Fundamentals of Actuarial Mathematics

Size: px

Start display at page:

Download "Fundamentals of Actuarial Mathematics"

Edwin Gilmore
5 years ago
Views:

1 Fundamentals of Actuarial Mathematics Third Edition S. David Promislow

3 Fundamentals of Actuarial Mathematics

5 Fundamentals of Actuarial Mathematics Third Edition S. David Promislow York University, Toronto, Canada

6 This edition first published John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 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, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought Library of Congress Cataloging-in-Publication Data Promislow, S. David. Fundamentals of actuarial mathematics / S. David Promislow. Third edition. pages cm Includes bibliographical references and index. ISBN (hardback) 1. Insurance Mathematics. 2. Business mathematics. I. Title. HG8781.P dc A catalogue record for this book is available from the British Library. ISBN: Set in 10/12pt Times by Aptara Inc., New Delhi, India

7 To Georgia and Griffith

9 Contents Preface Acknowledgements About the companion website xix xxiii xxiv Part I THE DETERMINISTIC LIFE CONTINGENCIES MODEL 1 1 Introduction and motivation Risk and insurance Deterministic versus stochastic models Finance and investments Adequacy and equity Reassessment Conclusion 6 2 The basic deterministic model Cash flows An analogy with currencies Discount functions Calculating the discount function Interest and discount rates Constant interest Values and actuarial equivalence Vector notation Regular pattern cash flows Balances and reserves Basic concepts Relation between balances and reserves Prospective versus retrospective methods Recursion formulas Time shifting and the splitting identity 26

10 viii CONTENTS 2.11 Change of discount function Internal rates of return Forward prices and term structure Standard notation and terminology Standard notation for cash flows discounted with interest New notation Spreadsheet calculations 34 Notes and references 35 Exercises 35 3 The life table Basic definitions Probabilities Constructing the life table from the values of q x Life expectancy Choice of life tables Standard notation and terminology A sample table 45 Notes and references 45 Exercises 45 4 Life annuities Introduction Calculating annuity premiums The interest and survivorship discount function The basic definition Relations between y x for various values of x Guaranteed payments Deferred annuities with annual premiums Some practical considerations Gross premiums Gender aspects Standard notation and terminology Spreadsheet calculations 58 Exercises 59 5 Life insurance Introduction Calculating life insurance premiums Types of life insurance Combined insurance annuity benefits Insurances viewed as annuities Summary of formulas A general insurance annuity identity The general identity The endowment identity 71

11 CONTENTS ix 5.8 Standard notation and terminology Single-premium notation Annual-premium notation Identities Spreadsheet applications 74 Exercises 74 6 Insurance and annuity reserves Introduction to reserves The general pattern of reserves Recursion Detailed analysis of an insurance or annuity contract Gains and losses The risk savings decomposition Bases for reserves Nonforfeiture values Policies involving a return of the reserve Premium difference and paid-up formulas Premium difference formulas Paid-up formulas Level endowment reserves Standard notation and terminology Spreadsheet applications 93 Exercises 94 7 Fractional durations Introduction Cash flows discounted with interest only Life annuities paid mthly Uniform distribution of deaths Present value formulas Immediate annuities Approximation and computation Fractional period premiums and reserves Reserves at fractional durations Standard notation and terminology 109 Exercises Continuous payments Introduction to continuous annuities The force of discount The constant interest case Continuous life annuities Basic definition Evaluation Life expectancy revisited 117

12 x CONTENTS 8.5 The force of mortality Insurances payable at the moment of death Basic definitions Evaluation Premiums and reserves The general insurance annuity identity in the continuous case Differential equations for reserves Some examples of exact calculation Constant force of mortality Demoivre s law An example of the splitting identity Further approximations from the life table Standard actuarial notation and terminology 131 Notes and references 132 Exercises Select mortality Introduction Select and ultimate tables Changes in formulas Projections in annuity tables Further remarks 142 Exercises Multiple-life contracts Introduction The joint-life status Joint-life annuities and insurances Last-survivor annuities and insurances Basic results Reserves on second-death insurances Moment of death insurances The general two-life annuity contract The general two-life insurance contract Contingent insurances First-death contingent insurances Second-death contingent insurances Moment-of-death contingent insurances General contingent probabilities Duration problems Applications to annuity credit risk Standard notation and terminology Spreadsheet applications 161 Notes and references 161 Exercises 161

13 CONTENTS xi 11 Multiple-decrement theory Introduction The basic model The multiple-decrement table Quantities calculated from the multiple-decrement table Insurances Determining the model from the forces of decrement The analogy with joint-life statuses A machine analogy Method Method Associated single-decrement tables The main methods Forces of decrement in the associated single-decrement tables Conditions justifying the two methods Other approaches 180 Notes and references 181 Exercises Expenses and profits Introduction Effect on reserves Realistic reserve and balance calculations Profit measurement Advanced gain and loss analysis Gains by source Profit testing 193 Notes and references 196 Exercises 196 *13 Specialized topics Universal life Description of the contract Calculating account values Variable annuities Pension plans DB plans DC plans 206 Exercises 207 Part II THE STOCHASTIC LIFE CONTINGENCIES MODEL Survival distributions and failure times Introduction to survival distributions The discrete case 212

14 xii CONTENTS 14.3 The continuous case The basic functions Properties of μ Modes Examples Shifted distributions The standard approximation The stochastic life table Life expectancy in the stochastic model Stochastic interest rates 221 Notes and references 222 Exercises The stochastic approach to insurance and annuities Introduction The stochastic approach to insurance benefits The discrete case The continuous case Approximation Endowment insurances The stochastic approach to annuity benefits Discrete annuities Continuous annuities Deferred contracts The stochastic approach to reserves The stochastic approach to premiums The equivalence principle Percentile premiums Aggregate premiums General premium principles The variance of r L Standard notation and terminology 243 Notes and references 244 Exercises Simplifications under level benefit contracts Introduction Variance calculations in the continuous case Insurances Annuities Prospective losses Using equivalence principle premiums Variance calculations in the discrete case Exact distributions The distribution of Z The distribution of Ȳ 252

15 CONTENTS xiii The distribution of L The case where T is exponentially distributed Some non-level benefit examples Term insurance Deferred insurance An annual premium policy 255 Exercises The minimum failure time Introduction Joint distributions The distribution of T The general case The independent case The joint distribution of (T, J) The distribution function for (T, J) Density and survival functions for (T, J) The distribution of J Hazard functions for (T, J) The independent case Nonidentifiability Conditions for the independence of T and J Other problems The common shock model Copulas 273 Notes and references 276 Exercises 276 Part III ADVANCED STOCHASTIC MODELS An introduction to stochastic processes Introduction Markov chains Definitions Examples Martingales Finite-state Markov chains The transition matrix Multi-period transitions Distributions Limiting distributions Recurrent and transient states Introduction to continuous time processes Poisson processes Waiting times Nonhomogeneous Poisson processes 295

Co p y r i g h t e d Ma t e r i a l

Co p y r i g h t e d Ma t e r i a l i JWBK850-fm JWBK850-Hilpisch October 13, 2016 14:56 Printer Name: Trim: 244mm 170mm Listed Volatility and Variance Derivatives ii JWBK850-fm JWBK850-Hilpisch October 13, 2016 14:56 Printer Name: Trim: