Actuarial Mathematics

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Text of Actuarial Mathematics

Educacin

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The work of science is to substitute facts for appearances and demonstrationsfor impressions.---RUSKIN.

ACTUARIAL MATHEMATICSBY NEWTON L. BOWERS, JR. HANS U. GERBER JAMES C. HICKMAN DONALD A. JONES CECIL J. NESBITT

Pub1ished by THE SOCIETY OF ACTUARIES 1997

Library of Congress Cataloging-in-PublicationData

Actuarial mathematics / by Newton L. Bowers . . . [et al.]. p. cm. Includes bibliographical references and index. ISBN 0-938959-46-8 (alk. paper) 1. Insurance-Mathematics. I. Bowers, Newton L., 1933HG8781.A26 1997 368.01-d~21 97-8528 CIP

Copyright 0 1997, The Society of Actuaries, Schaumburg, Illinois. All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher.Second Edition Printed in the United States of America 03 02 01 00 5 4 3 2 Edited by David Anderson Society of Actuaries Liaisons: Judy F. Anderson, F.S.A., Staff Fellow Cathy Cimo, Practice Area Administrator John F. Carey, Scientific Publications Manager Cover and interior design by Word Management, Highland Park, Ill., and Stet Graphics Incorporated, Schaumburg, Ill. Typeset by Sally A. Groft at Pro-Image Corporation, York, Pa. Printed by Edwards Brothers, Inc., Ann Arbor, Mich.

Preface for the Society of Actuaries

The first edition of Actuarial Mathematics, published in 1986, has served the profession well by presenting the modern mathematical foundations underlying actuarial science and developing powerful tools for the actuary of the future. The text not only introduced a stochastic approach to complement the deterministic approach of earlier texts but also integrated contingency theory with risk theory and population theory. As a result, actuaries now have the methods and skills to quantify the variance of random events as well as the expected value. Since the publication of the first edition, the technology available to actuaries has changed rapidly and the practice has advanced. To reflect these changes, three of the original five authors agreed to update and revise the text. The Education and Examination Committee of the Society has directed the project. Richard s. Mattison helped to define the specifications and establish the needed funding. Robert C. Campbell, Warren Luckner, Robert A. Conover, and Sandra L. Rosen have all made major contributions to the project. Richard Lambert organized the peer review process for the text. He was ably assisted by Bonnie Averback, Jeffrey Beckley, Keith Chun, Janis Cole, Nancy Davis, Roy Goldman, Jeff Groves, Curtis Huntington, Andre LEsperance, Graham Lord, Esther Portnoy, David Promislow, Elias Shiu, McKenzie Smith, and Judy Strachan. The Education and Examination Committee, the Peer Review Team, and all others involved in the revision of this text are most appreciative of the efforts of Professors Bowers, Hickman and Jones toward the education of actuaries. DAVID M. HOLLAND President ALAN D. FORD General Chairperson Education and Examination Steering and Coordinating Committee

Preface

v

Table of Contents

Authors Biographies ...................................................................

xv

Authors Introductions and Guide to Study ..................................... xvii .. Introduction to First Edition...................................................... xvii Introduction to Second Edition ................................................... xxi .. xxii Guide to Study...................................................................... 1 The Economics of Insurance ........................................................ 1.1 Introduction ...................................................................... 1.2 Utility Theory .................................................................... 1.3 Insurance and Utility ........................................................... 1.4 Elements of Insurance .......................................................... 1.5 Optimal Insurance .............................................................. 1.6 Notes and References .......................................................... Appendix ............................................................................... Exercises ................................................................................ 2 Individual Risk Models for a Short Term ........................................ 2.1 Introduction ...................................................................... 2.2 Models for Individual Claim Random Variables .......................... 2.3 Sums of Independent Random Variables ................................... 2.4 Approximations for the Distribution of the Sum.......................... 2.5 Applications to Insurance ..................................................... 2.6 Notes and References .......................................................... Exercises ................................................................................ 3 Survival Distributions and Life Tables ............................................ 3.1 Introduction ...................................................................... 3.2 Probability for the Age.at.Death ............................................. 3.2.1 The Survival Function ................................................. -3.2.2 Time-until-Death for a Person Age x ............................... 3.2.3 Curtate-Future.Lifetimes ............................................... 3.2.4 Force of Mortality ...................................................... 1 1 3 7 15 16 18 19 20 27 27 28 34 39 40 46 47 51 51 52 52 52 54 55

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Con tents

vi i

3.3 Life Tables........................................................................ 3.3.1 Relation of Life Table Functions to the Survival Function .................................................... 3.3.2 Life Table Example ..................................................... 3.4 The Deterministic Survivorship Group ..................................... 3.5 Other Life Table Characteristics .............................................. 3.5.1 Characteristics ........................................................... 3.5.2 Recursion Formulas .................................................... 3.6 Assumptions for Fractional Ages ............................................ 3.7 Some Analytical Laws of Mortality .......................................... 3.8 Select and Ultimate Tables .................................................... 3.9 Notes and References .......................................................... Exercises ................................................................................

5858 59 66 68 68 73 74 77 79 83 84

4 Life Insurance .......................................................................... 93 4.1 Introduction ...................................................................... 93 94 4.2 Insurances Payable at the Moment of Death ............................... 94 4.2.1 Level Benefit Insurance ................................................ 101 4.2.2 Endowment Insurance ................................................. 4.2.3 Deferred Insurance ..................................................... 103 4.2.4 Varying Benefit Insurance ............................................. 105 108 4.3 Insurances Payable at the End of the Year of Death ..................... 4.4 Relationships between Insurances Payable at the Moment of Death and the End of the Year of Death .................. 119 4.5 Differential Equations for Insurances Payable at the 125 Moment of Death ............................................................ 126 4.6 Notes and References .......................................................... Exercises ................................................................................ 126

5 Life Annuities .......................................................................... 5.1 Introduction ...................................................................... 5.2 Continuous Life Annuities .................................................... 5.3 Discrete Life Annuities ......................................................... 5.4 Life Annuities with m-thly Payments ....................................... 5.5 Apportionable Annuities-Due and Complete Annuities-Immediate ........................................................ 5.6 Notes and References .......................................................... Exercises ................................................................................6 Benefit Premiums ..................................................................... 6.1 Introduction ...................................................................... 6.2 Fully Continuous Premiums .................................................. 6