The Handbook of Portfolio Mathematics: Formulas for Optimal Allocation & Leverage

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