Stochastic Dominance and Applications to Finance, Risk and Economics © 2010 by Taylor and Francis Group, LLC Stochastic Dominance and Applications to Finance, Risk and Economics Songsak Sriboonchitta Chiang Mai University Chiang Mai, Thailand Wing-Keung Wong Hong Kong Baptist University Hong Kong, People’s Republic of China Sompong Dhompongsa Chiang Mai University Chiang Mai, Thailand Hung T Nguyen New Mexico State University Las Cruces, New Mexico, U.S.A © 2010 by Taylor and Francis Group, LLC Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor and Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number: 978-1-4200-8266-1 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts 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and the CRC Press Web site at http://www.crcpress.com © 2010 by Taylor and Francis Group, LLC 2009031871 Contents Preface xi Utility in Decision Theory 1.1 Choice under certainty 1.2 Basic probability background 1.2.1 Probability measures and distributions 1.2.2 Integration 1.2.3 Notes on topological spaces 1.3 Choice under uncertainty 1.4 Utilities and risk attitudes 1.4.1 Qualitative representations of risk attitudes 1.4.2 Notes on convex functions 1.4.3 Jensen’s inequality 1.5 Exercises 1 5 17 31 34 48 48 53 54 55 Foundations of Stochastic Dominance 2.1 Some preliminary mathematics 2.1.1 Approximation of utility functions 2.1.2 A fundamental lemma 2.2 Deriving representations of preferences 2.2.1 Representation for risk neutral individuals 2.2.2 Representation for risk averse individuals 2.2.3 Representation for risk seeking individuals 2.2.4 A subclass of risk averse individuals 2.3 Stochastic dominance 2.3.1 First-order stochastic dominance 2.3.2 Second-order stochastic dominance 2.3.3 Third-order stochastic dominance 2.4 Exercises 59 59 60 65 66 66 68 69 70 76 77 81 83 85 v © 2010 by Taylor and Francis Group, LLC vi CONTENTS Issues in Stochastic Dominance 3.1 A closer look at the mean-variance rule 3.2 Multivariate stochastic dominance 3.3 Stochastic dominance via quantile functions 3.3.1 First-order stochastic dominance 3.3.2 Second-order stochastic dominance 3.3.3 Stochastic dominance rule for risk seekers 3.4 Exercises 89 89 93 96 99 99 101 104 Financial Risk Measures 4.1 The problem of risk modeling 4.2 Some popular risk measures 4.2.1 Variance 4.2.2 Value-at-risk 4.2.3 Tail value-at-risk 4.3 Desirable properties of risk measures 4.4 Exercises 107 107 110 110 111 114 119 123 Choquet Integrals as Risk Measures 5.1 Extended theory of measures 5.2 Capacities 5.3 The Choquet integral 5.4 Basic properties of the Choquet integral 5.5 Comonotonicity 5.6 Notes on copulas 5.7 A characterization theorem 5.8 A class of coherent risk measures 5.9 Consistency with stochastic dominance 5.10 Exercises 129 129 130 135 142 149 154 161 164 172 176 Foundational Statistics for Stochastic Dominance 6.1 From theory to applications 6.2 Structure of statistical inference 6.3 Generalities on statistical estimation 6.4 Nonparametric estimation 6.4.1 The Glivenko-Cantelli theorem 6.4.2 Estimating probability density functions 6.4.3 Method of excess mass 6.4.4 Nonparametric regression 6.4.5 Risk estimation 6.5 Basics of hypothesis testing 179 179 182 189 196 196 201 205 207 209 213 © 2010 by Taylor and Francis Group, LLC CONTENTS 215 217 218 221 222 226 Models and Data in Econometrics 7.1 Justifications of models 7.1.1 The logit model 7.1.2 Stochastic volatility 7.1.3 Financial risk models 7.2 Coarse data 7.2.1 Indirect observations in auctions 7.2.2 Game theory 7.2.3 Measurement-error data in linear models 7.2.4 Censored data 7.2.5 Missing data 7.3 Modeling dependence structure 7.3.1 Copulas 7.3.2 Copulas for sampling designs 7.3.3 Estimation of copulas 7.4 Some additional statistical tools 7.4.1 Bayesian statistics 7.4.2 From linear regression to filtering models 7.4.3 Recursive estimation 7.4.4 What is a filter? 7.4.5 The Kalman filter 7.5 Exercises 233 233 234 238 244 248 248 250 252 253 258 264 264 268 270 271 271 271 275 276 278 281 Applications to Finance 8.1 Diversification 8.1.1 Convex stochastic dominance 8.1.2 Diversification for risk averters and risk seekers 8.2 Diversification on convex combinations 8.3 Prospect and Markowitz SD 8.3.1 Illustration 8.4 Market rationality and e]ciency 8.4.1 Applications of SD to calendar anomalies 8.4.2 Data 8.4.3 Results 285 285 285 291 293 300 304 305 308 309 309 6.6 6.5.1 The Neyman-Pearson lemma 6.5.2 Consistent tests 6.5.3 The Kolmogorov-Smirnov statistic 6.5.4 Two-sample KS tests 6.5.5 Chi-squared testing Exercises vii © 2010 by Taylor and Francis Group, LLC viii CONTENTS 8.5 318 319 321 321 325 328 331 333 333 335 337 341 343 344 350 351 354 363 364 365 367 374 374 10 Applications to Economics 10.1 Indi^erence curves/location-scale family 10.1.1 Portfolio and expected utility 10.1.2 A dilemma in using the mean-variance criterion 10.2 LS family for n random seed sources 10.2.1 Location-scale expected utility 10.2.2 Indi^erence curves 10.2.3 Expected versus non-expected LS utility functions 10.2.4 Dominance relationships over the LS family 10.3 Elasticity of risk aversion and trade 10.3.1 International trade and uncertainty 10.3.2 LS parameter condition and elasticity 10.3.3 Risk and mean e^ects on international trade 10.4 Income inequality 10.5 Exercises 381 381 381 386 387 389 393 394 398 400 400 401 402 403 406 8.6 SD and rationality of momentum e^ect 8.5.1 Evidence on profitability of momentum strategies 8.5.2 Data and methodology 8.5.3 Profitability of momentum strategy 8.5.4 Results of stochastic dominance tests 8.5.5 Robustness checks Exercises Applications to Risk Management 9.1 Measures of profit/loss for risk analysis 9.1.1 SD criterion for decision-making in risk analysis 9.1.2 MV criterion for decision-making in risk analysis 9.2 REITs and stocks and fixed-income assets 9.2.1 Data and methodology 9.2.2 Empirical findings 9.2.3 Discussion 9.3 Evaluating hedge funds performance 9.3.1 Data and methodology 9.3.2 Discussion 9.4 Evaluating iShare performance 9.4.1 Data and methodology 9.4.2 Results 9.4.3 Discussion 9.5 Exercises © 2010 by Taylor and Francis Group, LLC CONTENTS Appendix Stochastic Dominance Tests A.1 CAPM statistics A.2 Testing equality of multiple Sharpe ratios A.3 Hypothesis testing A.4 Davidson-Duclos (DD) test A.4.1 Stochastic dominance tests for risk averters A.4.2 Stochastic dominance tests for risk seekers A.5 Barrett and Donald (BD) test A.5.1 Stochastic dominance tests for risk averters A.5.2 Stochastic dominance tests for risk seekers A.6 Linton, Maasoumi and Whang test A.6.1 Stochastic dominance tests for risk averters A.6.2 Stochastic dominance tests for risk seekers A.7 Stochastic dominance tests for MSD and PSD Bibliography © 2010 by Taylor and Francis Group, LLC ix 409 409 410 413 414 414 416 417 417 418 419 419 419 420 425 Preface This book is essentially a text for a course on stochastic dominance for beginners as well as a solid reference book for researchers The material in this book has been chosen to provide basic background on the topic of stochastic dominance for various areas of applications The material presented is drawn from many sources in the literature For further reading, there is a bibliography The text is designed for a one-semester course at the advanced undergraduate or beginning graduate level The minimum prerequisites are some calculus and some probability and statistics However, we start from the ground up, and background material will be reviewed at appropriate places Clearly the course is designed for students from fields such as economics, decision theory, statistics and engineering A great portion of this text (Chapters 1—7) is devoted to a systematic exposition of the topic of stochastic dominance, emphasizing rigor and generality The other portion of the text (Chapters 8—10) reports some new applications of stochastic dominance in finance, risk and economics The exercises at the end of each chapter will deepen the students’ understanding of the concepts and test their ability to make the necessary computations After completing the course, the students should be able to read more specialized and advanced journal papers or books on the subject This is an introduction to the topic of stochastic dominance in investment decision science Several points about this text are as follows This text is written for students, especially for students in economics Since the main applications are in financial economics, we have the students in mind in the sense that they should enjoy the text and be able to read through the text without tears This includes self study Even students in economics are aware of the fact that the underlying language of economics is largely mathematical, and thus we proceed to explain clearly at the beginning of any economic concepts why necessary mathematics xi © 2010 by Taylor and Francis Group, LLC xii PREFACE enters the picture Our pedagogy is this: We only bring in mathematics when needed and carefully review or present the mathematical background at the places where it is immediately needed We should not give the impression that economics is reserved for mathematicians! Our point of view is this: Students should think about the material as useful concepts and techniques for economics, rather than a burden of heavy mathematics We want students to be able to read the text, line by line, with enthusiasm and interest, so we wrote the text in its most elementary (and yet general and rigorous) and simple manner For example, if a result can be proved by several di^erent methods, we choose the simplest one, since students need a nice, simple proof that they can understand To assist them in their reading, we will not hesitate to provide elementary arguments, as well as to supply review material as needed Also, we trust that in reading the proofs of results, the students will learn ideas and proof techniques which are essential for their future studies and research In summary, this text is not a celebration of how great the mathematical contributions to economics are, but simply a friendly guide to help students build a useful repertoire of mathematical tools in decisionmaking under uncertainty, especially in investment science W.K Wong would like to thank Z Bai, H Liu, U Broll, J.E Wahl, R Chan, T.C Chiang, Y.F Chow, H.E Thompson, S.X Wei, M Egozcue, W.M Fong, D Gasbarro, J.K Zumwalt, P.L Leung, D Lien, H.H Lean, C.K Li, M McAleer, K.F Phoon, C Ma, R Smyth and B Xia for their contributions to Chapters 8, and 10 and Appendix A All the authors would like to express thanks to their families for their love and support during the writing of this book We thank graduate student Hien Tran at New Mexico State University for his contributions to many proof details in this text as well as for his patience in checking our writing We are grateful to Professor Carol Walker of New Mexico State University for assisting us in producing the manuscript for this book We especially thank Bob Stern, our editor, and his sta^ for their encouragement and help in preparing this text Songsak Sriboonchitta, Wing-Keung Wong, Sompong Dhompongsa and Hung T Nguyen Chiang Mai, Hong Kong, Chiang Mai and Las Cruces © 2010 by Taylor and Francis Group, LLC dG(x) EX (1( ,a) ) EY (1( ,a) ) or i.e., F (x) G(x) We are going to show that the conditions P (X A) P (Y A) for all upper sets A in Rd P (X A) P (Y A) for all lower sets A in Rd are also suQcient for FSD Let u U1 Let u+ = max(u, 0) and u = min(u, 0), the positive and negative parts of u, noting that they are both nonnegative functions, and u+ is nondecreasing, while u is nonincreasing + + For u+ we use the approximation above: u+ n u , with Eun (X) + Eun (Y ) for each n, we have, by the monotone convergence theorem, + + Eu+ (X) = lim Eu+ n (X) lim Eun (Y ) = Eu (Y ) n n ' n For u we can obtain a sequence un (x) = 1/2n n2 i=1 1Ai (x) where each Ai ’s is a lower set in Rd , so that un u Then Eun (X) Eun (Y ) for each n Again, by the monotone convergence theorem, Eu (X) = lim Eun (X) lim Eun (Y ) = Eu (Y ) n n and hence Eu(X) = Eu+ (X) Eu (X) Eu+ (Y ) Eu (Y ) = Eu(Y ) Remark 3.4 In portfolio optimization, there is a random vector of returns X = (X1 , X2 , , Xn ) where, say, Xi is the future return for stock i in our portfolio consisting of n diRerent stocks The investor needs to find an optimal allocation of her budget a to the n stocks © 2010 by Taylor and Francis Group, LLC 96 CHAPTER ISSUES IN STOCHASTIC DOMINANCE Suppose u is her utility function, then the problem is to find an optimal allocation (b1 , b2 , , bn ) maximizing n , bi X i Eu i=1 subject to n , bi = a i=1 Thus we are facing the univariate 'n case: comparing stochastically the random variables (not vectors) i=1 bi Xi for diRerent allocations (b1 , b2 , , bn ) 3.3 Stochastic dominance via quantile functions Although the distribution function of a random variable contains all information about the random evolution of the variable, its quantile function plays a useful role in many aspects of statistical analysis, such as in quantile regression problems in econometrics Here we merely mention that stochastic dominance rules can be expressed also in terms of quantile functions of random variables Definition 3.5 The quantile function or inverse distribution of a distribution F is defined for p (0, 1) as F (p) = inf{x : F (x) p} Remark 3.6 Some useful facts about quantile functions are F (F (x)) F (F (p)) x, x R p, p (0, 1) F (x) t if and only if x F F F F =F 1, F F (t) F = F , and (F F (x)