INTRODUCTORY R E P O R T

E X P E R I E N C E RATING AND C R E D I B I L I T Y

HANS BOHLMANN
Ziirich

I. The Classical Ratemaking Problem

Classical statistics deals with the following standard problem
of estimation:
Given: random variables X1, X 2 . . . Xn independent, identically
distributed, and
observations x l , x~ . . . x n ,
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
Given: risks Rt, R 2 . . . Rn no contagion, homogeneous group,
Find: 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 Solution
Insurance people are practically minded, and so they have in
m a n y circumstances come out with pragmatic solutions, even if
the theoreticians were not able to provide them with full theore-
tical justification for doing so. Within the classical setup such
circumstances were happening if
200 EXPERIENCE RATING AND CREDIBILITY

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 b y Harald Bohman,
"Experience rating when the company aims at increasing the volume
of its business". This paper shows that the scope of experience
rating m a y even be extended beyond "proper-rate making".
Aiming at maximum volume of business is the alternative treated
b y 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 b e y o n d 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 Probability 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 b y assu-
ming the risk parameter to be a random variable. This advantage,
b y the way, is not bought at the expense of an increasingly com-
plicated formula, a fact that for practical purposes will certainly
be appreciated as well.
Two papers, namely, "Experience Rating in Subsets of Risks",
b y Fritz Bichsel, and "Une note sur des syst&mes de tarification
bas6s sur des modules du type Poisson"l), b y 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 homo-
geneous classes. What do we actually mean b y homogeneity?
This is best illustrated b y the following array of random variables
X,~ = claim produced b y risk i in the accounting period (year) j.
Xll, Xle . . . . Xi•

Xml, Xm2 . . . . Xmn.

a) P u b l i s h e d in A s t i n B u l l e t i n vol. I V P a r t 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 m y knowledge, been done b y dropping homo-
geneity in mass but b y 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 m a y make a suggestion
it is this: the changes in time considered b y 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 con-
sistent--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 Fluctuating 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.
It 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 m a r k e t - - a n d this is
the remarkable f a c t - - t h e credibility formula has been found to do
an excellent job. Under such circumstances it does not come as
a surprise that m a n y actuaries have tried to prove the credi-
bility formula starting from more general principles. The first to
do so was Arthur Bailey, one of the most outstanding A m e r i c a n
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
m a n y 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
b y Hans Ammeter in Trieste. Derron computes the credibility
improvement for a Pareto-type distribution if the largest claim is
excluded. The result m a y indicate a new way of approach for
m a n y other estimation problems which we encounter in our work.
I mean that Marcel Derron has very clearly shown to us the impor-
tance of truncating or curtailing our basic data to get more reliable
information out of them. The importance of his work is underlined
by m a n y 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. E q u i l i b r i u m 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
m e d i a n is investigated I believe t h a t in the actuarial literature the
estimator function for ~(~) is always chosen to be equal to
E[~(~)/X1, X~ . . . . X n ] = 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) P u b l i s h e d in A s t i n Bulletin Vol. IV P a r t i.
 EXPERIENCE RATING AND CREDIBILITY 205

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 b y 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 try 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

It is worthwhile to note here that the credibility folmula used
b y 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 approxima-
tion 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 b y solving the following problem
206 EXPERIENCE RATING AND CREDIBILITY

F i n d : a, b such t h a t
X1 + X2 . . . . X n
a + b X 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 : If 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
t h e n ao + boX is also the best linear a p p r o x i m a t i o n to
E[t~O)/x,, . . . x2

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

x2}~ + E {E[v.O)/x,, . . . x3 --~(~)?

Since t h e second t e r m on the right h a n d side does not d e p e n d

on ao a n d bo it is clear t h a t t h e left h a n d side a n d the right
h a n d side are m i m i m i z e d b y the s a m e ao a n d bo q.e.d.

R e f o r m u l a t i n g the original p r o b l e m we hence w a n t to find a a n d

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

W e find t h a t the a b o v e left h a n d side c a n be w r i t t e n

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) ]
a n d ii) b 2 E ( X - - ~t(8)) 2 + (I - - b) z Var Err(S)] mini-
mum
var[~O)]
which leads to b =
Var[t~(b)] + E ( X - - ~ ( ~ ) ) ~
A s s u m i n g n o w independence a n d identical d i s t r i b u t i o n of the
X , we find
I I
E(X-~O))~= -
n
E(X, -- ~(~))~ = -n E [ ~ ( 0 ) ]

Hence the credibility relation

(I - - b)" E[~(8)] + b" X
 EXPERIENCE RATING AND CREDIBILITY 207

where
n
b----
n+k

k--
Var[vt(8)]

Remarks:

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. It amounts to replacing the relation
I
E ( X - - ~(~))2 = n E[~(~)I b y 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.