Introduction to the mathematics of finance 2ed 2013

An Introduction to the Mathematics of Finance A Deterministic Approach Second Edition S J Garrett Published for the Institute and Faculty of Actuaries (RC000243) http://www.actuaries.org.uk Amsterdam  Boston  Heidelberg  London New York  Oxford  Paris  San Diego San Francisco  Singapore  Sydney  Tokyo Butterworth-Heinemann is an Imprint of Elsevier Butterworth-Heinemann is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB 225 Wyman Street, Waltham, MA 02451, USA First edition 1989 Second edition 2013 Copyright Ó 2013 Institute and Faculty of Actuaries (RC000243) Published by Elsevier Ltd All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangement with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein) Notices Knowledge and best practice in this field are constantly changing As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-08-098240-3 For information on all Butterworth-Heinemann publications visit our website at store.elsevier.com Printed and bound in the United Kingdom 13 14 15 16 10 Dedication Dedicated to Adam and Matthew Garrett, my two greatest achievements Preface This book is a revision of the original An Introduction to the Mathematics of Finance by J.J McCutcheon and W.F Scott The subject of financial mathematics has expanded immensely since the publication of that first edition in the 1980s, and the aim of this second edition is to update the content for the modern audience Despite the recent advances in stochastic models within financial mathematics, the book remains concerned almost entirely with deterministic approaches The reason for this is twofold Firstly, many readers will find a solid understanding of deterministic methods within the classical theory of compound interest entirely sufficient for their needs This group of readers is likely to include economists, accountants, and general business practitioners Secondly, readers intending to study towards an advanced understanding of financial mathematics need to start with the fundamental concept of compound interest Such readers should treat this as an introductory text Care has been taken to point towards areas where stochastic concepts will likely be developed in later studies; indeed, Chapters 10, 11, and 12 are intended as an introduction to the fundamentals and application of modern financial mathematics in the broader sense The book is primarily aimed at readers who are preparing for university or professional examinations The material presented here now covers the entire CT1 syllabus of the Institute and Faculty of Actuaries (as at 2013) and also some material relevant to the CT8 and ST5 syllabuses This combination of material corresponds to the FM-Financial Mathematics syllabus of the Society of Actuaries Furthermore, students of the CFA Institute will find this book useful in support of various aspects of their studies With exam preparation in mind, this second edition includes many past examination questions from the Institute and Faculty of Actuaries and the CFA Institute, with worked solutions The book is necessarily mathematical, but I hope not too mathematical It is expected that readers have a solid understanding of calculus, linear algebra, and probability, but to a level no higher than would be expected from a strong first year undergraduate in a numerate subject That is not to say the material is easy, xi xii Preface rather the difficulty arises from the sheer breath of application and the perhaps unfamiliar real-world contexts Where appropriate, additional material in this edition has been based on core reading material from the Institute and Faculty of Actuaries, and I am grateful to Dr Trevor Watkins for permission to use this I am also grateful to Laura Clarke and Sally Calder of the Institute and Faculty of Actuaries for their help, not least in sourcing relevant past examination questions from their archives I am also grateful to Kathleen Paoni and Dr J Scott Bentley of Elsevier for supporting me in my first venture into the world of textbooks I also wish to acknowledge the entertaining company of my good friend and colleague Dr Andrew McMullan of the University of Leicester on the numerous coffee breaks between writing This edition has benefitted hugely from comments made by undergraduate and postgraduate students enrolled on my modules An Introduction to Actuarial Mathematics and Theory of Interest at the University of Leicester in 2012 Particular mention should be given to the eagle eyes of Fern Dyer, George HodgsonAbbott, Hitesh Gohel, Prashray Khaire, Yueh-Chin Lin, Jian Li, and Jianjian Shao, who pointed out numerous typos in previous drafts Any errors that remain are of course entirely my fault This list of acknowledgements would not be complete without special mention of my wife, Yvette, who puts up with my constant working and occasional grumpiness Yvette is a constant supporter of everything I do, and I could not have done this, or indeed much else, without her Dr Stephen J Garrett Department of Mathematics, University of Leicester January 2013 CHAPTER Introduction 1.1 THE CONCEPT OF INTEREST Interest may be regarded as a reward paid by one person or organization (the borrower) for the use of an asset, referred to as capital, belonging to another person or organization (the lender) The precise conditions of any transaction will be mutually agreed For example, after a stated period of time, the capital may be returned to the lender with the interest due Alternatively, several interest payments may be made before the borrower finally returns the asset Capital and interest need not be measured in terms of the same commodity, but throughout this book, which relates primarily to problems of a financial nature, we shall assume that both are measured in the monetary units of a given currency When expressed in monetary terms, capital is also referred to as principal CONTENTS 1.1 The Concept of Interest 1.2 Simple Interest 1.3 Compound Interest 1.4 Some Practical Illustrations Summary If there is some risk of default (i.e., loss of capital or non-payment of interest), a lender would expect to be paid a higher rate of interest than would otherwise be the case; this additional interest is known as the risk premium The additional interest in such a situation may be considered as a further reward for the lender’s acceptance of the increased risk For example, a person who uses his money to finance the drilling for oil in a previously unexplored region would expect a relatively high return on his investment if the drilling is successful, but might have to accept the loss of his capital if no oil were to be found A further factor that may influence the rate of interest on any transaction is an allowance for the possible depreciation or appreciation in the value of the currency in which the transaction is carried out This factor is obviously very important in times of high inflation It is convenient to describe the operation of interest within the familiar context of a savings account, held in a bank, building society, or other similar organization An investor who had opened such an account some time ago with an initial deposit of £100, and who had made no other payments to or from the account, would expect to withdraw more than £100 if he were now to close the account Suppose, for example, that he receives £106 on closing his account An Introduction to the Mathematics of Finance http://dx.doi.org/10.1016/B978-0-08-098240-3.00001-1 Ó 2013 Institute and Faculty of Actuaries (RC000243) Published by Elsevier Ltd All rights reserved CHAPTER 1: Introduction This sum may be regarded as consisting of £100 as the return of the initial deposit and £6 as interest The interest is a payment by the bank to the investor for the use of his capital over the duration of the account The most elementary concept is that of simple interest This naturally leads to the idea of compound interest, which is much more commonly found in practice in relation to all but short-term investments Both concepts are easily described within the framework of a savings account, as described in the following sections 1.2 SIMPLE INTEREST Suppose that an investor opens a savings account, which pays simple interest at the rate of 9% per annum, with a single deposit of £100 The account will be credited with £9 of interest for each complete year the money remains on deposit If the account is closed after year, the investor will receive £109; if the account is closed after years, he will receive £118, and so on This may be summarized more generally as follows If an amount C is deposited in an account that pays simple interest at the rate of i per annum and the account is closed after n years (there being no intervening payments to or from the account), then the amount paid to the investor when the account is closed will be C1 ỵ niị (1.2.1) This payment consists of a return of the initial deposit C, together with interest of amount niC (1.2.2) In our discussion so far, we have implicitly assumed that, in each of these last two expressions, n is an integer However, the normal commercial practice in relation to fractional periods of a year is to pay interest on a pro rata basis, so that Eqs 1.2.1 and 1.2.2 may be considered as applying for all non-negative values of n Note that if the annual rate of interest is 12%, then i ¼ 0.12 per annum; if the annual rate of interest is 9%, then i ¼ 0.09 per annum; and so on Note that in the solution to Example 1.2.1, we have assumed that months and 10 months are periods of 1/2 and 10/12 of year, respectively For accounts of duration less than year, it is usual to allow for the actual number of days an account is held, so, for example, two 6-month periods are not necessarily regarded as being of equal length In this case Eq 1.2.1 becomes 1.2 Simple Interest EXAMPLE 1.2.1 Suppose that £860 is deposited in a savings account that pays simple interest at the rate of 5:375% per annum Assuming that there are no subsequent payments to or from the account, find the amount finally withdrawn if the account is closed after (a) months, (b) 10 months, (c) year in Eq 1.2.1 with C ¼ 860 and i ¼ 0.05375, we obtain the answers (a) £883.11, (b) £898.52, (c) £906.23 In each case we have given the answer to two decimal places of one pound, rounded down This is quite common in commercial practice Solution The interest rate is given as a per annum value; therefore, n must be measured in years By letting n ¼ 6/12, 10/12, and EXAMPLE 1.2.2 Calculate the price of a 30-day £2,000 treasury bill issued by the government at a simple rate of discount of 5% per annum Solution   30  0:05 ¼ £1; 991:78 £2; 000 À 365 The investor has received interest of £8.22 under this transaction By issuing the treasury bill, the government is borrowing an amount equal to the price of the bill In return, it pays £2,000 after 30 days The price is given by   mi C 1ỵ 365 (1.2.3) where m is the duration of the account, measured in days, and i is the annual rate of interest The essential feature of simple interest, as expressed algebraically by Eq 1.2.1, is that interest, once credited to an account, does not itself earn further interest This leads to inconsistencies that are avoided by the application of compound interest theory, as discussed in Section 1.3 As a result of these inconsistencies, simple interest has limited practical use, and this book will, necessarily, focus on compound interest However, an important commercial application of simple interest is simple discount, which is commonly used for short-term loan transactions, i.e., up to year Under CHAPTER 1: Introduction simple discount, the amount lent is determined by subtracting a discount from the amount due at the later date If a lender bases his short-term transactions on a simple rate of discount d, then, in return for a repayment of X after a period t (typically t < 1), he will lend X(1 À td) at the start of the period In this situation, d is also known as a rate of commercial discount 1.3 COMPOUND INTEREST Suppose now that a certain type of savings account pays simple interest at the rate of i per annum Suppose further that this rate is guaranteed to apply throughout the next years and that accounts may be opened and closed at any time Consider an investor who opens an account at the present time (t ¼ 0) with an initial deposit of C The investor may close this account after year (t ¼ 1), at which time he will withdraw C(1 ỵ i) (see Eq 1.2.1) He may then place this sum on deposit in a new account and close this second account after one further year (t ¼ 2) When this latter account is closed, the sum withdrawn (again see Eq 1.2.1) will be ẵC1 ỵ iị ỵ iị ẳ C1 ỵ iị2 ẳ C1 ỵ 2i þ i2 Þ If, however, the investor chooses not to switch accounts after year and leaves his money in the original account, on closing this account after years, he will receive C(1 ỵ 2i) Therefore, simply by switching accounts in the middle of the 2-year period, the investor will receive an additional amount i2C at the end of the period This extra payment is, of course, equal to i(iC) and arises as interest paid (at t ¼ 2) on the interest credited to the original account at the end of the first year From a practical viewpoint, it would be difficult to prevent an investor switching accounts in the manner described here (or with even greater frequency) Furthermore, the investor, having closed his second account after year, could then deposit the entire amount withdrawn in yet another account Any bank would find it administratively very inconvenient to have to keep opening and closing accounts in the manner just described Moreover, on closing one account, the investor might choose to deposit his money elsewhere Therefore, partly to encourage long-term investment and partly for other practical reasons, it is common commercial practice (at least in relation to investments of duration greater than year) to pay compound interest on savings accounts Moreover, the concepts of compound interest are used in the assessment and evaluation of investments as discussed throughout this book The essential feature of compound interest is that interest itself earns interest The operation of compound interest may be described as follows: consider a savings account, which pays compound interest at rate i per annum, into 1.3 Compound Interest which is placed an initial deposit C at time t ¼ (We assume that there are no further payments to or from the account.) If the account is closed after year (t ¼ 1) the investor will receive C(1 ỵ i) More generally, let An be the amount that will be received by the investor if he closes the account after n years (t ¼ n) It is clear that A1 ẳ C(1 ỵ i) By denition, the amount received by the investor on closing the account at the end of any year is equal to the amount he would have received if he had closed the account year previously plus further interest of i times this amount The interest credited to the account up to the start of the final year itself earns interest (at rate i per annum) over the final year Expressed algebraically, this denition becomes Anỵ1 ẳ An ỵ iAn or Anỵ1 ẳ ỵ iịAn n!1 (1.3.1) Since, by denition, A1 ẳ C(1 ỵ i), Eq.1.3.1 implies that, for n ẳ 1, 2, , An ẳ C1 ỵ iÞn (1.3.2) Therefore, if the investor closes the account after n years, he will receive C1 ỵ iịn (1.3.3) This payment consists of a return of the initial deposit C, together with accumulated interest (i.e., interest which, if n > 1, has itself earned further interest) of amount Cẵ1 ỵ iÞn À 1Š (1.3.4) In our discussion so far, we have assumed that in both these last expressions n is an integer However, in Chapter we will widen the discussion and show that, under very general conditions, Eqs 1.3.3 and 1.3.4 remain valid for all nonnegative values of n Since ẵC1 ỵ iịt1 ỵ iịt2 ẳ C1 ỵ iịt1 ỵt2 an investor who is able to switch his money between two accounts, both of which pay compound interest at the same rate, is not able to profit by such action This is in contrast with the somewhat anomalous situation, described at the beginning of this section, which may occur if simple interest is paid Equations 1.3.3 and 1.3.4 should be compared with the corresponding expressions under the operation of simple interest (i.e., Eqs 1.2.1 and 1.2.2) If interest compounds (i.e., earns further interest), the effect on the accumulation of an account can be very significant, especially if the duration of the account or ... methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility To the fullest extent of the law, neither the Publisher... original An Introduction to the Mathematics of Finance by J.J McCutcheon and W.F Scott The subject of financial mathematics has expanded immensely since the publication of that first edition in the 1980s,... rtịdt a The rate of payment at any time is therefore simply the derivative of the total amount paid up to that time, and the total amount paid between any two times is the integral of the rate of