Contents

Preface    xiii

Introduction    x

PARTI    Theory    1

CHAPTER1  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-MutuelBetting    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

CHAPTER2  ProbabilityDistributions   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 theNormal    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’sDistribution    92
• The Multinomial Distribution    95
• The Stable Paretian Distribution    96

CHAPTER3  Reinvestment of Returns and Geometric GrowthConcepts   99

• To Reinvest Trading Profits or Not   99
• Measuring a Good System for Reinvestment—The Geometric Mean   103
• Estimating the GeometricMean    107
• How Best to Reinvest    109

CHAPTER4  Optimal f    117

• OptimalFixedFraction    117
• AsymmetricalLeverage    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

CHAPTER5  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 inSimultaneous 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

CHAPTER6  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 NewFramework    226

CHAPTER7  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

CHAPTER8  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

CHAPTER9  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
• ImportantNotes    321

CHAPTER10  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 then+1 Dimensional Landscape    351
• Drawdown Management and the New Framework    359

PARTII   Practice    365

CHAPTER11  What the Professionals Have Done    367

• Commonalities    368
• Differences    368
• Further Characteristics of Long-Term Trend Followers    369

CHAPTER12  The Leverage Space Portfolio Model in the RealWorld    377

• Postscript    415
• Index    417