The Handbook of Portfolio Mathematics "For the serious investor, trader, or money manager, this book takes a rewarding look into modern portfolio theory. Vince introduces a leverage-space portfolio model, tweaks it for the drawdown probability, and delivers a superior model. He even provides equations to maximize returns for a chosen level of risk. So if you're serious about making money in today's markets, buy this book. Read it. Profit from it." —Thomas N. Bulkowski, author, Encyclopedia of Chart Patterns "This is an important book. Though traders routinely speak of their 'edge' in the marketplace and ways of handling 'risk,' few can define and measure these accurately. In this book, Ralph Vince takes readers step by step through an understanding of the mathematical foundations of trading, significantly extending his earlier work and breaking important new ground. His lucid writing style and liberal use of practical examples make this book must reading." —Brett N. Steenbarger, PhD, author, The Psychology of Trading and Enhancing Trader Performance "Ralph Vince is one of the world's foremost authorities on quantitative portfolio analysis. In this masterly contribution, Ralph builds on his early pioneering findings to address the real-world concerns of money managers in the trenches-how to systematically maximize gains in relation to risk." —Nelson Freeburg, Editor, Formula Research "Gambling and investing may make strange bedfellows in the eyes of many, but not Ralph Vince, who once again demonstrates that an open mind is the investor's most valuable asset. What does bet sizing have to do with investing? The answer to that question and many more lie inside this iconoclastic work. Want to make the most of your investing skills Open this book." —John Bollinger, CFA, CMT, www.BollingerBands.com

Table of Contents:
Preface xiii Introduction xvii Part I Theory 1 Chapter 1 The Random Process and Gambling Theory 3 Independent versus Dependent Trials Processes 5 Mathematical Expectation 6 Exact Sequences, Possible Outcomes, and the Normal Distribution 8 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 2 Probability Distributions 43 The Basics of Probability Distributions 43 Descriptive Measures of Distributions 45 Moments of a Distribution 47 The Normal Distribution 52 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 3 Reinvestment of Returns and Geometric Growth Concepts 99 To Reinvest Trading Profits or Not 99 Measuring a Good System for Reinvestment—The Geometric Mean 103 Estimating the Geometric Mean 107 How Best to Reinvest 109 Chapter 4 Optimal f 117 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 127 Geometric Average Trade 127 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 5 Characteristics of Optimal f 175 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 6 Laws of Growth, Utility, and Finite Streams 207 Maximizing Expected Average Compound Growth 209 Utility Theory 217 The Expected Utility Theorem 218 Characteristics of Utility Preference Functions 218 Alternate Arguments to Classical Utility Theory 221 Finding Your Utility Preference Curve 222 Utility and the New Framework 226 Chapter 7 Classical Portfolio Construction 231 Modern Portfolio Theory 231 The Markowitz Model 232 Definition of the Problem 235 Solutions of Linear Systems Using Row-Equivalent Matrices 246 Interpreting the Results 252 Chapter 8 The Geometry of Mean Variance Portfolios 261 The Capital Market Lines (CMLs) 261 The Geometric Efficient Frontier 266 Unconstrained Portfolios 273 How Optimal f Fits In 277 Completing the Loop 281 Chapter 9 The Leverage Space Model 287 Why This New Framework Is Better 288 Multiple Simultaneous Plays 299 A Comparison to the Old Frameworks 302 Mathematical Optimization 303 The Objective Function 305 Mathematical Optimization versus Root Finding 312 Optimization Techniques 313 The Genetic Algorithm 317 Important Notes 321 Chapter 10 The Geometry of Leverage Space Portfolios 323 Dilution 323 Reallocation 333 Portfolio Insurance and Optimal f 335 Upside Limit on Active Equity and the Margin Constraint 341 f Shift and Constructing a Robust Portfolio 342 Tailoring a Trading Program through Reallocation 343 Gradient Trading and Continuous Dominance 345 Important Points to the Left of the Peak in the n + 1 Dimensional Landscape 351 Drawdown Management and the New Framework 359 Part II Practice 365 Chapter 11 What the Professionals Have Done 367 Commonalities 368 Differences 368 Further Characteristics of Long-Term Trend Followers 369 Chapter 12 The Leverage Space Portfolio Model in the Real World 377 Postscript 415 Index 417