Mathematics of banking and finance

The Mathematics of Banking and Finance Dennis Cox and Michael Cox The Mathematics of Banking and Finance For other titles in the Wiley Finance Series please see www.wiley.com/finance The Mathematics of Banking and Finance Dennis Cox and Michael Cox Copyright C 2006 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wiley.com All Rights Reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to (+44) 1243 770620 Designations used by companies to distinguish their products are often claimed as trademarks All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners The Publisher is not associated with any product or vendor mentioned in this book This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the Publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Library of Congress Cataloging-in-Publication Data Cox, Dennis W The mathematics of banking and finance / Dennis Cox and Michael Cox p cm ISBN-13: 978-0-470-01489-9 ISBN-10: 0-470-01489-X Business mathematics Banks and banking—Mathematics I Cox, Michael II Title HF5691.M335 2006 332.101 513—dc22 2006001400 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 13 978-0-470-01489-9 (HB) ISBN 10 0-470-01489-X (HB) Typeset in 10/12pt Times by TechBooks, New Delhi, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production Contents Introduction Introduction to How to Display Data and the Scatter Plot 1.1 Introduction 1.2 Scatter Plots 1.3 Data Identification 1.3.1 An example of salary against age 1.4 Why Draw a Scatter Plot? 1.5 Matrix Plots 1.5.1 An example of salary against age: Revisited xiii 1 2 Bar Charts 2.1 Introduction 2.2 Discrete Data 2.3 Relative Frequencies 2.4 Pie Charts 7 12 Histograms 3.1 Continuous Variables 3.2 Cumulative Frequency Polygon 3.3 Sturges’ Formula 13 13 14 20 Probability Theory 4.1 Introduction 4.2 Basic Probability Concepts 4.3 Estimation of Probabilities 4.4 Exclusive Events 4.5 Independent Events 4.6 Comparison of Exclusivity and Independence 4.7 Venn Diagrams 4.8 The Addition Rule for Probabilities 4.8.1 A simple probability example using a Venn diagram 4.9 Conditional Probability 4.9.1 An example of conditional probability 21 21 21 22 22 22 23 23 24 25 25 26 vi Contents 4.10 The Multiplication Rule for Probabilities 4.10.1 A classical example of conditional probability 4.11 Bayes’ Theorem 4.11.1 An example of Bayes’ theorem 4.11.2 Bayes’ theorem in action for more groups 4.11.3 Bayes’ theorem applied to insurance 4.12 Tree Diagram 4.12.1 An example of prediction of success 4.12.2 An example from an American game show: The Monty Hall Problem 4.13 Conclusion 26 27 27 28 29 29 30 30 34 35 Standard Terms in Statistics 5.1 Introduction 5.2 Maximum and Minimum 5.2.1 Mean 5.2.2 Median 5.2.3 Mode 5.3 Upper and Lower Quartile 5.4 MQMQM Plot 5.5 Skewness 5.6 Variance and Standard Deviation 5.7 Measures for Continuous Data 37 37 37 37 38 39 39 40 41 41 44 Sampling 6.1 Introduction 6.2 Planning Data Collection 6.3 Methods for Survey Analysis 6.3.1 Random samples 6.3.2 Systematic sampling 6.3.3 Stratified sampling 6.3.4 Multistage sampling 6.3.5 Quota sampling 6.3.6 Cluster sampling 6.4 How It Can Go Wrong 6.5 What Might Be In a Survey? 6.6 Cautionary Notes 47 47 47 48 49 49 49 50 50 50 50 51 51 Probability Distribution Functions 7.1 Introduction 7.2 Discrete Uniform Distribution 7.2.1 Counting techniques 7.2.2 Combination 7.2.3 Permutation 7.3 Binomial Distribution 7.3.1 Example of a binomial distribution 53 53 53 54 54 55 55 56 Contents 7.4 7.5 7.6 7.7 7.3.2 Pascal’s triangle 7.3.3 The use of the binomial distribution The Poisson Distribution 7.4.1 An example of the Poisson distribution 7.4.2 Uses of the Poisson distribution Uses of the Binomial and Poisson Distributions 7.5.1 Is suicide a Poisson process? Continuous Uniform Distribution Exponential Distribution Normal Distribution 8.1 Introduction 8.2 Normal Distribution 8.2.1 A simple example of normal probabilities 8.2.2 A second example of normal probabilities 8.3 Addition of Normal Variables 8.4 Central Limit Theorem 8.4.1 An example of the Central Limit Theorem 8.5 Confidence Intervals for the Population Mean 8.5.1 An example of confidence intervals for the population mean 8.6 Normal Approximation to the Binomial Distribution 8.6.1 An example of the normal approximation to the binomial distribution 8.7 Normal Approximation to the Poisson Distribution 8.7.1 An example of fitting a normal curve to the Poisson distribution vii 56 57 58 59 60 60 62 64 66 67 67 67 69 69 70 70 70 71 71 72 72 72 73 Comparison of the Means, Sample Sizes and Hypothesis Testing 9.1 Introduction 9.2 Estimation of the Mean 9.2.1 An example of estimating a confidence interval for an experimental mean 9.3 Choice of the Sample Size 9.3.1 An example of selecting sample size 9.4 Hypothesis Testing 9.4.1 An example of hypothesis testing 9.5 Comparison of Two Sample Means 9.5.1 An example of a two-sample t test 9.6 Type I and Type II Errors 9.6.1 An example of type I and type II errors 75 75 75 10 Comparison of Variances 10.1 Introduction 10.2 Chi-Squared Test 10.2.1 An example of the chi-squared test 10.3 F Test 10.3.1 An example of the F test 10.3.2 An example considering the normal distribution 83 83 83 83 85 85 85 76 77 77 77 78 79 79 80 80 Appendix 281 Less than (), greater than or equal to (≥) r Less than (