The handbook of portfolio mathematics ( PDFDrive.com )

JWDD035-FM JWDD035-Vince February 12, 2007 7:3 Char Count= The Handbook of Portfolio Mathematics i JWDD035-FM JWDD035-Vince February 12, 2007 7:3 Char Count= Founded in 1807, John Wiley & Sons is the oldest independent publishing company in the United States With offices in North America, Europe, Australia, and Asia, Wiley is globally committed to developing and marketing print and electronic products and services for our customers’ professional and personal knowledge and understanding The Wiley Trading series features books by traders who have survived the market’s ever-changing temperament and have prospered—some by reinventing systems, others by getting back to basics Whether a novice trader, professional or somewhere in between, these books will provide the advice and strategies needed to prosper today and well into the future For a list of available titles, visit our Web site at www.WileyFinance.com ii JWDD035-FM JWDD035-Vince February 12, 2007 7:3 Char Count= The Handbook of Portfolio Mathematics Formulas for Optimal Allocation & Leverage RALPH VINCE iii JWDD035-FM JWDD035-Vince Copyright C February 12, 2007 7:3 Char Count= 2007 by Ralph Vince All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada Chapters 1–10 contain revised material from three of the author’s previous books, Portfolio Management Formulas: Mathematical Trading Methods for the Futures, Options, and Stock Markets (1990), The Mathematics of Money Management: Risk Analysis Techniques for Traders (1992), and The New Money Management: A Framework for Asset Allocation (1995), all published by John Wiley & Sons, Inc Wiley Bicentennial Logo: Richard J Pacifico 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 as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646–8600, or on the Web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic format For more information about Wiley products, visit our web site at www.wiley.com Library of Congress Cataloging-in-Publication Data: Vince, Ralph, 1958– The handbook of portfolio mathematics : formulas for optimal allocation & leverage / Ralph Vince: p cm ISBN-13: 978-0-471-75768-9 (cloth) ISBN-10: 0-471-75768-3 (cloth) Portfolio management–Mathematical models Investments–Mathematical models I Title HG4529.5.V555 2007 332.601 51 – dc22 2006052577 Printed in the United States of America 10 iv JWDD035-FM JWDD035-Vince February 12, 2007 7:3 Char Count= “You must not be extending your empire while you are at war or run into unnecessary dangers I am more afraid of our own mistakes than our enemies’ designs.” —Pericles, in a speech to the Athenians during the Peloponnesian War, as represented by Thucydides v JWDD035-FM JWDD035-Vince February 12, 2007 vi 7:3 Char Count= JWDD035-FM JWDD035-Vince February 12, 2007 7:3 Char Count= Contents Preface xiii Introduction xvii PART I Theory CHAPTER The Random Process and Gambling Theory Independent versus Dependent Trials Processes Mathematical Expectation Exact Sequences, Possible Outcomes, and the Normal Distribution Possible Outcomes and Standard Deviations 11 The House Advantage 15 Mathematical Expectation Less than Zero Spells Disaster 18 Baccarat 19 Numbers 20 Pari-Mutuel Betting 21 Winning and Losing Streaks in the Random Process 24 Determining Dependency 25 The Runs Test, Z Scores, and Confidence Limits 27 The Linear Correlation Coefficient 32 CHAPTER 43 Probability Distributions The Basics of Probability Distributions 43 Descriptive Measures of Distributions 45 Moments of a Distribution 47 The Normal Distribution 52 vii JWDD035-FM JWDD035-Vince February 12, 2007 viii 7:3 Char Count= THE HANDBOOK OF PORTFOLIO MATHEMATICS The Central Limit Theorem 52 Working with the Normal Distribution 54 Normal Probabilities 59 Further Derivatives of the Normal 65 The Lognormal Distribution 67 The Uniform Distribution 69 The Bernoulli Distribution 71 The Binomial Distribution 72 The Geometric Distribution 78 The Hypergeometric Distribution 80 The Poisson Distribution 81 The Exponential Distribution 85 The Chi-Square Distribution 87 The Chi-Square “Test” 88 The Student’s Distribution 92 The Multinomial Distribution 95 The Stable Paretian Distribution 96 CHAPTER Reinvestment of Returns and Geometric Growth Concepts To Reinvest Trading Profits or Not 99 99 Measuring a Good System for Reinvestment—The Geometric Mean 103 Estimating the Geometric Mean 107 How Best to Reinvest 109 CHAPTER 117 Optimal f Optimal Fixed Fraction 117 Asymmetrical Leverage 118 Kelly 120 Finding the Optimal f by the Geometric Mean 122 To Summarize Thus Far 125 How to Figure the Geometric Mean Using Spreadsheet Logic Geometric Average Trade 127 127 JWDD035-FM JWDD035-Vince February 12, 2007 7:3 Char Count= ix CONTENTS A Simpler Method for Finding the Optimal f 128 The Virtues of the Optimal f 130 Why You Must Know Your Optimal f 132 Drawdown and Largest Loss with f 141 Consequences of Straying Too Far from the Optimal f 145 Equalizing Optimal f 151 Finding Optimal f via Parabolic Interpolation 157 The Next Step 161 Scenario Planning 162 Scenario Spectrums 173 CHAPTER 175 Characteristics of Optimal f Optimal f for Small Traders Just Starting Out 175 Threshold to Geometric 177 One Combined Bankroll versus Separate Bankrolls 180 Treat Each Play as If Infinitely Repeated 182 Efficiency Loss in Simultaneous Wagering or Portfolio Trading 185 Time Required to Reach a Specified Goal and the Trouble with Fractional f 188 Comparing Trading Systems 192 Too Much Sensitivity to the Biggest Loss 193 The Arc Sine Laws and Random Walks 194 Time Spent in a Drawdown 197 The Estimated Geometric Mean (or How the Dispersion of Outcomes Affects Geometric Growth) 198 The Fundamental Equation of Trading 202 Why Is f Optimal? 203 CHAPTER Laws of Growth, Utility, and Finite Streams Maximizing Expected Average Compound Growth 207 209 Utility Theory 217 The Expected Utility Theorem 218 Characteristics of Utility Preference Functions 218 JWDD035-12 JWDD035-Vince February 10, 2007 408 22:19 Char Count= THE HANDBOOK OF PORTFOLIO MATHEMATICS do{ generator.setSeed(random.nextLong()); long pn=(long)(generator.nextDouble()*(double)nopermutations); double hprpermutation[]=nPq i(counter,pn); double theB = (double)B(hprpermutation,usedrawdowninsteadofruin); if(theB>0.0){ theB *= probThisB; passed += theB; } sumOfProbs += probThisB; permutationcount++; }while(permutationcount