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Fundamentals of,Actuarial Mathematics,Fundamentals of. Actuarial Mathematics,Third Edition,S David Promislow. York University Toronto Canada,This edition first published 2015. 2015 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 www wiley com. 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 1988, 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. Includes bibliographical references and index,ISBN 978 1 118 78246 0 hardback. 1 Insurance Mathematics 2 Business mathematics I Title. HG8781 P76 2014,368 01 dc23,2014027082, A catalogue record for this book is available from the British Library. ISBN 9781118782460, Set in 10 12pt Times by Aptara Inc New Delhi India.
To Georgia and Griffith,Preface xix,Acknowledgements xxiii. About the companion website xxiv, Part I THE DETERMINISTIC LIFE CONTINGENCIES MODEL 1. 1 Introduction and motivation 3,1 1 Risk and insurance 3. 1 2 Deterministic versus stochastic models 4,1 3 Finance and investments 5. 1 4 Adequacy and equity 5,1 5 Reassessment 6,1 6 Conclusion 6.
2 The basic deterministic model 7,2 1 Cash flows 7. 2 2 An analogy with currencies 8,2 3 Discount functions 9. 2 4 Calculating the discount function 11,2 5 Interest and discount rates 12. 2 6 Constant interest 12,2 7 Values and actuarial equivalence 13. 2 8 Vector notation 17,2 9 Regular pattern cash flows 18.
2 10 Balances and reserves 20,2 10 1 Basic concepts 20. 2 10 2 Relation between balances and reserves 22, 2 10 3 Prospective versus retrospective methods 23. 2 10 4 Recursion formulas 24,2 11 Time shifting and the splitting identity 26. viii CONTENTS,2 11 Change of discount function 27,2 12 Internal rates of return 28. 2 13 Forward prices and term structure 30,2 14 Standard notation and terminology 33.
2 14 1 Standard notation for cash flows discounted with interest 33. 2 14 2 New notation 34,2 15 Spreadsheet calculations 34. Notes and references 35,Exercises 35,3 The life table 39. 3 1 Basic definitions 39,3 2 Probabilities 40, 3 3 Constructing the life table from the values of qx 41. 3 4 Life expectancy 42,3 5 Choice of life tables 44. 3 6 Standard notation and terminology 44,3 7 A sample table 45.
Notes and references 45,Exercises 45,4 Life annuities 47. 4 1 Introduction 47,4 2 Calculating annuity premiums 48. 4 3 The interest and survivorship discount function 50. 4 3 1 The basic definition 50, 4 3 2 Relations between yx for various values of x 52. 4 4 Guaranteed payments 53,4 5 Deferred annuities with annual premiums 55. 4 6 Some practical considerations 56,4 6 1 Gross premiums 56.
4 6 2 Gender aspects 56,4 7 Standard notation and terminology 57. 4 8 Spreadsheet calculations 58,Exercises 59,5 Life insurance 61. 5 1 Introduction 61,5 2 Calculating life insurance premiums 61. 5 3 Types of life insurance 64,5 4 Combined insurance annuity benefits 64. 5 5 Insurances viewed as annuities 69,5 6 Summary of formulas 70.
5 7 A general insurance annuity identity 70,5 7 1 The general identity 70. 5 7 2 The endowment identity 71,CONTENTS ix,5 8 Standard notation and terminology 72. 5 8 1 Single premium notation 72,5 8 2 Annual premium notation 73. 5 8 3 Identities 74,5 9 Spreadsheet applications 74. Exercises 74,6 Insurance and annuity reserves 78,6 1 Introduction to reserves 78.
6 2 The general pattern of reserves 81,6 3 Recursion 82. 6 4 Detailed analysis of an insurance or annuity contract 83. 6 4 1 Gains and losses 83,6 4 2 The risk savings decomposition 85. 6 5 Bases for reserves 87,6 6 Nonforfeiture values 88. 6 7 Policies involving a return of the reserve 88,6 8 Premium difference and paid up formulas 90. 6 8 1 Premium difference formulas 90,6 8 2 Paid up formulas 90.
6 8 3 Level endowment reserves 91,6 9 Standard notation and terminology 91. 6 10 Spreadsheet applications 93,Exercises 94,7 Fractional durations 98. 7 1 Introduction 98,7 2 Cash flows discounted with interest only 99. 7 3 Life annuities paid mthly 101,7 3 1 Uniform distribution of deaths 101. 7 3 2 Present value formulas 102,7 4 Immediate annuities 104.
7 5 Approximation and computation 105,7 6 Fractional period premiums and reserves 106. 7 7 Reserves at fractional durations 107,7 8 Standard notation and terminology 109. Exercises 109,8 Continuous payments 112,8 1 Introduction to continuous annuities 112. 8 2 The force of discount 113,8 3 The constant interest case 114. 8 4 Continuous life annuities 115,8 4 1 Basic definition 115.
8 4 2 Evaluation 116,8 4 3 Life expectancy revisited 117. x CONTENTS,8 5 The force of mortality 118,8 6 Insurances payable at the moment of death 119. 8 6 1 Basic definitions 119,8 6 2 Evaluation 120,8 7 Premiums and reserves 122. 8 8 The general insurance annuity identity in the continuous case 123. 8 9 Differential equations for reserves 124,8 10 Some examples of exact calculation 125. 8 10 1 Constant force of mortality 126,8 10 2 Demoivre s law 127.
8 10 3 An example of the splitting identity 128, 8 11 Further approximations from the life table 129. 8 12 Standard actuarial notation and terminology 131. Notes and references 132,Exercises 132,9 Select mortality 137. 9 1 Introduction 137,9 2 Select and ultimate tables 138. 9 3 Changes in formulas 139,9 4 Projections in annuity tables 141. 9 5 Further remarks 142,Exercises 142,10 Multiple life contracts 144.
10 1 Introduction 144,10 2 The joint life status 144. 10 3 Joint life annuities and insurances 146,10 4 Last survivor annuities and insurances 147. 10 4 1 Basic results 147,10 4 2 Reserves on second death insurances 148. 10 5 Moment of death insurances 149,10 6 The general two life annuity contract 150. 10 7 The general two life insurance contract 152,10 8 Contingent insurances 153.
10 8 1 First death contingent insurances 153,10 8 2 Second death contingent insurances 154. 10 8 3 Moment of death contingent insurances 155,10 8 4 General contingent probabilities 155. 10 9 Duration problems 156,10 10 Applications to annuity credit risk 159. 10 11 Standard notation and terminology 160,10 12 Spreadsheet applications 161. Notes and references 161,Exercises 161,CONTENTS xi.
11 Multiple decrement theory 166,11 1 Introduction 166. 11 2 The basic model 166,11 2 1 The multiple decrement table 167. 11 2 2 Quantities calculated from the multiple decrement table 168. 11 3 Insurances 169, 11 4 Determining the model from the forces of decrement 170. 11 5 The analogy with joint life statuses 171,11 6 A machine analogy 171. 11 6 1 Method 1 172,11 6 2 Method 2 173,11 7 Associated single decrement tables 175.
11 7 1 The main methods 175,11 7 2 Forces of decrement in the associated. single decrement tables 176,11 7 3 Conditions justifying the two methods 177. 11 7 4 Other approaches 180,Notes and references 181. Exercises 181,12 Expenses and profits 184,12 1 Introduction 184. 12 2 Effect on reserves 186, 12 3 Realistic reserve and balance calculations 187.
12 4 Profit measurement 189,12 4 1 Advanced gain and loss analysis 189. 12 4 2 Gains by source 191,12 4 3 Profit testing 193. Notes and references 196,Exercises 196,13 Specialized topics 199. 13 1 Universal life 199,13 1 1 Description of the contract 199. 13 1 2 Calculating account values 201,13 2 Variable annuities 203.
13 3 Pension plans 204,13 3 1 DB plans 204,13 3 2 DC plans 206. Exercises 207, Part II THE STOCHASTIC LIFE CONTINGENCIES MODEL 209. 14 Survival distributions and failure times 211,14 1 Introduction to survival distributions 211. 14 2 The discrete case 212,xii CONTENTS,14 3 The continuous case 213. 14 3 1 The basic functions 214,14 3 2 Properties of 214.
14 3 3 Modes 215,14 4 Examples 215,14 5 Shifted distributions 216. 14 6 The standard approximation 217,14 7 The stochastic life table 219. 14 8 Life expectancy in the stochastic model 220,14 9 Stochastic interest rates 221. Notes and references 222,Exercises 222, 15 The stochastic approach to insurance and annuities 224. 15 1 Introduction 224, 15 2 The stochastic approach to insurance benefits 225.
15 2 1 The discrete case 225,15 2 2 The continuous case 226. 15 2 3 Approximation 226,15 2 4 Endowment insurances 227. 15 3 The stochastic approach to annuity benefits 229. 15 3 1 Discrete annuities 229,15 3 2 Continuous annuities 231. 15 4 Deferred contracts 233,15 5 The stochastic approach to reserves 233. 15 6 The stochastic approach to premiums 235,15 6 1 The equivalence principle 235.
15 6 2 Percentile premiums 236,15 6 3 Aggregate premiums 237. 15 6 4 General premium principles 240,15 7 The variance of r L 241. 15 8 Standard notation and terminology 243,Notes and references 244. Exercises 244, 16 Simplifications under level benefit contracts 248. 16 1 Introduction 248, 16 2 Variance calculations in the continuous case 248.
16 2 1 Insurances 249,16 2 2 Annuities 249,16 2 3 Prospective losses 249. 16 2 4 Using equivalence principle premiums 249, 16 3 Variance calculations in the discrete case 250. 16 4 Exact distributions 252,16 4 1 The distribution of Z 252. 16 4 2 The distribution of Y 252,CONTENTS xiii,16 4 3 The distribution of L 252. 16 4 4 The case where T is exponentially distributed 253. 16 5 Some non level benefit examples 254,16 5 1 Term insurance 254.
16 5 2 Deferred insurance 254,16 5 3 An annual premium policy 255. Exercises 256,17 The minimum failure time 259,17 1 Introduction 259. 17 2 Joint distributions 259,17 3 The distribution of T 261. 17 3 1 The general case 261,17 3 2 The independent case 261. 17 4 The joint distribution of T J 261,17 4 1 The distribution function for T J 261.
17 4 2 Density and survival functions for T J 264,17 4 3 The distribution of J 265. 17 4 4 Hazard functions for T J 266,17 4 5 The independent case 266. 17 4 6 Nonidentifiability 268, 17 4 7 Conditions for the independence of T and J 269. 17 5 Other problems 270,17 6 The common shock model 271. 17 7 Copulas 273,Notes and references 276,Exercises 276.
Part III ADVANCED STOCHASTIC MODELS 279,18 An introduction to stochastic processes 281. 18 1 Introduction 281,18 2 Markov chains 283,18 2 1 Definitions 283. 18 2 2 Examples 284,18 3 Martingales 286,18 4 Finite state Markov chains 287. 18 4 1 The transition matrix 287,18 4 2 Multi period transitions 288. 18 4 3 Distributions 288,18 4 4 Limiting distributions 289.
18 4 5 Recurrent and transient states 290, 18 5 Introduction to continuous time processes 293. 18 6 Poisson processes 293,18 6 1 Waiting times 295. Fundamentals of Actuarial Mathematics Third Edition S David Promislow JWST504 fm JWST504 Promislow Printer YettoCome Trim 244mm 170mm October13 2014 7 17 ii JWST504 fm JWST504 Promislow Printer YettoCome Trim 244mm 170mm October13 2014 7 17 Fundamentalsof ActuarialMathematics i JWST504 fm JWST504 Promislow Printer YettoCome Trim 244mm 170mm October13 2014 7 17 ii JWST504 fm JWST504

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