Table Of ContentErwin Straub
Non-Life Insurance
Mathematics
With 12 Figures
Springer-Verlag Berlin Heidelberg GmbH
Professor Dr. Erwin Straub
Swiss Reinsurance Company
Mythenquai 50/60
P.O.Box
CH-8022 Zürich, Switzerland
The illustration that appears on the front cover is described in detail on page 48.
Mathematics Subject Classification (1980): 62P05
ISBN 978-3-642-05741-0
Library ofCongress Cataloging-in-Publication Data
Straub, Erwin
Non-life insurance mathematics/Erwin Straub. p. cm.
ISBN 978-3-642-05741-0 ISBN 978-3-662-03364-7 (eBook)
DOI 10.1007/978-3-662-03364-7
1.lnsurance-Mathematics. I. Title
HG8781.S75 1988 368'.01-dc19 88-11959 CIP
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© Springer-Verlag Berlin Heidelberg 1988
Originally published by Springer-Verlag Berlin Heidelberg New York in 1988
Softcover reprint of the hardcover 1s t edition 1988
Typesetting: Macmillan (lndia) Limited, Bangalore;
214113140-543210 Printed on acid-free paper
To the Reader
The present book is dedicated to the late w. Leimbacher.
It is based on lecture notes which were written between 1975 and 1985 when
I taught Non-Life Mathematics at the University of Berne. It is meant to be a
textbook on the one hand for university students with limited practical experi
ence and on the other for practitioners knowing mathematics from so me
distance only, but it is definitely too elementary and incomplete for a thorough
bred modern Non-Life Actuary.
The main purpose is to provide insight into some pertinent practical
problems and their possible theoretical solutions, for "there is nothing more
practical than a good theory". I do not remember who said this so I cannot give
a reference - by the way, references are only fragmentary; I have given only the
ones I was most impressed by, the large remaining part of the contents was
taken from different places and put together without mentioning the source.
The whole thing is admittedly incomplete, imperfect and sometimes perhaps
even inaccurate, both with regard to contents and deduction of results. There are
three reasons for this: 1) to stir up the reader's mind and emotions to stimulate
new work, 2) because all perfection, completeness and correctness is deadly
boring, 3) because 1) and 2) are such handy excuses.
I would like to express my thanks to the many who supported me in writing
this, yet I confine myself to mentioning only a few: Swiss Re, who generously
granted me money and time, Hans Bühlmann for being my exemplary teacher
for many years, Adette Harnisch and Nicola Chappuis who typed all the
twenty-seven versions fast and skillfully and last but not least my students in
Berne, Belgrade and Strasbourg.
Zürich, August 1988· Erwin Straub
Table of Contents
Chapter 1. Problems
Chapter 2. Tools ......................................... . 7
2.1. The Model ........................................ . 7
2.2. Distributions for K and X ............................. 16
2.3. Moments ......................................... . 21
2.4. The Total Claims Cost Z ............................ . 29
2.5. Cramer's Inequality 36
2.6. Dependent Variables 42
Chapter 3. Premiums ....................................... 52
3.1. Pragmatic Principles ................... . . . . . . . . . . . . .. 52
3.2. Theoretical Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54
3.3. Experience Rating ................................... 59
Chapter 4. Reinsurance ..................................... 68
Chapter 5. Retentions ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76
Chapter 6. Statistics ........................................ 89
Chapter 7. Reserves 102
Chapter 8. Solutions 116
8.1. Negative Binomial 116
8.2. Exact Credibility 119
8.3. Closing the Circle 126
References 131
Subject Index 133
Chapter 1. Problems
Actuaries are the people who deal with all kinds of mathematical and statistical
problems in insurance - that's why we speak of actuarial problems. With the
more recent application of actuarial methods also to Property and Liability
insurance, it has become customary to distinguish between Life and Non-Life
actuarial sciences. Although such a distinction does not always make sense-in
Health insurance, for instance, the two domains overlap widely - the present
book deals exclusively with so-called Non-Life problems and their possible
solutions.
Insurance companies accept risks, i.e. potential claims, from their clients, the
insureds, against a certain price called a premium. If a risk or a portfolio of risks
is too large for a company, it will pass on parts of it to one or several other
companies, its reinsurers, whereby that part which finally remains with the first
company is called its retention. That is very briefty how the insurance industry
operates. If a risk is very large, some of the reinsurers in their turn may have to
pass on parts of their acceptances to yet other companies, so that the original
risk or portfolio is in the end covered by an entire network of insurance and
reinsurance arrangements between a number of companies, each of them
carrying the retention deemed appropriate. On page 2, there is a picture of such a
network (Fig. 1).
In order to get quickly an overview of the landscape of Non-Life actuarial
problems arising in practice, let a layman very naively ask a question like,
"What is an appropriate retention for a given company and how do you
determine it?" To this the expert's answer will be, "Hmm-well, it depends
... you see ..." .
It certainly depends on many things. In the first place, a wealthy company
can clearly afford to retain more for its own account than a poor one, for it has
more capital to stake. Secondly, it depends whether management is at all willing
to take risks; a conservative manager will display little risk willingness, contrary
to a coutageous or even foolhardy entrepreneur. Furthermore, the underwriter
will have to make up his mind on the premium: Is the risk or are the policy
conditions likely to produce a profit if there are no unusual claims? (Whatever
that means!) And, last but not least, a very close look must of course be taken at
the risk itself, for there are better balanced risks and/or portfolios on the one
hand and highly unbalanced ones on the other.
After some pondering on these indeed very sketchy explanations regarding
the problem of how to fix retentions, our naive layman-even if only vaguely-
2 I. Problems
Their retro
cessionaires
35.7%
• A retrocessionaire "or secc)Od order"
Figure 1
understands by instinct that basically things must be rat her simple and
obvious, in that the retention of each company in the above network can or
must be all the higher if the three items "capital", "risk willingness" and "profit
margin" are high. It should be all the lower, however, the more unbalanced the
risk iso Thus he can conclude that
. capital x risk willingness x profit margin
retentIon = --==--------:---:--=---:-----'=-------=-
un balancedness
must be the magie formula, and he can furthermore immediately see that,
depending on which four of the five items appearing in this formula are given,
the problem of calculating the fifth one represents a fundamental actuarial task.
So, for instance, as the calculation of premiums is connected with the term
"profit margin" and the above can be rewritten as
. _ retention x unbalancedness
fi
pro t margm - . I . k ·11·
caplta x ns WI mgness
we have here an indication of how to assess a premium.
I. Problems 3
"So, if it is all that easy, why do we need actuaries at all?" we may ask
ourselves desperately. No need to worry, however, because the above rule of
thumb has just been written down by mere instinct and without any intelligent
proof whatsoever. It is precisely one of the objectives ofthis book to show that a
proper risk-theoretical approach to insurance phenomena leads straight to this
formula if we allow for maximum possible simplifications in the mathematical
model. Also, the. instinctive formula is purely qualitative for it does not tell us,
for instance, how to express risk willingness numerically. Finally, why should
the mutual relationship between the five items in the formula be a multiplicative
one and not of some other type?
Be that as it may, the formula is handy, even correct, under certain assump
tions, as we shall see later on. Within this first chapter, however, we shall only use
it to broadly describe die five main types of problems we meet in practice,
namely
(i) Rating, i.e. premium calculation characterised by bringing our instinctive
formula into the shape of
. _ retention x unbalancedness
fi
pro t margm - . I . k ·11·
caplta x ns Wl mgness
telling us among other things that in some way or other the loading contained in
a premium ought to be dependent on the degree of unbalancedness of the risk or
portfolio under consideration.
(ii) Assessing reserves in the sense of contingency reserves, risk capital or
catastrophe funds through
. I _ retention x unbalancedness
caplta - fi . . k ·11·
pro t margm x ns Wl mgness
(iii) Underwriting limits. How to fix retentions? Weil, somehow in accordance
with
. capital x profit margin x risk willingness
retentton =
--='----=---_:_~-=-_:_-----==-
unbalancedness
if we follow basic common sense and provided there are no other prevailing
criteria induced, for example, by external solvency prescriptions.
(iv) Risk behaviour in general, i.e. a company's entrepreneurial attitude
towards risk indicated by
. k ·11· retention x unbalancedness
ns Wl mgness = ---:-.- ---::----::----:---
capltal x profit margin
This may serve, for example, to measure how conservative or not a company is
or ought to be or to compare the risk behaviour of two or several companies.
4 I. Problems
(v) The maximum tolerable risk load a company may be able to bear on its
gross account, given its rate level, retention and financial strength as weIl as its
general behaviour towards risk, abbreviated by the formula
capital x profitability x risk willingness
un b a Ia nee d ness = .
retentIOn
This theoretically tolerated unbalancedness may be set against the actual
fluctuations of business results observed in the past.
The above may be viewed as a sketch of five global problems dealing with items
that concern the company or account as a wh oie, namely the general rate level
(i), the overall reserves (ii), the absolute retention (iii), the general risk willingness
or aversion (iv) and the potential for more or less extreme fluctuations of the
company's total result (v). Parallel to such "holistic" problems, there are
corresponding questions of how to differentiate for individual risk categories or
sub-portfolios, how to distribute or how to graduate certain overall quantities at
lower levels of the company structure. Here is a short indication of wh at may be
called relative problems as opposed to the above-mentioned holistic or absolute
ones:
(i) Experience rating. There is this basic dilemma in every practical rating
situation: On the one hand, according to the principle of fair premiums, each
risk ought to finance its own claims in the long run (say asymptotically), while
on the other hand it is the very idea of insurance that a given portfolio forms a
collective of risks "with equal rights", each of them paying the same pure risk
premium equal to the portfolio's average yearly loss costs. Now these two
principles are in perfect harmony in the case of an ideally homogeneous risk
collective, but what if, as always in reallife, there are different risk categories or
even different individual risks (as in a Motor Liability portfolio, for example)?
How reliable is the individual's claims experience compared to the portfolio
average? How should premium rates be varied by risk category? How should the
degree of heterogeneity of a portfolio be measured? - Such questions are an
swered by the so-called credibility theory, one ofthe oldest branches ofNon-Life
mathematics.
(ii) Allocating reserves. Onee a given part ofthe company's total equity is ear
marked as risk capital, Le. as an overall shock-absorber against fluctuations of
the yearly overall result, one may need to break it down into individuallines of
business. Furthermore, it is not apriori evident that the sum of per line risk
capitals must equal the overall risk capital. They may either overlap or be
supplemented by reserves for common catastrophes.
(iii) Relative retentions. Quite obviously, it would not be optimal to have the
same amount for the company's own account on each risk, but how should the
Concluding Remarks and Exercises 5
retention by risk class be graduated reasonably? What appropriate statistical
tools are there to answer this question?
(iv) Risk behaviour. There may be different degrees of risk acceptance or
aversion as we move from one sub-portfolio to another.
(v) Unbalancedness. The overall unbalancedness is generated by the tluctua
tions of claims costs of individual lines andjor risk classes and here again "the
sum must not necessarily be equal to the total". There are furthermore different
types of tluctuations or rather different sources: Random tluctuations due to the
occurrencejnon-occurrence of (exceptionally large) claims, cydical market tluc
tuations and last but not least the financial risk of a company due to changes of
the value of the total of its investments.
With the above .attempt to sketch five overall and five relative Non-Life
actuarial tasks, the whole field is of course not completely covered - it should
nevertheless give a good first survey.
Concluding Remarks and Exercises
Insurance is a business of risks (not the only one, however) but then what is a
risk? - It is the danger of losing something, a loss potential, the loss may be
human lives, health or wealth. Observe that the loss depends on the viewpoint,
on the level so to speak: If a house bums down, for instance, nothing is lost from
nature's point of view. There is only a transformation - a transformation from
wood into ash and co al and smoke, from walls into debris and dust, from living
into dead matter, from potential energy into heat and light and so on, by a
simple chemical reaction called oxydation or fire, to put it more plainly. Nothing
is lost whatsoever from an overall point of view. If, however, the house that
burned down was my house, then it may mean a tremendous loss to me - unless
it was weil insured andjor I am a wise philosopher.
Exercise J
Make up a list of at least twenty risks an average family is confronted with.
Every person, every company and institution, every country, mankind as a
wh oie, each life, as a matter of fact everything is exposed to a number of specific
risks, has to live with these risks and has to apply his specific kind of risk
management usually by fight or tlight, i.e. so me type of protection against or
avoidance of risk.
