The Handbook of Portfolio Mathematics: Formulas for Optimal Allocation & Leverage (Wiley Trading)

445 Pages·2007·7.63 MB·English

by  Ralph Vince

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The
Handbook of
Portfolio
Mathematics
i
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The
Handbook of
Portfolio
Mathematics
Formulas for Optimal
Allocation & Leverage
RALPH VINCE
iii
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Copyright(cid:2)C 2007byRalphVince.Allrightsreserved.
PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey.
PublishedsimultaneouslyinCanada.
Chapters1–10containrevisedmaterialfromthreeoftheauthor’spreviousbooks,Portfolio
ManagementFormulas:MathematicalTradingMethodsfortheFutures,Options,and
StockMarkets(1990),TheMathematicsofMoneyManagement:RiskAnalysisTechniques
forTraders(1992),andTheNewMoneyManagement:AFrameworkforAssetAllocation
(1995),allpublishedbyJohnWiley&Sons,Inc.
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LibraryofCongressCataloging-in-PublicationData:
Vince,Ralph,1958–
Thehandbookofportfoliomathematics:formulasforoptimalallocation&leverage/
RalphVince:
p. cm.
ISBN-13:978-0-471-75768-9(cloth)
ISBN-10:0-471-75768-3(cloth)
1.Portfoliomanagement–Mathematicalmodels. 2.Investments–Mathematicalmodels.
I.Title.
HG4529.5.V5552007
332.601(cid:3)51–dc22 2006052577
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
iv
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“Youmustnotbeextendingyourempirewhileyouareatwarorrun
intounnecessarydangers.Iammoreafraidofourownmistakes
thanourenemies’designs.”
—Pericles,inaspeechtotheAtheniansduringthePeloponnesian
War,asrepresentedbyThucydides
v
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vi
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Contents
Preface xiii
Introduction xvii
PART I Theory 1
CHAPTER 1 The Random Process and Gambling Theory 3
IndependentversusDependentTrialsProcesses 5
MathematicalExpectation 6
ExactSequences,PossibleOutcomes,andthe
NormalDistribution 8
PossibleOutcomesandStandardDeviations 11
TheHouseAdvantage 15
MathematicalExpectationLessthanZeroSpellsDisaster 18
Baccarat 19
Numbers 20
Pari-MutuelBetting 21
WinningandLosingStreaksintheRandomProcess 24
DeterminingDependency 25
TheRunsTest,ZScores,andConfidenceLimits 27
TheLinearCorrelationCoefficient 32
CHAPTER 2 Probability Distributions 43
TheBasicsofProbabilityDistributions 43
DescriptiveMeasuresofDistributions 45
MomentsofaDistribution 47
TheNormalDistribution 52
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viii THEHANDBOOKOFPORTFOLIOMATHEMATICS
TheCentralLimitTheorem 52
WorkingwiththeNormalDistribution 54
NormalProbabilities 59
FurtherDerivativesoftheNormal 65
TheLognormalDistribution 67
TheUniformDistribution 69
TheBernoulliDistribution 71
TheBinomialDistribution 72
TheGeometricDistribution 78
TheHypergeometricDistribution 80
ThePoissonDistribution 81
TheExponentialDistribution 85
TheChi-SquareDistribution 87
TheChi-Square“Test” 88
TheStudent’sDistribution 92
TheMultinomialDistribution 95
TheStableParetianDistribution 96
CHAPTER 3 Reinvestment of Returns and Geometric
Growth Concepts 99
ToReinvestTradingProfitsorNot 99
MeasuringaGoodSystemforReinvestment—The
GeometricMean 103
EstimatingtheGeometricMean 107
HowBesttoReinvest 109
CHAPTER 4 Optimal f 117
OptimalFixedFraction 117
AsymmetricalLeverage 118
Kelly 120
FindingtheOptimalfbytheGeometricMean 122
ToSummarizeThusFar 125
HowtoFiguretheGeometricMeanUsing
SpreadsheetLogic 127
GeometricAverageTrade 127
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CONTENTS ix
ASimplerMethodforFindingtheOptimalf 128
TheVirtuesoftheOptimalf 130
WhyYouMustKnowYourOptimalf 132
DrawdownandLargestLosswithf 141
ConsequencesofStrayingTooFar
fromtheOptimalf 145
EqualizingOptimalf 151
FindingOptimalfviaParabolicInterpolation 157
TheNextStep 161
ScenarioPlanning 162
ScenarioSpectrums 173
CHAPTER 5 Characteristics of Optimal f 175
OptimalfforSmallTradersJustStartingOut 175
ThresholdtoGeometric 177
OneCombinedBankrollversusSeparateBankrolls 180
TreatEachPlayasIfInfinitelyRepeated 182
EfficiencyLossinSimultaneousWageringor
PortfolioTrading 185
TimeRequiredtoReachaSpecifiedGoalandthe
TroublewithFractionalf 188
ComparingTradingSystems 192
TooMuchSensitivitytotheBiggestLoss 193
TheArcSineLawsandRandomWalks 194
TimeSpentinaDrawdown 197
TheEstimatedGeometricMean(orHowtheDispersion
ofOutcomesAffectsGeometricGrowth) 198
TheFundamentalEquationofTrading 202
WhyIsfOptimal? 203
CHAPTER 6 Laws of Growth, Utility, and Finite
Streams 207
MaximizingExpectedAverageCompoundGrowth 209
UtilityTheory 217
TheExpectedUtilityTheorem 218
CharacteristicsofUtilityPreferenceFunctions 218