Loss Reserving: An Actuarial Perspective

395 Pages·2000·6.56 MB·English

by  Greg Taylor (auth.)

LOSS RESERVING
An Actuarial Perspective
Huebner International Series on
Risk, Insurance, and Economic
Security
J. David Cummins, Editor
The Wharton School
University of Pennsylvania
Philadelphia, Pennsylvania, USA
Series Advisors:
Dr. Phelim P. Boyle
University of Waterloo, Canada
Dr. Jean Lemaire
University of Pennsylvania, USA
Professor Akihiko Tsuboi
Kagawa University, Japan
Dr. Richard Zeckhauser
Harvard University, USA
Other books in the series:
Cummins, J. David and Derrig, Richard A.: Classical
Insurance Solvency Theory
Borba, Philip S. and Appel, David: Benefits, Costs, and
Cycles in Workers' Compensation
Cummins, J. David and Derrig, Richard A.: Financial Models
of Insurance Solvency
Williams, C. Arthur: An International Comparison of
Workers' Compensation
Cummins, J. David and Derrig, Richard A.: Managing the
Insolvency Risk of Insurance Companies
Dionne, Georges: Contributions to Insurance Economics
Dionne, Georges and Harrington, Scott E.: Foundations of
Insurance Economics
Klugman, Stuart A.: Bayesian Statistics in Actuarial Science
Durbin, David and Borba, Philip: Workers' Compensation Insurance:
Claim Costs, Prices and Regulation
Cummins, J. David: Financial Management of Life Insurance
Companies
Gustavson, Sandra G. and Harrington, Scott E.: Insurance,
Risk Management, and Public Policy
Lemaire, Jean: Bonus-Malus Systems in Automobile Insurance
Dionne, Georges and Laberge-Nadeau: Automobile Insurance:
Road Safety, New Drivers, Risks, Insurance Fraud and Regulation
LOSS RESERVING
An Actuarial Perspective
Greg Taylor
Taylor Fry, Australia
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library of Congress Cataloging-in-Publication Data
Taylor, G.C. (Gregory Clive), 1945-
Loss reserving: an actuarial perspective / by Greg Taylor,
p. cm.—(Huebner international series on risk, insurance, and economic
security)
Includes bibliographical references and index.
ISBN 978-1-4613-7070-3 ISBN 978-1-4615-4583-5 (eBook)
DOI 10.1007/978-1-4615-4583-5
1.Insurance—Reserves. I. Title.II. Series.
HG8106 .T39 2000
368'.01—dc21 99-057631
Copyright © 2000 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 2000
Softcover reprint of the hardcover 1st edition 2000
All rights reserved. No part of this publication may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, mechanical, photo
copying, recording, or otherwise, without the prior written permission of the
publisher, Springer Science+Business Media, LLC
Printed on acid-free paper.
to Eden and Luke
- the other silent contributors
CONTENTS
PREFACE xi
PART I DETERMINISTIC MODELS
BASIC CONCEPTS 3
1.1 The Claims Process 3
1.2 Estimates of Outstanding Loss Liability 7
1.3 Loss Reserving 13
1.4 Data 14
REFERENCES 16
2 CLAIM COUNTS 17
2.1 Exposure 17
2.2 IBNR Claims 18
2.3 Claim Frequency 38
2.4 Further Models 41
REFERENCES 41
3 CLAIM AMOUNTS -SIMPLE MODELS 43
3.1 Case Estimation 43
3.2 Chain Ladder 48
3.3 Separation Method 73
REFERENCES 86
4 CLAIM AMOUNTS -OTHER DETERMINISTIC MODELS 87
4.1 Introduction 87
4.2 Payment Based Models 88
4.3 Claim Closure Based Models 97
4.4 Case Estimate Based Models 128
REFERENCES 150
5 COMBINATION OF DETERMINISTIC ESTIMATES OF LIABILITY 151
5.1 Background 151
5.2 Comparative Analytical Properties of the Models 152
5.3 Average Claim Sizes 153
5.4 Relation of Estimated Outstanding Liability to Case Estimates 155
5.5 Combining the Results of the Different Models 157
5.6 Allowance for Prior Expectations 158
5.7 Commentary 164
REFERENCES 165
PART II STOCHASTIC MODELS 167
6 STOCHASTIC TECHNIQUES 169
6.1 Introduction 169
6.2 Generalised Linear Models (GLMs) 169
6.3 Credibility Theory 173
6.4 Kalman Filter 184
6.5 Bootstrap 190
6.6 Prediction Error 192
REFERENCES 194
7 STOCHASTIC CHAIN LADDER 195
7.1 Introduction 195
7.2 Log-Linear Models 195
7.3 Parametric Chain Ladder 196
7.4 Non-Parametric Chain Ladder 203
7.5 Gamma Cell Distributions 223
7.6 Another Related Model 225
REFERENCES 228
8 STOCHASTIC MODELS WITH A GLM BASIS 229
8.1 Log-Linear Models 229
8.2 Linear Models 259
REFERENCES 262
9 CREDIBILITY MODELS 263
9.1 General Model 263
9.2 Single Parameter Periods of Origin 264
9.3 Multiple Parameter Periods of Origin 275
9.4 Second Moments 296
REFERENCES 302
10 KALMAN FILTER 303
10.1 Motivation 303
10.2 Payments Per Claim Incurred Example 304
10.3 More General Applications of the Filter 312
10.4 Claim Closure Example 319
REFERENCES 328
Vlll
II BOOTSTRAP 329
11.1 General 329
11.2 Framework of the Example 329
11.3 Pseudo-Data 330
11.4 Pseudo-Estimates 332
11.5 Co lIation of Bootstrap Results 341
12 FINAL ESTIMATES OF LIABILITY 345
12.1 General 345
12.2 Model Blending 345
12.3 Reinsurance Recoveries 365
REFERENCES 382
APPENDIX A -Notation 383
APPENDIX B -Data for Numerical Example 385
INDEX 387
ix
PREFACE
This book grew out of a graduate lecture course in Loss Reserving given at the
University of Lisbon in March 1998.
Having been asked to provide such a course, my first thought concerned the extent
to which my 1986 treatise on the same subject might serve as a basis. The decision was
very quick. While I had been aware that the years were steadily overtaking the book,
I had not realised how dramatic its obsolescence was. A brief literature review revealed
that the amount of published research on the subject had roughly doubled in the decade
or so since the earlier volume, which was, as a result, totally inadequate.
In addition, there were a couple of aspects of that volume which were somewhat
unsatisfactory in retrospect. First, it did not illustrate its methodology numerically.
Second, it presented the chain ladder as just one more approach to loss reserving,
without recognition of its pre-eminent role in practice.
In the present volume, I have attempted to rectify the first of these defects by the
inclusion of copious numerical examples. A single set of data is included as an
appendix, and used to illustrate numerically the great majority of techniques discussed
in the book.
Opinions continue to differ between individuals, sometimes heatedly, on the proper
role of the chain ladder, but it survives as the most widely applied loss reserving
procedure. In view of this, it is fitting that the procedure receive reasonably
comprehensive theoretical treatment. I have attempted this in Chapters 2 and 3,
preferring to present the facts as neutrally as possible rather than entering the debate on
the merits or otherwise of the procedure.
I have also dealt with stochastic versions of the chain ladder in Chapters 7,8,9 and
11.
In view of these changes, this book cannot be regarded as a second edition of the
earlier one, but rather as a new book altogether.
Readers will [md omitted some topics which might have been included. These fall
into two categories. The first comprises those which I have judged not yet sufficiently
mainstream for inclusion. An example would be Markov Chain Monte Carlo (MCMC)
techniques, which have begun to appear in the actuarial literature in the last few years.
The second omission, and a very substantial one, is loss reserving for inwards
reinsurance portfolios. This is not the place to go into the reasons why these require
different treatment; suffice to assert that they do. While it is true that some of the
techniques covered in this book are also applicable to such portfolios, no coherent
treatment of reserving for them is attempted. This is because such a treatment would
have required a considerable expansion, virtually warranting a separate volume.
Undoubtedly, the book will contain some personal biases. For example, Chapters
1 to 5 can be taken as reasonably representative of the way in which loss reserving is
xi
carried out in my own country. Elsewhere, many of the same concepts and procedures
occur, but perhaps with shifts of emphasis or changes of terminology here and there.
Because this part of the book reflects practice as I know it, there are many
individuals, beyond those in the reference list, who have influenced its evolution. At
the risk of omission, I mention a few of those who have most influenced me or from
whom I have derived particular benefit.
In my earliest encounters with property and casualty insurance, I had the great good
fortune to fall under the brief tutelage of the late Professor Bobbie Beard. Bobbie, who
could fairly be regarded as the forefather of actuarial involvement in this sphere in the
UK, was then in the process of establishing loss reserving supervisory procedures in the
Department of Trade. His encyclopaedic knowledge, and his unfailing enthusiasm,
provided my initial orientation in this field.
At the same time in Australia, Richard Cumpston and Roger Sawkins occupied
dominant roles in the very immature actuarial endeavour in property and casualty
insurance. As a newcomer to the field, I had many lively discussions with them,
sometimes vigorous but always informative. Their teachings included many elements
of practicality.
A good part of the content of Chapters 4 and 5 evolved in the 1980s, and certainly
represents to a large extent my own "standard procedure" as a consultant during the
period. Over this decade (and beyond) I was the professional partner of Chris Latham.
Any parts of those chapters in whose development I might have played a part were
developed with the benefit of his commentary and steadying influence.
In the later 1980s and 1990s stochastic methodology has gained prominence, even
ifas yet insufficiently. Here, illumination has often been proved by Ben Zehnwirth and
Frank Ashe. Latterly, Alan Greenfield has spent many hours working with me on
practical implementations of the Kalman filter. My resulting discussions with him have
been of great benefit in clarifying ideas.
Amanda Clarke took on the difficult task of typing this book, and also prepared the
camera ready form. It is typical of her professionalism that, despite the difficulties and
frustrations that undoubtedly must have occurred, the whole process appeared to take
place effortlessly. Meegen Gamson provided occasional typing assistance as well as
an invaluable contribution to fmal proofreading.
I am grateful to Tillinghast-Towers Perrin for providing the facilities to produce
the manuscript.
My wife Rhonda has displayed her usual patience as yet another project has
developed from seemingly innocuous beginnings into a major consumer of time.
Greg Taylor
Sydney, March 1999
xii