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关键词：Management workpaper Portfolio Managemen Portfoli Management Portfolio workpaper

Portfolio Management Formulas

Preface
Introduction xi
Scope of This Book xi
Some Prevalent Misconceptions xu
Worst-Case Scenarios and Strategy xvi
Mathematics Notation xviii
Synthetic Constructs in This Text xviii
Optimal Trading Quantities and Optimal f xxi
V
1 The Empirical Techniques
Deciding on Quantity I
Basic Concepts 4
The Runs Test 5
Serial Correlation 9
Common Dependency Errors 14
Mathematical Expectation 16
To Reinvest Trading Profits or Not 20
Measuring a Good System for Reinvestment: The Geometric Mean 21
How Best to Reinvest 25
Optimal Fixed Fractional Trading 26
Kelly Formulas 27
Finding the Optimal f by the Geometric Mean 30
vii
To Summarize Thus Far 32
Geometric Average Trade 34 i
Why You Must Know Your Optimal f 35
The Severity of Drawdowu 38
Modem Portfolio Theory 39
The Markowitz Model 40
The Geometric Mean Portfolio Strategy 45
Daily Procedures for Using Optimal Portfolios 46
Allocations Greater Than 100% 49
How the Dispersion of Outcomes Affects Geometric Growth 53
The Fundamental Equation of Trading 58
2 Characteristics of Fixed Fractional Trading
and Salutary Techniques
Optima1 f for Small Traders Just Starting Out 63
Threshold to Geometric 65
One Combined Bankroll versus Separate Bankrolls 68
Treat Each Play As If Infinitely Repeated 7 1
Efficiency Loss in Simultaneous Wagering or Portfolio Trading
Time Required to Reach a Specified Goal and
the Trouble with Fractional f 76
Comparing Trading Systems 80
Too Much Sensitivity to the Biggest Loss 82
Equalizing Optimal f 83
Dollar Averaging and Share Averaging Ideas 89
The Arc Sine Laws and Random Walks 92
Time Spent in a Drawdown 95
3 Parametric Optimal f on the Normal Distribution
The Basics of Probability Distributions 98
Descriptive Measures of Distributions 100
Moments of a Distribution 103
The Normal Distribution 108
The Central Limit Theorem 109
Working with the Normal Distribution 111
Normal Probabilities 115
The Lognormal Distribution 124
The Parametric Optimal f 125
Finding the Optimal f on the Normal Distribution 132
4 Parametric Techniques on Other Distributions
73
63
9 8
The Kolmogorov-Smimov (K-S) Test 1 4 9
Creating Our Own Characteristic Distribution Function 1 5 3
Fitting the Parameters of the Distribution 1 6 0
Using the Parameters to Find the Optimal f I68
Performing ‘What Ifs” 175
Equalizing f 176
Optimal f on Other Distributions and Fitted Curves 177
Scenario Planning 178
Optimal f on Binned Data 1 9 0
Which is the Best Optimal f? 192
5 Introduction to Multiple Simultaneous
Positions under the Parametric Approach
Estimating Volatility 194
Ruin, Risk, and Reality 1 9 7
Option Pricing Models 199
A European Options Pricing Model for All Distributions 208
The Single Long Option and Optimal f 2 1 3
The Single Short Option 224
The Single Position in the Underlying Instrument 225
Multiple Simultaneous Positions with a Causal Relationship 228
Multiple Simultaneous Positions with a Random Relationship 233
193
6 Correlative Relationships and the
Derivation of the Efficient Frontier 237
Definition of the Problem 238
Solutions of Linear Systems Using Row-Equivalent Matrices 250
Interpreting the Results 258
7 The Geometry of Portfolios
The Capital Market Lines (CMLs) 266
The Geometric Efficient Frontier 271
Unconstrained Portfolios 278
How Optimal f Fits with Optimal Portfolios 283
Threshold to the Geometric for Portfolios 287
Completing the Loop 287
266
149
8 Risk Management 294
Asset Allocation 294
Reallocation: Four Methods 302
Why Reallocate? 311
Portfolio Insurance-The Fourth Reallocation Technique 312
The Margin Constraint 320
Rotating Markets 324
To Summarize 326
Application to Stock Trading 327
A Closing Comment 328
Introduction
Appendixes
A The Chi-Square Test 331
B Other Common Distributions
The Uniform Distribution 337
The Bernoulli Distribution 339
The Binomial Distribution 341
The Geometric Distribution 345
The Hypergeometric Distribution 347
The Poisson Distribution 348
The Exponential Distribution 352
The Chi-Square Distribution 354
The Student’s Distribution 356
The Multinomid Distribution 358
The Stable Paretian Distribution 359
C Further on Dependency: The Turning Points and
Phase Length Tests
Bibliography and Suggested Reading
Index
336

Credit Portfolio Management
CHAPTER 1
The Revolution in Credit—Capital Is the Key 1
The Credit Function Is Changing 1
Capital Is the Key 6
Economic Capital 8
Regulatory Capital 11
APPENDIX TO CHAPTER 1: A Credit Portfolio Model Inside
the IRB Risk Weights 21
Note 23
PART ONE
The Credit Portfolio Management Process 25
CHAPTER 2
Modern Portfolio Theory and Elements of the Portfolio 27
Modeling Process
Modern Portfolio Theory 27
Challenges in Applying Modern Portfolio Theory to
Portfolios of Credit Assets 34
Elements of the Credit Portfolio Modeling Process 38
Note 40
CHAPTER 3
Data Requirements and Sources for Credit
Portfolio Management 41
Probabilities of Default 41
Recovery and Utilization in the Event of Default 92
Correlation of Defaults 102
Notes 107
CHAPTER 4
Credit Portfolio Models 109
Structural Models 110
Explicit Factor Models 133
Actuarial Models 141
Analytical Comparison of the Credit Portfolio Models 148
Empirical Comparison of the Credit Portfolio Models 153
What Models Are Financial Institutions Using? 161
Notes 161
APPENDIX TO CHAPTER 4: Technical Discussion of Moody’s–
KMV Portfolio Manager Mattia Filiaci 162
Default Correlation 162
Facility Valuation 163
Generating the Portfolio Value Distribution 174
Outputs 176
Notes 178
PART TWO
Tools to Manage a Portfolio of Credit Assets 181
CHAPTER 5
Loan Sales and Trading 183
Primary Syndication Market 183
Secondary Loan Market 191
Note 192
CHAPTER 6
Credit Derivatives with Gregory Hayt 193
Taxonomy of Credit Derivatives 193
The Credit Derivatives Market 201
Using Credit Derivatives to Manage a Portfolio of Credit Assets 203
Pricing Credit Derivatives 209
Notes 224
CHAPTER 7
Securitization 225
Elements of a CDO 225
“Traditional” and “Synthetic” CDO Structures 229
Applications of CDOs 233
To What Extent and Why Are Financial Institutions
Using Securitizations? 236
Regulatory Treatment 237
Note 240
PART THREE
Capital Attribution and Allocation 241
CHAPTER 8
Capital Attribution and Allocation 243
Measuring Total Economic Capital 243
Attributing Capital to Business Units 247
Attributing Capital to Transactions 252
Performance Measures—The Necessary Precondition
to Capital Allocation 258
Optimizing the Allocation of Capital 267
Notes 269
APPENDIX TO CHAPTER 8: Quantifying Operational Risk 270
Process Approaches 274
Factor Approaches 274
Actuarial Approaches 275
Notes 276
APPENDIX
Statistics for Credit Portfolio Management Mattia Filiaci 277
Basic Statistics 278
Applications of Basic Statistics 306
Important Probability Distributions 314
Notes 324
References 327
Index 333

Capital Markets and Portfolio
Theory
Table of Contents
PART I Standard (One Period) Portfolio Theory . . . . . . . . . . . . . . . . . . . . . 1
1 Portfolio Choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.A Framework and notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.A.i No Risk-free Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.A.ii With Risk-free Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.B Efficient portfolio in absence of a risk-free asset . . . . . . . . . . . . . . . . . . . . . . 6
1.B.i Efficiency criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.B.ii Efficient portfolio and risk averse investors . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.B.iii Efficient set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.B.iv Two funds separation (Black) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.C Efficient portfolio with a risk-free asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.D HARA preferences and Cass-Stiglitz 2 fund separation . . . . . . . . . . . . . . 14
1.D.i HARA (Hyperbolic Absolute Risk Aversion) . . . . . . . . . . . . . . . . . . . . . . . . 14
1.D.ii Cass and Stiglitz separation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Capital Market Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.A CAPM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.A.i The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.A.ii Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.A.iii CAPM as a Pricing and Equilibrium Model . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.A.iv Testing the CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.B Factor Models and APT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.B.i K-factor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.B.ii APT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.B.iii Arbitrage and Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.B.iv References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
PART II Multiperiod Capital Market Theory : the
Probabilistic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.A Probability Space and Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.B Asset Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.B.i DeÞnitions and Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.C Portfolio Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.C.i Notation: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.C.ii Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.C.iii Continuous Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 AoA, Attainability and Completeness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.A DeÞnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.B Propositions on AoA and Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.B.i Correspondance between Q and Π : Main Results . . . . . . . . . . . . . . . . . . . 35
4.B.ii Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Alternative SpeciÞcations of Asset Prices . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.A Ito Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.B Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.C Diffusion state variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.D Theory in the Ito-Diffusion Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.D.i Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.D.ii Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.D.iii Redundancy and Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.D.iv Criteria for Recognizing a Complete Market . . . . . . . . . . . . . . . . . . . . . . . . 44
PART III State Variables Models: the PDE Approach. . . . . . . . . . . . . . . . 45
6 Framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7 Discounting Under Uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.A Ito’s lemma and the Dynkin Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.B The Feynman-Kac Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
8 The PDE Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8.A Continuous Time APT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8.A.i Alternative decompositions of a return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8.A.ii The APT Model (continuous time version). . . . . . . . . . . . . . . . . . . . . . . . . . 51
8.B One Factor Interest Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.C Discounting Under Uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
9 Links Between Probabilistic and PDE Approaches . . . . . . . . . . . . . . . 55
9.A Probability Changes and the Radon-Nikodym Derivative . . . . . . . . . . . 55
9.B Girsanov Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
9.C Risk Adjusted Drifts: Application of Girsanov Theorem . . . . . . . . . . . . 56
PART IV The Numeraire Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
10 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
11 Numeraire and Probability Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
11.AFramework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
11.A.i Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
.BCorrespondence Between Numeraires and Martingale Probabilities . 62
11.B.i Numeraire → Martingale Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
11.B.ii Probability → Numeraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
11.CSummary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
12 The Numeraire (Growth Optimal) Portfolio . . . . . . . . . . . . . . . . . . . . . . . 65
12.ADeÞnition and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12.A.i DeÞnition of the Numeraire (h,H). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
12.A.ii Characterization and Composition of (h,H) . . . . . . . . . . . . . . . . . . . . . . . . 65
12.A.iii The Numeraire Portfolio and Radon-Nikodym Derivatives . . . . . . . . . . . . 69
12.BFirst Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
12.B.i CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
12.B.ii Valuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
PART V Continuous Time Portfolio Optimization. . . . . . . . . . . . . . . . . . . . 72
13 Dynamic Consumption and Portfolio Choices (The Merton
Model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
13.AFramework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
13.A.i The Capital Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
13.A.ii The Investors (Consumers)’ Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
13.BThe Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
13.B.i Sketch of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
13.B.ii Optimal portfolios and L+2 funds separation . . . . . . . . . . . . . . . . . . . . . . 77
13.B.iii Intertemporal CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
14 THE ”EQUIVALENT” STATIC PROBLEM (Cox-Huang,
Karatzas approach) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
14.ATransforming the dynamic into a static problem . . . . . . . . . . . . . . . . . . . . 80
14.A.i The pure portfolio problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
14.A.ii The consumption-portfolio problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
14.BThe solution in the case of complete markets. . . . . . . . . . . . . . . . . . . . . . . . 83
14.B.i Solution of the pure portfolio problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
14.B.ii Examples of speciÞc utility functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
14.B.iii Solution of the consumption-portfolio problem . . . . . . . . . . . . . . . . . . . . . . 86
14.B.iv General method for obtaining the optimal strategy x∗∗ . . . . . . . . . . . . . . . 87
14.CEquilibrium: the consumption based CAPM . . . . . . . . . . . . . . . . . . . . . . . . 88
PART VI STRATEGIC ASSET ALLOCATION . . . . . . . . . . . . . . . . . . . . . . . 90
15 The problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
16 The optimal terminal wealt