ASTIN BULLETIN Volume 2, Number 2 - International Actuarial

137 Pages·2015·5.19 MB·English

THE ASTIN BULLETIN
PUBLICATION OF THE ASTIN SECTION OF THIS
PERMANENT CO~4MIT.FEE FOR
IN'I ERNe~T.ONAL ACE UARIAI~ CONGRESSES
VOL. II, PART I
S E P T E M B E R 1 9 6 2
CONTENTS
Dedication to Mr. Thdate . . . . . . . . . . . . . . . I77
Note by Chairman . . . . . . . . . . . . . . . . . . . 178
Editorial Notes . . . . . . . . . . . . . . . . . . . . 18o
Papers . . . . . . . . . . . . . . . . . . . . . . . . I82
Some Analyses of Car Insurance Claim Rates by J. H.
Munden, London . . . . . . . . . . . . . . . . . . I82
On the Use of Linear Discriminant Functions in the Reahn
of Industrial Accident Insurance by J. van Klinken,
Amsterdam . . . . . . . . . . . . . . . . . . . . 198
Application of Game Theory to Some Problems in Automo-
bile Insurance by K. Borch, Bergen . . . . . . . . . 208
Sur les dv6nements en chaine et la distribution binomiale
ndgative gdn~ralis~e par C. Campagne, Amsterdam . 222
Le test de la majoritd confirm4e et son utilisation dans Fin-
fdrence statistique par E. Franckx, Bruxelles . . . . . 241
Minimum Variance Reinsurance by S. Vajda, Londo~ . 257
Experience Rating--A New Application of the Collective
Theory of Risk by fir. Ammeter, Zurich . . . . . . . . 261
Assurances Automobiles et Revenus Financiers par J.
Soussdier, Paris . . . . . . . . . . . . . . . . . . 271
Note sur la gestion automatique de la reserve pour
sinistres en assurance automobile par J..Petit, Paris. 287
Reviews . . . . . . . . . . . . . . . . . . . . . . . 297
Biographical Notes . . . . . . . . . . . . . . . . . . . 31o
The Committee is not responsible for statements made or opinions ex-
pressed in the articles, criticisms and discussions published in this Bulletin.
Le Comit6 rappelle que selll l'auteur de chaque publicatioa est responsable
des fair qu'it expose et des opinions qu'il exprime.
Le comit6 d'ASTIN a d6cid6 lors de sa r6union h Juan les Pins
de d6dier les publications de l'ann6e I962 en hommage k Monsieur
THEATE. Cette marque de particuli~re attention lui est rendue par
tousles membres d'ASTIN, en reconnaissance de son intervention
d6cisive en faveur de la creation d'une nouvelle section du Comit6
Permanent.
A l'heure actuelle, par l'ensemble de ses discussions et de ses
travaux dans le domaine de la recherche scientifique et pratique
appliqu~e aux assurances ,,non life", ASTIN est fibre de justifier
la confiance dont le PrEsident du Comit~ Permanent a bien voulu
l'honorer.
Au moment oh, pour des raisons personnelles, Monsieur THEATE
a d~cid~ de renoncer ~ la charge qu'il occupait avec une ~minente
autorit~, tous les membres d'ASTIN s'associent pour exprimer
Monsieur THEATE leurs sentiments de profonde gratitude pour
l'oeuvre particuli~rement distinguee qu'il a accomplie pendant plus
de trente ans au Comit~ Permanent.
I3
NOTE BY CHAIRMAN
London, 3oth June 1962
ASTIN will be officially 5 years old in November 1962 and the
brief account of the formal business dealt with at the committee
meeting held during the Colloquium at Juan-les-Pins which appears
elsewhere in this issue of the Bulletin records the steady growth
of the first special section of the Permanent Committee for Inter-
national Actuarial Congresses since the I5th Congress in New York.
Like most vigorous children, a young organisation needs help
and guidance and we owe a considerable debt of gratitude to our
Chairmen who have so ably guided our steps in these early years
and created a solid basis for building for the future. In an organisa-
tion such as ours, the Secretary is very much aware of the responsa-
bilities which devolve upon the Chairman and I am thus specially
aware of the very significant contributions to the success of ASTIN
made by Paul Johansen, Chairman from 1957 to 196o and by Marcel
Henry, Chairman from 196o to 1962.
The committee decided at one of its early meetings that the term
of office of the Chairman should be short and I am very conscious
of the honour of being elected Chairman and of the challenge
presented to me of maintaining and furthering the tradition so
soundly created by my predecessors in office.
We are fortunate in being able to draw on the experience of
Edouard Franckx in our Committee deliberations, and his election
as Chairman of Honour describes succinctly the regard felt for
his part of the early work which led to our successful foundation
in 1957 .
The Bulletin has now taken its place among scientific publications
and our full appreciation is due to the untiring efforts of our Editor
(now also Vice Chairman), Hans Ammeter, for the very high
standard he maintains. A word of thanks too for the printers and
the care they give to this very important link between our members.
Behind the scenes the rest of the Committee share their part of
the burden, but I must not fail to mention the many members
NOTE BY CHAIRMAN 179
who have contributed the papers which have generated such
excellent discussions at the colloquia and which form the backbone
of the Bulletin.
Among enthusiasts, it is easy to be enthusiastic and I am looking
forward to my period of office based on the strong bonds that have
been built out of our common interests and on the support of our
active and understanding committee.
EDITORIAL NOTES
Lors du dernier Colloque, le Comit6 a proc6d6 A quelques changements
importants.
Monsieur Marcel Henry, President de i'ASTIN s'est d6mis de son impor-
tante fonction. Monsieur M. Henry remet sa charge apr~s avoir pr~sid~ ~ la
destin~e de notre association pendant deux ans, de I96O k I962. Tant durant
cette trop brave p~riode qu'auparavant, Monsieur M. Henry s'est d~vou~
sans compter ni son temps ni sa peine. Sa contribution au d6veloppement et
l'affermissement de I'ASTIN est considerable. Parmi ses nombreuses
initiatives, remarquons l'organisation de colloques: celui de 1959 qui se
tint ~ La Baule et celui de cette annie sur la C6te d'Azur. Tous ceux qui
ont particip~ ~ ces manifestations en gardent un excellent souvenir. Parall~le-
ment ~/ ses nombreuses activit6s, Monsieur M. Henry a toujours pr~sid~
avec competence les r~unions qu'il ~tait appeM ~ diriger. Sa participation
active aux travaux et discussions doit 8tre un exemple pour chacun. Le
Comit~ exprime sa gratitude ~ Monsieur M. Henry et le remercie de son
d6vouement ~ notre association.
Monsieur Robert Eric Beard, jusqu'alors vice-pr6sident, succ~de/~ Monsieur
M. Henry k la tote de I'ASTIN, tandis que Monsieur Hans Ammeter succ~de
Monsieur Beard comme vice-pr~sident.
I1 est certain que Monsieur Edouard Franckx dont la renomm~e n'est plus
faire aurait ~t~ un candidat s6rieux k cette ~lection. Trbs connu dans le
cadre de notre association dont i est membre fondateur, il ne l'est pas
moins dans le monde scientifique. Monsieur Ed. Franckx a ~t~ appel6
r~cemment k assumer les charges les plus hautes et les plus d~licates: il
preside le Comit6 Permanent, fonction dans laquelle il excelle. C'est pourquoi
le Comit~ n'a pu offrir ~ Monsieur Franckx la pr~sidence d'une sous-section
du Comit~ Permanent. En t~moignage de reconnaissance le Comit~ a ~lu
Monsieur Franckx President honoraire pour la p~riode I962-I964.
A meeting of the Committee was held on 24th May during the colloquium
at Juan-les-Pins and the following notes will bring members up to date on
the various aspects of ASTIN activities.
The membership, spread over 28 countries, now numbers 434 of whom
36o are ordinary members, 2 special members and 72 corporate members.
Whilst the current financial position was satisfactory, the balance at the
bank of B. Frs. 2o8,ooo had fallen by some B. Frs. 56,ooo from the figure
at 3I.I2.6i, largely due to the cost of Vol. II, Part I of the Bulletin. The
major part of the expenses of ASTIN are, of course, represented by the cost
of printing the Bulletin and if the supply of papers continues at the present
rate it might become necessary to consider the adequacy of the existing
annual subscription of B. Frs. 2o0 for ordinary members at the time of the
next general meeting in May 1964. This will be held in London during the
I7th International Congress, preliminary details of which will have been
received by all members and for which Subject 4--Practical applications
of modern statistical techniques in motor and other non-life insurance
branches and in life assurance--will provide ample scope for members to
EDITORIAL NOTES I8I
submit papers. Members of ASTIN who wish to avail themselves of the
opportunity for informal discussion are requested to communicate with the
Secretary of ASTIN regarding the subject they would like to be discussed.
At this general meeting the formal business will include the election of a new
committee in accordance with the ASTIN rules.
Some progress has been made with the preparation of a dictionary of
terms arising from the statistical aspects of non-life insurance; equivalents
of a number of terms were prepared for the four Scandinavian languages
and these have now been extended to include English. Extension to other
languages will be started shortly.
Members will also be interested to learn that the preparation of a text book
on risk theory is also well in hand. Messrs. PentikAinen and Pesonen have
revised the original Finnish draft and the first draft of an English version is
in being, the English collaborator being Mr. R. E. Beard.
As regards future events an invitation has been received for the next
ASTIN colloquium to be held in Trieste in 1963. Full details regarding timing
and subject matters will be circulated so soon as they become available.
Following the International Congress in May 1964 the Casualty Actuarial
Society will be celebrating their 5oth Anniversary in November 1964.
Any members who may be in a position to attend the meeting of the Casualty
Actuarial Society are requested to communicate with the Secretary of ASTIN.
SOME ANALYSES OF CAR INSURANCE CLAIM-RATES
BY
J. M. MUNDEN
London
Editorial Comment
We have printed the foregoing paper by Mr. Munden as submitted
because we think that the detailed analysis will be of interest to all
who have interests in the field of motor insurance rating. Of necessity, the
data does not lend itself to analysis with respect to some of the known
variables and we are conscious that some of the conclusions are controversial ;
some factors have also emerged from the discussions within ASTIN on motor
insurance and it is therefore hoped that the following comments will be of
value in relation to the paper.
It is of the utmost importance that a clear distinction is drawn between the
concept of accident proneness and the heterogeneity shown from observations
of claim frequencies under insurance policies. As the discussion at La Baule
brought out, the first conclusion to be derived when a compound Poisson
distribution emerged is that there is a degree of heterogeneity in the data.
This might be due to differences in accident probabilities of the underlying
risks, but it could be due, for example, to different exposures of similar risks.
Lanteli's paper to the R~ttvik colloquium showed a substantial variation
of claims experience with annual mileage and thus without an analysis
controlled with respect to mileage the conclusion that a proneness factor is
solely involved must be suspect.
Another factor which must have some effect is the incidence of "no claim
discount" or, to use the term first introduced by Carl Philipson, "hunger for
bonus". What evidence is available from studies of claims distributions by
amounts, shows a drop in claim frequency of about the expected amount in
the region of smaller claims. Probably there is also a relationship with the
age of car, as minor damage is more likely to be the subject of a claim on a
new car than an old. These factors may underly the increase in frequency
shown at durations 6 and 7, as this feature has been noted in experiences to
which the explanation in the paper would not apply.
Finally it would seem that the third party and comprehensive policies
experience must be treated with caution as there are conflicting influences
involved. New cars tend to be insured under comprehensive policies and old
cars for third party risks only. There is also a tendency for young owners to
first acquire an old car. The result of these tendencies, coupled with the
known improvement in claim frequency in the early years of driving ex-
perience and the tendency for mileage to be heaviest in the earliest years of
acquisition of a car must be a very complex pattern if the overall frequencies
are related to duration alone. There must also be some element of switching
RAC ECNARUSNI SETAR-MIALC I83
from one class of policy to another, which would introduce further difficulties
in regarding the heterogeneity in policy claims experience as due to proneness
of drivers.
In addition to the references in the paper ew would also refer readers to the
correspondence in J.I.A. ,48 pp. I23/4, .8591
We hope that the publication of .rM Munden's paper will stimulate further
investigations into this very difficult statistical field and that it will be
possible to devise methods for control of the factors ew have mentioned.
I. NOITCUDORTNI
1.1. Although much is known about the effect of factors such as
road layout and regulations on the frequency of road accidents,
comparatively little is known about the personal factors such as,
for instance, driving experience and accident-proneness. It is
difficult to study these from the official accident records, but motor
insurance policies and the claims made on them provide useful
material for the analysis of some of these factors. This paper presents
the analysis made by the Road Research Laboratory, Great Britain
of claim records kindly supplied by an insurance company and is
mainly concerned with the effects of age and experience, and with
claim-repeaters. An earlier analysis of some of the data has been
made by Johnson and Garwood (I).
2. ATAD
2.1. The data relate to 2 765 policies which were renewed between
October, I954 and December, i955 and which had run continuously
since their inception. Every policyholder, therefore, had had at
least one year's exposure to risk and a few had had more than
thirty years' exposure. The details of each policy used in this
study include the date of inception, the extent of cover, the number
of drivers covered, the class of use, the place where the car was
normally garaged, and the age and sex of the policyholder.
2.2. Eighty-nine per cent of the policies studied were issued for
'any driver' and most of the remainder were issued for 'owner only
driving', while 79 per cent provided 'comprehensive' cover. Seventy-
four per cent were issued for 'use Class I' (private and personal
business use only) and 23 per cent were issued for 'use Class II'
(all business purposes excluding commercial travelling and motor
trade). Seventy per cent of the policyholders garaged their cars in
London and the Home Counties.
184 CAR INSURANCE CLAIM-RATES
2.3. For purposes of identification of claims, only the dates on
which they were made have been used. A 'claim' is defined in this
study as any incident known to the company, excluding fire and
theft, whether it was made known by the policyholder or the third
party, and includes cases where the incident was reported but no
claim was made.
3. VARIATION OF CLAIM-RATE BY CALENDAR YEARS
3.1. The average annual claim-rates of the sample of drivers in three
periods between 1933 and 1954 are given in Table I for each sex
separately. The rates are the number of claims made in the particu-
lar period divided by the number of years of exposure to risk of all
policyholders during that period.
TABLE I
Average claim-rates by calendar periods
Period Males Females All policyholders
1933-39 o.22 o.19 o.22
194o-45 o.io o.o9 o.Io
1946-54 o.15 o.13 o.15
3.2. The claim-rates of female policyholders are slightly lower
than those of male policyholders. The post-war (World War II)
claim-rate is about 30 per cent lower than the pre-war rate. The
differences in the average age and average experience of drivers
in these two periods were slight and would not have accounted for
much of the reduction. Analysis of individual post-war years
shows that at no time did the claim-rate reach that of the pre-war
years.
4. AGE AND EXPERIENCE OF POLICYHOLDERS
4.1. There is considerable variation with age of policyholder in the
claim-rate per policy-year. Two curves of claim-rate against age
when exposed to risk are shown in Fig. I; one curve was obtained
using all the available data and the other shows the average
claim-rate during the first year of experience * of policy-
* Experience in this paper means experience with the company concerned
and does not necessarily mean driving experience.