Life Insurance Mathematics

224 Pages·1997·5.11 MB·English

by  Professor Hans U. Gerber (auth.)

LifeInsurance Mathematics
Springer-Verlag Berlin Heidelberg GmbH
Hans U.Gerber
Life Insurance
Mathematics
with exercises contributed by Samuel H.Cox
Third Edition 1997
Springer
SwissAssociation ofActuaries Zurich
Professor Hans U Gerber ProfessorSamuelH.Cox,ES.A.
Ecoledes H.E.C. Georgia State University
Universire de Lausanne Dept.of RiskManagement
CH-IOlS Lausanne and Insurance
Switzerland Atlanta,GA30303-3083
e-mail: [email protected] USA
e-mail: [email protected]
Translatorof the first edition:
Professor Walther Neuhaus
University of Oslo
Norway
CIPdataappliedfor
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Gerber, Hans U.:
Lire insurance mathematics/ Hans U. Gerber.With exercises
conlributed bySamuel H. Cox.Swiss Association of Actuaries,
Zürich. [Trans!. of thefirsl cd.:Walther Neuhaus].- 3.ed.
Berlin ; Heidelberg ; New York ;Barcelona ; Budapest ;Hong
Kong ; London ; Milan ; Paris ;Santa Clara ;Singapore ;
Tokyo :Springer, 1997
Dt, Ausg. u.d.T.: Lebensversicherungsmathematik
MathematicsSubjectClassification(1991):62P05
ISBN978-3-642-08285-6 ISBN978-3-662-03460-6(eBook)
DOI10.1007/978-3-662-03460-6
This work is subject to copyright. All right are reserved, whether the whole or part of
the material is concerned,specifically the rightsoftranslation, repnnring, reuse ofillustrations,
recitation, broadcasting,reproduction on microfilm or in any other way,and storagein data banks.
Duplication ofthispublication orpartsthercofispermittedonlyundertheprovisionsoftheGerman
Copyright LawofSeptember9,1965,in itscurrentversion,andpermissionfor usemusta1waysbe
obtainedfromSpringer-Verlag.ViolalationsareliableforprosecutionundertheGermanCopyrightLaw.
©Springer-VerlagBerlinHeidelberg1990,1995,1997
OriginallypublishedbySpringer-VerlagBerlinHeidelbergNewYorkin1997,
Softcoverreprintofthehardcover3rdedition1997
The useofgeneral descriptivenames,registerednames,trademarksetc.inthispublicariondoes not
imply,even in the absence ofaspecific statement,thatsuch namesareexemptfiom the relevant
protecrivelawsandregulationsandthereforefreefürgeneraluse.
Typesetting:Camera-readycopyfromtheauthors
SPIN:11335009 41/3111- 109876 -Printedonacid-free paper
To Cecil Nesbitt
Foreword
Halley's Comet has been prominently displayed in many newspapers during
the last few months. For the first time in 76 years it appeared this winter,
clearly visible against the nocturnal sky. This is an appropriate occasion to
pointoutthe fact thatSirEdmundHalleyalsoconstructedtheworld's first life
tablein 1693,thus creatingthe scientificfoundation oflifeinsurance. Halley's
lifetable and its successorswereviewedasdeterministiclaws, i.e. thenumber
of deaths in any given group and year was considered to be a well defined
number that could be calculatedby means ofa lifetable. However,in reality
this number is random. Thus any mathematical treatment of life insurance
willhave to rely more and more on probability theory.
By sponsoring this monograph the SwissAssociation of Actuaries wishes
to supportthe "modem" probabilisticviewoflifecontingencies. Weare fortu
nate that Professor Gerber, an internationally renowned expert, has assumed
the task of writing the monograph. We thank the Springer-Verlag and hope
that this monograph willbe thefirst in a successfulseries ofactuarial texts.
ZUrich, March 1986 Hans Biihlmann
President
Swiss Associationof Actuaries
Preface
Two major developments have influenced the environment ofactuarial math
ematics. One is the arrival of powerful and affordable computers; the once
important problem ofnumerical calculationhasbecomealmost trivialinmany
instances. The other is the fact that today's generation isquite familiar with
probability theory in an intuitive sense;the basic concepts ofprobability the
ory are taught at many high schools. These two factors should be taken into
account in the teaching and learning ofactuarial mathematics. A first conse
quence is,for example, that a recursive algorithm (fora solution) isas useful
as a solution expressed in terms ofcommutation functions. In many casesthe
calculations are easy; thus the question "why" a calculation is done is much
more important than the question "how" it isdone. The second consequence
is that the somewhat embarrassing deterministic model can be abandoned;
nowadays nothing speaks against the use ofthe stochastic model, which bet
ter reflects the mechanisms of insurance. Thus the discussion does not have
to be limited to expected values; it can be extended to the deviations from
the expected values, thereby quantifying the risk in the proper sense.
The book has been written in this spirit. It is addressed to the young
reader (where "young" should be understood in the senseofoperationaltime)
who likes applied mathematics and is looking for an introduction into the
basic concepts oflifeinsurance mathematics.
Inthe first chapteranoverviewofthe theory ofcompoundinterestisgiven.
In Chapters2-6 variousformsofinsurance and theirmechanismsare discussed
in the basic model. Herethe keyelementisthe future lifetimeofa lifeaged x,
which is denoted byT and whichis (ofcourse!) a random variable. In Chap
ter 7the model isextended to multiple decrements,where different causes for
departure (forexample death and disability) are introduced. In Chapter 8in
surancepoliciesare consideredwherethe benefits are contingentonmorethan
onelife(forexample widows'and orphans' pensions). Inallthese chaptersthe
discussion focuseson a singlepolicy,whichispossibleinthe stochastic model,
as opposed to the deterministic model, where each policy is considered as a
member of a large group of identical policies. In Chapter 9 the risk arising
fromagroup ofpolicies(a portfolio) isexamined. Itisshownhowthe distribu
tion ofthe aggregate claims can be calculated recursively. Information about
x Preface
this distribution is indispensable when reinsurance is purchased. Thetopic of
Chapter 10is ofgreat practical importance;forsimplicity ofpresentation the
expenseloadingisconsideredonly in thischapter. Chapter 11examines some
statistical problems, for instance, how to estimate the distribution ofT from
observations. Thebook has been writtenwithoutmuch compromise; however,
the appendix should be a sign of the conciliatory nature of the author. For
thevery samereason the basic probabilityspace (fl,:F,P) shall be mentioned
at least once: now!
The publicationofthisbook wasmadepossible bythesupportofthe Fund
for the Encouragement of Actuarial Mathematics of the Swiss Association of
Actuaries; my sincere thanks go to the members of its committee, not in
the least for the freedom granted to me. I would like to thank in particular
Professor Biihlmann and Professor Leepin for their valuable comments and
suggestions. Of course I am responsible for any remainingflaws.
For some years now a team of authors has been working on a compre
hensive text, which was commissioned by the Society of Actuaries and will
be published in 1987 in its definitive form. The cooperation with the coau
thors Professors Bowers,Hickman, Jones and Nesbitt has been an enormously
valuable experience for me.
FinallyI wouldliketo thankmyassistant, MarkusLienhard, forthecareful
perusalofthegalleyproofsand Springer-Verlagfortheirexcellent cooperation.
Lausanne, March 1986 Hans U. Gerber
Acknowledgement
I am indebted to my colleague, Dr. Walther Neuhaus (University of Oslo),
who translated the text into English and carried out the project in a very
competent and efficient way. We are also very grateful to Professor Hendrik
Boom (University of Manitoba) for his expert advice.
Lausanne and Winnipeg, April 1990 Hans U. Gerber
Acknowledgement
The second edition contains a rich collection of exercises, which have been
preparedby Professor Samuel H. Cox of Georgia State University of Atlanta,
who is an experienced teacher of the subject. I would like to express my
sincere thanks to my American colleague: due to his contribution, the book
will not only find readersbut it willfind users!
Lausanne, August 1995 Hans U. Gerber
XI
Acknowledgement
The second edition has been sold out rapidly. This led to the present third
edition,in which several misprintshave been corrected. Iam thankful to Sam
Cox, ChengShixue, WolfgangQuapp, Andre Dubeyand JeanCochetfortheir
valuable advice.
At thisoccasionIwouldliketo thankSpringerandtheSwissAssociationof
Actuaries for authorising the Chinese, Slovenian and Russian editions of Life
Insurance Mathematics. I am indebted to Cheng Shixue, Van Ying, Darko
Medved andValery Mishkin. From myownexperienceIknowthattranslating
a scientific text is a challengingtask.
Lausanne, January 1997 Hans U. Gerber