he ﬁr st group life insuranc insurance e for employ employe ees was
*James *James C. Hickman, Hickm an, F.S F.S.A., .A., A.C.A.S., A.C.A.S., Ph.D., is Emerit us Profes rof ess sor and Dean, Dea n, University University of Wiscons Wisconsin, in, School of Busines usiness s, 975 University University Avenue, Madison, Wisconsin 53706. †Linda †Linda Hea Heacox cox is Public Public Relatio elatio ns Specialis Specialistt at th e Society Society of ActuActuarie aries s, 475 N. Martingale Martingale Road, oad, Suite 800, Schaumburg, chaumburg, Illinois Illinois 60173, e-mail, [email protected][email protected]
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issued in the United States in 1911. Starting about the same time, the states enacted workers’ compensation laws. These laws were based on the new principle of liability without fault for occupational injuries and diseases. Employers were compell ed to ﬁnance the beneﬁts. Depending on the individual state, the ﬁnancing could be done by purchasing an insurance poli cy, by self-insuring, or by obtaining insurance from a monopolistic or competi ti ve state fund. The same path of development was foll owed in Canada. Suddenly, a new line of essentially compulsory group insurance had been created. Workers’ compensation not only changed tort law, it created a host of new actuarial problems. There were institutional and intellectual responses t o t hese chall enges. The Casualt y Act uari al Society was an inst it uti onal response, and credibili ty an int ell ectual response. Rodermund (1990), in the section entitled ‘‘Days to Remember,’’ retells the story of the efforts to develop an experience-rati ng process for workers’ compensation insurance. From these efforts came t he name, t he basic equation, and even the notation of credibility theory. The original account of these developments is by Whit ney (1918). The i dea of midcourse correcti ons has been around as long as humans have made journeys. Cont rol systems to maintain stability in the operation of a process were an import ant component that technology developed t o harness t he steam engine for transportation and production. The need for such adjustments in a l arge and almost completely new insurance system is obvious. An extr a force pushing for premium adjustments in workers’ compensation insurance was the social desir abili ty t o charge l ower premiums t o those employers who by safety engineeri ng reduced the cost of industrial accidents. Premium reductions could provide an economic incentive to safety.
3. THE I DEA The plan suggested for the periodic revision of group insurance premium rates used a li near adjust ment function: New Rat e Z
Observed Rat e
where 0 Z 1 and was an increasing function of the exposure size. The item Z is t he credibili ty factor. Clearly many functions have the desired properties. One example of such a function is:
where k 0 and n is a measure of exposure size. The remaining issues were to develop a consistent and coherent theory for the premium experi ence ad justment procedure so t hat it could be used i n a variety of applications and to ﬁnd practical ways to estimate the parameters of the credibilit y factor. In facing t he ﬁ rst of t hese t wo i ssues, i t soon becomes clear that one cannot avoid choosing sides on a fundamental phil osophic issue. Perhaps it is t he fact t hat a very practical business problem is found to require a phil osophic foundat ion t hat accounts for t he fascinat ion of credibili ty theory. Probably no subject wit hin actuarial science has att ract ed more att enti on fr om an array of economists, mathematicians, philosophers of science, and statisticians than has credibilit y theory. The task of describing in a few sentences the sali ent at tr ibutes of t he t wo philosophic foundat ions of credibili ty theory is daunting. It is acknowledged that there are int ermediate posit ions to t he t wo ext remes, and students of credibilit y t heory differ in the language they use. Nevertheless, to trace the history of the credibility idea it is useful to try to characterize the two basic positions: (1) Those starti ng from this posit ion interpret probabilit y as a relati ve frequency and assert that random vari ables, such as total claims i n an accounting period, have probabilit y distr ibutions wit h ﬁxed paramet ers. Pri or informat ion enters statistical model building only in the selection of a class of models. (2) Those starti ng from this posit ion interpret probability as a measure of the degree of belief in a proposit ion. Relati ve fr equencies are relevant to estimating probability distributions, but other information is also relevant. Prior information, that is, information avail able before recent information, is to be used in the form of probability distribution about various models or parameters. In statistics, position (1) is called the sam pling theory approach and positi on (2) is t he Bayesian ap proach. The Bayesian approach draws its name from Thomas Bayes (1702–1761), who made t he ﬁr st st atements of position (2). Amazingly, almost from t he beginning, perhaps wit hout digging down to the bedrock of a philosophic posit ion, both schools of t hought were represented by
CREDIBILITY THEORY: THE CORNERSTONE
actuaries working on development of credibili ty t heory. The superb set of references and t he secti on enti tl ed ‘‘Further Topics’’ provided by Venter (1990) permits one to trace the remarkable intellectual history of credibilit y theory and its competing philosophic foundat ions. Ideas do not respect organizati onal boundaries. Alt hough some would say that credibili ty is a product of casualt y act uari es, one of t he import ant mil est ones in t he history appeared i n t he Transactions of the Actuarial Society of America. The paper was by Keffer (1929) and adopted a Bayesian foundati on for experience rati ng in group life insurance.
4 . A REM ARKABLE ACTUARY In Secti on I of t his essay, Thomas Carl son was quoted about the singular contri buti on of Art hur Bailey to credibility theory. In Rodermund’s (1990) ‘‘Introduction’’ in Founda tions of Casu alty Actuarial Science, the section on Arthur Bailey is entitled ‘‘An Actuary t o Remember.’’ The nat ural questions are who was Art hur Bail ey and why was his contri bution important? In the int erviews t hat follow, bot h of t hese questions will be addressed. To provide context for the interview, it is helpful to review brieﬂy the controversy t hat was underway about the foundat ion of stati stics at the time that the Society of Actuaries was founded and Arthur Bailey was writing. The sampling theory and Bayesian foundations for statist ical inference had been present at least since the ti me of Thomas Bayes. In t he ﬁrst half of the twentieth century, rapid progress, built on a sampling theory foundation, had greatly increased the range of appli cat ion of st at isti cal methods. Names such asFisher, Neyman, E. Pearson, and K. Pearson are associated with the progress. The Bayesian approach received little attention. Writing in 1950, Bailey described the situation very well. ‘‘ The st atisti cal methods, developed by t he mathematici ans and avail able in t he st andard t ext books on st at isti cal procedures, deal wit h t he evaluati on of the indi cat ions of a group of observat ions, but under t he tacit or implicit assumption that no knowledge existed prior to the making of those particular observations. The credibility procedures, used in the revisions of casualt y rates, have been developed by casualt y actuaries t o give consistent weighti ngs to additional knowledge in it s combinati on with already existing knowledge.’’
What was amazing was that, at the same t ime, several phi losophers of science and statist ici answere saying almost exactly the same t hing. This group included Jeffreys, Barnard, Ramsey, de Finett i (also a contr ibut or to actuari al science), Savage, and Li ndley. What made Bail ey remarkable was t hat he came t o his crit icism of sampling theory and support of t he Bayesian approach because of int ensive study of a very practical business problem. Bail ey’s papers (1942, 1945, 1950a, 1950b) are not necessari ly easy t o r ead. He di d not use t he not ation and language of modern Bayesian analysis. Yet you wil l ﬁnd powerful arguments i n support of t he actuarial use of the Bayesian approach, a review of the satisfying examples of Bayesian conjugate analysis that reproduces st andard credibil it y formul as, and even a r eview of using least squares estimat es of next peri od’s expected claims, using a linear function of past claims and some pri or i nformati on about t he claims process.
5. THE I NTERVI EW S The int erviews wil l be wit h Robert A. Bail ey (1959, 1961), son of Arthur Bail ey and hi mself a contr ibutor t o credibili ty theory; Charl es C. Hewit t (1965, 1970), a contri butor to credibili ty theory whose actuarial career spans t hese developments; and Hans Bu ¨ hlmann, Swiss professor and actuary. Dr . Bu ¨ hlmann ( 1967) is a major contri butor to credibilit y theory, especially the idea of a least square’s linear estimate depending on prior data and parameters t o esti mate next year claims. Dr. Bu ¨ hlmann wil l be asked t o pl ace t his segment of American actuarial history in a worldwide perspecti ve.
ROBERT A. BAILEY Robert Bailey, F.C.A.S., says it has been hi s good fortune to work for some outstanding actuaries. He began hi s career with a j ob recommended by his famous father at the National Bur eau of Casualt y Underwri t ers. His boss was Thomas Carl son. From t here, he went t o Sentry Insurance, formerl y Hardware Mutuals, where he worked for Ruth Salzmann and Norton Masterson. He also served in the actuarial department of the Insurance Company of North America in Philadelphia working for Laurence Longley-Cook. Lat er, he was brought to the Michigan Insurance Bureau by Allen Mayerson, who was Insurance Commi ssioner for t he state. He holds a Mast er of Science degree i n mat hemat ical stati st ics from the Universit y of Iowa, where Robert Hogg was his thesis advisor. He now li ves
in Lansing, Michigan and has been retired from the Michigan Insurance Bureau since June 1997. Can you give us the highlight s of your fat her’s career? He graduated from the University of Michigan actuari al and statistical program i n 1928 and worked i n the stati sti cal depart ment at t he Unit ed Fruit Company i n Boston unti l the Depression, when the department was eliminated. Then he worked for The Alliance of Mutual Insurers in New York City, where he was in charge of making rates for the Mutual cartel. In those days, it was legal for cartels to make rates in concert, not a violation of anti-trust laws. That was until the SEUA case came along and caused t he l aw to be changed. SEUA day was the same day as D-Day. When it made that decision, the U.S. Supreme Court made my father’s job il legal. I remember t hat day. That’s what we talked about at the dinner table. He was also casualty actuary for the New York St at e Insurance Depart ment and he was at Lumberman’s when he died in 1954. He was 49 years old. W here did he get his creative ideas? Well, I don’t know, I can only guess. But he did have a questioning attitude toward everything. And his father had a simi lar at ti tude, const ructi ve, but never took anything for granted. Did his employment permit study? His ﬁ rst job duri ng which he became an Associate of the Casualty Actuarial Society was with the Alliance. His boss t here did not allow any company t ime for st udying or even to t ake t he act uari al exams. So my father decided to become a member of the Actuarial Society. He was reading and studying all t he ti me anyway. He passed his ACAS exams in one sitting on his vacation. He never took another exam. He didn’t need to, he had what he wanted—to be able t o writ e papers and at tend meeti ngs. It wasn’t unti l quit e a whil e later the CAS asked i f he would accept an honorary Fell owship on the basis of his papers. Were t he seeds of his ideas plant ed in college? A l ot of hi s coll eagues complained about the k ind of math he used. The notation was archaic. It was t he notation he was taught at the University of Michigan and that he used at the United Fruit Company. In the bibliography to James Hickman’s essay there is menti on of one of his papers, ‘‘Credibilit y Procedures:
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LaPlace’s Generalizati on of Bayes’ Rule.’’ He was working wit h Bayes’ rule before he got int o t he insurance business. I have a book he was given in 1940 called Facsimiles of Two Papers by Bayes. It was published by a colleague of his at the United Fruit Company, W. Edwards Deming. So I know my father was working on Bayes’ rule before he got int o t he insurance business. Did he read some of the literature on statistics? I don’t know. I used some of his textbooks when I was in high school just to do some of the exercises. What he gave me would have been on st ati stics. But he read widely, a lot more than just actuarial literature. I have two of his books. One is Human Knowledge: Its Scope and Limits, Bertrand Russell, 1948. The other is The Philosophy of Physical Science, Sir Arthur Eddington, 1949. What was the response of his colleagues to his ideas? His coll eagues usually complained about his math. It was hard t o underst and and the notati on was difﬁ cult. But they all understood what he said when he spoke. I have a 20-page lett er from John Carl et on, about the 1950 paper ‘‘ Credibili ty Procedures: LaPlace’s Generali zation of Bayes’ Rule,’’ wri t ten about a month and one-half before my father deli vered his paper. Carl eton was aware of it s contents, so obviously my father had been discussing it with him. Carleton asked my father t o keep hi s lett er personal and conﬁdenti al, but it was eventually publi shed in 1993 by t he Casualt y Actuarial Society. It was published as a paper in the Casualty Actuarial Society Forum, Summer 1993, and called ‘‘True Inherent Hazards and the Futil it y Thereof.’’ I recommend i t because i t is a fri endly review of the content of my father’s paper and contains some genuine actuarial humor. On the ﬁr st page John says, ‘‘ Ignorance is an asset which the i nsurance industry should not dissipate thoughtlessly.’’ W hat was the response t o his ideas from ot her st at ist icians or philosophers of science? If you l ook at t he reviews of t he 1950 paper ‘‘Credibil it y Procedures,’’ it had several reviewers who had nothing to do with the insurance business. They were former coll eagues or people that he knew and had been in touch wit h on mathemati cal statistics, not insurance.
CREDIBILITY THEORY: THE CORNERSTONE
Can you talk about your work in credibility? I think the paper of mine, which I wrote jointly with LeRoy Simon, ‘ ‘An Actuari al Note on t he Credibili ty of Experi ence of a Single Pri vate Passenger Car,’’ is the oldest paper on the CAS reading list. It was a fun paper, not very mathematical. At the time, merit rating was regarded as unsound because the insurance for a single car did not have adequate credibili ty. I was then working with INA, which was independent, so I was free and LeRoy was free to tr y to demonstr ate that t he poli t icall y correct idea was incorrect. My contri buti on is more in t he applicati on area rather t han t he theoretical area. How did you get involved with credibility? Well, my father was making rates for insurance companies and he noticed that procedures they were using were inconsistent wit h what he had learned at the University of Michigan and had used at the United Fruit Company. His quest ion was ‘ ‘Why are they doing it that way? It doesn’t make sense.’’ And my ﬁrst job at t he Nat ional Bureau was making rates for the stock insurance companies. I was busy using cr edibi li ty procedures, so I was facing t he same questions. What is credibili ty? How are we using i t ?
H ANS BU¨ HLM ANN Professor Bu ¨ hlmann has been associated wit h ETH Zurich since 1966 when he j oined the universit y. He served as i t s president from 1987–1990 and is currentl y Professor Emeri tus. At Swiss Re, he was employed as an actuary from 1958–1966 and is currently a member of its Board of Directors. In addition, Professor Bu ¨ hlmann has held visit ing appoint ments at several universit ies i n Asia, Canada, Europe, and the Unit ed States. How did you get acquainted with credibility? In 1963, Swiss Re asked me to explore whether mathematics could be of use on the nonlife side of insurance. In part icular, I was t o famil iarize myself wit h t he role of casualty actuaries in the United States. I had just ﬁnished an 18-month leave at t he stat isti cs depart ment at the Universit y of California at Berkeley. On my way back to Zurich, I stopped at the National Bureau of Casualty Underwri t ers’ San Francisco and New York ofﬁces. I n New York, Thomas Carl son gave me Longley-Cook’s An Introduction to Credibility (1962), which immediately caught my int erest .
I lik ed t he concept, but I also felt that the mathematical reasoning behind it, as presented by LongleyCook, was not convinci ng. Hi s argument reli ed completely on frequenti st t hinki ng. Fir st , he deri ved t he necessary volume of ri sks t o obt ain full credibili t y. This was done by constructing a conﬁdence interval. Then, he used a standard deviati on argument to reduce full credibility to partial credibility for smaller volumes. But why should a conﬁdence interval, which by deﬁnition covered the true value only with probability less than one, guarantee full credibility? Back in Zurich, I started to work on the concept. I remember that I had a special drawer in my desk cont aini ng notes which I ulti mat ely publi shed (‘ ‘Optimale Pra ¨ mienstufensyst eme,’’ Swiss Actuarial Journal 1964). In the beginning, I was rather int erest ed in t he concept of experi ence r at ing. How should observed claims inﬂ uence premiums? As a reinsurer, we had all kinds of devices in operation where the issue came up: proﬁt commissions, slidi ng scale premiums, spread loss t reat ies, and so on. Were t hese devices actuari all y sound or just gimmicks i nvented by t he sales force? Slowly, my mathemati cal reasoning found i t s r oute. Premiums should not only depend on the individual risk but also on the collective from which the risk was chosen. This led to the type of reasoning known as Bayesian, where besides the indi vidual ri sk distr ibution, the collective is reﬂected by the probability distr ibution of t he risk parameters (called prior distr ibution by the Bayesian school). When I tried how this concept would work in a case where i ndividual numbers of cl aims are Poisson di st ri buted and the Poisson paramet er follows a Gamma distr ibuti on, I found to my surpri se one of the formulae in Longley-Cook’s t reati se ( actually t he one that he could only explain by a numerical approximati on). What a l ucky coincidence! Of course, later I reali zed t hat my Ameri can colleague Allen Mayerson had done the same calculat ion (Mayerson 1964). All en visited us in Zuri ch quit e regularly, his wife Dorli being from Zurich. My surprise became even bigger when later on I came across Art hur Bailey’s paper ‘‘Credibilit y Procedures— LaPlace’s Generalization of Bayes’ Rule and the Combinati on of Coll ateral Knowledge wit h Observed Data’’ (Bail ey 1950a). What a great paper explaining the concept of credibili ty. I have asked many Ameri can colleagues why thi s achievement of Arth ur Bail ey has been so l it t le valued and for a long t ime has been forgott en. They have usuall y answered t hat very few, if any, actuari es understood Bailey’s lectur es, and the mat hemat ical st at isticians fail ed t o t ake i nterest in Bailey’s paper. Nevertheless, one of the latter group,
Professor Richard von Mises, made a visionary statement ( Proceedings of the Casualty Actuarial Society, Vol. 37, Part 9, 1950). ‘‘It is to be hoped that those and similar problems will ﬁnd the attention of competent statist ici ans as the unjusti ﬁed and unr easonable att acks on the Bayes theory, init iated by R.A. Fisher, will fade out.’’ You asked whether t he earl y development of credibility is mainly due to Americans. My answer is a convinced yes. Even if Ove Lundberg i n Sweden ( ‘‘ On Random Processes and Their Appli cat ions to Sickness and Accident St atisti cs,’’ Thesis Uppsala, 1940, pri nt ed by Aluquist and Wiksell s) had alr eady in 1940 appli ed Bayesian modeling, he did not pursue the subject to arri ve at the very i ntui ti ve formula of a weighted average between indi vidual experi ence and collecti ve mean. What was the contribution of Continental European actuaries? In Nort h Ameri ca, credibili ty entered insurance via work ers’ compensati on. Subsequently, i t was used in other types of insurance, part icularly automobil e i nsurance. But on t he Conti nental European scene, it was ﬁrst introduced via auto insurance. In the 1960s, most automobi le companies suffered h eavy losses and a new rating system was absolutely needed. The soluti on arri ved in the form of Bonus Malus systems on a contractual basis for each individual driver. The French had t he ideas (Pierr e Delaporte, ‘ ‘Princi pes de Tari ﬁcation de l ’Assurance Automobil e par la Pri me Modele ´ e sur le Risque,’’ Transactions of the ICA, Vol. III, 1964). But the Swiss were t he ﬁrst to put them int o pract ice. The formula was not a weighted average, but a t able which Fri tz Bichsel had calculated using the Poisson Gamma model which I mentioned before ( Sw iss Actuarial Journal 1964). At that time in Switzerl and all companies appli ed a single t ari ff, and in 1964, the common tariff based on Bichsel’s table was int roduced. Many other Cont inental European countries followed with similar tables. It has been my privil ege to bring t his Conti nental European t hinki ng and the early Ameri can development s t ogether. Yet , there is sti ll one fundamental difference in concept. Whereas earl y Bayesian stat isti cians used the prior distr ibution of risk parameters as a means t o express judgment (which in insurance we would call underwriting judgment), our interpretation thinks of the probability distribution of the risk parameters as having an objective meaning. It hence needs to be extr acted fr om t he data gat hered about t he collect ive. (Only in the case of a l ack of such data might one
NORTH AM ERICAN ACTUARIAL JOURNAL, V OLUME 3, NUM BER 2
accept the subjecti ve view faute de mieux). For t his reason, I have always i nsisted on speaki ng about the str uctural dist ri bution of ri sk parameters, avoiding the standard Bayesian terminology, ‘‘ prior dist ri bution.’’ I think that the early American pioneers in credibili t y were tr uly Bayesians. On the other hand, we European Continentals insisting on the objective meaning of the str uctural distr ibution may for that reason have avoided t he mi d-century heated debat e in mathemati cal stati st ics confronti ng frequenti sts on one side and Bayesians on the other. It is interesting that in the United States statistical community, Professor Robbins (‘ ‘An Empir ical Bayes Approach t o St atistics,’’ Third Berkeley Symposium, 1955) has taken the same view of the prior distribution. Do you have any thoughts in conclusion? Credibi li t y, as developed by Ameri can actuari es, has provided us wit h a very powerful and int uit ive formula. The European Continental School has contributed to it s int erpretation. The concepts are fundamental to insurance and will continue to be most important in the future. I ﬁnd i t deplorable that the world of ﬁ nance has not yet reali zed t he importance of coll ateral knowledge far beyond insurance and the power of credibility-type formulae.
CHARLES C. HEW ITT Charl es Hewit t , F.C.A.S., graduat ed wit h honors in math from Princeton University in 1940. His career began at Met ropoli tan Li fe Insurance Company where he was recruit ed by Gilbert Fit zhugh. During World War II, Hewitt’s career was interrupted by service in the Army Air Corps, where he studied and taught meteorol ogy and served as a weather ofﬁ cer. His four-year stint incl uded serving i n Shanghai, China as the st ati on weather ofﬁcer. After his mili tary service, he swit ched to casualt y insurance, working at New Jersey Manufacturers Casualt y Insurance Company, Trenton, from 1946–1957. The company’s main line was workers’ compensat ion. He left t o work as a consult ant wit h Bowles, Andrews & Towne, now Tillinghast Towers-Perri n, t hen went to Ameri can Int ernational Underwriters (now AIG) and Allstate before returning to work for Metropolitan. There, he helped form Metr opolit an’s casualt y company as a vice president. When Metropolitan created its reinsurance company, Hewit t tr ansferr ed t o t hat enti t y and became it s president and CEO. He also served on the boards of both Met ropoli t an’s casualt y and reinsurance companies.
CREDIBILITY THEORY: THE CORNERSTONE
Hewitt is a past vice president of the American Academy of Actuaries and has served the profession in a number of capacities. He is the author of several papers and was the ﬁ rst winner of t he Dorweil er Pri ze in 1971 for his paper ‘‘Credibility for Severity.’’ In your discussion of Mayerson’s credibility paper you comment on the lack of research on risk theory in North America. Do you have an explanation? I think there are a couple of fairl y important ideas that contri buted t o this sit uati on. We li ved for 500 years, unt il t he development of nucl ear physics, in a worl d where everyt hing was exact and determini st ic. When you get int o t he study of nucl ear physics, you ﬁnd that what happens is probabilistic and is not det ermi ned absolut ely. That det ermi nat ion encouraged an envir onment in which people refused t o use a priori probabili ty. There was a stati sti cal school t hat predominated when I was being trained called the Neyman-Pearson school. It said you shouldn’t make any assumptions about probabili ti es before you did your analysis. You examined your observat ions and made a determinati on as t o whether or not the assumption was justiﬁed on the basis of statistical evidence. All you did was say ‘‘Wit hin an agreed range, this assumption is believable or not believable.’’ About the t ime Hans Bu ¨ hlmann wrote his paper, there were two Harvard professors, Raiffa and Schlaiffer, and a Yale professor, James Savage, who revived an interest in the theorem of t he English minister Thomas Bayes in which he did make assumptions about the a priori probabilit ies. His system was to make assumptions, then get data, then look at the observations and determine a posteriori probabilities. That’s Bayesian analysis. That revival took pl ace in the late 1950s and earl y 1960s, and when Bu ¨ hlmann came along wit h his paper in 1967 the idea obtained respectability. Did you know Arthur Bailey? Do you have any recollections of him? I know Robert Bail ey and I knew hi s fat her t angentially. I was told by others that he was a very religious man who carri ed a Bible wit h him. In meeti ngs, duri ng downtime when t he rest of t he att endees were otherwise engaged, he’d be reading his Bible. I do k now a story about hi m t hat shows his sense of humor. In the late 1940s, the auto liability insurance rating bureau for the old line stock companies found t hat t here were differences i n drivers, namely, t hat young dri vers are a bigger ri sk t han older, more sedat e
drivers. Now i n those days, there were two rati ng bureaus that made the auto insurance rates—one bureau rated liabili ty and one rated colli sion/ comprehensive. The li abili ty rating bureau reali zed young drivers were bad ri sks. And t he ot her rating bureau didn’t catch up with this for a few years. I remember Arthur, who was casualt y actuary of t he New York St ate insurance depart ment, got up in a meet ing and said he couldn’ t understand how young drivers could do all thi s damage to other people’s cars without doing any damage to their own. Did you see the papers by Bailey and Mayerson before they were published? If so what was your view? I did see the Mayerson paper before it was publi shed but not the Bail ey paper. I discovered a mi nor mispri nt in the Mayerson paper before i t was published. It was before Bu ¨ hlmann’s paper. Mayerson’s ideas were an eye-opener for me and prepared my mind for t he Bu ¨ hlmann paper later on. The Bu ¨ hlmann paper was a great paper and got me pointed in the right direction. In your earlier education, had you learned about the Bayesian approach? This would be before t he Art hur Bailey and Allen M ayerson papers. When I ﬁrst got into the casualt y business, I knew there was a better way of making rates because of my mat h background. I was myst iﬁ ed that no one had come up with a solution. By the time the Mayerson paper came out, I was beginning to narrow down my t hink ing. When I read t he Bu ¨ hlmann paper i n 1967, it was li ke a giant li ght bulb li ghting up over my head. The Bu¨ hlmann paper was t he answer when one i s t hinki ng about the problem of making i nsurance rates or judgments; i nsurance uses mathematical stati st ics. I became a crusader. I became somebody who wanted to carry the gospel to everyone else. At the time, did actuaries recognize the fundament al nat ure of t he issues being presented by Bailey and then Mayerson? I think they didn’t. Many actuaries still don’t understand the fundamental nature of credibili t y. The t raining of the Neyman-Pearson school was partly responsibl e. I’ ve been in meetings where people have gott en up very ir ri tated wit h t he rest of us and t hey say, ‘‘ You cannot make a priori assumptions. You’re not allowed to.’’ In the math and statistical training of some of
NORTH AM ERICAN ACTUARIAL JOURNAL, V OLUME 3, NUM BER 2
these people they are taught you cannot make a pre judgment of probabili ti es, of somethi ng t hat’ s going to happen. Of course, I think some of them may not understand t he math. It ’s not that complicated but that may be part of it. Did you use credibility theory in your career? The answer i s yes though I couldn’t cit e you any references. I’m using it now in a completely different appli cation. For instance, I have often beaten the odds on football games in t he Las Vegas spreads, not for money, just for fun. I will say one thing. There are a lot of people that have gone way beyond my math ability in using credibil it y theory. Not all actuaries are t heoret ical actuari es. I just wish more people would understand t he principle. It’s been a great joy to me to see the idea. That has been one of the hi ghlights of t he mat hematical and theoretical aspects of my career.
REFERENCES BAILEY, A.L. 1942. ‘‘Sampling Theory i n Casualty I nsurance,’’ Proceedings of the Casu alty Actuarial Society 29, no. 59: 50–93. BAILEY, A.L. 1945. ‘‘A Generalized Theory of Credibility,’’ Proceedings of the Casualty Actuarial Society 32, no. 62: 13–20. BAILEY, A.L. 1950a. ‘‘Credibility Procedures: LaPlace’s Generalization of Bayes’ Rule and the Combination of Collateral Knowledge wit h Observed Data,’’ Proceedings of the Casualty Actuarial Society 37, no. 67:7–23. BAILEY, A.L. 1950b. Discussion, Journal of the Am erican Teachers of Insurance (now The Journal of Risk and Insurance) 17:17–24. BAILEY, R.A., AND SIMON, L.J. 1959. ‘‘An Actuarial Note on the Credibil it y of Experi ence of a Single Pri vate Passenger
Car,’’ Proceedings of the Casualty Actuarial Society 46, no. 85:159–64. BAILEY, R.A. 1961. ‘‘ Experience Rati ng Reassessed,’’ Proceedings of the Cas ualty Actuarial Society 48, no. 89:60–82. ¨ HLMANN, H. 1967. ‘‘Experience Rating and Credibility,’’ The BU ASTIN Bulletin 4, Part III:199–207. CARLETON, J.W. 1950. ‘‘True Inherent Hazards and the Futility Thereof,’’ Casualty Actuarial Society Forum, Summer 1993:285. HERZOG, T.N. 1994. Introduction to Credibility Theory. Winsted, Conn: Actex Publications. HEWITT, C.C. 1965. Discussion of ‘‘A Bayesian View of Credibility,’’ Proceedings of the Casu alty Actuarial Society 52: 121–27. HEWITT, C.C. 1970. ‘‘Credibili ty for Severit y,’’ Proceedings of the Casualty Actuarial Society 57, nos. 97, 98:148–71. KEFFER, R. 1929. ‘‘An Experience Rating Formula,’’ Transactions of the Actuarial Society of America 30:130–39. KLUGMAN, S.A., PANJER, H.H., AND WILLMOT, G.E. 1998. Loss Models: From Data to Decisions. Somerset, N.J.: John Wiley & Sons. LONGLEY-COOK, L.H. 1962. ‘‘An Int roducti on to Credibili ty Theory,’’ Proceedings of the Casualty Actuarial Society 49, nos. 91, 92:194–226. MAYERSON, A.L. 1964. ‘‘A Bayesian View of Credibili ty,’’ Proceedings of the Casualty Actuarial Society 51, nos. 95, 96: 85–104. MOORHEAD , E.J. 1989. Our Yesterdays: The History of the Actuarial Profession in North America 1809–1979. Schaumburg, Ill.: Society of Actuaries. RODERMUND, M. 1990. ‘‘Introduction,’’ in Foundations of Casualty Actuarial Science. New York: Casualty Actuari al Society. VENTER, G.G. 1990. ‘‘Credibility,’’ in Foundations of Casualty Actuarial Science. New York: Casualty Actuarial Society. WHITNEY, A.W. 1918. ‘‘The Theory of Experience Rating,’’ Proceedings of the Casualty Actuarial Society 4, nos. 9, 10: 274–92.