# An Introduction to the Mathematics of Finance

## 2nd Edition

### A Deterministic Approach

**Authors:**Stephen Garrett

**eBook ISBN:**9780080982755

**Hardcover ISBN:**9780080982403

**Paperback ISBN:**9780081013021

**Imprint:**Butterworth-Heinemann

**Published Date:**19th June 2013

**Page Count:**464

## Description

*An Introduction to the Mathematics of Finance: A Deterministic Approach, 2e,* offers a highly illustrated introduction to mathematical finance, with a special emphasis on interest rates. This revision of the McCutcheon-Scott classic follows the core subjects covered by the first professional exam required of UK actuaries, the CT1 exam. It realigns the table of contents with the CT1 exam and includes sample questions from past exams of both The Actuarial Profession and the CFA Institute. With a wealth of solved problems and interesting applications, *An Introduction to the Mathematics of Finance* stands alone in* *its ability to address the needs of its primary target audience, the actuarial student.

## Key Features

- Closely follows the syllabus for the CT1 exam of The Institute and Faculty of Actuaries
- Features new content and more examples
- Online supplements available: http://booksite.elsevier.com/9780080982403/
- Includes past exam questions from The Institute and Faculty of Actuaries and the CFA Institute

## Readership

The primary audience is upper-division undergraduates and graduate students in the UK studying financial mathematics. They will typically be taking courses such as "Principles of Finance" and "Actuarial Finance." The secondary audience is upper-division undergraduates and graduate students studying financial mathematics.

## Table of Contents

Dedication

Preface

Chapter 1. Introduction

1.1 The Concept of *Interest*

1.2 Simple Interest

1.3 Compound Interest

1.4 Some Practical Illustrations

Summary

Chapter 2. Theory of Interest Rates

2.1 The Rate of Interest

2.2 Nominal Rates of Interest

2.3 Accumulation Factors

2.4 The Force of Interest

2.5 Present Values

2.6 Present Values of Cash Flows

2.7 Valuing Cash Flows

2.8 Interest Income

2.9 Capital Gains and Losses, and Taxation

Summary

Exercises

Chapter 3. The Basic Compound Interest Functions

3.1 Interest Rate Quantities

3.2 The Equation of Value

3.3 Annuities-certain: Present Values and Accumulations

3.4 Deferred Annuities

3.5 Continuously Payable Annuities

3.6 Varying Annuities

3.7 Uncertain Payments

Summary

Exercises

Chapter 4. Further Compound Interest Functions

4.1 Interest Payable *p*thly

4.2 Annuities Payable *p*thly: Present Values and Accumulations

4.3 Annuities Payable at Intervals of Time *r*, Where *r* > 1

4.4 Definition of for Non-integer Values of *n*

Summary

Exercises

Chapter 5. Loan Repayment Schedules

5.1 The General Loan Schedule

5.2 The Loan Schedule for a Level Annuity

5.3 The Loan Schedule for a *p*thly Annuity

5.4 Consumer Credit Legislation

Summary

Exercises

Chapter 6. Project Appraisal and Investment Performance

6.1 Net Cash Flows

6.2 Net Present Values and Yields

6.3 The Comparison of Two Investment Projects

6.4 Different Interest Rates for Lending and Borrowing

6.5 Payback Periods

6.6 The Effects of Inflation

6.7 Measurement of Investment Fund Performance

Summary

Exercises

Chapter 7. The Valuation of Securities

7.1 Fixed-Interest Securities

7.2 Related Assets

7.3 Prices and Yields

7.4 Perpetuities

7.5 Makeham’s Formula

7.6 The Effect of the Term to Redemption on the Yield

7.7 Optional Redemption Dates

7.8 Valuation between Two Interest Dates: More Complicated Examples

7.9 Real Returns and Index-linked Stocks

Summary

Exercises

Chapter 8. Capital Gains Tax

8.1 Valuing a Loan with Allowance for Capital Gains Tax

8.2 Capital Gains Tax When the Redemption Price or the Rate of Tax Is Not Constant

8.3 Finding the Yield When There Is Capital Gains Tax

8.4 Optional Redemption Dates

8.5 Offsetting Capital Losses Against Capital Gains

Summary

Exercises

Chapter 9. Term Structures and Immunization

9.1 Spot and Forward Rates

9.2 Theories of the Term Structure of Interest Rates

9.3 The Discounted Mean Term of a Project

9.4 Volatility

9.5 The Volatility of Particular Fixed-interest Securities

9.6 The Matching of Assets and Liabilities

9.7 Redington’s Theory of Immunization

9.8 Full Immunization

Summary

Exercises

Chapter 10. An Introduction to Derivative Pricing: Forwards and Futures

10.1 Futures Contracts

10.2 Margins and Clearinghouses

10.3 Uses of Futures

10.4 Forwards

10.5 Arbitrage

10.6 Calculating the Forward Price

10.7 Calculating the Value of a Forward Contract Prior to Maturity

10.8 Eliminating the Risk to the Short Position

Summary

Exercises

Chapter 11. Further Derivatives: Swaps and Options

11.1 Swaps

11.2 Options

11.3 Option Payoff and Profit

11.4 An Introduction to European Option Pricing

11.5 The Black–Scholes Model

11.6 Trading Strategies Involving European Options

Summary

Exercises

Chapter 12. An Introduction to Stochastic Interest Rate Models

12.1 Introductory Examples

12.2 Independent Annual Rates of Return

12.3 The Log-Normal Distribution

12.4 Simulation Techniques

12.5 Random Number Generation

12.6 Dependent Annual Rates of Return

12.7 An Introduction to the Application of Brownian Motion

Summary

Exercises

Appendix 1. Theorem Proofs

Proof of Theorem 2.4.1

Proof of Theorem 3.2.1

Appendix 2. The Solution of Non-linear Equations

Bisection Method

Secant Method (Also Known as the Modified Regula Falsi Method)

Newton–Raphson Method

Exercises

Solutions

Appendix 3. Solutions to Exercises

Chapter 2 Exercises

Chapter 3 Exercises

Chapter 4 Exercises

Chapter 5 Exercises

Chapter 6 Exercises

Chapter 7 Exercises

Chapter 8 Exercises

Chapter 9 Exercises

Chapter 10 Exercises

Chapter 11 Exercises

Chapter 12 Exercises

Appendix 4. Compound Interest Tables

Additional Reading

Index

## Details

- No. of pages:
- 464

- Language:
- English

- Copyright:
- © Butterworth-Heinemann 2013

- Published:
- 19th June 2013

- Imprint:
- Butterworth-Heinemann

- eBook ISBN:
- 9780080982755

- Hardcover ISBN:
- 9780080982403

- Paperback ISBN:
- 9780081013021

## About the Author

### Stephen Garrett

Prof. Stephen Garrett is Professor of Mathematical Sciences at the University of Leicester in the UK. He is currently Head of Actuarial Science in the Department of Mathematics, and also Head of the Thermofluids Research Group in the Department of Engineering. These two distinct responsibilities reflect his background and achievements in both actuarial science education and fluid mechanics research. Stephen is a Fellow of the Royal Aeronautical Society, the highest grade attainable in the world's foremost aerospace institution.

### Affiliations and Expertise

Professor of Mathematical Sciences, University of Leicester, UK

## Reviews

"The main focus is the theory of compound interest, which is called deterministic financial mathematics by the author...well written and the materials therein are well organized." --**Zentralblatt MATH, An Introduction to the Mathematics of Finance **

"…it offers some very clear explanations of the fundamental building blocks of actuarial work, such as compound interest functions, term structures and discounting…The use of examples and exam questions from the IFoA and the CFA Institute makes this a very valuable study aide…as a primer, it is certainly a success and one which I hope is used by a great many students in the future." --**Annals of Actuarial Science, 2014**

"Stephen Garrett’s new edition of Introduction to the Mathematics of Finance gives an excellent, concise, and thorough treatment of the fundamentals of financial mathematics. By updating the original edition with more emphasis on derivative pricing, this book has become an up-to-date first class textbook on this topic." **--P.M. Barrieu, London School of Economics**

"This 2^{nd} edition will give students excellent support when tackling the Actuarial Profession’s examination in Subject CT1. It is written in a clear and concise way, and a wide range of realistic and relevant examples are provided which make the subject come alive. I will be recommending it to my students."** --Ben Rickayzen, Cass Business School **

"This edition is a timely update to a textbook that for many years was essential reading for actuarial students. It should prove to be a valuable resource for current students taking the CT1 exam." **--John Millett, University of Kent **

"This book contains a set of subjects that will be very close to most actuaries' hearts, being a text book aimed at covering the CT1 syllabus. As an update to McCutcheon and Scott's 1989 An Introduction to the Mathematics of Finance, it offers some very clear explanations of the fundamental building blocks of actuarial work, such as compound interest functions, term structures, and discounting. As a text for the beginner, this book is perfect....The use of examples and exam questions from both the IFoA and the CFA Institute makes this a very valuable study aide. The fact that the solutions to the large number of exercise questions are also given further increases its usefulness as a primary textbook." **--Annals of Actuarial Science 8:1, 2014**