Statistical and Probabilistic Methods in Actuarial Science

362 Pages·2007·2.304 MB·English

by  Boland, Philip J

I n t e rd i s c i p l i n a r y S t a t i s t i c s
STATISTICAL and PROBABILISTIC
METHODS
in
ACTUARIAL SCIENCE
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CHAPMAN & HALL/CRC
Interdisciplinary Statistics Series
Series editors: N. Keiding, B. Morgan, T. Speed, P. van der Heijden
AN INVARIANT APPROACH TO S. Lele and J. Richtsmeier
STATISTICAL ANALYSIS OF SHAPES
ASTROSTATISTICS G. Babu and E. Feigelson
BIOEQUIVALENCE AND S. Patterson and
STATISTICS IN CLINICAL B. Jones
PHARMACOLOGY
CLINICAL TRIALS IN ONCOLOGY J. Crowley, S. Green,
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DESIGN AND ANALYSIS OF D.L. Fairclough
QUALITY OF LIFE STUDIES
IN CLINICAL TRIALS
DYNAMICAL SEARCH L. Pronzato, H. Wynn, and
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GENERALIZED LATENT VARIABLE A. Skrondal and
MODELING: MULTILEVEL, S. Rabe-Hesketh
LONGITUDINAL, AND STRUCTURAL
EQUATION MODELS
GRAPHICAL ANALYSIS OF K. Basford and J. Tukey
MULTI-RESPONSE DATA
INTRODUCTION TO M. Waterman
COMPUTATIONAL BIOLOGY:
MAPS, SEQUENCES, AND GENOMES
MARKOV CHAIN MONTE CARLO W. Gilks, S. Richardson,
IN PRACTICE and D. Spiegelhalter
MEASUREMENT ERROR AND P. Gustafson
MISCLASSIFICATION IN STATISTICS
AND EPIDEMIOLOGY: IMPACTS AND
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STATISTICAL ANALYSIS OF GENE T. Speed
EXPRESSION MICROARRAY DATA
STATISTICAL CONCEPTS J. Aitchison, J.W. Kay,
AND APPLICATIONS IN and I.J. Lauder
CLINICAL MEDICINE
STATISTICAL AND PROBABILISTIC Philip J. Boland
METHODS IN ACTUARIAL SCIENCE
STATISTICS FOR ENVIRONMENTAL A. Bailer and W. Piegorsch
BIOLOGY AND TOXICOLOGY
STATISTICS FOR FISSION R.F. Galbraith
TRACK ANALYSIS
STATISTICS IN MUSICOLOGY J. Beran
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I n t e rd i s c i p l i n a r y S t a t i s t i c s
STATISTICAL and PROBABILISTIC
METHODS
in
ACTUARIAL SCIENCE
Philip J. Boland
University College Dublin
Ireland
Boca Raton London New York
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Dedication
To my wife Elizabeth, and my children Daniel and Katherine.
v

Preface
This book covers many of the diverse methods in applied probability and
statistics for students aspiring to careers in insurance, actuarial science and
finance. Itshouldalsoserveasavaluabletextandreferencefortheinsurance
analyst who commonly uses probabilistic and statistical techniques in prac-
tice. The reader will build on an existing basic knowledge of probability and
statistics and establish a solid and thorough understanding of these methods,
but it should be pointed out that the emphasis here is on the wide variety of
practical situations in insurance and actuarial science where these techniques
may be used. In particular, applications to many areas of general insurance,
including models for losses and collective risk, reserving and experience rat-
ing, credibility estimation, and measures of security for risk are emphasized.
The text also provides relevant and basic introductions to generalized linear
models, decision-making and game theory.
There are eight chapters on a variety of topics in the book. Although
there are obvious links between many of the chapters, some of them may be
studied quite independently of the others. Chapter 1 stands on its own, but
at the same time provides a good introduction to claims reserving via the
deterministic chain ladder technique and related methods. Chapters 2,3 and
4 are closely linked, studying loss distributions, risk models in a fixed period
of time, and then a more stochastic approach studying surplus processes and
the concept of ruin. Chapter 5 provides a comprehensive introduction to the
concept of credibility, where collateral and sample information are brought
togethertoprovidereasonablemethodsofestimation. TheBayesianapproach
to statistics plays a key role in the establishment of these methods. The final
three chapters are quite independent of the previous chapters, but provide
solid introductions to methods that any insurance analyst or actuary should
know. Experience rating via no claim discount schemes for motor insurance
in Chapter 6 provides an interesting application of Markov chain methods.
Chapter 7 introduces the powerful techniques of generalized linear models,
while Chapter 8 includes a basic introduction to decision and game theory.
There are many worked examples and problems in each of the chapters,
with a particular emphasis being placed on those of a more numerical and
practical nature. Solutions to selected problems are given in an appendix.
Therearealsoappendicesonprobabilitydistributions,Bayesianstatisticsand
basic tools in probability and statistics. Readers of the text are encouraged
(in checking examples and doing problems) to make use of the very versatile
and free statistical software package R.
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viii PREFACE
The material for this book has emerged from lecture notes prepared for
variouscoursesinactuarialstatisticsgivenatUniversityCollegeDublin(The
National University of Ireland – Dublin) over the past 15 years, both at the
upper undergraduate and first year postgraduate level. I am grateful to all
mycolleaguesinStatisticsandActuarialScienceatUCDfortheirassistance,
but particularly to Marie Doyle, Gareth Colgan, John Connolly and David
Williams. The Department of Statistics at Trinity College Dublin kindly
provided me with accommodation during a sabbatical year used to prepare
this material. I also wish to acknowledge encouragement from the Society
of Actuaries in Ireland, which has been supportive of both this venture and
our program in Actuarial Science at UCD since its inception in 1991. Patrick
Grealy inparticular providedvery useful advice and examples on the topic of
run-off triangles and reserving. John Caslin, Paul Duffy and Shane Whelan
were helpful with references and data.
Ihavebeenfortunatetohavehadmanyexcellentstudentsinbothstatistics
and actuarial science over the years, and I thank them for the assistance and
inspiration they have given me both in general and in preparing this text.
Particular thanks go to John Ferguson, Donal McMahon, Santos Faundez
Sekirkin, Adrian O’Hagan and Barry Maher. Many others were helpful in
reading drafts and revisions, including Una Scallon, Kevin McDaid and Rob
Stapleton. Finally, I wish to thank my family and many friends who along
the path to completing this book have been a constant source of support and
encouragement.
Introduction
In spite of the stochastic nature of most of this book, the first chapter is
rather deterministic in nature, and deals with Claims Reserving and Pricing
with Run-off Triangles. In running-off a triangle of claims experience, one
studieshowclaimsarisingfromdifferentyearshavedeveloped,andthenmakes
use of ratios (development factors and/or grossing-up factors) to predict how
future claims will evolve. Methods for dealing with past and future inflation
in estimating reserves for future claims are considered. The average cost per
claim method is a popular tool which takes account of the numbers of claims
as well as the amounts. The Bornhuetter–Ferguson method uses additional
informationsuchasexpectedlossratios(lossesrelativetopremiums)together
withthechainladdertechniquetoestimatenecessaryreserves. Delaytriangles
of claims experience can also be useful in pricing new business.
Modeling the size of a claim or loss is of crucial importance for an insurer.
In the chapter on Loss Distributions, we study many of the classic probabil-
ity distributions used to model losses in insurance and finance, such as the
exponential, gamma, Weibull, lognormal and Pareto. Particular attention is
paid to studying the (right) tail of the distribution, since it is important to
not underestimate the size (and frequency) of large losses. Given a data set
of claims, there is often a natural desire to fit a probability distribution with
reasonablytractablemathematicalpropertiestosuchadataset. Exploratory
data analysis can be very useful in searching for a good fit, including basic
descriptive statistics (such as the mean, median, mode, standard deviation,
skewness,kurtosisandvariousquantiles)andplots. Themethodofmaximum
likelihood is often used to estimate parameters of possible distributions, and
various tests may be used to assess the fit of a proposed model (for exam-
ple, the Kolmogorov–Smirnoff, and χ2 goodness-of-fit). Often one may find
that a mixture of various distributions may be appropriate to model losses
due to the varying characteristics of both the policies and policyholders. We
also consider the impact of inflation, deductibles, excesses and reinsurance
arrangements on the amount of a loss a company is liable for.
Followingonfromastudyofprobabilitydistributionsforlossesandclaims,
thechapteronRiskTheoryinvestigatesvariousmodelsfortheriskconsisting
of the total or aggregate amount of claims S payable by a company over a
relatively short and fixed period of time. Emphasis is placed on two types of
models for the aggregate claims S. In the collective risk model for S, claims
are aggregated as they are reported during the time period under consider-
ation, while in the individual risk model there is a term for each individual
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