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Fundamentals of Actuarial Mathematics PDF

Fundamentals of Actuarial Mathematics – This book provides a comprehensive introduction to actuarial mathematics, covering both deterministic and stochastic models of life contingencies, as well as more advanced topics such as risk theory, credibility theory and multi-state models.

This new edition includes additional material on credibility theory, continuous time multi-state models, more complex types of contingent insurances, flexible contracts such as universal life, the risk measures VaR and TVaR.

Key Features: Fundamentals of Actuarial Mathematics

  • Covers much of the syllabus material on the modeling examinations of the Society of Actuaries, Canadian Institute of Actuaries and the Casualty Actuarial Society. (SOA-CIA exams MLC and C, CSA exams 3L and 4.)
  • Extensively revised and updated with new material.
  • Orders the topics specifically to facilitate learning.
  • Provides a streamlined approach to actuarial notation.
  • Employs modern computational methods.
  • Contains a variety of exercises, both computational and theoretical, together with answers, enabling use for self-study.

An ideal text for students planning for a professional career as actuaries, providing a solid preparation for the modeling examinations of the major North American actuarial associations. Furthermore, this book is highly suitable reference for those wanting a sound introduction to the subject, and for those working in insurance, annuities and pensions

Contents – Fundamentals of Actuarial Mathematics

Preface xvii
Acknowledgements xxi
Notation index xxiii
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 Cashflows 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 Regular pattern cashflows 17
2.9 Balances and reserves 19
2.9.1 Basic concepts 19
2.9.2 Relationship between balances and reserves 21
2.9.3 Prospective versus retrospective methods 22
2.9.4 Recursion formulas 23
2.10 Time shifting and the splitting identity 24

Fundamentals of Actuarial Mathematics

*2.11 Change of discount function 26
*2.12 Internal rates of return 27
*2.13 Forward prices and term structure 29
2.14 Standard notation and terminology 31
2.14.1 Standard notation for cashflows discounted with interest 31
2.14.2 New notation 32
2.15 Spreadsheet calculations 33
2.16 Notes and references 33
2.17 Exercises 33
3 The life table 37
3.1 Basic definitions 37
3.2 Probabilities 38
3.3 Constructing the life table from the values of qx 39
3.4 Life expectancy 40
3.5 Choice of life tables 42
3.6 Standard notation and terminology 42
3.7 A sample table 43
3.8 Notes and references 43
3.9 Exercises 43
4 Life annuities 45
4.1 Introduction 45
4.2 Calculating annuity premiums 46
4.3 The interest and survivorship discount function 48
4.3.1 The basic definition 48
4.3.2 Relations between yx for various values of x 50
4.3.3 Tontines 51
4.4 Guaranteed payments 52
4.5 Deferred annuities with annual premiums 53
4.6 Some practical considerations 54
4.6.1 Gross premiums 54
4.6.2 Gender aspects 55
4.7 Standard notation and terminology 55
4.8 Spreadsheet calculations 56
4.9 Exercises 57
5 Life insurance 60
5.1 Introduction 60
5.2 Calculating life insurance premiums 60
5.3 Types of life insurance 63
5.4 Combined insurance–annuity benefits 63
5.5 Insurances viewed as annuities 67
5.6 Summary of formulas 68
5.7 A general insurance–annuity identity 69
5.7.1 The main theorem 69
5.7.2 The endowment identity 69

Fundamentals of Actuarial Mathematics

5.8 Standard notation and terminology 71
5.8.1 Single premium notation 71
5.8.2 Annual premium notation 72
5.8.3 Identities 72
5.9 Spreadsheet applications 72
5.10 Exercises 73
6 Insurance and annuity reserves 76
6.1 Introduction to reserves 76
6.2 The general pattern of reserves 79
6.3 Recursion 80
6.4 Detailed analysis of an insurance or annuity contract 81
6.4.1 Gains and losses 81
6.4.2 The risk–savings decomposition 83
6.5 Interest and mortality bases for reserves 84
6.6 Nonforfeiture values 86
6.7 Policies involving a ‘return of the reserve’ 87
6.8 Premium difference and paid-up formulas 88
6.8.1 Premium difference formulas 88
6.8.2 Paid-up formulas 89
6.8.3 Level endowment reserves 89
*6.9 Universal life and variable annuities 89
6.9.1 Universal life 90
6.9.2 Variable annuities 93
6.10 Standard notation and terminology 94
6.11 Spreadsheet applications 95
6.12 Exercises 96
7 Fractional durations 101
7.1 Introduction 101
7.2 Cashflows discounted with interest only 102
7.3 Life annuities paid mthly 104
7.3.1 Uniform distribution of deaths 104
7.3.2 Present value formulas 105
7.4 Immediate annuities 106
7.5 Approximation and computation 107
*7.6 Fractional period premiums and reserves 109
7.7 Reserves at fractional durations 110
7.8 Notes and references 112
7.9 Exercises 112
8 Continuous payments 115
8.1 Introduction to continuous annuities 115
8.2 The force of discount 116
8.3 The constant interest case 117
8.4 Continuous life annuities 118
8.4.1 Basic definition 118

Fundamentals of Actuarial Mathematics

8.4.2 Evaluation 119
8.4.3 Life expectancy revisited 120
8.5 The force of mortality 121
8.6 Insurances payable at the moment of death 122
8.6.1 Basic definitions 122
8.6.2 Evaluation 123
8.7 Premiums and reserves 125
8.8 The general insurance–annuity identity in the continuous case 126
8.9 Differential equations for reserves 127
8.10 Some examples of exact calculation 128
8.10.1 Constant force of mortality 128
8.10.2 Demoivre’s law 129
8.10.3 An example of the splitting identity 130
8.11 Standard actuarial notation and terminology 131
8.12 Notes and references 131
8.13 Exercises 132
9 Select mortality 136
9.1 Introduction 136
9.2 Select and ultimate tables 137
9.3 Changes in formulas 138
9.4 Projections in annuity tables 140
9.5 Further remarks 141
9.6 Exercises 141
10 Multiple-life contracts 143
10.1 Introduction 143
10.2 The joint-life status 143
10.3 Joint-life annuities and insurances 145
10.4 Last-survivor annuities and insurances 146
10.5 Moment of death insurances 147
10.6 The general two-life annuity contract 149
10.7 The general two-life insurance contract 150
10.8 Contingent insurances 151
10.8.1 First-death contingent insurances 151
10.8.2 Second-death contingent insurances 152
10.8.3 Moment-of-death contingent insurances 153
10.8.4 General contingent probabilities 153
10.9 Duration problems 154
10.10 Applications to annuity credit risk 157
10.11 Standard notation and terminology 158
10.12 Spreadsheet applications 159
10.13 Notes and references 159
10.14 Exercises 159
11 Multiple-decrement theory 164
11.1 Introduction 164

11.2 The basic model 164
11.2.1 The multiple-decrement table 165
11.2.2 Quantities calculated from the multiple-decrement table 166
11.3 Insurances 167
11.4 Determining the model from the forces of decrement 168
11.5 The analogy with joint-life statuses 169
11.6 A machine analogy 169
11.6.1 Method 1 170
11.6.2 Method 2 171
11.7 Associated single-decrement tables 173
11.7.1 The main methods 173
11.7.2 Forces of decrement in the associated
single-decrement tables 174
11.7.3 Conditions justifying the two methods 175
11.7.4 Other approaches 178
11.8 Notes and references 179
11.9 Exercises 179
12 Expenses 182
12.1 Introduction 182
12.2 Effect on reserves 184
12.3 Realistic reserve and balance calculations 185
12.4 Notes and references 187
12.5 Exercises 187
13 Survival distributions and failure times 191
13.1 Introduction to survival distributions 191
13.2 The discrete case 192
13.3 The continuous case 193
13.3.1 The basic functions 194
13.3.2 Properties of µ 194
13.3.3 Modes 195
13.4 Examples 195
13.4.1 The exponential distribution 195
13.4.2 The uniform distribution 195
13.4.3 The Gompertz–Makeham distribution 196
13.5 Shifted distributions 197
13.6 The standard approximation 198
13.7 The stochastic life table 199
13.8 Life expectancy in the stochastic model 201
13.9 Stochastic interest rates 202
13.10 Notes and references 202
13.11 Exercises 203 Fundamentals of Actuarial Mathematics

14 The stochastic approach to insurance and annuities 205
14.1 Introduction 205
14.2 The stochastic approach to insurance benefits 206
14.2.1 The discrete case 206
14.2.2 The continuous case 206
14.2.3 Approximation 207
14.2.4 Endowment insurances 208
14.3 The stochastic approach to annuity benefits 209
14.3.1 Discrete annuities 209
14.3.2 Continuous annuities 212
*14.4 Deferred contracts 213
14.5 The stochastic approach to reserves 214
14.6 The stochastic approach to premiums 215
14.6.1 The equivalence principle 215
14.6.2 Percentile premiums 216
14.6.3 Aggregate premiums 217
14.6.4 General premium principles 220
14.7 The variance of r L 220
14.8 Standard notation and terminology 223
14.9 Notes and references 223
14.10 Exercises 224
15 Simplifications under level benefit contracts 228
15.1 Introduction 228
15.2 Variance calculations in the continuous case 228
15.2.1 Insurances 228
15.2.2 Annuities 229
15.2.3 Prospective losses 229
15.2.4 Using equivalence principle premiums 229
15.3 Variance calculations in the discrete case 230
15.4 Exact distributions 231
15.4.1 The distribution of Z¯ 231
15.4.2 The distribution of Y¯ 231
15.4.3 The distribution of L 232
15.4.4 The case where T is exponentially distributed 232
15.5 Non-level benefit examples 233
15.5.1 Term insurance 233
15.5.2 Deferred insurance 234
15.5.3 An annual premium policy 234
15.6 Exercises 235
16 The minimum failure time 238
16.1 Introduction 238
16.2 Joint distributions 238
16.3 The distribution of T 240
16.3.1 The general case 240
16.3.2 The independent case 240 Fundamentals of Actuarial Mathematics

16.4 The joint distribution of (T, J ) 240
16.4.1 The distribution function for (T, J ) 240
16.4.2 Density and survival functions for (T, J ) 243
16.4.3 The distribution of J 244
16.4.4 Hazard functions for (T, J ) 245
16.4.5 The independent case 245
16.4.6 Nonidentifiability 247
16.4.7 Conditions for the independence of T and J 248
16.5 Other problems 249
16.6 The common shock model 250
16.7 Copulas 252
16.8 Notes and references 255
16.9 Exercises 255
17 The collective risk model 261
17.1 Introduction 261
17.2 The mean and variance of S 263
17.3 Generating functions 264
17.4 Exact distribution of S 265
17.5 Choosing a frequency distribution 265
17.5.1 The binomial distribution 266
17.5.2 The Poisson distribution 267
17.5.3 The negative binomial distribution 267
17.6 Choosing a severity distribution 269
17.6.1 The normal distribution 269
17.6.2 The gamma distribution 269
17.6.3 The lognormal distribution 270
17.6.4 The Pareto distribution 271
17.7 Handling the point mass at 0 271
17.8 Counting claims of a particular type 272
17.8.1 One special class 272
17.8.2 Special classes in the Poisson case 274
17.9 The sum of two compound Poisson distributions 275
17.10 Deductibles and other modifications 275
17.10.1 The nature of a deductible 276
17.10.2 Some calculations in the discrete case 277
17.10.3 Some calculations in the continuous case 278
17.10.4 The effect on aggregate claims 280
17.10.5 Other modifications 281
17.11 A recursion formula for S 281
17.11.1 The positive-valued case 281
17.11.2 The case with claims of zero amount 285
17.12 Notes and references 286
17.13 Exercises 287 Fundamentals of Actuarial Mathematics

18 Risk Assessment 291
18.1 Introduction 291
18.2 Utility theory 291
18.3 Convex and concave functions: Jensen’s inequality 294
18.3.1 Basic definitions 294
18.3.2 Jensen’s inequality 295
18.4 A general comparison method 296
18.5 Risk measures for capital adequacy 301
18.5.1 The general notion of a risk measure 301
18.5.2 Value-at-risk 301
18.5.3 Tail-value-at-risk 302
18.5.4 Distortion risk measures 305
18.6 Notes and references 305
18.7 Exercises 306
19 An introduction to stochastic processes 308
19.1 Introduction 308
19.2 Markov chains 310
19.2.1 Definition 310
19.2.2 Examples 311
19.3 Martingales 313
19.4 Finite-state Markov chains 314
19.4.1 The transition matrix 314
19.4.2 Multi-period transitions 315
19.4.3 Distributions 315
*19.4.4 Limiting distributions 316
*19.4.5 Recurrent and transient states 317
19.4.6 The nonstationary case 320
19.5 Notes and references 321
19.6 Exercises 321
20 Poisson processes 324
20.1 Introduction 324
20.2 Definition of a Poisson process 325
20.3 Waiting times 326
*20.4 Some properties of a Poisson process 326
20.4.1 Ordering of events 326
20.4.2 Distribution of arrival times 327
20.5 Nonhomogeneous Poisson processes 328
20.6 Compound Poisson processes 329
20.7 Notes and references 329
20.8 Exercises 329
21 Ruin models 332
21.1 Introduction 332
21.2 A functional equation approach 334 Fundamentals of Actuarial Mathematics

21.3 The martingale approach to ruin theory 336
21.3.1 Stopping times 336
21.3.2 The optional stopping theorem and its consequences 337
21.3.3 The adjustment coefficient 341
21.3.4 The main conclusions 343
21.4 Distribution of the deficit at ruin 345
21.5 Recursion formulas 346
21.5.1 Recursive calculation of ruin probabilities 346
21.5.2 The distribution of D(u) 348
21.6 The compound Poisson surplus process 350
21.6.1 The basic set-up 350
21.6.2 The probability of eventual ruin 351
21.6.3 The value of ψ(0) 352
21.6.4 The distribution of D(0) 352
21.6.5 The case when X is exponentially distributed 352
21.7 The maximal aggregate loss 353
21.8 Notes and references 357
21.9 Exercises 357
22 Credibility theory 361
22.1 Introduction 361
22.1.1 The nature of credibility theory 361
22.1.2 Information assessment 361
22.2 Conditional expectation and variance 365
22.2.1 Conditional expectation 365
22.2.2 Conditional variance 368
22.3 General framework for Bayesian credibility 370
22.4 Classical examples 372
22.5 Approximations 375
22.5.1 A general case 375
22.5.2 The Buhlman model 376 ¨
22.5.3 Buhlman-Straub model 377 ¨
22.6 Conditions for exactness 379
22.7 Estimation 382
22.7.1 Unbiased estimators 382
22.7.2 Calculating Var(X¯ ) in the credibility model 383
22.7.3 Estimation of the Bulhman parameters 384 ¨
22.7.4 Estimation in the Bulhman–Straub model 385 ¨
22.8 Notes and references 386
22.9 Exercises 387
23 Multi-state models 389
23.1 Introduction 389
23.2 The discrete-time model 390
23.2.1 Discrete-time multi-state insurances 390
23.2.2 Multi-state annuities 393 Fundamentals of Actuarial Mathematics

23.3 The continuous-time model 395
23.3.1 Forces of transition 395
23.3.2 Stationary continuous-time processes 399
23.3.3 Some methods for nonstationary processes 401
23.3.4 Insurance and annuity applications 402
23.3.5 Semi-Markov models 403
23.4 Notes and references 403
23.5 Exercises 403
Appendix A review of probability theory 406
A.1 Introduction 406
A.2 Sample spaces and probability measures 406
A.3 Conditioning and independence 408
A.4 Random variables 408
A.5 Distributions 409
A.6 Expectations and moments 410
A.7 Expectation in terms of the distribution function 411
A.8 The normal distribution 412
A.9 Joint distributions 413
A.10 Conditioning and independence for random variables 415
A.11 Convolution 416
A.11.1 The discrete case 416
A.11.2 The continuous case 418
A.11.3 Notation and remarks 420
A.12 Moment generating functions 420
A.13 Probability generating functions 423
A.14 Mixtures 425
Answers to exercises 427
References 443
Index 445

Preface – Fundamentals of Actuarial Mathematics

Several factors motivated the writing of this book. After teaching undergraduate actuarial courses for many years, it became clear that there was a definite need for more instructional material in this area. In most undergraduate courses, students who had problems reading a particular text could go to the library and find dozens of other references which might assist them. By comparison there were very few resources dealing with actuarial mathematics.

In addition, there was need for a book which would give full recognition to modern computing methods and techniques. Many existing books still emphasize material which was developed in a time when calculations were done by hand. At present, basic actuarial calculations are easily done using computer spreadsheets, and I felt it was time for a text which would develop the ideas and methods with this in mind. The book covers two fundamental topics in actuarial mathematics. These are life contingencies and risk theory, including the basics of ruin theory. The modern approach towards life contingencies is through a stochastic model, as opposed to the older deterministic viewpoint.

I certainly agree that mastering the stochastic model is the desirable end. However, my classroom experience has convinced me that this is not the right place to begin the instruction. I find that students are much better able to learn the new ideas, the new notation, the new ways of thinking involved in this subject, when done first in the simplest possible setting, namely a deterministic discrete model, and I have followed this approach in this book. After the main ideas are presented in this fashion, continuous models are introduced. In Part II of the book, the full stochastic model can then be dealt with in reasonably quick fashion.Fundamentals of Actuarial Mathematics

The book covers a great deal of the material on the modeling exams of the Society of Actuaries and the Casualty Actuarial Society. A major audience for the book will be students preparing for these exams. The order of topics, however, provides a degree of flexibility, so that the book can be of interest to different readers. Part I of the book will serve the needs of those who want only an introduction to the subject, without necessarily specializing in it. The only mathematical background required for this material is some elementary linear algebra and probability theory, and, beginning in Chapter 8, some basic calculus. A more advanced knowledge of probability theory is needed from Chapter 13 onward. All of this material is summarized in Appendix A. Basic concepts of stochastic processes are used in Part III of the book, which deals with the collective risk model. These are developed in the text in Chapters 18 and 19.

For the most part, we do not include statistical aspects of the subject, unlike for example Klugman et al. (2008). Rather, the emphasis is on methods of using the information that the statistician would produce. No prior knowledge of statistical inference, as opposed to probability theory, is required. A usual prerequisite for this type of material is a course in the theory of interest. Although this may be useful, it is not strictly required.Fundamentals of Actuarial Mathematics

All the interest theory that is needed is presented as a particular case of the general deterministic actuarial model in Chapter 2. A major source of difficulty for many students in learning actuarial mathematics is to master the rather complex system of actuarial notation. We have introduced some notational innovations, which tie in well with modern calculation procedures as well as allowing us to greatly simplify the notation that is required. We have, however, included all the standard notation in separate sections, at the end of the relevant chapters, which can be read by those readers who desire this material. The book is intended to cover the material at a basic level and is not as encyclopedic as a work like Bowers et al. (1997).Fundamentals of Actuarial Mathematics

To meet this goal, and to keep the length reasonable, we have necessarily had to omit certain important topics. The most notable of these is stochastic interest rates. There is a brief discussion of this idea, but for the most part interest rates are taken as deterministic. There is more of an emphasis on life insurance and annuities as opposed to casualty insurance. Some important casualty topics, such as loss reserving, are not covered here. Keeping in mind the nature of the book and its intended audience, we have avoided excessive mathematical rigor. Nonetheless, careful proofs are given in all cases where these are thought to be accessible to the typical senior undergraduate mathematics student.Fundamentals of Actuarial Mathematics

For the few proofs not given in their entirety, mainly those involving continuous-time stochastic processes, we have tried at least to provide some motivation and intuitive reasoning for the results. Exercises appear at the end of each chapter. In Parts I and II these are divided up into different types. Type A exercises generally are those which involve direct calculation from the formulas in the book. Type B involve problems where more thought is involved. Derivations and problems which involve symbols rather than numeric calculation are normally included in Type B problems. A third type is spreadsheet exercises which themselves are divided into two subtypes. The first of these ask the reader to solve problems using a spreadsheet. Detailed descriptions of applicable Microsoft ExcelR spreadsheets are given at the end of the relevant chapters. Readers of course are free to modify these or construct their own. Fundamentals of Actuarial Mathematics

The second subtype does not ask specific questions but instead asks the reader to modify the given spreadsheets to handle additional tasks. Answers to most of the calculation-type exercises appear at the end of the book. Sections marked with an asterisk * deal with more advanced material, or with special topics that are not used elsewhere in the book. They can be omitted on first reading. The exercises dealing with such sections are likewise marked with *. The material in the book comprises approximately three semesters of work in the typical North American university. A rough guide would be to do Chapters 1–8 in the first semester, Chapters 9–16 in the second semester, and Chapters 17–23 in the third. Part III is for the most part independent of Parts I and II. A major exception is Chapter 23, which generalizes material in Chapter 11, and can be read immediately after that chapter, for the reader with a basic knowledge of Markov chains, as presented in Chapter 19. Another exception is Section 20.4.1 which alludes to previous material. Chapters 7 (except for Section 7.3.1), 9 and 12 deal with topics that are important in applications, but which are not used in other parts of the book. They could be omitted without loss of continuity

Changes in the second edition

There are several additions and changes for the second edition. The most important of these of these are the inclusion of three new chapters, and substantial modifications to a few others. A chapter on credibility theory, has been added. This is a major actuarial topic which was not addressed in the first edition. While the emphasis of the book is still on the life and pension side of actuarial science, this chapter provides additional material for those whose main interest is in casualty insurance. A chapter was added on risk assessment, another major subject area which received only minimal coverage in the first edition. The theme here is the comparison and measurement of risk in random alternatives, and the chapter introduces such topics as utility theory, a stochastic ordering method, and risk measures, with a concentration on VaR and TailVaR. The subject of multi-state models, has proved to be an effective way of unifying much of actuarial theory. Some aspects of the discrete model were included in the first edition as an application of Markov chain theory. This material has been extended and combined with the continuous time model, to form a new chapter on this topic. Chapter 10 has been extended to include situations involving a a duration that runs from a death of an individual rather than from time zero. This provides additional techniques, and equips the reader to handle a greater variety of multiple-life contracts. The chapter also includes a section outlining applications to credit risk in annuities. In the first edition, the multiple-decrement theory was contained in two chapters, the classical model in Chapter 11 and a more general treatment in Chapter 16. In this edition, much of the Chapter 16 material has been rewritten and moved back to Chapter 11, so that this earlier chapter now contains a more complete exposition of the subject. Other changes include the following:  In Chapter 2 there is some additional material dealing with forward prices and term structure for bonds.  A section has been added to Chapter 6, outlining the provisions of some modern types of contracts such as universal life and variable annuities.  In Chapter 9 on select mortality, there is a new section illustrating how projections in annuity tables fit into the select framework.  The method of presentation of some of the preliminary material has been changed, and time diagrams are introduced as a visual aid for depicting insurance and annuity contracts.  The spreadsheets covering the early chapters have been modified to improve efficiency of use.  Additional examples and exercises have been added to several chapters. This book includes an accompanying website. Please visit for more information.


Several individuals assisted in the completion of this project. A special debt of gratitude is owed to Virginia Young for her work on the first edition. She read large portions of the manuscript, worked nearly all of the exercises, and made several suggestions for improvement. Many people found misprints in the first edition and earlier drafts. These include Valerie Michkine, Jacques Labelle, Karen Antonio, Kristen Moore, as well as students at York University and the University of Michigan. Moshe Milevsky provided enlightening comments on annuities and it was his ideas that motivated the credit risk applications in Chapter 10, as well as some of the material on generational annuity tables in Chapter 9. Elias Shiu suggested some interesting exercises. Christian Hess asked some questions which led to the inclusion of Example 17.10 to clear up an ambiguous point. Exercise 19.13 was motivated by Bob Jewett’s progressive practice routines for pool. My son Michael, a life insurance actuary, provided valuable advice on several practical aspects of the material. I would like to thank the editorial and production teams at Wiley, for their much appreciated assistance. Finally, I would like to thank my wife Shirley who provided support and encouragement throughout the writing of both editions of this book.

Review – Fundamentals of Actuarial Mathematics

“An ideal text for students planning for a professional career as actuaries, providing a solid preparation for the modeling examinations of the major North American actuarial associations.” (Mathematical Reviews, 2011)

“This second edition adds several chapters, including coverage of credibility theory, risk assessment, and multi-state models.” (Book News, 1 March 2011)

About the Author

S. David Promislow is the author of Fundamentals of Actuarial Mathematics, 3rd Edition, published by Wiley.

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