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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Provides a comprehensive coverage of both the deterministic and stochastic models of life contingencies, risk theory, credibility theory, multi-state models, and an introduction to modern mathematical finance.
- New edition restructures the material to fit into modern computational methods and provides several spreadsheet examples throughout.
- Covers the syllabus for the Institute of Actuaries subject CT5, Contingencies
- Includes new chapters covering stochastic investments returns, universal life insurance. Elements of option pricing and the Black-Scholes formula will be introduced.
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Fundamentals of Actuarial Mathematics
Third Edition
S. David Promislow
York University, Toronto, Canada
This edition first published 2015 © 2015 John Wiley & Sons, Ltd
Registered officeJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom
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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 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
To Georgia and Griffith
Contents
Preface
Acknowledgements
About the companion website
Part I THE DETERMINISTIC LIFE CONTINGENCIES 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 Cash flows
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 Vector notation
2.9 Regular pattern cash flows
2.10 Balances and reserves
2.11 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
Notes and references
Exercises
3 The life table
3.1 Basic definitions
3.2 Probabilities
3.3 Constructing the life table from the values of
q
x
3.4 Life expectancy
3.5 Choice of life tables
3.6 Standard notation and terminology
3.7 A sample table
Notes and references
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
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
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 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 Standard notation and terminology
6.10 Spreadsheet applications
Exercises
7 Fractional durations
7.1 Introduction
7.2 Cash flows discounted with interest only
7.3 Life annuities paid
m
thly
7.4 Immediate annuities
7.5 Approximation and computation
*7.6 Fractional period premiums and reserves
7.7 Reserves at fractional durations
7.8 Standard notation and terminology
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 Further approximations from the life table
8.12 Standard actuarial notation and terminology
Notes and references
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
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
Notes and references
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
Notes and references
Exercises
12 Expenses and profits
12.1 Introduction
12.2 Effect on reserves
12.3 Realistic reserve and balance calculations
12.4 Profit measurement
Notes and references
Exercises
*13 Specialized topics
13.1 Universal life
13.2 Variable annuities
13.3 Pension plans
Exercises
Part II THE STOCHASTIC LIFE CONTINGENCIES MODEL
14 Survival distributions and failure times
14.1 Introduction to survival distributions
14.2 The discrete case
14.3 The continuous case
14.4 Examples
14.5 Shifted distributions
14.6 The standard approximation
14.7 The stochastic life table
14.8 Life expectancy in the stochastic model
14.9 Stochastic interest rates
Notes and references
Exercises
15 The stochastic approach to insurance and annuities
15.1 Introduction
15.2 The stochastic approach to insurance benefits
15.3 The stochastic approach to annuity benefits
*15.4 Deferred contracts
15.5 The stochastic approach to reserves
15.6 The stochastic approach to premiums
15.7 The variance of
r
L
15.8 Standard notation and terminology
Notes and references
Exercises
16 Simplifications under level benefit contracts
16.1 Introduction
16.2 Variance calculations in the continuous case
16.3 Variance calculations in the discrete case
16.4 Exact distributions
16.5 Some non-level benefit examples
Exercises
17 The minimum failure time
17.1 Introduction
17.2 Joint distributions
17.3 The distribution of
T
17.4 The joint distribution of (
T
,
J
)
17.5 Other problems
17.6 The common shock model
17.7 Copulas
Notes and references
Exercises
Part III ADVANCED STOCHASTIC MODELS
18 An introduction to stochastic processes
18.1 Introduction
18.2 Markov chains
18.3 Martingales
18.4 Finite-state Markov chains
18.5 Introduction to continuous time processes
18.6 Poisson processes
18.7 Brownian motion
Notes and references
Exercises
19 Multi-state models
19.1 Introduction
19.2 The discrete-time model
19.3 The continuous-time model
19.4 Recursion and differential equations for multi-state reserves
19.5 Profit testing in multi-state models
19.6 Semi-Markov models
Notes and references
Exercises
20 Introduction to the Mathematics of Financial Markets
20.1 Introduction
20.2 Modelling prices in financial markets
20.3 Arbitrage
20.4 Option contracts
20.5 Option prices in the one-period binomial model
20.6 The multi-period binomial model
20.7 American options
20.8 A general financial market
20.9 Arbitrage-free condition
20.10 Existence and uniqueness of risk-neutral measures
20.11 Completeness of markets
20.12 The Black–Scholes–Merton formula
20.13 Bond markets
Notes and references
Exercises
Part IV RISK THEORY
21 Compound distributions
21.1 Introduction
21.2 The mean and variance of
S
21.3 Generating functions
21.4 Exact distribution of
S
21.5 Choosing a frequency distribution
21.6 Choosing a severity distribution
21.7 Handling the point mass at 0
21.8 Counting claims of a particular type
21.9 The sum of two compound Poisson distributions
21.10 Deductibles and other modifications
21.11 A recursion formula for
S
Notes and references
Exercises
22 Risk assessment
22.1 Introduction
22.2 Utility theory
22.3 Convex and concave functions: Jensen’s inequality
22.4 A general comparison method
22.5 Risk measures for capital adequacy
Notes and references
Exercises
23 Ruin models:
23.1 Introduction
23.2 A functional equation approach
23.3 The martingale approach to ruin theory
23.4 Distribution of the deficit at ruin
23.5 Recursion formulas
23.6 The compound Poisson surplus process
23.7 The maximal aggregate loss
Notes and references
Exercises
24 Credibility theory:
24.1 Introductory material
24.2 Conditional expectation and variance with respect to another random variable
24.3 General framework for Bayesian credibility
24.4 Classical examples
24.5 Approximations
24.6 Conditions for exactness
24.7 Estimation
Notes and References
Exercises
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
Chapter 24
Appendix A review of probability theory
A.1 Sample spaces and probability measures
A.2 Conditioning and independence
A.3 Random variables
A.4 Distributions
A.5 Expectations and moments
A.6 Expectation in terms of the distribution function
A.7 Joint distributions
A.8 Conditioning and independence for random variables
A.9 Moment generating functions
A.10 Probability generating functions
A.11 Some standard distributions
A.12 Convolution
A.13 Mixtures
References
Notation index
Index
End User License Agreement
Guide
Cover
Table of Contents
Preface
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Preface
The third edition of this book continues the objective of providing coverage of actuarial mathematics in a flexible manner that meets the needs of several audiences. These range from those who want only a basic knowledge of the subject, to those preparing for careers as professional actuaries. All this is carried out with a streamlined system of notation, and a modern approach to computation involving spreadsheets.
The text is divided into four parts. The first two cover the subject of life contingencies. The modern approach towards this subject is through a stochastic model, as opposed to the older deterministic viewpoint. I certainly agree that mastering the stochastic model is the desirable goal. 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. After the main ideas are presented in this fashion, continuous models are introduced. In Part II of the book, the full stochastic model of life contingencies can be dealt with in a reasonably quick fashion.
Another innovation in Part II is to depart from the conventional treatment of life contingencies as dealing essentially with patterns of mortality or disability in a group of human lives. Throughout Part II, we deal with general failure times which makes the theory more widely adaptable.
Part III deals with more advanced stochastic models. Following an introduction to stochastic processes, there is a chapter covering multi-state theory, an approach which unifies many of the ideas in Parts I and II. The final chapter in Part III is an introduction to modern financial mathematics.
Part IV deals with the subject of risk theory, sometime referred to as loss models. It includes an extensive coverage of classical ruin theory, a topic that originated in actuarial science but recently has found many applications in financial economics. It also includes credibility theory, which will appeal to the reader interested more in the casualty side of actuarial mathematics.
This book will meet the needs of those preparing for the examinations of many of the major professional actuarial organizations. Parts I to III of this new third edition covers all of the material on the current syllabuses of Exam MLC of the Society of Actuaries and Canadian Institute of Actuaries and Exam LC of the Casualty Actuarial Society, and covers most of the topics on the current syllabus of Exam CT5 of the British Institute of Actuaries. In addition, Part IV of the book covers a great deal of the material on Exam C of the Society of Actuaries and Canadian Institute of Actuaries, including the topics of Frequency, Severity and Aggregate Models, Risk Measures, and Credibility Theory.
The mathematical prerequisites for Part 1 are relatively modest. comprising 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, and this material summarized in Appendix A. A usual prerequisite for actuarial mathematics 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 allow 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.
Keeping in mind the nature of the book and its intended audience, we have avoided excessive mathematical rigour. 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 asks 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 *, as are a few other exercises which are of above average difficulty.
There are various ways of using the text for university courses geared to third or fourth year undergraduates, or beginning graduate students. Chapters 1 to 8 could form the basis of a one-semester introductory course. Part IV is for the most part independent of the first three parts, except for the background material on stochastic processes given in Chapter 18 and would constitute another one-semester course. The rest of the book constitutes roughly another two semesters worth of material, with possibly some omissions; Chapter 13 is not needed for the rest of the book. Chapters 7 (except for Section 7.3.1), 9 and 12 deal with topics that are important in applications, but which are used minimally in other parts of the text. They could be omitted without loss of continuity.
CHANGES IN THE THIRD EDITION
There are several additions and changes to the third edition.
The most notable is a new Chapter 20 providing an introduction to the mathematics of financial markets. It has been long recognized that knowledge of this subject is essential to the management of financial risk that faces the actuary of today.
Other additions include the following:
Chapter 12, on expenses, has been considerably enlarged to include the topic of profit testing.
The chapter on multi-state models has been expanded to include discussion of reserves and profit testing in such models, as well as several additional techniques for continuous-time problems.
Some extra numerical procedures have been included, such as Euler’s method for differential equations, and the three-term Woolhouse formulas for fractional annuity approximations.
An introduction to Brownian motion has been added to the material on continuous-time stochastic processes.
The previous material on universal life and variable annuities has been rewritten and included in a new chapter dealing with miscellaneous topics. A brief discussion of pension plans is included here as well.
Additional examples, exercises, and clarification have been added to various chapters.
As well as the changes there has been a reorganization in the material The previous two chapters on stochastic processes have been combined into one and now appear earlier in the book as background for the multi-state and financial markets chapters. In the current Part IV, the detailed descriptions of the various distributions have been removed and added as a section to the Appendix on probability theory.
Acknowledgements
Several individuals have assisted in the various editions of this book. I am particularly indebted to two people who have made a significant contribution by providing a number of helpful comments, corrections, and suggestions. They are Virginia Young for her work on the first edition, and Elias Shiu for his help with the third edition.
There are many others who deserve thanks. 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. Several 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. Christian Hess asked some questions which led to the inclusion of Example 21.10 to clear up an ambiguous point. Exercise 18.13 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. Thanks go to the editorial and production teams at Wiley for their much appreciated assistance. Finally, I thank my wife Shirley who provided support and encouragement throughout the writing of all three editions.
About the companion website
This book is accompanied by a companion website:
www.wiley.com/go/promislow/actuarial
The website includes:
A variety of exercises, both computational and theoretical
Answers, enabling use for self-study.
Part ITHE DETERMINISTIC LIFE CONTINGENCIES MODEL
1Introduction 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 should 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 neighbours 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 100 000. 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 1 000 000 and have precisely enough to cover the 100 000 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 to 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 intuitively 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 come 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 non-diversifiable, 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 consequences of the payments of money at various times. A much more elaborate treatment of financial matters, incorporating randomness, is presented in Chapter 20.
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 this 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 to 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 look at more advanced stochastic models and introduce the mathematics of financial markets. In Part IV, we consider models that encompass an entire portfolio of insurance contracts.
