Actuarial Mathematics I - Complete Lecture Notes PDF

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Lecture NotesMTH5124: Actuarial Mathematics IDr Adrian Baule School of Mathematical Sciences Queen Mary University of LondonNovember 27, 2020 1 Investment project appraisal 1.10 Payback periods 1.10 Yield 1 Immunisation of cash flows 2 Fixed Interest Securities and Other Investments 2 Fixed Interest...

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Lecture Notes MTH5124: Actuarial Mathematics I Dr Adrian Baule School of Mathematical Sciences Queen Mary University of London November 27, 2020

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Contents 0 Prologue 0.1 What is an actuary? 0.2 About this course . 0.3 About these notes . 0.4 Life Tables . . . . . 0.5 Books and tables . 0.6 Acknowledgements

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1 Compound interest 1.1 Two types of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Simple interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Compound interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Nominal and effective interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Accumulation factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Nominal interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Effective interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Force of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Time-dependent interest rates . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Force of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Special case of constant force of interest . . . . . . . . . . . . . . . . . . . . 1.4 Rates of discount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Relation between nominal rates of discount and interest . . . . . . . . . . . . 1.5 Discounting Cash Flows or Present Values . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Discrete cash flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Annuities-certain: introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Immediate annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Annuity-due . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Perpetuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4 Deferred annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.5 Increasing annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Annuities-certain: more variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Annuities payable p-thly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Annuities payable continuously . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.3 Accumulated values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Continuous cash flows (not examinable) . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Continuous cash flow with variable force of interest . . . . . . . . . . . . . . 1.9 Repayment of Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Schedule of payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Consolidating loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.10 Investment project appraisal . 1.10.1 Payback periods . . . 1.10.2 Yield . . . . . . . . . 1.11 Immunisation of cash flows .

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2 Fixed Interest Securities and Other Investments 2.1 Fixed Interest Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Bond Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Example of a Government Security . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Types of Corporate Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Valuation of a Bond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Cash including Treasury Bills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Inflation Linked Bonds and Real Returns . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Calculating the Real Rate of Return . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Valuation of Index-Linked Bonds . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Equities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Valuation of Shares by Discounting Future Dividends . . . . . . . . . . . . . 2.4.2 Characteristics of Preference Shares . . . . . . . . . . . . . . . . . . . . . . 2.5 Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Taxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Income Tax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Tax on Capital Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The term structure of interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Spot rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Forward rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52 52 53 53 54 55 56 57 59 60 61 62 63 64 64 65 66 67

3 Life tables and life-table functions 3.1 Lifetime as a random variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Future lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Basic life-table functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Definitions of life-table function . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Relation between lx and s(x) . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Basic life-table functions in terms of lx . . . . . . . . . . . . . . . . . . . . 3.3 Force of mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Relation between µ(x) and other functions . . . . . . . . . . . . . . . . . . 3.3.2 The curve of deaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Analytical laws of mortality (not examinable) . . . . . . . . . . . . . . . . . . . . . 3.5 The expectation of life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 The complete expectation of life . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 The curtate expectation of life . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Interpolation for fractional ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Linear interpolation on s(x) and lx . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Derivation of relation between complete and curtate expectations of life . . . 3.6.3 Other interpolation schemes on s(x) and lx . . . . . . . . . . . . . . . . . . 3.6.4 Assumptions to obtain µ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Select mortality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 70 71 71 73 74 76 77 78 78 79 79 80 82 83 85 86 86 88

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4 Life insurance and related functions 95 4.1 Introduction to life assurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.1.1 Commutation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.2 Whole-life assurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2.1 Death benefit payable at instant of death . . . . . . . . . . . . . . . . . . . 97 4.2.2 Death benefit payable at end of year of death . . . . . . . . . . . . . . . . . 99 4.3 Whole-life annuities payable annually . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3.1 Whole-life annuity-due . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.3.2 Whole-life immediate annuity . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.3.3 Life assurance premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.4 Policies of duration n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.4.1 n-year life assurance for life aged x . . . . . . . . . . . . . . . . . . . . . . 107 4.4.2 n-year pure endowment for life aged x . . . . . . . . . . . . . . . . . . . . . 108 4.4.3 n-year endowment policy for life aged x . . . . . . . . . . . . . . . . . . . . 110 4.4.4 n-year life annuities for life aged x . . . . . . . . . . . . . . . . . . . . . . . 110 4.5 p-thly payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.5.1 Whole-life assurance paid p-thly . . . . . . . . . . . . . . . . . . . . . . . . 112 4.5.2 Whole-life p-thly annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.5.3 Alternative approximation for Ax(p) . . . . . . . . . . . . . . . . . . . . . . . 116

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Chapter 0

Prologue 0.1

What is an actuary?

“An actuary is someone who expects everyone to be dead on time.” is one definition you can find on the internet.1 In fact, actuaries do much more than deal with the probabilities of people dying! In short, they use financial and statistical theories to quantify and manage risk in all areas of business. Traditionally actuaries specialize in consultancy, investment, life and general insurance or pensions. However, their analytical skills are also increasingly valuable in other areas—after all, “risk management” is of wide importance in financially turbulent times. Actuaries are highly-regarded and (well-paid!) professionals. For more details of career paths, etc., see http://www.actuaries.org.uk/becoming-actuary/.

0.2

About this course

The idea of this course is to introduce you to some of the basic mathematical ideas used in actuarial work. MTH5124 is designed for second or third year undergraduates and assumes a background in basic Calculus and Probability.2 All practicalities about the course itself (timetable, coursework, assessment details), etc., can be found on the course QM+ website http://qmplus.qmul.ac.uk/course/view.php?id=767. Important announcements and corrections will also be posted there. The course uses familiar mathematical concepts (e.g., geometric series, probability distributions) but in an unfamiliar context. I hope you will be interested to see how maths you already know can be used effectively in the “real world”. However, one of the problems in applying maths to a different field is that you often have to learn new terminology, notation, etc. in order to communicate with specialists in that area. This is certainly the case here; indeed you will soon be introduced to a whole zoo of complicated actuarial symbols and new vocabulary. It is crucial for success that you learn this “new language” so that the familiar mathematical objects do not become lost in the fog of unfamiliar actuarial terminology. The course has four main “chapters”: 1. Compound interest: Here we will cover various (interrelated) ways to quantify how compound interest is added to a loan/investment. You will learn how to calculate accumulations given the 1

A quick search with Google yields a variety of other humorous, and not-so-humorous, definitions as well as more useful career descriptions. 2 Formal prerequisites are Calculus II and Introduction to Probability; contact the lecturer if you have difficulty meeting these.

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force of interest (or related parameters) and how to deal with series of payments, in particular annuities-certain and perpetuities. 2. Fixed interest securities and other investments: You will learn about the most common forms of investments and how to value investments using compound interest. 3. Life tables and life-table functions: Life tables are the actuary’s basic tool. You will learn how to interpret them in order to find various probabilities related to life and death. 4. Life insurance and related functions: Here we will combine material from the first two chapters to deal with situations involving payments (with compound interest) whose value and/or timing may depend on a person’s survival or death! Life insurance is the classic example here.

0.3

About these notes

These notes will cover the material in roughly the same order as the lectures but their style will probably be slightly different. In particular, the printed notes may contain some longer explanations and extra examples which time prevents me covering in class. The lectures, however will be important for emphasizing the main points and giving exam tips—I strongly recommend that you attend or at least follow the recordings on QM+! To help you with revision, all important actuarial terms are printed in bold. You must be sure to understand what these mean, both in everyday language (could you explain them to your grandmother or your next-door neighbour?) and in terms of the associated mathematical formulation. The notes will also contain a number of “examples” and “exercises”. For the former you will find full details of the working; for the latter (usually) only the answers. A good way to check your own understanding would be to read the text and try the associated exercises (contact me if you have any difficulty getting the stated answers). I intend the unstarred exercises to correspond roughly to the Key Objectives for the course (i.e., everyone should be able to do them); starred exercises will be somewhat harder and should be attempted by those aiming for a high grade.

0.4

Life Tables

The examples in this document are based on the following life tables: • English Life Table No 17, • AMC00 mortality table for assured lives. These are available on the QM+ page for MTH5124. Copies will be provided in the examination. Full details of mortality tables published by the CMI can be found at https://www.actuaries.org.uk/learn-and-develop/continuous-mortality-investigation/cmi-mortality-andmorbidity-tables. The CMI is a subsidiary of the Institute and Faculty of Actuaries, and was previously known as The Continuous Mortality Investigation. CMI tables relate to the mortality experience of life insurance policyholders and the members of pension schemes. The English Life Tables represent the mortality experience of the population of England and Wales. Tables are published by the Office of National Statistics (ONS); further information on ELT17 can be found at 8

https://www.ons.gov.uk/peoplepopulationandcommunity/birthsdeathsandmarriages/lifeexpectancies/bulletins/englishlifetablesno17/2015-09-01.

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Books and tables

The course is designed to be fairly self-contained and does not follow any one textbook. However, you may find the following useful for background reading: • S. J. Garrett An Introduction to the Mathematics of Finance (Butterworth-Heinemann); – cited in these notes as [Gar13]. • J. J. McCutcheon & W. F. Scott An Introduction to the Mathematics of Finance (ButterworthHeinemann); – cited in these notes as [MS86]. • D. C. M. Dickson, Mary R. Hardy & Howard R. Waters Actuarial Mathematics for Life Contingent Risks (Cambridge University Press); – cited in these notes as [DHW13]. • A. Neill Life Contingencies (Heinemann); – cited in these notes as [Nei77]. • N. L. Bowers, H. U. Gerber, J. C. Hickman, D. A. Jones and C. J. Nesbitt Actuarial Mathematics (Society of Actuaries); – cited in these notes as [BGH+ 97].

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Acknowledgements

These notes have been produced by Jim Webber from notes for the predecessor module MTH6100 Actuarial Mathematics, a module which covered a very similar syllabus. Great credit and thanks to Dr Rosemary Harris for her excellent work in producing the original draft of the MTH6100 notes. Also thanks to Dr D. Stark and Dr W. Just for their additions and amendments to the notes since the original draft. In her original introduction, Dr Harris gave credit to the previous notes of Prof. B. Khoruzhenko and also the work of another previous lecturer, Dr L. Rass. The changes I have made have been limited, but I take full responsibility for this edition of the notes and the mistakes and contradictions that students will inevitably find. Please alert me to any mistake that you find by sending an email to: [email protected].

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Bibliography [Ber89]

J. Bernoulli. Tractatus de seriebus infinitis. manuscript, 1689.

[BGH+ 97] N. L. Bowers, H. U. Gerber, J. C. Hickman, D. A. Jones, and C. J. Nesbitt. Actuarial Mathematics. Society of Actuaries, Schaumburg, 1997. [DHW13] D. C. M. Dickson, M. R. Hardy, and H. R. Waters. Actuarial Mathematics for Life Contingent Risks. Cambridge University Press, Cambridge, 2013. [Gar13]

S. J. Garrett. An Introduction to the Mathematics of Finance. Butterworth-Heinemann, Oxford, 2013.

[Hal93]

E. Halley. An estimate of the degrees of mortality of mankind, drawn from the curious tables of the births and funerals at the city of Breslaw, with an attempt to ascertain the price of annuities upon lives. Philosophical Transactions, 17:596–610, 1693.

[MS86]

J. J. McCutcheon and W. F. Scott. An Introduction to the Mathematics of Finance. Butterworth-Heinemann, Oxford, 1986.

[Nei77]

A. Neill. Life Contingencies. Heinemann, London, 1977.

[Pac94]

L. Pacioli. Summa de Arithmetica. Venice, 1494.

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Chapter 1

Compound interest The material in this Chapter is covered very well in ([Gar13]). It is also covered in Chapters 1–4 of ([MS86].)

1.1

Two types of interest

Often in the course of daily life and business, people need (or choose) to borrow money. For example, you might have a student loan or, later in life, need a mortgage or business loan. On the other hand, if you happen to have “spare” money you can lend it to a bank, for example, by investing in a savings account or fixed term bond. In general, in all these situations the money lender receives a kind of “reward” for lending the money; you can also think of this as the charging of “rent” for the use of the money. To be more specific the original loan/investment is called the capital (or principal) and the “reward” to the lender/investor is the interest. The time-dependent value of the investment, i.e., the original loan plus the interest, is known as the accumulated amount (or accumulation). Interest is expressed as a rate in two senses: per unit capital and per unit time. In practice, the interest rate is often quoted in percent and usually, but not always the basic time unit is one year. Note that “p.a.” is often used as an abbreviation for per annum (i.e., each year). To avoid confusion you should always state the basic time period when giving an interest rate. The interest rate on a transaction is affected by various factors including: • the market rate for similar loans; • the risk involved in the use to which the borrower puts the money (cf. mor...