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THE ASTIN BULLETIN

PUBLICATION OF THE ASTIN SECT]iON OF THE PERMANENT COMMITTEE FOR

INTERNATIONAL ACTUARIAL CONGRESSES

VOL, IV, PART I l l

33 , ' ' ~ '

J 11 L Y 1 9 6 7

CONTENTS page

In 3{emoriam Frederik Esscher . . . . . . . . . . . . I93

Notes by Chairman . . . . . . . . . . . . . . . . . . 195

A slin Colloquium I965 Lucerne Subject two

Experience rating and credibility by Hans Biihlmann, Ziirich 199

Experience rating when the company aims to increase the volume of its business by Herald Bohman, Stockholm 208

Experience rating in subsets of risks by Fritz BichseI, Berne . 2Io

Asfin Colloquium 2-965 Lucerne subject three

Contr61e des op6rations d'assurance dans les branches non- vie par F. Bichsel, Berne . . . . . . . . . . . . . . . 23I

On the solvency of insurance companies by T. Pentikdinen, Helsinki . . . . . . . . . . . . . . . . . . . . . . 236

Magnitude control of technical reserves in Finland by Erkki Pesonen, Helsinki . . . . . . . . . . . . . . . . . . 248

The economic theory of insurance by Karl Borch, Bergen . 252

Markov chains and the determination of fair premiums by Stefan Vajda, Birmingham . . . . . . . . . . . . . . 265

Reviews . . . . . . . . . . . . . . . . . . . . . . . 269

Edi tor : C, P. Welten, Brans & Co., Koninginneweg x5 I, Amsterdam-Z, Holland.

IN MEMORIAM F R E D R I K ESSCHER

For many Astin members Esscher's name is associated with his method for the approximation of a distribution function for which the corresponding characteristic function is known. This method was in 1932 introduced for the Poisson process and in 1963 extended to a general distribution function. In 1963-1964 Esscher and Boh- man compared the results of this approximation for a Poisson and for a Polya process with the results of another evaluation with known precision over a wide field of the variable and of the process parameter. In a large domain of this field the agreement between the compared functions was found to be excellent. In the paper of 1932 Esscher also gave the exact expression for the distribution functions defining a Poisson process in the case, where the size of a change is dependent on the parameter point at which the change has occurred. In 1965 Esscher published a paper on stop loss premiums for large portfolios under different assumptions for the risk distribution of the process. The papers mentioned above were all published in Skandinavisk Aktuarietidskrift, and the last one, in addition, summarized in the Astin Bulletin 1967 .

The chief field of Esscher's activity was Life Insurance. From 1921 to 1928 he was actuary in Allm~inna Pensionsf6rs~ikrings- bolaget and from 1928 to 1959 chief actuary in Skandia and affilia- ted companies. In his inaugural dissertation (192o) and in the 25 years Jubilee Volume of the first mentioned company (1925) he developed statistical methods for the estimation of the force of mortality, which have had a material influence of later Swedish studies of mortality. In Skandias 75 years Jubilee Volume (193o) Esscher published two theoretical papers, one on interpolation and the other on graduation with certain polynomials according to the principle of least squares. In both papers the results were applied to different functions commonly used in life insurance technique. On several occasions Esscher took a prominent part in the prepara- tion of new technical bases for the calculation of premiums, sur- render values and office values, from 1928 common to all Swedish offices transacting ordinary life insurance. For his own companies

13

he elaborated a system for the distribution of dividends in accordance with the principle of equity provided in the Insurance Act of 1948, which system has formed a model for many life offices. From I95o--when the life offices established the Research Council for Actuarial Science and Insurance Stat ist ics--to 1965, Esscher was chairman of the council. He was a prominent member also of other joint committees, and for a long time member of the board of the Swedish Society of Actuaries, where he later was elected as an honorary member.

Also in other fields Esscher's contributions have been highly valuable. The charge for administration costs in the premiums of Third Par ty Liability Insurance, since 1929 compulsory for the owners of motor vehicles in traffic, has since the beginning been calculated according to ideas propounded by Esscher in I928. In 1956 he published a paper for the Nordic Conference of Casualty Insurance on rating problems in Motor Insurance with particular regard to no-claim bonus. The treaty of the pool of Nordic Accident Insurance, which treaty implies individual excess cover for accidents involving more than one injured person, was in 1963 amended in main conformity with a draft given by Esscher. He also studied internal problems within Management Science, for his companies, e.g. problems connected with underwriting ans selling costs.

In an article of 1959 Harald Cram6r pronounced about Fredrik Esscher that in him theoretical learning and practical ability had formed a union of rare harmony and, that he in debates used to brandish a swordblade of sharpedged steel. By the Swedish actuaries Esscher's lucid spirit and unusual ingenuity will always be com- memorated.

mai 1967 C. PHILIPSON

NOTES BY CHAIRMAN

This year, I967, marks the Ioth anniversary of ASTIN. As president of the Casualty Actuarial Society, I had the distinct privilege of being chairman of the inaugural meeting of ASTIN, The day was October x6, i957 during the XVth International Congress of Actuaries meeting in New York. Subsequent sessions of this XVth Congress followed in Washington, D.C. and Toronto, Canada. The first order of business at this historic meeting was my introduction of and presentation of Sir George Maddex, the official member of the Permanent Committee on the ASTIN Com- mittee. He called upon Prof. Edouard Franckx to present the report of the ASTIN provisional committee. With the adoption of this report by Prof. Franckx and the subsequent election of the first ASTIN Committee, we launched a most distinctive and distinguished international organization.

Lest we forget--after xo years of growth and progress--that first committee consisted of: Sir George Maddex as the official member of the Permanent Committee, Prof. Edouard Franckx, B. Monic, Robert E. Beard, Paul Johansen, Carl Philipson and Francis Perryman. These men were not only distinguished actuaries, but they had worked diligently to launch ASTIN following the XIVth Congress in Madrid in ~954.

We have convened for six colloquia during this first decade m La Baule I959, Rattvik x96i, Juan-les-Pins I962, Trieste I963, Lucerne I965 and Arnhem I966. In addition to these ASTIN colloquia we participated in International Congresses in Brussels I96o and in London--Edinburgh I964. We look forward to x968 and the next Congress meeting in Munich and the ASTIN meeting in West Berlin.

I have related our history in terms of places. But equally important is a recognition of persons. Prof. Edouard Franckx as honorary chairman and my four predecessors--Paul Johansen, Marcel Henry, Robert Beard, and Hans Ammeter furnished direction and quality of leadership which advanced ASTIN.

ASTIN has paralleled the Space Age. It was just twelve days

prior to our inaugural meeting in 1957 that the first sputnik of the Soviet Union was launched into orbit.

Is ASTIN doing well ? This question has provided discussion, excitement, and sole-searching as we approach the end of our Ioth year. My answer to the question is YES-- fo r its first decade. There has been definite progress in non-life insurance dialogue between my country and Western Europe, There has been an accelerated rate of exchange of papers, ideas, discussion among national actuarial groups and between individual actuaries. But for the second decade, we have a responsibility to continue to make progress to get favorable responses to another question: Is ASTIN doing better ?

Norton E. MASTERSON

ASTIN COLLOQUIUM 1965 LUCERNE

SUBJECT TWO

EXPERIENCE RATING AND CREDIBILITY

INTRODUCTORY REPORT

EXPERIENCE RATING AND CREDIBILITY

HANS BOHLMANN Ziirich

I . T h e C l a s s i c a l R a t e m a k i n g P r o b l e m

Classical statistics deals with the following standard problem of estimation:

G i v e n : random variables X1, X 2 . . . Xn independent, identically distributed, and observations x l , x~ . . . xn ,

E s t i m a t e : parameter (or function thereof) of the distribution function common to all X,.

It is not surprising that the "classical actuary" has mostly been involved in solving the actuarial equivalent of this problem in insurance, namely

G i v e n : risks Rt, R 2 . . . Rn no contagion, homogeneous group, F i n d : the proper (common) rate for all risks in the given class.

There have, of course, always been actuaries who have ques- tioned the assumptions of independence (no contagion) and/or identical distribution (homogeneity). As long as ratemaking is considered equivalent to the determination of the mean, there seem to be no additional difficulties if the hypothesis of independence is dropped. But is there a way to drop the condition of homogeneity (identical distribution) ?

2. P r a g m a t i c S o l u t i o n

Insurance people are practically minded, and so they have in many circumstances come out with pragmatic solutions, even if the theoreticians were not able to provide them with full theoretical justification for doing so. Within the classical setup such circumstances were happening if

2 0 0 E X P E R I E N C E R A T I N G A N D C R E D I B I L I T Y

a) you had to rate a risk with no or rather dubious observational data on the class to which it belonged,

b) you had to rate a risk which could not be grouped into a homogeneous class except for one so small that statistical inference drawn from it had little significance.

The pragmatic devices to handle such cases are known to all of us, namely

- - participation in profits - - no claim bonus - - premium scales (bonus-malus) - - sliding scale premiums and/or commissions in reinsurance.

All these devices are characterized by the fact that the rate originally fixed is gradually altered during the contractual period of risk coverage. This is called experience rating. It is done, yes, but what about its justification ? What does the actuary have to say about it ?

3. Report on Papers Submitted

Seven papers have been sent in oil the subject "Credibility and Experience Rating", and it is interesting to note how actuarial techniques vary in treating this subject.

Let me first speak on the contribution by Harald Bohman, "Experience rating when the company aims at increasing the volume of its business". This paper shows that the scope of experience rating may even be extended beyond "proper-rate making". Aiming at maximum volume of business is the alternative treated by Harald Bohman. His main result, namely that the maximum volume is generally not achieved at the lowest possible premium rates, is certainly of great practical value.

The other six papers treat the problem of experience rating with the aim of finding the correct rate. As all the authors use some form of sequential statistical techniques to tackle the problem, I shall restrict myself to the discussion of such sequential techniques as a means of coping with experience rating.

It is noteworthy that the classical sequential methods, such as those developed by Abraham Wald, do not permit to go beyond the testing of a whole tariff class. Having the two cases above in mind,

EXPERIENCE RATING AND CREDIBILITY 20I

where practical circumstances force some form of experience rating upon us, I would say that Wald's Sequential Probabili ty Ratio Test can be of help in case a), but unfortunately fails to solve the problem in case b). Lars Benckert brings this out even in the title of his paper "Testing of a Tariff", and I judge it very important to stress the word tariff as against individual risk. Indeed, Lars Benckert shows us how to control the adequacy of a tariff rate for a whole class using the logarithmic normal distribution for individual claim amounts.

To get a classification of risks inside a tariff class, it is necessary to treat the risk parameters no longer as constants, but as random variables. Only if you take this step of generalization, then you can "hunt accident prones", as Paul Johansen does. I am very glad that he has contributed this paper to the subject under discussion, since it shows very clearly the advantage which can be gained by assuming the risk parameter to be a random variable. This advantage, by the way, is not bought at the expense of an increasingly complicated formula, a fact that for practical purposes will certainly be appreciated as well.

Two papers, namely, "Experience Rating in Subsets of Risks", by Fritz Bichsel, and "Une note sur des syst&mes de tarification bas6s sur des modules du type Poisson"l), by Ove Lundberg, center around the application of the Bayes rule for experience rating purposes. Since Neo-Bayesian techniques seem to really be quite well adapted to experience rating problems, let me expound in greater detail on this approach.

We have said earlier that we would like to have a method which no longer requires the grouping of individual risks into homogeneous classes. What do we actually mean b y homogeneity? This is best illustrated by the following array of random variables X,~ = claim produced by risk i in the accounting period (year) j.

Xll, Xle . . . . Xi•

Xml, Xm2 . . . . Xmn.

a) Publ i shed in As t in Bu l l e t in vol . I V Part I.

202 EXPERIENCE RATING AND CREDIBILITY

The above array represents all "claims random variables" under observation from a class of m risks during n years.

i) This class would be called homogeneous in the mass of risks, if Xlj, X,~, Xsj . . . . Xmj are identically distributed for all fixed j.

ii) Each individual risk would be called homogeneous in time if X~i, X,2 . . . . X ,n are identically distributed for fixed i.

Dropping the requirement of homogeneity means either dropping i) or ii) or both at the same time. All actuarial work so far has, at least according to my knowledge, been done by dropping homogeneity in mass but by still requiring homogeneity in time of the individual risk *). Fritz Bichsel now also tackles the experience rating problem under changes of individual risks in time. He has thus added an additional dimension to the experience rating problem, a dimension which I consider most important from both theoretical and practical viewpoints. If I may make a suggestion it is this: the changes in time considered by Bichsel are random changes; it would be most valuable to treat the problem also under the aspect of trend type changes.

Ove Lundberg seems to have been the first actuary to realize the importance of Bayes procedures for experience rating. The basic principle is already mentioned in his I94O book.

Now that we other actuaries are understanding more and more the importance of his early work, we particularly welcome his new contribution "Une note sur des syst~mes de tarification bas6s sur des modules du type Poisson compos6". His main result consists in the proof of the consistency of the Bayes estimate derived from a Poisson process. In contrast to the Bayes estimate, Ore Lundberg shows that any estimate based on the time of absence is not consistent--a very powerful theoretical undermining of the super- iority of the bonus--malus system against the simple bonus system.

*) In the discussion Carl Philipson has pointed at some earlier researches of Ammeter ("A Generalization of the Collective Theory of Risk in Regard to Fluctuat ing Basic Probabilities", Skand. Akt. Tidskrift x948) and Philipson ("Einige Bemerkungen zur Bonusfrage in der Kraftversicherung" BDGVM I963; Eine Bemerkung zu Bichsels Herleitung der bedingten zuktinftigen Schadenh~ufigkeit einer Polya-Verteilung" MVSVM 1964).

EXPERIENCE RATING AND CREDIBILITY 203

But now let's turn to the second term in the heading of the subject discussed: credibility.

I t is quite remarkable that our American colleagues have been well advised in solving most of their Experience Rating problems in applying the by now famous credibility formula:

Pn = (i - - ~¢) pn-l+o~.rn-1

p~ : rate for period k r , = loss ratio for period k ~t is called credibility and assumed to be a function of the

volume V, mostly

V I if V >~ Vo

= if V < Vo

V0 is called the volume offidl credibility.

Hetereogeneity in mass, in time, rate changes for big groups and small groups as well, even some refund formulas are treated by the credibility method on the United States marke t - -and this is the remarkable fac t - - the credibility formula has been found to do an excellent job. Under such circumstances it does not come as a surprise that many actuaries have tried to prove the credibility formula starting from more general principles. The first to do so was Arthur Bailey, one of the most outstanding American actuaries in the mid-century to whose work Bruno de Finetti has drawn our attention at the Trieste colloquium. Since then I am sure that many of us have made our own personal attempts. Edouard Franckx in his paper "La ratification et son adaptation exp6riment- ale dans le cadre d'une classe de tarif" arrives at the credibility formula by starting from the principle of least squares. It seems to me that in doing so he is the first author to prove the credibility relation independent of the distribution function which governs the individual risks (but still dependent on the prior distribution of the risk parameter or parameters). This would justify our American colleagues' using credibility procedures beyond the assessment of claims frequencies--a point that since Arthur Bailey has worried many of our very best colleagues across the Atlantic.

I regret that there has been only one paper sent in which specifi-

204 EXPERIENCE RATING AND CREDIBILITY

cally deals with credibility techniques. Marcel Derron with his contribution "Credibility Betterment Through Exclusion of the Largest Claims ''1) takes up some very interesting thoughts originated by Hans Ammeter in Trieste. Derron computes the credibility improvement for a Pareto-type distribution if the largest claim is excluded. The result may indicate a new way of approach for many other estimation problems which we encounter in our work. I mean that Marcel Derron has very clearly shown to us the importance of truncating or curtailing our basic data to get more reliable information out of them. The importance of his work is underlined by many theoretical statisticians who are presently working on robust estimation methods, and who propose also the exclusion of extreme values to make the information more reliable.

With this I conclude my report on the papers submitted. Permit me though to add two rather special technical remarks as my personal contribution to the discussion.

4. Equ i l i b r ium in an experience rated portfolio

Consider a set @ of risk parameters 8. Each individual risk (characterized by a value of 8) can be observed on a number of random variables X,, i = I . . . n (To fix the ideas think of claim frequencies or claim sums produced by an individual risk in the year i).

As is usually done I assume the Xt's to be independent and identically distributed (homogeneity in time, but not in mass). Call the common distribution function Fo(x) with mean ~(~) and variance ,2(~). Experience rating is then understood as a sequence of estimates for ~(~) based on the observations of X1, X2 . . . . Xn.

Except for one trial on Delaporte's side where the a posteriori median is investigated I believe that in the actuarial literature the estimator function for ~(~) is always chosen to be equal to

E[~(~)/X1, X~ . . . . Xn] = a posteriori mean.

Justification for this choice is usually found in the fact that among all functions f depending on the observations only (and integrable of course) the expected square deviation.

J" {~z(b)--fCxl . . . . xn)} a dEs (xl)dF~(x2) . . . dV~(xn) dS(b)

1) Publ i shed in Ast in Bullet in Vol. IV P a r t i .


where S(B) denotes the "a priori distribution" of the risk parameter (structural function of portfolio), is smallest for the choice

f = E[tx(O~)/X1 . . . . Xn].

I feel that the least expected square deviation is no sufficient justification for the choice of the a posteriori mean as estimator function. Permit me therefore to set forth another justification which Paul Thyrion has indicated to me in a recent letter.

The basic idea can be described by the postulate of equilibrium in all those subclasses of a portfolio which are characterized by experience only. In other words it is postulated that each class of risks with equal observed risk performance should pay its own way. Let us t ry to put this into mathematical language. Let

(9 = set of all possible parameters

X = set of all possible observed risk performances (Xi, X2, . . . Xn) X' = subset of X, c(X') = cylinder in (9 × X with base X' c X

P = probability on (9 × X

Equilibrium for any subset X ' means

f ~(~)dP = / f(xl, x~ . . . . xn) dP for all cylinders c(X')

The above relation is exactly Kolmogoroff's definition of the conditional expectation

E[tz(~)/x, , . . . x,]

5. A Distribution Free Credibility Formula

I t is worthwhile to note here that the credibility folmula used by our American colleagues is nothing but a linearization of the above estimator function E[~(~)IX1 . . . . Xn].

I am giving here the least expected square deviation approximation to this a posteriori mean. (Observe that E E .J means expectation with respect to the probability P on the product space ® × X, E~[.] means expectation on X given the individual risk ~).

The best linear approximation to

E[W(~)/x 1, . . . x,~ is found by solving the following problem

2 0 6 E X P E R I E N C E R A T I N G A N D C R E D I B I L I T Y

F i n d : a, b such t h a t

X1 + X2 . . . . X n a + bX where X7 =

n

a p p r o x i m a t e s E[~(8) /x , , . . . x ~ best

i.e. E {E[~(8) /x , , x , , . . . x,~ - - (a + bX)} 2 = m i n i m u m

L e m m a : I f E(ao + b o X - ~(8)~) ~ E(a + b X - - ~ ( b ) 2) for a r b i t r a r y a and b

then ao + boX is also the best l inear a p p r o x i m a t i o n to

E [ t ~ O ) / x , , . . . x 2

Proof: E (ao + b o x - - ~(~))~ = E {ao + boX - - E[v (b ) / x , , . . . .

x 2 } ~ + E { E [ v . O ) / x , , . . . x 3 - - ~ ( ~ ) ?

Since the second t e r m on the r ight hand side does not depend on ao and bo it is clear t h a t the left hand side and the r ight hand side are mimimized b y the same ao and bo q.e.d.

Re fo rmula t ing the original p rob lem we hence w a n t to f ind a and

b such t h a t

E[a + b X - - ~t(O~)] 2 = m i n i m u m

We find t h a t the above left h a n d side can be wr i t t en

E [ b ( X - - tx(8))] ~ + EIa - - ( I - - b)~(0")] 2

which is m i n i m u m if i) a = (I - - b)E[vt0) ]

and ii) b2E(X - - ~t(8)) 2 + (I - - b) z Var Err(S)] mini- m u m

var[~O)] which leads to b =

Var[t~(b)] + E ( X - - ~ ( ~ ) ) ~

Assuming now independence and ident ical d i s t r ibu t ion of the

X , we find

I I - E ( X , - - ~ ( ~ ) ) ~ = - E [ ~ ( 0 ) ] E(X-~O))~ = n n

Hence the credibility relation

(I - - b)" E[~(8)] + b" X


where

n b - - - -

n + k

k - - Var[vt(8)]

R e m a r k s :

I) This relation makes no assumption as to the type of distribution function governing the individual risk or the a priori (structural) distribution function of the parameters.

2) The hypothesis of independence and identical distribution of the observational random variables of the same individual risk could easily be dropped. I t amounts to replacing the relation

I E ( X - - ~(~))2 = n E[~(~)I by some other function of n.

3) Since the above relation is generally true it is of great interest to estimate

E[a2(~)] and Var [~(~)]

directly from certain "a priori observations". This problem has not yet been attacked.

EXPERIENCE RATING WHEN THE COMPANY AIMS TO INCREASE

THE VOLUME OF ITS BUSINESS

HARALD BOHMAN

Stockholm

An insurance company aims to increase the volume of its business. The volume is measured by the premium income during one year.

The company makes use of N tariff groups.

Pn ---- premium charged by the company per unit of insurance in tariff group n.

qn = premium per unit of insurance in tariff group n corresponding to the statistical experience in the said group.

Ln ---- The size of the market in tariff group n. If all insurances in the group pay the same premium and if the company is able to cover the whole of its potential market then Ln is equal to the number of policies. If the premium per policy is equal to Pn times the sum insured and if the company is able to cover the whole of its potential market then Ln is equal to the total sum insured.

Fn(p) = the proportion of Ln that actually buys insurance in tariff group n if the company charges the premium p per unit of insurance.

The total premium P paid to the company for the portfolio will be equal to

/v

P = Z Ln .Fn(pn) .pn t

Administration costs and claims paid by the company for the portfolio will be equal to

A = Z Ln.Fn(pn) .qn 1

The aim of the company is to make P as large as possible while P is kept larger than A, or

P = k.A wi thk >/ I

EXPERIENCE RATING AT INCREASING VOLUME 209

If we put d P = 0 we get

Fn(pn) P*' = F 'n (Pn) for n = I, 2 . . . . . N,

choosing the root of the equation which makes Pn .Fn(pn) as large as possible and if these values inserted in

P k - -

A

give a k-value larger than I then pl, p~ . . . . . PN are a solution to the problem. Otherwise we procede as follows. Put t ing Pn = qn for n = 1 ,2 . . . . . N w e g e t

P = ]~ L n . F n ( q n ) . q n = Po

which means that P-----A. I t is then possible to find a solution pl, p2 . . . . . pN which makes P as large as possible while k / > I and this P is a fo r t io r i larger than Po. This solution may be found by some suitable numerical method.

We will not follow this mathematical model into more detail and draw only the following general conclusions from the preceding discussion. An insurance company which wants to fix its premium rates in such a way that the total premium income is both sufficient and as large as possible

shall probably not put Pn = qn for n = I, 2 . . . . . N.

Making the premium income as large as possible means here of course what is achievable by a mere change of the premium rates. Nothing is said of other actions which the company may take with the same intentions or actions taken by competing companies. Such effects are not studied with the aid of the mathematical model described here.

74

EXPERIENCE RATING IN SUBSETS OF RISKS

FRITZ BICHSEL

B e r n e

0 . - - P R E L I M I N A R Y "

00. - - S u m m a r y

We consider a set of risks B, which is divided into subsets B, according to some property as profession, region etc. The subsets B, are thought to be too small to permit a calculation of a premium from their own experience. On the other hand we assume that there are differences between the B,. This paper tries to give a solution to the problem how the experience of the single Bt should be com- bined with that of the whole B.

The problem is treated only with respect to frequencies of claims. The amounts of the claims are not taken into consideration. I suppose that an extension of the theory to amounts is possible and highly desirable for practical applications. Lack of time did not permit me however to proceed to such an extension in this paper.

01. - - Practical appl icat ion

I think that the methods developed in the sequel might be applied to the following practical problems:

- - rate making - - e x p e r i e n c e rating in the restrictive sense of the word, i.e. ad-

justing periodically the premium or granting premium refunds according to the experience of every single Bl

- - judging the experience of a B~.

02. - - References

I do not give a list of references which would have to be very long. The basic idea was, as far as I know, first developed by Lundberg.

In order to give a complete and systematic account, this paper

E X P E R I E N C E RATING IN SUBSETS OF RISKS 211

includes the derivation of some basic results which have been known for a long time.

What is new, are, as far as I know, the results of paragraphs IX, 211 and 221-225.

03. ~ A c c u r a c y

It is a principle of this paper to take into consideration only the mean and variance of the occurring distributions, neglecting moments of higher order. Thus we shall, whenever this will prove convenient, replace any distribution by any other distribution having the same mean and standard deviation.

I think that this manner of proceeding is adequate to the nature of the problem, at least for a first approach.

04. - - Subject ive probabi l i ty

The terminology used, especially in paragraph I, is that of the so-called subjective probability theory. I think that the more extensive interpretation which the "subjectivists" give to the concept of probability, is more adequate to the problems treated in this paper than that of the " f r e q u e n t i s t s " .

The proof of the pudding lies in the eating. I hope that the pieces offered in the sequel will be to the taste of some of the readers.

05. ~ Nota t ion

0 3 0 . - - We shall always carefully distinguish between a random variable ~, ~ etc. and the variables x, y etc. or n, m etc. occurring in the mathematical expression for its distribution or frequency functions.

Thus we shall write

P ( ~ = n ) = = ( n ; a )

(compare o6o for the meaning of ~ (n; a))

for a random variable ~ following the simple Poisson-distribution. The meaning of ~, ~ etc. is related to the problem treated whereas

x, y . . . . . n, m . . . only have a purely mathematical meaning. 0 5 1 . - In order to give the frequency function of a continuous

random variable we shall write

P (~ ~ x ) = f ( x ) .

2 1 2 EXPERIENCE RATING IN SUBSETS OF RISKS

Thus if ~ follows a P-distribution with parameters a and b we shall write (compare o6i)

P (~ ~ x ) = ~ (x; a, b)

I apologize for introducing this new notation, but it has proved useful to treat the problem.

Note that P (~ ~ x) may be greater than I.

0 5 2 . - - As usual we shall write

P ( ~ = n l ~ = y )

to denote the conditional probability that the random variable assumes the value n, granted that another random variable

has the value y.

I think that the notation

P ( ~ ~ x l ~ = y )

derived from that of o51 needs no further explanation.

0 5 3 . - - A c c o r d i n g to the principle exposed in o3, we shall occasionally write

F (x;a, b)

to denote any function being the distribution function of a random variable with mean a and variance b.

06. - - Abbreviations

We shall use the following abbreviations:

a n e - a 0 6 0 x (n ; a) - - n !

This is the frequency function of the (simple) Poisson-distribution.

t t

- - e - x / O 061 y (x ; a, b) = P(a/b)

This is the frequency function of the F-distribution.

062 qb (n; a, b) = \ ~ - ~ / ~ F (a/b)n!

EXPERIENCE RATING IN SUBSETS OF RISKS 2 1 3

This is the f requency funct ion of a composed Poisson-distri- but ion, having the dis t r ibut ion o61 as s t ruc tura l funct ion (see 073 ).

0 7 . - - L e m m a s

0 7 0 . - - I f P (~ = n) = ~ (n; a), then ~ (~) = a,

~s (~) = a.

0 7 1 . - If P (~ ~ x) = -~ (x; a, b), then bt (~) = a

a s (~) = ab.

(Note the difference to the nota t ion of o53 ) .

0 7 2 . - If P (~ ~ x) = "~ (x; a, b) then

073. - - If

P (c~ ~ x) = T ( x ; c a , cb).

P (~ = n i t = y) = ~ (n; y) and P ('~ ~ Y) = T ( y ; a , b),

then P (~ = n) = jv ~ (n; y) "T (Y; a, b) d y @

=- ~b (n ; a, b) with ~ (~) == a

~s (~ ) = a ( ~ + b )

Note tha t ~ (~) and as (~) also follow from o7o and o71 b y an applicat ion of L e m m a o75.

0 7 4 . - - + (n; a, b) = Y x', ~ , ~ + b

075. - - We use the nota t ion explained in o53, writ ing F1, F~ etc., instead of F in order to distinguish different functions.

If P ( ~ ~ x [ ~ q = y A ~ = z ) = F l ( x ; y , z ) and P(~q ~ y ) = F s ( y ; v t y , asu)

P ( ~ ~< z) = F3 [z; a s z (Y) , * 2 z(Y)]

a% (y) and .sz (y) being a l l o w e d t o depend on y ,

then t~ (~) = try

with *x=-2 ~ *~x (Y) dF2 (y, Vtv, ~sv) 0

214 EXPERIENCE RATING IN SUBSETS OF RISKS

and thus P (4 ~ x) = F4 (x; ~y, ~ x + trey)

Proof for a e (~):

~e (4) = ~ [ [ ( x - - ~)e dF, (x; r, z) aF~ (y; ~ , ~e) dF3 (z; ~ , ~e~) @ - c o - ~

= J" ~ E~ + ( y - ~y)~l riFe (y; ~v, .~y) rife (z; . e , . e ) Q

+ ~ ( y _ ~y)e dF~ (y; ~v, .~y) = eez + .~y

1 . - - SINGLE RISKS

In this paragraph we treat single risks or sets of risks which are considered as one single risk. We derive two theorems which are fundamental for the sequel. The first has been known for a long time, the second is, as far as I know, new.

In both theorems we assume that the risk under consideration has been observed during time unity and that k claims occurred. (Throughout the whole paper we take the period of observation as time unity. Every result may, by simple transformations, be extended to other time scales).

From this observation, we wish to determine the " t rue" claims frequency k of the risk, which we do not know exactly, but of which we presume to have some a priori knowledge. We assume that this a priori knowledge can be expressed by saying that ~ is a random variable following a certain distribution of which we know the mean and the standard deviation. In this paragraph we do not ask wherefrom we know this parameters. We simply consider them as given. According to the principle of 03, we shall assume that ~ follows a F-distribution.

In the first theorem, X is supposed to be constant though unknown. In the second theorem we assume that k changes at random from one time interval to the other following the distribution:

P (X ~ x l X = y) = ~(x; y, be)

of which we know the parameter be. In both theorems we denote by S the number of claims in time

unity, considered as a random variable.

EXPERIENCE RATING IN SUBSETS OF RISKS 2 1 5

10. - - Theorem I

If P ( s = n I x = x) = ~ ( n ; x), P (x ~ x) = y (x; a, b)

then E (X IS = k) -- a + b k

i + b

E (X[ S = k) denoting the est imate of X after the observation of k claims.

Proof:

By Bayes ' theorem we have

P ( S = k fx = x) P ( X ~ x) P(X ~ x S = k ) = P ( S = k )

This gives, applying 073 to the denominator ,

(k; x) ~, (x; a, b) P(X m x [ S = k ) = + (k; a, b)

f rom which we get by o74

e ( x ~ ' , x l S = k ) = T x; I + b '

For the purposes oi insurance we have to take the mean as the best estimate, so tha t we get

a + k b E (X IS = k) -- q.e.d.

I + b

11. - - Theorem 2

We denote by ~ the mean value of X for all t ime intervals, and we look for an es t imate E (~ I S = k) on the basis of the observation of one interval.

If P ( S = n [ X = x ) =z~(n;x), P ( X ~ x J X = y ) = y ( x ; y , b ~ ) ,

P (X ~ y) = y ( y ;a , b~).


t h e n E (X ] S = k ) - -

P~roof :

B y 073 we have

112

wi th

a + c b l k

I -]- cbl

I with c - -

x + b 2

P (S = n f ~ . = x) = + ( n ; x . b~)

( S l X = x) = x, .5 ( S t X = x) = x ( i + b~).

We replace this by ano the r d is t r ibut ion wi th the same mean and variance, namely by the dis t r ibut ion defined b y

113 P (c S = n [ X = x) = ~ (n, cx)

I where c - - - - , as a l ready del ined above.

I + b ~

Here we make the f ict ion t ha t S assumes only values such tha t c S is an integer. This f ict ion is allowed because of the principle

of 03.

We have by 070

f rom which we get

( c S l X = x ) = c x

.5 (c ,S IX = x) = cx.

( s i x = x) = x ~2 ( S [ ~ = X) = X/C = X (I + b~)

so t ha t the dis t r ibut ion defined by 113 real ly has the same mean and var iance as tha t given by 112.

Now we have

P ( c S = n l X = x ) = ~ ( n ; c x )

For y = cx, this gives

114 P (cS = n J c X = y) = x ( n , y)


On the other hand, by o72 we get

115 P (c X ~ y) = 7 (Y, ca, cbl).

Now we apply theorem I to 114 and 115, which gives

ca + c 2 b~ k E (c X l c S = c k) - - I + cbl

from which follows

a + cbik E (X IS = k) -- , q.e.d.

I + cbl

2. - - SETS OF RISKS

21. - - Set B consis t ing of single r i sks B ,

We suppose that the true claims frequencies X, of the risks B, follow a certain distribution, and we estimate the a posteriori value of a k, after the observation of the corresponding B, during a time interval of length unity. As mentioned before, the result may easily be extended to intervals of any length.

The result of this paragraph may for instance be applied to merit rating in automobile insurance.

Notation:

r = number of risks in B k, = number of claims in B, observed k = k i + . . . + kr S, = number of claims in B,, considered as a random variable.

210 . - - N o change o f the Z, in t ime

We suppose that the X, are, though unknown, constant in time. According to the principle of o3, we presume that the a priori- distribution of the X, is a F-distribution.

The application of theorem I gives directly

a + b k , E (X,[S, = k,) --

~ + b


Es t ima t ion of the pa ramete r s a and b:

We have

P (S, = k~) = ~ (k~;a, b)

~. (St) = a ~ ( S t ) = a (~ + b)

F rom all the observed k,, we es t imate ~ (St) and ,3 (St) as follows :

( s t ) = k/r

as (S~) - r - - I

This gives

a = k/r

X (k , - - a)~ b - -

( r - - I) a

211. - - The X, change at random in time

We suppose tha t the X, change f rom one t ime in terva l to the o ther following the dis t r ibut ion given b y

P (X, ~ x I Xt = y) = v (x; y, b~) *

b, being the same for all i. We wish to es t imate X,, the mean of the X, over all t ime intervals.

The appl icat ion of theorem 2 gives di rect ly

a + c bl k, 2110 E (~,, [ S, = k , ) - -

I + c b l

I with c - -

I + b ~

Es t ima t ion of the parameters :

According to the condit ions o f theorem 2, we have

P (St = n f x~ = x) = ~ (n ; x)

P (x, ~ x l~,, = y) = "r (x; y, b~) P (x~ ~ y) = v ( y ; a , bl)

*) These dis t r ibut ions are supposed to be mutua l ly independent .


By a repeated application of 075, we get from this

P ( S , ~ n ) = F ( n ; a , a + a b l +abe)

If we put b~ + be = b,

we m a y determine a and b in the same way as in 21o.

In order to es t imate be, we have, of course, to use the observations of two different t ime intervals.

If we denote by ~S, the number of claims in a first and by ,S, t ha t number in a second interval, the difference

is, by 073, a random variable wi th mean o and variance

2 y, (I + be)

Thus we have

[(~S, I X, = Y * ) - (,S, r X* = Y*) P [ . ~ x ] = F E x ; o , 2 ( l + b 2 ) ~

Using all the observed values lk, and ,k, and summing over i, we get from this

• (lk, _ 2r (I + be) Y~

o r

2111 b, = 2r y,

Now we do not know the y,. To overcome this difficulty, we m a y use a = kit as a first approximat ion for the y, and determine a first approximat ion of b, from 2111. By 2110 we get now a second approximat ion of the y, = X,, f rom which we get a second approximat ion of b2, etc.

22. - - Set B consisting of subsets B,

We are coming now to the main problem of this paper. The results of this paragraph are subject to two very impor tan t

conditions:

2 2 0 E X P E R I E N C E RATING IN SUBSETS OF RISKS

A. We presume that we have no a priori conjectures about the values of a particular ~,. If for instance the B, correspond to different regions in automobile insurance and if we know that in one particular B,, circulation is much denser than in other regions, we must not apply the following formulae to that Bf.

B. If the B, follow a certain order and if the observed values of the claims frequencies of the B, follow an order which shows a connection with the order of the B,, the following results must not be applied. Thus the theory cannot be used for the calculation of premium rates for different classes of horse power in automobile insurance.

These two conditions are in accordance with the subjective interpretation of the concept of probability.

For practical purpose they may be neglected for experience rating in the restrictive sense of the word as mentioned in oi.

Of course the two conditions also apply to the results of paragraph 21. If they have not been mentioned already there, it was because the results of that paragraph will. in practice, only be used for experience rating in the restrictive sense.

= set of risks = subsets of B = single risk belonging to B, = numbers of claims in time unity in B, B, and B,j,

conceived as random variables. k, k, and k,j = numbers of claims observed in time unity X, X~ and ~j = claims frequencies per risk.

r, = number of risks in B, s = number of subsets B, r = r l + . . . + r ~

2 2 0 . - - X~j constant in time

We suppose that the X, follow a certain distribution. According to the principle of 03, we presume that this is a F-distribution:

P (x, ~ x) = -~ (x; a, b)

The parameters a and b are considered as unknown. Their value will be estimated from the observed k,.

Notation:

B B,

B,j S, S, and S,1

E X P E R I E N C E RATING IN SUBSETS OF RISKS 2 2 1

By 072 we have

P ( r , X , ~ x ) = T (x ; r, a, r, b)

Now we apply theorem I with the following subst i tut ions:

S ---~ S, k -+ k~

a - + r~ a

b-+ r~ b

o r

This gives

E (r,X, [ S , = k , ) = r,a + r, bk,

I + rib

E (X, I S, = k,) -- a + b k ,

I -~- r,b

Est imat ion of parameters :

We have P (S~ = n) = ~b (n; tea, r~b) t~ (S,) = r~a a 2 (S , ) = r,a (I + r ,b )

Subst i tu t ing for the S, the observed k, and summing over all i. we get

k = r a

Z ( k , - - r , a ) 2 = X r,a (I + r,b)

From this we obtain

~ = k / r

X (k , - - r~a)~ - - k b =

a ~ r f ~

(We neglect the fact tha t the a of the second relation is calcula ted f rom the first relation).

2 2 1 . - X O constant in time, with complete exchange of risks

In this paragraph we suppose tha t every B, is replaced, from one


t ime interval to the other, by a new set B** containing the same number of risks with the same characteristic i.

After observation of one B,, we do not wish to determine the ~, of this part icular B,, bu t the average claims frequency of sets B, with the property i, which we denote by At and which is supposed to follow the distr ibution

P(A~ ~ y ) = y ( y ; a , bl)

The exchange of the risks f rom one t ime interval to the other has the effect of a r andom fluctuat ion of the X,, so tha t we can apply theorem 2.

We perform the following subst i tu t ion in tha t theorem

S--~ S, k - d - k ,

X ~ r , Xf

X ~ r, A , bz ~ r,b2,, a --~ r ia

bl --+ r,bl

so t ha t it reads:

If

P(r~X~ ~ x i r ~ A ~ P (r, Ai

P (S, = n i t s X,

then

= y) = y (x; y, r,b2,,) y) = y (y; r,a, rfbl)

= x) = ~ ( n ; x)

E (r, A, IS , = k,) --

F rom this we get

a + c, bl k,

I + Ci r f bl

r # +c , r ,b lk ,

I + c,r, bl

with c, -- I + r ,b , , ,

Est imat ion of the parameters :

We put

2 2 1 0 P (XO m x lX, = y) = 7 (x; y, d,)

E X P E R I E N C E RATING IN SUBSETS OF RISKS 22 3

The d, can be ca lcula ted as in 21o:

Z (k,j - - k,/r,)*

2211 d, = ~ - - I

( r , - I) k,[r,

Now we h a v e a p p r o x i m a t e l y

2212 P (X, / ~ x [A, = y) = P (X,j ~ x [ X, = y) = y (x; y, d,)

and thus

and

P (X, ~ x [ A , = y) = y (x; y, d,/r,)

P ( r , X , ~ x l r , A , = y ) = y ( x ; y , d , )

On the o ther hand, we have

P (r~ X~ ~ x [ r~ A, = y) = y (x; y, r,b~,~)

F r o m the last two re la t ions follows

b~,~ = d,/r,

so t h a t we get

X (k , j - - k,/r,)* 2213 b2,, = ~ I

(r• - - I ) k , r f

In order to de t e rmine a and b 1, we use the re la t ion

P (S, • n) = F (n; r,a, r,a + r*, abl + r, ~ ab,,,)

following f rom 075.

Subs t i tu t ing for the S, the observed k, and s u m m i n g over all i, we get

a = k/r

(k, - - r , a ) 2 - - a X r, 2 b ~ , , - - k 31 -~- a ~ r f z

222. - - X 0 constant in t ime, with part ial exchange o f r isks

We suppose t h a t we t lave observed B, dur ing a t ime in te rva l I , and we wish to e s t ima te the c la ims f requency in a subsequen t


in te rva l 2, of a set B** which is formed of a pa r t of B, and new risks with the same character is t ic i.

This case can ev ident ly be t r ea ted as a combina t ion of 220 and 22I.

223. - - k,j variable in t ime, no exchange o f r i sks

We denote by 7,, t and X, tile means of the k,j and X, over all t ime intervals and pu t

2231 P (kit ~ x iX~ 1 = y) = y (x; y, bs,,)

b~,, being the same for all j , and supposing tha t these dis tr ibut ions are mutua l ly independent for all X O.

Subst i tu t ions similar to those of 221 give the following result for the est imat ion of ~,:

a + c, bl k~ E (~.,[ S, : k,) =

I with c, - -

I + r, bz,,

Est imat ion of the paramete rs :

First we determine the ba, i of 2231 according to the m e th o d of 211. On the o ther hand, summing 2231 over all j gives

P (r~ X, ~ x [ ~,1 -~ y l A . . . A~*ri : Yr,)

: y (x; yl + . . . + Yr~, b,,,)

As the dis t r ibut ion on the r ight side depends only on the sum of the Yl and not on thei r individual values, the condi t ion on the left side can be replaced by

r, X, = y l + . . . + Yr,

y t + . . . + Yr, = Y, Put t ing

this gives P (r, ~, ~ x [ r , X, : y) ~ 7 (x; y, b3,1)

Comparing this with the condit ions for the applicat ion of theorem 2, we get

be,, = b3,~/r~

The est imat ion of a and b now is exac t ly the same as in 221.


2 2 4 . - - X,~ variable in time, complete exchange of risks

This is the general case. Notation for claims frequencies:

X,~ risk observed, time interval observed X,~ risk observed, mean over all intervals X, set observed, interval observed ~,, set observed, mean over all intervals A, arbitrary set of risks with characteristic i,

arbitrary time interval

This case is perfectly analagous to that of 221. We only have to replace X, by X, in 2210 and 2212. Consequently, we can use the formulae of 221 for the estimation of the parameters a, bl and b2,~.

This may seem paradoxical at first sight. The explanation lays in the fact the the k~j in 2211 and 2213, if there is a variation of the X~j in time, are influenced both by that variation and the variation of the X~ 1 about the X~.

As the case of the present paragraph and that of 221 can be treated exactly in the same way, it is not neces sa ry - - in the case of complete exchange of r isks-- to know whether there exists a variation in time of the X~ or not.

2 2 5 . - - X ~ variable in time, partial exchange of risks

This is a combination of 223 and 224.

Remarks added before printing

A) Jan Jung told me that, instead of applying the principle of accuracy of 03, I could have obtained my results by linear regression.

Thus, my theorem 2 can be derived as a linear regression of X on S as follows:

coy (X, S) ~ , * = E (X) + v a t s ( k - - E ( S ) )

cov (X, S) = E [ ( X - a) ( S - a)]

= E l ( X - - a ) ( S - - X + X - - a ) ] = E E(X - - a) (S - - X)] + E (X - - a)~

x5

226 E X P E R I E N C E RATING IN SUBSETS OF RISKS

E E(X--a) (S--X)] = o

(We integrate first over S with fixed X. As E (S[X) = X, this gives o.)

Thus

coy (~, S) = E (X--a) 2 = abl.

Further, by 075:

v a r S = a ( i + b l + b ~ )

Finally

abl X* = a + a (~ + b~ + b~) ( k - - a )

a + cbl k I -- with c --

I + cbl I + b~

which corresponds to my theorem 2. This derivation is much simpler than that given in my paper. For judging the practical applicability in concrete cases, I think it is useful to know that the theorem can be based either on linear regression or on my principle of accuracy.

There might be some doubt whether it makes sense to speak of a linear regression of X on S. I think it makes sense. The case is exactly that of an urn containing slips with a pair of values of X and S inscribed on each. The random experiment consists in drawing a slip, and the task is to estimate X from S after having read S but not X. It is, however, a peculiarity of our problem that X never will be directly observable, and that therefore special methods must be developed to determine the parameters of the linear regression. This is done in my paper.

B) My theorem 2 as well as, of course, theorem i, can also be derived as a special case of the credibility relation

(I - - b). E [~(0)] + b. X

derived in the paper "Experience Rating and Credibility" by Hans Bfihlmann, appearing in this same number of the Astin Bulletin.

EXPERIENCE RATING IN SUBSETS OF RISKS 227

The credibility relation of B/ihlmann too can be derived from the formula for linear regression.

C) In discussions about my paper I realized that "the "exchange of risks" treated in paragraphs z2I, 222, 224 and 225 was not easily understood. It may therefore be useful to give a concrete example for these cases.

Such an example is workmen's compensation with

B = portfolio of policies B, = single policy i, i.e. single factory i B, 1 = single worker j of single factory i

The exchange of risks here means the dismissal of old workers and their replacement by new ones. It is obvious that the weight of past experience must be diminished by such an exchange.

Another example would be a class of risks where some of the policies expire and new ones are contracted.

April I965

ASTIN COLLOQUIUM I965 LUCERNE

SUBJECT THREE

CONTROL OF NON-LIFE BUSINESS

RAPPORT INTRODUCTIF

CONTROLE DES OP]~RATIONS D'ASSURANCE DANS LES BRANCHES NON-VIE

F . BICHSEL Berne

Les 5 contributions suivantes ont ~td pr6sent6es sur le th~me no 3:

par R. E. Beard: 2 contributions intituldes ,,Calculation of Reserves for Non-Life Insurance" (Le calcul des r6serves pour les branches non-vie)

par Karl Borch: 1) ,,Control of a Portfolio of Insurance Contracts" (Le contr61e d 'un portefeuille d'assurances)

par B. H. Jongh: 1) "The Insurer's Ruin" (La ruine de l'assureur)

par T. PentikAinen, Helsinki: ,,On the Solvency of Insurance Companies" (De la solvabilit~ des compagnies d'assurance)

par Erkki Pesonen, Helsinki "Magnitude Control of Technical Reserves in Finland" (Le contr61e du montant des r6serves techniques en Finlande)

Les questions traitfies par ces auteurs peuvent gtre groupdes comme suit:

I. Donn6e la situation d'une compagnie d'assurance, quelles sont les affaires qu'elle devrait accepter ?

2. Donn~e la situation d'une compagnie d'assurance, dans quelle mesure doit-eUe r6assurer son portefeuille ?

3. Le calcul des r~serves techniques, c'est-k-dire de la r6serve pour risques en cours et pour les sinistres non r6gl6s.

4. La d6termination des r6serves de s6curit~ n6cessaires en plus des r6serves techniques proprement dites.

1) Publ ished in Ast in Bul le t in Vol. IV P a r t I.

232 CONTR6LE DANS LES BRANCHES NON-VIE

Les deux premieres questions ont 6t6 trait6es seulement par M. Karl Borch. Dans sa tr~s intdressante contribution, M. Borch formule le principe suivant pour la gestion d'une compagnie d'assurance:

L'esp6rance math~matique de la valeur actuelle des dividendes futurs doit ~tre maximale, 6tant entendu que la compagnie cesse d'exister d~s que son capital devient n6gatif.

M. Botch mentionne d'abord que, d'apr~s un th6or~me de Shubik-Thompson, pour un portefeuille d'assurances donnE, la meilleure politique est d'employer d 'abord tout profit ~ l'aug- mentat ion de la rdserve de s6curitd jusqu'~ ce que celle-ci atteigne un montant ddtermin~ et de distribuer ensuite ce qui exc~de ce montant comme dividende.

I. En ce qui concerne la question de savoir quelles affaires une compagnie pourrait accepter en addition au portefeuille donn~, M. Botch arrive ~t la conclusion que dans certaines conditions une compagnie pourrait avoir intdr~t ~ accepter une affaire dont l'esp~- rance math6matique des sinistres d6passe la prime de risque. Ce paradoxe s'explique par le fait que l 'acceptation d'une telle affaire pourrait, si tout va bien, rallonger consid~rablement la vie de la compagnie et ainsi beaucoup augmenter l'esp~rance math6matique des dividendes futurs. Je pense que ce r6sultat est tr~s discutable du point de vue pratique. I1 est certainement correct du point de rue math~matique dans les hypotheses faites par M. florch, mais on peut se demander s'il est juste de prendre en consideration seulement l'esp~rance mathdmatique de la valeur totale actuelle des dividendes futurs et de n~gliger la variance de ce total.

2. La deuxi&me question, celle de la r6assurance d'un portefeuille donn6, a 6galement 6t6 traitde seulement par M. Borch. Sous les conditions que je viens de citer, il traite le cas d 'un module extr~- mement simple oh il arrive ~ des conclusions qui me semblent tr~s paradoxales du point de vue pratique.

3. Le calcul des reserves techniques, c'est-A-dire des r6serves pour risques en cours et pour les sinistres non r6gl6s, est largement trait6 dans les deux rapports de M. Beard.

L'un a ~t6 prdpar6 par M. Beard ~ la demande de la sixi~me Conference europ6enne des services de contr61e des assurances

CONTR~LE DANS LES BRANCHES NON-VIE 2 3 3

privies. M. Beard dit dans cet expos6 qu'il a exprim~ ses vues personnelles, mais que le sujet serait discut6 h Lucerne.

Je me limiterai ~ relever quelques points qui me semblent parti- culi~rement int~ressants dans les deux expos6s de M. Beard.

a) Pour le calcul de la r6serve pour risques en cours, il est pr6f~ra- ble de se fonder plut6t sur les sinistres survenus dans le pass6 que sur la prime, celle-ci pouvant ~tre insuffisante et conduire ainsi k des r6serves insuffisantes. En d'autres termes, il vaut mieux employer une m6thode prospective qu'une m6thode r6trospective. Evidemment, cette mani~re de proc~der n'est indiqu6e que si le portefeuille est assez grand et si une ~tude statistique a montr6 que la difference entre les sinistres survenus et la prime de risque est significative. Ici, la seule fr6quence des sinistres donne des crit~res beaucoup plus sensibles que leur montant global.

b) Pour la mesure de la fr6quence des sinistres il importe de tenir compte des sinistres sans suite qui dans certaines branches peuvent ~tre assez nombreux. I1 faut aussi prendre en consideration les sinistres survenus mais pas encore annonc6s.

c) La r6serve pour risques en cours doit aussi couvrir les frais d'administration futurs pour les risques en question.

d) Pour d6terminer les r6serves pour risques en cours et sinistres non r~gl~s, il ne suffit pas de calculer l'esp~rance math~matique de ces grandeurs, n faut ajouter une marge de s6curit6 pour tenir compte de leurs fluctuations al6atoires.

e) La d~termination de la distribution des sinistres d'apr~s leur montant est rendue plus difficile par les circonstances suivantes:

changement du volume et de la composition du portefeuille; changement de la valeur de la monnaie et de l 'at t i tude des tr ibunaux ; d~lai entre la survenance et le r~glement d 'un sinistre, d~lai qui est en g~n6ral plus long pour les sinistres d 'un montant 61ev6.

Ces changements ont pour effet que la distribution des montants des sinistres rdglds pendant une ann6e n'est pas la m~me que la distribution des montants des sinistres survenus pendant une annie. Evidemment c'est cette derni&re qu'il faut connaitre.

234 CONTR()LE DANS LES BRANCHES NON-VIE

f) Pour la d6termination de la r6serve pour sinistres non r6glds il y a deux m6thodes:

la m6thode individuelle et la mfthode collective. M. Beard constate que, d'apr~s les exp6riences faites, la m6thode collective qui travaille avec le nombre et le montant moyen des sinistres donne des r6sultats plus exacts que la mfthode de l 'estimation individuelle.

4. D6termination des r6serves de sdcurit6.

J 'entends ici les termes ,,r6serves de sdcurit6" dans un sens math6matique, c'est-k-dire que je comprends par ,,r6serves de s6curit6" toute r6serve destin6e ~ parer k des fluctuations al6a- toires, y compris le capital et les r6serves libres.

Toutes les contributions prdsentdes s'occupent de la question des r6serves de s6curit6 qui peut ~tre divis6e de la mani~re suivante:

a) r6serves de s6curitd n~cessaires pour parer aux fluctuations al6atoires du montant n~cessaire au r~glement des risques en cours et des sinistres en suspens.

Cette question a 6t6 trait6e, comme je l'ai d6jk dit, par M. Beard. b) R6serves de s6curit6 pour parer aux fluctuations al6atoires

en rapport avec les risques souscrits pendant l'ann6e ~ venir:

Comme l'exposent MM. Pentik~iinen et Pesonen, c'est la r6serve de s6curit6 que doit fixer l 'autofit~ de surveillance pour permettre la continuation des op6rations d'une compagnie pour une ann6e.

Cette r6serve pent en principe ~tre calcul6e si l 'on donne la proba- bilit6 de ruine ¢, c'est-k-dire la probabilit6 que la reserve de s~curit~ ne sera pas suffisante. Dans ce calcul, il faut aussi prendre en consideration les fluctuations qui peuvent survenir dans les probabilitds de base pour la survenance d'un sinistre.

Le calcul exact pent ~tre tr&s compliqu6. C'est la raison pour laqueUe MM. Pentikliinen et Pesonen donnent des approximations par des formules relativement simples. L'approximation de M. Pesonen peut ~tre d~crite comme suit:

Le portefeuiUe donn6 est remplac6 par un autre portefeuille qui a la m~me esp~rance math6matique du rnontant des sinistres, mais oh tous les sinistres ont le m~me montant, celui-ci ~tant ~gal au montant maximum d'un sinistre individuel du portefeuille originel.

CONTR~LE DANS LES BRANCHES NON-VIE 2 3 5

Pour ce portefeuille fictif la r6serve de s6curit6 n6cessaire est calcul6e par application de la distribution simple de Poisson.

M. Pentikltinen relate que des 6tudes approfondies ont conduit 6tablir les r6gles suivantes pour la constitution des r6serves de

s~curit~ en Finlande, les montants ~tant exprim6s en L. Si la prime annuelle est inf6rieure ~ 270,000 L, la r6serve dolt se monter k 13,ooo L plus 20% de la prime annuelle. Si la prime annuelle est sup6rieure k 270,000 L, la r6serve doit se monter & 67,ooo L plus lO% de la prime annuelle. J ' a joute ici que M. Pentik~inen donne un aper~u sur tousles facteurs qui doivent 4tre pris en considdration pour juger de la solvabilit6 d'une compagnie d'assurance.

c) Finalement, on peut consid~rer la rdserve de s6curit6 n~cessaire pour parer aux fluctuations al~atoires qui r6sulteront d 'un portefeuille donn6 si celui-ci subsiste un certain nombre d'ann6es, fini ou infini. C'est le point de vue de la th6orie du risque proprement dite. Dans son expos6 M. Jongh pr~sente tr6s clairement le calcul de la probabilit6 de ruine pour une r6serve de s~curit6 initiale ddtermin~e. I1 fait remarquer que pour l 'application pratique de cette th6orie, il faut connaitre tr6s exactement la fonction de distribution des gains ou pertes annuels du portefeuille donn6, ce qui est assez difficile et n~cessite un mat6riel de statistique assez volumineux.

Tous les auteurs eit6s jusqu'ici calculent des r6serves de s6curit6 sur la base d'une probabilit6 de ruine donn6e. I1 est clair que le choix de cette probabilit6 constitue un 616ment arbitraire. M. Borch 4vite cet 616ment arbitraire par le proc6d6 suivant :

Comme je l'ai d~jh mentionn6, il formule le prineipe que l'esp6- rance math6matique de la valeur actuelle des paiements futurs de dividendes doit ~tre maximale. Pour un portefeuille donn~, il arrive ainsi ~ d6terminer une r~serve de s6curit6 qui doit ~tre accumul~e avant que des dividendes soient pay6s. M. Borch traite des cas extr4mement simples. I1 semble que l'application de sa th4orie ~ des cas pratiques conduira ~ des probl6mes math6matiques tr~s difficiles.

ON THE SOLVENCY OF INSURANCE COMPANIES

T. PENTIKAINEN

Helsinki

I. WHAT IS SOLVENCY ?

This report is a contribution to the discussion on the solvency problem, which has been taking place at ASTIN-meetings. In his report in Edinburgh x964 Beard referred to many aspects which are closely connected with the problem. Such aspects are

I. the evaluation of liabilities; 2. the evaluation of assets; 3. the level of the premiums of long term policies and 4. reinsurance.

If all of these are not in order, there is no sense in speaking about solvency. E.g. a solvency margin defined as the difference between assets and the expected value of liabilities would not be a reliable measure of the financial state of an insurance company, if either of these---or maybe both--are not evaluated in a reliable way. The fixing of solvency margins is not an isolated problem, on the contrary it is only part of the security measures which must all be managed at the same time. The ultimate purpose of the security system prescribed by legislation must be to safeguard policyholders and claimants against losses.

However, if the problem of solvency is understood in as wide a sense as is mentioned above, the subject has apparently grown so much that it would be inpracticable to discuss the whole of it at one meeting. That is why it seems to be advisable to limit the scope to the solvency problem "in a narrower meaning", i.e. to the solvency margin question only and to give up items 1- 3 mentioned above and also partially item 4 and let them be discussed at some other meeting or in some other organisation. The more so because already now in most countries these subjects may be, in a very detailed way, prescribed by Insurance Company Acts and the

ON THE SOLVENCY OF INSURANCE COMPANIES 237

s u p e r v i s i n g a u t h o r i t i e s p a y a g r e a t d e a l of a t t e n t i o n t o c h e c k i n g

t h e i r f u l f i l m e n t w i t h e a c h i n s u r a n c e c o m p a n y .

Some remarks. E v e n if i tems I-4 are not discussed in this paper we are going to give some few comment s on them, especially concerning some definit ions which are needed la te r on or concerning some aspects which Beard has ment ioned in his report .

i . Mathemat ica l reserves (including the reserve of ou t s t and ing claims). In some countr ies the rules applied provide fair ly exac t eva lua t ion of out- s tanding liabilities e.g. as ma thema t i ca l capi ta l values of fu ture claims. I n o ther countries, in addi t ion to that , some addi t ional amoun t s are al lowed (or even expected) to mee t unfavourable future f luc tuat ions and unexpec t - edly high t e m p o r a r y risks and claims. If " a f luc tua t ion reserve" or " a n ad ju s tmen t reserve" of this kind is included in the ma thema t i ca l reserves i t m a y be qui te correct to take i t into account , a t least to a cer ta in degree, as a proper pa r t of the solvency margin defined la ter on.

2. Evalua t ion o f assets. The assets mus t always be es t imated and t aken into balance sheets in a caut ious way. In f a c t - - d u e to inf la t ion and other r e a sons - - t he actual value of assets m a y of ten be m u c h grea ter t han the book-keeping value, the difference being an invisible reserve. I t m a y be reasonable t h a t this difference should be t aken into account as a pa r t of the company ' s actual secur i ty margin. E.g. the Finnish Insurance Company Act permi ts this policy.

An i m p o r t a n t ques t ion to be defined is wha t is the m a x i m u m acceptab le book-keeping va lue of assets. P robab ly i t mus t be the sales value, wi th some except ions concerning long t e rm business.

Beard discussed in his repor t the problems which can appear when a company is being wound up. I t can be qui te possible t ha t in cer ta in condi- t ions the sales va lue falls below the expected marke t value, thus causing loss. There are, however , some o ther aspects which counte rac t this risk and make i ts omission leasable. I t seems to be usual in insurance pract ice t h a t if a company is on the br ink of l iquidat ion, its direct ion, as a f inal measure, endeavours to find another company which is willing to t ake it over. If i t succeeds then no sale of assets is needed. There is one aspect which helps to find companies will ing to take over. The insurance portfol io represents a cer ta in capital , i.e. the acquis i t ion cost of bui lding up a portfol io can be considerable and the company tak ing over can calcula te t h a t i t will be prof i table in the fu ture when incompora ted in the company ' s own por t - folio. This fact can render it qui te reasonable to t ake over, even if minor defici ts appear in the assets or reserves. This reasoning provides, among o ther things, an adequa te level of long t e rm insurance (life assurance etc.), as ment ioned in i tem 3.

2. MEASURING SOLVENCY

T h e d e f i n i t i o n o f s o l v e n c y c a n b e l o o k e d a t i n t w o d i f f e r e n t w a y s :

a) F r o m t h e p o i n t o f v i e w o f t h e m a n a g e m e n t o f t h e c o m p a n y : T h e c o n t -

i n u a t i o n of t h e f u n c t i o n a n d e x i s t e n c e o f t h e c o m p a n y m u s t be s e c u r e d .

238 ON THE SOLVENCY OF INSURANCE COMPANIES

b) From the point of view of the supervising authorities: The benefits of the claimants and policyholders must be secured

Apparently definition b is narrower. I t does not demand the continuation of the company in all circumstances but also allows it to be wound up. However, also in the case of winding up, the liabilities due to policyholders must be secured either by means of the liquitation of assets and liabilities or the taking over of both by some other company. Definition b can oe approved as a basis of the legal security system. If this is done, then the care of the company's existence can be left to the management of each company, which can be carried out by means of adequate reserves, security loadings of premiums, reinsurance and other means. This means, in principle, that the supervising authorities and the legal security measures shall be restricted to the minimum i.e. to secure the insured benefits only, but otherwise each company shall have freedom to develop its function as it itself desires.

Dr. Pesonen in his paper "Solvency Measurement" (Edinburgh, I7th Congress of Actuaries) expressed the definition of security on these lines as follows:

"The reserve, when the accounts of a certain year are closed, is the amount the company would need in addition to future premiums in order to be capable, with a probability of I - - ¢ of meeting its present and future liabilities i f the company ceased to make new contracts after one year."

The period of one year is the same as the normal accountancy period of the companies. The status of each company can be observed only once a year. If it is then stated to be solvent, the continuation of its activity is allowed for the following year. If the company has not an adequate status, winding up will be immediately enforced if solvency is not re-established in a very short time by means of additional capital, additional reinsurance or by other means.

This definition is so general that it takes into account all kinds of risks without limitation to only some few categories of risks, as is the case in some other definitions.

3. How CAN SOLVENCY BE TESTED ?

The application of the definition given above provides an analysis of the different risks which can threaten an insurance company.


I) Random fluctuation of claims. This phenomenon is the object of the s tudy of the theory of risk.

2) The fluctuation of the basic probabilities of the claims and their trends. The cause, of fluctuations of this kind may be e.g. weather variations in the field of fire insurance, epidemic diseases in the field of life assurance etc. It is well known that economic conditions have an influence upon the loss ratio of many branches of the non-life business. The period of such fluctuations may be sometimes short (weather) and sometimes long, even several consecutive years (economic depressions).

This phenomenon may be estimated to a certain degree by means of the theory of risk, but to a large extent it must be estimated by very rough methods, on the basis of the behaviour of claim ratios observed in times passed.

3) Losses on investment. Losses of this kind can be caused by many reasons. It can be e.g. the bankruptcy of a loan holder in eases where the valuation of the securities has been too optimistic. Further reasons may be the reduction of the value of equities on the general market, the loss of the value of some real estate caused by some special condition, careless action in the valuation of securities or in holding them etc.

4) Miscellaneous risks. It is probably impossible to record thoroughly all kinds of risks which can affect the status of insurance institutions. Some of them can, however, be mentioned here.

a) Natural catastrophes like hurricanes, earthquakes, land- slides.

b) Failure of reinsurance. The reason can be a human error, e.g. the reinsurance of a large risk is omitted or the risk of conflagration is miscalculated. The insolvency of the reinsurer can also give trouble.

c) Emplezzlement or other misappropriation of the company's resources. This risk cannot be completely avoided even by the most competent audit or supervision.

d) Riots, sabotage and other disturbances. Ordinary war risks may be settled by special legislation in various countries and they need not be considered here. We can also presume

2 4 0 ON THE SOLVENCY OF INSURANCE COMPANIES

that atomic risks are dealt with by various special measures in an adequate way.

Many of the risks mentioned above are of such a nature that they cannot be reliably estimated in advance, especially risks (4). We must keep in mind that the legal, or any other precautionary measures, can never give absolute safety. If we took into account every, even the utmost improbable, chances of risk, security margins and other measures would become intolerably heavy. All we can do, is try and weigh the risks and security measures on a "common sense" basis, and take into account everything which we know by experience has some realistic probability of occuring and neglect risks of a more theoretical nature, which have small likelihood of ever appearing. The circumstances of course vary very much from country to country e.g. concerning items (a) and (d), which appears to make it impossible to find an international standard for a security margin to cover all cases. Probably the only thing to be done is to develop reinsurance so that it covers as many risks as possible and carefully exclude in companies' insurance contracts responsibility for any risk which could be overwhelming. The duty of the state supervision is to check that these measures are observed in every insurance institution and that the internal control and checking is sufficient to guarantee security in this respect as well.

I t seems advisable to leave risks 3 and 4--so far they cannot be excluded or covered by reinsurance--to be covered by an appropriate security margin. This will be discussed later on in par. 4. Risks I and 2 cannot be covered by a security margin only; instead a solvent combination of reinsurance and solvency margin is necessary. We are now coming to the question of how the adequacy of such a combination should be tested, i.e. in what conditions a company is secured by a probability I - ¢ against the fluctuations I and 2 mentioned above.

From the theoretical point of view probably the most natural way to procede would be to prescribe that the direction of each company be responsible for proving the company's solvency to the supervising authorities by means of actuarial calculations. In principle this is exactly the same method which is already applied

ON THE SOLVENCY OF I N S U R A N C E COMPANIES 241

concerning e.g. the evaluation of assets, evaluation of mathematical reserves, checking of the adequacy of the reassurance etc. The calculation of the security numbers, which is the same as the calculation of the probability of ruin, may be, however, a novel task for most actuaries, besides which there are very many non-life companies which have no actuary at all. This question has been much studied in Finland and Dr. Pesonen has developed methods which seem to be applicable to any company. It is true that a rather exact calculation is possible by means of very complicated methods of the theory of risk (also the periodical fluctuation and trends of the basic probabilities must be taken into account). Electronic computers may often be necessary. In practice this is, fortunately, not too formidable a job, because insurance companies can develop a joint programming for the task and then only some few very simple statistics and pieces of information are necessary as input to the computer; as output we can get the probabili ty numbers demanded. In Finland this method is being experimented with and it also seems to be quite feasible for practical purposes.

Fortunately the necessary probabili ty numbers can in most cases be calculated without any computers-- i t is possible to develop approximations which give very simple rules. These rules can also be accepted as a basis for the official supervision. Only in those very few cases where the solvency of the company cannot be proved by means of these simple rules has the company the right to show its solvency by means of more complicated methods e.g. by means of computers.

To illustrate the approximation method we can present as all example a formula which is much used in Finland for similar purposes. In fact the ruin p robabi l i ty , depends on the other variables and distribution of the theory of risk as follows

= F (U, P, SM(Z), X) (I)

where U is the solvency margin, P the premium income on the company's own retention, M the maximum net retention applied for a single claim, SM(Z) the distribution function of the size of one claim, which depends on the reinsurance and on the maximum net retention M, and X the safety loading included in the premiums. This equation can be expressed by the well-known generalised

I6

242 ON THE SOLVENCY OF INSURANCE COMPANIES

Poisson function, where the fluctuations and trends of the basic probabilities must also be taken into account. I t seems probable that this equations can, for most actual cases, be approximated by a formula as follows

Umin = a P + b VMPP (2)

where the equat ion is solved in respect of U after fixing ~ at some appropriate level and a and b are constants. The first term of the formula covers the fluctuation of the basic probabilities. If, from experience, for example the oscillation and trend (in short periods) of the basic probabilities can be expected to be say 30 percent and if normally the safety loading X is ~ o.i, the constant a may be 0 . 3 - - o . I = o.2. The second term covers the pure random fluctuations (category I above) and the constant b depends, among other things, on the chosen ruin probability ~. Often a value b ---- 2 or 3 may be appropriate. This formula was presented by the author in the ASTIN Bulletin Vol. II, Part I, Jan. 1962 (formula (18)) and earlier in Skandinavisk Aktuarietidskrift 1952. Dr. Pesonen has later on further developed it, replacing the second term by a function M y (v) where

(1 + q) P v - -

M

the transformed expected number of claims and y(v) a function representing the excess of the number of claims which can be ready tabulated. The constant q takes into account the changes of the basic probabilities. For larger numbers of v the Pesonen formula gives about the same results as formula (2) whereas for small numbers it gives a more accurate approximation.

Formula (2) is probably suitable for most companies. However, in special conditions, for example when a non-proportional reassurance is used, it must be replaced by other formulas. Also the constants a and b depend on the special conditions of each company and may vary considerably.

We will not discuss here any further details concerning solvency testing, instead we refer to the papers mentioned above and especially to Dr. Pesonen's paper presented to the 17th International Conference of Actuaries in Edinburgh and to his reports at this


colloquium. I think his methods and his principal lines of study are worth attention.

We see that in solvency testing in fact there are two free variables, the solvency margin U and the maximum net retention M (and of course also generally speaking the form of reassurance etc). If the actual margin is small, then also the maximum net retention M must be small and vice versa. If only the risk categories I and 2 above are taken into account, they, in principle, do not fix any absolute minimum amount for the security margin, they only link these two variables to each other by means of the general security equation (I), which expresses the Pesonen solvency definition by means of a formula. An appropriate way of proceding may be first to fix some suitable minimum standard as a solvency margin. For this purpose not only the risk categories I and 2 but also all others mentioned above must be taken into account. We shall discuss this question in par. 4. When a minimum for the solvency margin is arrived at, then it is left to the reassurance to secure, finally, the solvency of the company. To test this the formulas and methods mentioned above, or other similar methods, are available.

4- SOLVENCY MARGIN

When speaking of the solvency margin we understand, as mentioned above, the difference between the actual assets and liabilities of the company. There are reasons, as shown in the previous paragraph, for setting some minimum amount as the solvency margin in the legislation. Also the question of establishing some international standard has been discussed, as is well known. Such standards can probably be motivated even though we must always keep in mind that the solvency margin is only a part of the general solvency problem and the existence of an actual solvency margin exceeding the standardised minimum does not by any means alone guarantee the company's solvency.

To find a rule for the minimum amount of the solvency margin all the risks mentioned in paragraph 3 must be taken into account and the margin ought to be, with a large probability, sufficient to cover the risks mentioned in regard to reassurance and other safety measures and arrangements discussed above. We start again from

244 ON T H E S O L V E N C Y OF I N S U R A N C E C O M P A N I E S

the items I and 2 concerning random fluctuations and the fluctuation of the basic probabilities.

By means oi the theory of risk it is possible to calculate, or at least estimate, for each actual case a solvency margin U by means of the formula (I). We shall not present the formulas in detail and refer instead to publications mentioned above. We only state here that for actual computations it is necessary to know, or to define, for example the reinsurance method, the maximum net retention M and to have some idea of the magnitude of the safety loading ;~. Also the ruin probability ¢ must be fixed. To get a universal measure, which would be applicable to all companies, it is advisable to compute numerous examples on the basis of actual risk distributions and the conventional methods of reinsurance and maximum net retentions and make different assumptions concerning k and s. An at tempt at such a s tudy-- in practice very comprehensive-- was made ill Fintand when the new Insurance Company Act was prepared in 1951-1952 (the study exists completely only in the Finnish language, but some main points are published in the article mentioned above).

I t is apparent that nowadays, when electronic computers are available, the corresponding computations could easily be enlarged and done in a more efficient and accurate way.

The study mentioned above showed clearly that the solvency margin U is very largely dependent on the assumption concerning the safety loading X. In the figure we have shown the two main types. Figure I shows the case where k is positive and fig. 2 shows the case where X is zero or negative.

In most actual cases the safety loading X is positive, but there are also cases where X, at least temporarily, can be non-positive (due to variations of the basic probabilities, competition, excessive management expenses etc.). Because the legal margins must be constructed especially to cover the weak cases, it seems to be appropriate to preassume X non-positive. This means that the solvency margin must in some way or other be dependant on the size of the company, which implies some increasing function of the premium income P, perhaps having the same shape as shown by formula (2).

On the other tland the study showed that the results were not

O N T H E S O L V E N C Y O F I N S U R A N C E C O M P A N I E S 245

very dependant on the assumption concerning tile risk distribution SM(z), if the maximum net retention M was not very large. This means that if by insurance the top risks are cut out, the remaining risk distribution on the company's own retention is less affected by the total ruin probability numbers than was previously expected. This does not hold good of course if the maximum M is large, but we have definitely preassumed that the question of adequate reinsurance is already settled.

From the extensive studies the following rule (converted into English pounds) was obtained

U = ~ 1 3 o o o + o . 2 P f o r P < £ 2 7 o o o o = £ 4 o o o o + o . I P f o r P > £ 2 7 o o o o (3)

P here being the gros premium income of the company.

~ = o .o 5

~k ~ - 0 . 1 0 - - o . o 5

Fig. I Fig. 2

~ P

This rule is very similar to that applied in the United Kingdom, where the margin is IO percent of the premium income, subject to a minimum of £ 5 ° ooo.

On the other hand in most countries the minima of the security margin are fixed sums and do not depend, contrary to formula (3), on the size of the company. It seems to the author that dependence on size is more appropriate. It seems to be very difficult to find a fixed amount which is neither too small for large companies nor too large for small ones, and many of the risks are apparently apt to cause larger losses for larger portfolios.

As yet we have considered rule (3) mainly in terms of random fluctuation and the fluctuation of basic probabilities. However, other kinds of risks, mentioned in paragraph 3, must also be taken

246 ON TH E SOLVENCY OF INSURANCE COMPANIES

into account. Varied opinions exist as to whether for each category of risk, for example losses on investment etc. a separate security margin should be prescribed or whether it is enough to have a joint margin for all kinds of risks. In regard to the very small probability of large losses (assuming the supervision of the companies to be adequate, as mentioned in paragraph 2) it may be sufficient to have only one joint security margin, which is intended to be used in cases of emergency for all losses. This means, in other words, that we rely on the great probability, that, for example, a very large random loss and a very large loss on investments can never occur in the same year. The demand for several margins, or security funds, or reserves would, in practice, be unnecessarily burdensome for the companies and would give rise to considerable extra cost if such had to be collected and maintained. Of course in special circumstances, or for special branches, extra precautions can be motivated. For example in many countries a special "security fund" is required for life assurance to cover the risk of future decrease of the rate of interest or other deterioration of the basis of calculation. Such security funds are, however, "semi-obligatory", which means that the non-existence of a security fund does not cause the winding up of the company. Only a compulsory allocation of some portion of the profit may be prescribed in the legislation when the security fund is less than some prescribed minimum amount. We are not here considering these special semi-obligatory funds.

When the special risks mentioned in paragraphs 2 (items 3 and 4) and 3 are studied, it becomes apparent that some of them are also of the kind where the amount of possible loss depends on the size of the company. Even though the fixing of a margin for these very heterogeneous risks is extremely difficult and experience in different countries may vary very much, it seems to the author that the margin mentioned above is sufficient even for these different risk moments. At least the Finnish committee preparing the Insurance Company Act from the experience in Finland and in some other countries which was available to it came to the conclusion that this formula is sufficient to cover the actual need. Of course a comprehensive discussion on an international basis would be highly desirable to collect all the experience available

ON THE SOLVENCY OF I N S U R A N C E COMPANIES 247

and to fix a suitable minimum standard for the solvency margin. The author's opinion is, that the British and Finnish rule would be applicable at least as a basis for further discussion. In addition to this a method of testing solvency in general, concerning reinsurance and other security measures is needed as presented in par. 3.

MAGNITUDE CONTROL OF TECHNICAL RESERVES

IN FINLAND

ERKKI PESONEN

Helsinki

This paper deals with some requirements put forward by the Finnish Supervisory Service on technical reserves mainly for solvency reasons. Consequently only one but nevertheless an important aspect of solvency problem is touched upon. The solvency question in its entirety is discussed in a paper of Dr. Pentikfiinen at this same colloquium.

According to the instructions of the Supervisory Service, given by virtue of the Insurance Companies Act, insurance companies are obliged to calculate a so-called equalisation reserve in a specific way. Simplifying slightly, this equalisation reserve is that part of the technical reserves which exceeds the premium reserve and the claims reserve calculated according to classical actuarial principles; that is to say, the present value of the expectation value of the net insurance liabilities. The Insurance Companies Act calls for this reserve because conventional technical reserves do not take into account the random feature of claims amount.

A nonlife company has to verify that its equalisation reserve (E, say) is between a minimum amount (Emin) and a maximum amount (Emax) both defined in instructions given by the Super- visory Service. The former bound is linked to the solvency requirements implying that in case of E >~ Em|n the company is allowed to continue its activity at least one more year, whereas the latter amount, Emax, is associated with taxation. This upper bound is considered indispensable owing to the deductability of transfers to the technical reserves.

Let X be the total amount of claims after deducting cessions and retrocessions to reinsurers; let P be the annual earned premiums after deducting reinsurers' share and administration costs. Further let C be the capital and free reserves. Then the

MAGNITUDE CONTROL IN FINLAND 249

minimum amount Emln is de*ined in principle by the equation

P {I.o5(Emtn + C) + 1/I~O5 ( V - X) ~ o} = -99, (I)

where the factor I.o5 refers to the rate of interest. There is, however, a further restriction,

Emtn ~ Max {o, M - C},

where M is the greatest realistically possible size of a single claim (after deduction of cessions and retrocessions). There are some specific instructions on how to estimate M if the company under- writes Stop Loss reinsurance or any direct insurance in classes with unlimited liability.

Consequently the distribution function F(x) of the random variable X is needed for the calculation of Emtn . In order to elim- inate speculations in estimation of the function F(x) certain schematic instructions are given which contain some simplifications of reality.

The portfolio is partitioned in a given way into separate classes (k) of insurance so that X = Z Xk, where claims amounts X~ are supposed to be mutually independent. If the meai1 number of claims in class k is equal to n~ and if St(x) is the distribution function of the size of one claim in class k (after deduction of cessions and retrocessions), then the distribution function Fk(x) of the random variable Xk is defined in the instructions by means of equation

-(l+q~)n~ ~ [I +qr) n/c]" Sr , (X), Fk(x) = e r!

0

where q~ is a fluctuation constant which depends on the class k. The purpose of this constant is to pay attention to the well-known fact that in estimating Fk(x) the ordinary generalised Poisson function as such does not correspond to the real situation. In most classes q~ is betweeen .2 and .4, but e.g. in forest insurance it is as high as 6.0. The introduction of this kind of technique instead of a compound Poisson distribution is not induced merely from a practical point of view. It is also motivated in that in general, an exceptionally high number of claims appears in many companies at the same time.

2 5 0 MAGNITUDE CONTROL IN FINLAND

The Supervisory Service does not consider a system desirable if it renders possible many cases of ruin during the same year.

If for simplicity's sake the special case of Stop Loss reinsurance is excluded, it then follows that

~ r F(x) ---- II*Fk(x) = e-n r! Sr*(x)'

0

where n = ( I + qk)nk and S - - ( I + qk) nkSk. nA.

Consequently if a company has to calculate the function F(x) it must estimate the numbers n~ and the distribution functions Sk. In order to facilitate estimation a large set of common claims statistics has been collected, particularly useful for estimation of the tails of the functions Sk. Specific instructions are needed for reinsurance accepted and for cessions.

The direct calculation of E m i n c a n however usually be avoided. This is done by use of the following simple test. If the actual amount of equalisation reserve fulfills the condition

E ~>MAxl °' I I (m y(-r) - - P) - - C (2)

where v = E{X}/M and where y(.) is the smallest integer ~ 2 satisfying

U - - I

e-" ~ ) - 9 9 , 0

then E >~ Emln. With all the more reason this is true if ~ I

~r Z (I + q~)P~ (Z P~ = P) since in practice it may always be

assumed that E{X} ~ Z (I + qk)Pk. This simple test is based on a proposition 1) that for values of the argument greater than the mean value the ordinary Poisson function with parameter EIX}/M

x) Since t he proof of th i s p ropos i t ion is k n o w n only in a special case the t e r m h y p o t h e s i s is more appropr ia t e . More de ta i l s are found in t he p a p e r " O n t h e Ca lcu la t ion of the Genera l i sed Poisson F u n c t i o n " b y t he p re sen t au tho r .

MAGNITUDE CONTROL IN FINLAND 251

and with the size M of one claim is more dangerous than F(x) if the disturbance caused by steps in the Poisson function is removed. Since the function y(~) is given as a table in the instructions, the test calculation becomes elementary.

The maximum amount Emax is based on the condition that the equalisation reserve may remain at most with a probability of 99 per cent positive during the period of the next five years assuming that each year n = E (I + q~)nk. Consequently Emax is determined by means of a five fold integral which is easily calculated as soon as F(x) is known (this kind of integral and also more complicated five year assumptions can be treated conveniently by a Monte Carlo method). In practice, however, the calculation of Emax also becomes necessary only exceptionally, because in this case too it is possible to construct elementary tests which normally verify directly that E ~ Emax.

In the rare event that a company cannot avoid the calculation of F(x) the main task is the estimation of functions Sk. As soon as these are known there is a standard computer program which calculates F(x) fairly quickly and consequently at a low price. By means of this standard program companies are released from the necessity of having experts familiar with mathematical problems in connection with the calculation of generalised Poisson functions.

THE ECONOMIC THEORY OF INSURANCE

KARL BORCH Bergen

(Notes for an informal discussion in Edinburgh, i June 1964)

1. Introduction

1 . 1 . - - U n d e r Subject 4 at this Congress we have discussed the practical application of modern statistical techniques in different branches of insurance. During the last decades, there has been an almost explosive development in theoretical statistics and related branches of mathematics. I think it has been very useful to survey the techniques, which have been developed, and find out if they can be used in insurance. 1 . 2 . - There may, however, be some danger in this approach. When new means become available, we should of course have an open mind, and examine these means in order to see if they can serve our ends. We should, however, not get so excited over the power of new techniques, that we distort our ends just for the sake of being able to apply the means.

Linear programming, to take an example, is a powerful tool, which has proved extremely useful in many, apparently very different fields. There is, however, little point in using this technique in insurance, unless we have problems which consist of determining the maximum of a linear expression, subject to linear restraints. If there are problems in insurance which can be cast in this form, with sufficient approximation, then linear programming is obviously useful. If, however, we lose something essential by reformulating our problems in this way, linear programming may become a dangerous temptation, which we should resist. 1 . 3 . - In this paper I shall take a different approach. I shall t ry to take a good and hard look at the ends, with the hope that this will enable us to specify the means which we require. If these means already exist, all is well. If we cannot find any suitable techniques in the mathematician's armoury, we will have to do our own basic research, and develop the tools we need.

THE ECONOMIC THEORY OF INSURANCE 253

A generation ago, the subject "applied mathematics" consisted mainly of techniques which had proved extremely useful in classical physics. These techniques were used with considerable enthusiasm and little success in economics and other social sciences. The new statistical techniques which excite us to day, have to a large extent been developed to solve problems in quantum mechanics and telecommunications. We may therefore ask ourselves if we have any reason to expect these techniques to be useful in actuarial work.

This point has been made with considerable force by Von Neu- mann and Morgenstern ([51 P 6) who make the blunt statement: "It is unlikely that a mere repetition of the tricks which served us so well in physics, will do for the social phenomena too". They sum up their view: "It is therefore be to expected--or feared-- tha t mathematical discoveries of a stature comparable to that of calculus will be needed in order to produce decisive success in this field (i.e. economics)".

It is in this spirit we shall t ry to analyse the ends and means of actuarial science.

2. The Principle of Equivalence

2 . 1 . - To illustrate the point which I want to make, we shall begin by discussing an extremely simple example.

Consider an insurance contract under which the only possible payment is an amount of one monetary unit. We shall assume that this amount becomes payable if, and only if an event with probability p should occur.

This contract will define the following claim distribution:

o with probability I - - p I with probability p

The net premium of the contract is by definition p.

2.2. - - We shall next assume that an insurance company offers the contract we have described to the public, at a premium x > p. We shall assume that there is a demand for the insurance cover given by this contract, and that demand depends on the premium. We shall formalize this by assuming that the company will be able

2 5 4 THE ECONOMIC THEORY OF INSURANCE

to sell n = n(x) contracts if the premium is set at x. I t is natural to assume that n(x) will increase with decreasing x.

The problem is now to determine the premium x at which the company should offer this insurance contract in the market. This seems to be a very simple problem, and we ought to solve it in a satisfactory manner, before we tackle more complicated problems, or embark on the more ambitious task of constructing a general theory of insurance.

2.3. - - In classical theory our simple problem is solved by applying the Principle of Equivalence. According to this principle, the premium should be equal to expected claim payments + adminis- trative costs. This means that x should be determined by

I - C(n) x = P + n

where C(n) is the cost involved in selling and managing a portfolio of n contracts. If we assume that costs can be split up into "f ixed" and "variable" costs, we can write

C(n) = C1 + nC~

The premium will then be given by the equation

C1 x - - p + C~ + n(x--)

We have assumed that n(x) decreases with increasing x. This means that both sides of the equation will increase with x, so that the equation may have any number of solutions, depending on the shape of the function n(x).

2 . 4 . - The Principle of Equivalence gives a neat solution to our simple problem--if we are prepared to disregard the somewhat academic question about existence and uniqueness of the roots of the main equation. To solve the problem in practice, we have to know:

( i ) The basic probability p (ii) The cost elements C1 and C2 (iii) The function n(x)

To obtain this knowledge, we wtll usually have to resort to


statistical methods, or to be more precise, the techniques of statistical estimation.

The traditional task of the actuary is to provide the best possible estimate of p. He is also frequently called upon to supply estimates of C1 and C2, since this often requires statistical analysis.

The determination of the last element, the function n(x) is usually considered as being outside the duties of the actuary.

In most cases it will probably be the sales manager of the company or a market research department, who is responsible for guessing or estimating the shape of n(x).

2 . 5 . - - The function n(x) represents the demand or the market for the insurance contract under consideration. These are economic concepts, and this indicates that our problem cannot be satis- factorily solved, unless we bring in some elements of economic theory.

In some cases it may be possible to determine the "correct" premium without knowing the number of contracts which will be sold. Ttlis will be the case if n(x) is approximately constant, or in terms of economics, if insurance has a "low price elasticity". I t may seem fairly safe to assume that this actually is the case, if there is no evidence that lower premiums will lead to a significant increase in sales. One should, however, bear in mind that a reduction in the premium or an increase in the agent's commission, come to the same thing for the company, but that they may have very different effects on sales. If we ignore, or "assume away" n(x) in our calculations, we may therefore lose something which is essential to the problem we set out to study.

3. - - Operational Research and the Theory of Risk

3.1. - - If an insurance contract is offered to the public at a premium determined by the principle of equivalence, the expected profits on this transaction will be zero. The absence of profits is unpleasant in business, but this is not the point which we want to discuss here.

If an insurance company consistently makes losses on its opera- tions, the company will sooner or later be unable to fulfill its part of the insurance contracts. This means of course that the "insur-


ance" contracts do no longer serve the very purpose for which they were designed, i.e. to provide almost absolute security to the insured persons.

These considerations indicate that the premium must be set higher than dictated by the principle of equivalence. It is, however, all open question how much higher the premium should be, so that the simple problem discussed in para 2.2. is still unsolved.

3 . 2 . - - T h e simple problem is rarely explicitly formulated in actuarial literature, and no general solutions have been suggested. I t is, however, undeniable that the problem exists, and it has not been completely ignored. I think we can distinguish at least three different ways in which authors have tried to at tack the problem:

(i) The problem can be dismissed as too simple. It is obvious that the problems we meet in practice are vastly more complicated, and "practical" men may well claim that they have to spend their time solving these more "serious" problems. It is most likely that they have to make their decisions without full knowledge of the true probability p and of the exact shape of the demand function n(x), and that they claim that these decisions are rational or correct. However, if the simple problem is put aside for such reasons, the implications are that the problem becomes easier to solve if we bring in complications, and that ignorance can help us to make the right decisions.

(ii) One can add a safety loading to the premium determined by the principle of equivalence, so that expected profit becomes positive. This idea has probably originated in economic theory, where it is felt that expected profits should be greater the greater the "risks" are. However, economic theory has not so far, been very successful in defining the concept of "risk" and establish its relations to expected profits.

(iii) One can take the probability of ruin as a starting point. In our simple example this means that we consider the probability that the company will suffer a loss if the insurance contract is offered at a premium x. This approach is usually taken in actuarial literature; it is often referred to as the Theory of Risk. This theory is in many ways very attractive, but it has found few applications in practice. The reason is - - I bel ieve--that the theory does not


come to grips wi th the real problems as practising actuar ies see - - o r f ee l - - them.

3 . 3 . - - We shall now t r y a different approach to the problem, and in doing so we shall ignore the cost elements. This involves no loss of generali ty, since these elements can be b rough t in expl ic i t ly at any stage in the argument .

If an insurance c o m p a n y has underwr i t t en n cont rac t s against a p remium x, the ou tcome can be any resul t be tween the two ex t remes :

(i) A loss of n ( i - - x), if all cont rac ts lead to a claim. (ii) A prof i t of nx, if no claims are made.

In general the profi t z (positive or negative) will have a proba- bil i ty d is t r ibut ion de te rmined by:

Pr ( z . ~ n x - - y ) = ~ ( ~ ) p J ( I - - p ) ~ 4 JaY

where n depends on x.

3.4. - - F r om these considerat ions we see tha t the decision to offer an insurance cont rac t to the public a t a p remium x will give the com pa ny a profi t which is a s tochast ic variable. The probabi l i ty d is t r ibut ion of this variable will depend on x, the claim dis t r ibut ion and the demand function. This means t ha t the choice of a marke t p remium x implies the choice of a profit distribution.

If we now assume tha t an insurance compan y has some rules which enable it to decide on the p remium at which the con t rac t should be offered, it mus t also have a rule which makes it possible to pick out the best or the most preferred among the obtainable profi t distr ibutions. This rule will represent the company ' s willingness to assume risk, or its risk policy, or to use still ano the r term, the objectives, which the company wants to pursue.

3 . 5 . - The choice of policy or object ives is by its v e ry na tu re a subject ive decision. I t is not possible to s ta te categorical ly tha t it is r ight or wrong if the company underwri tes a given risk. I t may, however, be possible to s ta te whether a par t icu lar underwri t ing decision is consistent or not wi th the overall object ives of the company.

z7


In order to formalize these ideas, we shall assume:

(i) An insurance company has a complete preference ordering over the set of all profit distributions.

This ordering will represent the company's policy, and in every situation the company will seek to make the decision which leads to the most preferred among the attainable profit distributions.

(ii) The company's preference ordering is consistent. This term obviously requires a precise definition, a point which we shall not take up here. The different possible definitions have been studied in detail by a number of authors, i.a. Savage [61, and the application to insurance has been discussed in another paper C21.

3.6. From these assumptions it follows trivially that it is possible to assign a real number or an index U (F) to any profit distribution F(z) so that

if and only if F(z) is preferred to G(z). It follows further that there exists a real valued function u(z)

such that +®

u (v} = f u(z) dF(z)

This result is far from trivial. It was first proved by Von Neu- mann and Morgenstern [5~ in 1947. Since then a number of other proofs have been published, for instance in the book by Savage [61 already referred to.

The implication of this result to our problem is that any consistent rule for determining the premium for an insurance contract can be represented by a function u(z). This function is usually referred to as a utility function, because it can be interpreted as the utility assigned to an amount of money equal to z. The concept "ut i l i ty of money" plays a central part in classical economic theory, and it is interesting to note that this concept also appears necessary for further development of the theory of insurance.

3.7. - - In order to illustrate the application of the ideas, developed in the preceding paragraph, we shall s tudy an example, slightly less trivial than the one introduced in para 2.1. We shall find it convenient to make some changes of notation.

THE ECONOMIC T H E O R Y OF I N S U R A N C E 259

We shall consider an insurance company and assume:

i) The company's policy can be represented by a util i ty function u(x).

(ii) The company's initial capital (or free reserves) is S. (iii) The company considers offering the public an insurance

contract with a claim distribution F(x). (iv) The premium for this contract is fixed as P, for instance

by tariff-agreement or Government regulation. (v) If the company spends an amount s on advertising and sales

promotion, it will be able to sell n = n(s) contracts.

The problem is then to determine the optimal amount s which should be spent on sales promotion.

By a straight forward application of the results in para 3. 6. we find that s should be determined, so that the following expression is maximized:

u(s + n p - s - x) dFCn (x) 0

where F~'0 (x) is the n-th convolution of F(x) with itself. This value of s will lead to the profit distribution which according

to the company's policy, is considered the best attainable.

3 . 8 . - In the example above we have reformulated our original Problem so that our task finally was reduced to maximizing a mathematical expression. This approach to a problem is typical of Operational Research. This term is often used loosely about a group of more or less interrelated mathematical techniques. How- ever, the essential idea, and the real art of operational research lies, not in solving a particular class of mathematical problems, but in formulating the problem so that these mathematical techniques can be applied.

3 . 9 . - If the formulation given in para 3-7. comes to grips with the problem as practising actuaries see it, they should be able to decide in general terms on the kind of mathematical techniques which are required to solve the problem. The choice of specific techniques can probably best be made in each particular case, depending on the nature of the three functions u(x), n(s) and F(x).

Of these three functions, F(x) is well known to any actuary, and

260 T H E E C O N O M I C T H E O R Y OF I N S U R A N C E

n(s) represents a concept which should be familiar to company actuaries who keep in contact their colleagues in the market research department. The utility function u(x) may, however, seem strange and unfamiliar to many actuaries. The function represents the company's policy, and so far, little is known about the general shape of these utility functions. The main reason for this lack of knowledge is that few companies are very specific when they make public statements about their policy. This may mean that companies simply do not have a well-defined policy. It may, however, also mean that companies consider their policy a business secret. The companies may have good reasons for doing this. For instance in negotiations over a reinsurance treaty, it must be important for a company to hide that its real policy is to obtain cover almost at any cost.

We shall not pursue this subject any further. The possible shape of the utility function is discussed in some detail in another paper [I], and the problem has recently been studied by Welten [7]- 3 . 1 0 . - - O u r formulation does of course oversimplify the real problem, and this may mean that we have lost something which is essential--or to put it another way- - tha t we have solved the wrong problem--a problem which cannot occur in practice.

The two most serious aspects of our simplifying assumptions appear to be:

(i) We have studied an isolated decision to be taken once and for all. This means that we have ignored any implications the decision may have on the future of the company.

To meet objections on this point, we can formulate the problem in terms of a dynamic model. An at tempt in this direction has been made in another paper [4].

(ii) We have assumed that the company was alone in the market, or that our company reached its decision without considering the decisions or actions which competing companies might take.

We shall discuss this point in the following chapter, and we shall see that this leads us towards an economic theory of insurance.

4. Risk and Economic Theory

4 . 1 . - In para 3-7- we assumed that there existed a function n(s)


which determined the number of insurance contracts n which our company could sell if an amount s was spent on sales promotion. This function represented the market situation which confronted the company.

If more than one company operates in the market, the situation cannot be represented by a single function of one variable. If there are k companies, we may get an adequate description of the situation by specifying k functions

n , ( s l . . . s ~ ) ( i = I , 2 . . . k )

Here n, is the number of contracts which company i will sell if the k companies spend the amounts sl . . . s, . . . s , to promote their sales.

4 . 2 . - - In this model the task of company i will still be to maximize a mathematical expression of the same form as the one we considered at the end of para 3.7. However, this expression will now depend on the k variables s l . . . sk, and company i controls only one of these. The remaining k - - I variables are controlled by the other companies, and they will seek to use this control to pursue objectives which may be different, and even directly opposed to those of the company under consideration. This means that company i cannot select an optimal s, without knowing or guessing the values which the other companies will select for s z . . . s,.~, s , + ~ . . , s,. These other companies will, however, be in exactly the same kind of dilemma, so the whole situation becomes a game as to who can outguess whom.

4 . 3 . - I t is obvious that the situation we have described is essen- tially different and more complicated than the situations which we analysed in chapter 3. I t is also obvious that the companies in this situation cannot reduce their problems to the simple maximizing problem considered in para 3.8. Such a reduction of the problem is the very essence of the approach which leads to operational research. If this reduction is impossible, we must look for a different approach.

4 . 4 . - - T h e situation we have described is not very different from the classical model of a market where several sellers or producers compete for the favour of a large number of buyers or consumers.

262 THE ECONOMIC THEORY OF INSURANCE

Classical economic theory has been able to analyse such markets in a rather satisfactory manner, and it is natural to t ry if this theory can provide an approach which leads to a solution to our problem.

This leads us to consider insurance cover as a commodity for which there is a demand, depending on the price. We must then assume that some persons or institutions are willing and able to supply this commodity, and that the amount they will supply depends on the price.

If the supply and demand functions meet certain conditions, there exists a unique price which will make total supply in the market equal to total demand.

This price is referred to as the equilibrium price.

4 . 5 . - - T h e basic assumption of the classical market theory is that the traders behave in a passive manner, in the sense that they take the price as given and unchangeable, and decide how much they want to buy or sell at this price. If the traders make their decisions on the basis of a price different from the equilibrium price, supply and demand will be unequal, and this will generate forces which push the price towards the equilibrium price.

The crowning achievement of the classical theory was to prove that if all traders made their decisions on the basis of the equilibrium price, the market would reach a Pareto-optimal state. This means roughly that the market is in a state where no trader can improve his situation, except at the expense of others. This means that the price mechanism establishes a rational arrangement in a market which initially seemed to be a chaos of conflicting interests. This again led classical economists to claim that free competition would lead to "the best of all possible worlds".

4 . 6 . - If we t ry to apply the ideas of classical economic theory to an insurance market, we will run into difficulties almost immediately. One of the first difficulties is that there is no natural unit of insurance cover, so that it seems impossible to define price in a meaningful way. There are other difficulties of an even more fundamental nature, but I shah not deal with them here, since they have been discussed in detail in another paper [3].

Even if many of the basic concepts of classical economic theory


are meaningless or inapplicable in an insurance market, the most fundamental of them all, Pareto-optimality, can be defined fairly easily. It is therefore natural to take this concept as our starting point. This leads us to the Theory of Games E51, which in this context must be seen as a far-reaching generalization of the more orthodox economic theories.

4 . 7 . - The basic assumption in the theory of n-person games is that rational players will somehow come to a Pareto-optimal arrangement. This leads to another difficulty, since there usually will be an infinity of such arrangements.

To illustrate this, we can again consider the example of paras 4.i .--4.2. In this example there may well be a unique advertising expenditure which will be optimal for the k companies, seen as a group. There will, however, be infinitly many ways in which this expenditure and its fruits can be divided among the k companies.

To obtain a determinate solution to such problems, we must make additional assumptions about how the parties behave during negociations or bargaining. In game theory such behavioral assumptions concern the ways in which the players form coalitions in order to co-operate during the negociations towards a Pareto-optimal arrangement.

Classical economic theory reached a determinate solution, i.e. a unique equilibrium price by making the "additional assumption" that traders passively adjusted to prices, as if they were given by some deus ex machina. This assumption may be realistic or not, the point in the present context is that it has no meaning when applied to an insurance market. We must find other assumptions of about the same strength if we want to treat insurance as an economic activity, and analyse it within the framework of a general economic theory.

5. Concluding Remarks

5 . 1 . - The point I have tried to make in this paper, is that the ends should guide our choice of means. We should not adjust the ends in order to create new applications for means which happen to be fashionable.

A good actuary should of course explore new mathematical techniques and find out if they can be of help in his work. I do,

264 THE ECONOMIC THEORY OF INSURANCE

however, not believe that this is the most pressing need, neither in the actuarial profession nor in the insurance industry.

5 . 2 . - - I n chapter 3 I have tried to show that the methods of operational research can be successfully applied only in insurance companies which have a well-defined policy--or to put it tauto- logically--companies which can spell out their obiectives in an operational manner.

In chapter 4 I have indicated that there are situations in which the methods of operational research fall short. Mathematical methods which may prove useful in these situations, have been developed in game theory. The methods appear powerful, but we cannot hope to use them successfully unless we are quite clear about the object ives-- the obiectives of persons and companies, when they act individually, and when they act in groups where the members have partially conflicting interests.

5 . 3 . - The stress on objectives really means that we need more factual knowledge before we start experimenting with new mathematical techniques. We need to know more about man's need for security and willingness to take risks before we devise the insurance which will solve his problems.

It may be fitting to terminate this paper by quoting the conclusions. Von Neumann and Morgenstern reached in their analysis of the application of mathematical methods in economic theory: "The underlying vagueness and ignorance has not been dispelled by the inadequate and inappropriate use of a powerful instrument that is very difficult to handle" ([5] PP 4-5).

REFERENCES

[11 BORCH, K.: Reciproca l R e i n s u r ance Treat ies . The A S T I N Bulletin, Vol. I, pp. 17o-191.

[2] BORCH, K. : The U t i l i t y Concep t Appl ied to t he T h e o r y of Insu rance . The A S T I N Bulletin, Vol. I, pp. 245-255.

[3] BORCH, K. : E q u i l i b r i u m in a Re i n s u rance Marke t . Econometrica, Vol. 3 o, pp. 424-444 •

[43 BORCH, K. : P a y m e n t of D iv idend b y I n s u r a n c e Companies . Transactions of the i7th International Congress of Actuaries, Vol. I I I , pp. 527-54 o.

[51 NEUMANN, J. voN and O. MORGENSTERN : Theory of Games and Economic Behavior, 2rid Edi t ion , P r i n c e t o n 1947.

[6~ SAVAGE, L. J. : The Foundations of Statistics, New York 1954. [7] WELTEN, C. P. : R e i n s u r ance O p t i m i z a t i o n b y Means of U t i l i t y F u n c t i o n s ,

Actuaridle Studidn, F e b r u a r y 1964, pp. 166-175.

MARKOV CHAINS AND THE DETERMINATION OF FAIR PREMIUMS

STEFAN VAJDA Birmingham

The relationships between actuarial and pure mathematics are curious. Actuaries have contributed to the development of mathematical theory: it is sufficient to mention, as examples, Fredholm of an earlier, and Cram6r of a more recent generation. Scandinavian mathematicians, in particular, have been concerned with a very special type of stochastic process, reflected in the collective theory of risk, and the work of Philipson, Ammeter and others in this field is well known to readers of this Bulletin. However, the main stream of the theory of stochastic processes has little contact with actuarial applications.

On the other hand, many actuaries have studied and assimilated pure mathematics and have thrown light on actuarial matters by describing their own preoccupations in the terminology of modern, often abstract, mathematics. E. Franckx is one of their number.

The Insti tuto di Matematica Finanziaria of the University of Trieste (Faculty of Economics and Commerce) has published a booklet entitled

Essai d'une th~orie op6rationnelle des risques Markoviens which contains three lectures delivered by Professor Franckx in Trieste and a contribution which he presented to the ITth Congress of Actuaries, held in London in 1964 .

The central concept in these lectures is that of a Markov chain, a special stochastic process. I t is assumed that, at any given time, an item can be in any one of n states, and that the probabilities of passing, within the next unit of time, from state i (provided it is now there) to state j are known. They are denoted by p,j. The matrix (p,t) of these probabilities is referred to as a Markov matrix, or a transition matrix.

I t is easily seen that the transitions of individuals aged x into the state of being of age x + I, or into the state of having died can

266 M A R K O V C H A I N S A N D D E T E R M I N A T I O N

be described in terms of Markov chains. (The last named state is 'absorbing', i.e. the probability of remaining in that state is unity, and that of passing into another state is zero.)

This is discussed in the first lecture, where liabilities and premiums are introduced as payments connected with the transitions: typically, a person aged x gives up his premium reserve and pays a premium in order to receive either the premium reserve of the next higher age, with probability Px, or the sum assured with probability q~.

The second lecture extends these ideas to more general risks, taking into account, for instance, transitions from the active to the disabled state, with varying probabilities dependent on age.

Lecture No. 3 deals with non-life assurance. In many ways this is a simpler case than that of life assurance. But it is here where we find the investigation which might be considered the core of this course of lectures.

To introduce the problem, we might think of motor insurance. An underwriter who accepts a particular risk has only incomplete information on which to base the computation of a fair premium. He will, in many cases, charge the same premium for risks which, after one year's experience, may turn out to belong into quite different categories.

Let us now assume that we know the values Ply which describe the probabilities that a risk which, in a given year, has produced a claim level i, will during the next year produce a claim level j. The underwriter will place his risks into different categories, dependent on the first year's claim level. Which premium should he now charge ? If it is, for instance, not unlikely that a risk with high claims in the first year may show low claims in later years, then one might suspect that it is, after all, fair to charge the same premium to all categories (i.e. whatever the experience, perhaps misleading, of the first year).

To fix our ideas, let us assume that the underwriter places his risks into two categories, according to the experience in the first year, which might have shown a claim C1 for those in the first, and of C2 for those in the second category. Let the known transition probabilities between claim experiences be as follows:

MARKOV CHAINS AND DETERMINATION 267

claim Ct C2 next year

claim this C1 1/3 2/3 year C2 3/4 1/4

As a numerical illustration we shall assume that Ct = o and C~ = I. Then a risk belonging into category I should, for the next year, be charged a premium of C~/3 + 2C2/3 = 2/3, say, and a risk in the second category should be charged a premium of 3Ct/4 + C~/4 = I/4, say.

Let us now compute the premium for the third year. It is (I/3 × I/3 + 2/3 x 3/4) C1 + (1/3 × 2/3 + 2/3 × 1/4) C~ = 7/18 say for the first category, and (3/4 X I/3 + 1/4 × 3/4) C1 + (3/4 × 2/3 + 1/4 × I/4) C~ = 9/16, say for the second. The coefficients are the probabilities that claim C~ or C~ will become due for the respective categories. In Matrix notation we can write the premium for the second year as the rows of the matrix product

1/3 2/3~ ( C 1 ) = M.C, say 3/4 1/4] C2

and those for the year after that as the rows of

3/4 1/4] \3/4 1/4] C,

It emerges that the fair premium for the t-th year after the first year will be given by the rows of the matrix product Mt.C.

We notice that the premiums for the two categories, viz. 2/3 and 1/4 in the second, and 7/18 and 9/16 in the third year and so on get closer as the years proceed. Franckx proves--rather nea t ly - - that this concentration is always the case if the matrix M has row totals I, which is always the case for Markov matrices. He then asks under what conditions the powers of such a matrix converge to a matrix whose rows are all equal. This is an important question, because if this is so, then all categories will, in the long run, be charged identical premiums, and Franckx advances this as a justification for all risks to be charged identical premiums ab initio, when their risk structure is not yet known. The risks attaching to the categories with such a Markov matrix are called 'normal'.

268 M A R K O V C H A I N S A N D D E T E R M I N A T I O N

It is known (from a theorem due to Frobenius) that the powers of a Markov matrix converge to a matrix with identical rows if there exists a power M k (with finite k) such that all its elements are positive (and none is zero).

I t is natural to ask here how one recognizes whether a given matrix has such a power. The author points out that this is certainly the case if P11 > o for all j and also pal > o for all i. He calls a risk 'good' if, moreover, p n is large.

The risk level denoted by I is that where no claim is to be paid. I t follows easily that all 'good' risks are 'normal'.

I t will be noticed that the computation of premiums depends, in this study, on expected values. No matters appertaining to the theory of risk are touched upon.

The fourth lecture extends these considerations to the case of non-stationary matrices, i.e. those whose elements P~k change with time.

Unfortunately, the publication contains a number of misprints which make it difficult to read. However, the reviewer is confident that he has correctly presented Professor Franckx's interesting adaptation of the theory of positive non-decomposable matrices to a topical actuarial problem.

R E V I E W S

L E S L O I S E X P O N E N T I E L L E S COMPOSITES

P. Thyrion, Bul l e t i n de l 'Assoc ia t ion Roya le des Ac tua i re s Belges, no 62, 1964

L ' a u t e u r chos i t la loi exponen t ie l l e de fonc t ion de r 6 p a r t i t i o n :

F ( x ) = i - - e--x~,

oh X > o es t une cons t an te , c o m m e premi6re a p p r o x i m a t i o n pou r la loi de d i s t r i b u t i o n de la va r i ab l e X, cofit d ' u n sinis tre . Pa rce que ce t t e loi ne r e n d pas la r6ali t6 de fa~on s a t i s f a i s a n t e - - c o m m e le m o n t r e la r e l a t ion s u i v a n t e en t r e la d6v ia t ion s t a n d a r d e et l ' e sp6rance m a t h 6 m a t i q u e

(x) E (X) -- z,

qui ne t r a d u i t pas les ca rac t6 r i s t iques essentiel les d'e l ' exp6r ience - - on en a d6dui t , p a r analogie avec les lois de Poisson compos6es, les lois exponen t i e l l e s compos6es, qu i son t caract6r is6es p a r la fonc t ion de r6pa r t i t i on :

F ( x ) = i - - f e - X ~ dU(X), oh U(X) est la fonc t ion de s t ruc tu re , c 'est-X-dire que le p a r a m 6 t r e X es t lui- m6me une va r i ab l e a16atoire d a n s le d o m a i n e (o, ~). i1 en r6sul te a ins i une nouve l le forme de loi avec

. (x) ~ (x---~ > ~'

qui co r respond mieux A la r6alit6. L a fonc t ion de s t r u c t u r e U(X) e n g e n d r e les d i s t r i b u t i o n s exponen t ie l l e s

compos6es, c o m m e le m o n t r e n t que lques exemples couran t s . E n t r e a u t r e s l ' a u t e u r m e n t i o n n e qu ' i l n ' e s t pas n6cessaire de c o n n a i t r e U(X), F(x) p o u v a n t ~tre d6finie A l ' a ide de ses propr i6 t6s i m p o r t a n t e s ainsi que M. H o f m a n n l ' a fa i t pou r les lois de Pc i sson compos6es.

Apr6s que lques g6n6ra l i sa t ions les lois exponen t ie l l e s compos6es d 'o r ig ine n o n nul le son t d6finies p a r la loi de r 6pa r t i t i on :

G(x) = F(x--a) ,

oh x => a. La fo rmule pou r les m o m e n t s , la cond i t i on p o u r que

E - - > I

e t un exemple son t 6ga lemen t cit6s. A la f in de cet expos6 tr6s i n t 6 r e s s a n t l ' a u t e u r soul igne que la loi expo-

nen t ie l l e p u r e d o m i n e t o u t e loi exponen t i e l l e compos6e d 'o r ig ine nul le pou r t o u t x > o e t que la fami l le des d i s t r i b u t i o n s exponent ie l l es compos6es e t s t a b l e p a r r a p p o r t ~t son e x t r e m e inf6rieur.

270 REVIEWS

L ' A d a p t a t i o n de la T a r i f i c a t i o n p a r l ' exp lo i ta t ion de l ' i n f o r m a t i o n compld- men ta i re , E . F r a n c k x .

Le probl~me qui est repr is ici, parce qu ' i l se r appor t e ~ la recherche op6rat ionnel le , es t des plus actuels.

La ques t ion pr incipale est la su ivan te : C o m m e n t la p r ime initiale, qui ~tait d~termin~e sur la base de la p r ime mo y en n e e t , des in fo rmat ions pri- maires, est-elle A corriger ind iv idue l lement k l 'a ide des in format ions second- aires pour ar r iver A la pol i t ique de bonus-malus ? I I n e faut pas oubl ier que, lors de ce t te adap t a t i on , une l imite fix6e a priori ne doi t pas ~tre d6pass6e, car l ' id6e &assu rance pe rd ra i t son sens.

Un proc~d6 & a d a p t a t i o n progressive peu t se r6aliser pa r un a lgor i thme qui fait usage des d6cisions ant6rieures et des r6sul ta ts pr6c6dents :

dn+ 1 = f (d~ . . . . . dn ; x l . . . . . xn) ,

off f es t une fonction, di est la d6cision a v a n t la i ~me exper ience et x, es t le r6sul ta t de la i ~me exp6rience. Parmi les a lgor i thmes markoviens , qui son t tous de la forme

dn+ t = f ( d n , xn),

l ' au teur choisi t la classe part iculi~re;

n + ~ - - I I dn+ l - - . d n + - - . Xn,

n + X n + X

off X >_ o est le pa ram~t re de l ' a lgor i thme. L ' a lgo r i thme markov ien est compl~- t e m e n t d6fini si X et d~ sont fix6s. La d6cision init iale d~ doi t ~tre une valeur m o y e n n e des exp6riences pr61iminaires. La formule pr6c6dente a 6t6 sugg6r6e par M. Dubois de M o n t r e y n a u d pour la p r ime model6e. R e m a r q u o n s que plus le pa ram~t re X est grand, plus l '6volut ion vers la p r ime individuel le est ralent ie . Dans le cas X = cc la pr ime in i t ia le reste cons tan te . Donc X peu t ~tre choisi quelconque, mais na tu re l l ement au mieux des int6r~ts de la collectivit~ des assur6s. Par exemple M. Delapor te a est im6 k par la m6thode de Bayes.

A la fin de l 'ar t ic le un exemple 6Mgant avec les poids ~, = 20 et X = 4 ° m o n t r e l ' app l ica t ion des coefficients de mul t ip l ica t ion . Enf in il es t 6v ident que les a lgor i thmes markov iens d o n n e n t un proc6d~ s imple et souple pour a d a p t e r la p r ime initiale en ut i l i sant l ' i n fo rmat ion compl6menta i re dos6e ~t volont~.

S u r la t r a n s f o r m a t i o n d ' u n p roces sus de po i s son composd (selon les mdthodes de P . T h y r i o n et F . Esscher) , Car l P h i l i p s o n .

Au d~but deux fonct ions t r ans fo rmat r i ces sont d6termin6es, pa r lesquelles un processus de Poisson compos~, s t a t ionna i re ou non-s ta t ionna i re , et un Processus de Poisson compos6 au sens res t re in t sont d6finis. A l 'a ide de la fonc t ion g~n6ratrice quelques d is t r ibu t ions - - comme par exemple une d i s t r i bu t ion de Poisson compos~e par grappes - - sont aussi d~finies. Une d6mons t r a t i on de Thyr ion (1959) d ' une propos i t ion 6tablie par R. Consael • es t ~galement ment ionn~e.

Dans le p remier des deux th6or~mes l ' au teur ~nonce que les d i s t r ibu t ions qui d~finissent un processus de Poisson compose p e u v e n t ~tre t ransform6es en une d i s t r ibu t ion de Poisson pa r grappes de grappes g6n~ralis6e, et que celles qui d~finissent un processus de Poisson compos6 au sens res t re in t

REVIEWS 2 7 I

p e u v e n t 6 t re t r an s fo rm4es en une d i s t r i b u t i o n de Poisson p a r g rappes g6n4- ralis~e. D a n s le deuxi~me, il p r o u v e l ' ex i s t ence - - sous ce r t a ines cond i t i ons - - & e x p r e s s i o n s expl ic i tes pou r les fonc t ions de r 6pa r t i t i on , express ions qui d4f in i s sen t les d i s t r i b u t i o n s t r ans fo rm6es pa r le p r em ie r th4or~me. Les deux d 4 m o n s t r a t i o n s son t expl iqu4es s o m m a i r e m e n t e t des r6f6rences ~ la l i t t4- r a t u r e son t indiqu6es . E n f i n cet a r t ic le t r a i t e de la t r a n s f o r m a t i o n d 'Essche r .

Cet expose, t r~s concent r6 , se t e r m i n e p a r la r e m a r q u e que les e s t i m a t i o n s selon la m 4 t h o d e de B o h m a n (I96O) e t les e s t i m a t i o n s bas6es sur les t r a n s - form4es d ' E s s c h e r o n t une conco rdance r e m a r q u a b l e dans p r e s q u e t o u t le d o m a i n e 6tudi6.

R. RUTZ

Reinsurance optimization by means of utility functions, C. P. Welten, Actuar i~le S tud i~n Afl. 6, f eb rua r i 1964, ' s -Gravenhage .

The a u t h o r gives a n app l i ca t i on of t he u t i l i t y concep t in c o n n e c t i o n w i t h r e in su rance p rob l ems as i n t r o d u c e d b y Borch . I n p r inc ip le he cons iders t h e se t of all poss ible r e i n s u r ance t r ea t i e s {t,}. T he genera l p r o b l e m cons is t s in d e t e r m i n i n g t he t~ for wh ich t he u t i l i t y

V 0 + P

Utc = Y u(V)d~G(V), Vx = V o + P- -S¢¢

is a m a x i m u m . N e x t t he a u t h o r m a k e s t h e r e s t r i c t ions :

a. t h e u t i l i t y f u n c t i o n is of t h e e x p o n e n t i a l t y p e

u( V) = a - - b e *v,

b. t he d i s t r i b u t i o n func t ion of t h e t o t a l costs, St, belongs to a specif ied class of c o m p o u n d Po i s son-d i s t r ibu t ions .

Fo r th i s l imi ted se t of poss ibi l i t ies he d e m o n s t r a t e s t h a t t he p r o b l e m can be solved in a r a t h e r s imple w a y w i t h o u t m u c h c o m p u t a t i o n a l work.

J. v. KLZNKEN

THE ASTIN BULLETIN

THE ASTIN BULLETIN

PUBLICATION OF THE ASTIN SECTION OF THE PERMANENT COMMITTEE FOR

INTERNATIONAL ACTUARIAL CONGRESSES

VOL. IV

T H E ASTIN B U L L E T I N

I N D E X TO VOLUME IV

Papers

Almer B.

Modern General Risk Theory . . . . . . . . . . . . I36

Andreasson G.

Distribution free approximations in applied Risk Theory I I

Benktander G. and Ohlin J.

A combination of Surplus and Excess Reinsurance of a Fire Portfolio . . . . . . . . . . . . . . . . . . . 177

Bichsel F.

Experience rating in Subsets of Risks . . . . . . . . 2xo Contr61e des op6rations d'assurance dans les branches non-vie . . . . . . . . . . . . . . . . . . . . . 23 x

Bohman H.

Experience rating when the Company aims to increase the volume of its Business . . . . . . . . . . . . . 208

Borch K.

Control of a Portfolio of Insurance Contracts . . . . . 51 The economic Theory of Insurance . . . . . . . . . 252

Bfihlmann H.

Experience rating and Credibility . . . . . . . . . . 199

VI I N D E X TO V O L U M E IV

Derron M.

A Study in credibility Bet terment through Exclusion of the largest Claim . . . . . . . . . . . . . . . . . 39

Esscher F.

Some Problems Connected with the Calculation of Stop Loss Premiums for Large Portfolios . . . . . . . . . I7o

Hovinen E.

A Procedure to Compute Values of the Generalised Pois- son Function . . . . . . . . . . . . . . . . . . . I29

Jongh B. H. de

The insurer's ruin . . . . . . . . . . . . . . . . . 72

Kupper J.

The Recent Development of Risk Theory and its Applic- ations . . . . . . . . . . . . . . . . . . . . . . IO6

Lundberg O.

Une note sur des Syst~mes de Tarification basks sur des ModUles du type Poisson compos6 . . . . . . . . . . 49

Ohlin J. and Benktander G.

A Combination of Surplus and Excess Reinsurance of a Fire Portfolio . . . . . . . . . . . . . . . . . . . I77

Pentik~iinen T.

On the Solvency of Insurance Companies . . . . . . . 236

Pesonen E.

On the Calculation of the Generalised Poisson Function I25 On Optimal Properties of the Stop Loss Reinsurance . I78 Magnitude Control of technical Reserves in Finland 240

INDEX TO VOLUME IV VII

Philipson C.

Note on the Relation between Compound and Composed Poisson Processes . . . . . . . . . . . . . . . . . 191

Seal H. L.

The Random Walk of Simple Risk Business . . . . . 19

Thyrion P.

Regard sur le D6ve loppemen t r6cent de la Th6orie du

Risque . . . . . . . . . . . . . . . . . . . . . . 87

Va jda S.

Markov chains and the De t e rm i na t i on of fair p r emiums 265

Verbeek H. G.

On Op t ima l Reinsurance . . . . . . . . . . . . . . 29

Wolff K. H.

Collective Theo ry of Risk and Ut i l i ty Func t ions . 6

Secretar ial Notes . . . . . . . . . . . . . . . . . I , 81

Notes by Cha i rman . . . . . . . . . . . . . . . . . . . I95

Biographical Notes . . . . . . . . . . . . . . 5, 84, 193

Rules for the As t in Section of the P e r m a n e n t Commi t t ee . 3

S t a tu t s pou r la Sect ion Ast in du Comit6 P e r m a n e n t . . . . 4

Reviews . . . . . . . . . . . . . . . . . . . . . . . 269

Chairman

Vice Chairman

Treasurer

Members

Secretary

COMMITTEE OF ASTIN

MASTERSON Norton E., Stevens Point

SOUSSELIER Jean, Paris

THYRION Paul, Brussels

AMMETER Hans, Zurich

JOHANSEN Paul, Copenhagen

OTTAVlANI Guiseppe, Rome

STERNBERG Ingvar, Malm~

WELTEN C. P. , Amsterdam

BEARD Robert Eric,

PEARL Assurance Company Ltd.,

252, High Holborn,

London W.C. I

Honorary Chairman

Past Chairman

FRANCKX Edouard, Brussels

1957-196o I96o-I962

1962-1964

1964-1966

JOHANSEN Paul, Copenhagen

HENRY Marcel, Paris

BEARD Robert Eric, London

A~I~ETER Hans, Zurich

The Committee is not responsible for s ta tements made or opinions expressed in the articles, criticisms and discussions published in this Bulletin.

Le Comit6 rapelle que seul l ' auteur de chaque publication est responsable des faJts qu'il expose et des opinions qu' i l exprime.

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