Stochastic Dominance (eBook)

Prof. Levy was born in Jerusalem in 1939. He received his PhD from the Hebrew University in 1969 and in 1976 was promoted to full professorship. He developed a new field of financial economics called Stochastic Dominance, and developed economic models for risk-management, especially risk-reduction in investment, by means of international diversification and mergers and acquisitions. He served as economic advisor to the Bank of Israel; the Israeli Ministry of Finance; Ministry of Industry, Trade and Labor; and Ministry of National Infrastructures, among other government offices. His many awards include the Hebrew University's Prize for Excellence in Research for 1996. The two 1990 Nobel Prize winners in Economics stated that to a large extent their work draws on Prof. Levy's pioneering writings.

Preface 8
The Structure of the Book 11
The Main Changes in the Third Edition 13
Audience 14
Acknowledgements 15
Contents 16
Chapter 1: Risk: Is There a Unique Objective Measure? 24
1.1 What Is Risk? 24
1.2 Measures of Risk 27
a)?Domar and Musgrave Risk Indexes 27
b)?Roy´s Safety First Rule 29
c)?Dispersion as a Risk Index: Variance and Standard Deviation 31
d)?Semi-Variance (SV) as an Index of Risk 33
e)?Beta as a Measure of Risk 34
f)?Baumol´s Risk Measure 34
g)?Value at Risk-VaR(?) 36
h)?Shortfall VaR 36
i)?Loss as an Alternative Cost: The Minimax Regret 37
j)?Expected Utility and Risk 39
k)?Risk Perception Versus Actual Risk Behavioral Economic Approach
l)?The ``Fear Index´´ 40
1.3 Summary 41
Chapter 2: Expected Utility Theory 43
2.1 Introduction 43
2.2 Investment Criteria 44
a) The Maximum Return Criterion (MRC) 44
b) The Maximum Expected Return Criterion (MERC) 46
2.3 The Axioms and Proof of the Maximum Expected Utility Criterion (MEUC) 48
a) The Payoff of the Investments 49
b) The Axioms 49
c) Proof That the Maximum Expected Utility Criterion (MEUC) Is Optimal Decision Rule 51
2.4 The Properties of Utility Function 53
a) Preference and Expected Utility 53
b) Is U(x) a Probability Function or a Utility Function? 55
2.5 The Meaning of the Utility Units 57
2.6 MRC, MERC as Special Cases of MEUC 60
2.7 Utility, Wealth and Change of Wealth 61
2.8 Summary 62
Chapter 3: Stochastic Dominance Decision Rules 63
3.1 Partial Ordering: Efficient and Inefficient Sets 63
3.2 First Degree Stochastic Dominance (FSD) 66
a) Probability Function, Density Function and Cumulative Probability Function 66
b) The FSD Rule 69
c) Graphical Exposition of the FSD Rule 73
d) FSD: A Numerical Example of FSD 74
e) The Intuitive Explanation of FSD 76
3.3 Optimal Rule, Sufficient Rules and Necessary Rules for FSD 77
a) Sufficient Rules 79
b) Necessary Rules 81
3.4 FSD, Correlation and Arbitrage 83
3.5 Type I and Type II Errors When Sufficient Rules or Necessary Rules Are Employed 85
3.6 Second Degree Stochastic Dominance (SSD) 87
a) Risk Aversion 87
b) The SSD Investment Decision Rule 89
c) Graphical Exposition of SSD 92
d) An Intuitive Explanation of SSD 97
3.7 Sufficient Rules and Necessary Rules for SSD 100
a) Sufficient Rules 100
b) Necessary Rules 101
3.8 Third Degree Stochastic Dominance (TSD) 102
a) A Preference for Positive Skewness as a Motivation for TSD 102
b) The Definition of Skewness 103
c) Lottery, Insurance and Preference for Positive Skewness 105
d) Empirical Studies and Positive Skewness Preference (or U0) 106
e) Decreasing Absolute Risk Aversion (DARA), and Positive Skewness Preferences (or U0) 109
f) The Third Degree Stochastic Dominance (TSD) Investment Rule 109
g) Graphical Exposition of TSD 115
h) The Intuitive Explanation of TSD 121
3.9 Sufficient Rules and Necessary Rules for UU3 126
a) Sufficient Rules 126
b) Necessary Rules 126
3.10 Decreasing Absolute Risk Aversion (DARA) Stochastic Dominance (DSD) 127
a) DARA Utility Functions 127
b) DSD with Equal Mean Distributions 129
3.11 Risk-Seeking Stochastic Dominance (RSSD): The Rule 132
a) The Risk-Seeking Stochastic Dominance (RSSD) Rule 132
b) Graphical Exposition of 134
c) The Relationship Between SSD and 135
d) The Relationship Between FSD, SSD and 136
3.12 Nth Order Stochastic Dominance 137
3.13 Stochastic Dominance Rules: Extension to Discrete Distributions 138
3.14 The Role of the Mean and Variance in Stochastic Dominance Rules 143
3.15 Summary 145
Chapter 4: Stochastic Dominance: The Quantile Approach 147
4.1?The Quantile Function 147
4.2?Stochastic Dominance Rules: The Quantile Approach 151
a)?The FSD Rule with Quantiles 152
b)?The SSD Rule with Quantiles 155
4.3?Stochastic Dominance Rules with a Riskless Asset: A Perfect Capital Market 159
a)?FSD with a Riskless Asset: The FSDR Rule 159
b)?Graphical Illustration of the FSDR Rule 163
c)?SSD with a Riskless Asset: The SSDR Rule 165
d)?The SD and SDR Efficient Sets 171
4.4?Stochastic Dominance Rules with a Riskless Asset: An Imperfect Capital Market 171
4.5?Summary 174
Chapter 5: Algorithms for Stochastic Dominance 176
5.1?Using the Necessary Conditions and Transitivity to Reduce the Number of Comparisons 177
5.2?The FSD Algorithm 180
5.3?The SSD Algorithm 181
5.4?The TSD Algorithm 185
5.5?A Numerical Example Showing the Flaw in Existing TSD Algorithm 190
5.6?The Empirical Results 191
5.7?The SDR Algorithm 193
a)?FSDR Algorithm 193
b)?SSDR Algorithm 194
5.8?Summary 195
Chapter 6: Stochastic Dominance with Specific Distributions 197
6.1?Normal Distributions 198
a)?Properties of the Normal Distribution 198
b)?Dominance Without a Riskless Asset 200
c)?Dominance with a Riskless Asset 203
6.2?Lognormal Distributions 205
a)?Properties of the Lognormal Distribution 205
b)?Dominance Without a Riskless Asset 207
c)?Dominance with a Riskless Asset 209
6.3?Truncated Normal Distributions 211
a)?Symmetrical Truncation 211
b)?Non-symmetrical Truncation 215
6.4?Distributions That Intercept Once 217
6.5 Summary 219
Chapter 7: Almost Stochastic Dominance (ASD) 220
7.1?The Possible Paradoxes 221
7.2?FSD* Criterion Corresponding to U1*(epsi) 225
7.3?The SSD* Criterion Corresponding to U2*(epsi) 229
7.4?The Effectiveness of the Almost SD Rules 237
7.5?Application of FSD* to Investment Choices: Stocks Versus Bonds 238
a)?The Decrease in the Violation Area as the Horizon Increases 238
b)?Moshe Levy´s Study: The Preference Set May Decrease Rather Than Increase with the Increase in the Horizon 240
7.6?ASD: Experimental Results 241
7.7?Summary 244
Chapter 8: Stochastic Dominance and Risk Measures 245
8.1?When Is One Investment Riskier Than Another Investment? 246
8.2?Mean Preserving Spread (MPS) 247
8.3?Unequal Means and ``Riskier Than´´ with the Riskless Asset 250
8.4?``Riskier Than´´ and DARA Utility Function: Mean Preserving Antispread 253
a)?Spread and Antispread 254
b)?Increasing Risk and DARA 255
8.5?Summary 256
Chapter 9: Stochastic Dominance and Diversification 257
9.1?Arrow´s Conditions for Diversification and SD Rules 258
a)?Diversification with One Risky and the Riskless Asset 258
b)?The Effect of Shifts in Parameters or Diversification 264
9.2?Extension of the SD Analyses to the Case of Two Risky Assets 265
9.3?Diversification and Expected Utility: Some Common Utility Functions 269
a)?Shift in r 270
b)?Shift in X 271
c)?MPS Shifts 272
d)?MPA Shifts 273
e)?MPSA Shifts 274
9.4?Improving Diversification: The Marginal Conditional Stochastic Dominance (MCSD) Approach 274
9.5?Linear Programing Approach and Efficient SSD Diversification 278
9.6?The Mean Gini Diversification Model 279
9.7?Summary 280
Chapter 10: The CAPM and Stochastic Dominance 282
10.1?The CAPM with Heterogeneous Investment Horizons 283
a)?Quadratic Utility Function 284
b)?Single-Period Normal Distributions 285
c)?Multi-period Normal Distributions 287
d)?Log-Normal Distributions 288
(1) ?Stationary Distributions 288
(2)?Non-stationary Distributions of Returns 295
10.2?Summary 296
Chapter 11: The Empirical Studies: Dominance and Significance Tests 298
11.1?The Effectiveness of the Various Decision Rules: A Perfect Market 300
11.2?The Effectiveness of the Various Decision Rules: An Imperfect Market 305
11.3?The Performance of Mutual Funds with Transaction Costs 307
11.4?Further Reduction in the Efficient Sets: Convex Stochastic Dominance (CSD) 310
a)?FSD, CSD with Three Assets in the Efficient Set (N=3) 311
b)?Extension to N Assets in the FSD Efficient Set 312
11.5?Sampling Errors: Test for Significance of SD 315
a)?Kolmogorov-Smirnov: One Sample Test 315
b)?Kolmogorov-Smirnov: Two-Sample Test 316
c)?The First Phase of Statistical Studies: Pairwise Comparisons Without Diversification 318
d)?The Second Phase of Studies: Income Inequality and Diversification 321
11.6?Summary 324
Chapter 12: Applications of Stochastic Dominance Rules 325
12.1?Capital Structure and the Value of the Firm 325
12.2?Production, Saving and Diversification 328
12.3?Estimating the Probability of Bankruptcy 330
12.4?Option Evaluation, Insurance Premium and Portfolio Insurance 332
12.5?Application of SD Rules in Agricultural Economics 335
12.6?Application of SD Rules in Medicine 336
a)?Stochastic Dominance Rules and Medical Decision 336
b)?Employing SD Rules in the Small Abdominal Aortic Aneurysms Case: Actual Data 342
12.7?Measuring, Welfare, Poverty and Income Inequality 345
12.8?Summary 348
Chapter 13: Mean-Variance, Stochastic Dominance and the Investment Horizon 349
13.1?Tobin´s MV Multi-period Analysis 350
13.2?Sharpe´s Reward-to-Variability Ratio and the Investment Horizon 352
13.3?The Effect of the Investment Horizon on Correlations 355
13.4?The Effect of the Investment Horizon on the Composition of MV Portfolios 358
13.5?The Effect of the Investment Horizon on Beta 361
13.6?Stochastic Dominance and the Investment Horizon 364
13.7?Contrasting the Size of the MV and SD Efficient Set 367
13.8?Summary 369
Chapter 14: Stocks Versus Bonds: A Stochastic Dominance Approach 370
14.1?The Geometric Mean Investment Rule for the Very Long Horizon 371
14.2?The MGM Portfolio and Expected Utility 377
a)?The Contradiction Between MGM Rule and the Myopic Utility Functions 377
b)?A Suggested Resolution of the MGM Rule and Expected Utility Contradictory Results 379
14.3?Long But Finite Horizon: FSD and Almost FSD with Log-Normal Distributions 381
14.4?The Empirical Evidence 388
a)?Investment for the Long Run: Ibbotson´s Data 388
b)?The AFSD in the Long Run: The Study of Bali et al. 390
14.5?The MV and the Log-Normal Efficient Frontiers 394
14.6?Summary 400
Chapter 15: Non-expected Utility and Stochastic Dominance 404
15.1?The Expected Utility: Some Paradoxes 406
a)?The Allais Paradox 406
b)?The Ellsberg Paradox: Ambiguity Aversion 408
15.2?Non-expected Utility Theory 409
a)?Probability Weighting 410
b)?PT´s Decision Weights 412
c)?CPT´s Decision Weights: No FSD Violation 413
d)?Rank Dependent Expected Utility (RDEU) and FSD 414
e)?Configural Decision Weights 416
f)?Regret Theory 416
15.3?FSD Violations: Decision Weights or Bounded Rationality? 418
15.4?Temporary and Permanent Attitude Toward Risk 424
15.5?Summary 428
Chapter 16: Stochastic Dominance and Prospect Theory 430
16.1?CPT, Expected Utility and FSD Rule 432
16.2?Prospect Stochastic Dominance (PSD) 433
16.3?Markowitz´s Stochastic Dominance 440
16.4?CPT, MV and the CAPM 445
16.5?Experimental Testing the Competing Theories: SD Approach 448
a) The Certainty Equivalent Approach 448
b) The Stochastic Dominance Approach 450
c) Are People Risk Averse? (SSD Tests) 450
d) Is CPT Valid Theory? (PSD Tests) 451
16.6?SSD, PSD, MSD Rules and the Efficiency of the Market Portfolio 452
16.7?Summary 455
Chapter 17: Bivariate FSD (BFSD) 456
17.1?The Suggested Bivariate Preferences 458
a)?The Suggested Bivariate Preference by Abel 458
b)?The Ultimatum Game Experiments and the Suggested Bivariate Preferences 459
17.2?Bivariate First Degree Stochastic Dominance 462
17.3?The Cross Derivative and Attitude Toward Correlation 472
17.4?Summary 479
Chapter 18: Future Research 481
18.1?Portfolio Construction and Stochastic Dominance Equilibrium 481
18.2?Risk Attitude and Equilibrium 485
18.3?The Stochastic Dominance Rules and the Length of the Investment Horizon 487
18.4?Uncertain Investment Horizon 490
18.5?Risk Index 490
18.6?Stochastic Dominance and Increasing Interest Rate 491
18.7?Truncated Distributions and Stochastic Dominance 491
18.8?Employing Stochastic Dominance Criteria in Other Research Areas 492
18.9?Refining the Stochastic Dominance Criteria 493
18.10?Stochastic Dominance and Option Valuation 494
18.11?Experimental Stochastic Dominance Criteria 494
18.12?Multivariate Stochastic Dominance 495
18.13?Conditional Dominance (Monotonicity) 495
Bibliography 496
Index 513