Fundamentals of Actuarial Mathematics E-Book

S. David Promislow

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 851

Veröffentlichungsjahr: 2011

Beschreibung

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:

• 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.

Leseprobe

Contents

Cover

Half Title Page

Title Page

Copyright

Dedication

Preface

Changes in the second edition

Acknowledgements

Notation Index

Part I: THE DETERMINISTIC MODEL

1: Introduction and motivation

1.1 Risk and Insurance

1.2 Deterministic Versus Stochastic Models

1.3 Finance and Investments

1.4 Adequacy and Equity

1.5 Reassessment

1.6 Conclusion

2: The basic deterministic model

2.1 Cashflows

2.2 An Analogy with Currencies

2.3 Discount Functions

2.4 Calculating the Discount Function

2.5 Interest and Discount Rates

2.6 Constant Interest

2.7 Values and Actuarial Equivalence

2.8 Regular Pattern Cashflows

2.9 Balances and Reserves

2.10 Time Shifting and the Splitting Identity

2.11 Change of Discount Function

*2.12 Internal Rates of Return

*2.13 Forward Prices and Term Structure

2.14 Standard Notation and Terminology

2.15 Spreadsheet Calculations

2.16 Notes and References

2.17 Exercises

3: The life table

3.1 Basic Definitions

3.2 Probabilities

3.3 Constructing the Life Table from the Values of qx

3.4 Life Expectancy

3.5 Choice of Life Tables

3.6 Standard Notation and Terminology

3.7 A Sample Table

3.8 Notes and References

3.9 Exercises

4: Life annuities

4.1 Introduction

4.2 Calculating Annuity Premiums

4.3 The Interest and Survivorship Discount Function

4.4 Guaranteed Payments

4.5 Deferred Annuities with Annual Premiums

4.6 Some Practical Considerations

4.7 Standard Notation and Terminology

4.8 Spreadsheet Calculations

4.9 Exercises

5: Life insurance

5.1 Introduction

5.2 Calculating Life Insurance Premiums

5.3 Types of Life Insurance

5.4 Combined Insurance–Annuity Benefits

5.5 Insurances Viewed as Annuities

5.6 Summary of Formulas

5.7 A General Insurance–Annuity Identity

5.8 Standard Notation and Terminology

5.9 Spreadsheet Applications

5.10 Exercises

6: Insurance and annuity reserves

6.1 Introduction to Reserves

6.2 The General Pattern of Reserves

6.3 Recursion

6.4 Detailed Analysis of an Insurance or Annuity Contract

6.5 Interest and Mortality Bases for Reserves

6.6 Nonforfeiture Values

6.7 Policies Involving a ‘Return of the Reserve’

6.8 Premium Difference and Paid-up Formulas

*6.9 Universal Life and Variable Annuities

6.10 Standard notation and Terminology

6.11 Spreadsheet Applications

6.12 Exercises

7: Fractional durations

7.1 Introduction

7.2 Cashflows Discounted with Interest Only

7.3 Life Annuities Paid mthly

7.4 Immediate Annuities

7.5 Approximation and Computation

*7.6 Fractional Period Premiums and Reserves

7.7 Reserves at Fractional Durations

7.8 Notes and References

7.9 Exercises

8: Continuous payments

8.1 Introduction to Continuous Annuities

8.2 The Force of Discount

8.3 The Constant Interest Case

8.4 Continuous Life Annuities

8.5 The Force of Mortality

8.6 Insurances Payable at the Moment of Death

8.7 Premiums and Reserves

8.8 The General Insurance–Annuity Identity in the Continuous Case

8.9 Differential Equations for Reserves

8.10 Some Examples of Exact Calculation

8.11 Standard Actuarial Notation and Terminology

8.12 Notes and References

8.13 Exercises

9: Select mortality

9.1 Introduction

9.2 Select and Ultimate Tables

9.3 Changes in Formulas

9.4 Projections in Annuity Tables

9.5 Further Remarks

9.6 Exercises

10: Multiple-life contracts

10.1 Introduction

10.2 The Joint-Life Status

10.3 Joint-Life Annuities and Insurances

10.4 Last-Survivor Annuities and Insurances

10.5 Moment of Death Insurances

10.6 The General Two-Life Annuity Contract

10.7 The General Two-Life Insurance Contract

10.8 Contingent Insurances

10.9 Duration Problems

10.10 Applications to Annuity Credit Risk

10.11 Standard Notation and Terminology

10.12 Spreadsheet Applications

10.13 Notes and References

10.14 Exercises

11: Multiple-decrement theory

11.1 Introduction

11.2 The Basic Model

11.3 Insurances

11.4 Determining the Model from the Forces of Decrement

11.5 The Analogy with Joint-Life Statuses

11.6 A Machine Analogy

11.7 Associated Single-Decrement Tables

11.8 Notes and References

11.9 Exercises

12: Expenses

12.1 Introduction

12.2 Effect on Reserves

12.3 Realistic Reserve and Balance Calculations

12.4 Notes and References

12.5 Exercises

Part II: THE STOCHASTIC MODEL

13: Survival distributions and failure times

13.1 Introduction to Survival Distributions

13.2 The Discrete Case

13.3 The Continuous Case

13.4 Examples

13.5 Shifted Distributions

13.6 The Standard Approximation

13.7 The Stochastic Life Table

13.8 Life Expectancy in the Stochastic Model

13.9 Stochastic Interest Rates

13.10 Notes and References

13.11 Exercises

14: The stochastic approach to insurance and annuities

14.1 Introduction

14.2 The Stochastic Approach to Insurance Benefits

14.3 The Stochastic Approach to Annuity Benefits

*14.4 Deferred Contracts

14.5 The Stochastic Approach to Reserves

14.6 The Stochastic Approach to Premiums

14.7 The Variance of rL

14.8 Standard Notation and Terminology

14.9 Notes and References

14.10 Exercises

15: Simplifications under level benefit contracts

15.1 Introduction

15.2 Variance Calculations in the Continuous Case

15.3 Variance Calculations in the Discrete Case

15.4 Exact Distributions

15.5 Non-Level Benefit Examples

15.6 Exercises

16: The minimum failure time

16.1 Introduction

16.2 Joint Distributions

16.3 The Distribution of T

16.4 The Joint Distribution of (T, J)

16.5 Other Problems

16.6 The Common Shock Model

16.7 Copulas

16.8 Notes and References

16.9 Exercises

Part III: RISK THEORY

17: The collective risk model

17.1 Introduction

17.2 The Mean and Variance of S

17.3 Generating Functions

17.4 Exact Distribution of S

17.5 Choosing a Frequency Distribution

17.6 Choosing a Severity Distribution

17.7 Handling the Point Mass at 0

17.8 Counting Claims of a Particular Type

17.9 The Sum of two Compound Poisson Distributions

17.10 Deductibles and Other Modifications

17.11 A Recursion Formula for S

17.12 Notes and References

17.13 Exercises

18: Risk assessment

18.1 Introduction

18.2 Utility Theory

18.3 Convex and Concave Functions: Jensen's Inequality

18.4 A General Comparison Method

18.5 Risk Measures for Capital Adequacy

18.6 Notes and References

18.7 Exercises

19: An introduction to stochastic processes

19.1 Introduction

19.2 Markov Chains

19.3 Martingales

19.4 Finite-State Markov Chains

19.5 Notes and References

19.6 Exercises

20: Poisson processes

20.1 Introduction

20.2 Definition of a Poisson Process

20.3 Waiting Times

*20.4 Some Properties of a Poisson Process

20.5 Nonhomogeneous Poisson Processes

20.6 Compound Poisson Processes

20.7 Notes and References

20.8 Exercises

21: Ruin models

21.1 Introduction

21.2 A Functional Equation Approach

21.3 The Martingale Approach to Ruin Theory

21.4 Distribution of the Deficit at Ruin

21.5 Recursion Formulas

21.6 The Compound Poisson Surplus Process

21.7 The Maximal Aggregate Loss

21.8 Notes and References

21.9 Exercises

22: Credibility theory

22.1 Introduction

22.2 Conditional Expectation and Variance

22.3 General Framework for Bayesian Credibility

22.4 Classical Examples

22.5 Approximations

22.6 Conditions for Exactness

22.7 Estimation

22.8 Notes and References

22.9 Exercises

23: Multi-state models

23.1 Introduction

23.2 The Discrete-Time Model

23.3 The Continuous-Time Model

23.4 Notes and References

23.5 Exercises

Appendix: A review of probability theory

A.1 Introduction

A.2 Sample spaces and probability measures

A.3 Conditioning and Independence

A.4 Random variables

A.5 Distributions

A.6 Expectations and moments

A.7 Expectation in terms of the distribution function

A.8 The normal distribution

A.9 Joint Distributions

A.10 Conditioning and independence for random variables

A.11 Convolution

A.12 Moment generating functions

A.13 Probability generating functions

A.14 Mixtures

Answers to exercises

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

Chapter 21

Chapter 22

Chapter 23

References

Index

Fundamentals of Actuarial Mathematics

This edition first published 2011

© 2011 John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data

Promislow, S. David.

Fundamentals of actuarial mathematics/S. David Promislow. -- 2nd ed.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-68411-5 (cloth)

1. Insurance—Mathematics. 2. Business mathematics. I. Title.

HG8781.P76 2010

368′.01—dc22

2010029552

A catalogue record for this book is available from the British Library.

Print ISBN: 978-0-470-68411-5

ePDF ISBN: 978-0-470-97784-2

ePub ISBN: 978-0-470-97807-8

To Michael, Corinne, Natalie and Ruth

Preface

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.

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. 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). 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. 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 Excel® spreadsheets are given at the end of the relevant chapters. Readers of course are free to modify these or construct their own. 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 19.4.1 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 www.wiley.com/go/actuarial for more information.

Acknowledgements

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.

Notation index

The following is a list of the major symbols which are used in the book. For the most part, the first page they appear on is listed. An exception is the notation in Appendix A, where the first appearance in that chapter is noted. This list excludes that part of the standard actuarial notation which is not used in the main body of the text. The latter can be found in the approriate sections of Chapters 2–6, 8 and 10 entitled ‘Standard Notation and Terminology’.

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 19

Chapter 21

Chapter 22

Chapter 23

Appendix A

Part I

THE DETERMINISTIC MODEL

1

Introduction and motivation

1.1 Risk and Insurance

In this book we deal with certain mathematical models. This opening chapter, however, is a nontechnical introduction, designed to provide background and motivation. In particular, we are concerned with models used by actuaries, so we might first try to describe exactly what it is that actuaries do. This can be difficult, because a typical actuary is concerned with many issues, but we can identify two major themes dealt with by this profession.

The first is risk, a word that itself can be defined in different ways. A commonly accepted definition in our context is that risk is the possibility that something bad happens. Of course, many bad things can happen, but in particular we are interested in occurrences that result in financial loss. A person dies, depriving family of earned income or business partners of expertise. Someone becomes ill, necessitating large medical expenses. A home is destroyed by fire or an automobile is damaged in an accident. No matter what precautions you take, you cannot rid yourself completely of the possibility of such unfortunate events, but what you can do is take steps to mitigate the financial loss involved. One of the most commonly used measures is to purchase insurance.

Insurance involves a sharing or pooling of risks among a large group of people. The origins go back many years and can be traced to members of a community helping out others who suffered loss in some form or other. For example, people would help out neighbours who had suffered a death or illness in the family. While such aid was in many cases no doubt due to altruistic feelings, there was also a motivation of self-interest. You would be prepared to help out a neighbour who suffered some calamity, since you or your family could similarly be aided by others when you required such assistance. This eventually became more formalized, giving rise to the insurance companies we know today.

With the institution of insurance companies, sharing is no longer confined to the scope of neighbors or community members one knows, but it could be among all those who chose to purchase insurance from a particular company. Although there are many different types of insurance, the basic principle is similar. A company known as the insurer agrees to pay out money, which we will refer to as benefits, at specified times, upon the occurrence of specified events causing financial loss. In return, the person purchasing insurance, known as the insured, agrees to make payments of prescribed amounts to the company. These payments are typically known as premiums. The contract between the insurer and the insured is often referred to as the insurance policy.

The risk is thereby transferred from the individuals facing the loss to the insurer. The insurer in turn reduces its risk by insuring a sufficiently large number of individuals, so that the losses can be accurately predicted. Consider the following example, which is admittedly vastly oversimplified but designed to illustrate the basic idea.

Suppose that a certain type of event is unlikely to occur but if so, causes a financial loss of 100000. The insurer estimates that about 1 out of every 100 individuals who face the possibility of such loss will actually experience it. If it insures 1000 people, it can then expect 10 losses. Based on this model, the insurer would charge each person a premium of 1000. (We are ignoring certain factors such as expenses and profits.) It would collect a total of 1000000 and have precisely enough to cover the 100000 loss for each of the 10 individuals who experience this. Each individual has eliminated his or her risk, and in so far as the estimate of 10 losses is correct, the insurer has likewise eliminated its own risk. (We comment further on this statement in the next section.)

We conclude this section with a few words on the connection between insurance and gambling. Many people believe that insurance is really a form of the latter, but in fact it is exactly the opposite. Gambling trades certainty for uncertainty. The amount of money you have in your pocket is there with certainty if you do not gamble, but it is subject to uncertainty if you decide to place a bet. On the other hand, insurance trades uncertainty for certainty. The uncertain drain on your wealth, due the possibility of a financial loss, is converted to the certainty of the much smaller drain of the premium payments if you insure against the loss.

1.2 Deterministic Versus Stochastic Models

The example in Section 1.1 illustrates what is known as a deterministic model. The insurer in effect pretends it will know exactly how much it will pay out in benefits and then charges premiums to match this amount. Of course, the insurer knows that it cannot really predict these amounts precisely. By selling a large number of policies they hope to benefit from the diversification effect. They are really relying on the statistical concept known as the ‘law of large numbers’, which in this context intutitively says that if a sufficiently large number of individuals are insured, then the total number of losses will likely be close to the predicted figure.

To look at this idea in more detail, it may help to give an analogy with flipping coins. If we flip 100 fair coins, we cannot predict exactly the number of them that will comes up heads, but we expect that most of the time this number should be close to 50. But ‘most of the time’ does not mean always. It is possible for example, that we may get only 37 heads, or as many as 63, or even more extreme outcomes. In the example given in the last section, the number of losses may well turn out to be more than the expected number of 10. We would like to know just how unlikely these rare events are. In other words, we would like to quantify more precisely just what the words ‘most of the time’ mean. To achieve this greater sophistication a stochastic model for insurance claims is needed, which will assign probabilities to the occurrence of various numbers of losses. This will allow adjustment of premiums in order to allow for the risk that the actual number of losses will deviate from that expected. We will however begin the study of actuarial mathematics by first developing a deterministic approach, as this seems to be the best way of learning the basic concepts. After mastering this, it is not difficult to turn to the more realistic stochastic setting.

We will not get into all the complications that can arise. In actual coin flipping it seems clear that the results of each toss are independent of the others. The fact that one coin comes up heads, is not going to affect the outcomes of the others. It is this independence which is behind the law of large numbers, and which results in outcomes that are usually close to what is expected. There are some risks, often referred to as systematic or nondiversifiable, where the independence assumption fails, and which can adversely affect all or a large number of members of a group at the same time. For example, a spreading epidemic could cause life or health insurers to pay more in claims than they expected. Selling more policies in order to diversify would not help their financial situation. It could in fact make it worse, if the premiums were not sufficient to cover the extra losses. Severe climatic disturbances causing storms could impact property insurance in the same way. In 2008, falling real estate prices in the United States affected mortgage lenders and those who insured mortgage lenders against bad debts, to the extent that this helped trigger a global financial crisis. A detailed discussion of these matters is not within the scope of this work, and for the most part, the stochastic model we present will confine attention to the usual insurance model where the risks are considered as independent. It should be kept in mind however that the detection and avoidance of systematic risk are matters that the actuary must always be aware of.

1.3 Finance and Investments

The second theme involved in an actuary's work is finance and investments. In most of the types of insurance that we focus on in this book, an additional complicating factor is the long-term nature of the contracts. Benefits may not be paid until several years after premiums are collected. This is certainly true in life insurance, where the loss is occasioned by the death of an individual. Premiums received are invested and the resulting earnings can be used to help provide the benefits. Consider the simple example given above, and suppose further that the benefits do not have to be paid until 1 year after the premiums are collected. If the insurer can invest the money at, say, 5% interest for the year, then it does not need to charge the full 1000 in premium, but can collect only 1000/1.05 from each person. When invested, this amount will provide the necessary 1000 to cover the losses. Again, this example is oversimplified and there are many more complications. We will, in the next chapter, consider a mathematical model that deals with the the consequences of the payments of money at various times.

1.4 Adequacy and Equity

We can now give a general description of the responsibilities of an actuary. The overriding task is to ensure that the premiums, together with investment earnings, are adequate to provide for the payment of the benefits. If this is not true, then it will not be possible for the insurer to meet its obligations and some of the insureds will necessarily not receive compensation for their losses. The challenge in meeting this goal arises from the several areas of uncertainty. The amount and timing of the benefits that will have to be paid, as well as the investment earnings, are unknown and subject to random fluctuations. The actuary makes substantial use of probabilistic methods to handle this uncertainty.

Another goal is to achieve equity in setting premiums. If an insurer is to attract purchasers, it must charge rates that are perceived as being fair. Here also, the randomness means that it is not obvious how to define equity in this context. It cannot mean that two individuals who are charged the same amount in premiums will receive exactly the same back in benefits, for that would negate the sharing arrangement inherent in the insurance idea. While there are different possible viewpoints, equity in insurance is generally expected to mean that the mathematical expectation of these two individuals should be the same.

1.5 Reassessment

Actuaries design insurance contracts and must initially calculate premiums that will fulfill the goals of adequacy and equity, but that is not the end of the story. No matter how carefully one makes an initial assessment of risks, there are too many variables to be able to achieve complete accuracy. Such assessments must be continually re-evaluated, and herein lies the real expertise of the actuary. This work may be compared with sailing a ship in a stormy sea. It is impossible to avoid being blown off course occasionally. The skill is to detect when this occurs and to take the necessary steps to continue in the right direction. This continual monitoring and reassessing is an important part of the actuary's work. A large part of this involves calculating quantities known as reserves. We introduce this concept in Chapter 2 and then develop it more fully in Chapter 6.

1.6 Conclusion

We can now summarize the material found in the subsequent chapters of the book. We will describe the mathematical models used by the actuary to ensure that an insurer will be able to meet its promised benefits payments and that the respective purchasers of its contracts are treated equitably. In Part I, we deal with a strictly deterministic model. This enables us to focus on the main principles while keeping the required mathematics reasonably simple. In Part II, we look at the stochastic model for an individual insurance contract. In Part III, we consider models that encompass an entire portfolio of insurance contracts.

2

The basic deterministic model

2.1 Cashflows

As indicated in the previous chapter, a basic application of actuarial mathematics is to model the transfer of money. Insurance companies, banks and other financial institutions engage in transactions that involve accepting sums of money at certain times, and paying out sums of money at other times.

To construct a model for describing this situation, we will first fix a time unit. This can be arbitrary, but in most applications it will be taken as some familiar interval of time. For convenience we will assume that time is measured in years, unless we indicate otherwise. We will let time 0 refer to the present time, and time will then denote time units in the future. We also select an arbitrary unit of capital. In this chapter, we assume that all funds are paid out or received at integer time points, that is, at time 0, 1, 2,…. The amount of money received or paid out at time will be called the at time and denoted by . A positive value of denotes that a sum is to be received, while a negative value indicates that a sum is paid out. The entire transaction is then described by listing the sequence of cashflows. We will refer to this as a ,