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Rao’s Quadratic Entropy, Risk Management andPortfolio Theory
Thèse
Nettey Boevi Gilles Koumou
Doctorat en économiquePhilosophiæ doctor (Ph.D.)
Québec, Canada
© Nettey Boevi Gilles Koumou, 2017
Rao’s Quadratic Entropy, Risk Management andPortfolio Theory
Thèse
Nettey Boevi Gilles Koumou
Sous la direction de:
Benoît Carmichael, directeur de rechercheKevin Moran, codirecteur de recherche
Résumé
Cette thèse porte sur le concept de la diversification et sa mesure en théorie des choix deportefeuille. La diversification est un concept clé en finance et en économique, et est au cœurde la théorie des choix de portefeuille. Elle représente l’un des plus importants outils degestion du risque. Ainsi, plusieurs mesures de diversification de portefeuille ont été proposées,mais aucune ne s’est révélée totalement satisfaisante et la discipline recherche toujours uneapproche unifiée et cohérente de mesure et gestion de la diversification.
Cette thèse répond à ce besoin et développe une nouvelle classe de mesures de diversificationde portefeuille en adaptant à l’économie financière l’entropie quadratique de Rao, une mesurede diversité proposée par Rao et utilisée en statistique, en biodiversité, en écologie et dansplusieurs autres domaines. La thèse démontre que si l’entropie quadratique de Rao est biencalibrée, elle devient une classe valide de mesures de diversification de portefeuille résumant,de manière simple, les caractéristiques complexes de la diversification de portefeuille, et offranten même temps une théorie unifiée qui englobe de nombreuses contributions antérieures.
Ensuite, la thèse présente deux applications de la classe de mesures proposée. La premièreapplication s’est intéressée à la stratégie de diversification de portefeuille maximum diversi-fication (MD) développée par Choueifaty and Coignard (2008). Elle propose de nouvellesformulations de cette dernière en se basant sur la classe de mesures proposée. Ces nouvellesformulations permettent de donner un fondement théorique à la stratégie MD et d’améliorerses performances.
La deuxième application s’est intéressée au modèle moyenne-variance de Markowitz (1952).Elle propose une nouvelle formulation de ce dernier en se basant sur la classe de mesures pro-posée. Cette nouvelle formulation améliore significativement la compréhension du modèle, enparticulier le processus de rémunération des actifs. Elle offre également de nouvelles possibilitésd’amélioration des performances de ce dernier sans coûts d’implementation supplémentaires.
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Abstract
This thesis is about the concept of diversification and its measurement in portfolio theory.Diversification is one of the major components of portfolio theory. It helps to reduce or ulti-mately to eliminate portfolio risk. Thus, its measurement and management is of fundamentalimportance in finance and insurance domains as risk measurement and management. Conse-quently, several measures of portfolio diversification were proposed, each based on a differentcriterion . Unfortunately, none of them has proven totally satisfactory. All have drawbacksand limited applications. Developing a coherent measure of portfolio diversification is thereforean active research area in investment management.
In this thesis, a novel, coherent, general and rigorous theoretical framework to manage andquantify portfolio diversification inspiring from Rao (1982a)’s Quadratic Entropy (RQE), ageneral approach to measuring diversity, is proposed. More precisely, this thesis demonstratesthat when RQE is judiciously calibrated it becomes a valid class of portfolio diversificationmeasures summarizing complex features of portfolio diversification in a simple manner andprovides at the same time a unified theory that includes many previous contributions.
Next, this thesis presents two applications of the proposed class of portfolio diversificationmeasures. In the first application, new formulations of maximum diversification strategy ofChoueifaty and Coignard (2008) is provided based on the proposed class of measures. Thesenew formalizations clarify the investment problem behind the MD strategy, help identify thesource of its strong out-of-sample performance relative to other diversified portfolios, andsuggest new directions along which its out-of-sample performance can be improved.
In the second application, a novel and useful formulation of the mean-variance utility functionis provided based on the proposed class of measures. This new formulation significantlyimproves the mean-variance model understanding, in particular in terms of asset pricing. Italso offers new directions along which the mean-variance model can be improved withoutadditional computational costs.
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Contents
Résumé iii
Abstract iv
Contents v
List of Tables ix
List of Figures x
Remerciements xiii
Acronyms xv
Notation xvii
Introduction 1
I Background 9
1 Mathematics Material 101.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.1 Some Special Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.2 Definiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.3 Dissimilarity and Similarity . . . . . . . . . . . . . . . . . . . . . . . 121.1.4 Euclidean Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.5 Pseudo-Euclidean Behaviour . . . . . . . . . . . . . . . . . . . . . . 15
1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.1 Generalized Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.2 Lipschitz continuous function . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Rao’s Quadratic Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Diversification And Portfolio Theory: Review 212.1 Portfolio Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.1 Pre-Modern Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.1.2 Modern Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.2.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.1.2.1.1 Markowitz (1952) . . . . . . . . . . . . . . . . . . 27
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2.1.2.1.2 Tobin (1958)’s Extension . . . . . . . . . . . . . . 302.1.2.1.3 Capital Asset Pricing Model . . . . . . . . . . . . 302.1.2.1.4 Expected Utility Theory . . . . . . . . . . . . . . 332.1.2.1.5 Risk Budgeting Models . . . . . . . . . . . . . . . 34
2.1.2.2 Diversification Principle . . . . . . . . . . . . . . . . . . . . 362.1.2.2.1 Markowitz (1952); Tobin (1958)’s Diversification
Principle . . . . . . . . . . . . . . . . . . . . . . . 362.1.2.2.2 CAPM Diversification Principle . . . . . . . . . . 392.1.2.2.3 Expected Utility Diversification Principle . . . . . 392.1.2.2.4 Risk Budgeting Diversification Principle . . . . . . 43
2.2 Diversification Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.2.1 Law of Large Numbers Diversification Measures . . . . . . . . . . . . 452.2.2 Correlation Diversification Measures . . . . . . . . . . . . . . . . . . 46
2.2.2.1 Embrechts et al. (2009)’s Class of Measures . . . . . . . . . 462.2.2.2 Diversification Ratio . . . . . . . . . . . . . . . . . . . . . . 482.2.2.3 Diversification Return . . . . . . . . . . . . . . . . . . . . . 502.2.2.4 Frahm and Wiechers (2013)’s Measure . . . . . . . . . . . . 51
2.2.3 CAPM Diversification Measures . . . . . . . . . . . . . . . . . . . . 522.2.3.1 Portfolio Size . . . . . . . . . . . . . . . . . . . . . . . . . . 522.2.3.2 Sharpe (1972)’s Measure . . . . . . . . . . . . . . . . . . . 532.2.3.3 Coefficient of Determination or R-squared . . . . . . . . . . 532.2.3.4 Tracking Error . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2.4 Risk Contribution Diversification Measures . . . . . . . . . . . . . . 542.2.4.1 Effective Number of Correlated and Uncorrelated Bets . . . 542.2.4.2 Portfolio Diversification Index . . . . . . . . . . . . . . . . 54
2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
II A Class Of Measures Of Portfolio Diversification Based On Rao’sQuadratic Entropy 57
3 Minimum Desirable Properties Of Coherent Measure Of Portfolio Di-versification 583.1 Minimum Desirable Properties of Portfolio Diversification Measures . . . . . 593.2 Compatibility With the Preference For Diversification in the Mean-Variance
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.3 Coherence of Most Currently Used Methods of Measuring Portfolio Diver-
sification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 Rao’s Quadratic Entropy, Diversification And Portfolio Theory 784.1 Rao’s Quadratic Entropy and Portfolio Diversification . . . . . . . . . . . . 79
4.1.1 Calibration of P, X and P . . . . . . . . . . . . . . . . . . . . . . . 794.1.2 Calibration of D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Examples of Choice of D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2.1 Uncertainty Diversification . . . . . . . . . . . . . . . . . . . . . . . 834.2.2 Factor Risks Diversification . . . . . . . . . . . . . . . . . . . . . . . 864.2.3 Risk Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
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4.2.4 Targeted Diversification . . . . . . . . . . . . . . . . . . . . . . . . . 874.2.5 Diversification When no Information is Available . . . . . . . . . . . 884.2.6 Diversification When Partial Information is Available . . . . . . . . . 89
4.3 Properties of Portfolio RQE . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.3.1 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.3.2 Other Properties of PRQE . . . . . . . . . . . . . . . . . . . . . . . 92
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5 Properties of Rao’s Quadratic Entropy Portfolios 945.1 Theoretical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1.1 Existence and Uniqueness of RQEP . . . . . . . . . . . . . . . . . . 945.1.2 Interior Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.1.3 Monotonicity and Duplication Invariance . . . . . . . . . . . . . . . 965.1.4 Equally Weighted Portfolio . . . . . . . . . . . . . . . . . . . . . . . 975.1.5 Mean-Variance Optimality . . . . . . . . . . . . . . . . . . . . . . . . 985.1.6 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Empirical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.2.1 Fama-French 49 Industry Portfolios Daily Returns . . . . . 1065.2.2.2 Fama-French 100 Portfolios Formed on Size and Book-to-
Market Daily Returns World . . . . . . . . . . . . . . . . . 1125.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
IIIApplications 119
6 Rao’s Quadratic Entropy And Maximum Diversification 1206.1 Maximum Diversification Meets Rao’s Quadratic Entropy . . . . . . . . . . 1216.2 An Alternative Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.3 Improving MDP Out-of-Sample Performance . . . . . . . . . . . . . . . . . 126
6.3.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3.2.1 Scenario I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.3.2.2 Scenario II . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.3.2.3 Scenario III . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.4 Robustness Checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.4.1 Portfolio Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.4.2 Estimation Window . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.4.3 Covariance Matrix Estimation . . . . . . . . . . . . . . . . . . . . . . 136
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7 Rao’s Quadratic Entropy And Mean-Variance Models 1387.1 Rao Meets Markowitz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.1.1 Link Between MVP and RQEP . . . . . . . . . . . . . . . . . . . . . 1397.1.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.2 Rao Meets Sharpe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.2.1 The Capital Market Equilibrium . . . . . . . . . . . . . . . . . . . . 146
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7.2.2 Asset Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.3 Mean-Variance Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.3.1 Heuristic Mean-Semivariance Approaches . . . . . . . . . . . . . . . 1537.3.1.1 Nawrocki (1991)’s Model . . . . . . . . . . . . . . . . . . . 1547.3.1.2 Estrada (2008)’s Model . . . . . . . . . . . . . . . . . . . . 1557.3.1.3 Cumova and Nawrocki (2011)’s Model . . . . . . . . . . . . 156
7.3.2 Estimation Risk Approaches . . . . . . . . . . . . . . . . . . . . . . . 1577.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Conclusion 160
Bibliography 163
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List of Tables
1.1 Some Schoenberg Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 Test of Considered Portfolio Diversification Measures . . . . . . . . . . . . . . . 76
5.1 List of Considered Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2 List of Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.3 Characteristics of Portfolios-49Ind . . . . . . . . . . . . . . . . . . . . . . . . . 1085.4 Other Characteristics of Portfolios-49Ind . . . . . . . . . . . . . . . . . . . . . . 1095.5 Annualized Returns During Bear Markets-49Ind . . . . . . . . . . . . . . . . . . 1105.6 Other Characteristics of Portfolios During Bear Markets-49Ind . . . . . . . . . 1115.7 Characteristics of Portfolios-World100FF . . . . . . . . . . . . . . . . . . . . . . 1135.8 Other Characteristics of Portfolios-World100FF . . . . . . . . . . . . . . . . . . 1145.9 Annualized Returns During Bear Markets-World100FF . . . . . . . . . . . . . . 1155.10 Other Characteristics of Portfolios During Bear Markets-World100FF . . . . . 1165.11 Regression of RQEPDρ
on the MDP . . . . . . . . . . . . . . . . . . . . . . . . 116
6.1 List of Portfolios Considered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.2 Scenario I: Performance of Portfolios . . . . . . . . . . . . . . . . . . . . . . . . 1306.3 Scenario II: Performance of Portfolios . . . . . . . . . . . . . . . . . . . . . . . 1316.4 Scenario III: Performance of Portfolios . . . . . . . . . . . . . . . . . . . . . . . 1336.5 Portfolio Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.6 Sharpe Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
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List of Figures
2.1 Mean-Variance Portfolio Frontier (Without short sales) . . . . . . . . . . . . . . 292.2 Mean-Variance Efficient Frontier With Risk-Free Asset . . . . . . . . . . . . . . 312.3 Relationship Between Portfolio Size and Risk . . . . . . . . . . . . . . . . . . . 38
3.1 Representation of the Effective Number of Correlated Bets for N = 2, υ = 2and % = σ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2 Representation of the Effective Number of Bets for N = 2, υ = 2 and % = σ2 . 74
5.1 R-squared of the Regression of ξi(D) on ξmin(S(∞)) and Slope of the Regressionof∣∣ξ2(D)
∣∣ on ξmin(S(∞)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.2 U.S. Market Portfolio Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1 Scenario I: Portfolio Cumulative Returns . . . . . . . . . . . . . . . . . . . . . . 1296.2 Scenario I: Performance Metrics Depending on Transaction Costs Parameters κ 1306.3 Scenario I: Diversification Level Comparison . . . . . . . . . . . . . . . . . . . . 1316.4 Scenario II: Portfolio Cumulative Returns . . . . . . . . . . . . . . . . . . . . . 1326.5 Scenario II: Performance Metrics Depending on Transaction Costs Parameters κ 1336.6 Scenario III: Portfolio Cumulative Returns . . . . . . . . . . . . . . . . . . . . . 1346.7 Scenario III: Performance Metrics Depending on Transaction Costs Parameters κ 135
7.1 Mean-Variance Portfolio Frontier in the Space(σ2, HD, µ
)(when asset are
risky and short sales are allowed) . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.2 Security Market Plane (SMP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
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À la mémoire de ma méreAnani Kokoe Djiffa Delphine
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An investor who knew futurereturns with certainty wouldinvest in only one security,namely the one with the highestfuture return. If several securitieshad the same, highest, futurereturn then the investor would beindifferent between any of these,or any combination of these. Inno case would the investoractually prefer a diversifiedportfolio. But diversification is acommon and reasonableinvestment practice. Why? Toreduce uncertainty!
Markowitz (1991, pp. 279-280)
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Remerciements
Sans le soutien et l’aide de certaines personnes cette thèse n’aura jamais vu le jour. Je tiensà adresser mes sincères remerciements à toutes ces personnes. En particulier à/au:
Mon directeur de thèse Prof. Benoît Carmichael et à mon codirecteur de thèse Prof. KevinMoran pour leur encadrement, leur disponibilité et leurs précieux conseils;
Prof. Sylvain Dessy pour sa disponibilité et ses précisieux conseils;
Prof. Markus Hermann pour m’avoir offert mon premier travail en tant qu’assistant derecherche et également pour ses conseils;
Prof. Carlos Criado pour m’avoir offert une bourse d’étude et également pour ses conseils;
Université Laval, au département d’économique et aux centres de recherche CIRPÉE et CRE-ATE pour leur soutien financier;
Fonds de recherche sur la société et la culture de Québec pour m’avoir octroyé une bourse dedoctorat en recherche pour étudiants étrangers d’une durée de trois ans;
Tous mes camarades de promotion: Setou Diarra, Ali Yedan, Jean Armand Gnagne, MbéaBell, Aimé Simplice Nono et Isaora Diahali pour toutes les discussions, académiques ou non,la camaraderie et l’amitié que nous avons partagées ensemble;
Mon ami André-Marie Taptue pour son accueil chaleureux à mon arrivée à Québec et ses pré-cieux conseils. Je remercie également sa femme Jeannette son soutien pendant la préparationde mes examens de synthèse;
Leonnie Teki pour son soutien à ma famille à Québec durant ces années d’études;
Roger, le mari de la ma sœur, pour son soutien financier et moral;
Ma famille au Togo, en particulier mon père et ma tante, mes frères Thomas, Assama, Kuevi,Anoumou, Assion et ma soeur Sankou pour leur soutien financier et moral, leur encouragementet leur sacrifice;
Ma conjointe Hermance et à mes enfants Bel-ange et Michael pour leur patience, leur soutien
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moral et financier et leur sacrifice.
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Acronyms
bcbs Basel Committee on Banking Supervisionbm Bouchaud’s measurecalpers California Public Employees’ Retirement Systemcapm Capital Asset Pricing Modelcd Coefficient of Determinationcebs Committee of European Banking Supervisorsceiops Committee of European Insurance and Occupational Pension Supervisorscnd Conditional Negative Definitecr Cumulative Returncrmcr Committee on Risk Management and Capital Requirementscsnd Conditional Strictly Negative Definitecvar Conditional Value-at-Riskcvarr Conditional Value-at-Risk Ratiodd Delta Diversificationdr Diversification Ratioeiopa European Insurance and Occupational Pension Authorityenb Effective Number of Uncorrelated Betsenc Effective Number of Constituentsencb Effective Number of Correlated Betserc Equally Risk Contributionercp Equally Risk Contribution Portfolioeut Expected Utility Theorymvp Mean-Variance Portfolioewp Equally Weighted Portfolioftse Financial Times Stock Exchangegs Gini-Simpsonmd Maximum Diversificationmdd Maximum Drawdown Riskmddr Maximum Drawdown Risk Ratiomdp Most Diversified Portfolioml Maximum Lossmp Market Portfoliompt Modern Portfolio Theorymv Mean-Variancenc2p 2-Norm Constrained Minimum-Variance Portfoliopd Positive Definitepdi Portfolio Diversification Indexprqe Portfolio Rao’s Quadratic Entropyps portfolio Sizepsd Positive Semi-Definitept Portfolio Theoryrb Risk Budgetingrc Risk Contributionrp Risk Parity
xv
rqe Rao’s Quadratic Entropyrqep Rao’s Quadratic Entropy Portfoliosr Sharpe Ratiotobam Think Out of the Box Asset Managementtrn Turnovervar Value-at-Riskvp Minimum-Variance Portfolio
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Notation
Basic Sets, Spaces and Operators
|a| absolute value of a ∈ RCov(.) covariance operatorE(.) expectation operator(RN , ‖.‖2
)Euclidean space
〈., .〉 inner product〈., .〉X inner product on space a ∈ X‖A‖∞ infinity norm of matrix A ∈ RN . ‖x‖∞ = max
1≤j≤N
∑Ni=1 |aij |
‖A‖p Lp norm of matrix A ∈ RN . ‖A‖p =(∑N
i=1 |aij |p)1/p
‖x‖∞ infinity norm of x ∈ RN . ‖x‖∞ = max1≤j≤N
∑Ni=1 |xi|
‖x‖p Lp norm of x ∈ RN . ‖x‖p =(∑N
i=1 |xi|p)1/p
‖.‖ normRN N -dimensional real vector spaceN set of natural numbersA universe of N assets. A = AiNi=1 where Ai is asset i(R(p,q), ‖.‖E
)pseudo-Euclidean space
R vector space of bounded real-valued random variables. R = L∞(Ω,F , P ) isthe vector space of bounded real-valued random variables on a probability space(Ω,F , P ) with Ω is the set of states of nature F
R set of real numbersR− set of negative real numbersR+ set of positive real numbersVar(.) variance operatorW set of long-only portfolios. W = w = (w1, ..., wN )|w ∈ RN+W− set of long-short portfolios. W− = w = (w1, ..., wN )|w ∈ RN
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Vectors and Matrices
ρ correlation matrix. ρ = (ρij)Ni,j=1
D dissimilarity matrix. D = (dij)Ni,j=1
EA eigenvectors matrix of a matrix A. w = (wi, ..., wN )>
µ vector column of asset expected return. µ = (µ1, ..., µN )>
r matrix of asset observed return. r = (ri, ..., rN )>
R vector column of asset future return. R = (R1, ..., RN )>
ri vector column of asset observed return. ri = (ri1, ..., riT )>
σ2 vector column of asset variance. σ2 = (σ21, ..., σ
2N )>
Σ covariance matrix. Σ = (σij)Ni,j=1
σ vector column of asset volatility. σ = (σ1, ..., σN )>
S similarity matrix. S = (sij)Ni,j=1
w portfolio. w = (wi, ..., wN )>
Scalars and Functions
σ2(w) portfolio w weighted average variance. σ2(w) = w>σ2
βi covariance between asset i and the market. βi = σimσ2m
ρij correlation between asset i and j. ρij =σijσiσj
ξi(A) i-th eigenvalue of a square matrix A.ξmax(A) largest eigenvalue of a square matrix A.ξmin(A) lowest eigenvalue of a square matrix A.ξ−(A) number of negative eigenvalue of a square matrix A.ξ+(A) number of positive eigenvalues of a square matrix A.R(w) portfolio w future return. R(w) = w>R
µi asset expected return. µi = E(Ri)
Ri asset i future return (random variable). Ri ∈ Rrit observation of Ri at period t. rit ∈ Rµ(w) portfolio w expected return. µ(w) = E(w>R)
µN+1 = Rf risk free rate or return.a a = µ>Σ−1µ.b b = µ>Σ−11 = 1>Σ−1µ.c c = 1>Σ−11.d d = a c− b2.e e = µ>D−11 = 1>D−1µ.f f = 1>D−11.g g = (σ2)>D−11 = 1>D−1σ2.h h = µ>D−1µ.
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i i = (σ2)>D−1σ2.σi asset volatility or standard deviation. σi =
√Var(Ri)
σ2i asset volatility or standard deviation. σ2
i = Var(Ri)
σ2(w) portfolio w volatility. σ2(w) = w>Σw
σij covariance between asset i and j. σij = Cov(Ri, Rj)
σ(w) portfolio w volatility. σ(w) =√
w>Σw
j j = µ>D−1σ2 = (σ2)>D−1µ.τ risk aversion coefficient.τ risk tolerance coefficient.ς preference for diversification coefficient.% risk measure other than variance. % : R → R
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Introduction
A consumer faces two important economic decisions. First, how to determine his or herconsumption and how to allocate it among goods and services. Second, how to invest hisor her saving among various assets. The first problem is known as the consumption-savingdecision and the second as the portfolio selection decision. The two problems can be analyzedseparately (see Deaton, 2012; Markowitz, 1952, 1959) or jointly (see Merton, 1969, 1971;Samuelson, 1969; Fama, 1970). This thesis is about the portfolio selection decision. It isdevoted to the study of the concept of diversification and its measurement. It develops a novel,coherent, flexible, unified, computational efficient and rigorous approach for diversificationmeasurement in portfolio theory using Rao (1982a)’s Quadratic Entropy.
Motivation and Statement
Portfolio theory is a part of the microeconomics of action under uncertainty. Its primaryconcern is portfolio selection problem and asset pricing theory. Its Achilles’ heel is portfoliodiversification.
Portfolio diversification consists in investing in variety assets and “do not put all your eggsin one basket”. It helps to minimize both the probability of portfolio loss and its severity,through a multilateral insurance in which each asset is insured by the remaining assets. Thekey to the success of this multilateral insurance lies in the interdependence or more generallyin the dissimilarity between assets. More assets are dissimilar, more the probability that theydo poorly at the same time in the same proportion is low and the better is the protectionoffered by this multilateral insurance which is the diversification.
Diversification is a well-established risk management technique in economic and finance longbefore the birth of portfolio theory. Mention of it can be found in Babylonian Talmud: TractateBaba Mezia, folio 42a 1 (see DeMiguel et al., 2009c, pp. 1914; Sullivan, 2011, pp. 1914)
A man should always keep his wealth in three forms: one-third in real estate,another in merchandise, and the remainder in liquid assets.
1http://juchre.org/talmud/babametzia/babametzia.htm.
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in Bernoulli (1738/1954)
It is advisable to divide goods which are exposed to some small danger into severalsmall portions rather than to risk them all together.
and more recently in Leavens (see 1945, pp. 473)
An important question is the extent of diversification that is desirable. Table 1 in-dicates that diversification among 10 items greatly narrows the range of probableresults as compared with 1 item. Adding another 10 will not give as much rela-tive improvement. In fact, it may be shown that in general the improvement bydiversification, in narrowing the spread between probable losses and gains, variesas the square root of the number of items. For example, it would take 40 issuesto give results twice as good (that is, with half the spread) as 10 issues.
Thus, after a reasonable diversification, which naturally can (and must, becauseof inadequate supplies) be larger for a portfolio measured in millions of dollarsthan for one measured in thousands of dollars, the advantage of any further diver-sification hardly balances the difficulty of choosing (and watching) a great varietyof additional securities.
The assumption, mentioned earlier, that each security is acted upon by indepen-dent causes, is important, although it cannot always be fully met in practice.Diversification among companies in one industry cannot protect against unfavor-able factors that may affect the whole industry; additional diversification amongindustries is needed for that purpose. Nor can diversification among industriesprotect against cyclical factors that may depress all industries at the same time.Diversification is only one principle of investment management; it is primarilyto offset lack of full knowledge. The investor must still use all the informationand judgment that he can muster in choosing specific securities and in timing hispurchases and sales.
What was missing, as mentioned in Markowitz (1999), “was an adequate theory of investmentthat covered the effects of diversification when risks are correlated, distinguished betweenefficient and inefficient portfolios, and analyzed risk-return trade-off on the portfolio as awhole”. Portfolio theory contribution therefore has been to provide a set of adequate theories ofinvestment that covered the effects of diversification. The cornerstone of this set is the mean-variance model of Markowitz (1952, 1959) also known under the name of modern portfoliotheory or efficient diversification. This model marks the beginning of portfolio theory andconstitutes therefore the first mathematical foundations of the idea of portfolio diversification.It suggests that a risk-averse investor must construct his or her portfolio maximizing portfolioreturn for a given level of portfolio risk, where return is measured by expected return, risk byvariance and the level of risk is determined by the investor’s ability to assume the risk or theinvestor’s risk aversion coefficient.
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As pointed out in Kolm et al. (2014), Markowitz’s work has had a major impact both onacademic research and the financial industry as a whole. It is the foundation of investmenteducation for chartered financial planner (CFP), for chartered financial analysts (CFA) and formaster of business administration (MBA) (Kitces, 2012). The funds under its management isestimated to around $7 trillion (Solin, 2012). The number of articles in Google Scholar citingMarkowitz’s original paper Portfolio Selection is about 25,857. The paper is also the mostcited articles of all time in The Journal of Finance. When you search for modern portfoliotheory, you obtain about 4,230,000 results. Not surprisingly, Markowitz won a Nobel Prize inEconomics in 1990.
However, the 2008-2009 financial crisis has raised a large number of questions about the capa-bility of (efficient) diversification to protect well against loss. Critics argue that diversificationfails to adequately protect against loss during the 2008-2009 financial crisis, because correla-tions have the tendency to peak during bear markets. For example, Thomas Kieselstein, CIOand managing partner at Quoniam, a quantitative asset management firm based in Frankfurt,in Fabozzi et al. (see 2014, pp. 28-35) says
The financial crisis has clearly shown that when you need diversification most, itmay not work. Historical correlations may simply be wrong. Different liquidity ofdifferent asset classes may mean that some less risky assets may still be punishedbecause they are tradable. We need better management of such extreme situations.
Robert Brown, Ph.D., CFA, with Genworth Financial Asset Management in Encino, Califor-nia, in Holton (2009, pp. 21-22) says
It doesn’t work and it doesn’t really have much of any connection to the realworld...I began to watch it closely starting in the 1980s, and with the passageof each year I’ve appreciated more and more that it’s really nothing more than amarvelous academic concept that has been heavily popularized within the financialplanning community. It’s nothing more than an abstraction from reality thatdoesn’t have a lot of client-based relevance.
As a result, a new portfolio diversification strategy was proposed. This new diversificationstrategy, known under the name of risk contribution diversification or risk parity (see Qian,2011; Maillard et al., 2010), considers that a portfolio is well-diversified if and only if as-set risk contributions are equal. The superiority of the risk contribution diversification overMarkowitz’s diversification remains an open question (see Maxey, 2015; Laise, 2010; Corkeryet al., 2010).
For the defenders, diversification does not fail during the 2008-2009 financial crisis. It ismisunderstood. Ilmanen and Kizer (2012, pp. 15) say
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Yet, diversification has come under attack after the 2007-2009 financial crisis,when diversification seemed to fail as virtually all long-only asset classes, otherthan high-quality sovereign debt, moved in the same direction (down). We arguethat the attacks are undeserved. Instead, we believe that the problem is "usererror"; most investors were never as diversified as they thought they were. Thereis ample room for improvement by shifting the focus from asset class diversificationto factor diversification.
Statman (2013, pp. 11) says
Even those who believe that Markowitz is wrong continue to use the languageof mean-variance portfolio theory. We have gained Markowitz’s mathematicalformulation of diversification and adopted its language of correlations, but in theprocess many have lost the intuition underlying diversification’s benefits.
Miccolis and Goodman (2012, pp. 45) say
Why was our industry not successful in mitigating the effects of the market crashof 2008 for its clients? One key reason was a naive understanding of diversification.
Markowitz et al. (2009) say
It is sometimes said that portfolio theory fails during financial crises because:
- All asset classes go down;
- All correlations go up.
These statements are true, roughly, but should be preceded by the phrase “Aspredicted by portfolio theory” and followed by the phrase, “which is why oneshould use MPT, Modern Portfolio Theory.”
This thesis believes that diversification is not dead. However, it argues that what the 2008-2009 financial crisis highlighted is 1) the need of a risk management regulation taking intoaccount diversification; 2) the need of a rigorous framework to manage and quantify portfo-lio diversification. Since the 2008-2009 financial crisis, considerable efforts were made by theregulators, in particular in insurance and banking industries, to take into account diversifica-tion in risk management regulation (see CEBS, 2010; Laas and Siegel, 2016; CEIOPS, 2010a;CRMCR, 2016; BCBS, 2010, 2013; EIOPA, 2014; CEIOPS, 2010b). However, less is done forthe development of a rigorous framework to manage and quantify portfolio diversification.
Of course, since Markowitz (1952), several portfolio diversification measures were proposed,each based on a different criterion (see Evans and Archer, 1968; Sharpe, 1972; Fernholz andShay, 1982; Booth and Fama, 1992; Woerheide and Persson, 1993; Statman and Scheid, 2005;
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Goetzmann et al., 2005; Rudin and Morgan, 2006; Choueifaty and Coignard, 2008; Meucci,2009; Meucci et al., 2014; Vermorken et al., 2012). Unfortunately, none of them has proventotally satisfactory. All have drawbacks and limited applications. To make matters worse,there is no clear objective criteria that allow to distinguish between coherent and incoher-ent measures. In the absence of clear objective criteria, researchers or portfolio managershave usually based their choice on convenience, familiarity, empirical properties, or on vaguemethodological grounds. This can have major implications in terms of risk and consequentlysocial welfare if they are mistaken. Thus, as in the case of inequality (see Sen, 1976; Atkinson,1987; Allison, 1978), polarization (Esteban and Ray, 1994) and risk (see Wang et al., 1997;Rockafellar et al., 2006a; Frittelli and Gianin, 2005; Stone, 1973; Follmer and Schied, 2010;van der Hoek and Sherris, 2001; Pedersen and Satchell, 1998; Artzner et al., 1999) measure-ment, the necessity is not to have a measure of portfolio diversification, but to have a coherentframework to manage and quantify portfolio diversification. This is the goal of this thesis.
The thesis develops a novel, coherent, flexible, unified, computational efficient and rigorousapproach to portfolio diversification measurement. The suggested diversification statisticsinspiring from Rao’s Quadratic Entropy (RQE) summarizes complex features of portfoliodiversification in a simple manner and provides at the same time a unified theory that includesmany previous contributions.
RQE is a general approach to measuring diversity introduced by Rao (1982a,b) and usedextensively in fields such as statistics (see Rao, 1982b,a; Nayak, 1986b,a) and ecology (seeChampely and Chessel, 2002; Pavoine et al., 2005; Pavoine and Bonsall, 2009; Pavoine, 2012;Ricotta and Szeidl, 2006; Zhao and Naik, 2012). It has also been used in energy policy (seeStirling, 2010) and in income inequality analysis (see Nayak and Gastwirth, 1989). Thisthesis adapts and extends its use to portfolio theory as a new class of portfolio diversificationmeasures.
Contributions
Principal Contribution
Based on RQE, the thesis develops a novel, coherent, flexible, unified, computational efficientand rigorous approach to portfolio diversification measurement called portfolio RQE (PRQE)(Chapter 4) which presents the following advantages:
1. It is easy to interpret. It can be interpreted as a dependence measure or as a multivariateexpected utility;
2. It has low computational cost i.e. computational efficient;3. It is easy to implement;4. It meets ex-ante desirable properties of a coherent portfolio diversification measure when
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asset dissimilarity matrix is homogeneous, translate-invariant and conditionally negativedefinite;
5. It covers both the law of large numbers and the correlation diversification strategies;6. It captures the diversification benefit of the risk-free asset;7. It is extremely flexible. This flexibility allows it to easily:
a) diversify according to any characteristic of assets;b) take into account asset linear and non-linear dependence separately or jointly;c) take into account estimation errors;d) handle sensitivity problems;e) perform targeted diversification;f) perform factors risk diversification;
8. It embeds the portfolio diversification measures such as Gini-Simpson’s index and di-versification return of Booth and Fama (1992) or excess growth rate of Fernholz (2010)offering therefore a novel and useful interpretation of these measures;
9. It governs the diversification in the measures such as diversification ratio of Choueifatyand Coignard (2008), Goetzmann and Kumar (2008)’s measure and Frahm and Wiechers(2013)’s measure offering therefore a novel and useful interpretation of these measures;
10. It governs the diversification in the utility functions such as the mean-variance utilityand Bouchaud et al. (1997)’s general free utility offering therefore a novel and usefulinterpretation of these utility functions.
The thesis also provides some theoretical properties of the optimal portfolio of RQE calledRQE portfolio (RQEP) (Chapter 5, Section 5.1), which is obtained by maximization. Theproperties of RQEP examined are:
1. The existence (Proposition 5.1.1);2. The uniqueness (Proposition 5.1.2);3. The conditions under which it is an interior solution (Proposition 5.1.3);4. The monotonicity property (Proposition 5.1.4);5. The duplication invariance property (Proposition 5.1.5);6. The conditions under which it coincides with the equally weighted portfolio (Proposi-
tion 5.1.6);7. The conditions under which it is mean-variance optimal (Proposition 5.1.8);8. The sensibility relative to the dissimilarity matrix (Propositions 5.1.10 and 5.1.11).
It also provides some empirical properties of RQEP examining the performance of twentyRQEP out-of-sample, using different performance metrics, on two different empirical datasets,against four standard diversified portfolios which are: the equally weighted portfolio, theequally risk contribution portfolio, the most diversified portfolios and the market portfolio(Chapter 5, Section 5.2). The results confirm the superiority of RQEP both in terms returnsand risk, in particular during bear markets. They also provide guidelines in the calibration of
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the dissimilarity matrix.
Secondary Contributions
This thesis also makes other important contributions to portfolio theory.
Literature Review
It provides, for the first time, a review on the concept of diversification and its measurement inportfolio selection (Chapter 2). In this review, four diversification strategies are distinguished:the law of large numbers, the correlation, the CAPM (capital asset pricing model) and therisk contribution diversification strategies. The main portfolio theory models covered thesefour diversification strategies are presented. These models are the mean-variance model, itsTobin (1958) and Sharpe (1964)’s extension, the expected utility theory and the risk budgetingapproach. The portfolio diversification measures covered these four diversification strategiesare also presented, and their advantages and limits are discussed.
Coherent Portfolio Diversification Measure
The thesis also provides, for the first time, a definition of a coherent portfolio diversificationmeasure (Chapter 3). This definition is established in two steps. First, nine desirable proper-ties are postulated formalizing some intuitions of diversification and adapting some propertiesof risk measures in portfolio theory (Section 3.1). The measures satisfying these propertiesare called coherent (Section 3.2, Definition 3.1.3). Next, the compatibility of these proper-ties with investors’ preference for diversification in the mean-variance model is demonstrated(Section 3.2, Propositions 3.2.1). Finally, a list of portfolio diversification measures (reviewedin Chapter 2) is examined against these properties (Section 3.3, Propositions 3.3.1 and 3.3.2).
Maximum Diversification
The thesis also formally establishes the principles at play behind the maximum diversification(MD) approach developed by Choueifaty and Coignard (2008) and used to manage 8 billiondollars U.S by the firm Think Out of the Box Asset Management (TOBAM) (Chapter 6).This is done by showing that the optimal portfolio of the MD strategy maximizes the ratio ofportfolio RQE to portfolio variance or, alternatively, minimizes portfolio variance subject todiversification constraint, where the diversification is measured by RQE. These new formal-izations clarify the investment problem behind the MD strategy and help identify the sourceof its strong out-of-sample performance relative to other diversified portfolios. As a result,the funds under management of TOBAM are not systematically at risk as suggested by thecriticism from Lee (2011) and Taliaferro (2012). Moreover, these new formulations suggestnew directions along which its out-of-sample performance can be improved, and it shows thatthese improvements are economically meaningful.
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Mean-Variance Model
The thesis also provides important contributions in the mean-variance (MV) model. First,it definitely clarifies diversification measurement in the mean-variance model. Contrary toFernholz (2010), it shows that there is a specific measure of portfolio diversification in theMV model, which is the diversification return (Chapter 7, Section 3.2). As a result, sincethe diversification return is a portfolio RQE, diversification in the MV model is RQE diver-sification. Second, it provides a new and useful equivalent formulation of the MV utilityfunction based on RQE (Chapter 7, Section 7.1). This new formulation significantly improvesthe mean-variance model understanding, in particular in terms of asset pricing (Section 7.2).It also offers new directions along which the mean-variance model can be improved withoutadditional computational costs (Section 7.3).
Outline
The remainder of this thesis is organized as follows. Chapter 1 presents a brief review ofsome of mathematics notions and results that are used in this thesis. Chapter 2 reviews theconcept of diversification and its measurement in portfolio selection. Chapter 3 presents theminimum desirable properties for a measure of diversification to be considered as coherent.Chapter 4 adapts and extends the use of Rao’s Quadratic Entropy (RQE) to portfolio theoryas measure of portfolio diversification. The resulting new class of portfolio diversificationmeasures is called portfolio RQE (PRQE) and its optimal portfolio is called RQE portfolio(RQEP). Chapter 5 analyzes the theoretical and empirical properties of REQP. Chapter 6deeply analyzes the relationship between PRQE and diversification ratio of Choueifaty andCoignard (2008). Chapter 7 deeply analyzes the relationship between PRQE and the mean-variance models. In the last chapter, the conclusions of the thesis and future directions areprovided. Chapters 1 and 2 constitute the first part of the thesis, Chapters 3 to 5 the secondpart and Chapters 6 and 7 the third part.
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Part I
Background
9
Chapter 1
Mathematics Material
This first part reviews some background of mathematics and portfolio diversification. The firstchapter presents a brief review of some of mathematics notions and results that are used inthis thesis. This review concerns some notions and results about matrices (Section 1.1), gen-eralized convexity, Lipschitz continuous function (Section 1.2) and Rao’s Quadratic Entropy(Section 1.3). The following notations are adopted. The real, real positive, natural numbersets are denoted R, R+ and N, respectively. The derivatives of univariate function f(.) aredenoted in the usual fashion by f ′ , f ′′ and so forth. Higher-order derivatives are denoted byf (n) for the nth derivative. Partial derivatives of multivariate function are denoted by ∂f(x,y)
∂x
or ∂xf(x, y) and ∂f(x,y)∂x∂y or ∂x yf(x, y). The closed intervals are denoted by brackets, open
intervals by parentheses. For example, x ∈ [a, b]⇔ a ≤ x ≤ b and x ∈ [a, b)⇔ a ≤ x < b witha, b ∈ R.
1.1 Matrices
It is assumed that the reader is familiar with the basic notions of matrix manipulations.Vectors and matrices are written as boldface letters. The identity matrix is denoted by I
and the null vector or matrix by 0. 1 is a vector of ones. δi is a vector with zeros for allelements except the ith, which is one. Transpose operator is denoted by >. Vectors are,unless otherwise specified, column vectors, transposed vectors are row vectors. The inverseof a square matrix A is denoted by A−1. In the case where the spectral decomposition ofA exists, the eigenvectors matrix is denoted by EA and the eigenvalues diagonal matrix byΞA with the diagonal consisting of eigenvalues of A ranked in descending order. The i-theigenvalue of a square matrix A is denoted by ξi(A). Assuming that dimension of A is N ,ξi(A) are ordered such as ζ1(A) ≥ ... ≥ ξi(A) ≥ ... ≥ ξN (A). ξ1(A), the largest eigenvalue,is denoted ξmax(A), while ξN (A), the lowest eigenvalue, is denoted ξmin(A). The number ofpositive eigenvalues of A is denoted by ξ+(A) and that of negative eigenvalues by ξ−(A). Thedeterminant of A is denoted by det(A) and its rank by rank(A).
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Some of matrix operations that will be useful are outlined next.
1.1.1 Some Special Matrices
Definition 1.1.1. Consider an N ×N matrix A = (aij)Ni,j=1.
1) A is non-negative (positive) matrix if and only if aij ≥ 0 (aij > 0) (Non-negativity(Positivity));
2) A is symmetric if and only if for all i, j = 1, ..., N , aij = aji (Symmetric);3) A is hollow if and only if for all i = 1, ..., N , aii = 0 (Hollow);4) A is definite if and only if aij = 0 implies i = j (Definite);5) A verifies a triangle inequality if and only if for all i, j, k = 1, ..., N , aij ≤ aik + akj
(Triangle inequality);6) A verifies a strong triangle inequality if and only if for all i, j, k = 1, ..., N , aij ≤
max(aik, akj) (Strong triangle inequality);7) A is orthogonal if and only if A>A = I;8) A matrix A− is a generalized inverse of A if and only if A A−A = A.9) A is reducible (decomposable) if one may partition 1, ..., N into two non-empty subsets
I1, I2 such that aij = 0, ∀ i ∈ I1 and j ∈ I2, otherwise A is irreducible (indecomposable);10) An M ×M matrix B, M ≤ N is said to be a leading principal sub-matrix of order M if
B is obtained by deleting the last N −M rows and columns of A. The determinant ofB is called the k-th order leading principal minor of A and B is denoted A(M).
1.1.2 Definiteness
Definition 1.1.2. Consider an N ×N symmetric matrix A = (aij)Ni,j=1.
1) A is said to be positive semi-definite (positive definite) if and only if x>Ax ≥ 0 for allx ∈ RN (x>Ax > 0 for all x ∈ RN and x 6= 0) and x>Ax = 0 for some x 6= 0;
2) A is said to be (strictly) positive subdefinite if for all x ∈ RN
x>A x < 0 =⇒ A x ≤ (<) 0 or A x ≥ (>) 0; (1.1)
3) A is said to be conditionally (positive (CPD)) negative definite (CND) if and only ifx>A x ≤ (≥)0 for all x ∈ RN such that
∑Ni=1 xi = 0;
4) A is said to be conditionally strictly (positive (CSPD)) negative definite (CSND) if andonly if x>A x < (>)0 for all x ∈ RN such that
∑Ni=1 xi = 0 and x 6= 0.
Proposition 1.1.1 (Farebrother (1977): Theorems 1 and 2). Let A be a real symmet-ric matrix, and B be a real matrix. Then x>Ax is positive whenever Bx = 0 and x 6= 0 i.e.conditionally strictly definite positive if and only if A+ θB>B is a symmetric positive definitematrix for all θ ≥ θ∗, for some positive scalar θ∗.
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Proposition 1.1.2 (Bavaud (2010): Theorem 1). Let A be a symmetric matrix. For anyα ∈ RN+ such that
∑Ni=1 αi = 1,
GA(α) = −1
2J(α) A J(α)> is positive semi-definite (PSD) ⇔ A is CND. (1.2)
where J(α) = I− 1α> is the N ×N centring matrix and GA(α) is the N ×N Gram matrixassociated to A.
Proposition 1.1.3 (Pekalska and Duin (2005): Remark 3.6). If there is α ∈ RN+ suchthat
∑Ni=1 αi = 1 and GA(α) is PSD, then GA(α) is PSD for all α ∈ RN+ such that
∑Ni=1 αi =
1.
Proposition 1.1.4 (Lau (1985): Theorem 2.1). Let A be a symmetric matrix. If A isCND, then Aκ is CND for any κ ∈ [0, 1].
Proposition 1.1.5 (From Berman and Shaked-Monderer (2003): Proposition 1.3).Any principal sub-matrix of a PSD matrix is also PSD.
Proposition 1.1.6 (Lancaster (1969): Theorem 3.63, pp. 119). If A is a real symmet-ric matrix with eigenvalues ξ1(A) ≥ ξ2(A) ≥ ... ≥ ξN (A), B is a positive semi-definite matrixof rank such that 1 ≤ rank(B) = r ≤ N , and A+B has eigenvalues ξ1(A+B) ≥ ξ2(A+B) ≥... ≥ ξN (A + B), then
1) ξi(A + B) ≥ ξi(A), i = 1, 2, ..., N ;2) ξj−r(A) ≥ ξj(A + B), j = rank(B) + 1, ..., N ;
1.1.3 Dissimilarity and Similarity
Let X be a finite set of cardinality N > 1. Consider a function d : X × X → R.
Definition 1.1.3. d(., .) is said to be a dissimilarity function if and only if d(., .) is non-negative, symmetric and hollow.
Definition 1.1.4. An N ×N matrix D = (d(i, j))Ni,j=1 is called a dissimilarity matrix.
This definition is a special case of that proposed in Orozco (2004), and it is equivalent to thedefinition of quasimetric in Pekalska and Duin (2005). In this thesis a dissimilarity matrixwill be denoted D = (dij)
Ni,j=1 with dij ≡ d(i, j).
Definition 1.1.5. Let D = (dij)Ni,j=1 be an N × N dissimilarity matrix. D is a metric if it
is definite and verified the triangular inequality.
Definition 1.1.6. Let D = (dij)Ni,j=1 be an N × N dissimilarity matrix. D is a ultrametric
if it is definite and verified the strong triangular inequality.
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Now, assume that the dissimilarity function is defined based on real characteristic of individualsof X .
Definition 1.1.7. D = (dij)Ni,j=1 is said to be
(i) homogeneous if and only if d(b i, b j) = |b|κd(i, j);(ii) translate-invariant if and only if d(i+ a, j + a) = d(i, j),
where a, b ∈ R.
For the complement of dissimilarity (similarity), the following definition is adopted.
Definition 1.1.8 (Similarity function). By a similarity matrix will be meant any matrixS = (sij)
Ni,j=1 verifying the following properties
1) 0 ≤ sij ≤ smax;2) sij = sji (symmetry);3) sij = smax ⇐⇒ i = j (Strong reflexivity).
1.1.4 Euclidean Behaviour
Definition 1.1.9. An N ×N dissimilarity matrix D is said to be Euclidean if and only if D
can be embedded in an Euclidean space (RM , ‖.‖2) i.e. there is N points x1, ...,xN in space(RM , ‖.‖2) such that dij = ‖xi − xj‖22. The points x1, ...,xN are obtained through the spectraldecomposition of the Gram matrix GD(α). The point xi is the i-th column of the matrix X
such thatX = EGD
Ξ12GD
. (1.3)
Proposition 1.1.7 (Pekalska and Duin (2005): Theorem 3.13). A dissimilarity matrixD is Euclidean if and only if D2 is CND.
Proposition 1.1.8 (Lemin (2001): Theorem 1.1). Every ultrametric is Euclidean.
Definition 1.1.10 (Schoenberg Transformations). A Schoenberg transformation is a func-tion φ(.) from R+ to R+ of the form (Schoenberg, 1938a)
φ(x) =
∫ ∞0
1− exp(−αx)
αg(α)dα, (1.4)
where g(α)dα is a non-negative measure on R+ and∫∞
1g(α)α dα <∞.
Table 1.1 presents some examples of Schoenberg transformations. By construction, Schoen-berg transformations are characterized by φ(x) ≥ 0 with φ(0) = 0, positive odd derivativesφ(n)(x) ≥ 0 with n odd and negative even derivatives φ(n)(x) ≤ 0 with n even.
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Table 1.1 – Some Schoenberg Transformations
function g(α) transformation φ(x)
g(α) = λ exp(−λα) φ(x) = xα(α+x) , α > 0
g(α) = αΓ(1−α)λ
−α φ(x) = xα, 0 < α < 1
g(α) = δ(λ− α) φ(x) = 1−exp(−αx)α , α ≥ 0
g(α) = exp(−λα) φ(x) = ln(1 + x
α
), α > 0
Berg et al. (see 2008) φ(x) = xα
1+xα , 0 < α < 1
Notes. Γ(.) is a gamma function.
Proposition 1.1.9 (Fundamental Property of Schoenberg Transformations (Bavaud, 2011)).Let φ(.) be a Schoenberg transformation. If D is an Euclidean dissimilarity matrix, thenφ(D) = (φ(dij))
Ni,j=1 is an Euclidean dissimilarity matrix.
Readers are referred to Schoenberg (1938a) and Bavaud (2011) for more details on Schoenbergtransformations.
Proposition 1.1.10 (From Cailliez (1983)). If D is a dissimilarity matrix, then there isconstant d∗ such that for all d ≥ d∗ the matrix with elements dij + d, i 6= j is Euclidean. d∗
is the largest eigenvalue of the matrix(0N×N −J
(1N
)D2 J
(1N
)−IN×N 2 J
(1N
)D J
(1N
) )
Proposition 1.1.11 (Gower and Legendre (1986): Theorem 6, pp. 10). If S is a PSDsimilarity matrix with elements sij such that 0 ≤ sij ≤ 1 and sii = 1, then the dissimilaritymatrix D =
√11> − S with elements dij =
√1− sij is Euclidean.
Definition 1.1.11 (Circum-Euclidean matrix (Tarazaga et al., 1996)). A Circum-Euclideanmatrix is an Euclidean distance matrices generated by points lying on a hypersphere. Formally,consider D = (dij)
Ni,j=1 an Euclidean distance matrix. From Definition 1.1.9, there is N points
x1, ...,xN in space (RM , ‖.‖2) such that dij = ‖xi − xj‖22. If ‖x1‖2 =, ...,= ‖xN‖2, then D isCircum-Euclidean.
The following results characterize a Circum-Euclidean matrix.
Proposition 1.1.12 (Test I for Circum-Euclidean behaviour (Gower, 1985)). An Eu-clidean distance matrix D is said to be Circum-Euclidean if and only 1>D− 1 6= 0, where D−
is a generalized inverse.
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Proposition 1.1.13 (Test II for Circum-Euclidean behaviour (Gower, 1985)). An Eu-clidean distance matrix D is said to be Circum-Euclidean if and only if rank(D) = rank(GD(1/N))+
1.
For other tests for a Circum-Euclidean distance matrix, readers are referred to Tarazaga et al.(1996).
1.1.5 Pseudo-Euclidean Behaviour
Definition 1.1.12 (Pseudo-Euclidean Spaces (Pekalska and Duin, 2005)). A Pseudo-Euclidean Spaces E = R(p,q) is a real vector space equipped with a non-degenerate, indefi-nite inner product 〈., .〉E . E admits a direct orthogonal decomposition E = E+ ⊕ E−, whereE+ = Rp and E− = Rq and the inner product is positive definite on E+ and negative definiteon E−. A vector x ∈ E is represented as an ordered pair of two real vectors: x = (x+,x−).The inner product in E is defined as 〈x,y〉E = x>Ip qy =
∑p
i=1 x+i y
+i −
∑q
i=1 x−i y−i , where
Ip q = [Ip×p 0; 0 Iq×q]. As a result, norm of x ∈ E denoted by ‖.‖2E is defined as ‖x‖2E =
x>Ip qx =∑p
i=1 x+i x
+i −
∑pi=1 x
−i x−i .
Definition 1.1.13 (Pseudo-Euclidean Embedding (Pekalska and Duin, 2005)). An N×N dissimilarity matrix D is said to be Pseudo-Euclidean if it can be embedded in a Pseudo-Euclidean space
(R(p,q), ‖.‖E
)i.e. there is N points x1, ...,xN in space
(R(p,q), ‖.‖E
)such
that dij = ‖xi − xj‖2E . The points x1, ...,xN are obtained through the spectral decompositionof the Gram matrix GD(α). The point xi is the i-th column of the matrix X such that
X = EGD|ΞGD
|12 . (1.5)
Proposition 1.1.14 (Pekalska et al. (2001)). Any dissimilarity matrix is Pseudo-Euclidean.
1.2 Functions
This section first reviews the definition of concave, quasi-concave and pseudo-concave functionsand some of their characteristics. Next, it reviews the definition of Lipschitz continuousfunction and some of their characteristics.
1.2.1 Generalized Convexity
Definition 1.2.1 (Cambini and Martein (2009); Avriel et al. (2010)). Let f be a func-tion defined on the convex set X ⊂ RN . For every x1 ∈ X, x2 ∈ X and α ∈ [0, 1]
(i) f is (strictly) concave if f(αx1 + (1−α) x2) ≥ (>) α f(x1) + (1−α) f(x2) (α ∈ (0, 1)).
15
(ii) f is (strictly) quasi-concave if f(αx1 + (1 − α) x2) ≥ (>) min(f(x1), f(x2)) (x1 6=x2, α ∈ (0, 1)).
Definition 1.2.2 (Cambini and Martein (2009); Avriel et al. (2010)). Let f be a func-tion defined on an open convex set X ⊂ RN . For every x1 ∈ X, x2 ∈ X and α ∈ [0, 1], f is(strictly) pseudo-concave if f(x1) > (≥) f(x2)⇒ (x1−x2)>∇f(x2) > 0 (x1 6= x2), where ∇fis the gradient of f .
The definitions of (strictly) convex, (strictly) quasi-convex and (strictly) pseudo-convex canbe obtained replacing f by −f .
Proposition 1.2.1 (Avriel et al. (2010): Corollary 6.4). Let X be an open convex set inRN and let X denote the closure of X. Then F (x) = 1
2 x>A x + a> x is quasi-concave on X ifand only if F (x) is pseudo-concave on X.
Proposition 1.2.2 (Avriel et al. (2010): Corollary 6.16). Let F (x) = x>A x be non-concave on RN . F (x) is pseudo-concave on RN+ if and only if the following statements aresatisfied
(i) A ≥ 0 i.e. aij ≥ 0, ∀ i, j = 1, ..., N .(ii) ξ+(A) = 1.
Proposition 1.2.3 (Avriel et al. (2010): Theorem 6.20). Let F (x) = x>A x be non-concave on RN . Then F (x) is strictly pseudo-concave on RN+ if and only if the followingstatements are satisfied
(i) A ≥ 0.(ii) (−1)idet(A(i)) < 0, ∀ i = 2, ..., N , where A(i) is the leading principal sub-matrix of order
i of A.
Proposition 1.2.4 (Cambini and Martein (2009): Theorem 3.2.10). Consider the ra-tio h(x) = f(x)
g(x) where f and g are differentiable functions defined on an open convex setX ⊆ RN .
(i) If f is convex and g is positive and affine, then h is pseudo-convex;.(ii) If f is non-negative and convex, and g is positive and strictly concave, then h is strictly
pseudo-convex.
Proposition 1.2.5 (Cambini and Martein (2009): Theorem 3.2.11). Let f : X ⊆ RN →R be a pseudo-convex (strictly pseudo-convex) function on an open convex set X and letφ : R → R be a differentiable function such that φ′(x) > 0,∀x ∈ R. Then, the compositefunction φ f is pseudo-convex (strictly pseudo-convex).
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1.2.2 Lipschitz continuous function
Definition 1.2.3 (Klatte (1985); Han et al. (2015)). A function g : X ⊂ RN → R issaid to be Lipschitzian on X if there is a constant γ1 such that
|g(x1)− g(x2)| ≤ γ1‖x1 − x2‖, ∀ x1,x2 ∈ X. (1.6)
where γ1 is a Lipschitz constant for the function g and ‖.‖ is a norm defined on X and |.| isabsolute value operator.
Definition 1.2.4 (Bonnans and Shapiro (1998)). Consider the following optimization prob-lem
minx∈X⊂RN
f(x). (1.7)
Denote X∗ the set of optimal solutions. f satisfies the growth condition of order κ > 0 if thereexists a constant γ2 > 0 such that
f(x)− f(x∗) ≥ γ2 (dist(x,X∗))κ , (1.8)
where dist(x,X∗) = infx∗∈X∗
‖x− x∗‖.
Definition 1.2.5 (Guigues (2011): Definition 3.1, pp. 561). For any symmetric matrixA, let γ(A) be such that the quadratic function x>A x is γ(A)-strongly convex with respect to‖.‖1 i.e.
γ(A) = infx 6=0
x>A x
‖x‖21(1.9)
Proposition 1.2.6 (Guigues (2011): Proposition 3.1, pp. 561). Consider the two op-timization problems
minx∈X
f1(x) (1.10)
minx∈X
f2(x) (1.11)
where f1, f2 : X ⊆ RN → R. Let X∗1 be the set of solutions of problem (1.10) and let x∗2 be asolution of problem (1.11). If (i) f1 satisfies a second-order growth condition on X and (ii) thefunction f2 − f1 is Lipschitz continuous with modulus γ1 on X, then there is a constant γ2
such thatdist (x∗2,X∗1) ≤ γ1
γ2(1.12)
Proposition 1.2.7 (Guigues (2011): Lemma 3.1, pp. 561). Let A be a real symmetricmatrix, then sup
x∈RN+ ,x>1=1
‖Ax‖2 = maxi‖Ci(A)‖2, where Ci(A) is the i-th column of A.
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1.3 Rao’s Quadratic Entropy
Rao’s Quadratic Entropy (RQE), also known under the names of Diversity Coefficient (Rao,1982b) or Quadratic Entropy (Rao and Nayak, 1985), is a general approach to measuringdiversity introduced by Rao (1982a,b). Given a population of individuals P, it is definedas the average difference between two randomly drawn individuals from P. More formally,suppose that each individual in P is characterized by a set of measurement X and denote byP the probability distribution function of X. RQE of P is defined as
HDP (P ) =
∫d(X1, X2)P (dX1)P (dX2), (1.13)
where the non-negative, symmetric dissimilarity function d(., .) expresses the difference be-tween two individuals from P.
Suppose that one individual is drawn from a population P1 and another from a populationP2. Then
HDP1×P2(P1, P2) =
∫d(X1, X2)P1(dX1)P2(dX2). (1.14)
When X is a discrete random variable, (1.13) and (1.14) become
HDP (p) =
|P|∑i,j=1
dP,ijpi pj , (1.15)
HDP1×P2(p1,p2) =
|P1|∑i=1
|P2|∑j=1
dP1×P2,ijp1i p2j , (1.16)
(1.17)
where p = (p1, ..., pN )> and pk = (pk1, ..., pkN )> with pi = P (X = xi), pki = P (Xk =
xki), k = 1, 2, |P| is a cardinal of P, DP = (dP,ij)|P|i,j=1 is a dissimilarity matrix with dP,ij the
dissimilarity between individual i and j from P and DP1×P2 = (dP1×P2,ij)(|P1|,|P2|)(i=1,j=1) is a dis-
similarity matrix with dP1×P2,ij the dissimilarity between individual i form P1 and individualj from P2. The interpretation of RQE is straightforward: the higher HDP (p) is, the higherthe diversity of P is; the higher HDP1×P2
(p1,p2) is, the higher the difference between P1 andP2 is.
RQE has been used extensively in fields such as statistics (see Rao, 1982b,a; Nayak, 1986b,a)and ecology (see Champely and Chessel, 2002; Pavoine et al., 2005; Pavoine and Bonsall, 2009;Pavoine, 2012; Ricotta and Szeidl, 2006; Zhao and Naik, 2012). It has also been used in energypolicy (see Stirling, 2010) and in income inequality analysis (see Nayak and Gastwirth, 1989).Several generalization of RQE were also proposed (see Leinster and Cobbold, 2012; Stirling,2010; Ricotta and Szeidl, 2006; Guiasu and Guiasu, 2011). This thesis proposes to adapt andto extend its use to portfolio theory as a new class of portfolio diversification measures. The
18
following results will be necessary for this achievement. When it is not necessary to distinguishbetween population of individuals, the notation HDP (p) will be replaced by HD(p).
Proposition 1.3.1 (Rao (1982a): pp. 7). HD(p) is concave if and only if the (N − 1)×(N − 1) matrix
(diN + djN − dij) , i, j = 1, ..., N − 1 (1.18)
is PSD.
Rao (2010) shows that Proposition 1.3.1 is equivalent to the following one.
Proposition 1.3.2 (Rao (2010): Lemma 3.1, pp. 76). HD(p) is concave if and only ifD is CND.
Pavoine (2012) establishes the condition under which HD(p) is strictly concave.
Proposition 1.3.3 (Pavoine (2012): pp. 515). HD(p) is strictly concave if and only ifrank
(GD
(1N
))= N − 1.
Proposition 1.3.4 (Rao and Nayak (1985)). If P1, ..., PM are the distributions of X inpopulations P1, ...,PM with a priori probabilities α1, ..., αM , then the distribution in the mixturepopulation P0 ≡
∑Mi=1 αiPi is P0 =
∑Mi=1 αiPi. If RQE is concave, then
HDP0(P0) =
M∑i=1
αiHDPi(Pi) +
M∑i,j=1
αi αjDHDPi×Pj(Pi, Pj), (1.19)
where∑M
i,j=1 αi αjDHDPi×Pj(Pi, Pj) is called the Jensen difference and DHDPi×Pj
(Pi, Pj) iscalled the directed divergence or cross entropy (Rao and Nayak, 1985) defined as follows
DHDPi×Pj(Pi, Pj) = 2HDPi×Pj
(Pi, Pj)−HDPi(Pi)−HDPj
(Pj) (1.20)
Rao and Nayak (1985) also show that√DHDPi×Pj
(., .) is a metric on the space of multinomial
distributions when D is CND.
Proposition 1.3.5 (Rao and Nayak (1985): Theorem 3.1, pp. 592).√DH(., .) is a met-
ric on the space of multinomial distributions when D is CND.
Shimatani (2001, pp. 140) also shows that HD (p) can be decomposed as follows.
Proposition 1.3.6 (Shimatani (2001): pp. 140). For any D and p, HD(p) can be de-composed as follows
HD (p) =1
2G(p)×A (D) +
1
2B (p,D) , (1.21)
19
where
G(p) =
|P|∑i 6=j=1
pi pj (1.22)
A (D) =2
|P|(|P| − 1)DT (D) (1.23)
B (p,D) =
|P|∑i 6=j=1
(d(xi, xj)−A (D))(pi pj − G(p)
)(1.24)
with G(p) = 2|P|(|P|−1)G(p) and DT (D) =
∑|P|i=1 d(xi, xj).
The optimal distribution p is obtained maximizing HD (p). Denote p∗ the optimal distribu-tion. Pavoine and Bonsall (2009) provide some interesting results on p∗.
Proposition 1.3.7 (Pavoine and Bonsall (2009), pp. 155). Consider the optimization prob-lem
maxp
HD (p) (1.25)
1) The problem (1.25) can have several solutions.2) Let p∗1 and p∗2 two solutions of (1.25). Then, for all α ∈ [0, 1], αp∗1 + (1 − α)p∗2 is a
solution of (1.25).3) Let p∗1 and p∗2 two solutions of (1.25). Then, DH(p∗1,p
∗2) = 0.
4) If√
D is ultrametric, thena) The problem (1.25) has a unique solution on R and
p∗2 =D−11
1>D−11. (1.26)
b) Dp∗
1>D−11= HD (p∗) 1.
c) DHDP×P(p,p∗) = (HD (p∗)−HD (p)) /2.
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Chapter 2
Diversification And Portfolio Theory:Review
This chapter has two goals: 1) stress the importance of diversification in portfolio selection;2) provide a selective review of the methods for its measurement.
To achieve the first goal, Section 2.1 reviews some pre-modern and modern portfolio selectionrules and their diversification principle. These rules cover the four existing portfolio diver-sification strategies which are the law of large numbers, the correlation, the CAPM (capitalasset pricing model) and the risk contribution strategies. The pre-modern rules are a list ofadvice and prescriptions. The modern rules are the mean-variance models (including Tobin’sextension and the CAPM), the expected utility theory and the risk budgeting approaches.
Section 2.2 focuses on the second goal. A selective review of the methods for measuring portfo-lio diversification is provided. Only the most popular measures are considered. These measuresare grouped in four categories according to the diversification strategies. The advantages andshortcomings of each considered measure are also discussed.
2.1 Portfolio Selection Rules
Consider an investor with a given initial wealth denotedW . The problem faces by this investoris how to allocate his wealth W among N alternative investment opportunities. This problemis known under the name of portfolio selection problem. More formally, the portfolio selectionproblem is to find the optimal allocation or portfolio w = (w1, ..., wN )> which best suits theneeds of the investor, where wi is the share of W invested in asset i such that
∑Ni=1wi = 1.
In the case where the investment opportunities or assets are risk-free, the solution of thisproblem is trivial. The optimal allocation is to put the total wealth W on the asset withthe highest future return. However, in the case where some assets are risky, it is no longeroptimal to concentrate. Diversification becomes recommended. Therefore, several rules based
21
on diversification have been proposed to solve portfolio selection problem. In what follows,some of them are reviewed as well as their diversification principle.
A portfolio w will be said to be diversified if wi = 0 for at least one i and wi > 0 for at leasttwo i. A portfolio w will be said to be completely diversified if wi > 0 for all i. A portfoliow will be said to be well-diversified if it is an optimal portfolio of a measure of portfoliodiversification or of a diversification strategy. A portfolio w will be said to be concentratedif wi = 1 for some i. The concentrated portfolio is also called a single asset portfolio. It isassumed that all asset are not risk-free. and the initial wealth W is normalized to 1, unlessotherwise specified.
There are two types of portfolio selection problems: the one-period and the multi-periodproblems. In the one-period specification, the investor constructs his or her portfolio at thebeginning of the period (time 0) and get the pay off at the end (time T ). In a multi-periodproblem, the investor constructs his or her portfolio at the beginning of the period (time 0)and thereafter restructured it at t = 1, ..., T − 1 before he obtains his/her reward after thefinal period, at time T . This thesis concerns the one-period problem. For the multi-periodproblem readers are referred to Samuelson (1969), Merton (1969), Steinbach (2001), Li andNg (2000) and Markowitz (1959).
2.1.1 Pre-Modern Rules
The situation prior to Markowitz (1952)’s seminal work Portfolio selection published in TheJournal of Finance, is described by Bernstein (2005) and Markowitz (1999) itself.
Bernstein (2005, pp. 55)
Before Markowitz, we had no genuine theory of portfolio construction, only rulesof thumb and folklore.
Markowitz (1999, pp. 5)
What was lacking prior to 1952 was an adequate theory of investment that coveredthe effects of diversification when risks are correlated, distinguished between effi-cient and inefficient portfolios, and analyzed risk-return trade-offs on the portfolioas a whole.
The following reports some of these rules of thumb and folklore.
Shakespeare and Phelps (c1923, Act I, Scene 1, pp.) (see also Markowitz, 1999; Rubinstein,2002)
My ventures are not in one bottom trusted, Nor to one place; nor is my whole
22
estate Upon the fortune of this present year; Therefore, my merchandise makesme not sad.
Babylonian Talmud: Tractate Baba Mezia, folio 42a 1 (DeMiguel et al., 2009c, pp. 1914; Sul-livan, 2011, pp. 1914)
A man should always keep his wealth in three forms: one-third in real estate,another in merchandise, and the remainder in liquid assets.
Bernoulli (1738/1954) (see also Sullivan, 2011, pp. 74)
It is advisable to divide goods which are exposed to some small danger into severalsmall portions rather than to risk them all together.
Keynes (1934)2 (see Keynes and (Grande-Bretagne), 1971)
As time goes on I get more and more convinced that the right method in investmentis to put fairly large sums into enterprises which one thinks one knows somethingabout and in the management of which one thoroughly believes. It is a mistaketo think that one limits one’s risk by spreading too much between enterprisesabout which one knows little and has no reason for special confidence. [...] One’sknowledge and experience are definitely limited and there are seldom more thantwo or three enterprises at any given time in which I personally feel myself entitledto put full confidence.
Loeb (1965)
When an investment is made, its prospect must be so good that placing a ratherlarge proportion of one’s total funds in such a single situation will not seem ex-cessively risky. At the same time, the potential gain must be so large that onlya moderate portion of total capital need invested to get the desired percentageappreciation on total funds. Expressing the matter in different way, this meansthat diversification is undesirable. One or two, or at most three or four securitiesshould be bought. And the should be so well selected, their purchase so expertlytimed and their profit possibilities so large that it will never be necessary to riskin any of them a large proportion of available capital. Under this policy, only thebest is bought at the best possible time. Risk are reduced in two ways-first bythe care used in selection and, second, by the maintenance of a large cash reserve.Concentration of investments in a minimum of stocks insures that enough timewill be given to the choice of each so that every important detail about them
1http://juchre.org/talmud/babametzia/babametzia.htm.2From Keynes’ letter to F. C. Scott, 15 August, 1934 (see Keynes and (Grande-Bretagne), 1971). This
letter is available at http://www.berkshirehathaway.com/letters/1991.html.
23
will be known. This policy involves not only avoiding diversification but also attimes holding one’s capital uninvested for long periods of time (see Loeb, 1965,pp. 41-42).
The beginner needs diversification until he learns the ropes. For those who areaccomplished I think most accounts have entirely too much diversification of thewrong sort and not enough of the right. [...] Some geographical diversificationmight be justified for large funds. This sort of thing might be necessary whencapital reaches an unwieldy total, or it might be necessary where no intelligentsupervision is likely. Otherwise, it is an admission of not knowing what to do andan effort to strike an average. The intelligent and safe way to handle capital isto concentrate. If things are not clear, do nothing. When something comes up,follow it to the limit, subject to the method of procedure that follows. If it’s notworth following to the limit, it is not worth following at all. My thought, of course,is always start with a large cash reserve; next, begin in one issue in a small way.If it does not develop, close out and get back to cash. But if it does do what isexpected of it, expand your position in this one issue on a scale up. After, butnot before, it has safely drawn away from your highest purchase price, then youmight consider a second issue. The greatest safety lies in putting all your eggs inone basket and watching the basket. You simply cannot afford to be careless orwrong. Hence, you act with much more deliberation (see Loeb, 1965, pp. 119).
Graham and Dodd (1951, Chap. 31, Summary, pp. 403)
In this discussion we have described various current approaches to common-stockinvestment and have expressed our own views as to their relative possibilities anddifficulties. Our preference is either for the simple purchase of a diversified listof primary issues at reasonable price levels- a task well within the competence ofa defensive investor- or else for the effort, by means of skillful security analysis,to find common stocks selling well below their intrinsic value. The student hasbeen sufficiently warned, we hope, that our judgement on these matters is notnecessarily shared by the majority of experienced investors or practicing securityanalysis.
Graham (1949)
The defensive investor who follows our suggestions will purchase only UnitedStates government bonds plus a diversified list of leading common stocks. [...] Itis natural for the investor to inquire whether we believe his list should be limitedto so-called industrial companies, or whether it could or should include represen-tative common stocks of railroads, public utilities, and financial enterprises. Inour view there is no superior advantage in any fixed pattern of diversification. Itis not essential to spread risk around, in pre-established proportions, so that eachof the major categories of American enterprise is included. What is essential is
24
that a reasonable diversity of industries be achieved , so that the investor can feelhe has his stake in a fairly good cross section of the economy (see Graham, 1949,Chap. XVI, pp. 147-156).
Diversification is an established tenet of conservative investment. By accepting itso universally, investors are really demonstrating their acceptance of the margin-of-safety principle, to which diversification is the companion. (see Graham, 1949,Chap. XVI, pp. 258).
Williams (see 1956, pp. 67-69)
The customary way to find the value of a risky security has always been to add apremium for risk to the pure interest rate, and then use the sum as the interestrate for discounting future receipts.[...] Strictly speaking, however, there is no riskin buying the bond in question if its price is right. Given adequate diversification,gains on such purchases will offset losses, and a return at the pure interest ratewill be obtained. Thus the net risk turns to be nil.
Leavens (see 1945, pp. 473)
An important question is the extent of diversification that is desirable. Table 1 in-dicates that diversification among 10 items greatly narrows the range of probableresults as compared with 1 item. Adding another 10 will not give as much rela-tive improvement. In fact, it may be shown that in general the improvement bydiversification, in narrowing the spread between probable losses and gains, variesas the square root of the number of items. For example, it would take 40 issuesto give results twice as good (that is, with half the spread) as 10 issues.
Thus, after a reasonable diversification, which naturally can (and must, becauseof inadequate supplies) be larger for a portfolio measured in millions of dollarsthan for one measured in thousands of dollars, the advantage of any further diver-sification hardly balances the difficulty of choosing (and watching) a great varietyof additional securities.
The assumption, mentioned earlier, that each security is acted upon by indepen-dent causes, is important, although it cannot always be fully met in practice.Diversification among companies in one industry cannot protect against unfavor-able factors that may affect the whole industry; additional diversification amongindustries is needed for that purpose. Nor can diversification among industriesprotect against cyclical factors that may depress all industries at the same time.Diversification is only one principle of investment management; it is primarilyto offset lack of full knowledge. The investor must still use all the informationand judgment that he can muster in choosing specific securities and in timing hispurchases and sales.
25
In sum, portfolio diversification as risk reduction strategy was a well-established practicelong before the portfolio theory (PT). For example, Bernoulli (1738/1954) and Williams (see1956, pp. 67-69)’s advice can be related to the law of large numbers diversification strategy,and Shakespeare and Phelps (c1923, Act I, Scene 1, pp.), Leavens (see 1945, pp. 473) andBabylonian Talmud’s advice to the correlation diversification strategy. Only the CAPM andthe risk contribution diversification strategies do not exist before the PT. The correlationdiversification strategy was mathematically formalized, for the first time, by Markowitz (1952)and will be presented in the next section. The law of large numbers diversification strategy isroughly defined as follows.
Definition 2.1.1 (Law of Large Numbers Diversification Strategy). The law of largenumbers (LLN) diversification strategy consists in investing a small fraction of wealth in eachof a large number of assets.
In other words, the LLN diversification is a diversification in terms of portfolio weights andsize. In this diversification strategy, the third dimension of diversification, which is portfoliorisk, is not explicitly taken into account. The effect of the LLN diversification is covered inthe arbitrage pricing theory (see Ross, 1976; Chamberlain, 1983b; Ingersoll, 1984) and thecapital asset pricing model (see Sharpe, 1964). The most used LLN diversification strategy isthe naive diversification (or equally weighted portfolio in practice), which consists in investingthe same amount of wealth in each available asset. This diversification strategy was provedunder some conditions to outperform a sophisticate portfolio selection rules (see DeMiguelet al., 2009b; Pflug et al., 2012). Consequently, several exchange traded and mutual funds arelaunched based on this diversification strategy by several firms (see for example GuggenheimInvestments, Invesco).
2.1.2 Modern Rules
The year 1952 saw the publication of Markowitz’s seminal paper Portfolio selection in TheJournal of Finance. Its publication transforms portfolio selection and marks the beginning ofPT. Consequently, Markowitz is often considered as the father of PT. An honour shared withRoy (1952)3 (Markowitz, 1999, pp. 5). In 1990, he was also awarded the Nobel Prize in Eco-nomics for his work in PT. This section first reviews Markowitz (1952)’s mean-variance (MV)model and its extensions proposed in Tobin (1958) and Sharpe (1964), the expected utilitytheory and the risk budgeting approaches. Second, it reviews the diversification principle ineach model.
3About three months after the publication of Markowitz (1952)’s paper, Roy (1952) also proposed aportfolio selection model known under name of “safety-first”. This approach consists to minimize the probabilityof the portfolio’s return falling below a minimum desirable level. When minimum desirable level of return equalto the risk-free rate and portfolio returns is normally distributed, Roy (1952)’s coincides with the Sharpe ratiomaximization
26
2.1.2.1 Models
2.1.2.1.1 Markowitz (1952)The following is based upon Markowitz (1952, 1959). Readers are also referred to Steinbach(2001) for more details. Markowitz (1952)’s approach is based on the idea that, in the presenceof uncertainty, a risk-averse investor does not only want portfolio return to be high, but italso wants portfolio return to be “dependable, stable, not subject to uncertainty” (Markowitz,1959, Part I, Chap. I, pp. 6). Markowitz argues therefore that an investor must always holda portfolio chosen among the set of efficient portfolios i.e. those with maximum return for agiven level of uncertainty, where the level of uncertainty is determined by the willingness andthe ability of the investor to assume uncertainty.
Formally, consider an universe of N risky assets. Denote Ri the future return of asset i. Onecommon needs for all investors is portfolio future return defined as
R(w) = w>R, (2.1)
where R = (R1, ..., RN )> is the N × 1 vector of asset future returns. Considering Ri, ∀ i =
1, ..., N as a random variable. Assume that Ri ∈ R where R = L∞(Ω,F , P ) is the vectorspace of bounded real-valued random variables on a probability space (Ω,F , P ), with Ω isthe set of states of nature, F is the σ−algebra of events, and P is a σ−additive probabilitymeasure on (Ω,F). Denote E(.), Cov(.) and Var(.) the expectation, covariance and varianceoperators, respectively. Markowitz predicts R(w) using portfolio expected return
µ(w) = w>µ (2.2)
and measures the uncertainty of this return in the future using portfolio variance
σ2(w) = w>Σ w, (2.3)
where µ = (µ1, ..., µN )> is the N × 1 vector of asset expected returns with µi = E(Ri)
and Σ = (σij)Ni,j=1 the N × N asset returns covariances matrix with Cov(Ri, Rj) = σij the
covariance between asset i and j and Cov(Ri, Ri) = Var(Ri) = σii ≡ σ2i the variance of
asset i. Portfolio variance characterizes how far, on average, portfolio return deviates fromthe portfolio’s average return. Markowitz also interprets it as a measure of portfolio riskexplicitly, and uses it to quantify the diversification effect in the reduction of uncertainty orrisk.
Given the beliefs, µ and Σ, the investor finds his or her efficient portfolio resolving the followingoptimization problem
wMV ∈ minw∈W
w>Σ w (2.4)
s.t w>µ ≥ µ, (2.5)
27
where the set W =
w ∈ RN+∣∣w>1 = 1
defines the investor budget constraint and short
sales restriction with 1 the N × 1 vector of ones. The threshold µ is specified by the investorin accordance with his or her willingness and ability to assume uncertainty.
Alternative formulations of the problem (2.4)-(2.5) are obtained by either maximizing theexpected return subject to an upper limit on the portfolio variance
wMV ∈ maxw∈W
w>µ (2.6)
s.t w>Σ w ≤ σ2 (2.7)
or by maximizing the linear combination of portfolio expected return and variance or therisk-adjusted expected return
wMV ∈ minw∈W
w> µ− 1
τw>Σ w (2.8)
where 1τ ≥ 0 is the parameter measuring the degree of aversion of the investor towards un-
certainty which measures how the investor views the tradeoff between risk or uncertainty andreturn: the higher is 1
τ , the greater is the aversion of the investor towards uncertainty. Theparameter τ therefore measures the degree of tolerance of the investor towards uncertainty. τis specified by the investor, as well as σ2, in accordance with his or her willingness and abilityto assume uncertainty or risk. This thesis focuses only on the formulation (2.8).
To solve the problem (2.4)-(2.5), Markowitz (1956, 1959, 1987) propose a critical line algorithmwhich was later improved by others (see Perold, 1984; Jacobs et al., 2005; Steuer et al.,2006; Jacobs et al., 2006; Babaitsev et al., 2012). Since the problem (2.4)-(2.5) are quadraticprogram, it can also be solved using standard numerical optimization software.
By solving problem (2.4)-(2.5) for all possible values of µ from µmin = miniµi to µmax =
maxi
µi, one can numerical obtain the MV efficient frontier. Figure 2.1 shows an example ofMV efficient frontier when short sales are restricted. The dashed curve is the MV inefficientfrontier, while the solid one is the efficient frontier. The dashed and the solid curves representtogether the MV portfolio frontier, which is obtained solving the following problem
minw∈W
1
2w>Σ w
s.t w>µ = µ
for all possible values of µ from µmin to µmax. The MV frontier is therefore the part of theportfolio frontier which is undominated.
The efficient frontier can also be generated by solving problem (2.6)-(2.7) for all possible valuesof σ2 from 0 to the variance of asset with higher expected return, or by solving problem (2.8)for all possible values of τ from 0 to +∞.
28
Figure 2.1 – Mean-Variance Portfolio Frontier (Without short sales)
0 200 400 600 800 1,000 1,200 1,400 1,600 1,800
4
6
8
10
12
Variance
Exp
ectedreturn
Efficient FrontierInefficient Frontier
Note: this figure is not based on real data.
An analytical expression of the MV optimal portfolio (wMV ) and portfolio frontier can beobtained when short sales are unrestricted. Define
a =µ>Σ−1µ (2.9)
b =µ>Σ−11 = 1>Σ−1µ (2.10)
c =1>Σ−11 (2.11)
d =a c− b2. (2.12)
From the problem (2.8), one can show that (see Merton, 1972)
wMV =2τ
(Σ−1µ− b
cΣ−11
)+
Σ−11
c(2.13)
µ(wMV
)=2τ
d
c+b
c(2.14)
σ2(wMV
)=4τ2d
c+
1
c. (2.15)
From (2.14)-(2.15), the expression of the frontier portfolio is parabola is the space (σ2, µ)
σ2(wMV
)=c µ2
(wMV
)− 2b µ
(wMV
)+ a
d(2.16)
and the efficient frontier is
µ(wMV
)=b
c+
1
c
√d c
(σ2 (wMV )− 1
c
). (2.17)
29
2.1.2.1.2 Tobin (1958)’s ExtensionMarkowitz’s work had no immediate major impact on finance until after Tobin (1958)’s work.Tobin uses the MV model to the analysis of the preference for liquidity as behaviour towardsrisk. Assuming the MV model with N risky assets and one risk-free asset, cash, the authorprovides an extremely important extension of Markowitz model. He shows that, in the presenceof a risk-free asset, a risk-averse investor can find his or her efficient portfolio in two separatedsteps. Denote wf the share of wealth invested in the risk-free asset. The first step consistsin finding the allocation w of 1− wf , the share of wealth invested in the risky assets, amongrisky assets maximizing the Sharpe ratio of the risky assets portfolio
maxw∈W−
µ(w)−Rfσ(w)
, (2.18)
where Rf is the risk-free asset return and W− =
w ∈ RN∣∣w>1 = 1
the set of long-short
portfolios. The solution of the problem (2.18) is called the tangent portfolio and will bedenoted wTG. The second step consists in finding wf
1− wf = 2τµ (w)−Rfσ2 (w)
. (2.19)
This result is known under the name of Tobin’s separation theorem. The first step is deter-mined by µ and Σ, and the second step by the investor risk tolerance coefficient, τ . Theimplications of Tobin separation theorem are that
(i) The efficient frontier becomes a line with equation (see Figure 2.2)
µ(wMV
)= Rf +
(µ(wTG
)−Rf
σ (wTG)
)σ(wTG
)(2.20)
and every efficient portfolio is a mixture of a portfolio consisting only of risk-free asset,wf , and tangent portfolio, wTG
wMV =(wf , (1− wf ) wTG
). (2.21)
(ii) All investors hold the same risky portfolio, a tangent portfolio, but in different propor-tions.
2.1.2.1.3 Capital Asset Pricing ModelThe capital asset pricing model (CAPM) is the most famous model in asset pricing. It wasdeveloped separately by Sharpe (1964), Lintner (1965) and Mossin (1966). The model assumesan economy of K investors and N risky assets plus one risk-free asset in which each investork allocating his or her wealth Wk among available assets following Markowitz (1952)’s MVmodel. For convenience, Wk, will be expressed here as a proportion of aggregate investedwealth. The typical investor k problem is therefore
maxwk∈W−
w>k µ−1
τkw>k Σ wk, (2.22)
30
Figure 2.2 – Mean-Variance Efficient Frontier With Risk-Free Asset
4.6 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7
8
8.1
8.2
8.3
8.4
8.5
8.6
8.7
Variance
Exp
ectedReturn
Risky Portfolios FrontierEfficient Frontier
Note: this figure is not based on real data.
where wk =(wk1, ..., wk(N+1)
)> is the vector of investor k portfolio shares wki allocated amongavailable assets with wk(N+1) = wf , µ = (µ1, ..., µN+1)> is the vector of assets expected returnswith µN+1 = Rf , and τk is the investor k’s risk tolerance coefficient.
This economy is populated by atomistic price taker investors. They all share the same infor-mation on the relevant assets, and this information is provided to them costlessly. Moreover,there is no transaction costs, no capital or income taxes, and all assets are perfectly divisibleand liquid. Finally, there are no short sales restrictions on any asset and the risk-free assetreturn is an exogenous variable. Readers are referred to Jensen (1972), Harris (1980) and Levy(2012) for full details on the assumptions of the standard CAPM.
In this framework, Sharpe (1964), Lintner (1965) and Mossin (1966) demonstrate that investork efficient portfolio can be expressed as a weighted combination of the risk free asset and aportfolio, wm = (wm1, ..., wmN ), made of risky assets only and with typical elements wmi =∑K
k=1Wk wki. Moreover, wm is the tangent market portfolio. This result is the consequenceof Tobin (1958)’s separation theorem. More importantly, the authors show that
µi = Rf + βi(µm −Rf ), ∀ i = 1, ..., N (2.23)
where µm ≡∑N
i=1wmiµi = µ (wm) is the expected market return and βi = σmi/σ2m, the mar-
ket beta of asset i, is the covariance of its return with the market return, σmi ≡ Cov(Ri, R(wm)),divided by the variance of the market return, σ2
m ≡ σ2 (wm). Equation (2.23) states that the
31
expected return on asset i is equal to the risk-free asset return plus a risk premium, with thelatter being equal to the market premium λm = µm − Rf times asset i market beta βi. βiis commonly interpreted as the marginal contribution of asset i to the market portfolio risk.The relation (2.23) represents the CAPM risk-return equilibrium trade-off and it is referredto as the Security Market Line (SML).
From (2.23), the authors derive the relationship between the return of asset i, Ri, and that ofmarket, Rm ≡ R(wm),
Ri = Rf + βi (Rm −Rf ) + ei, ∀ i = 1, ..., N (2.24)
where ei is uncorrelated with Rm and has expected value equal to zero. From (2.24), thevariance of asset i can be decomposed in two components
σ2i = β2
i σ2m + σ2
ei , ∀ i = 1, ..., N. (2.25)
The first term, β2i σ
2m, is identified as asset i systematic risk and represents that part of
the total risk (of investing in asset i) associated with the market as a whole. The secondcomponent, σ2
ei , is the idiosyncratic or unsystematic risk of asset i. When N is large andwmi is evenly distributed4, it is claimed that only the systematic risk is remunerated by themarket. This is because the unsystematic risks should be eliminated via the law of largenumbers diversification strategy.
In the absence of the risk-free asset, Black (1972) provides a more general version of theCAPM, known as the Black’s version. The author demonstrates that each investor holds anefficient portfolio according to his or her risk aversion coefficient, which can be expressed asa weighted combination of the market portfolio and the zero-β portfolio. The latter (zero-βportfolio) is defined as the minimum variance portfolio among all portfolios uncorrelated withthe market portfolio. The equilibrium risk-return trade-off becomes
µi = µz + βi(µm − µz), ∀ i = 1, ..., N (2.26)
where µz ≡ µ (wz) refers to the expected return of the zero-β portfolio with wz the zero-βportfolio. Equation (2.26) is the same as (2.23) except for the zero-β portfolio return thatreplaces the risk-free asset return.
This thesis provides a new formulation of Markowitz (1952)’s utility function based on theproposed class of portfolio diversification measures (see Chapter 7), and discusses its eco-nomic implications in terms of the MV optimal portfolios, the CAPM and the MV modelgeneralization.
4These assumptions imply that the term σ2i wmi in βi =
σ2i wmi+
∑N−1j 6=i σijwmj
Σmcan be neglected, which is
not the case when N is small or wmi is not evenly distributed.
32
2.1.2.1.4 Expected Utility TheoryOne of the limits of the MV model is its non-applicability when the investor’s utility functioncannot be approximated by a quadratic utility function, which is the case when asset returnsdistribution are very far from elliptical distribution (see Chamberlain, 1983a), or when riskinessis not limited or returns distribution belongs to a family of compact or small risk distributionor when the length of the trading interval is not close to zero (see Samuelson, 1970). Thus,alternative new portfolio selection approaches have been proposed in order to improve the MVmodel. One of these approaches is the expected utility theory (EUT).
Expected utility theory (EUT) is the most famous theory of choice under risk and uncertainty.It was first introduced by Bernoulli (1738/1954) to solve the so-called St.Petersburg paradoxas an alternative of the expected return theory. The theoretical foundation of the EUT, basedon axiomatic approach, was provided later by Von Neumann and Morgenstern (2007). Werefer readers to Von Neumann and Morgenstern (2007) and Kreps (1988) for more details onthe axiomatic definition of the EUT.
The use of the EUT in portfolio selection is natural, since assets future returns are uncertain.It is first suggested by Tobin (1958) and Markowitz (1959). Applying the EUT, a rationalrisk-averse investor must solve the following problem
wEU ∈ arg Maxw∈W
Eu(W (1 + w>R)
), (2.27)
where u(.) is a utility function. The attitude of investors towards risk is captured by the shapeof the function u(.). An investor is risk-averse if and only if u(.) is concave. An investor isrisk-neutral if and only if u(.) is linear. An investor is risk-loving if and only if u(.) is convex.In finance, and particularly in portfolio theory, it is assumed that investors are risk-averse. Itis also assumed that u(.) is an increasing function. This property is known in the literatureas nonsatiation i.e. a higher level of wealth should induce a higher utility.
When u(.) is quadratic or the distribution of returns belongs to the elliptic family, the EUTcoincides with the MV model. When the distribution of asset returns is not far from ellipticaldistribution, or when riskiness is limited or returns distribution does not belong to a familyof compact or small risk distribution or when the length of the trading interval is close tozero, the MV model was shown to be a good approximation of the EUT (see Samuelson, 1970;Markowitz, 2014).
Despite its theoretical appeal, the EUT presents some important drawbacks. Readers arereferred to Martellini and Ziemann (2010), Jurczenko and Maillet (2006), Samuelson (1970),Markowitz (2014), Benartzi and Thaler (1995), Levy and Levy (2004), De Giorgi and Hens(2006), Adler and Kritzman (2007), Bernard and Ghossoub (2010), He and Zhou (2011) andPirvu and Schulze (2012) for more details.
33
2.1.2.1.5 Risk Budgeting ModelsAnother major limit of the MV model is it does not take into account estimation risks. Indeed,the MV model is in general implemented setting the input parameters µ and Σ at theirestimated values using the historical data, ignoring therefore estimation risks. However, theserisks are known to have a huge impact on the MV optimal portfolio (MVP). More precisely,they are the source of undesirable extreme weights and the poor out-of-sample performanceof the MVP (see Chopra and Ziemba, 1993; Michaud, 1989). Additionally, the MVP arehigher sensitive: small changes in the parameter estimates can lead to drastic changes in theMVP weights (see Best and Grauer, 1992). To overcome theses problems various solutionshave been proposed. For example, Barry (1974); Klein and Bawa (1976, 1977) propose toestimate µ and Σ using the Bayesian approach with diffuse prior. Frost and Savarino (1986)suggest empirical Bayesian approach to estimate µ et Σ. Jorion (1986) proposes a Bayes-Stein shrinkage approach to estimate µ. Ledoit and Wolf (2004a,b) propose a shrinkageestimation for Σ. Goldfarb and Iyengar (2003), Garlappi et al. (2007) and Tutuncu andKoenig (2004) suggest robust portfolio allocation rules. Michaud (1989) suggests resamplingapproach. Ma and Jagannathan (2003), Fastrich et al. (2015) and DeMiguel et al. (2009a)used weight constraint. Black and Litterman (1992) combine two sources of information(equilibrium returns and investors’ views) to obtain predictive returns. Fernandes et al. (2012)combine Black and Litterman (1992) and Michaud (1989)’s approaches. Kan and Zhou (2007)propose a combination of several portfolio weights. Since the 2008-2009 financial crisis, anotherroute has been explored by considering some heuristic methods like the risk-based approach.This approach, also called µ-free strategies (see Braga, 2016, Chap. 3), determines portfolioallocation by relying only on the risk characteristics of assets, avoiding the need to provideestimates for expected returns and thus eliminating any loss associated with estimation errorsof µ. That is because it is more difficult to accurately estimate µ than Σ (Merton, 1980) anderrors in estimates of µ have larger impact on portfolio weights than errors in Σ (Chopra andZiemba, 1993). The risk-based approach includes: the risk budgeting, the equally-weighted,the minimum-variance and the most diversified portfolio approaches. This section focuses onthe risk budgeting approach, because, contrary to all proposed solutions and mentioned above,it covers a diversification strategy different from the law of larger numbers, the correlationand the CAPM strategies.
Pearson (2002, pp. 4) defines risk budgeting (RB) as
The process of decomposing the aggregate risk of a portfolio into its constituents,using these risk measures to allocate assets, setting limits in terms of these mea-sures, and then using the limits to monitor the asset allocations and portfoliomanagers.
The process is therefore a risk allocation which involves the choice of risk measure, the de-termination of the sources of risk decomposing the risk of a portfolio into its constituents,
34
measurement of the risk contributions, setting limits on the risk contributions (risk budget),the asset allocation and afterwards the comparison of the realized risk contributions with thedesired ones to remain within the limits (Unger, 2015).
Formally, let % : R → R denotes a risk measure. Assume that %(w>R) can be additivelydecomposed
%(w>R) =
N∑i=1
ARCi(w>R), (2.28)
where ARCi is the absolute risk contribution of asset i. Tasche (2000, 2008) show that thedecomposition (2.28) can be obtained at the level of assets using Euler principle when the riskmeasure is homogeneous of degree κ ∈ R. In that case, ARCi is defined as follows
ARCi(w>R) =
w>∂w %(w>R)
κ, ∀ i = 1, ..., N (2.29)
where ∂w %(w>R) =(∂%(w>R)∂w1
, ..., ∂%(w>R)∂wN
)is a vector of asset marginal risk contributions
and ∂w is a partial derivative operator in respect of w. In the case where κ = 1, Qian (2006)provides an intuitive financial interpretation of ARCi as a contribution to a given loss of thetotal portfolio.
The decomposition (2.28) can also be obtained at the level of risk factors. Meucci (2009);Meucci et al. (2014) provide an example of risk factors decomposition when risk is measuredusing portfolio variance or volatility. Meucci (2009) shows that, extracting the risk factors(uncorrelated) using the principal component analysis, ARCi becomes
ARCi(w>R) = w2
i ξ2i (Σ), ∀ i = 1, ..., N (2.30)
where ξ(Σ) = (ξ1(Σ), ..., ξN (Σ))> is the vector of eigenvalues of Σ, w = E−1Σ w with EΣ
the eigenvectors matrix of Σ. Meucci et al. (2014) show that, extracting the risk factors(uncorrelated) using the minimum torsion (MT) linear transformation, ARCi becomes
ARC ≡ (t−1)>w ⊗ tΣ w, ∀ i = 1, ..., N (2.31)
where t = dg(σ)πc−1dg(σ)−1 with π is a perturbation matrix, dg(.) is a diagonal matrixoperator, c is the Riccati root of asset returns correlation matrix ρ and ⊗ denotes the term-by-term product. Two other risk factors decompositions are the Hoeffding decomposition (seeRosen and Saunders, 2010) and Fama-French decomposition (see Carli et al., 2014; Roncalliand Weisang, 2016). Note that these risk factors decompositions, with the exception of Fama-French decomposition, are consistent with the view of Sharpe (2002, pp. 80) who argues thata measure of risk is additive only if risks are independent.
35
Consider a set of given risk budgets %1, ..., %
N∈ R. The RB approach is defined by the
following constraints:
ARCi(w>R) = %
1...
ARCi(w>R) = %
i...
ARCi(w>R) = %
N
(2.32)
For practical purposes, portfolio risk can be normalized to 1. In that case, %iis in a range
between 0 and 1 such that∑N
i=1 %i = 1, and ARC becomes the percentage contribution torisk (PCR).
The RB portfolio is obtained solving the following problem
minw∈W
N∑i=1
(ARCi(w
>R)− %i
)2(2.33)
using the sequential quadratic programming (SQP) algorithm. An alternative formulation of(2.33) is
minw∈RN+
%(w>R) (2.34)
s.t %> ln(w) ≥ %, (2.35)
where % ∈ R is the diversification (in terms of weight) threshold. For the existence anduniqueness of the RB portfolio readers are referred to Bruder and Roncalli (2012).
2.1.2.2 Diversification Principle
This section reviews the diversification principle in Markowitz (1952), Tobin (1958) and Sharpe(1964)’s model, in the EUT and in the RB approach.
2.1.2.2.1 Markowitz (1952); Tobin (1958)’s Diversification PrincipleMarkowitz (1952)’s model is considered as the first mathematical formulation of the idea ofdiversification of investments (see Rubinstein, 2002). However, it is important to stress thatMarkowitz (1952) does not provide neither a proper definition of well-diversified portfolio nor aspecific measure of portfolio diversification (see Fernholz, 2010). It only provides a descriptionof the diversification principle in his model as follows
The adequacy of diversification is not thought by investors to depend solely onthe number of different securities held. A portfolio with sixty different railwaysecurities, for example, would not be as well diversified as the same size portfoliowith some railroad, some public utility, mining, various sort of manufacturing, etc.
36
The reason is that it is generally more likely for firms within the same industryto do poorly at the same time than for firms in dissimilar industries. (Markowitz,1952, pp.89)
The core of the diversification principle in Markowitz (1952)’s model is therefore asset covari-ances. As a result, the diversification principle in Markowitz (1952)’s model is the correlationdiversification strategy which can be defined as follows.
Definition 2.1.2 (Correlation diversification). The orrelation diversification consists inexploiting the correlation (or more generally the interdependence) between asset returns inorder to reduce portfolio risk.
The philosophy behind the correlation strategy is as follows: diversification helps to minimizeboth the probability of portfolio loss and its severity, through a multilateral insurance inwhich each asset is insured by the remaining assets. The key to the success of this multilateralinsurance lies in the correlation between assets. Less assets are (positively) correlated, more theprobability that they do poorly at the same time in the same proportion is low and the betteris the protection offered by this multilateral insurance which is the diversification. Therefore,in the presence of correlation, it becomes dangerous to follow the LLN diversification strategy.
The observed diversification effect is captured through the reduction of portfolio risk measuredby portfolio variance. Consider the following decomposition of portfolio variance
σ2(w) =N∑i=1
w2i σ
2i +
N∑i,j=1
wiwjσij (2.36)
The first component,∑N
i=1w2i σ
2i , captures the effect of the law of large numbers strategy. The
second component,∑N
i,j=1wiwjσij , captures the effect of the correlation strategy. To see this,Markowitz (1959) considers the particular case of the naive portfolio, where all N assets havethe same weight 1/N . The portfolio variance becomes
σ2
(1
N
)=
1
N
(N∑i=1
σ2i
N
)+N − 1
N
N∑i,j=1
σijN(N − 1)
. (2.37)
If N increases (N →∞), the portfolio variance converges to the mean of asset covariances
σ2
(1
N
)→
N∑i,j=1
σijN(N − 1)
. (2.38)
It follows that portfolio risk cannot be totally eliminated by the law of large numbers diversi-fication strategy, unless assets are uncorrelated. Figure 2.3 presents an illustration.
37
Figure 2.3 – Relationship Between Portfolio Size and Risk
0 10 20 30 40 50 60 70 80 90 100 1100
2 · 10−2
4 · 10−2
6 · 10−2
8 · 10−2
0.1
0.12
0.14
0.16
Portfolio size (N)
Portfolio
risk
(Variance)
ρ = 0.3
ρ = 0.5
ρ = 0.8
Systematic risk level
Now, consider the case where the risk-free asset is available. In that case the variance of thenaive portfolio becomes
σ2
(1
N + 1
)=
N
(N + 1)2
(N∑i=1
σ2i
N
)+N(N − 1)
(N + 1)2
N∑i,j=1
σijN(N − 1)
. (2.39)
The contribution of the risk-free asset to portfolio diversification in the mean-variance modelcan be evaluated by the difference σ2
(1
N+1
)− σ2
(1N
)σ2
(1
N + 1
)− σ2
(1
N
)=
(N∑i=1
σ2i
N
)(N
(N + 1)2− 1
N
)︸ ︷︷ ︸
A
+
N∑i,j=1
σijN(N − 1)
(N(N − 1)
(N + 1)2− N − 1
N
)︸ ︷︷ ︸
B
. (2.40)
It is straightforward to verify that the factors A and B are negative. Therefore the differenceσ2(
1N+1
)− σ2
(1N
)is negative. As a result, the risk-free asset contributes both to the law of
large numbers and the correlation diversification in the MV model.
This thesis makes two important contributions to the diversification in Markowitz (1952)’smodel. First, it demonstrates that there is a specific portfolio diversification measure in
38
Markowitz (1952)’s model contrary to Fernholz (2010). This measure is the diversification re-turn (see Section 2.2.2.3) and a particular case of the proposed class of portfolio diversificationmeasures. Second, the thesis establishes the conditions under which a MV optimal portfoliois well-diversified in terms of correlation diversification (see Section 5.1.5).
2.1.2.2.2 CAPM Diversification PrincipleDespite the fact that the CAPM is based on Markowitz (1952)’s model, its diversificationprinciple is totally different. In the CAPM, a well-diversified portfolio is defined as follows.
Definition 2.1.3 (CAPM’s diversification). A portfolio is well-diversified if it is not ex-posed to the unsystematic risk.
Definition 2.1.3 implies that the market portfolio is the only well-diversified portfolio accordingto the CAPM diversification strategy.
The fundamental difference between the CAPM diversification strategy and that of Markowitz(1952) is that the CAPM strategy does not take into account the diversification in β. Thislimit was pointed out by Sharpe (1972) itself
There are other considerations. For example, a portfolio of ten chemical securitiesis likely to offer less effective diversification than one of ten securities, each froma different industry. This type of difference is difficult to capture in a simpleformula. But differences in relative holdings and securities’ non-market risks canbe accommodated quite easily.
2.1.2.2.3 Expected Utility Diversification PrincipleThe diversification in the expected utility approach was first studied by Samuelson (1967).The author provides the conditions under the diversification in the sense of the law of largenumbers strategy is optimal in the expected utility approach. The core of these conditions isthe interdependence between portfolio assets and the difference in asset expected return. Formore details, we refer readers to Samuelson (1967, Theorem I, II, III and Corollary I, II).
Brumelle (1974) generalizes the results of Theorem III of Samuelson (1967). However, Dekel(1989) generalizes Samuelson (1967)’work providing the definition of the preference for diver-sification in the EUT.
Definition 2.1.4 (Preference for diversification). The preference relation exhibits pref-erence for diversification if for any Ri, i = 1, ..., N ∈ R and αi ∈ [0, 1], i = 1, ..., N such that∑N
i=1 αi = 1,
R1 ∼ R2 ∼ ... ∼ RN ⇒N∑i=1
αiRi Rj ∀ j = 1, ..., N. (2.41)
39
Dekel (1989) argues that “if an investor is an expected utility maximizer, then he or she isrisk-averse if and only if he or she exhibits a preference for portfolio diversification.” This resultis proved by Chateauneuf and Tallon (2002). Chateauneuf and Tallon (2002); Chateauneufand Lakhnati (2007) also introduce other diversification concepts such as: sure diversification,strong diversification and comonotone diversification. The authors show that all these notionsof diversification are equivalent to that of Dekel (1989) in the EUT. Readers are also referredto De Giorgi and Mahmoud (2016) for more details on diversification in the EUT.
The diversification principle in the EUT can also be described, independently of the LLNdiversification, using the notion of correlation aversion. The notion of correlation aversion wasintroduced by Epstein and Tanny (1980). However, this concept was first studied by Richard(1975) but under the name of multivariate risk aversion.
Consider a decision maker (DM) whose preference is represented by a bivariate utility functionv(y, z), y, z ∈ R. Assume that v(y, z) is twice continuously differentiable in y and z.
Definition 2.1.5 (From Richard (1975): Theorem 1). A DM is correlation averse if andonly if ∀ (y, z) ∈ R× R,
∂2v(y, z)
∂y ∂z≤ 0, (2.42)
with strict inequality for at least one pair (y, z) ∈ R× R.
A DM is said to be correlation lover if ∂2v(y,z)∂y ∂z ≥ 0 with strict inequality for at least one pair
(y, z) ∈ R× R. His or her is said to be correlation neutral if ∂2v(y,z)∂y ∂z = 0.
In the case where the DM preference is represented by a multivariate utility function v(y1, ..., yN ), yi ∈R, the condition (2.42) is replaced by the following one (see Richard, 1975)
∂2v(y)
∂yi ∂yj≤ 0. (2.43)
For example, consider v(y1, y2, y3) = −(y1y2)−1 +√y3. The mixed second order partial deriva-
tives are
∂2v(y)
∂y1 ∂y2= −(y1y2)−2 (2.44)
∂2v(y)
∂y1 ∂y3= 0 (2.45)
∂2v(y)
∂y2 ∂y2= 0 (2.46)
It follows that v exhibits correlation aversion in y1 and y2 and v is correlation neutral in y1
and y3 and in y2 and y3.
40
When v(y, z) is not twice continuously differentiable, the condition (2.42) is replaced by thesubmodularity condition
v(y ∨ z) + v(y ∧ z) ≤ v(y) + v(z), ∀ (y, z) ∈ R× R (2.47)
where y ∨ z is the componentwise maximum of y and z, y ∧ z is the componentwise minimumof y and z. Consider the following two lotteries
L1 =
(y − y, z − z) with probability 0.5
(y, z) with probability 0.5
L2 =
(y − y, z) with probability 0.5
(y, z − z) with probability 0.5
where y, z are positive. Richard (1975) and recently Eeckhoudt et al. (2007) show that a DMexhibits correlation aversion if he or she prefers L2 to L1. This implies that a correlationaverse DM prefers the diversification, more precisely the correlation diversification principle,than the concentration (see Epstein and Tanny, 1980). It is important to note that thecorrelation aversion concept is totally different to the risk aversion concept is the EUT. Riskaversion behaviour does not imply necessary correlation aversion behaviour and vice versa.For example, consider v(y, z) = (yz)1/4 for y, z > 0, then ∂2v(y,z)
∂y ∂z = 116(yz)−3/4 > 0, ∂
2v(y,z)∂y2 =
− 316y−7/4z1/4 < 0 and ∂2v(y,z)
∂z2 = − 316y−/4z−7/4 < 0.
Without a loss of generality, assume that W = 1. u (W ) can be rewritten as a bivariatefunction
u (W ) = u
1 + wiRi + (1− wi)N∑j 6=i
wjRj
≡ uRi, N∑
j 6=iwjRj
(2.48)
or as a multivariate function
u (W ) = u
(1 +
N∑i=1
wiRi
)≡ u (R1, ..., RN ) . (2.49)
The partial derivative of u (W ) with respect of Ri and R−i =∑N
j 6=iwjRj gives
∂2u(Ri, R−i)
∂Ri∂R−i= wi(1− wi)
∂2u(W )
∂W 2, ∀ i = 1, ..., N (2.50)
The mixed second order partial derivatives of u (W ) are
∂2u(R1, ..., RN )
∂Ri∂Rj= wiwj
∂2u(W )
∂W 2, ∀ i, j = 1, ..., N. (2.51)
It is observable that a risk-averse investor whose preference can be represented by the EUTexhibits a correlation aversion. As a result, the diversification principle in the EUT follows a
41
correlation diversification principle and is guided by the concavity of the function u. This resultremains valid in the presence of the risk-free asset, but not when short sales are unrestricted.
We end this section by providing some results on correlation aversion necessary for the defi-nition of our new class of portfolio diversification measures.
Definition 2.1.6 (Correlation Order (Dhaene et al., 2009)). Consider the random cou-ples (Z1, Z2) and (Y1, Y2). Then (Z1, Z2) is said to be less correlated than (Y1, Y2), notation(Z1, Z2) corr (Y1, Y2), if any of the following equivalent conditions holds true:
i) Z1d= Y1 and Z2
d= Y2 and Cov (f(Z1), g(Z2)) ≤ Cov (f(Y1), g(Y2)) holds for all non-
decreasing functions f : R→ R and g : R→ R for which the covariances exist;ii) Z1
d= Y1 and Z2
d= Y2 and P (Z1 ≤ y1, Z2 ≤ y2) ≤ P (Y1 ≤ y1, Y2 ≤ y2) holds for all
y1, y2 ∈ R;iii) Z1
d= Y1 and Z2
d= Y2 and P (Z1 ≥ y1, Z2 ≥ y2) ≤ P (Y1 ≥ y1, Y2 ≥ y2) holds for all
y1, y2 ∈ R;iv) E (f(Z1, Z2)) ≥ E (f(Y1, Y2)) holds for all submodular functions f : R2 → R for which
the expectations exist;v) E (f(Z1, Z2)) ≥ E (f(Y1, Y2)) holds for all twice differentiable functions f : R2 → R
satisfying ∂f∂y1∂y2
≤ 0 and such that the expectations exist.
where d= stands for “equally distribution”
Definition 2.1.7 (Submodular Order (Dhaene et al., 2009)). Let Z = (Z1, ..., ZN ) andY = (Y1, ..., YN ) be two N -dimensional random vectors. Then Z is said to be smaller than Yin the supermodular order, notation Z SM Y, if any of the following equivalent conditionsholds true:
i) E (f(Z)) ≥ E (f(Y)) holds for all submodular functions f : RN → R for which theexpectations exist;
ii) E (f(Z)) ≥ E (f(Y)) holds for all twice differentiable functions f : RN → R satisfying∂f
∂yi∂yj≤ 0 for every 1 ≤ i < j ≤ N and such that the expectations exist.
Proposition 2.1.1 (Dhaene et al. (2009): Implication of Definition 3).
Z SM Y =⇒ (Zi, Zj) corr (Yi, Yj) ∀ i 6= j. (2.52)
Proposition 2.1.2 (Denuit et al. (2005): Property 6.2.14). Consider the random cou-ples (Z1, Z2) and (Y1, Y2) with Z1
d= Y1 and Z2
d= Y2 and (Z1, Z2) corr (Y1, Y2). Then
i) ρ(Z1, Z2) ≤ ρ(Y1, Y2);ii) ρk(Z1, Z2) ≤ ρk(Y1, Y2);iii) ρs(Z1, Z2) ≤ ρs(Y1, Y2).
42
where ρ, ρk and ρs are Pearson’s, Kendall rank and Spearman’s rank correlation coefficients,respectively.
Dhaene et al. (2009) make the link between correlation order (corr) and diversification in thecase of capital requirement risk measure.
Definition 2.1.8 (Dhaene et al. (2009): Definition 4, 5: Diversification Benefit).
i) Consider the vector of random losses Y = (Y1, ..., YN ) with respective capitals C =
(C1, ..., CN ). The diversification benefit DC(Y) is defined as
DC(Y) =N∑i=1
(Yi − Ci)+ +
(N∑i=1
(Yi − Ci)
)+
. (2.53)
ii) Consider the vector of random losses Y = (Y1, ..., YN ) and a capital requirement %. Thediversification benefit D%(Y) is defined as
D%(Y) =
N∑i=1
(Yi − %(Yi))+ +
(N∑i=1
Yi − %
(N∑i=1
Yi
))+
. (2.54)
where (Y )+ = maxY, 0.
Proposition 2.1.3 (Dhaene et al. (2009): Theorems 10, 12).
i) Consider two couples of random losses Y = (Y1, Y2) and Z = (Z1, Z2). Furthermore, letC = (C1, C2). Then
Y corr Z =⇒ EDC(Y) ≥ EDC(Z). (2.55)
ii) Assume that the capital requirement % is law invariant, translation invariant and posi-tively homogeneous. Furthermore, consider two couples of random losses Y = (Y1, Y2)
and Z = (Z1, Z2). Assume that their sums Y1 + Y2 and Z1 + Z2 belong to the samelocation-scale family of distributions with finite variances. Then one has that
Y corr Z =⇒ ED%(Y) ≥ ED%(Z). (2.56)
2.1.2.2.4 Risk Budgeting Diversification PrincipleThe diversification in the RB approach is defined in terms of asset absolute or percentage riskcontribution as follows.
Definition 2.1.9 (Risk Contribution Diversification). A portfolio is said well-diversifiedif all assets have the same absolute risk contribution i.e.
ARCi(w>R) = ARCj(w
>R) ∀ i, j = 1, ..., N. (2.57)
43
This portfolio is obtained setting uniform budget %i
= 1/N, ∀ i = 1, ..., N . In this case, the RBis known under the name of risk parity (RP) or equally risk contribution (ERC) approach (seeMaillard et al., 2010; Bhansali et al., 2012; Roncalli, 2014; Roncalli and Weisang, 2016). Thisdiversification strategy is qualified by Qian (2011) as a “true diversification”. Moreover, manyempirical studies report desirable properties of the ERC compared to the 60/40 benchmark(equity/bond), the equally weighted portfolio, the most diversified portfolio and the minimum-variance portfolio, but these results depend on the data used, the estimation window, etc. (seeBraga, 2015; Chaves et al., 2011; Maillard et al., 2010; Bhansali et al., 2012; Chow et al., 2011;Clarke et al., 2013; Steiner, 2012; Bhansali, 2012; Lohre et al., 2012). Furthermore, the ERChas some important shortcomings. Bhansali et al. (2012, pp. 103) argue that
the traditional asset-based risk parity, whether naïve or otherwise improved upon,can still be highly concentrated in only one or two true risk exposures (especiallyin equity risk), and therefore be under-diversified in other risk exposures. Thisis true even if the direct capital allocation to equities appears small. We showevidence that an asset-class based approach to risk parity fails to achieve riskparity in the true underlying risk factor exposures. Having diversification in riskcontribution from assets is generally not the same as having diversification in theprimitive sources of risk underlying asset returns. An easy way to understand thisargument is to think of assets as foods and risk factors as nutrients. While thebody consumes foods, it actually needs the underlying nutrients to build bonesand muscles. A healthy diet is not necessarily one which contains a diversifiedbasket of foods but a diversified basket of nutrients. The advantage of using riskdrivers or risk factors to define and understand risk parity is that we properlyaccount for the essential nutrients of our investment portfolios. This factor-basedframework allows for a clearer insight and hence healthier portfolio construction
Despite its shortcomings the ERC portfolios are a popular investment strategy. Many assetmanagement firms launch a risk parity fund to its clients. Bridgewater Associates were thefirst in 1996 with the AllWeather fund (see Bridgewater Associates, 2000). Nowadays, to namea few, we have AQR Capital 5, Aquila Capital 6, PanAgora Asset Management 7, PutnamInvestments 8 and Invesco 9). There are also equally risk contribution indices available (EUROiSTOXX 50 Equal Risk10, Lyxor SmartIX ERC 11 and Salient Risk Parity Index 12.
5https://www.aqr.com/6http://www.aquila-capital.de/en7https://www.panagora.com/8https://www.putnam.com/9http://www.invesco.com/corporate
10https://www.stoxx.com/index-details?symbol=SXERCE11http://www.morningstar.co.uk/uk/etf/snapshot/snapshot.aspx?id=0P0000WCJO12http://www.salientindices.com/risk-parity.html
44
2.2 Diversification Measures
Given the importance of diversification in portfolio selection as highlighted in the previoussection, several portfolio diversification measures were proposed. This section reviews themost popular of them. Their advantages and weaknesses are also discussed.
2.2.1 Law of Large Numbers Diversification Measures
This section reviews measures that are considered to capture the effect of law of large numbersdiversification strategy. Most of them can be related to the effective number of constituents(ENC), a class of measures of portfolio diversification analyzed in Deguest et al. (2013) andCarli et al. (2014). The ENC is defined in terms of the norm of portfolio weight w
ENCυ(w) = ‖w‖υ
1−υυ =
(N∑i=1
wυi
) 11−υ
, υ ≥ 0, υ 6= 1. (2.58)
The ENC can also be viewed as the weighted generalized or power of mean of the inverse ofportfolio weight 1
w =(
1w1, ..., 1
wN
), where the weight vector is w
ENCυ(w) =
(N∑i=1
wi
(1
wi
)1−υ) 1
1−υ
, υ ≥ 0, υ 6= 1. (2.59)
When υ = 2, ENC2 coincides with the inverse of the Herfindahl or Gini-Simpson index
ENC2(w) =1
HI(w), (2.60)
where HI(w) =∑N
i=1w2i is the Herfindahl index. The Herfindahl index was introduced as a
measure of portfolio diversification by Woerheide and Persson (1993) and much used in theliterature (see Zhou et al., 2013; DeMiguel et al., 2009a).
It can also be shown that when υ converges to 1, ENC1 coincides with the exponential of theShannon index or entropy
ENC1(w) = exp
(−
N∑i=1
wi ln (wi)
)(2.61)
with ln(.) the natural logarithm function. The Shannon index or entropy is another popularmeasure of portfolio diversification (see Zhou et al., 2013).
Moreover, ENCυ is also related to Bouchaud et al. (1997)’s class of measures of portfoliodiversification (BMυ), another weight-based measures, as follows
BMυ(w) =1− (ENCυ(w))1−υ
υ − 1, υ ≥ 0, υ 6= 1. (2.62)
45
In the case where υ converges to 1, the relation (2.62) becomes
BM1(w) = ln (ENC1(w)) . (2.63)
It is straightforward to check that, for any long-only portfolio w, ENCυ reaches a mini-mum equal to 1 if the portfolio is fully concentrated in a single asset, and a maximumequal to N if the portfolio coincides with the naive portfolio or equally weighted portfolio,w = (1/N, ..., 1/N)>.
In practice, the choice of the value of υ remains an open question. Carli et al. (2014) rec-ommend using ENC2 when dealing with the long-short portfolios and both ENC1 and ENC2
when dealing with the long-only portfolios. The principal advantage of the ENC is that it issimple to implement and easy to interpret. Moreover, it captures perfectly portfolio weightsconcentration and can be applied to any universe of assets (risky or not).
However, as well as all measures inspired from the law of large numbers diversification strategy,the ENC suffers from some important shortcomings. These shortcomings are essentially due tothe fact that the ENC takes into account only two dimensions of portfolio diversification, whichare portfolio size and weights, but ignores the risk dimension. This thesis mentions only two ofthem. Readers are referred to Frahm andWiechers (2011), Carli et al. (2014) and Deguest et al.(2013) for other shortcomings. On the one hand, the ENC is symmetric. Consider a universeA = A1, A2, A3 of three assets. The symmetry of the ENC implies that the portfolios(wA1 , wA2 , wA3)> and (wA2 , wA1 , wA3)> have the same degree of diversification. However, theassets A1 and A2 are not identical. On the second hand, the ENC can be deceiving whenapplied to an universe with duplicate assets. Consider a universe A = A1, A2, A2 of threeassets, where asset A2 is duplicated. The ENC implies that the well-diversified portfolio is(wA1 = 1
3 , wA2 = 13 , wA2 = 1
3
)>. However, intuitively, the well-diversified portfolio must be(wA1 ,
1−wA12 ,
1−wA12
)>.
2.2.2 Correlation Diversification Measures
This section reviews measures that capture the effect of the correlation diversification strategy.
2.2.2.1 Embrechts et al. (2009)’s Class of Measures
Let R1, ..., RN be a sequence of random variable. Embrechts et al. (2009)’s class of diversifi-cation measures is defined as
EFK%(R) =N∑i=1
% (Ri)− %
(N∑i=1
Ri
), (2.64)
where % is a risk measure. Applied to portfolio w, the following class of measures is obtained
EFK%(w) =
N∑i=1
% (wiRi)− %(w>R
). (2.65)
46
An intuitive interpretation can be provided to EFK%(w) when % is homogeneous of degree onei.e. %(bR) = b%(R) for all b > 0. In that case, EFK%(w) can be rewritten as follows
EFK%(w) =
N∑i=1
wi
(% (Ri)− %
(w>R
)). (2.66)
The term in the parenthesis, % (Ri) − %(w>R
), measures the benefit of diversification, in
terms of risk reduction, to hold portfolio w instead of to concentrate on asset i. It follows thatEFK%(w) measures the average benefit of diversification, in terms of risk reduction, to holdportfolio w instead of holding a single asset portfolio. Tasche (2006) analyzes the normalizedversion of EFK% which applied to portfolio w is defined as follows
DF%(w) =%(w>R
)∑Ni=1 %(wiRi)
. (2.67)
In general, EFK% and DF% are viewed as adequate portfolio diversification measures if %satisfies the followings properties:
1) Positive homogeneity : for b > 0 and X ∈ R, %(bX) = b%(X);2) Sub-additivity : for all X,Y ∈ R, λ ∈ [0, 1], % (λX + (1− λ)Y ) ≤ λ % (X)+(1−λ) % (Y );3) Comonotonic additivity : for comonotonic X,Y ∈ R i.e. there is Z ∈ R and f1 and f2
non decreasing functions such as X = f1(Z) and Y = f2(Z), % (X + Y ) = % (X)+% (Y );4) Non-additivity for independent : for independent X,Y ∈ R, % (X + Y ) 6= % (X) + % (Y ).
The diversification delta (DD) introduced by Vermorken et al. (2012) can be viewed as aspecial case of EFK%. Assume that % is entropy
%(R) ≡ H(R) = −∫dFR(r) ln (dFR(r)) dr. (2.68)
ThenDD(w) = 1− exp (EFKH(w)) . (2.69)
Salazar Flores et al. (2017) show that the DD is inadequate measure of diversification sinceH is not positive homogeneity, sub-addtitivity and comonotonic additivity. Moreover, H isnot left-bounded. To correct these shortcomings of DD, Salazar Flores et al. (2017) suggest toreplace H by the exponential entropy exp(H) which satisfies these desirable properties, anddefine a new measure which can be viewed as a special case of DF%
DD∗(w) = 1−DFexp(H)(w). (2.70)
The principal advantage of Embrechts et al. (2009) or Tasche (2006)’s class measures is theyconsider the joint distribution of asset returns. Their major limit is the computational costand it is only applicable to a portfolio of risky assets. Moreover, it requires risk measurespecification and cannot be used directly with interdependence measure.
47
2.2.2.2 Diversification Ratio
The diversification ratio is a portfolio diversification measure developed by Choueifaty andCoignard (2008). It is defined as the ratio of the weighted average of asset volatilities to theportfolio volatility
DR(w) =w> σ√w>Σ w
, (2.71)
where σ = (σ1, ..., σN )> is the vector of asset volatilities. It can be related to Tasche (2006)’smeasure when risk measure is volatility
DR(w) =1
DFσ(w). (2.72)
The optimal portfolio of DR is obtained by maximization and is called the most diversifiedportfolio (MDP). According Choueifaty et al. (2013), the MDP can also be obtained from thefollowing optimization problem
maxw∈R+
w>Σ w (2.73)
s.t. w>σ = 1. (2.74)
From (2.73)-(2.74), the MDP can be obtained in two steps. In the first step, the followingoptimization problem is solved
maxw∈W
w>ρw, (2.75)
where ρ is asset correlation matrix Denote w∗ the solution of the problem (2.75), the secondstep is to rescale w∗ using asset volatilities
wMDi =
w∗iσi/
N∑i=1
w∗iσi, ∀ i = 1, ..., N (2.76)
where wMD is the weight vector of the MDP. The portfolio construction based on DR is knownunder the name of maximum diversification (MD) strategy, which is considered as one of themost popular risk-based investment strategy.
An intuitive interpretation can be given to the DR as the Sharpe ratio when each asset’svolatility is proportional of its expected premium
E(Ri)−Rf = δσi, (2.77)
where δ > 0. The relation (2.77) is inconsistent with the CAPM, but can be justified theoret-ically by the results of Merton (1980) and Malkiel and Yexiao (2006)13. Martellini (2008) also
13Malkiel and Yexiao (2006)’s model is the generalization of Merton (1980)’s model. Malkiel and Yexiao(2006) show that when some investors cannot hold the market portfolio, the remaining investors will also beunable to hold the market portfolio. Therefore, idiosyncratic risk could also be priced to compensate rationalinvestors for inability to hold the market portfolio.
48
reports empirical evidence which support this relation14. However, other authors report em-pirical evidence that high-volatility stocks have long underperformed low-volatility stocks15.In that case, (2.77) is no longer held. Furthermore, Lee (2011, pp. 16) points out that (2.77)is not the relation of non arbitration. Accordingly, the investment problem which the DR isbuilt to solve is unclear. Lee (2011, pp. 15-16) reports that the MDP is constructed to max-imize the distance between two volatilities measures of the same portfolio. Taliaferro (2012,pp. 127) also reports that
More curious is the MD methodology, which does not seek risk minimization, re-turn maximization, Sharpe ratio maximization or the optimization of any othereconomically grounded measurement of investment performance. Instead, the MDmethodology seeks portfolios with the greatest difference between pre- and post-formation risk, without regard for the risk or return profile of the resulting port-folio. Consequently, an MD portfolio only has desirable properties by accident.
To clarify the investment problem behind the DR maximizing, Choueifaty et al. (2013) providetwo intuitive interpretations to the DR. The first is based on the following decomposition ofthe DR
DR(w) = [ρ(w)(1− σCR(w)) + σCR(w)]−12 , (2.78)
where ρ(w) is the volatility-weighted average correlation of the assets in the portfolio
ρ(w) =
∑Ni 6=j=1wiσiwjσjρij∑Ni 6=j=1wiσiwjσj
(2.79)
and σCR(w) is the volatility-weighted concentration ratio of the portfolio
σCR(w) =
∑Ni=1(wiσi)
2(∑Ni=1wiσi
)2 . (2.80)
The σCR(w) measures both the concentration of weights and risks. As can be seen, the DRincreases when ρ(w) and/or the σCR(w) decrease. This means that the DR is high when theportfolio is less concentrated in terms of weights and risks and when assets are less correlated.This decomposition, instead to better improve the interpretation of DR, obscures it.
14The author reports empirical evidence of the relationship between the total volatility of the stock and itsexpected return. He finds that stock total volatility is a model-free proxy for a stock’s excess return.
15Ang et al. (2006) show that stocks with high idiosyncratic risk have an extremely low expected returns.The same authors examine this relation in the international markets (Ang et al., 2009). They find that stockswith recent past high idiosyncratic volatility have low future average returns around the world (23 developedmarkets), after controlling for world market, size, and value factors. Baker et al. (2011) show that, regardlessof whether the risk is defined (beta or volatility) or whether all stocks or only large caps are considered, lowrisk portfolio consistently outperformed high risk portfolio over the period (January 1968-December 2008). Forexample, a dollar invested in the lowest-volatility portfolio grew to $ 59.55 ($ 10.12 in real terms), and a dollarinvested in the highest-volatility portfolio declined by 42 cents (90 cents in real terms).
49
The second interpretation considers the DR as a measure of degrees of freedom. Based onempirical evidence, the authors proclaim that DR2 can be interpreted as the number of inde-pendent risk factors, or degrees of freedom represented in the portfolio. This interpretationcan however yield to counter-intuitive results. To illustrate this, consider an asset’s universeof four assets. Assume that all the assets have the same volatility, and the correlation matrixis given by
ρ =
1
1 1
0 0 1
0 0 −1 1
. (2.81)
The optimal value of DR2 isDR2
(wMD
)= +∞. (2.82)
This drawback arises because DR does not treat the negative and the positive correlationsimilarly. Another limit of the DR is that it is based the covariance matrix of asset returns.However, asset variance is an adequate risk measure and asset correlation an adequate depen-dence measure only if asset returns are elliptical distributed, which is often not the case (seeEmbrechts et al., 1999). Moreover, the DR is applicable only to a portfolio of risky assets.
Despite its limitations, DR is currently used by TOBAM16 to manage over $8 billion (asof June 2016) via its Anti-Benchmarkr strategies in Equities and Fixed Income. TOBAMalso offers the TOBAM Maximum Diversification Index series, a set of 11 indices based onTOBAM’s original approach.17 This thesis will show that the diversification principle in theDR is driven by the proposed class of portfolio diversification measures (see Chapter 4 andChapter 6).
2.2.2.3 Diversification Return
The diversification return is a measure of portfolio diversification introduced by Booth andFama (1992). It is defined as the difference between portfolio compound return and theweighted average of each asset compound return
Dr(w) = ln(1 + w>R)−w> ln(1 + R), (2.83)
where ln(1 + R) = (ln(1 +R1), ..., ln(1 +RN ))> is a vector of asset compound return andln(1 + w>R) portfolio compound return. Let µG be the geometric mean. Since
ln(1 + µG) = E(ln(1 +R)), (2.84)16TOBAM (Think Out of the Box Asset Management) is a Paris-based asset management firm, formed
in 2005 by Yves Choueifaty, independent and employee-owned. It has two minority shareholders: CaliforniaPublic Employees’ Retirement System (CalPERS) since April 2011 and Amundi, since May 2012.
17(see http://www.tobam.fr/about-us/ for more details)
50
Dr(w) can be rewritten in terms of µG
Dr(w) = ln(1 + µG(w))−w> ln(1 + µG), (2.85)
where µG = (µG1 , ..., µGN )> is a vector of asset geometric mean and µG(w) the geometricmean of portfolio w. Booth and Fama (1992) approximate Dr(w) (Equation (2.83)) using thesecond order Taylor series expansion about the mean return. The approximation of Dr(w)
(Equation (2.83)) using the first order Taylor series expansion about zero leads to the excessgrowth rate
Dr(w) = µG(w)−w>µG, (2.86)
a measure of portfolio diversification analyzed by Fernholz and Shay (1982); Fernholz (2010)in stochastic portfolio theory.
Taking the second order Taylor series expansion of (2.84) about zero and neglecting the terms(µG)2 and (E(R))2 leads to
µG ≈ E(R)− Var(R)
2. (2.87)
Combining (2.86) and (2.87), one obtains another approximation Dr(w)
Dr(w) ≈ w>σ2 −w>Σ w. (2.88)
This approximation was analyzed in Willenbrock (2011), Fernholz (2010), Chambers andZdanowicz (2014), Bouchey et al. (2012) and Qian (2012). Its normalized version
GLR(w) =w>σ2
w>Σ w(2.89)
was analyzed in Goetzmann et al. (2005); Goetzmann and Kumar (2008).
This thesis refers to the approximation (2.88) as diversification return as in Willenbrock (2011).Dr and GLR have an advantage to be applicable to any portfolio. Their limit is that they arebased on the covariance matrix of asset returns. This thesis will show that the diversificationreturn is a specific diversification measure in the mean-variance model and is a member of thenew class of portfolio diversification measures analyzed.
2.2.2.4 Frahm and Wiechers (2013)’s Measure
Frahm and Wiechers (2011)’s measure is based on portfolio variance. It is defined as a ratioof the variance of the minimum-variance portfolio to portfolio variance
NV(w) =
(1>Σ 1
)−1
w>Σ w. (2.90)
Since for all w ∈ W(1>Σ 1
)−1 ≤ w>Σ w, NV(w) is in a range between mini
(1>Σ 1)−1
σ2i
and 1. NV(w) reaches the maximum value equal to 1 when w coincides with the minimum-variance portfolio. The problem with NV(w) is that is not a valid measure of portfolio
51
diversification, diversification implies risk reduction but the reverse is not true. Consequently,any risk measure cannot be used as a portfolio diversification measure. NV(w) has also adisadvantage to be applicable only to a portfolio of risky assets. Moreover, in the next chapter,it will be proved that NV(w) is not a coherent measure. It provides a ranking inconsistentwith investor preference for diversification in the MV model.
2.2.3 CAPM Diversification Measures
This section reviews measures that capture the effect of the CAPM diversification strategy.
2.2.3.1 Portfolio Size
Portfolio size can be defined formally as follows
PS(w) =
N∑i=1
1wi>0(wi), (2.91)
where 1wi>0(wi) is the indicator function. This measure was introduced implicitly by Evansand Archer (1968) inspiring from the CAPM. The authors examine “the rate at which thevariation of returns for randomly selected portfolios is reduced as a function of the numberof securities included in the portfolio”. As a result, they find a “relatively stable, predictableand decreasing relationship” between portfolio size and portfolio risk. Since portfolio returnsvariation reduction is due to the reduction of portfolio unsystematic risk and the reductionof portfolio unsystematic risk is interpreted as an increased of portfolio diversification inthe CAPM, this relationship means that the higher is portfolio size, the higher is portfoliodiversification degree.
Portfolio size is the most used measure of portfolio diversification in the literature, because itis straightforward to implement and to interpret. Moreover, it is applicable to any portfolio.However, it remains a crude and naive measure of portfolio diversification (see Sharpe, 1972;Goetzmann and Kumar, 2008), because it takes into account only one dimension of portfoliodiversification, which is portfolio size, ignoring therefore the two other dimensions which areportfolio weight and risk. It is adequate measure if and only if asset returns distribution isexchangeable or investors have no useful information about asset expected return and risk(see Markowitz, 1952; Samuelson, 1967). As examples of limits, consider portfolio w1 =
(0.8, 0.1, 0.1) and w2 = (1/3, 1/3, 1/3). Portfolio size will declare that w1 and w2 have samediversification degree. It fails to detect weight concentration. Consider universe of assetsA = B1, B2 and B = B1, B2, B2 and the corresponding portfolios wA = (1/2, 1/2) andwB = (1/3, 1/3, 1/3). Portfolio size will declare wB more diversified than wA. It fails todetect redundant information.
52
2.2.3.2 Sharpe (1972)’s Measure
Consider Equation (2.24) for a portfolio w,
R(w) = Rf + β (w) (Rm −Rf ) + e(w), (2.92)
where e(w) =∑N
i=1wi ei. Equation (2.92) implies that portfolio risk can also be decomposedas follows
σ2(w) = (β (w))2 σ2(wm) +N∑i=1
w2i σ
2ei (2.93)
Consider asset k as a reference asset. Define
σ2ei|k
=σ2ei
σ2ek
. (2.94)
σ2ei|k
is the relative asset i idiosyncratic risk. Sharpe (1972)’s measure is defined as follows
SH(w) =1∑N
i=1w2i σ
2ei|k
. (2.95)
When assets have the same idiosyncratic risk level, σ2ei = σ2
e0 , ∀ i = 1, ..., N , SH(w) is pro-portional or equivalent to the inverse of the Herfindhal index
SH(w) = σ2e0
1∑Ni=1w
2i
. (2.96)
The limit of this measure is that it only considers the difference in asset idiosyncratic risk.It ignores completely the difference in asset β as highlighted by Sharpe (1972) itself alreadymentioned in Section 2.1.2.2.2. Moreover, the measure is sensitive to the choice of the marketportfolio and applicable only to a portfolio of risky assets.
2.2.3.3 Coefficient of Determination or R-squared
The coefficient of determination is a measure introduced by Barnea and Logue (1973) to quan-tify corporate diversification. It is defined as a ratio of portfolio systematic risk to portfoliorisk
CD(w) =
(w>β
)2σ2(wm)
σ2(w). (2.97)
By definition, CD(w) is in a range between 0 and 1 i.e. 0 ≤ CD(w) ≤ 1. If w is well-diversified,then CD(w) = 1.
2.2.3.4 Tracking Error
The tracking error is another measure introduced by Barnea and Logue (1973) to quantifycorporate diversification. It is defined as follows
σe(w) =√
E(e(w)2). (2.98)
53
Contrary to CD(w), σe(w) is not normalized by a function depending on w. Therefore, thetwo measures do not give the same ranking. The two measures are mainly used in corporatefinance (Datta et al., 1991; Amihud and Baruch, 1999; Demsetz and Strahan, 1997; Amihudand Baruch, 1981; Jahera Jr. et al., 1987) than portfolio selection (Cresson, 2002; Byrne andLee, 2003), and are subject to the same criticism that Sharpe (1972)’s measure faces.
2.2.4 Risk Contribution Diversification Measures
This section reviews measures that capture the effect of the risk contribution diversificationstrategy.
2.2.4.1 Effective Number of Correlated and Uncorrelated Bets
Effective number of correlated bets (ENCB) is based on percentage risk contribution at thelevel of assets. Its general formulation provided by Carli et al. (2014) is
ENCBυ(w) =
(N∑i=1
(PRCi(w))υ) 1
1−υ
, (2.99)
where
PRCi(w) =
∂%(w>R)∂wi
%(w>R). (2.100)
Roncalli (2014) analyzed the case where % is portfolio variance or volatility and υ = 2 or υconverges to 1.
If PRCi(w) is defined at the level of risk factors, then ENCBυ(w) becomes the effective num-ber of uncorrelated bets (ENBυ(w)). Meucci (2009) analyzes ENBυ(w) where % is portfoliovariance or volatility, υ converges to 1 and factors are extracted using the principal componentanalysis (PCA). Meucci et al. (2014) extend the work of Meucci (2009). Instead to extractthe factors using the PCA, Meucci et al. (2014) extract the factors using the minimum tor-sion linear transformation. ENCB and ENB suffer from the shortcomings of risk contributiondiversification strategy (see Section 2.1.2.2.4). Moreover, they are applicable only to portfolioof risky assets. Readers can also see Deguest et al. (2013) and Carli et al. (2014) for moredetails.
2.2.4.2 Portfolio Diversification Index
Portfolio diversification index (PDI) was developed by Rudin and Morgan (2006) based on thePCA. The authors define PDI as a center of mass of the eigenvalues relative strengths vectorof assets return correlation matrix (ρ)
PDI = 2N∑i=1
iΛi − 1, (2.101)
54
where Λ = (Λ1, ...,ΛN )> is the vector eigenvalues relative strengths of ρ with Λi = ξi(ρ)∑Ni=1 ξi(ρ)
where ξ(ρ) = (ξ1(ρ), ..., ξN (ρ))> is the eigenvalues vector of ρ. PDI is in a range between1 and N (see Diyarbakirlioglu and Satman, 2013). For a completely non-diversified portfoliodominated by a single factor i.e. Λ1 = 1 and Λi = 0, i > 1, PDI = 1. In other words,when asset returns are perfectly correlated, PDI reaches its minimum value equal to 1. Foran ideally diversified portfolio i.e. Λi = 1/N for each i and PDI = N . In other words, whenasset returns are uncorrelated, PDI reaches its maximum value equal to N . PDI is also anon-increasing function of asset correlation degree. More explicitly, more assets are correlated(positively or negatively), more the value of PDI is low. For an application of PDI, readersare referred to Diyarbakirlioglu and Satman (2013); Oyenubi (2016).
The PDI has three main shortcomings. The first is it not depend on portfolio weight. Thenit not can be used to quantify the degree of diversification of a given portfolio w. It onlyuseful to quantify the potential diversification of an universe of assets. Second, it is unableto distinguish between negative and positive correlation. For example, consider the followingextreme case. Suppose that one has an universe of two perfectly positives correlated assets. Inthat case, it is straightforward to verify that ξ1(ρ) = 2 and ξ2(ρ) = 0. It follows that PDI = 1.Now, suppose that one has an universe of two perfectly negatives correlated assets. The resultremains the same, PDI = 1. However, an universe of two perfectly negatives correlated assetsis more diversified than an universe of two perfectly positives correlated assets. Third, it isapplicable only to a portfolio of risky assets.
2.3 Summary
The first goal of this chapter is to stress the importance of diversification in portfolio selection.No doubt, diversification is at the core of both pre-portfolio theory and portfolio theory. It isone of the most prominent elements of portfolio management as well as risk and return. It helpsto reduce or ultimately eliminate portfolio risk. Despite all criticisms made since the 2008-2009 financial crisis, it is still an important risk management tool (see Miccolis and Goodman,2012; Statman, 2013; Sandstrom, 2011; CEBS, 2010; Laas and Siegel, 2016; CEIOPS, 2010a;CRMCR, 2016; BCBS, 2010, 2013; EIOPA, 2014; CEIOPS, 2010b; Markowitz et al., 2009;Ilmanen and Kizer, 2012). Moreover, there is no unique definition of portfolio diversification.There are four definitions of portfolio diversification: the law of large numbers, the correlation,the CAPM and the risk contribution diversification strategies.
The second goal is to review some methods for its measurement. The following conclusionscould be drawn:
1. Several measures of portfolio diversification were proposed. These measures can begrouped in four categories according to the four diversification strategies identified.
2. In each category, there is no unique measure. There is a rich choice set which makes the
55
selection of measures difficult. Moreover, none of these measures has proven totally sat-isfactory. All have drawbacks and limited applications. To make matters worse, there isno general objective criteria that facilitate the choice of researchers, portfolio managers,financial planners or advisers. In the absence of clear objective criteria, they have usu-ally based their choice on convenience, familiarity, or on vague methodological grounds.This can have major implications in terms of risk and consequently social welfare ifthey are mistaken. Thus, as in the case of inequality (see Sen, 1976; Atkinson, 1987;Allison, 1978), polarization (Esteban and Ray, 1994) and risk (see Wang et al., 1997;Rockafellar et al., 2006a; Frittelli and Gianin, 2005; Stone, 1973; Follmer and Schied,2010; van der Hoek and Sherris, 2001; Pedersen and Satchell, 1998; Artzner et al., 1999)measurement, the necessity is not only to have a measure of portfolio diversification,but to have a coherent measure of portfolio diversification. In other words, a measureof portfolio diversification that is consistent with investor preference for diversificationin the rational choice theory under uncertainty. What constitutes a coherent measureof portfolio diversification? What are the minimum desirable properties a measure ofportfolio diversification should have in order to be considered as coherent? The nextchapter addresses these questions. It provides the definition of what we consider in thisthesis to be a coherent measure of portfolio diversification.
56
Part II
A Class Of Measures Of PortfolioDiversification Based On Rao’s
Quadratic Entropy
57
Chapter 3
Minimum Desirable Properties OfCoherent Measure Of PortfolioDiversification
This second part of the thesis is the main contribution. It adapts and extends the use of Rao(1982a)’s Quadratic Entropy to portfolio theory (PT) as a novel, coherent, flexible, unified,computational efficient and rigorous approach to manage and quantify portfolio diversification.This first chapter defines what it means by coherent portfolio diversification measure.
As outlined in Chapter 2, the necessity is not to have a measure of portfolio diversification,but to have a coherent measure of portfolio diversification, as in the case of inequality (see Sen,1976; Atkinson, 1987; Allison, 1978), polarization (see Esteban and Ray, 1994) and risk (seeWang et al., 1997; Rockafellar et al., 2006a; Frittelli and Gianin, 2005; Stone, 1973; Follmerand Schied, 2010; van der Hoek and Sherris, 2001; Pedersen and Satchell, 1998; Artzner et al.,1999) measurement. In other words, a measure of portfolio diversification that is consistentwith investors’ preference for diversification in the rational choice theory under uncertainty.
This chapter introduces the definition of a coherent measure of portfolio diversification usedin this thesis. More specifically, it defines the set of minimal desirable properties or axiomsthat a measure of portfolio diversification must satisfy in order to be considered as coherent.The definition is robust to portfolio diversification strategies.
The chapter is organized as follows. First (Section 3.1), formalizing some intuitions of diver-sification and adapting some properties of risk measures in portfolio theory, nine desirableproperties for a measure of portfolio diversification are given and justified. The measures ofportfolio diversification satisfying these properties are called coherent. Next (Section 3.2),the compatibility of these properties with investors’ preference for diversification in the mean-variance model is studied. Finally (Section 3.3), a list of portfolio diversification measures
58
(reviewed in Chapter 2) is examined against these properties. Throughout the chapter, theterms property and axiom are used equivalently.
3.1 Minimum Desirable Properties of Portfolio DiversificationMeasures
Before going into the presentation of our set of desirable axioms on measures of portfoliodiversification, let us first introduce the definition of a portfolio diversification measure.
How does one define a diversification measure of a portfolio w? Although there are essentiallyfour definitions or strategies of diversification in portfolio theory as discussed in Chapter 2,the diversification interest variable and benefit are unique, that is the distribution of portfolioweight w and risk or uncertainty reduction, respectively. Thus, let Φ be a measure of portfoliodiversification. It is natural to represent Φ as a mapping from WN into R conditional to assetrisk characteristics described by asset returns R = (R1, ..., RN )>
Φ: WN → R (3.1)
w 7→ Φ (w|R) , (3.2)
where WN is the set of long-only portfolios of size N . The form of the function Φ depends onthe definition of diversification and some properties of portfolio diversification measure.
In what follows, the set of desirable properties is introduced. It is assumed that the optimalportfolio of Φ (w|R), denoted w∗, is obtained by maximization. A single asset portfolio isdenoted δi = (δij)
Nj=1, where δij is Kronecker delta i.e. δii = 1 for all i = 1, ..., N and δij = 0
for all j 6= i. To distinguish between portfolios set in terms of size and assets universe, W willbe denoted WN referring to portfolios set of size N or sometimes WN
A referring to portfoliosset of size N associated to universe of assets A = A1, ..., AN with Ai a i-th asset. Portfoliow will also be denoted wA referring to a portfolio associated to the universe of assets Asometimes.
Axiom 1 (Concavity). For all w1,w2 ∈WN and α ∈ [0, 1],
Φ (αw1 + (1− α) w2|R) ≥ αΦ (w1|R) + (1− α) Φ (w2|R) (3.3)
and strict inequality for at least one α.
Axiom 1 implies that holding different assets increases total diversification. It also ensures thatthe diversification is always beneficial and can be decomposed across asset classes. Axiom 1can be replaced by a less restrictive one
Axiom 1’ (Quasi-concavity). For all w1,w2 ∈WN and α ∈ [0, 1],
Φ (αw1 + (1− α) w2|R) ≥ min (Φ (w1|R) ,Φ (w2|R)) (3.4)
59
and strict inequality for at least one α.
Axiom 2 (Degeneracy in N). There is a constant Φ ∈ R such that for all w ∈WN ,
Φ (δi|R) = Φ ≤ Φ (w|R) ∀ i = 1, ..., N. (3.5)
Axiom 2 states that all single asset portfolios have the same degree of diversification andare the least diversified portfolio. This implies that diversification is always better than fullconcentration or specialization. Axiom 2 is clearly necessary to prevent portfolio concentrationto remain undetected.
Axiom 3 (Degeneracy in R). Let A = AiNi=1 be an universe of N assets such that Ai =
A, ∀ i = 1, ..., N . Then, for all w ∈WN
Φ (w|R) = Φ. (3.6)
Axiom 3 ensures that there is no gain to diversify across perfectly similar assets. Such diver-sification is equivalent to portfolio full concentration. Axiom 3 is also necessary to preventportfolio concentration to remain undetected.
Before introducing Axiom 4, let us first recall the definition of comonotonic random variables.
Definition 3.1.1 (Comonotonicity: Dhaene et al. (2008)). A random vector R is comono-tonic if and only if there are non-decreasing functions fi, i = 1, ..., N and a random variableR such as R
d= (f1(R), ..., fN (R)), where d
= stands for “equal in distribution”.
Intuitively, the comonotonicity corresponds to an extreme form of positive dependency. Allreturns are driven linearity or nonlineraly by a unique factor, but positively. For other char-acterizations and more details about the concept of comonotonicity and its applications infinance, we refer readers to Dhaene et al. (2006); Deelstra et al. (2011); Dhaene et al. (2002b,a).
Axiom 4 (Relevance). Consider the equation Φ (w|R) = Φ on R for all w ∈WN such thatw 6= δi, i = 1, ..., N . Assume that R∗ is the solution. Then R∗ is comonotonic.
Axiom 4 is a complementary to Axiom 3. It is also necessary to prevent portfolio concentrationto remain undetected.
Axiom 5 (Duplication Invariance). Let A+ = A+i
N+1i=1 be a universe of assets such that
A+i = Ai, ∀ i = 1, ..., N and A+
N+1 = Ak, k ∈ 1, ..., N. Then
Φ(w∗A|RA
)=Φ(w∗A+ |RA+
)(3.7)
w∗Ai =w∗A+i∀ i 6= k (3.8)
w∗Ak =w∗A+k
+ w∗A+N+1
. (3.9)
60
Axiom 5 is seen desirable by Choueifaty et al. (2013). It ensures that the optimal diversifiedportfolio is not biased towards a more representative asset. This property allows to avoid riskconcentration.
Axiom 6 (Monotonicity in N). Let A++ = A++i
N+1i=1 be a universe of assets such that
A++i = Ai, ∀ i = 1, ..., N and A++
N+1 6= Ai, i = 1, ..., N . Then
Φ(w∗A++ |RA++
)≥ Φ
(w∗A|RA
). (3.10)
Axiom 6 is seen as natural in portfolio diversification literature (see Evans and Archer, 1968;Rudin and Morgan, 2006; Vermorken et al., 2012). It reveals that increasing portfolio sizedoes not decrease the degree of portfolio diversification. It also states that increasing portfoliosize does not systematically increase the degree of portfolio diversification.
The following assumptions are made for the next two axioms.
Assumption 3.1.1. % is translate-invariant i.e. for all a ∈ R, R ∈ R and κ ≥ 0
%(R+ a) = %(R)− κ a. (3.11)
Assumption 3.1.2. % is homogeneous i.e. for all κ ∈ R, R ∈ R and b ≥ 0
%(bR) = bκ%(R). (3.12)
Assumption 3.1.1 states that risk can be reducing adding cash, except in the case where κ = 0.Assumption 3.1.2 states that a linear increase of the return by a positive factor leads to a non-linear increase in risk, except in the case where κ = 1. Readers are referred to Sereda et al.(2010); Follmer and Schied (2010); Pedersen and Satchell (1998); Artzner et al. (1999) formore details.
Axiom 7 (Translation-invariance). Let A+ a = Ai + aNi=1 be a universe of assets suchas RAi+a = RAi + a, ∀ i = 1, ..., N, a ∈ R. Then for all w ∈WN ,
Φ (w|RA+a) = Φ(w|RA
). (3.13)
The desirability of Axiom 7 comes from the translation-invariance property of risk measure(see Assumption 3.1.1). It implies that adding a same amount of cash to asset returns doesnot change the degree of portfolio diversification. The intuition is as follows. Assume thatκ = 0. In that case, adding a same amount of cash to asset returns does not affect the degreeof portfolio risk. Then, the degree of portfolio diversification is not affected. Now, assumethat κ > 0. In that case, adding a same amount of cash to asset returns reduces portfoliorisk, but does not affect portfolio diversification degree. Because, since the cash is added toall assets, it can be considered as a systematic shock that affects the whole market.
61
However, when risk is defined as capital requirement or probability of loss (example expectedshort fall or conditional Value-at-Risk), Axiom 7 can be seen as counter-intuitive. To seethis, consider the particular case where risk % is defined as capital requirement verifyingAssumption 3.1.1. Assume that a =
%(w>R)κ with κ 6= 0. Then %
(w>R + a
)= 0, but
Φ (w|RA+a) = Φ (w|RA) ≥ 0. This counter-intuitive result can be viewed as an over diversi-fication, which can be interpreted as an extreme caution against extreme risk.
In the case where Φ is a normalized measure i.e. when Φ(w|R) can be rewritten as follows
Φ(w|R) =Ψ(w|R)
%(w>R), (3.14)
or equivalently as follows
Φ(w|R) =Ψ(w|R)− %(w>R)
%(w>R), (3.15)
with Ψ is a portfolio diversification measure such as Ψ(w|R + a) = Ψ(w|R), Axiom 7 mustbe replaced by the following one.
Axiom 7’ (Translation-invariance). Let A+ a = Ai + aNi=1 be a universe of assets suchthat RAi+a = RAi + a for all i = 1, ..., N, a ∈ R. Then
∂Φ(w|RA+a
)∂a
≥0, (3.16)
lima→−∞
Φ(w|RA+a
)=Φ, (3.17)
lima→+∞
Φ(w|RA+a
)=Φ, (3.18)
lima→ %(w|RA)
κ
a>%(w|RA)
κ
Φ(w|RA+a
)=−∞, (3.19)
lima→ %(w|RA)
κ
a<%(w|RA)
κ
Φ(w|RA+a
)=∞. (3.20)
The intuition behind Axiom 7’ is as follows. Equation (3.16) states that adding cash increasesthe diversification benefit. This due to the fact that adding cash reduces the total risk % anddoes not affect diversification Ψ. Equations (3.17) and (3.18) ensure that when cash convergesto +∞ or −∞, the diversification benefit vanished, because the whole system becomes ho-mogeneous. Equations (3.19) and (3.20) capture the over diversification behaviour of Φ whenrisk converges to 0 and diversification becomes unnecessary.
Axiom 8 (Homogeneity). Let bA = bAiNi=1 be a universe of assets such that RbAi =
bRAi , ∀ i = 1, ..., N, b ≥ 0. Under Assumption 3.1.2, there is κ ∈ R such that for all w ∈WN
Φ(w|RbA
)= bκΦ
(w|RA
). (3.21)
62
The desirability of Axiom 8 also comes naturally from the homogeneous property of riskmeasure (see Assumption 3.1.2). In the case where Φ is a normalized measure, Axiom 8 mustbe replaced by the following one.
Axiom 8’ (Homogeneity). Let bA = bAiNi=1 be a universe of assets such that RbAi =
bRAi , ∀ i = 1, ..., N, b ≥ 0. Under Assumption 3.1.2, for all w ∈WNA = WN
bA
Φ(w|RbA
)= Φ
(w|RA
). (3.22)
Axiom 8’ ensures that a normalized diversification measure must not be depended on thescalability.
Before introduce our last axiom, let us recall the definition of exchangeable random variables.
Definition 3.1.2 (Exchangeability). The random variables (R1, ..., RN ) are said to be ex-changeable if and only if their joint distribution FR(r1, ..., rN ) is symmetric.
A well-known example of exchangeable sequence of random variables is an independent andidentically distributed sequence of random variables. For more details on exchangeable randomvariables, we refer readers to Aldous (1985).
Axiom 9 (Exchangeability). If (R1, ..., RN ) are exchangeable, then w∗A = 1N .
The desirability of Axiom 9 comes from Samuelson (1967). The author shows that when R
is exchangeable the optimal portfolio maximizing the expected utility is the equally weightportfolio. This suggests that any portfolio diversification measure must have equally weightportfolio as well-diversified portfolio if R is exchangeable. Note that Axiom 9 is also seendesirable by Meucci et al. (2014).
For a measure that the optimal portfolio is obtained by minimization, Axioms 1, 1’, 2, 6become
Axiom 1 (Convexity). For all w1,w2 ∈WN and α ∈ [0, 1],
Φ (αw1 + (1− α) w2|R) ≤ αΦ (w1|R) + (1− α) Φ (w2|R) (3.23)
and strict inequality for at least one α.
Axiom 1’ (Quasi-convexity). For all w1,w2 ∈WN and α ∈ [0, 1],
Φ (αw1 + (1− α) w2|R) ≤ max (Φ (w1|R) ,Φ (w2|R)) (3.24)
and strict inequality for at least one α.
63
Axiom 2 (Degeneracy in N). There is a constant Φ ∈ R such that for all w ∈ WN andδi ∈WN ,
Φ (δi|R) = Φ ≥ Φ (w|R) . (3.25)
Axiom 6 (Monotonicity in N). Let A++ = A++i
N+1i=1 be a universe of assets such that
A++i = Ai, ∀ i = 1, ..., N and A++
N+1 6= Ai, i = 1, ..., N . Then
Φ(w∗A++ |RA++
)≤ Φ
(w∗A|RA
). (3.26)
Although we agree that our set of desirable properties is not exhaustive, we believe that itdescribes necessary and sufficient features for a measure of portfolio diversification to be con-sidered as coherent. In other words, it will define coherent portfolio diversification measures.
Definition 3.1.3 (Coherence). A portfolio diversification measure satisfying Axioms 1 or1’, 2 to 6, 7 or 7’, 8 or 8’ and 9 is said to be coherent. Otherwise, it is called as incoherent.
Notice that from Definition 3.1.3 a measure satisfying Axioms 1 or 1’, 2 to 6, 7 or 7’, 8 or 8’and 9 is not necessarily a portfolio diversification measure. This implies that our set of axioms,contrary to Artzner et al. (1999); Esteban and Ray (1994); Sen (1976), does not provide arepresentation or a characterization of coherent portfolio diversification measures or does notdetermine a unique class of portfolio diversification measures. The reason is simple. Our set ofaxioms is robust to portfolio diversification definitions. The choice of a portfolio diversificationmeasure based on Definition 3.1.3 does not imply a choice of a specific definition of portfoliodiversification, which is not the case of that of Artzner et al. (1999). The set of axioms ofArtzner et al. (1999) is only consistent with risk defined as a capital requirement.
3.2 Compatibility With the Preference For Diversification inthe Mean-Variance Model
Since a portfolio diversification measure has to be used in portfolio choice, it must take intoaccount investors’ attitudes towards diversification. Therefore, this section studies the compat-ibility or consistence of our set of desirable properties, discussed in the previous section, withinvestors’ preference for diversification (PFD) in the mean-variance (MV) model exploitingthe relation between the expected utility theory (EUT) and the MV model.
The definition of the PFD in the EUT Definition 2.1.4 can be rewritten as follows:
Definition 3.2.1. A Von Neumann Morgenstern utility u(.) exhibits preference for diversi-fication if for any Ri ∈ R, i = 1, ..., N and αi ∈ [0, 1], i = 1, ..., N such that
∑Ni=1 αi = 1,
Eu(R1) = ... = Eu(RN ) =⇒ Eu
(N∑i=1
αiRi
)≥ Eu(Rj) ∀ j = 1, ..., N. (3.27)
64
The gain of diversification in the EUT can therefore be measured by the difference
Eu (w|R) = Eu
(N∑i=1
wiRi
)−
N∑i=1
wiEu(Ri). (3.28)
It is well-known that the MV model is a special case of the EUT when utility function isthe exponential utility function and asset returns are elliptical distributed. Therefore, thedefinition of the PFD in the EUT is also valid for the MV model, since it is invariant to assetreturns distribution. It follows that the diversification in the MV model can be measured by
UMV (w|R) = UMV
(N∑i=1
wiRi
)−
N∑i=1
wiUMV (Ri), (3.29)
where UMV (.) is the MV utility function
UMV (R) = E(R)− 1
τVar(R). (3.30)
Substituting UMV (.) from (3.30) into (3.29), (3.29) becomes
UMV (w|R) =1
τ
(w>σ2 − σ2(w)
).
As a result, contrary to Fernholz (2010), there is a specific measure of portfolio diversification inthe MV model, which is the diversification return, a portfolio diversification measure discussedin Section 2.2.2.3.
The definition of compatibility of the proposed desirable properties with the PFD in the MVmodel adopted is follows.
Definition 3.2.2 (Compatibility with PFD in the MV model). The proposed set of ax-ioms are compatible with the PFD in the MV model if and only if UMV (w|R) or 1
τUMV (w|R)
satisfies Axioms 1 or 1’, 2 to 6, 7 or 7’, 8 or 8’ and 9.
Proposition 3.2.1 (Compatibility with PFD in the MV model). UMV (w|R) satisfiesAxioms 1 to 9.
Proof (of Proposition 3.2.1). Axiom 1) Since the first and second terms of UMV (w|R)
are linear and convex in w respectively, UMV (w|R) is concave on W.
Axiom 2) Let δi ∈W be a single asset portfolio. It is straightforward to show that
UMV (δi|R) =1
τ
(σ2i − σ2
i
)= 0. (3.31)
It is also straightforward to verify that UMV (w|R) can be rewritten as follows
UMV (w|R) =1
τ
N∑i=1
wi Var (Ri −R(w)) . (3.32)
Therefore,UMV (w|R) ≥ 0 = Φ = UMV (δi|R). (3.33)
65
Axiom 3) Since Ai = A, ∀ i, j = 1, ..., N , Ri = R, ∀ i = 1, ..., N . Then, σi = σ, ∀ i = 1, ..., N
and ρij = 1, ∀ i, j = 1, ..., N . It follows that
UMV (w|R) = σ2 − σ2
(N∑i=1
wi
)2
= 0 = Φ. (3.34)
Axiom 4) Since wi ≥ 0, i = 1, ..., N , from (3.32),
UMV (w|R) = 0⇔ Var (Ri −R(w)) = 0, ∀ i = 1, ..., N (3.35)
⇔ Ri = R(w), ∀ i = 1, ..., N (3.36)
⇔ Ri = Rj , ∀ i, j = 1, ..., N. (3.37)
The result follows.
Axiom 5) Since A+N+1 = Ak, A+
i = Ai, i = 1, ..., N ,
UMV (wA+ |RA+) =N+1∑i=1
wA+iσ2A+i−
N+1∑i,j=1
wA+iwA+
jσA+
iσA+
jρA+
i A+j
(3.38)
=N∑
i 6=k=1
wA+iσ2A+i
+(wA+
N+1+ wA+
k
)σ2A+k
(3.39)
−N∑
i,j 6=k=1
wA+iwA+
jσA+
iσA+
j−
N∑i 6=k=1
wA+i
(wA+
k+ wA+
N+1
)σA+
iσA+
kρA+
i A+k.
(3.40)
Let
w∗∗A =
(w∗A+
1, ..., w∗
A+k−1
, w∗A+k
+ w∗A+N+1
, w∗A+k+1
, ..., w∗A+N
)(3.41)
w∗∗A+ =
(w∗A1
, ..., w∗Ak−1,w∗Ak
2, w∗Ak+1
, ...,w∗AN+1
2
). (3.42)
It follows that
UMV (w∗A+ |RA+) = UMV (w∗∗A |RA) ≤ UMV (w∗A|RA) (3.43)
UMV (w∗A|RA) = UMV (w∗∗A+ |RA+) ≤ UMV (w∗A+ |RA+). (3.44)
Then
UMV (w∗A+ |RA+) = UMV (w∗A|RA) (3.45)
w∗Ai = w∗A+i, ∀ i 6= k (3.46)
w∗Ak = w∗A+k
+ w∗A+N+1
. (3.47)
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Axiom 6) Consider a portfolio wA++ =(w∗A, 0
). Portfolio wA++ is an element of WN+1
A++ , so
UMV
(w∗A++ |RA++
)≥ UMV
(wA++ |RA++
). (3.48)
Since UMV
(wA++ |RA++
)= UMV
(w∗A|RA
),
UMV
(w∗A++ |RA++
)≥ UMV
(w∗A|RA
). (3.49)
Axiom 7 and 8) Because covariance is translate-invariant and homogeneous of degree two.
Axiom 9) Since σi = σ, ∀ i = 1, ..., N and ρij = ρ, ∀ i, j = 1, ..., N when R is exchangeable,
UMV (w|R) = σ2 − σ2
N∑i=1
w2i + ρ
N∑i,j=1
wiwj
.
It is straightforward to show that 1/N maximizes UMV (w|R).
From Proposition 3.2.1, it follows that the set of desirable properties is compatible withinvestors’ PFD in the MV model. This result strengthens the desirability, reasonableness andrelevance of the postulated axioms. In turn, UMV (w|R) is seen to be coherent.
3.3 Coherence of Most Currently Used Methods of MeasuringPortfolio Diversification
This section studies whether popular portfolio diversification measures used by theoreticiansand practitioners are coherent with the above stated desirable properties. These measures are:
(a) Portfolio size;(b) Diversification return;(c) Goetzmann et al.’s measure;(d) Diversification ratio;(e) Frahm and Wiechers’s measure;(f) Embrechts et al.’s class of measures;(g) Tasche’s class of measures;(h) Effective number of constituents;(i) Effective number of correlated bets;(j) Effective number of uncorrelated bets.
The following assumption is required for Embrechts et al. and Tasche’s measures.
Assumption 3.3.1. The risk measure % satisfies the following properties: positive homogene-ity, translation-invariance, convexity and non-additivity for independent
67
As an example of a risk measure satisfying the above properties, there is concave distortionrisk measures (see Dhaene et al., 2006).
Proposition 3.3.1. Under Assumption 3.3.1, the following measures are coherent:
(a) Diversification ratio (DR);(b) Goetzmann et al.’s measure. (GLR);(c) Diversification return (Dr);(d) Embrechts et al.’s measures (EFK%);(e) Tasche’s measures (DF%).
Proof (of Proposition 3.3.1). (a) Diversification ratio (DR): DR satisfies Axioms 1’ and2 to 9:
Axiom 1’): Since portfolio volatility is convex and asset average volatility w>σ is linear onW, from Avriel et al. (2010), DR (w|R) is quasi-concave.
Axiom 2): DR (δi|R) = σi/σi = 1. Also, DR (w|R) ≥ 1 for all w ∈ W by definition. As aresult,
DR (δi|R) = 1 ≤ DR (w|R)
with Φ = 1.
Axiom 3): Ai = A implies that Ri = R, ∀ i = 1, ..., N . Consequently, σi = σj = σ andρij = 1, ∀ i, j = 1, ..., N , where σ is constant. It follows that for all w ∈W
DR(w|R) =σ
σ= 1 = Φ.
Axiom 4): One needs to solve the equation DR(w|R) = 1 for all R ∈ R given w ∈ W suchthat w 6= δi, i = 1, ..., N .
DR(w|R) = 1 =⇒ ρij = 1, ∀ i, j = 1, ..., N.
The solution is therefore the comonotonic random variance R∗ =(R1−µ1
σ1, ..., RN−µNσN
)such
that Ri−µiσi
= R, ∀ i = 1, ..., N with R a random variable.
Axiom 5): Follows the proof of Proposition 3.2.1. Readers can also see Choueifaty et al.(2013).
Axiom 6): Follows the proof of Proposition 3.2.1.
Axiom 7): Since covariance matrix is translate- invariant, DR (w|R) is also translate-invariant.
Axiom 8): Since portfolio and asset volatility are homogeneous of degree one, DR (w|R) ishomogeneous of degree zero.
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Axiom 9): Assume that R = (R1, ..., RN ) is exchangeable. In that case, it well-known thatσi = σ for all i = 1, ..., N and ρij = ρ for all i, j = 1, ...N , where σ and ρ are constants. Itfollows that
DR(w|R) =σ
σ(1− (1− ρ)∑N
i 6=j=1wiwj).
It is straightforward to verify that the equally weighted portfolio 1/N maximizes DR(w|R).
(b) Goetzmann et al.’s measure (GLR): Follows the proof of DR.
(c) Diversification return: See the proof of Proposition 3.2.1.
(d) Embrechts et al. (1999)’s measures (EFK%): EFK% satisfies Axioms 1 to 9:
Axiom 1): Since % is convex on R, % is convex on W. It follows that EFK%(w|R) is convexon W.
Axiom 2): EFK% (δi|R) = % (Ri) − % (Ri) = 0. Also, since % is convex, EFK%(w|R) ≥ 0. Itfollows that
EFK% (δi|R) = 0 ≤ EFK%(w|R)
with Φ = 0.
Axiom 3): Ai = A implies that Ri = R for all i = 1, ..., N . Then
EFK% (w|R) = % (Ri)− % (Ri) = 0 = Φ.
Axiom 4): One needs to solve the equation EFK%(w|R) = 0 for all R ∈ R given w ∈W suchthat w 6= δi, i = 1, ..., N . If % is comonotonic additive and non-additive for independent,
EFK%(w|R) = 0⇐⇒ R is comonotonic.
The result follows.
Axiom 5): Follows the proof of Proposition 3.2.1.
Axiom 6): Follows the proof of Proposition 3.2.1.
Axiom 7): If % is translate-invariant, then
EFK% (w|RA+a) =w>% (RA + a)− %(w>RA + a
)=w>% (RA)− κ a− %
(w>RA
)+ κ a
=EFK% (w|RA) .
69
Axiom 8): If % is homogeneous, then
EFK% (w|bR) =w>%(bR)− %(bw>R)
=bκ w>%(R)− bκ %(w>R)
=bκ EFK% (w|R) .
Axiom 9): Straightforward to proof.
(e) Tasche (2006)’s measures (DF%): DF% satisfies Axioms 1’, 2 to 6, 7’, 8’ and 9:
Axiom 1’): Since % is convex and∑N
i=1wi%(Ri) is linear on W, from Avriel et al. (2010),DF% (w|R) is quasi-concave.
Axiom 2): DF% (δi|R) = %(Ri)%(Ri)
= 1. Also, since % is convex, DF% (w|R) ≥ 1. As a result,
DF% (δi|R) = 1 ≤ DF% (w|R) , ∀ i = 1, ..., N
with Φ = 1.
Axiom 3): Ai = A implies that Ri = R for all i = 1, ..., N . Then
DF% (w|R) =%(R)
%(R)= 1 = Φ
Axiom 4): One needs to solve the equation DF%(w|R) = 1 for all R ∈ R given w ∈ W suchas w 6= δi, i = 1, ..., N . If % is comonotonic additive and non-additive for independent,
DF%(w|R) = 1⇐⇒ R is comonotonic.
The result follows.
Axiom 5): Follows the proof of Proposition 3.2.1.
Axiom 6): Follows the proof of Proposition 3.2.1.
Axiom 7’ and 7): If % is translate-invariant, then
DF% (w|RA+a) =w>% (RA)− κ a% (w>RA)− κ a
.
If κ = 0,DF% (w|RA+a) = DF% (w|RA) .
70
If κ 6= 0, then
∂DF%(w|RA+a
)∂a
≥0, (3.50)
lima→−∞
DF%(w|RA+a
)=1 = Φ, (3.51)
lima→+∞
DF%(w|RA+a
)=1 = Φ, (3.52)
lima→ %(w|RA)
κ
a>%(w|RA)
κ
DF%(w|RA+a
)=−∞, (3.53)
lima→ %(w|RA)
κ
a<%(w|RA)
κ
DF%(w|RA+a
)=∞. (3.54)
Axiom 8): If % is homogeneous, then
DF% (w|bR) =w>% (bR)
% (bw>R)
=bκ w>% (R)
bκ % (w>R)
=DF% (w|R)
Axiom 9): Straightforward to proof.
Proposition 3.3.2. The following measures are incoherent:
(a) Portfolio size (PS);(b) Effective number of constituents (ENCυ);(c) Frahm and Wiechers’s measure (NV);(d) Effective number of correlated bets (ENCBυ);(e) Effective number of bets (ENBυ).
Proof (of Proposition 3.3.2). (a) Portfolio size (PS): By definition, portfolio size failsAxioms 1 or 1’, 3, 5, 4, 8 and 9 and verifies Axioms 2, 6, 7.
(b) Effective number of constituents (ENCυ): ENCυ fails Axioms 3, 4, 5, 7, 8, 9 and satisfiesAxioms 1, 2, 6 .
Axioms 3, 5, 4, 7, 8, 9) by definition.
Axiom 1) From Patil and Taille (1982), it is known that BMυ(w|R) is concave if υ ≥ 1. Since
ENCυ(w|R) = h (BMυ(w|R)) ,
where h(x) = (1 − (υ − 1)x)1
1−υ if υ ≥ 0, υ 6= 1 and h(x) = exp(x) if υ = 1, ENCυ(w|R) isconcave if υ > 1 and quasi-concave if υ = 1.
71
Axiom 2) Straightforward to proof.
Axiom 6) Follows proof of Proposition 3.2.1.
(c) Frahm and Wiechers’s measure (NV): NV fails Axioms 2, 3, 4 and satisfies Axioms 1, 4,5, 6, 7, 8, 9.
Axiom 1) It is well-known that portfolio variance is convex.
Axiom 2) Consider two assets i and j such as σ2i < σ2
j . Then NV (δi|R) = σ2i < σ2
j =
σ2 (δj |R). The result follows.
Axiom 3) The result comes from Axiom 2.
Axiom 4) The result comes from Axiom 2.
Axiom 5) See Choueifaty et al. (2013).
Axiom 6) Follows the proof of Proposition 3.2.1.
Axiom 7) Since covariance matrix is translate-invariant, NV (w|R) is also translate-invariant.
Axiom 8) It is well-known that covariance matrix is homogeneous of degree two.
Axiom 9) Since σi = σ, ∀ i, j = 1, ..., N and ρij = ρ, ∀ i, j = 1, ..., N when R is exchangeable,
NV(w|R) = σ2
N∑i=1
w2i + ρ
N∑i,j=1
wiwj
.
It is straightforward to show that 1/N maximizes NV(w|R).
(f) Effective number of correlated bets (ENCBυ): ENCBυ fails Axioms 1 or 1’, 2, 3, 4, 5,satisfies Axioms 6, 8, 9 and satisfies Axiom 7 conditionally.
Axioms 1 or 1’) Representing graphically the effective number of bets in the case where N =
2, υ = 2 and risk is portfolio variance such that σ1 = σ2 = 1 and ρ12 = −0.5 (see Figure 3.1),it is observable that ENCBυ is neither concave nor quasi-concave.
Axiom 2) See Figure 3.1.
Axiom 3) Assume that risk is measured using portfolio volatility σ(w). Since Ai = A, ∀ i =
1, ..., N , Ri = R, ∀ i = 1, ..., N . Then ρij = 1, σi = σj = σ, ∀ i, j = 1, ..., N and
σ(w) =N∑i=1
wi σi.
The absolute risk contribution of asset i is therefore
PRCi(w) =σ
σ(w).
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Figure 3.1 – Representation of the Effective Number of Correlated Bets for N = 2, υ = 2 and% = σ2
0 0.2 0.4 0.6 0.8 1
0.8
1
1.2
1.4
1.6
1.8
2
w1/w2
ENCB
It follows thatENCBυ(w|R) = N > Φ = 1.
Axiom 4) The equation ENCBυ(w|R) = 1 on R with w 6= δi has no solution.
Axiom 5) See Pola (2014a).
Axiom 6) Follows the proof of Proposition 3.2.1.
Axiom 7) Assume that % is translate-invariant such that
%(w|R + a) = %(w|R).
In that casePRCi(w|R + a) = PRCi(w|R).
It follows that ENCBυ(w|R) is translate-invariant. Now, assume that risk measure % istranslate-invariant such that
%(w|R + a) = %(w|R)− κ a with κ 6= 0.
In that case
PRCi(w|R + a) =wi
(∂%(w|R)∂w − κ a
)%(w|R)− κ a
.
It follows that ENCBυ(w|R) is not translate-invariant.
Axiom 8) Assume that the risk measure % is homogeneous as follows
%(w|bR) = bκ%(w|R).
73
It is well-known that in this case the first partial derivative of % in respect of w is homogeneousof degree κ− 1
∂%(w|bR)
∂w= bκ−1∂%(w|R)
∂w.
Therefore
PRCi(w|bR) =bwi
∂%(bw|R)∂w
%(w|bR)=wi
∂%(w|R)∂w
%(w|R)= PRCi(w|R).
It follows that ENCBυ(w|R) is homogeneous of degree 0.
Axiom 9) Again assume risk is measured using portfolio volatility σ(w). R = (R1, ..., RN ) isexchangeable implies that σi = σj = σ and ρij = ρ for all i, j = 1, ..., N . It follows that
PRCi(w|R) =wi σ
2(1− ρ
)+ σ2ρ
σ(w), ∀ i = 1, ..., N.
At optimal, we havePRCi = PRCj , ∀ i, j = 1, ..., N.
Then wi = wj = 1/N, ∀ i, j = 1, ..., N .
(f) Effective number of bets (ENBυ): ENBυ fails Axioms 1 or 1’, 2, 5, 9 and satisfies Axioms 3,4, 6, 7, 8 when the factors are extracted using the principal component analysis (PCA). Whenthe factors are extracted using the minimal linear torsion (MLT), ENBυ fails Axioms 1 or 1’,2, satisfies Axioms 6, 7, 8, 9 and is not defined in the case of Axioms 3, 4, 5.
Axiom 1 or 1’) Representing graphically the effective number of bets in the case where N = 2
(see Figure 3.2), it is observable that ENBυ is neither concave nor quasi-concave.
Figure 3.2 – Representation of the Effective Number of Bets for N = 2, υ = 2 and % = σ2
0 0.2 0.4 0.6 0.8 1
1
1.2
1.4
1.6
1.8
2
w1/w2
ENB
PCAMLT
Axiom 2) See Figure 3.2.
74
Axiom 3) Assume that the factors are extracted using the PCA. Since Ai = A, ∀ i = 1, ..., N ,Ri = R, ∀ i = 1, ..., N . It follows that the number of extracted factors is equal to one. Then
PRC1 = 1, and PRCi 6=1 = 0
As a result ENBυ(w|R) = 1. Note that the factors cannot be extracted using the MLT.
Axiom 4) The equation ENBυ(w|R) = 1 on R with w 6= δi has no solution in the case wherefactors are extracted using the MLT transformation. In the case where factors are extractedusing the PCA, the solution of the equation ENBυ(w|R) = 1 on R with w 6= δi is thecomonotonic R = (R1, ..., RN ) with Ri = R.
Axiom 5) In the case where the factors are extracted using the PCA see Pola (2014a). In thecase where the factors are extracted using the MLT, ENBυ is not defined.
Axiom 6) Follows the proof of Proposition 3.2.1.
Axiom 7) Since covariance is translate-invariant, ENBυ(w|R) is translate-invariant.
Axiom 8) Since covariance is homogeneous of degree two,
PRCi(w|bR) = PRCi(w|R)
when the factors are extracted using the PCA. As a result ENB(w|R) is homogeneous ofdegree 0. The result remains the same when the factors are extracted using the MLT. As aresult, ENBυ(w|R) is homogeneous of degree 0.
Axiom 9) See Meucci et al. (2014, Appendix A.2).
Table 3.1 summarizes the results of Propositions 3.3.1 and 3.3.2. It is observable that onlymeasures related to the correlation diversification strategy are coherent. The effective numberof constituents, of correlated and uncorrelated bets, portfolio size and Frahm and Wiechers’smeasure are incoherent. Portfolio size is the most incoherent. Between the effective numbermeasures, the effective number of constituents is the most incoherent. Between the effectivenumber of correlated and uncorrelated bets, the latter is most coherent. Between the effectivenumber of uncorrelated bets, that based on the minimum linear torsion is the most coherent.As a result, it is less risky to follow the correlation diversification strategy than to followthe equal weighted or risk parity diversification strategies. Moreover, among the risk paritydiversification strategies, it is less risky to adopt the factor level approach than the asset levelone. In particular, it is better to adopt the factor level approach where the factors are extractedusing the minimum linear torsion. The results also confirm that portfolio size and the effectivenumber of constituents are crude measures. They can be used only under restricted conditions.For example, in the case where information about asset risk characteristics R is not useful.
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Table 3.1 – Test of Considered Portfolio Diversification Measures
Axioms1 2 3 4 5 6 7 8 91’ 7’ 8’
Portfolio size 5 5 5
Effective number of constituents 5 5 5 5
Diversification return 5 5 5 5 5 5 5 5 5
Diversification ratio 5 5 5 5 5 5 5 5 5
Goetzmann et al.’s measure 5 5 5 5 5 5 5 5 5
Embrechts et al.’s measures 5 5 5 5 5 5 5 5 5
Tasche’s measures 5 5 5 5 5 5 5 5 5
Frahm and Wiechers’s measure 5 5 5 5 5 5
Effective number of correlated bets 5 5
Effective number of bets (PCA) 5 5 5 5 5
Effective number of bets (MLT) nd nd nd 5 5 5 5
Notes. 5: satisfies; nd: not defined.
3.4 Summary
This chapter has introduced the definition of a coherent measure of portfolio diversificationused in this thesis. This definition is based on a set of nine desirable properties, which arerobust to portfolio diversification strategies and were proved compatible with the preferencefor diversification in the mean-variance model. Although we agree that our set of desirableproperties is not exhaustive, we believe that it describes necessary and sufficient features fora measure of portfolio diversification to be considered as coherent.
Testing a set of popular measures of portfolio diversification against this definition, it wasobservable that only diversification return, Goetzmann et al.’s measure, diversification ratio,Embrechts et al. (1999)’s measures, Tasche (2006)’s measures, which are based on the corre-lation diversification strategy, are coherent. The measures such as portfolio size, Frahm andWiechers’s measure, effective number of constituents, of correlated and uncorrelated bets werefound incoherent with the proposed definition. Portfolio size is the measure least compatiblewith the proposed definition. Between the effective number measures, the effective number ofconstituents is the most incoherent. Between the effective number of correlated and uncor-related bets, the latter is most coherent. Between the effective number of uncorrelated bets,that based on the minimum linear torsion is the most coherent.
As a result, it is less risky to follow the correlation diversification strategy than to followthe equally weighted or risk parity diversification strategies. Moreover, among the risk paritydiversification strategies, it is less risky to adopt the factor level approach than the assetlevel one. In particular, it is better to adopt the factor level approach where the factors areextracted using the minimum linear torsion approach. The results also confirm that portfolio
76
size and the effective number of constituents are crude measures. They can be used onlyunder restricted conditions. For example, in the case where information about asset riskcharacteristics R is not useful.
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Chapter 4
Rao’s Quadratic Entropy,Diversification And Portfolio Theory
This chapter is the main contribution of this thesis. It adapts and extends the use of Rao(1982a)’s Quadratic Entropy to portfolio theory (PT) as a novel, coherent, flexible, unified,computational efficient and rigorous approach to manage and quantify portfolio diversification.
As outlined in Chapter 2, portfolio diversification is at the core of portfolio selection, so itsmeasurement and management is of fundamental importance in finance and insurance do-mains as risk measurement and management. Consequently, since Markowitz (1952), severalmeasures of portfolio diversification were proposed, each based on a different criterion (Sec-tion 2.2). Unfortunately, none of them has proven totally satisfactory. All have drawbacksand limited applications (see Sections 2.2 and Chapters 3). Developing a coherent measure ofportfolio diversification is therefore an active research area in investment management.
In this chapter, a new class of portfolio diversification measures is proposed inspiring from Rao(1982a)’s Quadratic Entropy (RQE), a general approach to measuring diversity. The discretedefinition of RQE (see Equation (1.15)) is considered, because it is more naturally suited toportfolio selection.
The chapter is organized as follows. Section 4.1 demonstrates that RQE becomes a validclass of portfolio diversification measures when it is judiciously calibrated i.e. when P, X,P and D are judiciously defined. The resulting class of measures of portfolio diversificationis called RQE of a portfolio or portfolio RQE. Section 4.2 presents some examples of thedissimilarity matrix D. It also establishes the relationship between RQE and some existingportfolio diversification measures. Section 4.3 presents some properties of portfolio RQE andestablishes the conditions under which it is coherent. Section 4.4 summarizes the results.
78
4.1 Rao’s Quadratic Entropy and Portfolio Diversification
At first, calibration of P, X and P is discussed in Section 4.1.1. This defines the RQE of aportfolio or portfolio RQE. Next, calibration of D is discussed in Section 4.1.2.
4.1.1 Calibration of P, X and P
Consider an universe A = AiNi=1 of N assets (risky or not), and w a given long-only portfolioassociated to A so that each portfolio w can be viewed as a population of individuals, whereindividuals are assets. Next, define the random variable X to take the finite values 1, ..., N (Nassets) and its probability distribution P (X = i) = wi, ∀ i = 1, ..., N , so that it is associatedto the random experiment whereby assets are randomly selected (with replacement) from w.RQE of a long-only portfolio w is then defined as follows.
Definition 4.1.1 (RQE of portfolio). RQE of a long-only portfolio w is defined as half ofthe average difference between two randomly drawn (with replacement) assets from w i.e.
HD(w) =1
2
N∑i,j=1
dij wiwj (4.1)
or in vector form
HD(w) =1
2w>D w. (4.2)
Let XD be the discrete random variable equal to the dissimilarity between two assets drawn,so that XD takes a finite number of values dij , i, j = 1, ..., N . The probability distribution ofXD is P (XD = dij) = wiwj . RQE of w can also be interpreted as half of the expectation ofXD
HD(w) =1
2E (XD) . (4.3)
Definition 4.1.1 is still valid when short sales are allowed. However, this thesis focuses onHD(w) when short sales are restricted unless otherwise specified.
As can be noted, RQE of a portfolio differs from other entropy measures used in finance, suchas Shannon entropy and Tsallis Entropy (see Zhou et al., 2013), essentially by the fact that itincorporates not only portfolio size (N), portfolio weight (w), but also the intrinsic differencebetween asset quantified here by dij . This is the source of RQE attractiveness and flexibilityas a measure and as a tool of portfolio diversification.
In what follows PRQE (referring to portfolio RQE) denotes RQE of a portfolio. PRQE is aclass of measures (not necessary measures of portfolio diversification) and PRQED denotes aspecific RQE of a portfolio based on a dissimilarity matrix D. The next section discusses thecalibration of D . It provides the conditions on D under which PRQE can be considered as avalid class of portfolio diversification measures.
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4.1.2 Calibration of D
The choice of the dissimilarity matrix D specifies PRQE and determines if yes or no it canbe considered as a valid class of portfolio diversification measures. This thesis argues that thefollowing two conditions on D, taken together, are sufficient for PRQE to be considered as avalid class of portfolio diversification measures.
Condition 1. The dissimilarity matrix D must be chosen such that HD(w) is concave on theset of long-only portfolio (Concavity).
Condition 2. The dissimilarity matrix D must be chosen such that HD(w) is consistent withuncertainty or risk reduction (Uncertainty or risk reduction).
Condition 1 is recommended by Rao (1982a) and is necessary for a measure to be coherent(see Axiom 1 in Chapter 3). It ensures that the diversification is always beneficial and can bedecomposed across asset classes.
When N = 2, it is straightforward to verify that HD(w) always fulfils Condition 1. To checkCondition 1, when N > 2, one can perform the following tests.
Proposition 4.1.1 (Test I for Condition 1). HD(w) fulfils Condition 1 if and only if D
is conditional negative definite (CND).
Proof (of Proposition 4.1.1). See Proposition 1.3.2.
Proposition 4.1.2 (Test II for Condition 1). HD(w) fulfils Condition 1 if and only if forany α ∈W, GD(α) is positive semi-definite (PSD).
Proof (of Proposition 4.1.2). Proposition 1.1.2 implies that D is CND if and only ifGD(α) is PSD for any α ∈W. Thus, the result follows from Proposition 4.1.1.
Proposition 4.1.3 (Test III for Condition 1). HD(w) fulfils Condition 1 if and only ifthe (N − 1)× (N − 1) matrix
(diN + djN − dij) , i, j = 1, ..., N − 1, (4.4)
is PSD.
Proof (of Proposition 4.1.3). Follows from Proposition 1.3.1. This is a special case ofProposition 4.1.2 for α = δN . See also Lau (1985).
Proposition 4.1.4 (Test IV for Condition 1). HD(w) fulfils Condition 1 if one of thefollowing statements is respected
1) D is a metric and N ≤ 4;
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2) there is κ ∈ (0, 2] such that D1/κ is Euclidean;3) D is a ultrametric.
Proof (of Proposition 4.1.4). 1) It is straightforward to show that if D is a metricand N ≤ 4, the (N − 1)× (N − 1) matrix
(diN + djN − dij) , i, j = 1, ..., N − 1, (4.5)
is PSD (see Ricotta and Szeidl, 2006). Then, by Proposition 4.1.3, HD(w) is concave.2) Suppose that there is κ ∈ (0, 2] such as D1/κ is Euclidean. From Proposition 1.1.3,
D2/κ is CND. From Proposition 1.1.4, D2κ1/κ, ∀κ1 ∈ [0, 1] is CND. In particular, forκ1 = κ/2, D is CND. The result follows from Proposition 4.1.1.
3) Suppose that D is a ultrametric. By Proposition 1.1.8, D is Euclidean. From Proposi-tion 1.1.3, D2 is CND. From Proposition 1.1.4, Dκ/2, ∀κ ∈ [0, 1], is CDN. In particular,for κ = 1/2, D is CND. The result follows from Proposition 4.1.1.
In practice, test IV is easy to perform when D satisfies the requirement conditions. Otherwise,test II is most useful.
Condition 1 can be replaced by a less restrictive one.
Condition 1’. The dissimilarity matrix D must be chosen such that HD(w) is quasi-concave(Quasi-concavity).
Condition 1’ is also a necessary condition of a portfolio diversification measure to be consideredas coherent (see Axiom 1’ in Chapter 3). To check Condition 1’ one can perform the followingtests.
Proposition 4.1.5 (Tests I for Condition 1’). Let D(k) be the leading principal submatrixof order k of D. HD(w) fulfils Condition 1’ if and only if (−1)kdet
(D(k)
)< 0, ∀ k = 2, ..., N .
Proof (of Proposition 4.1.5). The result follows from Propositions 1.2.1 and 1.2.3.
Proposition 4.1.6 (Test II for Condition 1’). Let ξ+(D) be the number of positive eigen-values of D. HD(w) fulfils Condition 1’ if and only if ξ+(D) = 1.
Proof (of Proposition 4.1.6). The result follows from Proposition 1.2.1 and 1.2.2.
In practice, test II (Proposition 4.1.6) is easy to perform.
Condition 2 is also necessary and ensures the consistency of PRQE with the diversificationgoal, which is risk or uncertainty reduction. To see this, suppose that the dissimilarity matrixD is defined such that
dij = |wi − wj |, (4.6)
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where |.| is the absolute value operator. D is obviously conditionally negative definite. Then,HD(w) is concave. However, this dissimilarity is not related to assets or portfolio uncertaintyor risk.
There is no unique strict rule to follow concerning the choice of D to fulfil Condition 2 as inthe case of Condition 1 or Condition 1’. In what follows, a rule based on correlation aversionnotion is provided. The thesis suggests to define D such that
dij = E(d(Ri, Rj)
), (4.7)
where d(., .) is a dissimilarity function exhibiting an aversion of correlation. It follows, fromDefinition 2.1.6 and 2.1.8 and Proposition 2.1.2 and 2.1.3, that dij can be interpreted as adependence measure that implies diversification. As a result, HD(w) is a weighted averagedependence of portfolio w and therefore can be related to the correlation diversification strat-egy. The higher HD(w), D ∈ D is, the higher portfolio w diversification is, where D is the setof dissimilarity matrices that fulfil Conditions 1 and 2 or Conditions 1’ and 2. The intuitionis as follows: the more dissimilar are assets, the higher is the expected dissimilar of the assets(HD(w)) or the lower is the expected dependence of the assets, the less is the probabilitythat they do poorly at the same time in the same proportion and the more diversified is theportfolio w. Moreover, HD(w), D ∈ D can be interpreted as a (multivariate) expected utility
HD(w) = Eu(R) (4.8)
where u(R) =∑N
i,j=1wiwj d(Ri, Rj).
Therefore, RQE is arguably a valid class of portfolio diversification measures when it is spec-ified judiciously calibrated. Given two universes of assets A = AiNi=1 and B = BiNi=1, aportfolio wA is more diversified than a portfolio wB if HDA (wA) ≥ HDB (wB), where DA
and DB is based on the same dissimilarity formula. PRQE well-diversified portfolios (shortlyRQEP) is then defined as follows.
Definition 4.1.2 (PRQE well-diversified portfolio). Portfolio w is RQEP if and only ifit is a solution of the following constrained optimization problem:
maxw∈W
HD(w). (P(D))
The problem P(D) always has a solution, since W is compact and HD(w) is continuous. Thisensures the existence of RQEP. The main properties of RQEP are studied in the next chapter.
In practice, the interpretation of HD(w) is simplified by considering its standardized versionHD(w) = HD(w)/HD
(wRQE
), where wRQE denotes the weight vector of RQEP. However,
this normalization is unuseful when one compares two PRQE.
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4.2 Examples of Choice of D
This section presents some useful examples of D ∈ D. It is equivalent to the presentationsome examples of HD(w).
4.2.1 Uncertainty Diversification
Recall that Ri is the future return of asset i. Ri is a random variable. Let ri be investorprediction of Ri. In the mean-variance model ri = µi. The uncertainty of asset i is thendefined as Ri − ri. Define
d(Ri, Rj) =∣∣∣(Ri − ri)− (Rj − rj)
∣∣∣q, (4.9)
where q ≥ 0. It is well-known that d(Ri, Rj) is submodular when q ≥ 1 (see Simchi-Levi et al.,2005). Then it exhibits an aversion of correlation, consequently implies correlation diversifi-cation. d(Ri, Rj) can be interpreted as potential diversifiable uncertainty when asset i andj are combined. Using d(Ri, Rj), several dissimilarity matrices consistent with Conditions 1and 2 or Conditions 1’ and 2 can be proposed.
Consider a dissimilarity matrix D such that
dij =(
Ed(Ri, Rj)) lq. (4.10)
D can be rewritten as a function of Lq norm
dij =∥∥∥(Ri − ri)− (Rj − rj)
∥∥∥lq. (4.11)
When q = 1 or q = 2,∥∥∥(Ri − ri) − (Rj − rj)
∥∥∥qis an Euclidean distance. Then, from
Propositions 1.1.4 and 1.1.7, HD(w) is concave for 0 < l ≤ 2. It follows that D verifiedCondition 2 if q = 1 or q = 2 and 0 < l ≤ 2. However, when q ≥ 3 or q < 1, one needsto check after computing D if Condition 1 or Condition 1’ is verified. In the case whereq = l = 2, D captures the linear dependence between assets. Otherwise, D captures thenon-linear dependence between assets. Therefore, q and l can be interpreted as investors’preference for asset returns moments.
The following proposition shows that in the particular case where q = l = 2 and ri = µi, PRQEis related to portfolio variance, diversification return or excess growth rate and Goetzmannet al.’s measure.
Proposition 4.2.1. Let D defined as (4.10). If q = l = 2 and ri = µi, then RQE is relatedto:
1) Portfolio variance: σ2(w) = w>σ2 −HD(w);
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2) Diversification Return or Excess Growth Rate: Dr(w) = HD(w);3) Goetzmann et al.’s measure: GLR(w) = w>σ2
w>σ2−HD(w).
Proof (of Proposition 4.2.1). Consider D such that (4.10). Assume that q = l = 2 andri = µi. Then
dij = σ2i + σ2
j − 2σij . (4.12)
dij can therefore be related to the relatively negative expectation dependent (RNED) conceptintroduced by Wright (1987). To see this, Wright (1987, Corollary, pp. 116) shows that, when(Ri, Rj) is a joint normal variables, Ri is (strictly) RNED on Rj , which it is denoted (strict)RNED(Ri|Rj), if
Cov(Ri −Rj , Rj) ≤ (<)0. (4.13)
RNED(Ri|Rj) is not symmetric. One can define the symmetric version of RNED(Ri|Rj) asfollows: when (Ri, Rj) is a joint normal variables, Ri is (strictly) relatively negative expectationdependent on Rj , which it is denoted (strict) RNED(Ri, Rj), if
Cov(Ri −Rj , Rj) + Cov(Rj −Ri, Ri) ≤ (<)0. (4.14)
It is clearly observable that
RNED(Ri, Rj) =⇒ dij ≥ (>)0 (4.15)
This implies that dij can be considered as a generalization of the concept of RNED when(Ri, Rj) is a joint normal variables.
RQE based on D is
HD(w) =1
2
N∑i,j=1
(σ2i + σ2
j − 2σij)wiwj (4.16)
=1
2
N∑i=1
σ2iwi +
N∑j=1
σ2jwj − 2σ2(w)
(4.17)
=w>σ2 − σ2(w). (4.18)
The result follows.
Corollary 4.2.1 (RQE and MV model). The diversification in the MV model is RQE di-versification
Proof (of Corollary 4.2.1). Since the diversification return is a specific measure of port-folio diversification in the mean-variance (MV) model (see Section 3.2), Proposition 4.2.1 im-plies that the diversification in the MV model is RQE diversification. The relation betweenRQE and the MV model will deeply investigated in Chapter 7.
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One should stress that as defined in (4.9), d(Ri, Rj) is not scaling free. This may be prob-lematic in applications involving different time periods. One can however resolve this issue byworking with a normalized d(Ri, Rj). A natural way to do this is to use asset volatility
d(Ri, Rj) =
∣∣∣∣Ri − riσi− Rj − rj
σj
∣∣∣∣q. (4.19)
Other normalizations are possible. Interested readers are referred to Gower and Legendre(1986) for more details. When q = 2 and ri = µi, D is defined such that
dij = (2(1− ρij))l2 . (4.20)
D as defined in (4.20) is widely used in asset allocation based on clustering analysis (seeOnnela et al., 2003; Bonanno et al., 2004; Tola et al., 2008; Wang et al., 2016; Bonanno et al.,2000) when l = 1. D2 is at the core of the most diversified portfolio (see Section 2.2.2.2). Itdetermines the composition and diversification degree of the most diversified portfolio. To seethis,
HD2(w) =N∑
i,j=1
(1− ρij)wiwj (4.21)
It is straightforward to verify that HD2(w) in (4.21) is equivalent to the problem (2.75). Therelationship between RQE and the maximum diversification strategy with the most diversifiedportfolio as well-diversified portfolio will be deeply examined in Chapter 6.
D as defined in (4.20) can be extended to other dependence measures such as Kendall’s tau,Spearman’s rho, tail dependence, conditional correlation, etc (see Nelsen, 2006). This can beuseful when the standard correlation measure becomes inapplicable, which is the case whenasset returns do not have an elliptical distribution (see Embrechts et al., 1999). Considerρ = (ρij)
Ni,j=1 a dependent matrix other than the standard correlation. In the spirit of (4.20),
the dissimilarity matrix based on ρ can be defined such that
dij =(
2(1− ρij)) l
2. (4.22)
If ρ is semi-definite positive similarity matrix with 0 ≤ ρij ≤ 1 and ρii = 1, then the dissimilar-ity matrix D =
√11> − ρ is Euclidean (see Proposition 1.1.11). Thus, D = (11>− ρ)l for all
0 < l ≤ 2 is conditionally definite negative and consequently HD(w) is concave. D as definedin (4.22) is also used in asset allocation based on clustering analysis. For example, Wang et al.(2017) use it with ρ the matrix of tail dependence. See also De Luca and Zuccolotto (2014b,2015, 2014a); Durante et al. (2015) for other specifications of ρ.
In practice, dij must be estimated from a sample of observed data points. This can be doneas follows
dij =
(T∑i=1
1
T|(rit − ri)− (rjt − rj)|q
) lq
, (4.23)
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where rit, t = 1, ..., T is the historical observed values of the random variable Ri. Thissuggests that dij can be generalized as follows
dij =
(T∑i=1
ωt |(rit − ri)− (rjt − rj)|q) l
q
, (4.24)
where ωt is the weight of the period t. dij can also be modified to take into account asset returnscomplexity or behaviour. Interest readers are referred to Batista et al. (2014); Chouakria andNagabhushan (2007) for more details.
4.2.2 Factor Risks Diversification
Suppose that asset returns are generated by the linear factor model as in exact arbitrage pricingtheory (see Ross, 1976), or as in approximate arbitrage pricing theory (see Chamberlain, 1983b)
Ri = µi +
NK∑k=1
βkiFk + ei, ∀ i, j = 1, ..., N (4.25)
where Fk is the kth mean zero common factor, NK ≤ N is the number of factors, βki isthe factor loading for asset i on the kth factor and ei is the asset specific factor. For theassumptions on Fk and ei, readers are referred to Ross (1976) and Chamberlain (1983b).
The matrix D can be specified only using the NK first risk factors such that
dij =
(NK∑k=1
|βki − βkj |q) l
q
. (4.26)
HD(w) in that case measures diversification across factors (Fk) and β. The factor diversi-fication is measured by the number of factors considered NK and the beta diversification ismeasured by the difference or dissimilarity or distance between betas across factors |βki − βkj |q.Therefore, HD(w) can be used to perform a risk factor diversification and consequently toconstruct a well-diversified portfolio free from risk if there is no essential factor risk. We referreaders to Ingersoll (1984); Khan and Sun (2003) for details concerning the construction of awell-diversified portfolio free from risk.
When factors are extracted using the principal component analysis one can focus only onidiosyncratic risk diversification defining D taking into account only the last N −NK
dij =
N∑k=NK+1
|βki − βkj |q
lq
. (4.27)
By definition the matrices D (Equations (4.26) and (4.27)) verify Condition 2, but there is noevidence ex-ante that they verify Condition 1 or Condition 1’. However, in practice, this canbe checked easily after computing D.
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4.2.3 Risk Diversification
Consider the risk measure %. If % is subadditivity, then the dissimilarity matrix D can bedefined such that
dij = %(Ri) + %(Rj)− %(Ri +Rj). (4.28)
However, if % is convex, D can be defined such that
dij =
∫ 1
0[κ%(Ri) + (1− κ)%(Rj)− %(κRi + (1− κ)Rj)] d κ. (4.29)
One can easily verify that these matrices are dissimilarity matrices. By definition the matricesD (Equations (4.28) and (4.29)) verify Condition 2, but there is no evidence ex-ante thatthey verify Condition 1 or Condition 1’. However, in practice, this can be checked easily aftercomputing D.
4.2.4 Targeted Diversification
The dissimilarity matrix can also be designed with the purpose of diversifying on the subsetof the universe of assets. Consider any dissimilarity matrix D. Its corresponding targeteddissimilarity matrix, Dt = (dtij)
Ni,j=1, can be defined such that
dtij = dijψ(ci)ψ(cj), (4.30)
where ψ(.) is a strictly positive function and ci is asset i characteristic used to target thedesirable subset.
The following proposition shows that for some specifications of D, ψ(.) and ci, PRQE based onDt can be related to diversification ratio of Choueifaty and Coignard (2008) and Goetzmannet al.’s measure.
Proposition 4.2.2. 1) Diversification ratio: if ci = σiσ(w) , D is defined as (4.20) with
l = 2 and ψ(.) is identity function, then DR(w) =
√HDt (w)
2 + 1.
2) Goetzmann et al.’s measure: if ci = σiσ(w) , D is defined such that dij = σi
σj+
σjσi− ρij
and ψ(.) is identity function, then GLR(w) = HDt(w) + 1.
Proof (of Proposition 4.2.2). 1) Assume that ci = σiσ(w) , D is defined as (4.20) with
l = 2 and ψ(.) is identity function. The quantity σiσ(w) can be interpreted as the risk reduction
ratio of asset i. Thendtij = 2(1− ρij)
σi σjσ2(w)
. (4.31)
The subset of assets targeted is therefore the subset of high risk reduction ratios assets. It isstraightforward to show that PRQE based on Dt is
HDt(w)
2=
(w>σ
σ(w)
)2
− 1. (4.32)
The result follows.
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2) Now, assume that ci = σiσ(w) , ψ(.) is identity function and D is defined such that dij =
σiσj
+σjσi− ρij . Then
dtij =
(σiσj
+σjσi− 2ρij
)σi σjσ2(w)
. (4.33)
It is straightforward to show that PRQE based on Dt is
HDt(w) = GLR(w)− 1. (4.34)
Other interesting asset characteristics can also be used to select the subset of assets to betargeted such as: liquidity, book value, cash flow, sales, revenues, dividends, employment, etc.Moreover, several characteristics can be combined to select the subset of assets to be targeted
dtij = dijψ(ci)ψ(cj), (4.35)
where ci is a vector of asset characteristics.
If D verifies Condition 2, then so is Dt. If D verifies Condition 1 or Condition 1’, then Dt
verifies at least Condition 1’.
4.2.5 Diversification When no Information is Available
Now, suppose that no information is available. In that case, there is no reason to believe that,given any two pairs of assets (i, j) and (k, l), the dissimilarity between i and j is greater orlesser than the dissimilarity between k and l. In such a situation, the dissimilarity matrix D
can be specified such thatD = 11> − I. (4.36)
The following proposition shows that PRQE based on D is equivalent to the Gini-Simpsonand Herfindhal indices.
Proposition 4.2.3 (Gini-Simpson’s Index). If D = 11> − I, then
HD(w) =1
2(1−HI(w)) (4.37)
with HI(w) =∑N
i=1w2i .
Proof (of Proposition 4.2.3). Straightforward to verify.
As a result, portfolio RQE also cover the law of large numbers diversification strategy.
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4.2.6 Diversification When Partial Information is Available
In general, the dissimilarities dij , i, j = 1, ..., N are set at their estimated values using thehistorical data, ignoring estimation risks. However, these risks are known to have a hugeimpact in portfolio selection analysis. More precisely, they are the source of undesirableextreme weights and the poor out-of-sample performance of portfolios (Chopra and Ziemba,1993).
To take into account estimation risks, one can define the dissimilarity matrix Dα such that
Dα = (1− α)D + α (11> − I), (4.38)
where the parameter α ∈ [0, 1] takes into account estimation errors. When estimation errorsare equal to zero, α = 0 and Dα reduces to D, which is optimal. When estimation errors arevery high, however α = 1 and Dα reduces to 11> − I, which is optimal. This occurs because,under very high estimation errors, information becomes totally unuseful.
In such case dα,ij can be interpreted as the expected dissimilarity, and the parameter α repre-sents the probability that the dissimilarity between assets i and j is equal to dij , while 1− αrepresents the probability that the dissimilarity between assets i and j is equal to 1 − δij .It can also be interpreted as the shrinking dissimilarity, whereby D is the guess, 11> − I isthe shrinkage target, and α is the shrinkage constant. The parameter α can be calibratedfollowing Ledoit and Wolf (2003). When D is non-Euclidean dissimilarity matrix, α can bechosen such as one has a double dividend (see Proposition 1.1.10): 1) reduction of estimationerrors; 2) Dα becomes Euclidean.
The dissimilarity matrix Dα can also be used when a portfolio manager wants to handle futureuncertainty. Indeed, even if D is estimated without errors, since it is based on historical data,it only represents past performance. However, past performance is no guarantee of futuresuccess. In that case, α = 0 means that past performance is a guarantee of future success, sothere is no uncertainty going forward and the optimal dissimilarity matrix is the estimated D
matrix. However, α = 1 means that past performance is no guarantee of future success, whichmeans there is a total uncertainty about the future. The optimal choice of the dissimilaritymatrix in the latter case is 11> − I.
Another way to take into account estimation errors is to use Schoenberg transformations.Let φ(.) be a Schoenberg transformation as defined in Section 1.1.4. φ
(D)is concave and
consequently ∣∣∣φ(dij)− φ(dkl) ∣∣∣ ≤ ∣∣dij − dkl∣∣. (4.39)
Suppose thatdij = dij + εij , (4.40)
89
where εij is estimation errors. Equation (4.39) implies∣∣∣φ(dij)− φ(dkl) ∣∣∣ ≤ ∣∣dij − dkl∣∣+∣∣εij − εkl∣∣. (4.41)
This shows that estimation errors can be reduced replacing D by φ(D)
=(φ(dij
))Ni,j=1
.
Propositions 4.2.1 to 4.2.3 show clearly that portfolio RQE is a unified theory that includesmany previous contributions. This result strengthens our view that RQE is a valid class ofportfolio diversification measures when it is judiciously calibrated. Moreover, Corollary 4.2.1shows that the diversification in the MV model is RQE diversification. Finally, Proposi-tion 4.2.3 shows that portfolio RQE can also be used to capture the benefit of the law of largenumbers diversification strategy.
4.3 Properties of Portfolio RQE
This section presents some properties of PRQE. First, the coherence of PRQE is studied.Next, other properties of PRQE are presented.
4.3.1 Coherence
This section studies the coherence of portfolio RQE. It provides conditions on D under whichPRQED is a coherent portfolio diversification measure.
Proposition 4.3.1 (Portfolio RQE Coherence). PRQED, D ∈ D is coherent if and onlyif D is homogeneous and translate-invariant.
Proof (of Proposition 4.3.1). Axiom 1 or Axiom 1’) Since D must satisfy Condition 1or Condition 1’, HD (w|R) verifies Axiom 1 or Axiom 1’.
Axiom 2) First, since wi ≥ 0, ∀ i = 1, ..., N and dij ≥ 0, ∀ i, j = 1, ..., N , HD (w|R) ≥ 0.Then Φ = 0. Second, HD (δi|R) = dii = 0. Therefore,
HD (w|R) ≥ HD (δi|R) = 0, ∀ i = 1, ..., N. (4.42)
Axiom 3) Since Ai = Aj , ∀ i, j = 1, ..., N , dij = 0, ∀ i, j = 1, ..., N . Then
HD (w|R) = 0. (4.43)
Axiom 4) Since w ≥ 0, w 6= δi, i = 1, ..., N and dij ≥ 0, ∀ i, j = 1, ..., N ,
HD (w|R) = 0⇐⇒ dij = 0, ∀ i, j = 1, ..., N. (4.44)
Then R is comonotonic.
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Axiom 5) HD (w|RA+) can be decomposed as follows
HD (w|RA+) =
N∑i,j 6=k
d(Ai, Aj)wAiwAj +
N+1∑i=1
d(Ak, Ai)wAkwAi +
N+1∑i=1
d(A+N+1, Ai)wA+
N+1wAi .
(4.45)Since A+
N+1 = Ak, d(Ak, Ai) = d(A+N+1, Ai), ∀ i = 1, ..., N + 1. Then, (4.45) can be rewritten
as follows
HD (w|RA+) =N∑
i,j 6=kd(Ai, Aj)wAiwAj +
N+1∑i=1
d(Ak, Ai)wAi(wAk + wA+N+1
). (4.46)
Let
w1 =(wRQEA1
, ..., wRQEAk−1, w1,Ak , w
RQEAk+1
, ..., wRQEAN, w1,A+
N+1
)>(4.47)
w2 =
(wRQEA+
1
, ..., wRQEA+k−1
, wRQEA+k
+ wRQEA+N+1
, wRQEA+k+1
, ..., wRQEA+N
)>(4.48)
such that w1,Ak + w1,AN+1= wRQEAk
. It is straightforward to show that
HD
(wRQEA
∣∣∣RA) = HD (w1|RA+) = HD (w2|RA) = HD
(wRQEA+
∣∣∣RA+
). (4.49)
Equation (4.49) shows that PRQE and RQEP are both duplicate invariant.
Axiom 6) Consider a portfolio w =(wRQEA1
, ..., wRQEAN, 0)>
. Portfolio w is an element of
WN+1A+
, so
HD
(wRQEA+|RA+
)≥ HD
(w|RA+
). (4.50)
Since HD
(w|RA+
)= HD
(wRQEA |RA
),
HD
(wRQEA+|RA+
)≥ HD
(wRQEA |RA
). (4.51)
Axiom 7) PRQE is translate-invariant if and only if D is translate-invariant. It is straightfor-ward to show that the dissimilarity matrices discussed in Sections 4.2.1 to 4.2.3 are translate-invariant.
Axiom 8) PRQE is positive homogeneous if and only if D is positive homogeneous. It isstraightforward to show that the dissimilarity matrices discussed in Sections 4.2.1 to 4.2.3 arepositive homogeneity.
Axiom 9) If R is exchangeable, then the dissimilarities between assets are constant i.e. dij =
d, ∀ i, j = 1, ..., N . Thus, HD (w|R) is proportional to the Gini-Simpson or Herfindhal index
HD (w|R) =d
2(1−HI(w)) . (4.52)
As a result, wRQE = 1N .
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4.3.2 Other Properties of PRQE
In this section two additional properties are presented. The first property is that PRQE isdecomposable across asset classes.
Proposition 4.3.2 (Decomposition). Assume that HD (w) is concave. Then it can be de-composed as follows
HD(w) =
NK∑k=1
nkHDk(wk) +
NK∑k,l=1
nk nlDH(wk,wl), (4.53)
where NK is the number of asset classes, nk is the share of capital invested on the asset classk, Dk is the dissimilarity matrix of assets of class k, wk,i is the share of nk invested on theasset i of the asset class k (wi = nk wk,i) and DH(., .) is a dissimilarity defined on W or across-entropy induced by RQE such that
DH (wk,wl) = HDk×l (w1,w2)− 1
2HDk
(wk)−1
2HDl
(wl) . (4.54)
Proof (of Proposition 4.3.2). See Proposition 1.3.4.
The first term of (4.53) represents the intra-class diversification and the second term theinter-class diversification. PRQE is therefore useful tool to study diversification within andbetween asset classes and at the country, industry, assets class level simultaneously. Notethat in (4.53), the first stage of decomposition is at the single-asset level and the second stageis at the level of assets class. PRQE can also be decomposed at N levels (see Rao, 2004).Therefore, in applications, PRQE can be a useful tool to simultaneously study diversificationat the country, industry, assets class level, etc.
The second property is that PRQE is an increasing function of the portfolio total dissimilarityDT (D) =
∑Ni,j=1 dij .
Proposition 4.3.3 (Monotonicity in dissimilarity). Let A = AiNi=1 and B = BiNi=1.Then for all w ∈W such that wi > 0, i = 1, ..., N
DT (DA) ≤ DT (DB)⇔ HD (w|RA) ≤ HD (w|RB) . (4.55)
Proof (of Proposition 4.3.3). The result follows from Proposition 1.3.6 taking the partialderivative of HD(w) with respect to DT (D).
Proposition 4.3.3 implies that more assets of the universe are dissimilar or less dependent,more the universe is diversified. This property will help in the selection of the asset universe.
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4.4 Summary
This chapter has adapted and extended the use of Rao’s Quadratic Entropy (RQE) to portfoliotheory as a new class portfolio diversification measures. It has demonstrated that when RQEis judiciously calibrated it becomes a valid class of portfolio diversification measures, calledportfolio RQE (PRQE). The advantage of PRQE is that it summarizes complex features ofportfolio diversification in a simple manner and provides at the same time a unified theorythat includes many previous contributions. The list of its features presented in this chapter isas follows:
1. It is easy to interpret. It can be interpreted as a dependence measure or as a multivariateexpected utility;
2. It has low computational cost i.e. computational efficient;3. It is easy to implement;4. It meets ex-ante desirable properties of a portfolio diversification measure when asset
dissimilarity matrix is homogeneous, translate-invariant and conditionally negative def-inite;
5. It covers both the law of large numbers and the correlation diversification strategies;6. It captures the diversification benefit of the risk-free asset;7. It is extremely flexible. This flexibility allows it to easily:
a) diversify according any characteristic of assets;b) take into account asset linear and non-linear dependences separately or jointly;c) take into account estimation errors;d) perform targeted diversification;e) perform factor risks diversification;
8. It embeds the portfolio diversification measures such as Gini-Simpson’s index and di-versification return of Booth and Fama (1992) or excess growth rate of Fernholz (2010)offering therefore a novel and useful interpretation of these measures;
9. It governs the diversification in the measures such as diversification ratio of Choueifatyand Coignard (2008), Goetzmann and Kumar (2008)’s measure and Frahm and Wiechers(2013)’s measure offering therefore a novel and useful interpretation of these measures;
10. It governs the diversification in the mean-variance models.
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Chapter 5
Properties of Rao’s Quadratic EntropyPortfolios
In the previous chapter, a novel, coherent, flexible and computational efficient class of port-folio diversification measures named Rao’s Quadratic Entropy (RQE) of portfolio or portfolioRQE (PRQE) was developed inspiring from Rao’s Quadratic Entropy, a general approach tomeasuring diversity developed by Rao (1982a). In this chapter, some theoretical and empiricalproperties of its well-diversified portfolios named RQE portfolios (RQEP) are analyzed.
The chapter is organized as follows. Section 5.1 examines the theoretical properties. First,the existence and the uniqueness of RQEP and conditions under which RQEP is an interiorsolution are discussed. Second, the monotonicity and the duplication invariance properties ofRQEP are presented. Third, the conditions under which RQEP and the equally weighted port-folios coincide are established. Four, the mean-variance optimality of RQEP is investigated.Finally, the sensibility of RQEP relative to the dissimilarity matrix is analyzed. Section 5.2explores the empirical properties. Twenty RQEP are examined out-of-sample, on two differentempirical datasets, against four standard diversified portfolios which are: the equally weightedportfolio, the equally risk contribution portfolio, the most diversified portfolios and the marketportfolio.
5.1 Theoretical Properties
This section examines some theoretical properties of RQEP.
5.1.1 Existence and Uniqueness of RQEP
At first, the result of the existence of RQEP is presented.
Proposition 5.1.1 (Existence of RQEP). RQEP exists since W is a closed and boundedset and HD(w) is continuous on W.
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Proof (of Proposition 5.1.1). Straightforward to verify.
Proposition 5.1.2 (Uniqueness of RQEP). RQEP is unique if and only if HD(w) is strictlyconcave or quasi-concave on W.
Proof (of Proposition 5.1.2). Straightforward to verify.
From Proposition 1.3.3, HD(w) is strictly concave if and only if rank(GD
(1N
))= N−1. The
following result is obtained.
Corollary 5.1.1. if rank(GD
(1N
))= N − 1, then RQEP is unique.
Note from Proposition 1.3.7 that when√
D =(√
dij)Ni,j=1
is ultrametric rank(GD
(1N
))=
N − 1. Then RQEP is unique.
5.1.2 Interior Solution
Now, let us introduce the conditions of interior solution. These conditions are established onlyfor the case where D is CDN.
Proposition 5.1.3 (Interior solution). Suppose that D is CND. The problem P(D) hasan interior solution if D is circum-Euclidean and its associated circumsphere is the smallestenclosing hypersphere containing the generating points. The solution is
wRQE =D−1
1>D−1, (5.1)
where D− is a generalized inverse of D. When D−1 exist then RQEP is unique and
wRQE =D−11
1>D−11, (5.2)
Proof (of Proposition 5.1.3). Suppose that D is circum-Euclidean and its associatedcircumsphere is the smallest enclosing hypersphere containing the generating points. It followsthat the points that generated D lie on the surface of a hypersphere. The weight of the centerof hypersphere is w = D−1
1>D−1and is positive i.e. wi > 0,∀ i = 1, ..., N , and its radius is
1>D−1. It is straightforward to show that the weight w = D−11>D−1
is a solution of the problem
maxw∈W
HD(w), (5.3)
which is interior solution version of the problem P(D).
When the interior solution is impossible, the RQEP can be obtained numerically using thestandard algorithms for (quasi) convex quadratic programming.
When the problem P(D) has multiple solutions (which is the case when HD(w) is not strictly(quasi) concave), one can define additional criteria on their portfolio in order to select a bestRQEP.
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5.1.3 Monotonicity and Duplication Invariance
It is interesting to note that, in general, the weights of RQEP are an increasing function ofasset dissimilarity contribution.
Proposition 5.1.4. Let two assets i and j held in RQEP i.e. wRQEi , wRQEj > 0, M be thenumber of assets held and Di
(wRQE
)=∑M
j=1 dij be the dissimilarity contribution of an asseti held in the RQEP. Then
Di
(wRQE
)≤ Dj
(wRQE
)⇐⇒ wRQEi ≤ wRQEj (5.4)
with equality if and only if Di
(wRQE
)= Dj
(wRQE
).
Proof (of Proposition 5.1.4). Consider the first order conditions of problem P(D)
N∑j=1
dijwRQEj = η − νi, ∀ i = 1, ..., N, (5.5)
1>wRQE = 1, (5.6)
min(wRQEi , νi) = 0 ∀ i = 1, ..., N (5.7)
where η is the Lagrange multiplier of the budget constraint w>1 = 1 and νi, i = 1, ..., N
are the Lagrange multipliers of positive constraints. Since only assets held by RQEP areconsidered, (5.5) becomes
M∑j=1
dijwRQEj = η, ∀ i = 1, ...,M. (5.8)
Summing (5.8) over i gives
M∑j=1
wRQEj Dj
(wRQE
)= ηM, ∀ i = 1, ...,M. (5.9)
Equation (5.9) completes the proof of both the first and the second points, since η = 2HD
(wRQE
).
Proposition 5.1.4 implies that the RQEP spreads capital or money by putting more capitalin more dissimilar assets and fewer capital in less dissimilar assets. As a result, holding theRQEP can help reduce losses during bear markets or financial crises, since the more dissimilarare the assets, the less is the probability that they do poorly at the same time in the sameproportion.
It is also straightforward to show that RQEP is duplicate invariant.
Proposition 5.1.5 (Duplication invariance). Let A = AiNi=1 a universe of N assets andA+ = A+
i N+1i=1 a new universe of N + 1 assets such that asset A+
i = Ai, ∀ i = 1, ..., N and
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A+N+1 is a duplication of an asset Ak, k ∈ 1, ..., N. Let wRQE
A and wRQEA+ denote the weights
of RQEP associated to A and A+, respectively. Then
wRQEAi= wRQE
A+i
, i 6= k and wRQEAk= wRQE
A+k
+ wRQEA+N+1
. (5.10)
Proof (of Proposition 5.1.5). See the proof of Proposition 4.3.1 Axiom 6.
The duplicate invariance property also protects RQEP from mass destruction during bearmarket or financial crisis. It implies that RQEP is unbiased towards a multirepresentativeasset. Consequently, RQEP is less concentrate in terms of information, so less exposed to therisk systematic.
5.1.4 Equally Weighted Portfolio
This section establishes the conditions under which RQEP coincides with the equally weightedportfolio. The first condition is when N = 2.
Proposition 5.1.6. When N = 2, wRQE =(
12 ,
12
).
Proof (of Proposition 5.1.6). Suppose that N = 2. In that case,
HD(w) = d12w1(1− w1) (5.11)
PRQE optimization problem becomes
maxw1∈[0,1]
d12w1(1− w1). (5.12)
The first order conditions of (5.12) are
d12
(1− 2wRQE1
)= 0 (5.13)
0 ≤ wRQE1 ≤ 1. (5.14)
It follows that the unique RQEP is
wRQE =
(1
2,1
2
)(5.15)
The second condition is when asset dissimilarity contribution, Di =∑N
j=1 dij , ∀ i = 1, ..., N ,are equal.
Proposition 5.1.7. Let Di =∑N
j=1 dij be asset i dissimilarity contribution. Assume thatDi = Dj , ∀ i, j = 1, ..., N . Then wRQE = 1
N , where 1 is the vector column of ones.
Proof (of Proposition 5.1.7). It is straightforward to verify that portfolio 1N verifies the
first order conditions of problem P(D) (??)-(??) if and only if Di = Dj , ∀ i, j = 1, ..., N .
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5.1.5 Mean-Variance Optimality
This section investigates the mean-variance (MV) optimality of RQEP.
Proposition 5.1.8 (MV optimality of RQEP). Suppose that D is defined such that
dij = σ2i + σ2
j − 2σij . (5.16)
Then
1) when the risk-free asset is not available, RQEP is mean-variance optimal if and onlyif assets have the same variance i.e. σi = σj , ∀ i, j = 1, ..., N or assets have samemean-variance utility i.e. UMV (δi) = UMV (δj), ∀ i, j = 1, ..., N .
2) when the risk-free asset is available, RQEP is mean-variance optimal if and only if assetshave the same mean-variance utility i.e. UMV (δi) = UMV (δj), ∀ i, j = 1, ..., N − 1.
Proof (of Proposition 5.1.8). Suppose that D is defined by (5.16). Then from Sec-tion 4.2.1
HD(w) = w>σ2 − σ2(w). (5.17)
1) First, assume that the risk-free asset is not available:a) Suppose that σi = σj = σ, ∀ i, j = 1, ..., N . Then
HD(w) = σ2 − σ2(w). (5.18)
It follows that HD(w) is equivalent to σ2(w). Therefore, RQEP is mean-varianceoptimal.
b) Suppose that UMV (δi) = UMV (δj) = U, ∀ i, j = 1, ..., N . Then there is a risktolerance coefficient τ such that
µi = U +1
τσ2i , ∀ i = 1, ..., N. (5.19)
Consider the MV utility function with risk tolerance coefficient equal to τ
UMV (w) =
N∑i=1
wi
(µi −
1
τσ2i
)+
1
τ
(w>σ2 − σ2(w)
). (5.20)
Substituting (5.19) in (5.20) gives
UMV (w) = U +1
τHD(w). (5.21)
It follows that HD(w) is equivalent to UMV (w) with risk tolerance coefficient equalto τ . Therefore RQEP is mean-variance optimal.
2) Now, assume that risk-free asset is available, with rate Rf . Then, HD(w) can be rewrit-ten as follows
HD(w, wf ) = wf (1− wf )
N∑i=1
diN wi + (1− wf )2N∑
i,j=1
wiwj dij (5.22)
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The first term represents diversification contribution of the risk free asset and the sec-ond term represents diversification contribution of risky assets. Replacing D in (5.22),HD(w, wf ) becomes
HD(w, wf ) = wf (1− wf )
N∑i=1
σ2i wi + (1− wf )2
(N∑i=1
σ2i wi − σ2(w)
)(5.23)
Rearranging (5.23), one has
HD(w, wf ) = (1− wf )
N∑i=1
σ2i wi − (1− wf )2 σ2(w) (5.24)
Now suppose that there is τ > 0 such that µi −Rf =σ2iτ , i = 1, ..., N − 1. Then
HD(w, wf ) = UMV (w, wf )−Rf (5.25)
with τ risk aversion coefficient of UMV (w, wf ). It follows that RQEP is mean-varianceoptimal.
5.1.6 Sensitivity Analysis
This section studies the sensitivity of RQEP relative to the dissimilarity matrix D. Thecase where HD(w) is strictly concave and N ≥ 3 is considered, because when N = 2 byProposition 5.1.6 RQEP is insensitive to D. To do so, given the dissimilarity matrices Dk, k =
1, 2 and their corresponding RQEP weights wRQE1 and wRQE2 , following Guigues (2011),a upper-bound of
∥∥∥wRQE1 − wRQE2
∥∥∥1and
∥∥∥wRQE1 − wRQE2
∥∥∥2are provided in terms of
‖D1 − D2‖∞ or ‖D1 − D2‖2 respectively. Recall that ‖w‖q =(∑N
i=1 |wi|q)1/q
, ‖D‖∞ =
max1≤j≤N
∑Ni=1 |dij | and ‖D‖2 =
(∑Ni,j=1 |dij |2
)1/2.
Let us first introduce the dual characterization of RQE of portfolio.
Proposition 5.1.9 (Dual characterization of PRQE). Let D be a conditionally strictlynegative definite (CSND) dissimilarity matrix and S(θ) = (sij(θ))
Ni,j=1 be the matrix defined
from D as followsS(θ) = θ 11> −D. (5.26)
Then
1) There is θ∗ such as for all θ ≥ θ∗, S(θ) is positive definite;2) S(θ) is a similarity matrix;3) HS(θ)(w) = θ
2 −HD(w), where HS(θ)(w) = 12w>S(θ) w;
4) ξmin(S(θ)) ≤∣∣ξ2(D)
∣∣, where ξmin (S(θ)) is the minimum eigenvalue of S(θ) and ξ2(D)
is the second largest eigenvalue of D;
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5) ξmin(S(θ)) is an increasing function of θ.
Proof (of Proposition 5.1.9). Let D be a CSND dissimilarity matrix and S(θ) = (sij(θ))Ni,j=1
be the matrix such that S(θ) = θ 11> −D. Then
sij(θ) = θ − dij . (5.27)
1) From Proposition 1.1.1;
2) Straightforward to verify.
3) By definition
HS(θ)(w) =1
2
N∑i,j=1
(θ − dij)wiwj . (5.28)
Developing (5.28), one has
HS(θ)(w) =θ
2
N∑i,j=1
wiwj −HD(w). (5.29)
Since w>1 = 1, θ2∑N
i,j=1wiwj = θ2 . The result follows.
4) Set A = −D and B = θ11>. Then ξ1(A) =∣∣ξN (D)
∣∣ ≥ ... ≥ ξN−1(A) =∣∣ξ2(D)
∣∣ ≥ξN (A) = ξ1(D), rank(B) = 1 and S = B + A. Since 11> is PSD, θ11> is PSD. FromProposition 1.1.6, it is straightforward to show that∣∣ξ2(D)
∣∣ = ξN−1(A) ≥ ξN (S(θ)) = ξmin(S(θ)). (5.30)
5) Since the function f(θ) = θ x>11> xx> x
− x>Dxx> x
(Raleigh ratio of S(θ)) is an increasing functionof θ (because x>11> x
x> x≥ 0), ξmin(S(θ)) = min
x6=0f(θ) is also an increasing function of θ.
Proposition 5.1.9 implies that the function ξmin(S(θ)) has horizontal asymptote at ∞ i.e.
limθ→∞
ξmin(S(θ)) = ξmin(S(∞)). (5.31)
Proposition 5.1.9 also implies that RQEP is also a solution of the following optimizationproblem
minw∈W
HS(θ)(w), (P(S(θ)))
which is the dual of the problem P(D). The following proposition provides a sensitivityanalysis of problem P(D).
Proposition 5.1.10 (RQEP sensitivity). Let Dk, k = 1, 2 be two CSND dissimilarity ma-trices. Consider problem P(S(θ)) with S(θ) = S1(θ) and S(θ) = S2(θ), where Sk(θ) =
100
θ 11> −Dk, k = 1, 2 with θ ≥ maxk=1,2
(θ∗k), k = 1, 2. Then
∣∣∣HD1
(wRQE1
)−HD2
(wRQE2
) ∣∣∣ ≤ ‖D1 −D2‖∞2
(5.32)
∥∥∥wRQE1 −wRQE2
∥∥∥1≤ 2
‖D1 −D2‖∞max(γ(S1(θ)), γ(S2(θ)))
(5.33)
∥∥∥wRQE1 −wRQE2
∥∥∥2≤ 2
maxi=1,...,N
‖Ci(D1 −D2)‖2
max(ξmin(S1(θ)), ξmin(S2(θ)))(5.34)
where ξmin(Sk(θ)) = ξN (Sk(θ)) the lowest eigenvalue of Sk(θ), k = 1, 2, γ(Sk), k = 1, 2 isdefined such as HS(θ)(w) is γ(Sk(θ))-strongly convex with respect to ‖.‖1, and Ci(D1−D2) isthe i-th column of D1 −D2.
Proof (of Proposition 5.1.10). The proof of Proposition 5.1.10 follows Guigues (2011,Theorem 3.1, pp. 561).
1) Proof of (5.32): Without a loss of generality suppose that HD2
(wRQE2
)≥ HD1
(wRQE1
)(the other case is symmetric). In this case,∣∣∣HD2
(wRQE2
)−HD1
(wRQE1
) ∣∣∣ = HD2
(wRQE2
)−HD1
(wRQE2
)+HD1
(wRQE2
)−HD1
(wRQE1
)(5.35)
Since HD1
(wRQE2
)−HD1
(wRQE1
)< 0,∣∣∣HD2
(wRQE2
)−HD1
(wRQE1
) ∣∣∣ ≤ (wRQE2)>
(D2 −D1) wRQE2
2(5.36)
≤∥∥wRQE2
∥∥2
1
∥∥D2 −D1
∥∥∞
2(5.37)
Since∥∥wRQE2
∥∥2
1≤ 1, the result follows.
2) Proof of (5.33): First note that HS(θ)(w) satisfies a second-order growth condition. Indeed,a second-order Taylor series expansion of HS(θ)(w) at wRQE gives
HS(θ)(w) =HS(θ)
(wRQE
)+(w −wRQE
)>∇HS(θ)
(wRQE
)(5.38)
+1
2
(w −wRQE
)>∇2HS(θ)
(wRQE
) (w −wRQE
), (5.39)
where ∇HS(θ)
(wRQE
)= S(θ) wRQE and ∇2HS(θ)
(wRQE
)= S(θ). It is well-known that the
first-order optimality conditions give(w −wRQE
)>∇HS(θ)
(wRQE
)≥ 0, ∀w ∈W (5.40)
From Definition 1.2.5,(w −wRQE
)>∇2HS(θ)
(wRQE
) (w −wRQE
)≥ γ(S(θ))
∥∥w −wRQE∥∥2
1. (5.41)
101
Hence, HS(θ)(w) satisfies a second-order growth condition with c = γ(S(θ))2 and c > 0 since
S(θ) is positive definite. It remains to show that the function H(.) = HS1(θ)(.)−HS2(θ)(.) isLipschitz continuous on W. Indeed, since H is continuous and differentiable, the mean valuetheorem can be used to get∣∣∣H(w1)−H(w2)
∣∣∣ ≤ supw∈W
(∥∥∇H(w)∥∥∞)‖w1 −w2‖1, w1,w2 ∈W (5.42)
Further, for all w ∈W∥∥∥∇H(w)∥∥∥∞
=∥∥(S2(θ)− S1(θ))w
∥∥∞ ≤
∥∥S2(θ)− S1(θ)∥∥∞ = γ, (5.43)
with γ the Lipschitz constant. Applying Proposition 1.2.6, we obtain∥∥∥wRQE1 −wRQE2
∥∥∥1≤ 2
∥∥S2(θ)− S1(θ)∥∥∞
γ(S1(θ)). (5.44)
Exchanging the role of wRQE1 , HS1(θ), S1(θ) and wRQE2 , HS2(θ), S2(θ), it is straightforwardto show that ∥∥∥wRQE1 −wRQE2
∥∥∥1≤ 2
∥∥S2(θ)− S1(θ)∥∥∞
γ(S2(θ))(5.45)
and (5.33) follows.
3) Proof of (5.34): The proof of (5.34) can be obtained following that of (5.33) and applyingProposition 1.2.7.
Since ξmin(S1(θ)) is an increasing function of θ (see Proposition 5.1.9), (5.34) can be rewrittenas follows
∥∥∥wRQE1−wRQE2
∥∥∥2≤ 2
maxi=1,...,N
‖Ci(D1 −D2)‖2
max(ξmin(S1(∞)), ξmin(S2(∞)))≤ ... ≤ 2
maxi=1,...,N
‖Ci(D1 −D2)‖2
max(ξmin(S1(θ∗)), ξmin(S2(θ∗))).
(5.46)The sensitivity of RQEP depends therefore on the small eigenvalue of the similarity matrixS(∞). Without loss of generality, assume that ξmin(S1(∞)) < ξmin(S2(∞)). If, for 0 < ε < 1,
ξmin(S2(∞)) ≥2 maxi=1,...,N
‖Ci(D1 −D2)‖2
ε(5.47)
or equivalently
maxi=1,...,N
‖Ci(D1 −D2)‖2 ≤ε ξmin(S2(∞))
2, (5.48)
then∥∥∥wRQE1 −wRQE2
∥∥∥2≤ ε, consequently the stability of RQEP is guaranteed.
This result can also be expressed in terms of D. To do so, the relationship between ξmin(S(∞))
and the eigenvalues of D is investigated numerically through a simulation. Consider a set ofN , N = N0 = 10, N1, ...Ni, ..., N1000 = 1000, where Ni = Ni−1 + 10. For each value of
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Figure 5.1 – R-squared of the Regression of ξi(D) on ξmin(S(∞)) and Slope of the Regressionof∣∣ξ2(D)
∣∣ on ξmin(S(∞))
0 10 20 30 40 50 60 70 80 90 100 1100
0.2
0.4
0.6
0.8
1
1.2
1.4
N
R-squ
ared
-βR-squared of ξ2(D) on ξmin(S(∞))
Slope of∣∣ξ2(D)
∣∣ on ξmin(S(∞))
Average R-squared of ξi6=2(D) on ξmin(S(∞))
N ∈ N , D such that dij = σ2i +σ2−2σij is constructed using data randomly sampled form an
universe of 2000 variables. Next, the eigenvalues of D, ξi(D), i = 1, ..., N , and ξmin(S(∞)) arecomputed setting ∞ = 10000000. Finally, each ξi(D) is regressed on ξmin(S(∞)) separately.The universe of 2000 variables is constructed using the following three factor model
rit = λi +3∑j=1
fjtβij + eit, ∀ i = 1, ..., 2000 and t = 1, ..., 100, (5.49)
where f1 ∼ U(−0.3, 0.4), f2 ∼ U(−0.5, 0.4), f1 ∼ U(−0.3, 0.3), βi1 ∼ U(0, 3), βi2 ∼ U(−2, 2),βi3 ∼ U(−1, 1), λi ∼ U(−0.1, 0.1) and ei1 ∼ N(0, σi) with σi ∼ U(0.1, 0.3). The results of thesimulation are reported on Figure 5.1. The solid line represents the R-squared of the regressionof ζ2(D) on ξmin(S(∞)) denoted R2
ξ2(D). The dashed-dotted line represents the average of theR-squared of the regression of ξi(D), i 6= 2 on ξmin(S(∞)) denoted R2
ξi 6=2(D). The dottedline represents the slope of the regression of
∣∣ξ2(D)∣∣ on ξmin(S(∞)) denoted β∣∣ξ2(D)
∣∣. It is
observable that R2ξ2(D) is close to 1 and R2
ξi 6=2(D) is less than 0.3. It follows that ξmin(S(∞))
is totally determined by ξ2(D). Moreover, β∣∣ξ2(D)∣∣ is positive. As a result, the sensitivity of
RQEP depends on ξ2(D) or∣∣ξ2(D)
∣∣. The lower is ξ2(D) or the higher is∣∣ξ2(D)
∣∣, the higheris ξmin(S(∞)) and the more stable is RQEP. These results are summarized in the followingproposition.
103
Proposition 5.1.11 (Stability of RQEP). The lower is ξ2(D) or the higher is∣∣∣ξ2(D)
∣∣∣ orequivalently the higher is ξmin(S(∞)), the more stable is RQEP.
The question which remains is how to stabilize RQEP when it is unstable? In what follows,the stability of RQEP based on S(∞) or D are discussed. From S(∞), the stability of RQEPcan be increased replacing S(∞) by Sκ(∞) such that
Sκ(∞) = S(∞) + κI. (5.50)
It is well-known that ζmin(Sκ(∞)) = ζmin(S(∞)) + κ. Then choosing κ very large willguaranteed the stability of RQEP. Replace S(∞) by Sκ(∞) is equivalent to replace D byDκ = D + κ
(11> − I
). To see this, consider
Sκ(θ) = θ11> −Dκ. (5.51)
Replacing Dκ,Sκ(θ) = (θ − κ)11> −D + κI. (5.52)
Set θ =∞,Sκ(∞) = S(∞) + κI. (5.53)
5.2 Empirical Properties
This section examines some empirical properties of RQEP. Twenty RQEP are examined out-of-sample, on two different empirical datasets, against four standard diversified portfolios (thebenchmark portfolios) which are: the equally weighted portfolio (EWP), the equally riskcontribution portfolio (ERCP), the most diversified portfolios (MDP) and the market portfo-lio (MP)(see Table 5.1). The twenty RQEP includes eight non-targeted RQEP (RQEPDσ2
,RQEPDρ
, RQEPDρc, RQEPDβ
, RQEPDκσ2, RQEPDκ
ρ, RQEPDκ
ρc, RQEPDκ
β) and twelve tar-
geted RQEP (RQEPDiσ2, RQEPDi
ρ, RQEPDi
ρc, RQEPDi
β, i ∈ t, t+, t−). RQEP
Dt−σ2
targetsthe subset of assets with higher dissimilarity and lower expected shortfall or expected tailloss. RQEP
Dt+
σ2targets the subset of assets with higher dissimilarity and lower expected tail
loss. RQEPDt+
σ2targets the subset of assets with higher dissimilarity and higher ratio of ex-
pected tail gain and loss. The dissimilarity matrices in RQEPDσ2, RQEPDρ
, RQEPDβand
their corresponding targeted versions capture the linear dependence between assets. However,the dissimilarity matrix in RQEPDρc
and those in RQEPDκσ2, RQEPDκ
ρ, RQEPDκ
ρc, RQEPDκ
β
capture the non-linear dependence between assets.
5.2.1 Methodology
To conduct the comparisons, two different empirical Fama-French datasets are considered1:1The Dataset is available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_
library.html
104
Table 5.1 – List of Considered Portfolios
Portfolios Abbreviation
Rao’s Quadratic Entropy (RQE) Portfoliosdσ2,ij = σ2
i + σ2j − 2σij RQEPDσ2
dρ,ij = 1− ρij RQEPDρ
dρc,ij = 1− ρc,ij RQEPDρc
dβ,ij =∑Kk=1(βik − βjk)2 RQEPDβ
dtσ2,ij = (σ2i + σ2
j − 2σij)(
CVaR(−Ri)CVaR(Ri)
)(CVaR(−Rj)CVaR(Rj)
)RQEPDt
σ2
dtρ,ij = (1− ρij)(
CVaR(−Ri)CVaR(Ri)
)(CVaR(−Rj)CVaR(Rj)
)RQEPDt
ρ
dtρc,ij = (1− ρc,ij)(
CVaR(−Ri)CVaR(Ri)
)(CVaR(−Rj)CVaR(Rj)
)RQEPDDt
ρc
dtβ,ij =(∑K
k=1(βik − βjk)2)(
CVaR(−Ri)CVaR(Ri)
)(CVaR(−Rj)CVaR(Rj)
)RQEPDt
β
dt+
σ2,ij = (σ2i + σ2
j − 2σij)(
CVaR(−Ri)∑Ni=1 CVaR(−Ri)
)(CVaR(−Rj)∑Nj=1 CVaR(−Rj)
)RQEP
Dt+
σ2
dt+
ρ,ij = (1− ρij)(
CVaR(−Ri)∑Ni=1 CVaR(−Ri)
)(CVaR(−Rj)∑Nj=1 CVaR(−Rj)
)RQEPDt+
ρ
dt+
ρc,ij= (1− ρc,ij)
(CVaR(−Ri)∑Ni=1 CVaR(−Ri)
)(CVaR(−Rj)∑Nj=1 CVaR(−Rj)
)RQEPD
Dt+ρc
dt+
β,ij =(∑K
k=1(βik − βjk)2)(
CVaR(−Ri)∑Ni=1 CVaR(−Ri)
)(CVaR(−Rj)∑Nj=1 CVaR(−Rj)
)RQEP
Dt+
β
dt−
σ2,ij = (σ2i + σ2
j − 2σij)(∑N
i=1 CVaR(Ri)
CVaR(Ri)
)(∑Nj=1 CVaR(Rj)
CVaR(Rj)
)RQEP
Dt−σ2
dt−
ρ,ij = (1− ρij)(∑N
i=1 CVaR(Ri)
CVaR(Ri)
)(∑Nj=1 CVaR(Rj)
CVaR(Rj)
)RQEPDt−
ρ
dt−
ρc,ij= (1− ρc,ij)
(∑Ni=1 CVaR(Ri)
CVaR(Ri)
)(∑Nj=1 CVaR(Rj)
CVaR(Rj)
)RQEPD
Dt−ρc
dt−
β,ij =(∑K
k=1(βik − βjk)2)(∑N
i=1 CVaR(Ri)
CVaR(Ri)
)(∑Nj=1 CVaR(Rj)
CVaR(Rj)
)RQEP
Dt−β
dσ2,ij =(σ2i + σ2
j − 2σij)κ
RQEPDκσ2
dρ,ij = (1− ρij)κ RQEPDκρ
dρc,ij = (1− ρc,ij)κ RQEPDκρc
dβ,ij =(∑K
k=1(βik − βjk)2)κ
RQEPDκβ
Most Used Diversified PortfoliosEqually weight portfolio EWPEqually risk contribution portfolio ERCPMost diversified portfolio MDPMarket portfolio MP
(a) The Fama-French forty-nine industry portfolios dataset (49Ind) of equally weighted dailyasset average returns with observed periods from 07-01-1969 to 06-30-2016;
(b) The Fama-French twenty-five North American, Asia Pacific ex Japan, Japanese and Eu-
105
ropean Portfolios Formed on Size and Book-to-Market dataset (World100FF) of equallyweighted daily asset average returns with observed periods from 07-02-1990 to 06-30-2016.
The out-of-sample portfolio returns are generated using a rolling-window. The estimationwindow length is one year and portfolios are rebalanced yearly. At the end of each year (lasttrading day of the year), the portfolio weights are obtained by optimizing the portfolio withthe historical daily returns of the preceding year. Then, the portfolio is held one year andrebalanced again. The procedure generates a time series of T − t − 1 daily out-of-samplereturns: rjt+1 = wj>
t rt+1, t = t, ..., T −1, where rt+1 is the vector of period t+1 asset returns,wjt the portfolio weight vector at time t for strategy j and t the length of the estimation
window. Using the time series rjt+1, the performance of considered portfolios is evaluated.The performance metrics used are presented in Table 5.2. Moreover, portfolios performanceduring bear market periods are compared. The parameters K and κ are chosen to minimize,in-sample, the portfolios RQEPDβ
, RQEPDiβ, i ∈ t, t−, t+, RQEPDκ
β, RQEPDκ
σ2, RQEPDκ
ρ
and RQEPDκρc
average drawdown risk.
5.2.2 Results
5.2.2.1 Fama-French 49 Industry Portfolios Daily Returns
At first, the portfolios will be examined on the Fama-French forty-nine industry Portfoliosdataset of equally weighted daily asset average returns. This dataset represents U.S. stockmarket and it ranges from 07-01-1969 to 06-30-2016.
Table 5.3 compares the performance of RQEP and the benchmark portfolios in terms ofannualized mean, volatility (or standard deviation), Sharpe ratio, turnover, average number ofassets held and its standard deviation. RQEP which outperforms all the benchmark portfoliosis shaded in gray. It is observable that none of the benchmark portfolios has higher meanor volatility or Sharpe ratio than all RQEP considered. All RQEP have higher mean andSharpe ratio than the MP, except RQEP
Dt+
σ2and RQEP
Dt+ρc. All RQEP that have higher
mean or volatility or Sharpe ratio than all the benchmark portfolios are based on targeteddiversification. Across the targeted RQEP, it is those based on Dt+ which exhibits the worstoutcomes. The EWP has the lowest turnover.
Table 5.4 displays other risk and performance ratio measures. Again, it is observable thatnone of the benchmark portfolios has the best outcomes, regardless of the considered risk andperformance ratio measures. For each risk or performance ratio measures considered, there isat least one RQEP that dominates all the benchmark portfolios. In the case of the Rachev ra-tio (RCVaR), almost RQEP dominate all the benchmark portfolios. In particular, RQEP
Dt−β
dominates all the benchmark portfolios, regardless of the considered risk and performance ra-tios measures. Again, across RQEP, it is those based on targeted diversification which exhibits
106
Table 5.2 – List of Performance Metrics
Performance Metrics Formula
Return µj =1
T − t
T−1∑t=t
rjt+1
Standard Deviation σj =
√√√√ 1
T − t− 1
T−1∑t=t
(rjt+1 − µj
)2Sharpe Ratio (SR) SRj =
µj
σj
Annualized Portfolio total return µjy =
T−1∏t=t
(1 + wj>
t rt+1
)252/(T−t)
− 1
Annualized Standard Deviation σjy =
√√√√ 1
T − t− 1
T−1∑t=t
(rjt+1 − µj
)2×√
252× 100
Annualized Sharpe Ratio SRjy =
µjy
σjyMaximum Loss (ML) MLj = min(rjt+1)
Value at Risk VaRjα(w) = −infx|P (rj(w) ≤ x) ≥ α
Conditional Value at Risk CVaRjα(w) =
1
α
∫ α
0
VaRju (w) du
Maximum Drawdown Risk (MDD) MDDj = maxt1∈(t,T−1)
[max
t2∈(t,t1)
(rjt1+1 − r
jt2+1
)]CVaR Ratio (CVaRR) CVaRRj =
µj∣∣CVaRj∣∣
Maximum Drawdown Risk Ratio (MDDR) MDDRj =µj
MDDj× 100
Rachev RatioCVaR+j
CVaR−j=
CVaR(−R)j
CVaR(R)j
End period cumulative return (CR) CRj =
T−1∏t=t
(1 + wj>
t rt+1
)Turnover (TRN) TRNj =
1
T − t− 1
T−1∑t=t
N∑i=1
(|wji,t+1 − w
ji,t+ |
)Average number of assets held N(w∗)Standard Deviation of a number of assets held σN(w∗)
wji,t+ is the portfolio weight before rebalancing at t+ 1.
the best outcomes. Across the benchmark portfolios, it is the MDP which exhibits the bestoutcomes and the MP which exhibits the worst outcomes. For example, one dollar invested inthe MP in 07-01-1970 yields 115.975 dollars in 06-30-2016, while the same investment in theRQEPDt
ρcfor the same period yields 38183.235 dollars.
Tables 5.5 and 5.6 display considered portfolio performance during bear market periods. Thebear markets are characterized by periods of time when equity prices fall. This thesis measuresU.S. equity prices by market portfolio price and defines bear period as period where market
107
Table 5.3 – Characteristics of Portfolios-49Ind
µy σy SRy TRN N(w∗) σN(w∗)
RQEPDσ20.178 0.244 0.73 1.624 5.37 1.511
RQEPDtσ2
0.199 0.236 0.841 1.798 4.587 1.087RQEP
Dt+
σ20.148 0.286 0.519 1.648 3.435 1.068
RQEPDt−σ2
0.154 0.208 0.741 1.59 4.217 1.632
RQEPDκσ2
0.209 0.183 1.141 3.32 29.87 10.909RQEPDρ
0.224 0.164 1.365 1.844 12.630 3.785RQEPDt
ρ0.252 0.168 1.505 2.824 7.109 1.703
RQEPDt+ρ
0.181 0.249 0.726 1.808 4.739 1.512RQEP
Dt−ρ
0.227 0.113 2 1.445 6.652 2.368RQEPDκ
ρ0.223 0.149 1.501 1.921 30.087 12.771
RQEPDρc0.226 0.155 1.459 1.677 16.543 4.188
RQEPDtρc
0.257 0.165 1.564 3.332 8.065 2.462RQEP
Dt+ρc
0.168 0.256 0.655 1.522 4.739 1.437RQEP
Dt−ρc
0.22 0.108 2.033 1.555 6.674 1.477RQEPDκ
ρc0.228 0.146 1.561 1.489 32.348 12.212
RQEPDβ0.232 0.149 1.554 2.29 18.652 14.41
RQEPDtβ
0.255 0.16 1.599 3.009 7.326 3.163RQEP
Dt+β
0.176 0.237 0.74 1.79 4.391 1.483
RQEPDt−β
0.247 0.124 1.989 1.958 6.065 2.551
RQEPDκβ
0.23 0.15 1.532 2.27 20.087 14.365EWP 0.218 0.147 1.487 0.966 49 0MDP 0.234 0.142 1.654 3.837 12.630 3.785ERCP 0.223 0.138 1.609 2.063 48.957 0.206MP 0.109 0.164 0.661 - 49 0
portfolio price (or return) falling at least 20%. Figure 5.2 depicts the annual market portfolioreturns during the period 1971-2016. It is observable that only the periods 1973, 1974, 2002and 2008 can be considered as bear market periods. However, in this thesis, all periods wheremarket portfolio returns is negative (1973-1974 (dramatic rise in oil prices), 1977, 1981, 2000(Dot-com bubble), 2001 (September 11 attacks), 2002 (Internet bubble bursting), 2008 (globalfinancial crisis)) and the periods 1987 (Black Monday), 1990 (Early 1990s recession due toKuwait invasion by Iraq), 2011 (stock markets fall), 2015 (Chinese stock market crash) areconsidered as bear market periods.
Table 5.5 reports the annualized return of considered portfolios during each considered bearmarket. It is observable that for each, except for the periods 1977 which is not a proper bearmarket period and 2000, there is RQEP that exhibits higher return than all the benchmarkportfolios. In particular, in 2008, RQEP
Dt+β
exhibits positive annualized return.
108
Table 5.4 – Other Characteristics of Portfolios-49Ind
MDD CVaR ML CR CVaRR MDDR CVaRR+
RCVaRR−
RQEPDσ20.768 -0.035 -0.122 1894.421 0.022 0.1 1.058
RQEPDtσ2
0.643 -0.033 -0.122 4207.32 0.025 0.129 1.078RQEP
Dt+
σ20.859 -0.041 -0.152 586.817 0.017 0.083 1.063
RQEPDt−σ2
0.72 -0.03 -0.104 747.258 0.022 0.091 1.095
RQEPDκσ2
0.617 -0.027 -0.113 6196.148 0.03 0.133 0.992RQEPDρ
0.538 -0.024 -0.094 10866.648 0.036 0.159 1.004RQEPDt
ρ0.532 -0.024 -0.091 31639.898 0.04 0.178 1.071
RQEPDt+ρ
0.754 -0.035 -0.123 2094.518 0.022 0.104 1.062RQEP
Dt−ρ
0.447 -0.017 -0.083 12149.803 0.051 0.187 1.006RQEPDκ
ρ0.546 -0.022 -0.101 10708.181 0.038 0.155 0.966
RQEPDρc0.52 -0.023 -0.09 11741.152 0.038 0.164 0.984
RQEPDtρc
0.607 -0.023 -0.094 38183.235 0.041 0.159 1.055RQEP
Dt+ρc
0.8 -0.037 -0.131 1269.887 0.02 0.093 1.055RQEP
Dt−ρc
0.501 -0.016 -0.082 9477.489 0.051 0.162 0.983RQEPDκ
ρc0.528 -0.022 -0.095 12690.857 0.039 0.162 0.957
RQEPDβ0.46 -0.021 -0.106 14718.374 0.041 0.189 1.021
RQEPDtβ
0.515 -0.022 -0.095 35346.509 0.043 0.185 1.083RQEP
Dt+β
0.78 -0.034 -0.11 1717.833 0.022 0.097 1.062
RQEPDt−β
0.405 -0.018 -0.089 26202.27 0.051 0.224 1.073
RQEPDκβ
0.52 -0.022 -0.106 13678.746 0.04 0.166 1.016EWP 0.59 -0.022 -0.109 8798.288 0.037 0.14 0.934MDP 0.495 -0.021 -0.096 16232.682 0.043 0.177 0.98ERCP 0.576 -0.021 -0.105 10472.874 0.04 0.145 0.931MP 0.547 -0.024 -0.174 115.975 0.019 0.085 0.978
Table 5.6 reports other risk and performance ratio measures. These performances are com-puted using the portfolios daily returns during the considered bear market periods. Again,for each performance measures, there is at least one RQEP that dominates all the benchmarkportfolios. Across the RQEP, it is those based on targeted diversification which exhibits thebest outcomes. In particular, RQEP
Dt−β
dominates all the benchmark portfolios, regardless ofthe considered risk and performance ratio measures.
109
Table 5.5 – Annualized Returns During Bear Markets-49Ind
1973 1974 1977 1981 1987 1990 2000 2001 2002 2008 2011 2015 µ σ SR
RQEPDσ2-0.215 -0.071 0.248 -0.268 0.295 -0.191 0.054 0.825 0.224 -0.264 -0.124 -0.522 -0.001 0.357 -0.003
RQEPDtσ2
-0.1 -0.037 0.194 -0.225 0.209 -0.192 0.126 0.975 0.253 -0.18 -0.128 -0.386 0.042 0.355 0.118RQEP
Dt+
σ2-0.381 -0.213 0.207 -0.273 0.353 -0.264 0.04 0.876 0.311 -0.372 -0.195 -0.609 -0.043 0.415 -0.104
RQEPDt−σ2
-0.335 -0.192 0.135 -0.219 0.122 -0.178 0.056 0.78 0.308 -0.239 -0.102 -0.28 -0.012 0.318 -0.038
RQEPDκσ2
-0.254 -0.089 0.273 -0.179 0.122 -0.098 0.127 0.527 0.119 -0.347 -0.122 -0.29 -0.018 0.257 -0.07RQEPDρ
-0.043 -0.051 0.29 -0.112 0.12 -0.077 0.265 0.493 0.168 -0.279 -0.041 -0.299 0.036 0.237 0.152RQEPDt
ρ0.061 0.019 0.279 -0.153 0.209 -0.086 0.151 0.894 0.007 -0.168 -0.01 -0.218 0.082 0.297 0.276
RQEPDt+ρ
-0.222 -0.091 0.205 -0.249 0.353 -0.199 0.142 0.9 0.232 -0.175 -0.132 -0.504 0.022 0.37 0.059RQEP
Dt−ρ
0.006 -0.17 0.255 0.237 0.016 -0.031 0.179 0.416 0.176 -0.213 0.008 0.073 0.079 0.182 0.434RQEPDκ
ρ-0.196 -0.101 0.293 -0.085 0.08 -0.085 0.234 0.456 0.131 -0.282 -0.053 -0.208 0.015 0.225 0.067
RQEPDρc-0.15 -0.093 0.282 -0.07 0.098 -0.123 0.234 0.429 0.122 -0.265 -0.069 -0.235 0.013 0.218 0.06
RQEPDtρc
0.021 -0.063 0.24 -0.154 0.171 -0.109 0.175 0.899 -0.016 -0.274 -0.045 -0.141 0.059 0.305 0.193RQEP
Dt+ρc
-0.282 -0.144 0.186 -0.256 0.357 -0.249 0.128 0.916 0.215 -0.276 -0.164 -0.546 -0.01 0.392 -0.026RQEP
Dt−ρc
-0.205 -0.147 0.228 0.237 0.003 -0.038 0.207 0.41 0.106 -0.313 0.017 0.039 0.045 0.207 0.217RQEPDκ
ρc-0.214 -0.103 0.267 -0.05 0.079 -0.088 0.185 0.409 0.126 -0.282 -0.071 -0.169 0.007 0.208 0.034
RQEPDβ-0.128 -0.085 0.294 -0.022 0.016 -0.098 0.222 0.457 0.118 -0.136 0.03 -0.121 0.046 0.191 0.241
RQEPDtβ
-0.084 -0.027 0.302 -0.129 0.155 -0.082 0.156 0.807 -0.02 -0.17 0.065 -0.068 0.075 0.268 0.28RQEP
Dt+β
-0.229 -0.054 0.206 -0.271 0.316 -0.219 0.131 0.924 0.254 0.208 -0.091 -0.547 0.052 0.378 0.138
RQEPDt−β
-0.055 -0.175 0.238 0.309 -0.032 -0.051 0.124 0.488 0.152 -0.18 0.064 0.021 0.075 0.199 0.377
RQEPDκβ
-0.14 -0.083 0.294 -0.017 0.016 -0.098 0.223 0.454 0.131 -0.213 0.008 -0.122 0.038 0.2 0.19EWP -0.324 -0.172 0.256 0.021 0.013 -0.078 0.132 0.416 0.039 -0.374 -0.084 -0.11 -0.022 0.224 -0.098MDP -0.033 -0.04 0.303 0.014 0.072 -0.07 0.275 0.395 0.121 -0.252 -0.011 -0.181 0.049 0.195 0.251ERCP -0.304 -0.173 0.256 0.055 0.022 -0.078 0.182 0.413 0.064 -0.363 -0.065 -0.084 -0.006 0.223 -0.027MP -0.196 -0.277 -0.032 -0.035 0.015 -0.061 -0.119 -0.113 -0.212 -0.365 0.005 0.001 -0.116 0.123 -0.943
110
Figure 5.2 – U.S. Market Portfolio Returns
1960 1980 2000 2020−40
−20
0
20
40
S&P
500pricechan
ge
Table 5.6 – Other Characteristics of Portfolios During Bear Markets-49Ind
MDD CVaR MDDR CVaRR CVaR+
CVaR−
RQEPDσ20.757 -0.045 -0.007 -0.001 0.987
RQEPDtσ2
0.673 -0.043 0.022 0.003 1.015RQEP
Dt+
σ20.855 -0.054 -0.031 -0.005 0.973
RQEPDt−σ2
0.619 -0.038 -0.014 -0.002 1
RQEPDκσ2
0.655 -0.037 -0.013 -0.002 0.895RQEPDρ
0.567 -0.031 0.022 0.004 0.893RQEPDt
ρ0.552 -0.031 0.051 0.009 1.004
RQEPDt+ρ
0.738 -0.046 0.005 0.001 0.99RQEP
Dt−ρ
0.357 -0.022 0.081 0.013 0.934RQEPDκ
ρ0.551 -0.03 0.008 0.001 0.873
RQEPDρc0.548 -0.03 0.007 0.001 0.878
RQEPDtρc
0.613 -0.031 0.03 0.006 0.977RQEP
Dt+ρc
0.781 -0.048 -0.014 -0.002 0.975RQEP
Dt−ρc
0.458 -0.022 0.03 0.006 0.87RQEPDκ
ρc0.538 -0.029 0.003 0.001 0.875
RQEPDβ0.457 -0.027 0.04 0.007 0.912
RQEPDtβ
0.521 -0.029 0.052 0.009 1RQEP
Dt+β
0.646 -0.042 0.019 0.003 1.006
RQEPDt−β
0.372 -0.023 0.074 0.012 0.979
RQEPDκβ
0.515 -0.028 0.029 0.005 0.907EWP 0.582 -0.031 -0.02 -0.004 0.869MDP 0.501 -0.027 0.037 0.007 0.869ERCP 0.565 -0.029 -0.01 -0.002 0.858MP 0.836 -0.033 -0.051 -0.013 0.963
111
5.2.2.2 Fama-French 100 Portfolios Formed on Size and Book-to-Market DailyReturns World
In this section, the portfolios are examined on the dataset obtained combining the Fama-French twenty-five North American, Asia Pacific ex Japan, Japanese and European PortfoliosFormed on Size and Book-to-Market dataset (World100FF) of equally weighted daily assetaverage returns with observed periods from 07-02-1990 to 06-30-2016.
Table 5.7 compares the performance of RQEP and the benchmark portfolios in terms of mean,volatility (or standard deviation), Sharpe ratio, turnover, average number of assets held andits standard deviation. It is observable that none of the benchmark portfolios has higher meanor volatility or Sharpe ratio than all RQEP considered. For each performance measure (exceptN(w∗) and σN(w∗)), there is RQEP which dominates all the benchmark portfolios. Across theRQEP, it is those based on target diversification which exhibits the best outcomes. Acrossthe benchmark portfolios, it is the MDP which exhibits the best outcomes and the MP whichexhibits the worst outcomes.
Table 5.8 displays other risk and performance ratio measures of portfolios. Again, it is observ-able that none of the benchmark portfolios has the best outcomes, regardless of the consideredrisk and performance ratio measures. For each risk or performance ratio measures considered,there is at least one RQEP that dominates all the benchmark portfolios. In the case of theRachev ratio, almost RQEP dominate all the benchmark portfolios. Again, across RQEP, itis those based on targeted diversification which exhibits the best outcomes. Across the bench-mark portfolios, it is the MDP which exhibits the best outcomes and the MP which exhibitsthe worst outcomes. For example, one dollar invested in the MPJ in 07-01-1991 yields 0.4dollars in 06-30-2016, while the same investment in the RQEPDt−
ρcfor the same period yields
613.676 dollars.
Tables 5.10 and 5.9 display the performance of considered portfolios during bear market pe-riods. Table 5.9 reports the annualized return of considered portfolios during each consideredbear market. It is observable that for each, there is RQEP that exhibits higher return thanall the benchmark portfolios. Table 5.10 reports other risk and performance ratio measures.These performances are computed using the portfolios daily returns during the considered bearmarket periods. Again, for each performance measure, there is at least one RQEP that domi-nates all the benchmark portfolios. Across RQEP, it is those based on targeted diversificationwhich exhibits the best outcomes.
In sum, the RQEP and particular RQEP based on targeted diversification exhibit better out-comes than the standard diversified portfolios used on the marketplace. The best performanceof the MDP relative to other standard diversified portfolios is due essentially to its relation-ship with the RQEPDρ
. Indeed, it is straightforward to verify that the weight of RQEPDρ,
wRQEDρ , coincides with the optimal weight of the first step in the determination of the MDP.
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Table 5.7 – Characteristics of Portfolios-World100FF
µy σy SRy TRN N(w∗) σN(w∗)
RQEPDσ20.122 0.167 0.726 1.908 6.08 1.382
RQEPDtσ2
0.193 0.171 1.125 11.392 4.52 1.636RQEP
Dt+
σ20.131 0.182 0.719 1.8 3.64 1.036
RQEPDt−σ2
0.197 0.131 1.507 2.363 6.16 2.23
RQEPDκσ2
0.123 0.149 0.821 2.132 40.92 31.499RQEPDρ
0.159 0.12 1.329 3.673 16.560 4.388RQEPDt
ρ0.222 0.143 1.555 1.889 6.36 2.307
RQEPDt+ρ
0.152 0.172 0.886 2.032 4.88 1.563RQEP
Dt−ρ
0.277 0.098 2.825 1.392 6.48 1.584RQEPDκ
ρ0.159 0.12 1.325 2.084 38.6 23.934
RQEPDρc0.147 0.124 1.189 3.895 26.72 5.256
RQEPDtρc
0.227 0.145 1.566 1.924 8.52 3.429RQEP
Dt+ρc
0.12 0.172 0.697 1.839 6.8 2.236RQEP
Dt−ρc
0.281 0.103 2.743 1.428 8.68 2.428RQEPDκ
ρc0.144 0.122 1.179 3.067 58.6 29.869
RQEPDβ0.195 0.134 1.453 1.903 15.84 23.216
RQEPDtβ
0.272 0.158 1.723 1.842 4.92 2.565RQEP
Dt+β
0.185 0.176 1.052 2.519 4.6 2.769
RQEPDt−β
0.264 0.119 2.212 1.709 6.28 3.932
RQEPDκβ
0.19 0.133 1.424 1.851 19.28 23.849EW 0.124 0.124 0.999 2.038 100 0MDP 0.17 0.113 1.506 1.817 16.560 4.388ERC 0.14 0.119 1.179 1.964 100 0MPEU 0.067 0.184 0.365 - 25 0MPJ -0.035 0.223 -0.156 - 25 0MPAJ
0.088 0.175 0.505 - 25 0MPAN 0.088 0.174 0.505 - 25 0
More formally, wRQEDρ = w∗, where w∗ is the solution of the problem (2.75). Table 5.11reports the results of the regression of RQEPDρ
on the MDP. It is clear that the return ofRQEPDρ
explains more than 95% of variability of the return of the MDP.
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Table 5.8 – Other Characteristics of Portfolios-World100FF
MDD CVaR ML CR CVaRR MDDR CVaR+
CVaR−
RQEPDσ20.574 -0.024 -0.06 19.503 0.021 0.09 1.016
RQEPDtσ2
0.537 -0.024 -0.06 95.534 0.031 0.141 1.103RQEP
Dt+
σ20.573 -0.026 -0.063 24.26 0.022 0.098 1.067
RQEPDt−σ2
0.493 -0.019 -0.065 104.973 0.039 0.152 0.999
RQEPDκσ2
0.545 -0.022 -0.062 19.925 0.023 0.094 0.985RQEPDρ
0.468 -0.018 -0.059 45.654 0.034 0.132 0.941RQEPDt
ρ0.493 -0.021 -0.064 180.426 0.04 0.17 1.032
RQEPDt+ρ
0.551 -0.024 -0.06 39.143 0.026 0.113 1.062RQEP
Dt−ρ
0.431 -0.014 -0.066 564.605 0.069 0.23 0.966RQEPDκ
ρ0.491 -0.018 -0.057 45.415 0.034 0.126 0.931
RQEPDρc0.525 -0.019 -0.055 34.797 0.031 0.11 0.928
RQEPDtρc
0.502 -0.021 -0.069 200.523 0.04 0.171 1.028RQEP
Dt+ρc
0.558 -0.024 -0.063 18.697 0.021 0.092 1.042RQEP
Dt−ρc
0.471 -0.016 -0.079 613.676 0.065 0.213 0.933RQEPDκ
ρc0.522 -0.019 -0.054 32.332 0.031 0.109 0.919
RQEPDβ0.476 -0.02 -0.069 100.56 0.036 0.156 0.955
RQEPDtβ
0.486 -0.023 -0.071 507.91 0.043 0.207 1.045RQEP
Dt+β
0.52 -0.025 -0.061 80.373 0.029 0.141 1.074
RQEPDt−β
0.456 -0.018 -0.077 427.847 0.053 0.21 0.963
RQEPDκβ
0.476 -0.02 -0.069 90 0.036 0.153 0.953EWP 0.548 -0.019 -0.061 20.642 0.027 0.092 0.92MDP 0.454 -0.017 -0.068 58.872 0.038 0.144 0.926ERCP 0.54 -0.018 -0.057 29.733 0.03 0.103 0.907MPEU 0.624 -0.028 -0.09 5.397 0.013 0.056 0.931MPJ 0.598 -0.032 -0.109 0.4 0.005 0.025 0.984MPAJ
0.624 -0.027 -0.104 8.957 0.015 0.066 0.917MPAN 0.548 -0.026 -0.09 8.856 0.016 0.075 0.959
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Table 5.9 – Annualized Returns During Bear Markets-World100FF
2000 2001 2002 2008 2011 2015 µ σ SR
RQEPDσ2-0.26 0.11 -0.108 -0.338 0.009 0.103 -0.08 0.188 -0.428
RQEPDtσ2
-0.351 0.66 0.09 -0.291 0.043 0.27 0.07 0.373 0.188RQEP
Dt+
σ2-0.318 0.148 -0.088 -0.341 0.065 0.166 -0.061 0.226 -0.271
RQEPDt−σ2
-0.203 0.07 -0.051 -0.287 -0.023 0.268 -0.038 0.197 -0.191
RQEPDκσ2
-0.187 -0.016 -0.138 -0.363 -0.015 0.088 -0.105 0.16 -0.659RQEPDρ
-0.057 -0.031 -0.056 -0.249 -0.027 0.123 -0.049 0.119 -0.416RQEPDt
ρ-0.191 0.26 0.148 -0.242 -0.019 0.373 0.055 0.247 0.222
RQEPDt+ρ
-0.272 0.304 -0.013 -0.315 0.035 0.207 -0.009 0.249 -0.035RQEP
Dt−ρ
0.106 0.131 0.231 -0.262 0.025 0.202 0.072 0.179 0.403RQEPDκ
ρ-0.073 -0.003 -0.03 -0.294 -0.026 0.064 -0.06 0.123 -0.49
RQEPDρc-0.087 -0.009 -0.069 -0.336 -0.056 0.101 -0.076 0.144 -0.527
RQEPDtρc
-0.153 0.28 0.156 -0.287 -0.063 0.426 0.06 0.273 0.219RQEP
Dt+ρc
-0.307 0.148 -0.152 -0.343 -0.025 0.199 -0.08 0.228 -0.352RQEP
Dt−ρc
0.159 0.155 0.222 -0.3 0.023 0.219 0.08 0.199 0.4RQEPDκ
ρc-0.096 -0.022 -0.047 -0.333 -0.043 0.1 -0.074 0.143 -0.515
RQEPDβ-0.135 0.308 0.146 -0.288 -0.095 0.044 -0.003 0.213 -0.016
RQEPDtβ
-0.185 0.5 0.301 -0.193 -0.027 0.397 0.132 0.305 0.433RQEP
Dt+β
-0.354 0.559 0.052 -0.256 0.004 0.179 0.031 0.326 0.094
RQEPDt−β
0.025 0.264 0.289 -0.285 0.002 0.175 0.078 0.214 0.367
RQEPDκβ
-0.136 0.29 0.132 -0.288 -0.096 0.047 -0.009 0.207 -0.041EWP -0.073 -0.044 -0.046 -0.385 -0.042 0.058 -0.089 0.152 -0.584MDP -0.007 -0.017 -0.035 -0.241 -0.03 0.119 -0.035 0.116 -0.306ERCP -0.036 -0.033 -0.012 -0.375 -0.04 0.057 -0.073 0.152 -0.479MPEU -0.099 -0.199 -0.13 -0.445 -0.127 -0.005 -0.167 0.15 -1.118MPJ -0.324 -0.282 -0.068 -0.247 -0.097 0.12 -0.15 0.167 -0.898MPAJ
-0.113 -0.079 -0.059 -0.496 -0.145 -0.075 -0.161 0.167 -0.965MPAN -0.079 -0.109 -0.196 -0.365 -0.009 -0.015 -0.129 0.135 -0.957
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Table 5.10 – Other Characteristics of Portfolios During Bear Markets-World100FF
MDD CVaR MDDR CVaRR CVaR+
CVaR−
RQEPDσ20.65 -0.032 -0.048 -0.01 0.984
RQEPDtσ2
0.507 -0.033 0.035 0.005 1.157RQEP
Dt+
σ20.641 -0.033 -0.039 -0.008 1.032
RQEPDt−σ2
0.541 -0.027 -0.03 -0.006 0.893
RQEPDκσ2
0.676 -0.029 -0.063 -0.014 0.907RQEPDρ
0.533 -0.026 -0.033 -0.007 0.862RQEPDt
ρ0.47 -0.029 0.041 0.007 1.011
RQEPDt+ρ
0.541 -0.032 -0.008 -0.001 1.066RQEP
Dt−ρ
0.385 -0.021 0.066 0.012 0.798RQEPDκ
ρ0.535 -0.026 -0.042 -0.009 0.848
RQEPDρc0.586 -0.027 -0.052 -0.011 0.854
RQEPDtρc
0.495 -0.03 0.04 0.007 1.002RQEP
Dt+ρc
0.658 -0.033 -0.051 -0.01 1.006RQEP
Dt−ρc
0.422 -0.023 0.065 0.012 0.777RQEPDκ
ρc0.575 -0.027 -0.051 -0.011 0.837
RQEPDβ0.484 -0.028 -0.005 -0.001 0.94
RQEPDtβ
0.43 -0.033 0.108 0.014 1.06RQEP
Dt+β
0.489 -0.033 0.013 0.002 1.096
RQEPDt−β
0.401 -0.026 0.069 0.011 0.896
RQEPDκβ
0.489 -0.028 -0.009 -0.002 0.932EWP 0.616 -0.027 -0.06 -0.014 0.839MDP 0.48 -0.025 -0.025 -0.005 0.837ERCP 0.576 -0.027 -0.053 -0.011 0.825MPEU 0.742 -0.038 -0.09 -0.017 0.921MPJ 0.729 -0.039 -0.078 -0.015 0.938MPAJ
0.736 -0.036 -0.093 -0.019 0.843MPAN 0.691 -0.038 -0.067 -0.012 0.981
Table 5.11 – Regression of RQEPDρon the MDP
Parameters (49Ind)Intercept 9.707e-05wRQEDρ 9.273e-01R-squared 0.9891
Parameters (World100FF)Intercept 7.902e-05wRQEDρ 9.291e-01R-squared 0.9656
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5.3 Summary
This chapter has first analyzed some theoretical properties of the well-diversified portfoliosof portfolio Rao’s Quadratic Entropy named Rao’s Quadratic Entropy portfolio and denotedRQEP. The theoretical properties of RQEP examined are:
1. The existence (Proposition 5.1.1);2. The uniqueness (Proposition 5.1.2);3. The conditions under which it is an interior solution (Proposition 5.1.3);4. The monotonicity property (Proposition 5.1.4);5. The duplication invariance property (Proposition 5.1.5);6. The conditions under which it coincides with the equally weighted portfolio (Proposi-
tion 5.1.6);7. The conditions under which it is mean-variance optimal (Proposition 5.1.8);8. The sensibility relative to the dissimilarity matrix (Propositions 5.1.10 and 5.1.11).
The following results were obtained.
1. By definition, PRQE optimization problem has always a solution;2. REQP is unique when PRQE is strictly concave or quasi-concave. This is the case when
the dissimilarity matrix D is defined such that rank(GD
(1N
))= N − 1;
3. If D is circum-Euclidean and its associated circumsphere is the smallest enclosing hy-persphere containing the generating points, then RQEP is an interior solution;
4. If N > 2, then RQEP coincides with the equally weighted portfolio if and only if assettotal dissimilarity
∑Nj=1 dij , are equal. Otherwise i.e. if N = 2, RQEP is always equal
to the equally weighted portfolio;5. When the risk-free asset is not available, RQEP is mean-variance optimal if and only
if assets have the same variance or the same mean-variance utility. However, when therisk-free asset is available, RQEP is mean-variance optimal if and only if assets have thesame mean-variance utility;
6. RQEP is stable if and only if the second eigenvalue of D is low.
The second part of this chapter was devoted to the empirical properties of RQEP. TwentyRQEP were examined out-of-sample, on two different empirical datasets, against four standarddiversified portfolios which are: the equally weighted portfolio, the equally risk contributionportfolio, the most diversified portfolios and the market portfolio using different performancemetrics. The results have shown that regardless the performance metric, there is RQEPthat outperforms the standard diversified portfolios. In particular, RQEP based on targeteddissimilarity matrix have proved to be exhibited better performance. They also provide aguideline in the choice of the dissimilarity matrix in practice.
All these results strengthen the belief of this thesis that RQE provides the best framework to
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manage and quantify portfolio diversification.
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Part III
Applications
119
Chapter 6
Rao’s Quadratic Entropy AndMaximum Diversification
This part can be viewed as an application of the proposed portfolio diversification measure,portfolio Rao’s Entropy Quadratic (PRQE). It deeply analyzes the relationship between PRQEand maximum diversification (MD) strategy on the one hand and the mean-variance (MV)models on the second hand. This first chapter is devoted to the relationship between PRQEand MD strategy.
As mentioned in Section 2.1.2.1.5, risk-based indexing, also known as smart beta indexing (seeCazalet et al., 2014a), has recently become a popular alternative to traditional investmentstrategies based on the mean-variance models. This approach determines portfolio allocationsby relying only on the risk characteristics of assets, avoiding the need to provide estimates forexpected returns and thus eliminating any loss associated with the estimation of these returns(see Chopra and Ziemba, 1993).
One of the most popular risk-based investment strategy stems from Choueifaty and Coignard(2008)’s MD approach. It maximizes what the authors refer to as the diversification ratio(DR), and the portfolio that it generates is labelled the most diversified portfolio (MDP). TheMDP currently underlies the allocation choices of important managed funds worldwide: itforms the basis of TOBAM’s maximum diversification index which has attracted funds fromglobal players like California Public Employees’ Retirement System (CALPERS). In addition,the FTSE group recently launched a new family of indices that follow this strategy.
However, the source of the MDP’s desirable properties has not been formally established andas a result, their validity and robustness to unforeseen shocks have often been questioned. Forexample, Lee (2011, p. 15-16) argues that the diversification ratio forming the basis of thestrategy is a differential rather than an absolute diversification measure and as such cannot
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be used as a measure of portfolio diversification.1 In addition, he argues that no investmentobjective function is associated with the maximization of that ratio and that as a result, theinvestment problem underlying the MDP is undefined. Taliaferro (2012) also stresses thelack of clear investment objective underlying the MDP and suggests that its success has beencoincidental.
The present chapter contributes to this debate in two important and related ways. First, it for-mally establishes the principles at play behind the MD approach. This is done by showing thatMDP maximizes the ratio of PRQE to portfolio variance or, alternatively, minimizes portfoliovariance subject to diversification constraint, where the diversification is measured by PRQE.This formalization clarifies the investment problem behind the MDP and helps identify thesource of its strong out-of-sample performance relative to other diversified portfolios. Second,using these new formulations, new directions along which the out-of-sample performance ofthe MD strategies can be improved are suggested and it shows that these improvements areeconomically meaningful.
The chapter is organized as follows. Section 6.1 develops the new formulation of the MDindexation based on RQE as the ratio of PRQE to portfolio variance. Section 6.2 presentsan alternative formulation as a variance minimization subject to a diversification constraintmeasured by RQE. Section 6.3 provides an empirical investigation showing that improvementssuggested by the reformulation are economically important. Section 6.4 checks the robustnessof the empirical findings and Section 6.5 concludes.
6.1 Maximum Diversification Meets Rao’s Quadratic Entropy
This section reformulates the problem of the maximum diversification (MD) strategy in thecontext of Rao’s Quadratic Entropy (RQE).
In Section 4.2.1, it has been shown that in the two steps formulation of the MD strategy, thefirst step is the maximization of portfolio RQE with D such that
dij = 1− ρij . (6.1)
In Section 4.2.4, it has been shown that the objective function of the MD strategy, the diver-sification ratio (DR), is equivalent to the portfolio RQE with D such that
dij = (1− ρij)σi√
w>Σ w
σj√w>Σ w
. (6.2)
Here, a new formulation of the DR in terms of portfolio RQE is provided. Recall the formulaof DR
DR(w) =w> σ√w>Σ w
. (6.3)
1 DR is a differential diversification measure because it maximizes the difference between the volatility ofthe portfolio in an imaginary state (in which the correlation between all stocks is 1) and the volatility of thesame portfolio in a real state.
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The square of DR minus 1 gives
DR2(w)− 1 =
(w> σ
)2w>Σ w
− 1. (6.4)
Since (6.4) preserves the preference ordering on the set of long-only portfolios, the MDP canalso be obtained by maximizing DR2 − 1. The development of
(w> σ
)2 and w>Σ w are
(w> σ
)2=
N∑i=1
w2i σ
2i +
N∑i 6=j=1
wiwjσiσj . (6.5)
w>Σ w =N∑i=1
w2i σ
2i +
N∑i 6=j=1
wiwjρijσiσj . (6.6)
Therefore,
DR2(w)− 1 =
∑Ni,j=1(1− ρij)σiσjwiwj
w>Σ w=
w>DMD w
w>Σ w, (6.7)
where DMD = (dMD,ij)Ni,j=1 with
dMD,ij = (1− ρij)σi σj . (6.8)
The numerator w>DMD w is RQE of portfolio w with the dissimilarity matrix DMD. Thedissimilarity dMD,ij measures the difference in terms of correlation and volatility betweenassets i and j. This difference is high when assets have high or moderate volatility, andlow correlation. portfolio RQEDMD
therefore selects moderate to high volatility assets withlow correlation between them. Alternatively, diversification with HDMD
(w) can be viewed astargeted towards the subset of moderate to high volatility assets exhibiting lower correlations.
As a result, one can interpret the ratio (6.7) as a diversification-risk trade-off, with diversifi-cation (the numerator) defined by portfolio RQEDMD
and risk (the denominator) defined byportfolio variance. Since DR2 − 1 and DR represent the same preference ordering on W, theMDP investment problem can be rewritten as
wMD ∈ arg Maxw∈W
w>DMD w
w>Σ w. (6.9)
The formulation in (6.9) reveals that criticisms directed at the MDP may be unjustified:contrary to what Lee (2011) claims, the MDP implicitly solve a well-specified investmentproblem that aims to maximize an absolute portfolio diversification measure normalized by theportfolio variance. As such, and contrary to arguments made in Taliaferro (2012), the MDP’sdesirable properties are probably not coincidental, but rather a result of this objective. Forexample, the MDP is duplication invariant, because the minimum-variance portfolio (VP) i.e.portfolio that minimizes portfolio variance and RQE portfolio are both duplication invariant.The duplication invariance of the VP is proved in Choueifaty et al. (2013).
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Equation (6.9) also reveals that the MD indexation is a particular case of a more generalportfolio allocation problem wherein
wD ∈ arg Maxw∈W
w>D w
w>Σ w, (6.10)
where D is a dissimilarity matrix. This suggests that the out-of-sample performance of theMDP can potentially be improved using an alternative dissimilarity matrix. Section 6.3 pro-vides an empirical illustration that shows this is indeed the case. In the remainder of thischapter, problem (6.9) is referred as the Rao’s Quadratic Entropy ratio problem with dissimi-larity matrix D, or more compactly RQERD. Its optimal portfolio is denoted RQERPD. Thisimplies that the MD strategy and RQERDMD
refer to the same problem.
6.2 An Alternative Formulation
This section rewrites problem (6.9) as one of variance minimization subject to a diversificationconstraint measured by RQE. This alternative formulation of (6.9) facilitates the compari-son between the MDP and other diversified portfolios such as the equally risk contribution(ERCP), the minimum-variance (VP) and the 2-norm constrained minimum-variance (NC2P)(see DeMiguel et al., 2009a; Yanou, 2010; Cazalet et al., 2014a) portfolios; in turn, this willserve: to better identify the source of the out-of-sample performance of the MDP.
Consider the following optimization problem
wD (~D) ∈ arg Minw∈W
w>Σ w (6.11)
s.t w>D w ≥ ~D, (6.12)
where ~D denotes the portfolio manager’s minimum targeted level of diversification. Problem(6.11)-(6.12) can be rewritten as
wD (~D) ∈ arg Minw∈W
w>Σ w (6.13)
s.t w>S(θ) w ≤ θ − ~D, (6.14)
where S(θ) is a similarity matrix induced by D. In the case where D is chosen such asHD(w) is strictly concave, there is θ∗ such as S(θ) is positive definite for all θ ≥ θ∗ (seeSection 5.1.6). Problem (6.13)-(6.14) coincides therefore with the norm-constrained minimum-variance problem without short sales (see DeMiguel et al., 2009a).
It is straightforward to show that for dissimilarity matrix DMD, problems (6.9) and (6.11)-(6.12) are equivalent. In other words, the MD strategy problem is equivalent to (6.11)-(6.12)
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when D = DMD and ~D = ~DMD= HDMD
(wMD
)wMD
(HDMD
(wMD
))∈ arg Min
w∈Ww>Σ w (6.15)
s.t w>DMD w ≥ ~DMD, (6.16)
In what follows, problem (6.11)-(6.12) is referred as the constrained Rao’s Quadratic Entropyproblem or RQECD,~D , and its optimal portfolio is denoted RQECPD,~D . This implies thatthe MD strategy, RQERDMD
and RQECDMD,~DMDall refer to the same problem.
Formulating the MD strategy as RQECDMD,~DMDhas two advantages. The first is that it
helps clarify the fact that, similarly to the ERCP and the NC2P, the MDP is the solution toa minimum-variance problem subject to a diversification constraint. This similitude is madeclear when the different optimization problems underlying these portfolios are compared:
NC2P: wNC2 ∈ arg Minw∈W
w>Σ w (6.17)
s.tN∑
i 6=j=1
wiwj ≥ ν (6.18)
VP: wMV ∈ arg Minw∈W
w>Σ w, (6.19)
ERCP: wERC ∈ arg Minw∈W
w>Σ w (6.20)
s.tN∑i=1
1
Nln
(1/N
wi
)≤ − 1
N
(η +N ln(N)
). (6.21)
The left-hand sides of the diversification constraints (6.18) and (6.21) are respectively theGini-Simpson index and the Kullback-Leibler divergence of the naive portfolio relative to theportfolio w.
Comparing problems (6.17)-(6.18), (6.19) and (6.20)-(6.21) to RQECDMD,~DMDi.e. problem
(6.15)-(6.16), it is observable that the difference lies in the presence of RQEDMDin the diversi-
fication constraint (6.16). Therefore, the source of any difference in volatility, turnover, draw-down risk, concentration and Sharpe ratio (see Choueifaty and Coignard, 2008; Choueifatyet al., 2013; Demey et al., 2010; Chow et al., 2011; Leote et al., 2012; Clarke et al., 2013)between the MDP on the one hand and the NC2P, the VP and the ERCP on the other orig-inates from the portfolio RQEDMD
diversification constraint, which is characterized by thedissimilarity matrix DMD and the minimum targeted level of diversification ~DMD
.
It follows that if the diversification constraint (6.16) is supported by the data, a significantportion of the out-of-sample performance of the MDP arises from the returns to diversificationas measured by RQE.2 This result is consistent with Choueifaty and Coignard (2008):
2For example, the diversification constraint (6.16) is supported by the data when, as discussed in Propo-sition 4.3.3, each asset’s volatility is proportional to its expected excess return.
124
Furthermore, if we continuously rebalance the Most-Diversified Portfolio, and be-cause it is a market cap-independent methodology, the Most-Diversified Portfolioshould get a significant part of the benefits from diversification returns when com-pared to a pure buy-and-hold strategy...
Even when the diversification constraint (6.16) is not supported by the data, imposing it canhelp reduce the upward-biased estimation error of the correlation matrix through a shrinkage-like effect and therefore improve the out-of-sample performance of the MDP. 3 To see this,using (6.15)-(6.16), it is straightforward to demonstrate that the MDP solves the modifiedminimum-variance problem
wMD ∈ arg Minw∈W
w>Σ w,
where the true covariance matrix in (6.19) has been replaced by the matrix Σ defined as
Σ = σ>[ρ− ν(11> − ρ)
]σ,
with ν the Lagrange multiplier associated to the constraint (6.16) and ρ the correlation matrix.This shows, in the spirit of Ma and Jagannathan (2003), that the imposed RQE diversificationconstraint can be interpreted as a shrinking of the correlation matrix toward −11>, whichmay help reduce potentially upward-biased estimates. This reduction in turn means thateven when the diversification constraint is incorrectly specified, the MDP has the potential toproduce better out-of-sample performance. This advantage depends on the trade-off betweenthe reduction in sampling errors and the specification errors generated by the diversificationconstraint.
An additional advantage of the constrained RQECDMD,~DMDformulation is that one can
potentially improve the out-of-sample performance of the MDP, not only by replacing thedissimilarity matrix DMD by a more suitable one, but also by choosing a more appropriatelevel of diversification ~D, different from ~DMD
.
In summary, the new formulation (6.15)-(6.16) reveals that the superior out-of-sample perfor-mance of the MDP relative to the VP, the ERCP and the NC2P are due either to the returnsto diversification when the diversification constraint that is employed is correct (supported bythe data), or to the reduction of the upward-biased estimates in the correlation matrix when itis not. This leads us to the general conclusion that the funds under the MD indexation man-agement should not be considered systematically at risk. Moreover, when the diversificationconstraint is incorrect (i.e. not supported by the data) and the specification errors exceed thesample errors, the new formulations reveal that one can potentially improve the MDP’s out-of-sample performance by changing the dissimilarity matrix DMD and the minimum targetedlevel of diversification ~DMD
behind the MDP.3For example, the diversification constraint is not supported by the data when the data exhibits the low
volatility anomaly.
125
6.3 Improving MDP Out-of-Sample Performance
As demonstrated above, the MD strategy represents a special case of the more general un-constrained (RQERD,D = DMD) and constrained (RQECD,~D ,D = DMD and ~D = ~DMD
)problems based on portfolio Rao’s Quadratic Entropy. This implies that the approach underly-ing the MD strategy could potentially be improved through the use of alternative specificationfor the dissimilarity matrix D and the minimum targeted level of diversification ~D.
This section provides an empirical illustration that shows such improvements are indeedpresent and economically meaningful. To do so, the simulated out-of-sample performanceof five portfolios with those of the MDP are compared. Each portfolio is obtained from prob-lem (6.10) using different dissimilarity matrix D, or from problem (6.11)-(6.12), using differentdissimilarity matrix D and minimum targeted diversification level ~D.
6.3.1 Methodology
Three different procedures are considered to construct the alternative portfolios. The first(Scenario I, top panel of Table 6.1) looks at the effect of changing the minimum level ofdiversification ~DMD
while keeping the dissimilarity matrix of the MDP (DMD). FollowingDeMiguel et al. (2009a), ~D is chosen to maximize the portfolio return in the last period of theestimation window and the value of ~D drawn from this procedure is denoted ~D in Table 6.1.This scenario is implemented using the constrained problem RQECD,~D .
The second procedure (Scenario II, middle panel of Table 6.1) looks at the effect of changing thedissimilarity matrix using the unconstrained problem RQERD. Three dissimilarity matricesother than the benchmark DMD are considered: DGS , DMDσ+ and DMD
σ+/− . First, thematrix DGS assumes that assets are equi-dissimilar, so that the portfolio RQE based on DGS
is equivalent to the Gini-Simpson index i.e.
DGS = 1 1> − I. (6.22)
Next, the matrix DMDσ+ is defined by replacing the volatility in DMD by the upside-risk i.e.
dMDσ+ ,ij =(
1− ρij)
E(max(0, Ri)3)E(max(0, Rj)
3) (6.23)
Portfolio RQEDMDσ+
thus selects moderate to high upside-risk assets that are less correlated.Finally, the matrix DMD
σ+/− uses the ratio of upside to downside risk to replace the volatilityin DMD i.e.
dMDσ+ ,ij =
(1− ρij
)E(max(0, Ri)
3)
E(min(0, Ri)2)
E(max(0, Rj)3)
E(min(0, Rj)2)
.
(6.24)
Portfolio RQEDMDσ+/−
therefore selects assets with moderate to high ratios of upside todownside-risk which are less correlated.
126
Table 6.1 – List of Portfolios Considered
Dissimilarity matrix (D) Target Level of Diversification (~) Portfolios
Scenario I
dMD,ij = (1− ρij)σi σj ~DMDMDP†(RQECPDMD,~DMD
)
dMD,ij = (1− ρij)σi σj ~r RQECPDMD,~r
Scenario II
dMD,ij = (1− ρij)σi σj - MDP†(RQERPDMD)
dGS,ij = 1− δij - RQERPDGS
dMDσ+ ,ij =(
1− ρij)
E(max(0, Ri)3)E(max(0, Rj)
3) - RQERPDMDσ+
dMDσ+ ,ij =
(1− ρij
)E(max(0,Ri)
3)E(min(0,Ri)2)
E(max(0,Rj)3)
E(min(0,Rj)2)- RQERPDMD
σ+/−
Scenario III
dMD,ij = (1− ρij)σi σj ~r RQECP†DMD,~rdGS,ij = 1− δij ~r RQECPDGS ,~rdMDσ+ ,ij =
(1− ρij
)E(max(0, Ri)
3)E(max(0, Rj)3) ~r RQECPDMD
σ+,~r
dMDσ+ ,ij =
(1− ρij
)E(max(0,Ri)
3)E(min(0,Ri)2)
E(max(0,Rj)3)
E(min(0,Rj)2)~r RQECPDMD
σ+/− ,~r
Notes. † : benchmark portfolio.
127
Finally, a third procedure (Scenario III, bottom panel of Table 6.1) analyzes change to dis-similarity matrix DMD using the constrained problem RQECD,~D . The dissimilarity matricesconsidered remain DGS , DMDσ+ and DMD
σ+/− and ~D is chosen following DeMiguel et al.(2009a). The difference between the scenarios II and III thus lays in the specification of theminimum targeted level of diversification ~D: for scenario II, ~D = HD(wD) is specified im-plicitly such the diversification per unity of risk is maximized, while in scenario III, ~D isspecified explicitly following DeMiguel et al. (2009a). Identifying the best specification for agiven dissimilarity matrix D requires comparing RQECP
D,~Dwith RQERPD.
The empirical work is based on Fama and French’s 25 monthly portfolios returns sorted bysize and book-to-market covering the period running from July 1963 to December 2013. Thedataset is available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. Following DeMiguel et al. (2009a), portfolio returns are constructed using arolling sample procedure with an estimation window length t = 120 and monthly rebalancing.The procedure generates a time series of T − t − 1 monthly out-of-sample returns : rjt+1 =
wj>
t rt+1, t = t, ..., T − 1, where rt+1 is the vector of period t + 1 asset returns and wjt the
portfolio weight vector at time t for strategy j. Using this time series, performance is evaluatedby reporting each portfolio’s cumulative return (CRj), its variance,
((σj)2) its Sharpe ratio
(SRj) and, finally, its turnover (TRNj) (see Table 5.2). The impact of turnover on performancemetrics under non-zero transaction costs are also evaluated using the following portfolio returnnet of transaction cost
rjt+1 =
(1− κ
N∑i=1
|wji,t+1 − wji,t+|
)(1 + wj>
t rt+1)− 1, (6.25)
where κ is the transaction cost parameter (see DeMiguel et al., 2009b). Performance metricsare computed using portfolio returns net of proportional transaction costs (6.25) for differentvalues of κ ranging from 0.0001 to 0.005. The range 0.0001 to 0.005 is chosen followingBalduzzi and Lynch (1999, pp. 63) who argue that small investors probably face a proportionaltransaction cost closer to the 0.005, while large investors likely face costs greater than the0.0001. To test the out-of-sample variance and Sharpe ratio differences between two givenportfolios, the studentized circular block bootstrap proposed in Ledoit and Wolf (2011) andLedoit and Wolf (2008) are used respectively. The block length is set equal to 5 and iterationsequal to 1000 to compute the two-sided p-value. The null hypothesis is : H0 : 2 ln
(σi)−
2 ln(σj)
= 0 for the variance test, and H0 : µi/σi − µj/σj = 0 for the Sharpe ratio test. Thecode is available at http://www.econ.uzh.ch/en/faculty/wolf/publications.
6.3.2 Results
6.3.2.1 Scenario I
Figure 6.1 depicts the out-of-sample cumulative returns of the MDP and RQECPDMD,~DMD
.
128
It shows that RQECPDMD,~DMD
is never dominated by the MDP. One dollar invested inRQECP
DMD,~DMDin July 1973 yields 190 dollars in December 2013, while the same investment
in the MDP yields 81 dollars.
Figure 6.1 – Scenario I: Portfolio Cumulative Returns
Juil-73 Juil-83 Juil-93 Juil-03 Dec-13
0
20
40
60
80
100
120
140
160
180
200
Periods
CR
MDP†(RQECPDMD,~DMD)
RQECPDMD,~DMD
Note. † : benchmark portfolio.
Table 6.2 compares the performance of both portfolios in terms of variance, Sharpe ratio andturnover. It reveals that the MDP has essentially the same variance as RQECP
DMD,~DMD,
but significantly lower Sharpe ratio. From these evidences, one can conclude that the implicittarget level of diversification of the MDP (~DMD
) may not always be “optimal ”.
Table 6.2 also shows that RQECPDMD,~DMD
implies a significant more active managementpolicy, since turnover is higher. Figure 6.2 evaluates the impact of this higher turnover forour performance metrics when transaction costs are non-zero. Only results for Sharpe ratioand cumulative returns at the end-of-period are reported, but full results are available. It isobservable that RQECP
DMD,~DMDalways outperforms the MDP for investors facing small to
moderate transaction costs, like moderate and large funds.
The source of this difference of performance is the over-diversification. As can be observed
129
Table 6.2 – Scenario I: Performance of Portfolios
Portfolios σ2 SR TRN
MDP†(
RQECPDMD,~DMD
)0.00226 0.21537 0.06337
RQECPDMD,~DMD
0.00227 0.25329∗ 0.75830
† : benchmark portfolio.Significance Level: ∗= p-value ≤ 5% means that performancesof RQECPDMD,~DMD
is significantly different from that of thebenchmark.
Figure 6.2 – Scenario I: Performance Metrics Depending on Transaction Costs Parameters κ
0 2 4 60.15
0.2
0.25
0.3
0.35
0.4
κ × 1000
SR
Panal A: Sharpe ratio
MDP†(
RQECPDMD,~DMD
)RQECPDMD,~DMD
0 2 4 6
50
100
150
200
κ × 1000
CR
Panal B: Cumulative return (end-of-period)
MDP†(
RQECPDMD,~DMD
)RQECPDMD,~DMD
Note. † : benchmark portfolio.
on Figure 6.3, RQECPDMD,~DMD
is less diversified than the MDP. Consequently the MDP isover-diversified.
6.3.2.2 Scenario II
Figure 6.4 depicts the cumulative returns of the MDP, RQERPDMDσ+
, RQERPDMDσ+/−
andRQERPDGS
. It shows that RQERPDGSand RQERPDMD
σ+/−are never dominated by the
MDP, while the performance of RQERPDMDσ+
broadly coincides with that of the MDP.
One dollar invested in RQERPDGS
(respectivelyRQERPDMD
σ+/−
)in July 1973 yields, in
December 2013, 1.6 (respectively 1.7) times the cumulative return of one dollar invested inthe MDP.
Table 6.3 reports the portfolios performance in terms of variance, Sharpe ratio and turnover.
130
Figure 6.3 – Scenario I: Diversification Level Comparison
Juil-73 Juil-83 Juil-93 Juil-03 Dec-13
0
1
2
3
4·10−3
Periods
HDMD
( wMD) an
dH
DMD
( wDMD
( ~ DMD
))
MDP†(RQECPDMD,~DMD)
RQECPDMD,~DMD
Juil-73 Juil-83 Juil-93 Juil-03 Dec-13
0
0.1
0.2
0.3
0.4
PeriodsCum
ulativeH
DMD
( wMD) an
dH
DMD
( wDMD
( ~ DMD
))
MDP†(RQECPDMD,~DMD)
RQECPDMD,~DMD
Note. † : benchmark portfolio.
It shows that RQERPDMDσ+
offers essentially the same performance in terms of Sharpe ratioas the MDP, but with a significantly higher variance. RQERPDMD
σ+/−offers essentially the
same performance in terms of variance as the MDP, but with a significantly higher Sharperatio. RQERPDGS
has significantly lower variance and higher Sharpe ratio than the MDP.Again, these evidences suggest that the implicit dissimilarity matrix of the MDP is not always“optimal ”.
Table 6.3 – Scenario II: Performance of Portfolios
Portfolios σ2 SR TRN
MDP†(RQERPDMD) 0.00226 0.21537 0.06337
RQERPDGS0.00191∗ 0.25194∗ 0.03992
RQERPDMDσ+
0.00251∗ 0.21136 0.08780RQERPDMD
σ+/−0.00215 0.24239∗ 0.09046
Note. † : benchmark portfolio.Significance Level: ∗= p-value ≤ 5% means that performancesof RQECPDGS
, RQECPDMDσ+
and RQECPDMDσ+/−
is signif-
icantly different from that of the benchmark.
In addition, Table 6.3 also shows that RQERPDGSand RQERPDMD
σ+/−have respectively
lower and higher turnover than the MDP. The impact of the higher turnover of RQERPDMDσ+/−
131
Figure 6.4 – Scenario II: Portfolio Cumulative Returns
Juil-73 Juil-83 Juil-93 Juil-03 Dec-13
0
20
40
60
80
100
120
140
Periods
CR
MDP†(RQERPDMD
)RQERPDGS
RQERPDMDσ+
RQERPDMDσ+/−
Note. † : benchmark portfolio.
is evaluated when transaction costs are present. Results (Figure 6.5) show that RQERPDMDσ+/−
always outperforms the MDP. This means that the presence of non-zero transaction costs isnot sufficient to overturn our results.
6.3.2.3 Scenario III
Figure 6.6 depicts the cumulative returns of RQECPDMD,~DMD
, RQECPDGS ,~DGS
, RQECPDMD
σ+,~DMD
σ+
and RQECPDMD
σ+/− ,~DMDσ+/−
. It shows that RQECPDMD,~DMD
is dominated by RQECPDGS ,~DGS
,
RQECPDMD
σ+,~DMD
σ+
and RQECPDMD
σ+/− ,~DMDσ+/−
: One dollar invested in these three
strategies, in July 1973, yields 254, 312 dollars and 225 dollars respectively in December 2013.By contrast, one dollar invested in RQECP
DMD,~DMDat the same period delivers 197 dollars.
Table 6.4 shows that RQECPDMD,~DMD
has a significantly higher variance than RQECPDGS ,~DGS
and RQECPDMD
σ+/− ,~DMDσ+/−
but a lower one than RQECPDMD
σ+,~DMD
σ+
. In terms of
Sharpe ratio, RQECPDMD,~DMD
is slightly dominated by RQECPDGS ,~DGS
, RQECPDMD
σ+,~DMD
σ+
132
Figure 6.5 – Scenario II: Performance Metrics Depending on Transaction Costs Parameters κ
0 2 4 60.15
0.2
0.25
0.3
0.35
0.4
κ × 1000
SR
Panel A: Sharpe ratio
MDP†(RQERPDMD
)RQERPDGS
RQERPDMDσ+/−
0 2 4 6
100
150
200
κ × 1000
CR
Panal B: Cumulative return (end-of-period)
MDP†(RQERPDMD
)RQERPDGS
RQERPDMDσ+/−
Note. † : benchmark portfolio.
and RQECPDMD
σ+/− ,~DMDσ+/−
, but the difference is not significant. Again these lead us to
conclude that the implicit dissimilarity matrix of the MDP is not always “optimal”.
Table 6.4 – Scenario III: Performance of Portfolios
Portfolios σ2 SR TRN
RQECPDMD,~DMD
0.00227 0.25329 0.75830RQECP
DGS ,~DGS0.00211∗ 0.27229 0.80755
RQECPDMD
σ+,~DMD
σ+
0.00240∗ 0.26724 0.89158
RQECPDMD
σ+/− ,~DMDσ+/−
0.00209∗ 0.27303 0.72992
Note. † : benchmark portfolio.Significance Level: ∗= p-value ≤ 5% means that perfor-mances of RQECPDGS ,~DGS
, RQECPDMDσ+
,~DMDσ+
, and
RQECPDMDσ+/−
,~DMDσ+/−
is significantly different from that of the
benchmark.
Table 6.4 also shows that RQECPDMD
σ+/− ,~DMDσ+/−
has lower turnover than RQECPDMD,~DMD
,
while RQECPDGS ,~DGS
and RQECPDMD
σ+,~DMD
σ+
have higher turnover. The impact of the
higher turnover is evaluated when transaction costs are present in Figure 6.7, which shows thatRQECP
DGS ,~DGS, RQECP
DMDσ+
,~DMDσ+
and RQECPDMD
σ+/− ,~DMDσ+/−
always outperform
RQECPDMD,~DMD
.
133
Figure 6.6 – Scenario III: Portfolio Cumulative Returns
Juil-73 Juil-83 Juil-93 Juil-03 Dec-13
0
50
100
150
200
250
300
Periods
CR
RQECP†DMD,~DMDRQECPDGS ,~DGS
RQECPDMDσ+
,~DMDσ+RQECPDMD
σ+/−,~DMD
σ+/−
Note. † : benchmark portfolio.
In short, while the MD indexation at the basis of the MDP is equivalent to the problemsRQERDMD
and RQECDMD,~DMD
, the results show that one can significantly improve its out-of-sample performance using different dissimilarity matrices and/or by changing the minimumtargeted level of diversification. Note that the potential improvements are particularly impor-tant in terms of Sharpe ratio when asset excess returns are not perfectly proportional to itsvolatility, and low otherwise. The potential improvements can also be important in terms ofcumulative returns when asset returns are not normally distributed.
134
Figure 6.7 – Scenario III: Performance Metrics Depending on Transaction Costs Parametersκ
0 2 4 60.15
0.2
0.25
0.3
0.35
0.4
κ × 1000
SR
Panel A: Sharpe ratio
RQECP†DMD,~DMDRQECPDGS ,~DGS
RQECPDMDσ+
,~DMDσ+RQECPDMD
σ+/−,~DMD
σ+/−
0 2 4 6
100
200
300
κ × 1000C
R
Panel B: Cumulative return (end-of-period)
RQECP†DMD,~DMDRQECPDGS ,~DGS
RQECPDMDσ+
,~DMDσ+RQECPDMD
σ+/−,~DMD
σ+/−
† : benchmark portfolio.
6.4 Robustness Checks
In this section, the robustness of the empirical findings is checked by considering alternativeportfolio risk measures, estimation window, and covariance matrix estimation.
6.4.1 Portfolio Risk
To check robustness to portfolio risk, which was equated to portfolio variance or volatil-ity, alternative portfolio risk measures are considered such as: semivariance (SV), value atrisk (VaR), conditional value at risk or expected shortfall (CVaR) and maximum drawdown(MDD). The semivariance risk measure is defined as follows
SVj(w) =1
T − t− 1
T−1∑t=t
[min(0, rjt+1 − µ
j)]2. (6.26)
For the definition of VaR, CVaR and MDD see Table 5.2.
Table 6.5 reports the value of these different risk measures for our set of considered portfo-lios. In Scenario I, it is observable that RQECP
DMD,~DMDdominates the MDP for all risk
measures. In Scenario II, RQERPDMDσ+
is dominated by the MDP, which is itself dominatedby RQERPDGS
and RQERPDMDσ+/−
for all risk measures. In Scenario III, RQECPDGS ,~DGS
dominates the MDP for all risk measures. RQECPDMD
σ+/− ,~DMDσ+/−
dominates the MDP
for all risk measures, except for MDD. RQECPDMD
σ+,~DMD
σ+
dominates the MDP only for
VaR. As a result, our findings remain robust in terms of portfolio risk.
135
Table 6.5 – Portfolio Risks
Portfolios SV VaR0.05 CVaR0.05 MDD
Scenario I MDP†(
RQECDMD,~DMD
)0.00130 -0.07151 -0.10821 0.58557
RQECPDMD,~DMD
0.00122 -0.06478 -0.10116 0.54941
Scenario IIMDP†(RQERPDMD
) 0.00130 -0.07151 -0.10821 0.58557RQERPDGS
0.00108 -0.06039 -0.09806 0.52827RQERPDMD
σ+0.00144 -0.07789 -0.11410 0.58808
RQERPDMDσ+/−
0.00124 -0.06913 -0.10538 0.57478
Scenario IIIRQECP†
DMD,~DMD
0.00122 -0.06478 -0.10116 0.54941
RQECPDGS ,~DGS
0.00116 -0.06334 -0.09837 0.52396RQECP
DMDσ+
,~DMDσ+
0.00128 -0.06231 -0.10225 0.56233
RQECPDMD
σ+/− ,~DMDσ+/−
0.00115 -0.06080 -0.09786 0.55067
Note. † : benchmark portfolio.
6.4.2 Estimation Window
The empirical analysis assumes an estimation window of 120 months. To check whether theresults are driven by this assumption, alternative window lengths of 60 and 240 months areconsidered. The resulting Sharpe ratios are reported in Table 6.6. It is observable that, fort = 240 and t = 60, the dissimilarity matrix behind the MDP (DMD) is “sub-optimal ”, but theminimum level of diversification (~DMD
) seems “optimal ”. In short, the findings are mostlyrobust for t = 240 and t = 60.
6.4.3 Covariance Matrix Estimation
The results are based on the sample covariance matrix. To check whether an alternativespecification would be more appropriate, a covariance matrix is computed using Ledoit andWolf (2003)’s shrinkage estimation, where the shrinkage target is obtained from a one-factormodel where the factor is equal to the cross-sectional average of all the random variables.Computations are made with the code available at https://r-forge.r-project.org/scm/
viewvc.php/pkg/ExpectedReturns/man/?root=expectedreturns&pathrev=2. As the infor-mation in Table 6.6 indicates, the results are unchanged by this. In Scenario I, the MDP stillhas the lower out-of-sample Sharpe ratio. In Scenario II, we can find a dissimilarity matrixthat out-performs DMD (DGS and DMD
σ+/− ). In short, the results are robust to covariancematrix estimation.
136
Table 6.6 – Sharpe Ratios
Rolling Window Robustness (Σ)
Portfolios t = 60 t = 240 t = 120
Scenario I MDP†(
RQECPDMD,~DMD
)0.17171 0.20072 0.21859
RQECPDMD,~DMD
0.20247 0.22940 0.25765∗
Scenario IIMDP†
(RQERPDMD
)0.17171 0.20072 0.21859
RQERPDGS0.22667∗ 0.25425∗ 0.25164∗
RQERPDMDσ+
0.14337 0.17806 0.21323RQERPDMD
σ+/−0.21550 0.22132 0.24453∗
Scenario IIIRQECP†
DMD,~DMD
0.20247 0.22940 0.25765
RQECPDGS ,~DGS
0.22551 0.25210∗ 0.27229RQECP
DMDσ+
,~DMDσ+
0.18535 0.17961 0.27001
RQECPDMD
σ+/− ,~DMDσ+/−
0.22410 0.21184 0.27387
Note. Significance Level: ∗= p-value ≤ 5%.† : benchmark portfolio.
6.5 Summary
In this chapter, new formulations for the maximum diversification (MD) strategy using port-folio Rao’s Quadratic Entropy (RQE) were proposed. It has shown that the MD strategy isequivalent to the maximization of the ratio of portfolio RQE (PRQE) to portfolio variance or,said differently, the minimization of portfolio variance subject to a diversification constraint,where the diversification is measured by RQE. Contrary to Lee (2011), these new formulationsreveal that the MD strategy solves a well-defined investment objective and the funds underits management is not systematically at risk as suggested by Taliaferro (2012).
Moreover, the new formulations also suggest two new directions to improve the out-of-sampleperformance of the MDP: it has shown that the MDP’s out-of-sample performance can besignificantly improved by changing its dissimilarity matrix or/and its minimum level of di-versification. The robustness analysis has shown that these results are not dependent on riskmeasure, turnover, estimation window and covariance matrix estimation.
The portfolio allocation problem (6.11)-(6.12) constitutes therefore a very credible alternativeto the MD strategy, particularly when asset excess expected returns are not perfectly propor-tional to its volatility or when asset returns are not normally distributed. Additional researchcould compare the out-of-sample performance of the optimal portfolio of problem (6.11)-(6.12)across different empirical datasets for various dissimilarity matrices and minimum level of di-versification specifications.
137
Chapter 7
Rao’s Quadratic Entropy AndMean-Variance Models
This second chapter focuses on the relationship between portfolio RQE (PRQE) and themean-variance (MV) models (Markowitz, 1952; Tobin, 1958; Sharpe, 1964). As outlined inSection 2.1.2.2, Markowitz (1952)’s model is considered as the first mathematical formulationof the idea of diversification of investments. However, Markowitz (1952) does not provide aspecific measure of portfolio diversification. Moreover, Fernholz (2010) claims that there is noa specific measure of portfolio diversification in the MV model. In Section 3.2, exploiting thedefinition of the preference for the diversification in the expected utility theory, it has beendemonstrated that, contrary to Fernholz (2010), there is a specific diversification measure inthe MV model and this measure is the diversification return
Dr(w) = w>σ2 −w>Σ w. (7.1)
In Proposition 4.2.1, it has been shown that when D is defined such that
dij = σ2i + σ2
j − 2σij , (7.2)
PRQE coincides with the diversification return
HD(w) = Dr(w). (7.3)
Therefore, it has been deduced (see Corollary 4.2.1) that the diversification in the MV modelis RQE diversification.
These results have two principal economic implications. The first is that one can derive therelationship between risk and diversification in the MV model without any restriction. Thesecond important implication, which is the principal interest of this chapter, is that one canrewrite the MV model in terms of PRQE, providing therefore a novel and useful formulationof the MV utility function.
138
In Section 7.1, the new formulation of the MV utility function in terms of RQE is derived. Thisnew formulation is called RQE formulation. Next, the implications of this new formulationin terms of the MV optimal portfolios are discussed. Section 7.2 discusses the implicationsin terms of the capital asset pricing model. In Section 7.3, the implications in terms of thegeneralization of the MV model are presented. Section 7.4 summarizes the chapter.
7.1 Rao Meets Markowitz
Consider Markowitz (1952)’s mean-variance utility function
UMV (w) = w>µ− 1
τw>Σ w. (7.4)
To make the link with PRQE, one has to transform and rewrite (7.4) as
UMV (w) = w>µ− 1
τw>σ2 +
1
τ
(w>σ2 −w>Σ w
). (7.5)
From (7.1) and (7.3), the MV utility function can be rewritten equivalently as a trade-offbetween return, risk and diversification
UMV (w) = w>µ− 1
τw>σ2 +
1
τHD(w), (7.6)
where the return is measured by portfolio expected return, w>µ, the risk by assets weightedaverage variance, w>σ2, and the diversification by PRQE, HD(w). The formulation (7.6)is called RQE formulation. From (7.6), two important observations can be made. The firstobservation is that the risk aversion coefficient 1
τ also plays the role of the coefficient of thepreference for diversification in the MV model. The second observation is that the portfoliovariance, w>Σ w, represents the undiversified risk and portfolio RQE the diversified risk. Thesum of undiversified and diversified risks gives portfolio risk, which is assets weighted averagevariance
w>σ2 = HD(w) + w>Σ w. (7.7)
In the case where the risk-free asset is available, (7.6) becomes
UMV (w) = (1− wf )w>µ+ wf Rf −1
τ(1− wf )2w>σ2 +
1
τ(1− wf)2HD(w). (7.8)
The formulations (7.6) and (7.8) provide new insight of the MV model and offers new directionsfor its improvement. In the next section, economic implications of the formulations (7.6) and(7.8) in terms of the MV optimal portfolios (MVP) are discussed.
7.1.1 Link Between MVP and RQEP
From (7.6), the MV optimization problem can be reformulated equivalently as
maxw∈W
w>µ− 1
τw>σ2 +
1
τHD(w). (7.9)
139
The MV optimal portfolios (MVP) therefore can be related to the RQE optimal portfolio(RQEP). In what follows, this relationship is derived in the case where short sales are allowed.
Assume that assets are risky. Using Lagrange multiplier, the problem (7.9) can be rewrittenas
maxw∈W−
w>µ− τ w>σ2 + τ HD(w)− ν(1−w> 1), (7.10)
where ν is the Lagrange multiplier of the investment constraint and τ = 1τ is a risk aversion
coefficient. The first order conditions of (7.10) are
µ− τσ2 + τ DwMV + ν 1 = 0, (7.11)
1− 1>wMV = 0. (7.12)
From (7.11),
wMV = −D−1µ
τ− νD−11
τ+ D−1σ2. (7.13)
Substituting (7.13) in (7.12) yields
1 = −1>D−1µ
τ− ν 1>D−11
τ+ 1>D−1σ2. (7.14)
Define
e =µ>D−11 = 1>D−1µ (7.15)
f =1>D−11 (7.16)
g =(σ2)>D−11 = 1>D−1σ2 (7.17)
From (7.14) and (7.15)-(7.17),
ν =τ g − e− τ
f. (7.18)
Reporting ν from (7.18) into (7.13) yields
wMV = −D−1µ
τ−(τ g − e− τ
f
)D−11
τ+ D−1σ2. (7.19)
Since wRQE = D−111>D−11
(see Chapter 5),
wMV = −D−1µ
τ+ D−1σ2 −
(τ g − e− τ
)τ
wRQE . (7.20)
Denote wTG1 = D−1µ1>D−1µ
and wTG2 = D−1σ2
1>D−1σ2 .
Proposition 7.1.1. Portfolios wTG1 and wTG2 are solutions of the following problems re-spectively
maxw∈W−
w>µ+ τ1HD(w), (7.21)
maxw∈W−
−w>σ2 + τ2HD(w), (7.22)
140
with
τ1 = −e, (7.23)
τ2 = g. (7.24)
Proof (of Proposition 7.1.1). Consider the problem (7.21). The first order conditionsare
µ+ τ1 DwTG1 + ν 1 = 0 (7.25)
1− 1>wTG1 = 0. (7.26)
From (7.25)
wTG1 = −D−1µ
τ1
− ν D−11
τ1
. (7.27)
Substituting (7.27) in (7.26) yields
1 = − e
τ1
−ν f
τ1
. (7.28)
Equation (7.28) implies that
ν = −e+ τ1
f. (7.29)
Combining (7.27) and (7.29), one has
wTG1 = −D−1µ
τ1
+(e+ τ1)
f
D−11
τ1
(7.30)
Suppose that τ1 = −e. Then
wTG1 =D−1µ
e=
D−1µ
1>D−1µ(7.31)
The result follows. Following the same steps, it is straightforward to verify that wTG2 issolution of problem (7.22) with τ2 = g.
Portfolio wMV can therefore be rewritten as follows
wMV =
(− eτ
)wTG1 + gwTG2 +
(−τ g + e+ τ
)τ
wRQE . (7.32)
It follows that the expected return of the MVP is linearly related to that of RQEP
µ(wMV
)=
(− eτ
)µ(wTG1
)+ g µ
(wTG2
)+
(−τ g + e+ τ
)τ
µ(wRQE
). (7.33)
141
Also, the variance of the MVP and that of RQEP are related as follows
σ2(wMV
)=
(− eτ
)2
σ2(wTG1
)+ g2 σ2
(wTG2
)+
(−τ g + e+ τ
)2τ2
σ2(wRQE
)+ 2
(− eτ
)gCov
(R(wTG1
), R(wTG2
))+ 2
(− eτ
) (−τ g + e+ τ)
τCov
(R(wTG1
), R(wRQE
))(7.34)
+ 2g
(−τ g + e+ τ
)τ
Cov(R(wTG2
), R(wRQE
)).
In the case where a risk-free asset is available, it is well-known that
wf = 1−µ(wTG
)−Rf
2τσ2 (wTG), (7.35)
where wTG is the tangent portfolio obtained from the MV problem setting τ = b2 , with
b = µ>Σ 1. Thus, the risk-free asset weight wf can be expressed in terms of RQEP ex-pected returns and variance substituting µ
(wMV
)from (7.33) and σ2
(wMV
)from (7.34)
into (7.35) and setting τ = b2 . The above results are summarized in the following proposition
and corollary.
Proposition 7.1.2 (RQEP and MVP Relationship). Assume that assets are risky andshort sales are unrestricted. Then
wMV =
(− eτ
)wTG1 + gwTG2 +
(−τ g + e+ τ
)b2
wRQE (7.36)
with wTG1 = D−1µ1>D−1µ
and wTG2 = D−1σ2
1>D−1σ2 . When a risk-free asset is available, wMV
becomes
wTG =
(− e
b2
)wTG1 + gwTG2 +
(− b
2 g + e+ b2
)τ
wRQE (7.37)
and
wf = 1−µ(wTG
)−Rf
bσ2 (wTG)(7.38)
with
b =µ>Σ 1
e =µ>D−11 = 1>D−1µ
f =1>D−11
g =(σ2)>D−11 = 1>D−1σ2
142
Corollary 7.1.1 (µ(wRQE
), µ
(wMV
), σ2
(wRQE
)and σ2
(wMV
)Relationships). Assume
that assets are risky and short sales are unrestricted. Then
µ(wMV
)=
(− eτ
)µ(wTG1
)+ g µ
(wTG2
)+
(−τ g + e+ τ
)τ
µ(wRQE
)(7.39)
σ2(wMV
)=
(− eτ
)2
σ2(wTG1
)+ g2 σ2
(wTG2
)+
(−τ g + e+ τ
)2τ2
σ2(wRQE
)+ 2
(− eτ
)gCov
(R(wTG1
), R(wTG2
))+ 2
(− eτ
) (−τ g + e+ τ)
τCov
(R(wTG1
), R(wRQE
))(7.40)
+ 2g
(−τ g + e+ τ
)τ
Cov(R(wTG2
), R(wRQE
)).
When a risk-free asset is available, µ(wMV
)and σ2
(wMV
)become
µ(wMV
)= wf Rf +
(− eτ
)µ(wTG1
)+ g µ
(wTG2
)+
(−τ g + e+ τ
)τ
µ(wRQE
)(7.41)
σ2(wMV
)= (1− wf )2
((− eτ
)2
σ2(wTG1
)+ g2 σ2
(wTG2
)+
(−τ g + e+ τ
)2τ2
σ2(wRQE
)+ 2
(− eτ
)gCov
(R(wTG1
), R(wTG2
))+ 2
(− eτ
) (−τ g + e+ τ)
τCov
(R(wTG1
), R(wRQE
))(7.42)
+ 2g(−τ g + e+ τ)
τCov
(R(wTG2
), R(wRQE
)))with τ = b
2 .
The formulations (7.6) and (7.8) also imply that the MV frontier portfolio can be plottedin three-dimensional space
(σ2, HD, µ
). Figure 7.1 provides an illustration in the case where
assets are risky and short sales are allowed. Since the analytical expression of the MV portfoliofrontier(Equation (2.16)) in space
(σ2, µ
)is
σ2(wMV
)=c(µ(wMV
))2 − 2b µ(wMV
)+ a
d, (7.43)
the analytical expression of the MV portfolio frontier in space(σ2, HD, µ
)is therefore the
intersection of the curve of equation
σ2(wMV
)−HD
(wMV
)=c(µ(wMV
))2 − 2b µ(wMV
)+ a
d(7.44)
and the plane orthogonal to the plane µ = 0 at line HD
(wMV
)= −σ2
(wMV
).
143
Figure 7.1 – Mean-Variance Portfolio Frontier in the Space(σ2, HD, µ
)(when asset are risky
and short sales are allowed)
100
200
−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1
8
8.5
9
9.5
10
σ2
µ
−HD
σ2
7.1.2 Sensitivity Analysis
The problem (7.9) also suggests that the sensitivity of the MVP can be performed in termsµ, σ as usual but also in terms of D. From Proposition 5.1.9,
HD(w) =θ
2−HS(θ)(w). (7.45)
As mentioned in Section 5.1.6, there is θ∗ such that for all θ ≥ θ∗, S(θ) is positive definiteif D is conditional strictly negative definite(CSND). Replacing HD(w) from (7.45) into (7.9),the MV optimization problem becomes
maxw∈W
w>µ− 1
τw>σ2 − 1
τHS(θ)(w). (7.46)
Following Guigues (2011, Theorem 3.1), the following result is obtained.
Proposition 7.1.3 (MV sensitivity). Let Dk, k = 1, 2 be two CSND dissimilarity matri-ces. Consider problem (7.46) with S(θ) = S1(θ) and S(θ) = S2(θ), where Sk(θ) = θ 11> −
144
Dk, k = 1, 2 with θ ≥ maxk=1,2
(θ∗k), k = 1, 2. Then
∣∣∣UMV
(wMV1
)− UMV
(wMV2
) ∣∣∣ ≤ ‖D1 −D2‖∞2
+ τ
∥∥∥∥∥(µ1 −
1
τσ2
1
)−(µ2 −
1
τσ2
2
)∥∥∥∥∥∞(7.47)∥∥∥wMV1 −wMV2
∥∥∥1≤ 2
max(γ(S1(θ)), γ(S2(θ)))
(∥∥∥D1 −D2
∥∥∥∞
+ τ
∥∥∥∥(µ1 −1
τσ2
1
)−(µ2 −
1
τσ2
2
)∥∥∥∥∞
)(7.48)
∥∥∥wMV1 −wMV2
∥∥∥2≤ 2
max(ξmin(S1(θ)), ξmin(S2(θ)))
(max
i=1,...,N
∥∥Ci(D1 −D2)∥∥
2
+ τ
∥∥∥∥(µ1 −1
τσ2
1
)−(µ2 −
1
τσ2
2
)∥∥∥∥2
)(7.49)
where ξmin(Sk(θ)) = mini=1,...,N
(ξi(Sk(θ))), k = 1, 2 with ξi(Sk(θ)) the i-th eigenvalues of Sk(θ), k =
1, 2, γ(Sk(θ)) k = 1, 2 is defined such as HSk(θ)(w) is γ(Sk(θ))-strongly convex with respect to‖.‖1, and Ci(D1 −D2) is the i-th column of D1 −D2.
Proof (of Proposition 7.1.3). Follows the proof of Proposition 5.1.10.
As in the case of RQEP, the sensitivity of the MV model depends therefore on the small eigen-value of the similarity matrix S(∞). Without loss of generality, assume that ξmin(S1(∞)) <
ξmin(S2(∞)). If, for 0 < ε < 1,
ξmin(S2(∞)) ≥ 2
maxi=1,...,N
‖Ci(D1 −D2)‖2 + τ∥∥∥ (µ1 − 1
τσ21
)−(µ2 − 1
τσ22
) ∥∥∥2
ε(7.50)
or equivalently
maxi=1,...,N
‖Ci(D1 −D2)‖2 + τ
∥∥∥∥(µ1 −1
τσ2
1
)−(µ2 −
1
τσ2
2
)∥∥∥∥2
≤ ε ξmin(S2(∞))
2, (7.51)
then∥∥∥wMV1 −wMV2
∥∥∥2≤ ε, then the stability of the MVP is guaranteed. It follows that the
stability of the MVP is intrinsically related to that of RQEP. This result is summarized in thefollowing corollary.
Corollary 7.1.2. The MVP is stable if and only if the RQEP is stable.
Corollary 7.1.2 and the results on the stabilization of the RQEP (see Section 5.1.6) imply thatone can also stabilize the MV portfolio shrinking the matrix S(∞) toward the identity matrixI.
145
7.2 Rao Meets Sharpe
This section investigates the implications of RQE formulation in terms of the capital assetpricing model (CAPM).
7.2.1 The Capital Market Equilibrium
Consider the dual formulation (7.6) instead of the standard formulation (7.4). In that case,the investor k’s problem can therefore be rewritten as follows
maxwk∈W−
w>k
(µ− σ
2
τk
)+
w>k D wk
2τk+ νk
(1−
N∑i=1
wki
), (7.52)
where νk is the Lagrange multiplier of the investment constraint which can be interpreted asthe investor k’s marginal utility of wealth. To derive the equilibrium risk-return trade-off inthis framework, Sharpe (1991) is followed. First, consider the case of no risk-free asset. Thefirst order conditions of problem (7.52) can then be written as
µi −σ2i
τk+Dki
τk= νk ∀ i = 1, ..., N, (7.53)
where Dki =∑N
j=1wMVkj dij is an expression that measures the dissimilarity of asset i with
investor k’s optimal portfolio. Assuming that markets clear, the equilibrium relationshipsamong key variables can be examined by aggregating the conditions that must hold wheneach investor holds an optimal portfolio, taking into account the relative amounts of wealth,Wk, that each has invested. To do so, rewrite (7.53) as follows
τk µi − σ2i +Dki = νkτk ∀ i = 1, ..., N. (7.54)
Next, multiply (7.54) by Wk and sum the resulting equation over all investors to obtain a setof N conditions that must hold in equilibrium
K∑k=1
Wk τk µi −K∑k=1
Wk σ2i +
K∑k=1
WkDki =
K∑k=1
Wk νk τk ∀ i = 1, ..., N. (7.55)
Equation (7.55) can be rewritten as
µi −σ2i
τm+Dmi
τm= νm ∀ i = 1, ..., N, (7.56)
where τm =∑K
k=1Wk τk is the wealth-weighted market risk and/or diversification prefer-
ence coefficient, νm =∑Kk=1 Wk νkτk
τmis the societal marginal utility of wealth, and Dmi =∑N
j=1 dijwmj with wmi =∑K
k=1Wk wMVki is the dissimilarity of asset i with the market. Dmi
gives the diversification gain obtained at the margin by putting an additional unit of asset iin the market portfolio wm. This gain can be decomposed in two components
Dmi =
N∑j=1
(σ2i − σij
)wmj +
N∑j=1
(σ2j − σij
)wmj . (7.57)
146
The first component is the gain in terms of asset i risk reduction. While the second componentis the risk reduction coming out of all other assets.
Form (7.56), the expected return of asset i can be derived
µi = νm +σ2i
τm− Dmi
τm∀ i = 1, ..., N. (7.58)
Equation (7.58) can be rearranged as follows
µi = νm + λv βvi + λh βhi ∀ i = 1, ..., N, (7.59)
where
λv =σ2
m
τm(7.60)
βvi =σ2i
σ2m
(7.61)
λh = −2Hm
τm(7.62)
βhi =Dmi
2Hm(7.63)
with
σ2m = w>mσ
2 (7.64)
Hm = HD(wm) (7.65)
Equation (7.59) is RQE-CAPM equilibrium risk-return trade-off. The variables βvi and βhi areasset i total and diversifiable risks. These can also be interpreted as the contribution of asseti in market total risk and diversification, respectively. It can be shown that
∑Ni=1wmiβvi =∑N
i=1wmiβhi = 1.
The variable λv is the total risk (variance) premium per unit of βvi. While λh is the diversi-fication premium or the diversifiable risk premium per unit of βhi. Since Wk is positive, thesocietal risk tolerance or the societal diversification preference τm is also positive. Therefore,the premium λv is positive while λh is negative.
From (7.59), νm can also be interpreted as the return of a minimum variance portfolio wzz
such that βv(zz) =∑N
i=1wzzσ2i = 0 and βh(zz) =
∑Ni,j=1 dijw(zz)iwmj = 0. When N > 3, which
is generally the case, portfolio wzz always exists and is unique. In the rest of the section, νmwill be denoted µzz =
∑Ni=1w(zz)iµi.
Equation (7.59) also implies that
1
τm=
µm − µzzσ2
m − 2Hm
. (7.66)
147
Since τm is positive, it must be the case that µm−µzz and σ2m− 2Hm have the same sign. It
follows that the spread between the market and the zz portfolio returns depends on the extendof the diversification offered by the market portfolio. More specifically, when Hm > σ2
m/2
or equivalently σ2m ≡ σ2(wm) < σ2
m/2, the market portfolio offers good diversification (i.e.the portfolio as a whole is less risky than its parts) and the market expected return is lowerthan the expected return of portfolio wzz. Inversely, when Hm < σ2
m/2 or equivalentlyσ2m > σ2
m/2, the market portfolio offers poor diversification and, as a result, µm is higherthan µzz.
To help the comparison with standard CAPM, µi is expressed relative to µz, the zero-βportfolio returns. Replacing asset i by zero-β portfolio in (7.59), one has
µz = µzz + λv βvz + λh βhz. (7.67)
Subtracting (7.67) from (7.59), gives
µi = µz + λv (βvi − βvz) + λh (βhi − βhz) ∀ i = 1, ..., N. (7.68)
In the presence of the risk-free asset, since βv(N+1) = 0, (7.68) becomes
µi = µN+1 + λv βvi + λh (βhi − βh(N+1)) ∀ i = 1, ..., N. (7.69)
Equations (7.68) and (7.69) give asset i expected return in absence and in presence of a risk-freeasset. For instance, (7.68) states that the expected return on asset i is equal to the risk-freerate plus a two parts risk premium. The first part equal the market price of total risk, λv,times the quantity of total risk taken on board by asset i given by βvi. The second componentof asset i risk premium is the market price of diversification (i.e. λh) times (βhi − βh(N+1)),the quantity of diversification offered by asset i relative to the risk-free asset.
Since (see Sharpe, 1991)2
τm=µm − µzσ2m
, (7.70)
it is straightforward to show, from (7.68), that the classic CAPM premium, denoted here asλm, is related to λv and λh as follows
λm = 2λv + λh. (7.71)
It is also the case that the classic CAPM βi are linked to βvi and βhi as follows
βi = 2φ1 (βvi − βvz) + φ2 (βhi − βhz) ∀ i = 1, ..., N. (7.72)
where φ1 = σ2m
σ2m, φ2 = −Hm
σ2m
and φ1 + φ2 = 1. In presence of a risk-free asset, βv(N+1) = 0,and (7.72) becomes
βi = 2φ1 βvi + φ2 (βhi − βh(N+1)) ∀ i = 1, ..., N. (7.73)
These results are summarized in the following proposition.
148
Figure 7.2 – Security Market Plane (SMP)
βvi
βi
µi − µN+1 = 0
βhi − βh(N+1)
µi − µN+1
SML
SMP
Proposition 7.2.1 (RQE-CAPM). Consider the CAPM. The risk-return equilibrium trade-off can be rewritten as follows
µi = µz + λv (βvi − βvz) + λh (βhi − βhz) ∀ i = 1, ..., N (7.74)
with
βi = 2φ1 (βvi − βvz) + φ2 (βhi − βhz) ∀ i = 1, ..., N. (7.75)
λm = 2λv + λh, (7.76)
149
where
λv =σ2
m
τm(7.77)
βvi =σ2i
σ2m
(7.78)
λh = −2Hm
τm(7.79)
βhi =Dmi
2Hm(7.80)
φ1 =σ2
m
σ2m
(7.81)
φ2 = −Hm
σ2m
(7.82)
φ1 + φ2 = 1. (7.83)
In the presence of a risk-free asset
µz = µN+1 = R− f (7.84)
βvz = βv(N+1) = 0 (7.85)
Proposition 7.2.1 implies that, in equilibrium, all assets are located along the SML, which isitself located in a plane defined by the equation
z = µz + λv x+ λh y.
This plane is called the Security Market Plane (SMP). Figure 7.2 provides an illustration incase where the risk-free rate is available.
7.2.2 Asset Prices
This section discusses RQE-CAPM risk-return trade-off implications for the pricing of assets.For simplicity, the discussion is limited to the case where a risk-free asset is available. In thatcase
µi = µN+1 + λv βvi + λh (βhi − βhz) ∀ i = 1, ..., N (7.86)
Two conclusions can be drawn form (7.86) regarding the pricing of assets. Firstly, asset i totalrisk (measured by its variance) is priced positively and the premium λv can be interpretedas a gross premium. Secondly, asset i contribution to the efficient diversification relative tothe risk-free asset (measured by βhi − βh(N+1)) is priced negatively and the premium is λh.Thus, RQE-CAPM representation reveals that investors are first compensated for asset i totalrisk and then are willing to pay, in terms of a reduced return, for asset i contribution toefficient diversification. The premium in the classic CAPM can therefore be interpreted as
150
a net premium (see Equation (7.76)), which corresponds for the price of asset net risk (seeEquation (7.75)).
One implication of this result is that the idiosyncratic risk is priced twice: 1) it is positivelypriced because it is a risk; 2) it is taxed (negatively priced) because it is a diversifiablerisk. Moreover, in this RQE-CAPM, these two effects cancelled each other out, becausethe investor’s risk aversion and diversification preference coefficients are equal. To see this,consider the asset i total risk decomposition
σ2i = β2
i σ2m + σ2
ei . (7.87)
Since βvi =σ2i
σ2m, it must be the case that
βvi =β2i σ
2m
σ2m
+σ2ei
σ2m
. (7.88)
Similarly for βhi − βh(N+1),
βhi − βh(N+1) =β2i σ
2m
2Hm+
σ2ei
2Hm− 2σmi
2Hm, (7.89)
where σmi is the covariance between asset i and the market. Denote βIvi =σ2ei
σ2m, βIhi =
σ2ei
2Hm
the idiosyncratic component and βSvi =β2i σ
2m
σ2m, βShi =
β2i σ
2m
2Hmthe systematic component. The
premium of the idiosyncratic risk λIv is
λIv = λv βIvi =
1
τm(7.90)
and the diversification premium λIh is
λIh = λh βIhi = − 1
τm. (7.91)
We can see that the two premiums sum to zero. As a result, it is more appropriate to say thatidiosyncratic risks are priced, but do not determine assets expected return for two reasons:
1. It is priced both positively and negatively and the two price effects cancelled each otherout, because investors’ risk aversion and diversification preference coefficients are equal.
2. The effect of asset total risk is negligible in asset beta due to the fact that N is largeand there is no dominant wmi.
The first reason arises from the correlation diversification stressed in this thesis while thesecond reason is due to the law of large numbers diversification. In sum, the idiosyncratic riskdoes not determine asset expected return because it is diversified away both by the correlationand the law of large numbers diversifications. This result suggests that when
1. investors fail to diversify properly in terms of weights and size (law of large numbersdiversification), the idiosyncratic risk must be priced.
151
2. investors’ risk aversion and diversification preference coefficients are different, the id-iosyncratic risk must be priced.
The theoretical implications of a lack of the law of large numbers diversification has alreadybeen investigated in the literature, notably by Levy and Kroll (1978), Merton (1987) and morerecently by Malkiel and Yexiao (2006). The possibility that idiosyncratic risks must be pricedbecause investors may have different risk aversion and preference for diversification coefficientshas however never been discussed in the literature. It is one of the interesting implications ofcasting the Markowitz (1952)’s mean-variance (MV) utility with Rao’s Quadratic Entropy. Inwhat follows, an illustration is provided.
To explore the implications of letting the coefficient of the preference for diversification to bedifferent than the coefficient of risk aversion, consider the CAPM framework but assume thatinvestors’ utility function is defined as follows
U = w>µ− 1
τw>σ2 +
1
ςHD(w). (7.92)
Without loss of generality, assume that ς < τ . This restriction states that the investor’spreference for diversification is higher than its aversion for the risk. In other order, theinvestor’s willingness to reduce risk through diversification is higher than that to reduce riskthrough concentration. The first order conditions (7.53) become
µi −σ2i
τk+Dki
ςk= νk ∀ i = 1, ..., N, (7.93)
Equation (7.93) can be rewritten as follows
ςkµi −ςkτkσ2i +Dki = ςkνk ∀ i = 1, ..., N, (7.94)
Denoting ιk = ςkτk, multiplying (7.94) by Wk and summing the resulting equation over all
investors to obtain a set of N conditions that must hold in equilibrium
µi −ιmςmσ2i +
Dmi
ςm= νm ∀ i = 1, ..., N, (7.95)
where ςm =∑K
k=1Wk ςk is the wealth-weighted market diversification preference coefficient,
νm =∑Kk=1Wk νkςk
ςmis the societal marginal utility of wealth, and ιm =
∑Nj=1Wk ιk =
∑Nj=1Wk
ςkτk
the wealth-weighted market diversification and risk coefficient ratio.
From (7.95)µi = νm + λ∗v βvi + λ∗h βhi ∀ i = 1, ..., N, (7.96)
where λ∗v = ιmςmσ2
m, βvi =σ2i
σ2m, λ∗h = −2Hm
ςmand βhi = Dmi
2Hm. Equation (7.96) is RQE-
CAPM equilibrium risk-return trade-off when the investor’s risk aversion and preference fordiversification coefficients are not equal. The interpretation of the variables λ∗v, βvi, λ∗h andβhi remains the same.
152
Following (7.87)-(7.89), it is straightforward to prove that the premium of the idiosyncraticrisk λI∗v is
λI∗v = λ∗v β
Ivi =
ιmςm
(7.97)
and the diversification premium λI∗h is
λI∗h = λh β
Ihi = − 1
ςm.
It is observable that the two premiums do not sum to zero. Thus, the idiosyncratic risks arepriced by markets when the investor’s risk aversion and preference for diversification coeffi-cients are not equal.
7.3 Mean-Variance Generalization
RQE formulation also offers new directions along which the MV model can be improvedwithout additional computational cost. Indeed, from (7.6), the MV utility function can begeneralized as follows
UgMV (w) = w>µ− 1
τw>%+
1
ςHD(w), (7.98)
where % = (%(R1), ..., %(RN ))> is a vector any risk measure, ς the coefficient of the preferencefor diversification and D a dissimilarity matrix. This generalization can be used to handle theMV model limits such as: inadequacy of variance as risk measure, estimation errors, extremeweight concentration, etc. One of the important advantages of this generalization is that itcan be done with no additionally computational cost if τ and ς remain exogenous relativelyto w. In what follows, examples of specification of UgMV (w) already used in the literature areprovided to corroborate this claim.
7.3.1 Heuristic Mean-Semivariance Approaches
As mentioned in Chapter 2, one of the limits of the MV model is that the variance is anadequate measure of risk under strict conditions. These are in practice often not respected,because asset returns are not generally elliptically distributed. Moreover, variance has beenfound ineffective for measuring the risk of low-probability events. This led to the developmentof other risk measures. The first alternative risk measure is semivariance, which was pro-posed by Markowitz (1959) himself. Semivariance is like variance, except that it only countsdownward deviation, not up and down deviations as does variance. Its standard formula is asfollows
σ2µ(R(w)) = E (min (0, R(w)− µ(w)))2 . (7.99)
Its benchmark version is
σ2r (R(w)) = E (min (0, R(w)− r))2 , (7.100)
153
where r is a benchmark return. Since investors worry more about underperformance ratherthan overperformance, semivariance can be a more appropriate measure of investors’ risk thanvariance.
However, as highlight by Markowitz (1959), semivariance is more difficult to compute due tothe fact that the co-semivariance matrix is endogenous. To reduce a computational cost, someauthors (see Nawrocki, 1991; Estrada, 2008; Cumova and Nawrocki, 2011) have proposed toreplace the endogenous co-semivariance matrix by an exogenous one. This section shows thatwhen UgMV (w) is well specified, it coincides with heuristic mean-semivariance models proposedby Nawrocki (1991), Estrada (2008) and Cumova and Nawrocki (2011). It provides thereforea theoretical foundation of these heuristic mean-semivariance models.
7.3.1.1 Nawrocki (1991)’s Model
Let R−i = min(Ri − r, 0), ∀ i = 1, ..., N . Define a matrix Σ− = (σ−ij), with
σ−ij =
√E((R−i )2
)E(
(R−j )2)ρij . (7.101)
Nawrocki (1991) suggests to approximate the semivariance of portfolio w as follows
(σ−)2r(w) ≈ w>Σ−w. (7.102)
Nawrocki (1991)’s mean-semivariance utility function is therefore
U−MS(w) = w>µ− 1
τw>Σ−w. (7.103)
In what follows, it is shown that there is a risk measure % and a dissimilarity matrix D suchas U−MS(w) = UgMV (w). Define a matrix D− such that
d−ij = E((R−i )2
)+ E
((R−j )2
)− 2
√E((R−i )2
)√E(
(R−j )2)ρij . (7.104)
It is straightforward to verify that D− is a dissimilarity matrix. PRQE based on D− is
HD−(w) =1
2
N∑i,j=1
(E((R−i )2
)+ E
((R−j )2
)− 2
√E((R−i )2
)√E(
(R−j )2)ρij
)wiwj .
(7.105)The development of (7.105) gives
HD−(w) =
N∑i=1
E((R−i )2
)wi −w>Σ−w. (7.106)
Consider the generalized utility UgMV (w) with HD−(w) and % (Ri) = E((R−i )2
)UgMV (w) = w>µ− 1
τ
N∑i=1
wi E((R−i )2
)+
1
τHD−(w). (7.107)
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The development of UgMV (w) gives
UgMV (w) = w>µ− 1
τw>Σ−w. (7.108)
As a result, there is a risk measure % and a dissimilarity matrix D such as U−MS(w) = UgMV (w).The generalized utility UgMV (w) is therefore the objective function of the heuristic mean-semivariance approach of Nawrocki (1991) based on symmetric co-lower partial moment. Itprovides a theoretical foundation of Nawrocki (1991)’s heuristic mean-semivariance approach.
7.3.1.2 Estrada (2008)’s Model
Now, define a matrix Σ−− = (σ−−ij ), with
σ−−ij = E(R−i R
−j
). (7.109)
Estrada (2008) suggests to approximate the semivariance of portfolio w as follows
(σ−−)2r(w) ≈ w>
(Σ−−
)w. (7.110)
Estrada (2008)’s mean-semivariance utility function is therefore
U−−MS(w) = w>µ− 1
τw>
(Σ−−
)w. (7.111)
In what follows, it is shown that there is a risk measure % and a dissimilarity matrix D suchas U−−MS(w) = UgMV (w). Define a matrix D−− such that
d−−ij = E((R−i )2
)+ E
((R−j )2
)− 2 E
(R−i R
−j
). (7.112)
It is straightforward to verify that D−− is a dissimilarity matrix. PRQE based on D−− is
HD−−(w) =1
2
N∑i,j=1
(E((R−i )2
)+ E
((R−j )2
)− 2 E
(R−i R
−j
))wiwj . (7.113)
The development of (7.113) gives
HD−−(w) =
N∑i=1
E((R−i )2
)wi −w>
(Σ−−
)w. (7.114)
Consider the generalized utility UgMV (w) with HD−−(w) and % (Ri) = E((R−i )2
)UgMV (w) = w>µ− 1
τ
N∑i=1
wi E((R−i )2
)+
1
τHD−−(w). (7.115)
The development of UgMV (w) gives
UgMV (w) = w>µ− 1
τw>
(Σ−−
)w. (7.116)
As a result, there is a risk measure % and a dissimilarity matrix D such as U1MS(w) = UgMV (w).
The generalized utility UgMV (w) is therefore the objective function of the heuristic mean-semivariance approach of Estrada (2008) based on symmetric co-lower partial moment. Itprovides a theoretical foundation of Estrada (2008)’s heuristic mean-semivariance approach.
155
7.3.1.3 Cumova and Nawrocki (2011)’s Model
Let Ri = Ri − r, ∀ i = 1, ..., N . Define a matrix Σ− = (σ−ij), with
σ−ij = E(RiR
−j
)+ E
(Rj R
−i
). (7.117)
Cumova and Nawrocki (2011) suggests to approximate the semivariance of portfolio w asfollows
(σ−)2b(w) ≈ w>Σ−w. (7.118)
Cumova and Nawrocki (2011)’s mean-semivariance utility function is therefore
U−MS(w) = w>µ− 1
τw>Σ−w. (7.119)
In what follows, it is shown that there is a risk measure % and a dissimilarity matrix D suchas U−MS(w) = UgMV (w). Define a matrix D− such that
d−ij = E(RiR
−i
)+ E
(Rj R
−j
)− E
(RiR
−j
)− E
(Rj R
−i
). (7.120)
It is straightforward to verify that D− is a dissimilarity matrix. PRQE based on D− is
HD−(w) =1
2
N∑i,j=1
(E(RiR
−i
)+ E
(Rj R
−j
)− E
(RiR
−j
)− E
(Rj R
−i
))wiwj . (7.121)
The development of (7.121) gives
HD−(w) =
N∑i=1
E(RiR
−i
)wi −w>Σ−w. (7.122)
Consider the generalized utility UgMV (w) with HD−(w) and % (Ri) = E(RiR
−i
)UgMV (w) = w>µ− 1
τ
N∑i=1
wi E(RiR
−i
)+
1
τHD−(w). (7.123)
The development of UgMV (w) gives
UgMV (w) = w>µ− 1
τw>
(Σ−)w. (7.124)
As a result, there is a risk measure % and a dissimilarity matrix D such as U−MS(w) = UgMV (w).The generalized utility UgMV (w) is therefore the objective function of the heuristic mean-semivariance approach of Cumova and Nawrocki (2011) based on symmetric co-lower partialmoment. It provides a theoretical foundation of Cumova and Nawrocki (2011)’s heuristicmean-semivariance approach. These results are summarized in the following proposition.
Proposition 7.3.1 (PRQE and Heuristic Mean-Semivariance Approach). The approx-imated mean-semivariance utility functions of Nawrocki (1991), Estrada (2008) and Cumovaand Nawrocki (2011) are particular cases of RQE generalization of the mean-variance utilityfunction
UgMV (w) = w>µ− 1
τw>%+
1
ςHD(w). (7.125)
156
7.3.2 Estimation Risk Approaches
Another major limit of the MV model is that it not take into account estimation risks. Indeed,the MV model is in general implemented setting µ and Σ at their estimated values using thehistorical data, ignoring therefore estimation risks. However, these risks are known to havea huge impact on Markowitz’s portfolios. More precisely, they are the source of undesirableextreme weights and the poor out-of-sample performance of the MV portfolios (see Chopraand Ziemba, 1993; Michaud, 1989). Moreover, the MV portfolios are very often concentratedon a few assets. Due to the instability of the covariance matrix and average return in time,this leads to higher transaction costs. To overcome these problems several solutions wereproposed as mentioned in Section 2.1.2.1.5. In what follows, it is shown that a particular caseof the solution proposed by Bouchaud et al. (1997), the general free utility, can be viewedas a particular case of the generalized utility UgMV (w). Bouchaud et al. (1997)’s general freeutility is defined as follows
Fq(w) = UMV (w)− ν Yq(w)− 1
q − 1, (7.126)
where Yq(w) =∑N
i=1wqi is a concentration measure, −Yq(w)−1
q−1 is Patil and Taille (1982)’sdiversity measure and q an integer greater than one. This section shows that when UgMV (w)
is well specified, it coincides with Bouchaud et al. (1997)’s general free utility when q = 2,providing therefore a theoretical foundation of this general utility function.
When q = 2,
Y2(w)− 1
2− 1=
N∑i=1
w2i − 1 (7.127)
= −N∑
i 6=j=1
wiwj (7.128)
= −w>(11> − I
)w (7.129)
= −2H(11>N−I)(w). (7.130)
Bouchaud et al. (1997)’s general free utility can therefore be rewritten when q = 2 as follows
F2(w) = UMV (w) + 2ν H(11>−I)(w). (7.131)
Replacing UMV (w) from (7.4) into (7.131), (7.131) becomes
F2(w) = w>µ− 1
τw>σ2 +
1
τHD(w) + 2ν H(11>−I)(w), (7.132)
with D is defined by (7.2). Equation (7.132) can be rewritten as follows
F2(w) = w>µ− 1
τw>σ2 +
(1
τ+ 2ν
)(( 1τ
1τ + 2ν
)HD(w) +
(2ν
1τ + 2ν
)H(11>−I)(w)
).
(7.133)
157
Define
θ =1τ
1τ + 2ν
(7.134)
Dθ = θD + (1− θ)(11> − I
). (7.135)
Fq(w) can be rewritten as follows
F2(w) = w>µ− 1
τw>σ2 +
(1
τ+ 2ν
)HDθ
(w). (7.136)
As a result, when q = 2, Bouchaud et al. (1997)’s general free utility coincides with UgMV (w)
with HDθ(w), % ≡ σ2 and 1
ς = 1τ + 2ν. The generalized utility UgMV (w) provides there-
fore a theoretical foundation of Bouchaud et al. (1997)’s general free utility. This result issummarized in the following proposition.
Proposition 7.3.2 (PRQE and Bouchaud et al. (1997)’s Utility). Bouchaud et al. (1997)’sgeneral free utility with q = 2 is a particular case of RQE generalization of the mean-varianceutility function
UgMV (w) = w>µ− 1
τw>%+
1
ςHD(w). (7.137)
7.4 Summary
This chapter has first provided a novel and useful equivalent formulation of the mean-variance(MV) utility function based on Rao’s Entropy Quadratic of a portfolio (PRQE). Next, someeconomic implications in terms of the MV optimal portfolios, capital asset pricing model andmean-variance model generalization are discussed. The following results have been obtained.
1. The risk aversion coefficient in the MV model also plays the role of the coefficient ofpreference for the diversification;
2. Portfolio risk is represented by the weighted average of asset variances not by portfoliovariance. Portfolio variance represents the undiversified risk, while portfolio RQE repre-sents the diversified risk. Portfolio risk is the sum of the diversified and the undiversifiedrisk;
3. The MV optimal portfolio (MVP) is linearly related to RQE optimal portfolio (RQEP).Consequently, its expected return and variance are also related;
4. The MV portfolio frontier can be represented is three-dimensional space where the axesare represented by the weighted average of asset variances, portfolio return and portfolioRQE;
5. The MVP is stable if and only if RQEP is stable. Therefore, the MVP can also bestabilized using the stabilization techniques of RQEP;
6. A new equivalent capital asset pricing model (CAPM) was derived. This new CAPMclarifies the mechanism of asset pricing in the CAPM. It has shown that
158
a) asset total risk measured by its variance is positively priced;b) the idiosyncratic risk is priced twice. It is positively priced because it is a risk and
it is taxed (negatively priced) because it is a diversifiable risk. Moreover, the twoeffects cancelled each other, because risk aversion and preference for diversificationcoefficients are equal in the MV model. As a result, the idiosyncratic risk is notpriced in the MV model not only because of the law of large numbers diversifi-cation but also because risk aversion and preference for diversification coefficientsare equal. Therefore, the idiosyncratic risk will be priced when investors fail todiversify in terms of size and weight or when investors’ risk aversion and preferencefor diversification coefficients are not equal. Theoretical implications of a lack ofthe law of large numbers diversification has already been investigated in the lit-erature, notably by Levy and Kroll (1978), Merton (1987) and more recently byMalkiel and Yexiao (2006). The possibility that idiosyncratic risks must be pricedbecause investors may have different risk aversion and preference for diversificationcoefficients has however never been discussed in the literature. It is one of the inter-esting implications of casting Markowitz (1952)’s MV utility with Rao’s QuadraticEntropy. An illustration was provided;
7. A new and useful generalization of the MV model, without additional computationalcosts, was provided. It has proved that this new generalization provides a theoreticalfoundation of Nawrocki (1991), Estrada (2008) and Cumova and Nawrocki (2011)’sheurustic mean-semivariance approaches and of a special case of the Bouchaud et al.(1997)’s general free utility.
159
Conclusions and Future Work
Conclusions
This thesis was about the concept of diversification and its measurement in portfolio theory.It has adapted and extended the use of Rao’s Quadratic Entropy (RQE), a general approachto measuring diversity, to portfolio theory as a new class of portfolio diversification measures.
Portfolio diversification is at the core of portfolio theory. It helps to reduce or ultimately toeliminate portfolio risk. Thus, its measurement and management is of fundamental impor-tance in finance and insurance domains as risk measurement and management. Consequently,several measures of portfolio diversification were proposed, each based on a different criterion.Unfortunately, none of them has proven totally satisfactory. All have drawbacks and limitedapplications. Developing a coherent measure of portfolio diversification is therefore an activeresearch area in investment management.
In this thesis, a novel, coherent, flexible, unified, computational efficient and rigorous theo-retical framework to manage and quantify portfolio diversification inspiring from RQE wasproposed. More precisely, it has demonstrated that when RQE is judiciously calibrated itbecomes a valid class of portfolio diversification measures, called portfolio RQE (PRQE). Theadvantage of PRQE is that it summarizes complex features of portfolio diversification in asimple manner and provides at the same time a unified theory that includes many previouscontributions. The list of its features presented in this thesis is as follows:
1. It is easy to interpret. It can be interpreted as a dependence measure or as a multivariateexpected utility;
2. It has low computational cost i.e. computational efficient;3. It is easy to implement;4. It meets ex-ante desirable properties of a portfolio diversification measure when asset
dissimilarity matrix is homogeneous, translate-invariant and conditionally negative def-inite;
5. It covers both the law of large numbers and the correlation diversification strategies;6. It captures the diversification benefit of the risk-free asset;7. It is extremely flexible. This flexibility allows it to easily:
160
a) diversify according any characteristic of assets;b) take into account asset linear and non-linear dependences separately or jointly;c) take into account estimation errors;d) handle sensitivity problem;e) perform targeted diversification;f) perform factor risks diversification;
8. It embeds the portfolio diversification measures such as Gini-Simpson’s index and di-versification return of Booth and Fama (1992) or excess growth rate of Fernholz (2010)offering therefore a novel and useful interpretation of these measures;
9. It governs the diversification in the measures such as diversification ratio of Choueifatyand Coignard (2008), Goetzmann and Kumar (2008)’s measure and Frahm and Wiechers(2013)’s measure offering therefore a novel and useful interpretation of these measures;
10. It governs the diversification in the utility functions such as the mean-variance utilityand Bouchaud et al. (1997)’s general free utility offering therefore a novel and usefulinterpretation of these utility functions;
11. Testing against four standard diversified portfolios (the equally weighted portfolio, theequally risk contribution portfolio, the most diversified portfolios and the market port-folio) on two different empirical datasets using different performance metrics, the well-diversified portfolio of RQE exhibits interesting behaviour even in bear markets.
Next, this thesis has provided two applications of PRQE. In the first application, the thesis hasformally established the principles at play behind the maximum diversification (MD) approachdeveloped by Choueifaty and Coignard (2008) and used to manage 8 billion dollars U.S bythe firm Think Out of the Box Asset Management (TOBAM). This is done by providingnew formulations of the MD strategy in terms of RQE. These new formalizations clarify theinvestment problem behind the MD strategy and helps identify the source of its strong out-of-sample performance relative to other diversified portfolios. As a result, the funds undermanagement of TOBAM are not systematically at risk as suggested by the criticism from Lee(2011) and Taliaferro (2012). Moreover, using these new formulations, new directions alongwhich the out-of-sample performance of the MD strategy can be improved are suggested andit has shown that these improvements are economically meaningful.
In the second application, using RQE, a novel and useful equivalent formulation of the mean-variance utility function was provided. This new formulation significantly improves the mean-variance model understanding, in particular in terms of asset pricing. It also offers new direc-tions along which the mean-variance model can be improved without additional computationalcosts.
In adapting and extending the use of RQE to portfolio theory, the objective of this thesiswas to develop a more coherent, flexible and computation efficient framework of portfoliodiversification management and measurement. We believe that this thesis shows that RQE of
161
portfolio is a strong candidate for being such a framework.
Future Work
The results in this thesis lead to several future directions as follows.
1. In this thesis, the principal results on PRQE were proved under the conditions that shortsales are restricted and the dissimilarity matrix does not depend on portfolio weights. Infuture research it would be interesting to re-examine the properties of PRQE when short salesare allowed or/and the dissimilarity matrix depends on portfolio weights;
2. The dissimilarity matrix is at the core of PRQE. In Section 4.1.2, a guideline was providedfor its definition. In Section 5.2, empirical properties of some dissimilarity matrices werepresented and discussed. Unfortunately, much remains to be done, in particular empirically.In future research it would be interesting to re-examine empirical properties of PRQE fordifferent dissimilarity matrices in order to improve the choice of the dissimilarity matrix;
3. Pavoine (2005) mentions a link between RQE and Smallest enclosing ball problem (seeElzinga and Hearn, 1972a,b; Larsson and Kallberg, 2013; Yildirim, 2008; Zhou et al., 2005;Kallberg and Larsson, 2013; Gartner, 1999; Sylvester, 1857). In this thesis this link is notstudied. In future research, it would be interesting to analyze this relationship in the contextof portfolio selection.
4. RQE has also a link with multidimensional scaling (see Torgerson, 1952; Gower, 1982, 1985;Goldfarb, 1984; Pekalska et al., 2001; Bavaud, 2006, 2010, 2011). This link is not investigatedin this thesis. In future research, it would be interesting to analyze this relationship in thecontext of portfolio selection.
5. In Chapter 7, using RQE, a novel and useful equivalent formulation of the mean-varianceutility function was provided. This thesis has only discussed theoretical economic implica-tions of this new formulation. In future research, it would be interesting to discuss empiricaleconomic implications of this new formulation.
In addition to the application studied in this thesis, RQE have a much wider spectrum ofapplications. It can also be used in corporate finance to manage and quantify corporate diver-sification (Lang and Stulz, 1994; Berger and Ofek, 1995). It can also be used in robust controlanalysis in macroeconomics as an alternative of Kullback Leibler divergence or conditionalrelative entropy (Hansen and Sargent, 2001). It can also be used in ecological economy as aclass of measures of biodiversity (Polasky et al., 2005).
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Bibliography
Jahera Jr., J. S., W. P. Lloyd, and D. E. Page (1987). “The Relationship BetweenFinancial Performance and Stock Market Based Measures of Corporate Diversification,” TheFinancial Review, 22, 379, copyright - Copyright Eastern Finance Association Nov 1987;.
Acerbi, C. (2002). “Spectral measures of risk: A coherent representation of subjective riskaversion,” Journal of Banking & Finance, 26, 1505 – 1518.
Adler, T. and M. Kritzman (2007). “Mean-variance versus full-scale optimisation: In andout of sample,” Journal of Asset Management, 7, 302–311.
Ahmadi-Javid, A. (2012). “Entropic Value-at-Risk: A New Coherent Risk Measure,” Journalof Optimization Theory and Applications, 155, 1105–1123.
Aldous, D. J. (1985). “Exchangeability and related topics,” in École d’Été de Probabilités deSaint-Flour XIII-1983, Springer, 1–198.
Alexander, G. J. and A. M. Baptista (2002). “Economic implications of using a mean-VaR model for portfolio selection: A comparison with mean-variance analysis,” Journal ofEconomic Dynamics and Control, 26, 1159 – 1193, finance.
Ali, M. M. (1975). “Stochastic dominance and portfolio analysis,” Journal of Financial Eco-nomics, 2, 205 – 229.
Allais, M. (1953). “Le Comportement de l’Homme Rationnel devant le Risque: Critique desPostulats et Axiomes de l’Ecole Americaine,” Econometrica, 21, pp. 503–546.
——— (1979). The So-Called Allais Paradox and Rational Decisions under Uncertainty, Dor-drecht: Springer Netherlands, 437–681.
Allison, P. D. (1978). “Measures of Inequality,” American Sociological Review, 43, 865–880.
Amenc, N. and L. Martellini (2011). “In Diversification We Trust?” The Journal ofPortfolio Management, 37, 1–2.
Amihud, Y. and L. Baruch (1981). “Risk Reduction as a Managerial Motive for Conglom-erate Mergers,” The Bell Journal of Economics, 12, 605–617.
163
——— (1999). “Does Corporate Ownership Structure Affect Its Strategy towards Diversifica-tion?” Strategic Management Journal, 20, 1063–1069.
Amir Barnea, D. E. L. (1973). “Stock-Market Based Measures of Corporate Diversification,”The Journal of Industrial Economics, 22, 51–60.
Anderson, E. W., L. P. Hansen, and T. J. Sargent (2000). “Robustness, Detection andthe Price of Risk,” .
Ang, A., R. J. Hodrick, Y. Xing, and X. Zhang (2006). “The Cross-Section of Volatilityand Expected Returns,” Journal of Finance, 61, 259–299.
——— (2009). “High idiosyncratic volatility and low returns: International and further U.S.evidence,” Journal of Financial Economics, 91, 1–23.
Arditti, F. D. (1967). “Risk and the Required Return on Equity,” The Journal of Finance,22, pp. 19–36.
Arnott, R. D., J. Hsu, and P. Moore (2005). “Fundamental Indexation,” FinancialAnalysts Journal, 61, 83–99.
Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath (1999). “Coherent Measures ofRisk,” Mathematical Finance, 9, 203–228.
Atkinson, A. B. (1987). “On the Measurement of Poverty,” Econometrica, 55, 749–764.
Ausloos, M., N. Vandewalle, P. Boveroux, A. Minguet, and K. Ivanova (1999).“Applications of statistical physics to economic and financial topics,” Physica A: StatisticalMechanics and its Applications, 274, 229–240.
Avriel, M., W. Diewert, S. Schaible, and I. Zang (2010). Generalized Concavity,Society for Industrial and Applied Mathematics.
Babaitsev, V., A. Brailov, and V. Popov (2012). “Fast algorithm for the Markowitzcritical line method,” Mathematical Models and Computer Simulations, 4, 241–250.
Baker, M., B. Bradley, and J. Wurgler (2011). “Benchmarks as Limits to Arbitrage:Understanding the Low-Volatility Anomaly,” Financial Analysts Journal, 67, 1–15.
Baker, N. L. and R. A. Haugen (2012). “Low Risk Stocks Outperform within All Observ-able Markets of the World,” Tech. rep.
Balakrishna, N. and C. D. Lai (2009). Continuous Bivariate Distributions: Second Edi-tion, New York, NY: Springer New York, chap. Concepts of Stochastic Dependence, 105–140.
Balduzzi, P. and A. W. Lynch (1999). “Transaction costs and predictability: some utilitycost calculations,” Journal of Financial Economics, 52, 47 – 78.
164
Barnea, A. and D. E. Logue (1973). “Stock-Market Based Measures of Corporate Diver-sification,” Journal of Industrial Economics, 22, 51–60.
Barry, C. B. (1974). “Portfolio Analysis under Uncertain Means, Variances, and Covari-ances,” Journal of Finance, 29, 515–22.
Batista, G. E. A. P. A., E. J. Keogh, O. M. Tataw, and V. M. A. de Souza (2014).“CID: an efficient complexity-invariant distance for time series,” Data Mining and KnowledgeDiscovery, 28, 634–669.
Bavaud, F. (2006). Spectral Clustering and Multidimensional Scaling: A Unified View, Berlin,Heidelberg: Springer Berlin Heidelberg, 131–139.
——— (2010). Euclidean Distances, Soft and Spectral Clustering on Weighted Graphs, Berlin,Heidelberg: Springer Berlin Heidelberg, 103–118.
——— (2011). “On the Schoenberg Transformations in Data Analysis: Theory and Illustra-tions,” Journal of Classification, 28, 297–314.
Bawa, V. S. (1975). “Optimal rules for ordering uncertain prospects,” Journal of FinancialEconomics, 2, 95 – 121.
——— (1982). “Research Bibliography?Stochastic Dominance: A Research Bibliography,”Management Science, 28, 698 – 712.
Bawa, V. S., E. B. Lindenberg, and L. C. Rafsky (1979). “An Efficient Algorithm toDetermine Stochastic Dominance Admissible Sets,” Management Science, 25, 609 – 622.
BCBS (2010). “Developments in Modelling Risk Aggregation,” .
——— (2013). “Consultative Document October 2013. Fundamental review of the tradingbook: A revised market risk framework,” .
Benartzi, S. and R. H. Thaler (1995). “Myopic Loss Aversion and the Equity PremiumPuzzle,” The Quarterly Journal of Economics, 110, pp. 73–92.
Beninel, F. (1999). “Dissimilarites de type spherique et positionnement multidimensionnelnorme,” RAIRO - Operations Research - Recherche Operationnelle, 33, 569–581.
Bera, A. and S. Park (2008). “Optimal Portfolio Diversification Using the Maximum En-tropy Principle,” Econometric Reviews, 27, 484–512.
Berg, C., J. Mateu, and E. Porcu (2008). “The Dagum family of isotropic correlationfunctions,” Bernoulli, 14, 1134–1149.
Berger, P. G. and E. Ofek (1995). “Diversification’s effect on firm value,” Journal ofFinancial Economics, 37, 39 – 65, symposium on Corporate Focus.
165
Berman, A. and N. Shaked-Monderer (2003). Completely Positive Matrices, World Sci-entific.
Bernard, C. and M. Ghossoub (2010). “Static portfolio choice under Cumulative ProspectTheory,” Mathematics and Financial Economics, 2, 277–306.
Bernoulli, D. (1738/1954). “Exposition of a New Theory on the Measurement of Risk,”Econometrica, 22, 23–36.
Bernstein, P. L. (2005). “Capital Ideas: From the past to the Future,” Financial AnalystsJournal, 61, 55–59.
Best, M. J. and R. R. Grauer (1991). “On the Sensitivity of Mean-Variance-EfficientPortfolios to Changes in Asset Means: Some Analytical and Computational Results,” TheReview of Financial Studies, 4, 315–342.
——— (1992). “The analytics of sensitivity analysis for mean-variance portfolio problems,”International Review of Financial Analysis, 1, 17 – 37.
Bey, R. P. (1979). “Estimating the Optimal Stochastic Dominance Efficient Set with a Mean-Semivariance Algorithm,” The Journal of Financial and Quantitative Analysis, 14, pp. 1059–1070.
Bhansali, V. (2012). “Active Risk Parity,” Journal of Investing, 21, 88–92,8,10.
Bhansali, V., J. Davis, G. Rennison, J. C. Hsu, and F. Li (2012). “The Risk in RiskParity: A Factor Based Analysis of Asset Based Risk Parity,” Journal of investing, 21,102–110.
Bignozzi, V., T. Mao, B. Wang, and R. Wang (2016). “Diversification limit of quantilesunder dependence uncertainty,” Extremes, 19, 143–170.
Bird, R. and M. Tippett (1986). “Naive Diversification and Portfolio Risk-A Note,” Man-agement Science, 32, 244–251.
Black, F. (1972). “Capital Market Equilibrium with Restricted Borrowing,” The Journal ofBusiness, 45, 444–455.
Black, F. and R. Litterman (1992). “Global Portfolio Optimization,” Financial AnalystsJournal, 48, 28–43.
Bloomfield, T., R. Leftwich, and J. B. Long (1977). “Portfolio strategies and perfor-mance,” Journal of Financial Economics, 5, 201 – 218.
Bonanno, G., G. Caldarelli, F. Lillo, S. Micciché, N. Vandewalle, and R. N.
Mantegna (2004). “Networks of equities in financial markets,” The European PhysicalJournal B, 38, 363–371.
166
Bonanno, G., N. Vandewalle, and R. N. Mantegna (2000). “Taxonomy of stock marketindices,” Phys. Rev. E, 62, R7615–R7618.
Bonnans, J. F. and A. Shapiro (1998). “Optimization Problems with Perturbations: AGuided Tour,” SIAM Review, 40, 228–264.
Booth, D. G. and E. F. Fama (1992). “Diversification Returns and Asset Contributions,”Financial Analysts Journal, 48, 26–32.
Boubaker, H. and N. Sghaier (2013). “Portfolio optimization in the presence of dependentfinancial returns with long memory: A copula based approach,” Journal of Banking &Finance, 37, 361 – 377.
Bouchaud, J.-P. and M. Potters (2000). Theory of Financial Risk and Derivative Pricing:From Statistical Physics to Risk Management, Cambridge University Press.
Bouchaud, J.-P., M. Potters, and J.-P. Aguilar (1997). “Missing information and assetallocation,” Science & Finance (CFM) working paper archive 500045, Science & Finance,Capital Fund Management.
Bouchey, P., V. Nemtchinov, A. Paulsen, and D. M. Stein (2012). “Volatility Har-vesting: Why Does Diversifying and Rebalancing Create Portfolio Growth?” Journal ofWealth Management, 15, 26–35.
Bouye, E., V. Durrleman, A. Nikeghbali, G. Riboulet, and T. Roncalli (2000).“Copulas for Finance - A Reading Guide and Some Applications,” in working papers,available at https://ssrn.com/abstract=1032533orhttp://dx.doi.org/10.2139/ssrn.1032533.
Braga, M. D. (2015). “Risk parity versus other µ-free strategies: a comparison in a tripleview,” Investment Management and Financial Innovation, 12, 277–289.
——— (2016). Risk-Based Approaches to Asset Allocation: Concepts and Practical Applica-tions, Cham: Springer International Publishing.
Bridgewater Associates, L. (2000). “Engineering Targeted Returns and Risks.”in White papers, available at http://orcamgroup.com/wp-content/uploads/2012/10/
pmpt-engineering-targeted-returns-and-risks.pdf.
Bruder, B. and T. Roncalli (2012). “Managing Risk Exposures using the Risk BudgetingApproach,” SSRN Working Paper.
Brumelle, S. L. (1974). “When Does Diversification Between Two Investments Pay?” TheJournal of Financial and Quantitative Analysis, 9, 473–483.
167
Buckley, J. J. (1985). “Entropy Principles in Decision Making Under Risk,” Risk Analysis,5, 303–313.
BUETOW, G. (2010). “In Defense of Modern Portfolio Theory,” On Wall Street, 20, 38–40.
Buser, S. A. (1977). “Mean-Variance Portfolio Selection with Either a Singular or Non-singular Variance-Covariance Matrix,” Journal of Financial and Quantitative Analysis, 12,347–361.
Byrne, P. and S. Lee (2003). “An exploration of the relationship between size, diversificationand risk in UK real estate portfolios: 1989-1999,” Journal of Property Research, 20, 191–206.
Cailliez, F. (1983). “The analytical solution of the additive constant problem,” Psychome-trika, 48, 305–308.
Cambini, A. and L. Martein (2009). Generalized Convexity and Optimization Theory andApplications, vol. 616 of Lecture Notes in Economics and Mathematical Systems, SpringerBerlin Heidelberg, 10.1007/978-3-540-70876-6.
Campbell, J. Y., A. W. Lo, A. C. MacKinlay, and A. Y. Lo (1997). The econometricsof financial markets, Princeton University Press.
Carli, T., R. Deguest, and L. Martellini (2014). “Improved Risk Reporting with Factor-Based Diversification Measures,” EDHEC-Risk Institute Publications.
Carroll, K. (2015). “Voices: Kris Carroll, on Reassessing Modern Portfolio Theory; ’ForMPT to be effective, one has to be able to predict three inputs: variance, correlation andexpected return’,” The Wall Street Journal.
Cazalet, Z., P. Grison, and T. Roncalli (2014a). “The Smart Beta Indexing Puzzle,”The Journal of Index Investing, 5, 97–119.
——— (2014b). “The Smart Beta Indexing Puzzle,” The Journal of Index Investing, 5, 97–119.
CEBS (2010). “CEBS’s position paper on the recognition of diversification benefits underPillar 2,” .
CEIOPS (2010a). “CEIOPSâĂŹ Advice for Level 2 Implementing Measures on Solvency II:SCR STANDARD FORMULA Article 111(d) Correlations,” .
——— (2010b). “Solvency II Calibration Paper,” .
Chabrillac, Y. and J.-P. Crouzeix (1984). “Definiteness and semidefiniteness of quadraticforms revisited,” Linear Algebra and its Applications, 63, 283 – 292.
Chamberlain, G. (1983a). “A characterization of the distributions that imply mean-Varianceutility functions,” Journal of Economic Theory, 29, 185 – 201.
168
——— (1983b). “Funds, Factors, and Diversification in Arbitrage Pricing Models,” Economet-rica, 51, 1305–1323.
Chambers, D. and J. S. Zdanowicz (2014). “The Limitations of Diversification Return,”The Journal of Portfolio Management, 40, 65–76.
Champely, S. and D. Chessel (2002). “Measuring Biological Diversity Using EuclideanMetrics,” Environmental and Ecological Statistics, 9, 167–177.
Chateauneuf, A. and G. Lakhnati (2007). “From sure to strong diversification,” EconomicTheory, 32, 511–522.
Chateauneuf, A. and J.-M. Tallon (2002). “Diversification, convex preferences and non-empty core in the Choquet expected utility model,” Economic Theory, 19, 509–523.
Chaves, D., J. Hsu, F. Li, and O. Shakernia (2011). “Risk Parity Portfolio vs. OtherAsset Allocation Heuristic Portfolios,” Journal of Investing, 20, 108–118.
Chen, C., G. Iyengar, and C. C. Moallemi (2013). “An Axiomatic Approach to SystemicRisk,” Management Science, 59, 1373–1388.
Chen, Z. and L. G. Epstein (2002). “Ambiguity, risk and asset returns in continuous time,”Econometrica, 70, 1403?1443.
Chen, Z., Q. Hu, and R. Lin (2016). “Performance ratio-based coherent risk measure andits application,” Quantitative Finance, 16, 681–693.
Cheng, D., X. Hu, and C. Martin (2006). “On the Smallest Enclosing Balls,” Commun.Inf. Syst., 6, 137–160.
Chollete, L., V. de la Peña, and C.-C. Lu (2012). “International diversification: Anextreme value approach,” Journal of Banking & Finance, 36, 871 – 885, perspectives onInternational and Corporate Finance.
Chopra, V. K. and W. T. Ziemba (1993). “The Effect of Errors in Means, Variances, andCovariances on Optimal Portfolio Choice,” Journal of Portfolio Management, 19, 6–11.
Chouakria, A. D. and P. N. Nagabhushan (2007). “Adaptive dissimilarity index formeasuring time series proximity,” Advances in Data Analysis and Classification, 1, 5–21.
Choueifaty, Y. and Y. Coignard (2008). “Toward Maximum Diversification,” Journal ofPortfolio Management, 35, 40–51.
Choueifaty, Y., T. Froidure, and J. Reynier (2011). “Properties of the Most DiversifiedPortfolio,” in working papers, available at http://ssrn.com/abstract=1895459orhttp:
//dx.doi.org/10.2139/ssrn.1895459.
169
——— (2013). “Properties of the most diversified portfolio,” Journal of Investment Strategies,2, 49–70.
Chow, T.-m., J. Hsu, V. Kalesnik, and B. Little (2011). “A Survey of AlternativeEquity Index Strategies,” Financial Analysts Journal, 67, 37–57.
Clarke, R., H. de Silva, and S. Thorley (2013). “Risk Parity, Maximum Diversification,and Minimum Variance: An Analytic Perspective,” Journal of Portfolio Management, 39,39–53.
Corkery, M., C. Cui, K. Grind, and J. Chung (2010). “How Risky Are Those Low-RiskFunds?” The Wall Street Journal - Eastern Edition, 261.
Corlu, C. G., M. Meterelliyoz, and M. Tinic (2016). “Empirical distributions of dailyequity index returns: A comparison,” Expert Systems with Applications, 54, 170 – 192.
Cottle, R. W. and J. A. Ferland (1972). “Matrix-theoretic criteria for the quasi-convexityand pseudo-convexity of quadratic functions,” Linear Algebra and its Applications, 5, 123–136.
Coyle, T. (2012). “Advisers Question Modern Portfolio Theory,” The Wall Street Journal.
Cresson, J. E. (2002). “R2: A Market-Based Measure of Portfolio and Mutual Fund Diver-sification,” Quarterly Journal of Business and Economics, 41, 115–143.
Crezee, Dominiek, P. and L. A. P. Swinkels (2011). “High-conviction equity portfoliooptimization,” The Journal of Risk, 13, 57–70.
Critchley, F. and B. Fichet (1997). “On (super-)spherical distance matrices and tworesults from Schoenberg,” Linear Algebra and its Applications, 251, 145 – 165.
CRMCR (2016). “Risk Aggregation and Diversification,” .
Cumova, D. and D. Nawrocki (2011). “A symmetric LPM model for heuristicmean?semivariance analysis,” Journal of Economics and Business, 63, 217 – 236.
——— (2014). “Portfolio optimization in an upside potential and downside risk framework,”Journal of Economics and Business, 71, 68 – 89.
Curado, E. and C. . Tsallis (1991). “Generalized statistical mechanics: connection withthermodynamics,” Journal of Physics A: Mathematical and General, 24, L69–L72.
Datta, D. K., N. Rajagopalan, and A. M. A. Rasheed (1991). “Diversification andPerformance: Critical Review and Future Directions*,” Journal of Management Studies,28, 529–558.
170
de Athayde, G. M. and R. G. F. Jr. (2004). “Finding a maximum skewness portfolio?ageneral solution to three-moments portfolio choice,” Journal of Economic Dynamics andControl, 28, 1335 – 1352.
De Giorgi, E. (2002). “A Note on Portfolio Selection under Various Risk Measures,” IEW-Working Papers 122, Institute for Empirical Research in Economics-University of Zurich.
De Giorgi, E. and T. Hens (2006). “Making prospect theory fit for finance,” FinancialMarkets and Portfolio Management, 20, 339–360.
De Giorgi, E. G. and O. Mahmoud (2016). “Diversification preferences in the theory ofchoice,” Decisions in Economics and Finance, 39, 143–174.
De Luca, G. and P. Zuccolotto (2011). “A tail dependence-based dissimilarity measurefor financial time series clustering,” Advances in Data Analysis and Classification, 5, 323–340.
——— (2014a). Dynamic Clustering of Financial Assets, Cham: Springer International Pub-lishing, 103–111.
——— (2014b). Time Series Clustering on Lower Tail Dependence for Portfolio Selection,Cham: Springer International Publishing, 131–140.
——— (2015). “Dynamic tail dependence clustering of financial time series,” Statistical Papers,1–17.
de Melo Mendes, B. V. and D. S. Marques (2012). “Choosing an optimal investmentstrategy: The role of robust pair-copulas based portfolios,” Emerging Markets Review, 13,449 – 464.
De Soete, G. (1986). “Optimal variable weighting for ultrametric and additive tree cluster-ing,” Quality and Quantity, 20, 169–180.
Deaton, A. (2012). “Franco Modigliani and the life-cycle theory of consumption,” PSL Quar-terly Review, 0.
Deelstra, G., J. Dhaene, and M. Vanmaele (2011). “An overview of comonotonicity andits applications in finance and insurance,” in Advanced mathematical methods for finance,Springer, 155–179.
Deguest, R., L. Martellini, and A. Meucci (2013). “Risk Parity and Beyond-FromAsset Allocation to Risk Allocation Decisions,” SSRN Working Paper.
Dekel, E. (1989). “Asset Demand Without the Independence Axiom,” Econometrica, 57, pp.163–169.
171
Demey, P., S. Maillard, and T. Roncalli (2010). “Risk-Based Indexation,” White Pa-per 1, Lyxor.
DeMiguel, V., L. Garlappi, F. J. Nogales, and R. Uppal (2009a). “A GeneralizedApproach to Portfolio Optimization: Improving Performance by Constraining PortfolioNorms,” Management Science, 55, 798–812.
DeMiguel, V., L. Garlappi, and R. Uppal (2009b). “Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy?” The Review of Financial Studies, 22, 1915–1953.
——— (2009c). “Optimal Versus Naive Diversification: How Inefficient is the 1/N PortfolioStrategy?” Review of Financial Studies, 22, 1915–1953.
Demsetz, R. S. and P. E. Strahan (1997). “Diversification, Size, and Risk at Bank HoldingCompanies,” Journal of Money, Credit and Banking, 29, 300–313.
Dentcheva, D. and A. Ruszczynski (2003). “Optimization with Stochastic DominanceConstraints,” SIAM Journal on Optimization, 14, 548–566.
Denuit, M., J. Dhaene, M. Goovaerts, and R. Kaas (2005). Actuarial theory for de-pendent risks measures, orders and models, Hoboken, N.J: Wiley, bibliogr.
Detlefsen, K. and G. Scandolo (2005). “Conditional and dynamic convex risk measures,”Finance and Stochastics, 9, 539–561.
Deza, M. M. and E. Deza (2009). Encyclopedia of Distances, Springer.
Dhaene, J., M. Denuit, M. J. Goovaerts, R. Kaas, and D. Vyncke (2002a). “Theconcept of comonotonicity in actuarial science and finance: applications,” Insurance: Math-ematics and Economics, 31, 133–161.
——— (2002b). “The concept of comonotonicity in actuarial science and finance: theory,”Insurance: Mathematics and Economics, 31, 3–33.
Dhaene, J., M. Denuit, and S. Vanduffel (2009). “Correlation order, merging and di-versification,” Insurance: Mathematics and Economics, 45, 325 – 332.
Dhaene, J., S. Vanduffel, and M. J. Goovaerts (2008). Comonotonicity, John Wiley& Sons, Ltd.
Dhaene, J., S. Vanduffel, M. J. Goovaerts, R. Kaas, Q. Tang, and D. Vyncke
(2006). “Risk Measures and Comonotonicity: A Review,” Stochastic Models, 22, 573–606.
Dionisio, A., R. Menezes, and D. A. Mendes (2003). “Mutual information: a dependencemeasure for nonlinear time series,” Econometrics 0311003, EconWPA.
172
Diyarbakirlioglu, E. and M. H. Satman (2013). “The Maximum Diversification Index,”Journal of Asset Management, 14, 400–409.
Duchin, R. and H. Levy (2009). “Markowitz Versus the Talmudic Portfolio DiversificationStrategies,” Journal of Portfolio Management, 35, 71–74.
Duin, R. P. W., E. Pękalska, and M. Loog (2013). Non-Euclidean Dissimilarities:Causes, Embedding and Informativeness, London: Springer London, 13–44.
Durante, F., R. Pappadà, and N. Torelli (2015). “Clustering of time series via non-parametric tail dependence estimation,” Statistical Papers, 56, 701–721.
Ebrahimi, N. (1982). “The ordering of negative quadrant dependence,” Communications inStatistics - Theory and Methods, 11, 2389–2399.
Eeckhoudt, L., B. Rey, and H. Schlesinger (2007). “A Good Sign for Multivariate RiskTaking,” Management Science, 53, 117–124.
EIOPA (2014). “The underlying assumptions in the standard formula for the Solvency CapitalRequirement calculation,” .
El-Tamimi, A. A. and K. Al-Zahrani (2013). “An improved quadratic program for un-weighted Euclidean 1-center location problem,” Journal of King Saud University - Engi-neering Sciences, 25, 161 – 165.
Elton, E. J. and M. J. Gruber (1977). “Risk Reduction and Portfolio Size: An AnalyticalSolution,” The Journal of Business, 50, 415–37.
——— (1997). “Modern portfolio theory, 1950 to date,” Journal of Banking & Finance, 21,1743–1759.
Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann (2002). Modernportfolio theory and investment analysis, New York J. Wiley & Sons, 6th edition ed.
Elzinga, D. J. and D. W. Hearn (1972a). “The Minimum Covering Sphere Problem,”Management Science, 19, 96–104.
Elzinga, J. and D. W. Hearn (1972b). “Geometrical Solutions for Some Minimax LocationProblems,” Transportation Science, 6, 379–394.
Embrechts, P., H. Furrer, and R. Kaufmann (2009). Handbook of Financial TimeSeries, chap. Different Kinds of Risk, 729–751.
Embrechts, P., A. McNeil, and D. Straumann (1999). “Correlation And Dependence InRisk Management: Properties And Pitfalls,” in RISK MANAGEMENT: VALUE AT RISKAND BEYOND, Cambridge University Press, 176–223.
173
Embrechts, P., B. Wang, and R. Wang (2015). “Aggregation-robustness and model un-certainty of regulatory risk measures,” Finance and Stochastics, 19, 763–790.
Epstein, L. G. and M. Schneider (2010). “Ambiguity and Asset Markets,” NBER WorkingPapers 16181, National Bureau of Economic Research, Inc.
Epstein, L. G. and S. M. Tanny (1980). “Increasing Generalized Correlation: A Defi-nition and Some Economic Consequences,” The Canadian Journal of Economics / Revuecanadienne d’Economique, 13, 16–34.
Esteban, J.-M. and D. Ray (1994). “On the Measurement of Polarization,” Econometrica,62, 819–851.
Estrada, J. (2003). “Mean-semivariance behavior: A note,” Finance Letters, 1, 9 – 14.
——— (2008). “Mean-Semivariance Optimization: A Heuristic Approach,” Journal of AppliedFinance, 18, 57 – 72.
Evans, J. L. (1975). “An Examination of the Principle of Diversification,” Journal of BusinessFinance & Accounting, 2, 243–255.
Evans, J. L. and S. H. Archer (1968). “Diversification and the reduction of dispersion: anempirical analysis,” The Journal of Finance, 23, 761–767.
Evensky, H. (2009). “Is MPT Dead? Is Alpha Beta? And Other Interesting Questions,”Journal of Financial Planning, 22, 30–33.
Fabozzi, F. J., S. M. Focardi, and C. Jonas (2014). “Investment Management: A Scienceto Teach or an Art to Learn?” Research Foundation Books, 2014, 1–113.
Fama, E. F. (1965). “The Behavior of Stock-Market Prices,” The Review of Economic Studies,38, 34–105.
——— (1970). “Multiperiod Consumption-Investment Decisions,” The American EconomicReview, 60, 163–174.
Farebrother, R. (1977). “Necessary and sufficient conditions for a quadratic form to bepositive whenever a set of homogeneous linear constraints is satisfied,” Linear Algebra andits Applications, 16, 39 – 42.
Fastrich, B., S. Paterlini, and P. Winker (2015). “Constructing optimal sparse portfo-lios using regularization methods,” Computational Management Science, 12, 417–434.
Feldstein, M. S. (1969). “Mean-Variance Analysis in the Theory of Liquidity Preferenceand Portfolio Selection,” The Review of Economic Studies, 36, pp. 5–12.
174
Feng, Y. and D. P. Palomar (2015). “SCRIP: Successive Convex Optimization Methodsfor Risk Parity Portfolio Design,” IEEE Transactions on Signal Processing, 63, 5285–5300.
Ferland, J. A. (1980). “Positive subdefinite matrices,” Linear Algebra and its Applications,31, 233 – 244.
Fernandes, J. L. B., J. R. H. Ornelas, and O. A. M. Cusicanqui (2012). “Combiningequilibrium, resampling, and analyst’s views in portfolio optimization,” Journal of Banking& Finance, 36, 1354 – 1361.
Fernholz, R. (2010). Diversification, John Wiley & Sons, Ltd.
Fernholz, R. and B. Shay (1982). “Stochastic Portfolio Theory and Stock Market Equi-librium,” The Journal of Finance, 37, 615–624.
Fischer, K., B. Gärtner, and M. Kutz (2003). Fast Smallest-Enclosing-Ball Computationin High Dimensions, Berlin, Heidelberg: Springer Berlin Heidelberg, 630–641.
Fishburn, P. C. (1974). “Convex stochastic dominance with continuous distribution func-tions,” Journal of Economic Theory, 7, 143 – 158.
——— (1977). “Mean-Risk Analysis with Risk Associated with Below-Target Returns,” Amer-ican Economic Review, 67, 116–26.
——— (1980). “Stochastic Dominance and Moments of Distributions,” Mathematics of Oper-ations Research, 5, pp. 94–100.
——— (1982). “Foundations of risk measurement. II. Effects of gains on risk,” Journal ofMathematical Psychology, 25, 226 – 242.
——— (1984). “Foundations of Risk Measurement. I. Risk As Probable Loss,” ManagementScience, 30, 396–406.
Follmer, H. and A. Schied (2002). “Convex measures of risk and trading constraints,”Finance and Stochastics, 6, 429–447.
——— (2010). Convex Risk Measures, John Wiley & Sons, Ltd.
Fragkiskos, A. (2013). “What is Portfolio Diversification?” SSRN Working Paper.
Frahm, G. and C. Wiechers (2011). “On the diversification of portfolios of risky assets,”Discussion Papers in Statistics and Econometrics 2/11, University of Cologne, Departmentfor Economic and Social Statistics.
——— (2013). A Diversification Measure for Portfolios of Risky Assets, London: PalgraveMacmillan UK, 312–330.
175
Frittelli, M. and E. R. Gianin (2002). “Putting order in risk measures,” Journal ofBanking & Finance, 26, 1473 – 1486.
——— (2005). Law invariant convex risk measures, Tokyo: Springer Tokyo, 33–46.
Frost, P. A. and J. E. Savarino (1986). “An Empirical Bayes Approach to EfficientPortfolio Selection,” Journal of Financial and Quantitative Analysis, 21, 293–305.
——— (1988). “For better performance: Constrain portfolio weights,” Journal of PortfolioManagement, 15, 29–34.
Garlappi, L. and G. Skoulakis (2011). “Taylor series approximations to expected utilityand optimal portfolio choice,” Mathematics and Financial Economics, 5, 121–156.
Garlappi, L., R. Uppal, and T. Wang (2007). “Portfolio Selection with Parameter andModel Uncertainty: A Multi-Prior Approach,” Review of Financial Studies, 20, 41–81.
Gartner, B. (1999). “Fast and Robust Smallest Enclosing Balls,” in Proceedings of the 7thAnnual European Symposium on Algorithms, London, UK, UK: Springer-Verlag, ESA ’99,325–338.
Gärtner, B. and S. Schönherr (2000). “An Efficient, Exact, and Generic Quadratic Pro-gramming Solver for Geometric Optimization,” in Proceedings of the Sixteenth Annual Sym-posium on Computational Geometry, New York, NY, USA: ACM, SCG ’00, 110–118.
Gilboa, I. and D. Schmeidler (1989). “Maxmin expected utility with non-unique prior,”Journal of Mathematical Economics, 18, 141–153.
Goetzmann, W. N. and A. Kumar (2008). “Equity Portfolio Diversification,” Review ofFinance, 12, 433–463.
Goetzmann, W. N., L. Li, and K. G. Rouwenhorst (2005). “Long-Term Global MarketCorrelations,” The Journal of Business, 78, 1–38.
Goldfarb, D. and G. Iyengar (2003). “Robust Portfolio Selection Problems,” Mathematicsof Operations Research, 28, 1–38.
Goldfarb, L. (1984). “A unified approach to pattern recognition,” Pattern Recognition, 17,575?582.
Goovaerts, M. J., R. Kaas, J. Dhaene, and Q. Tang (2003). “A Unified Approach toGenerate Risk Measures,” ASTIN Bulletin, 33, 173–191.
——— (2004). “Some new classes of consistent risk measures,” Insurance: Mathematics andEconomics, 34, 505 – 516.
176
Gourieroux, C. and A. Monfort (2005). “The econometrics of efficient portfolios,” Jour-nal of Empirical Finance, 12, 1–41.
Gower, J. (1985). “Properties of Euclidean and non-Euclidean distance matrices,” LinearAlgebra and its Applications, 67, 81 – 97.
Gower, J. and P. Legendre (1986). “Metric and Euclidean properties of dissimilaritycoefficients,” Journal of Classification, 3, 5–48.
Gower, J. C. (1966). “Some distance properties of latent root and vector methods used inmultivariate analysis,” Biometrika, 53, 325–338.
——— (1982). “Euclidean distance geometry,” Mathematical Scientist, 7, 1–14.
Graham, B. (1949). The intelligent investor, a book of practical counsel, New York: Harper.
Graham, B. and D. L. Dodd (1951). Security analysis : principles and technique, NewYork: McGraw-Hill, 3rd ed ed.
Green, P. E. and A. Maheshwari (1969). “Common Stock Perception and Preference: AnApplication of Multidimensional Scaling,” The Journal of Business, 42, 439–57.
Green, R. C. and B. Hollifield (1992). “When Will Mean-Variance Efficient Portfoliosbe Well Diversified?” The Journal of Finance, 47, 1785–1809.
Groenena, P. J. and P. H. Franses (2000). “Visualizing time-varying correlations acrossstock markets,” Journal of Empirical Finance, 7, 155?172.
Grote, J. (2012). “Going to One: Is Diversification PassÃľ?” Journal of Financial Planning,25, 20–25.
Guiasu, R. C. and v. Guiasu (2011). “The weighted quadratic index of biodiversity forpairs of species: a generalization of Rao?s index,” Natural Science, 3, 707–719.
Guidolin, M. and F. Rinaldi (2013). “Ambiguity in asset pricing and portfolio choice: areview of the literature,” Theory and Decision, 74, 183–217.
Guigues, V. (2011). “Sensitivity analysis and calibration of the covariance matrix for stableportfolio selection,” Computational Optimization and Applications, 48, 553–579.
Haasdonk, B. (2005). “Feature space interpretation of SVMs with indefinite kernels,” IEEETransactions on Pattern Analysis and Machine Intelligence, 27, 482–492.
Hadar, J. and W. R. Russell (1969). “Rules for Ordering Uncertain Prospects,” AmericanEconomic Review, 59, 25–34.
177
Haezendonck, J. and M. Goovaerts (1982). “A new premium calculation principle basedon Orlicz norms,” Insurance: Mathematics and Economics, 1, 41 – 53.
Haff, L. R. (1980). “Empirical Bayes Estimation of the Multivariate Normal CovarianceMatrix,” The Annals of Statistics, 8, 586–597.
Han, Y.-p., Z.-p. Chen, and F. Zhang (2015). “Lipschitz continuity of the optimal valuefunction and KKT solution set in indefinite quadratic programs,” Applied Mathematics-AJournal of Chinese Universities, 30, 102–110.
Hanoch, G. and H. Levy (1969). “The Efficiency Analysis of Choices Involving Risk,”Review of Economic Studies, 36, 335–46.
Hansen, L. P. and T. J. Sargent (2001). “Robust Control and Model Uncertainty,” TheAmerican Economic Review, 91, 60–66.
Harris, R. G. (1980). “A General Equilibrium Analysis of the Capital Asset Pricing Model,”The Journal of Financial and Quantitative Analysis, 15, 99–122.
Harvey, C., J. Liechty, M. Liechty, and P. Muller (2010). “Portfolio selection withhigher moments,” Quantitative Finance, 10, 469–485.
Hatherley, A. and J. Alcock (2007). “Portfolio construction incorporating asymmetricdependence structures: a user’s guide,” Accounting & Finance, 47, 447–472.
Haugen, R. A. and N. L. Baker (Spring 1991). “The efficient market inefficiency ofcapitalization-weighted stock portfolios,” Journal of Portfolio Management, 17, 35–40.
He, X. D. and X. Y. Zhou (2011). “Portfolio Choice Under Cumulative Prospect Theory:An Analytical Treatment,” Management Science, 57, 315–331.
Hlawitschka, W. (1994). “The Empirical Nature of Taylor-Series Approximations to Ex-pected Utility,” The American Economic Review, 84, pp. 713–719.
Holthausen, D. M. (1981). “A Risk-Return Model with Risk and Return Measured asDeviations from a Target Return,” The American Economic Review, 71, pp. 182–188.
Holton, L. (2009). “Is Markowitz Wrong? Market Turmoil Fuels Nontraditional Approachesto Managing Investment Risk,” Journal of Financial Planning, 22, 20–26.
Honeine, P. and C. Richard (2010). “The angular kernel in machine learning for hyper-spectral data classification,” in Hyperspectral Image and Signal Processing: Evolution inRemote Sensing (WHISPERS), 2010 2nd Workshop on, 1–4.
Hsu, J. C. (2006). “Cap-Weighted Portfolios are Sub-Optimal Portfolios,” Journal of Invest-ment Management, 4, 1–10.
178
Huang, C. F. and R. H. Litzenberger (1988). Foundations for Financial Economics,New York: North Holland, Elsevier Science Publishing Co., Inc.
Huang, X. (2012). “An Entropy Method for Diversified Fuzzy Portfolio Selection,” Interna-tional Journal of Fuzzy System, 14, 160–165.
Ilmanen, A. and J. Kizer (2012). “The Death of Diversification Has Been Greatly Exag-gerated,” The Journal of Portfolio Management, 38, 15–27.
Ingersoll, J. E. (1984). “Some Results in the Theory of Arbitrage Pricing,” The Journal ofFinance, 39, 1021–1039.
Jacobs, B. I., K. N. Levy, and H. Markowitz (2006). “Trimability and Fast Optimizationof Long?Short Portfolios,” Financial Analysts Journal, 62, 36–46.
Jacobs, B. I., K. N. Levy, and H. M. Markowitz (2005). “Portfolio Optimization withFactors, Scenarios, and Realistic Short Positions,” Operations Research, 53, 586–599.
Javanmardi, L. and Y. Lawryshyn (2015). “A new rank dependent utility approach tomodel risk averse preferences in portfolio optimization,” Annals of Operations Research,1–16.
Jensen, M. C. (1972). “Capital Markets: Theory and Evidence,” Bell Journal of Economics,3, 357–398.
Jobson, J. D. and v. Korkie (1980). “Estimation for Markowitz Efficient Portfolios,” Jour-nal of the American Statistical Association, 75, 544–554.
Joe, H. (2001, c1997).Multivariate models and dependence concepts, Monographs on statisticsand applied probability, Boca Raton: Chapman & Hall/CRC, bibliogr.: p. [383]-394.
Johnson, K. and D. Shannon (1974). “A note on diversification and the reduction ofdispersion,” Journal of Financial Economics, 1, 365 – 372.
Joly, S. and G. Le Calve (1986). “Étude des puissances d’une distance,” Statistique etanalyse des données, 11, 30–50.
Jondeau, E. and M. Rockinger (2003). “How Higher Moments Affect the Allocation ofAssets,” Finance Letters, 1, pp. 354–371.
——— (2006). “Optimal Portfolio Allocation under Higher Moments,” European FinancialManagement, 12, 29–55.
Jorion, P. (1986). “Bayes-Stein Estimation for Portfolio Analysis,” Journal of Financial andQuantitative Analysis, 21, 279–292.
179
Jurczenko, E. and B. Maillet (2006). Multi-moment asset allocation and pricing models,Chichester, England ; Hoboken, NJ : John Wiley & Sons, Inc, includes bibliographicalreferences and index.
Kahneman, D. and A. Tversky (1979). “Prospect Theory: An Analysis of Decision underRisk,” Econometrica, 47, pp. 263–292.
Kallberg, L. and T. Larsson (2013). “Faster Approximation of Minimum Enclosing Ballsby Distance Filtering and GPU Parallelization,” Journal of Graphics Tools, 17, 67–84.
Kan, R. and G. Zhou (2007). “Optimal Portfolio Choice with Parameter Uncertainty,”Journal of Financial and Quantitative Analysis, 42, 621–656.
Kang, T., B. W. Brorsen, and B. D. Adam (1996). “A new efficiency criterion: Themean-separated target deviations risk model,” Journal of Economics and Business, 48, 47– 66.
Kavous, A. (2000). “Interpretations of the CAPM, Diversification, and Beta: Clarifications,”Humanomics, 16, 35–50.
Keynes, J. M. and R. E. S. (Grande-Bretagne) (1971). The collected writings of JohnMaynard Keynes, London: Macmillan.
Khan, M. and Y. Sun (2003). “Exact arbitrage, well-diversified portfolios and asset pricingin large markets,” Journal of Economic Theory, 110, 337 – 373.
Kijima, M. and M. Ohnishi (1993). “Mean-risk analysis of risk aversion and wealth effects onoptimal portfolios with multiple investment opportunities,” Annals of Operations Research,45, 147–163.
Kitces, M. E. (2012). “The Rise of Tactical Asset Allocation,” Journal of Financial Planning,25, 28–29.
Klatte, D. (1985). “On the Lipschitz behavior of optimal solutions in parametric problemsof quadratic optimization and linear complementarity,” Optimization, 16, 819–831.
Klein, R. W. and V. S. Bawa (1976). “The effect of estimation risk on optimal portfoliochoice,” Journal of Financial Economics, 3, 215–231.
——— (1977). “The effect of limited information and estimation risk on optimal portfoliodiversification,” Journal of Financial Economics, 5, 89–111.
Klemkosky, R. and J. Martin (1975). “The Effect of Market Risk on Portfolio Diversifi-cation,” The Journal of Finance, 30, 147–154.
180
Koliai, L. (2016). “Extreme Risk Modelling: An EVT?Pair-copulas Approach for FinancialStress Tests,” Journal of Banking & Finance, –.
Kolm, P. N., R. Tutuncu, and F. J. Fabozzi (2014). “60 Years of portfolio optimization:Practical challenges and current trends,” European Journal of Operational Research, 234,356 – 371, 60 years following Harry MarkowitzâĂŹs contribution to portfolio theory andoperations research.
KONNO, H. and R. YAMAMOTO (2005). “A MEAN-VARIANCE-SKEWNESS MODEL:ALGORITHM AND APPLICATIONS,” International Journal of Theoretical and AppliedFinance, 08, 409–423.
Konno, H. and H. Yamazaki (1991). “Mean-absolute Deviation Portfolio OptimizationModel and Its Applications to Tokyo Stock Market,” Manage. Sci., 37, 519–531.
Kopa, M. and P. Chovanec (2008). “A second-order stochastic dominance portfolio effi-ciency measure,” Kybernetika, 44, 243–258.
Kopa, M. and T. Post (2009). “A Portfolio Optimality Test Based on the First-OrderStochastic Dominance Criterion,” The Journal of Financial and Quantitative Analysis, 44,pp. 1103–1124.
——— (2014). “A general test for SSD portfolio efficiency,” OR Spectrum, 1–32.
Kraus, A. and R. H. Litzenberger (1976). “Skewness Preference and the Valuation ofRisk Assets,” The Journal of Finance, 31, pp. 1085–1100.
Kreps, D. (1988). Notes on the Theory of Choice, Westview press.
Kritzman, M. (2011). “The Graceful Aging of Mean-Variance Optimization,” The Journalof Portfolio Management, 37, 3–5.
Krokhmal, P., S. Uryasev, and J. Palmquist (2001). “Portfolio optimization with con-ditional value-at-risk objective and constraints,” Journal of risk, 4, 43–68.
Kroll, Y., H. Levy, and H. M. Markowitz (1984). “Mean-Variance Versus Direct UtilityMaximization,” The Journal of Finance, 39, pp. 47–61.
Kromer, E., L. Overbeck, and K. Zilch (2016). “Systemic risk measures on generalmeasurable spaces,” Mathematical Methods of Operations Research, 84, 323–357.
Kumar, P., J. S. B. Mitchell, and E. A. Yildirim (2003). “Approximate MinimumEnclosing Balls in High Dimensions Using Core-sets,” J. Exp. Algorithmics, 8.
Laas, D. and C. F. Siegel (2016). “Basel III Versus Solvency II: An Analysis of RegulatoryConsistency Under the New Capital Standards,” Journal of Risk and Insurance, n/a–n/a.
181
Laise, E. (2010). “How Risky Are Those Low-Risk Funds?” The Wall Street Journal - EasternEdition, 256.
Laliberte, E. and P. Legendre (2010). “A distance-based framework for measuring func-tional diversity from multiple traits,” Ecology, 91, 299–305.
Lancaster, P. (1969). Theory of matrices, New York: Academic Press.
Lang, L. H. P. and R. M. Stulz (1994). “Tobin’s q, Corporate Diversification, and FirmPerformance,” Journal of Political Economy, 102, 1248–1280.
Larsson, T. and L. Kallberg (2013). “Fast and Robust Approximation of Smallest En-closing Balls in Arbitrary Dimensions,” Computer Graphics Forum, 32, 93–101.
Lau, K.-S. (1985). “Characterization of Rao’s Quadratic Entropies,” Sankhy?: The IndianJournal of Statistics, Series A (1961-2002), 47, 295–309.
Leal, R. P. C. and B. V. M. Mendes (2013). “Assessing the effect of tail dependence inportfolio allocations,” Applied Financial Economics, 23, 1249–1256.
Leavens, D. H. (1945). “Diversification of investments,” Trusts & Estates, 80, 469–473.
Ledoit, O. and M. Wolf (2003). “Improved estimation of the covariance matrix of stockreturns with an application to portfolio selection,” Journal of Empirical Finance, 10, 603–621.
——— (2004a). “Honey, I Shrunk the Sample Covariance Matrix : Problems in Mean-VarianceOptimization,” Journal of Portfolio Management, 30, 110–119.
——— (2004b). “A well-conditioned estimator for large-dimensional covariance matrices,”Journal of Multivariate Analysis, 88, 365–411.
——— (2008). “Robust performance hypothesis testing with the Sharpe ratio,” Journal ofEmpirical Finance, 15, 850–859.
——— (2011). “Robust Performances Hypothesis Testing With the Variance,” Wilmott, 2011,86–89.
Lee, W. (2011). “Risk-Based Asset Allocation: A New Answer to an Old Question?” JournalOf Portfolio Management, 37, 11–28.
Leinster, T. and C. A. Cobbold (2012). “Measuring diversity: the importance of speciessimilarity,” Ecology, 93, 477–489.
182
Lemin, A. J. (2001). “Isometric Embedding of Ultrametric (non-Archimedean) Spaces inHilbert Space and Lebesgue Space,” in p-adic functional analysis proceedings of the sixthinternational conference, ed. by A. K. Katsaras, W. H. Schikhof, and L. Van Hamme, NewYork, NY: Marcel Dekker, Inc., chap. 16.
Leote, R., X. Lu, and P. Moulin (2012). “Demystifying Equity Risk-Based Strategies: ASimple Alpha plus Beta Description,” Journal of Portfolio Management, 38, 56–70.
Levy, H. (1969). “A Utility Function Depending on the First Three Moments,” The Journalof Finance, 24, pp. 715–719.
——— (1992). “Stochastic Dominance and Expected Utility: Survey and Analysis,” Manage-ment Science, 38, pp. 555–593.
——— (2006). “Stochastic Dominance : Investment Decision Making under Uncertainty,” inStudies in Risk and Uncertainty, ed. by H. Levy, Springer US, vol. 2 of 12, 2 ed.
——— (2012). The Capital Asset Pricing Model in the 21st Century, Cambridge UniversityPress.
Levy, H. and R. Duchin (2010). Markowitz’s Mean–Variance Rule and the Talmudic Di-versification Recommendation, Boston, MA: Springer US, 97–123.
Levy, H. and G. Hanoch (1970). “Relative Effectiveness of Efficiency Criteria for PortfolioSelection,” The Journal of Financial and Quantitative Analysis, 5, pp. 63–76.
Levy, H. and Y. Kroll (1978). “ORDERING UNCERTAIN OPTIONS WITH BORROW-ING AND LENDING,” The Journal of Finance, 33, 553–574.
Levy, H. and M. Levy (2004). “Prospect Theory and Mean-Variance Analysis,” The Reviewof Financial Studies, 17, pp. 1015–1041.
Levy, H. and H. M. Markowitz (1979). “Approximating Expected Utility by a Functionof Mean and Variance,” The American Economic Review, 69, pp. 308–317.
Levy, H. and M. Sarnat (1970). “Alternative Efficiency Criteria: An Empirical Analysis,”The Journal of Finance, 25, pp. 1153–1158.
Li, D. and W.-L. Ng (2000). “Optimal Dynamic Portfolio Selection: Multiperiod Mean-Variance Formulation,” Mathematical Finance, 10, 387–406.
Li, Y. and W. T. Ziemba (1990). “Rules for diversification for all risk averters,” Journal ofEconomics and Business, 42, 165 – 170.
Lin, H.-t. and C.-J. Lin (2003). “A Study on Sigmoid Kernels for SVM and the Training ofnon-PSD Kernels by SMO-type Methods,” Tech. rep.
183
Lingoes, J. (1971). “Some boundary conditions for a monotone analysis of symmetric matri-ces,” Psychometrika, 36, 195–203.
Lintner, J. (1965). “The Valuation of Risk Assets and the Selection of Risky Investments inStock Portfolios and Capital Budgets,” The Review of Economics and Statistics, 47, 13–37.
Linzmeier, D. (2011). “Risk Balanced Portfolio Construction,” in working papers, avail-able at http://www.quoniam.de/fileadmin/Web-Files/Kundenservice/Publikationen/RiskBalancedPortfolioConstruction_QAM_DL.pdf.
Liu, Y.-F., R. Diao, F. Ye, and H.-W. Liu (2016). “An Efficient Inexact Newton-CGAlgorithm for the Smallest Enclosing Ball Problem of Large Dimensions,” Journal of theOperations Research Society of China, 4, 167–191.
Lloyd, W. P., J. H. Hand, and N. K. Modani (1981). “The Effect of Portfolio Construc-tion Rules on the Relationship between Portfolio Size and Effective Diversification,” Journalof Financial Research, 4, 183–193.
Loeb, G. M. (1943). “The Battle for Investment Survival,” Copyright - Copyright Dow Jones& Company Inc Aug 16, 1943;.
——— (1965). The Battle for Investment Survival, Box 494, Burlington, Vermont 05402:Fraser Publishing Company Burlington, Vermont.
Lohre, H., U. Neugebauer, and C. Zimmer (2012). “Diversified Risk Parity Strategiesfor Equity Portfolio Selection,” Journal of Investing, 21, 111–128,10.
Loistl, O. (1976). “The Erroneous Approximation of Expected Utility by Means of a Taylor’sSeries Expansion: Analytic and Computational Results,” The American Economic Review,66, pp. 904–910.
Loomes, G. and R. Sugden (1982). “Regret Theory: An Alternative Theory of RationalChoice Under Uncertainty,” The Economic Journal, 92, 805–824.
Loosli, G., S. Canu, and C. S. Ong (2016). “Learning SVM in Krein Spaces,” IEEETransactions on Pattern Analysis and Machine Intelligence, 38, 1204–1216.
Lozano, S. and E. Gutierrez (2008). “Data envelopment analysis of mutual funds basedon second-order stochastic dominance,” European Journal of Operational Research, 189, 230– 244.
Ma, T. and R. Jagannathan (2003). “Risk Reduction in Large Portfolios: Why Imposingthe Wrong Constraints Helps,” Journal of Finance, 58, 57–72.
Maccheroni, F., M. Marinacci, and A. Rustichini (2006). “Ambiguity Aversion, Ro-bustness, and the Variational Representation of Preferences,” Econometrica, 74, 1447–1498.
184
Machadoa, J. T., F. B. Duarte, and G. M. Duartec (2011). “Analysis of stock mar-ket indices through multidimensional scaling,” Commun Nonlinear Sci Numer Simulat, 16,4610?4618.
Machina, M. J. (1982). “"Expected Utility" Analysis without the Independence Axiom,”Econometrica, 50, 277–323.
MacMinn, R. D. (1984). “A General Diversification Theorem: A Note,” The Journal ofFinance, 39, 541–550.
Maillard, S e., T. Roncalli, and J. Teiletche (2010). “The Properties of EquallyWeighted Risk Contribution Portfolios,” Journal of Portfolio Management, 36, 60–70.
Mainik, G., G. Mitov, and L. Ruschendorf (2015). “Portfolio optimization for heavy-tailed assets: Extreme Risk Index vs. Markowitz,” Journal of Empirical Finance, 32, 115 –134.
Malkiel, B. G. and X. Yexiao (2006). “Idiosyncratic Risk and Security Returns,” Tech.rep., https://www.utdallas.edu/~yexiaoxu/IVOT_H.PDF.
Mandelbrot, B. (1963). “The Variation of Certain Speculative Prices,” The Review of Eco-nomic Studies, 36, 394–419.
Mantegna, R. N. and H. E. Stanley (2000). Introduction to Econophysics: Correlationsand Complexity in Finance, Cambridge University Press.
Markowitz, H. (1952). “Portfolio Selection,” The Journal of Finance, 7, 77–91.
——— (1956). “The optimization of a quadratic function subject to linear constraints,” NavalResearch Logistics Quarterly, 3, 111–133.
——— (1959). Portfolio Selection: Efficient Diversification of Investments, Malden, Mas-sachusetts: Blackwell Publishers, Inc., second edition ed.
——— (1987). Mean-variance analysis in portfolio choice and capital markets, Oxford ; NewYork : B. Blackwell.
——— (2014). “Mean-variance approximations to expected utility,” European Journal of Op-erational Research, 234, 346–355.
Markowitz, H., M. T. Hebner, and M. E. Brunson (2009). “Does Portfolio TheoryWork During Financial Crises?” .
Markowitz, H. M. (1991). “Foundations of Portfolio Theory,” The Journal of Finance, 46,469–477.
185
——— (1999). “The Early History of Portfolio Theory: 1600-1960,” Financial Analysts Jour-nal, 55, 5–16.
——— (2010). “Portfolio Theory: As I Still See It,” Annual Review of Financial Economics,2, 1–23.
Marschinski, R., P. Rossi, M. Tavoni, and F. Cocco (2007). “Portfolio selection withprobabilistic utility,” Annals of Operations Research, 151, 223–239.
Martellini, L. (2008). “Toward the Design of Better Equity Benchmarks: Rehabilitatingthe Tangency Portfolio from Modern Portfolio Theory,” Journal of Portfolio Management,34, 34–41.
Martellini, L. and V. Ziemann (2010). “Improved Estimates of Higher-Order Comomentsand Implications for Portfolio Selection,” Review of Financial Studies, 23, 1467–1502.
Martos, B. (1969). “Subdefinite Matrices and Quadratic Forms,” SIAM Journal on AppliedMathematics, 17, 1215–1223.
Matouv sek, J. and B. Gärtner (2007). Understanding and using linear programming,Springer.
Maxey, D. (2015). “Are Risk-Parity Funds a Better Strategy for Diversification?” The WallStreet Journal.
McNamara, J. R. (1998). “Portfolio Selection Using Stochastic Dominance Criteria,” Deci-sion Sciences, 29, 785–801.
McNeil, A. J. and R. Frey (2000). “Estimation of tail-related risk measures for het-eroscedastic financial time series: an extreme value approach,” Journal of Empirical Fi-nance, 7, 271 – 300, special issue on Risk Management.
Merton, R. (1971). “Optimum consumption and portfolio rules in a continuous-time model,”Journal of Economic Theory, 3, 373–413.
Merton, R. C. (1969). “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case,” The Review of Economics and Statistics, 51, 247–257.
——— (1972). “An Analytic Derivation of the Efficient Portfolio Frontier,” The Journal ofFinancial and Quantitative Analysis, 7, pp. 1851–1872.
——— (1980). “On Estimating the Expected Return on the Market: An Exploratory Investi-gation,” Journal of Financial Economics, 8, 323–361.
——— (1987). “A Simple Model of Capital Market Equilibrium with Incomplete Information,”Journal of Finance, 42, 41–81.
186
Meucci, A. (2009). “Managing Diversification,” Risk, 22, 74–79.
Meucci, A., A. Santangelo, and R. Deguest (2014). “Measuring Portfolio DiversificationBased on Optimized Uncorrelated Factors,” SSRN Working Paper.
Meyer, M. and B. Strulovici (2014). “Assessing Interdependence Using the SupermodularStochastic Ordering: Theory and Applications,” .
Miccolis, J. A. and M. Goodman (2012). “Next Generation Investment Risk Management:Putting the ’Modern’ Back in Modern Portfolio Theory,” Journal of Financial Planning, 25,44–51.
Michaud, R. O. (1989). “The Markowitz Optimization Enigma: Is ’Optimized’ Optimal?”Financial Analysts Journal, 45, 31–42.
Montero, P. and J. Vilar (2014). “TSclust: An R Package for Time Series Clustering,”Journal of Statistical Software, 62, 1–43.
Mossin, J. (1966). “Equilibrium in a Capital Asset Market,” Econometrica, 34, 768–783.
Nawrocki, D. N. (1991). “Optimal Algorithms And Lower Partial Moment: Ex-Post Re-sults,” Applied Economics, 23, 465–470.
Nayak, T. K. (1986a). “An analysis of Diversity Using Rao’s Quadratic Entropy,” IndianJournal of Statistics, 48, 315–330.
——— (1986b). “Sampling Distributions in Analysis of Diversity,” Indian Journal of Statistics,48, 1–9.
Nayak, T. K. and J. L. Gastwirth (1989). “The Use of Diversity Analysis to Assess theRelative Influence of Factors Affecting the Income Distribution,” Journal of Business &Economic Statistics, 7, 453–460.
Nelsen, R. B. (2006). An Introduction to Copulas (Springer Series in Statistics), Secaucus,NJ, USA: Springer-Verlag New York, Inc.
Nielsen, F. and R. Nock (2009). “Approximating Smallest Enclosing Balls with Applicationto Machine Learning,” International Journal of Computational Geometry & Applica-tions, 19, 389–414.
Oakley, R. A. (1973). “An application of multidimensional scaling to the construction ofpredictive portfolio selection models,” Ph.D. thesis, Ohio State University, Business Admin-istration, Ohio State University and OhioLINK, available at https://etd.ohiolink.edu/ap/10?0::NO:10:P10_ACCESSION_NUM:osu1261421392.
187
Ogryczak, W. and A. Ruszczynski (1999a). “From stochastic dominance to mean-riskmodels: Semideviations as risk measures,” European Journal of Operational Research, 116,33–50.
——— (1999b). “From stochastic dominance to mean-risk models: Semideviations as riskmeasures1,” European Journal of Operational Research, 116, 33 – 50.
——— (2001). “On consistency of stochastic dominance and mean?semideviation models,”Mathematical Programming, 89, 217–232.
——— (2002). “Dual Stochastic Dominance and Related Mean-Risk Models,” SIAM Journalon Optimization, 13, 60–78.
Ohlson, J. A. (1975). “The Asymptotic Validity of Ouadratic Utility as the Period-SpacingApproaches Zero,” in Stochastic Optimization Models in Finance, ed. by W. T. Ziemba andR. G. Vickson, New York: Academic Press, Inc.
Onnela, J. P., A. Chakraborti, K. Kaski, J. Kertész, and A. Kanto (2003). “Dy-namics of market correlations: Taxonomy and portfolio analysis,” Phys. Rev. E, 68, 56110.
Orozco, J. L. (2004). “Similarity and Dissimilarity Concepts in Machine Learning,” researchReport LSI-04-9-R, Universitat Politecnica De Catalunya, Spain.
Otranto, E. (2008). “Clustering heteroskedastic time series by model-based procedures,”Computational Statistics & Data Analysis, 52, 4685 – 4698.
Oyenubi, A. (2016). “Diversification Measures and the Optimal Number of Stocks in a Port-folio: An Information Theoretic Explanation,” Computational Economics, 1–29.
Patil, G. P. and C. Taille (1982). “Diversity as a Concept and its Measurement,” Journalof the American Statistical Association, 77, 548–561.
Patton, A. J. (2004). “On the Out-of-Sample Importance of Skewness and AsymmetricDependence for Asset Allocation,” Journal of Financial Econometrics, 2, 130–168.
Pavoine, S. (2005). “Methodes Statistiques pour la Mesure de la Biodiversite,” Ph.D. thesis,Universite Claude Bernard-Lyon I.
——— (2012). “Clarifying and developing analyses of biodiversity: towards a generalisationof current approaches,” Methods in Ecology and Evolution, 3, 509–518.
Pavoine, S. and M. B. Bonsall (2009). “Biological diversity: Distinct distributions canlead to the maximization of Rao’s quadratic entropy,” Theoretical Population Biology, 75,153–163.
188
Pavoine, S., S. Ollier, and D. Pontier (2005). “Measuring diversity from dissimilaritieswith Rao’s quadratic entropy : Are any dissimilarities suitable?” Theoretical PopulationBiology, 67, 231–239.
Pearson, N. D. (2002). Risk budgeting : portfolio problem solving with value-at-risk, Wileyfinance series, New York, Chichester, Weinheim: J. Wiley and sons.
Pedersen, S. C. and E. S. Satchell (1998). “An Extended Family of Financial-RiskMeasures,” The Geneva Papers on Risk and Insurance Theory, 23, 89–117.
Pekalska, E. and R. P. W. Duin (2005). The Dissimilarity Representation for Pat-tern Recognition: Foundations And Applications (Machine Perception and Artificial In-telligence), World Scientific Publishing Company.
Pekalska, E., P. Paclik, and R. P. Duin (2001). “A Generalized Kernel Approach toDissimilarity Based Classification,” Journal of Machine Learning Research, 2, 175–211.
Perold, A. F. (1984). “Large-Scale Portfolio Optimization,” Management Science, 30, pp.1143–1160.
Pfaff, B. (2012). Diversification Reconsidered, John Wiley & Sons, Ltd, 189–216.
——— (2013). Financial risk modelling and portfolio optimization with R, Statistics in prac-tice, Chichester, West Sussex: Wiley, bibliogr.
Pfeifer, D. (2013). Correlation, Tail Dependence and Diversification, Berlin, Heidelberg:Springer Berlin Heidelberg, 301–314.
Pflug, G. C., A. Pichler, and D. Wozabal (2012). “The 1/N investment strategy isoptimal under high model ambiguity,” Journal of Banking & Finance, 36, 410 – 417.
Piccolo, D. (1990). “A DISTANCE MEASURE FOR CLASSIFYING ARIMA MODELS,”Journal of Time Series Analysis, 11, 153–164.
Piotrowski, E. W. and J. Sladkowski (2000). “The thermodynamics of portfolios,” Pa-pers cond-mat/0011280, arXiv.org.
Pirvu, T. and K. Schulze (2012). “Multi-stock portfolio optimization under prospect the-ory,” Mathematics and Financial Economics, 6, 337–362.
Pola, G. (2014a). “Is your portfolio effectively diversified? Various perspectives on portfoliodiversification,” Amundi Working Paper WP-040-2014, Amundi Asset Management.
Pola, G. and Z. Ali (2014b). “Entropy, Diversification and the Inefficient Frontier,”Amundi Cross Asset Special Focus, Amundi Asset Management.
189
Polasky, S., C. Costello, and A. Solow (2005). “The Economics of Biodiversity,” inHandbook of Environmental Economics, ed. by K. G. MÃďler and J. R. Vincent, Elsevier,vol. 3 of Handbook of Environmental Economics, chap. 29, 1517–1560.
Porter, R. B. (1974a). “Semivariance and Stochastic Dominance: A Comparison,” TheAmerican Economic Review, 64, 200–204.
——— (1974b). “Semivariance and Stochastic Dominance: A Comparison,” The AmericanEconomic Review, 64, pp. 200–204.
Porter, R. B. and R. P. Bey (1974). “An Evaluation of the Empirical Significance ofOptimal Seeking Algorithms in Portfolio Selection,” The Journal of Finance, 29, pp. 1479–1490.
Porter, R. B. and J. E. Gaumnitz (1972). “Stochastic Dominance vs. Mean-VariancePortfolio Analysis: An Empirical Evaluation,” The American Economic Review, 62, pp.438–446.
Porter, R. B. and R. C. Pfaffenberger (1975). “Efficient Algorithms for ConductingStochastic Dominance Tests on Large Numbers of Portfolios: Reply,” The Journal of Fi-nancial and Quantitative Analysis, 10, pp. 181–185.
Porter, R. B., J. R. Wart, and D. L. Ferguson (1973). “Efficient Algorithms for Con-ducting Stochastic Dominance Tests on Large Numbers of Portfolios,” The Journal of Fi-nancial and Quantitative Analysis, 8, pp. 71–81.
Pulley, L. B. (1981). “A General Mean-Variance Approximation to Expected Utility forShort Holding Periods,” The Journal of Financial and Quantitative Analysis, 16, pp. 361–373.
——— (1983). “Mean-Variance Approximations to Expected Logarithmic Utility,” OperationsResearch, 31, pp. 685–696.
Qian, E. (2006). “On the Financial Interpretation of Risk Contribution: Risk Budgets DoAdd Up,” Journal of Investment Management, 4, 41–51.
——— (2011). “Risk Parity and Diversification,” The Journal of Investing, 20, 119–127.
——— (2012). “Diversification Return and Leveraged Portfolios,” Journal of Portfolio Man-agement, 38, 14–25.
Quiggin, J. (1982). “A theory of anticipated utility,” Journal of Economic Behavior & Or-ganization, 3, 323 – 343.
Quirk, J. P. and R. Saposnik (1962). “Admissibility and Measurable Utility Functions,”The Review of Economic Studies, 29, pp. 140–146.
190
RACHEV, S., S. ORTOBELLI, S. STOYANOV, F. J. FABOZZI, and A. BIGLOVA
(2008). “DESIRABLE PROPERTIES OF AN IDEAL RISK MEASURE IN PORTFOLIOTHEORY,” International Journal of Theoretical and Applied Finance, 11, 19–54.
Rachev, S. T., S. V. Stoyanov, A. Biglova, and F. J. Fabozzi (2005). Data Analysisand Decision Support, Berlin, Heidelberg: Springer Berlin Heidelberg, chap. An EmpiricalExamination of Daily Stock Return Distributions for U.S. Stocks, 269–281.
Rao, R. (1982a). “Diversity : Its Measurement, Decomposition, Apportionment and Analy-sis,” Indian Journal of Statistics, 44, 1–22.
Rao, R. and T. K. Nayak (1985). “Cross Entropy, Dissimilarity Measures, and Characteri-zations of Quadratic Entropy,” Information Theory, IEEE Transactions, 31, 589–593.
Rao, R. C. (1982b). “Diversity and Dissimilarity Coefficients: A Unified Approach,” Theo-retical Population Biology, 21, 24–43.
——— (2004). Rao’s Axiomatization of Diversity Measures, John Wiley & Sons, Inc.
——— (2010). “Quadratic Entropy and Analysis of Diversity,” Sankhy : The Indian Journalof Statistics, 72, 70–80.
Richard, S. F. (1975). “Multivariate Risk Aversion, Utility Independence and SeparableUtility Functions,” Management Science, 22, 12–21.
Ricotta, C. and L. Szeidl (2006). “Towards a unifying approch to diversity measures :Bridging the gap between the Shannon entropy and Rao’s quadratic index,” TheoreticalPopulation Biology, 70, 237–243.
——— (2009). “Diversity partitioning of Rao’s quadratic entropy,” Theoretical PopulationBiology, 76, 299–302.
Rockafellar, R. T. and S. Uryasev (2000). “Optimization of conditional value-at-risk,”Journal of risk, 2, 21–41.
Rockafellar, R. T., S. Uryasev, and M. Zabarankin (2006a). “Generalized deviationsin risk analysis,” Finance and Stochastics, 10, 51–74.
——— (2006b). “Master funds in portfolio analysis with general deviation measures,” Journalof Banking & Finance, 30, 743–778.
——— (2006c). “Optimality conditions in portfolio analysis with general deviation measures,”Mathematical Programming, 108, 515–540.
——— (2007). “Equilibrium with investors using a diversity of deviation measures,” Journalof Banking & Finance, 31, 3251 – 3268, risk Management and Quantitative Approaches inFinance.
191
Roll, R. (1977). “A critique of the asset pricing theory’s tests Part I: On past and potentialtestability of the theory,” Journal of Financial Economics, 4, 129 – 176.
——— (1980). “Orthogonal Portfolios,” The Journal of Financial and Quantitative Analysis,15, pp. 1005–1023.
Roncalli, T. (2014). “Introduction to risk parity and budgeting,” in Chapman & Hall/CRCfinancial mathematics series., ed. by B. R. T. . Francis.
Roncalli, T. and G. Weisang (2012). “Risk Parity Portfolios with Factors,” in workingpapers, available at http://ssrn.com/abstract=2155159orhttp://dx.doi.org/10.2139/ssrn.2155159.
——— (2016). “Risk parity portfolios with risk factors,” Quantitative Finance, 16, 377–388.
Rosen, D. and D. Saunders (2010). “Risk factor contributions in portfolio credit riskmodels,” Journal of Banking & Finance, 34, 336 – 349.
Ross, S. A. (1976). “The arbitrage theory of capital asset pricing,” Journal of EconomicTheory, 13, 341–360.
Rothschild, M. and J. E. Stiglitz (1970). “Increasing risk: I. A definition,” Journal ofEconomic Theory, 2, 225–243.
——— (1971). “Increasing risk II: Its economic consequences,” Journal of Economic Theory,3, 66–84.
Roy, A. D. (1952). “Safety First and the Holding of Assets,” Econometrica, 20, 431–449.
Rubinstein, M. (1973). “The Fundamental Theorem of Parameter-Preference Security Val-uation,” The Journal of Financial and Quantitative Analysis, 8, pp. 61–69.
——— (2002). “Markowitz’s "Portfolio Selection": A Fifty-Year Retrospective,” The Journalof Finance, 57, 1041–1045.
Rudin, A. M. and J. Morgan (2006). “A Portfolio Diversification Index,” The Journal ofPortfolio Management, 32, 81–89.
Salazar, Y., R. J. Bianchi, M. E. Drew, and S. Trück (2014). “The DiversificationDelta: A Different Perspective,” .
Salazar Flores, Y., R. J. Bianchi, M. E. Drew, and S. Trück (2017). “The Di-versification Delta: A Different Perspective,” The Journal of Portfolio Management, 43,112–124.
Samuelson, P. A. (1967). “General Proof that Diversification Pays,” The Journal of Finan-cial and Quantitative Analysis, 2, 1–13.
192
——— (1969). “Lifetime Portfolio Selection By Dynamic Stochastic Programming,” The Re-view of Economics and Statistics, 51, 239–246.
——— (1970). “The Fundamental Approximation Theorem of Portfolio Analysis in terms ofMeans, Variances and Higher Moments,” The Review of Economic Studies, 37, 537–542.
Sandstrom, A. (2011). Handbook of solvency for actuaries and risk managers: theory andpractice, 10, Chapman & Hall/CRC.
Scheffman, D. T. (1973). “The Diversification Problem in Portfolio Models,” Departmentof Economics Research Reports 7318, University of Western Ontario.
Scherer, B. (2007). “Can robust portfolio optimisation help to build better portfolios?”Journal of Asset Management, 7, 374?387.
Schneider, H. (1977). “The concepts of irreducibility and full indecomposability of a matrixin the works of Frobenius, Konig and Markov,” Linear Algebra and its Applications, 18, 139– 162.
Schoenberg, I. J. (1938a). “Metric Spaces and Completely Monotone Functions,” Annals ofMathematics, 39, 811–841.
Schoenberg, J. (1938b). “Metric spaces and positive definite functions,” Transactions of theAmerican Mathematical Society, 522–536.
Scholkopf, B. (1993). “The kernel trick for distances,” in TR MSR 2000-51, MicrosoftResearch, 5–3.
——— (1997). “Support Vector Learning,” .
Schölkopf, B., J. C. Platt, J. C. Shawe-Taylor, A. J. Smola, and R. C.
Williamson (2001). “Estimating the Support of a High-Dimensional Distribution,” NeuralComput., 13, 1443–1471.
Scholkopf, B., A. Smola, K.-R. Muller, C. Burges, and V. Vapnik (1998). “SupportVector Methods in Learning and Feature Extraction,” .
Scholkopf, B., R. C. Williamson, A. Smola, and J. Shawe-Taylor (1999). “SVEstimation of a Distribution’s Support,” .
Sen, A. (1976). “Poverty: An Ordinal Approach to Measurement,” Econometrica, 44, 219–231.
Sereda, E., E. Bronshtein, S. Rachev, F. Fabozzi, W. Sun, and S. Stoyanov (2010).“Distortion Risk Measures in Portfolio Optimization,” in Handbook of Portfolio Construc-tion, ed. by J. Guerard, JohnB., Springer US, 649–673.
193
Shakespeare, W. and W. L. Phelps (c1923). The merchant of Venice, The Yale Shake-speare, New Haven: Yale University Press, bibliogr. : p. [114].
Shalit, H. and S. Yitzhaki (1984). “Mean-Gini, Portfolio Theory, and the Pricing of RiskyAssets,” The Journal of Finance, 39, 1449–1468.
——— (2005). “The Mean-Gini Efficient Portfolio Frontier,” Journal of Financial Research,28, 59–75.
Sharpe, W. F. (1963). “A Simplified Model for Portfolio Analysis,” Management Science, 9,277–293.
——— (1964). “Capital Asset Prices : A Theory of Market Equilibrium under Conditions ofRisk,” The Journal of Finance, 19, 425–442.
——— (1967). “Portfolio Analysis,” The Journal of Financial and Quantitative Analysis, 2,pp. 76–84.
——— (1972). “Risk, Market Sensitivity and Diversification,” Financial Analysts Journal, 28,74–79.
——— (1991). “Capital Asset Prices with and without Negative Holdings,” The Journal ofFinance, 46, pp. 489–509.
——— (2002). “Budgeting and Monitoring Pension Fund Risk,” Financial Analysts Journal,58, 74–86.
Shaw, W. T. (2010). “Monte Carlo Portfolio Optimization for General Investor Risk-ReturnObjectives and Arbitrary Return Distributions: a Solution for Long-only Portfolios,” .
Shimatani, K. (2001). “On the measurement of species diversity incorporating species differ-ences,” Oikos, 93, 135–147.
Simaan, Y. (1993). “What is the Opportunity Cost of Mean-Variance Investment Strategies?”Management Science, 39, 578–587.
——— (2014). “The opportunity cost of mean?variance choice under estimation risk,” Euro-pean Journal of Operational Research, 234, 382 – 391, 60 years following Harry Markowitz?scontribution to portfolio theory and operations research.
Simchi-Levi, D., X. Chen, and J. Bramel (2005). Convexity and Supermodularity, NewYork, NY: Springer New York, 13–32.
Sklar, A. (1959). Fonctions de re?partition a? n dimensions et leurs marges.
194
——— (1996). Random variables, distribution functions, and copulas—a personal look back-ward and forward, Hayward, CA: Institute of Mathematical Statistics, vol. Volume 28 ofLecture Notes–Monograph Series, 1–14.
Skoulakis, G. (2012). “On the quality of Taylor approximations to expected utility,” AppliedFinancial Economics, 22, 863–876.
Smola, A. J., B. Scholkopf, and K.-R. Muller (1998). “The connection between regu-larization operators and support vector kernels,” Neural Networks, 11, 637 – 649.
Solin, D. (2012). “Why I Am Clinging to Failed Investment Strategies,” .
Statman, M. (1987). “How Many Stocks Make a Diversified Portfolio?” The Journal ofFinancial and Quantitative Analysis, 22, 353–363.
——— (2013). “Is Markowitz Wrong? Investment Lessons from the Financial Crisis,” TheJournal of Portfolio Management, 40, 8–11.
Statman, M. and J. Scheid (2005). “Global Diversification,” Journal of Investment Man-agement, 3, 1–11.
——— (2008). “Correlation, return gaps and the benefits of diversification,” Journal of Port-folio Management, 34, 132–139.
Steinbach, M. C. (2001). “Markowitz Revisited: Mean-Variance Models in Financial Port-folio Analysis,” SIAM Review, 43, 31–85.
Steiner, A. (2012). “Risk Parity for the Masses,” Journal of Investing, 21, 129–139,10.
Steuer, R., Y. Qi, and M. Hirschberger (2006). “Portfolio optimization: New capabili-ties and future methods,” Zeitschrift fur Betriebswirtschaft, 76, 199–220.
Stirling, A. (2007). “A general framework for analysing diversity in science, technology andsociety,” The Journal of The Royal Society, 4, 707–719.
——— (2010). “Multicriteria diversity analysis: A novel heuristic framework for appraisingenergy portfolios,” Energy Policy, 38, 1622–1634.
Stone, B. K. (1973). “A General Class of Three-Parameter Risk Measures,” The Journal ofFinance, 28, 675–685.
Sullivan, E. J. (2011). A.D. Roy: The Forgotten Father of Portfolio Theory, 73–82.
Swisher, P. (2009). “MPT Was Never Alive,” Journal of Financial Planning, Trends inInvesting, 16–18.
195
Sylvester, J. J. (1857). “A Question in the Geometry of Situation,” Quarterly Journal ofPure and Applied Mathematics, 1.
Taliaferro, R. (2012). “Understanding risk-based portfolios,” The Journal of InvestmentStrategies, 1, 119–131.
Tarazaga, P., T. Hayden, and J. Wells (1996). “Circum-Euclidean distance matricesand faces,” Linear Algebra and its Applications, 232, 77 – 96.
Tasche, D. (2000). “Risk Contributions and Performance Measurement,” Tech. rep., Researchpaper, Zentrum Mathematik (SCA.
——— (2006). “Measuring sectoral diversification in an asymptotic multifactor framework,”Journal of credit risk, 2.
——— (2008). “Capital Allocation to Business Units and Sub-Portfolios: the Euler Principle,”Papers, arXiv.org.
Tax, D. M. and R. P. Duin (2004). “Support Vector Data Description,” Machine Learning,54, 45–66.
Tax, D. M. J. (2001). “One-class classification: Concept-learning in the absence of counter-examples,” Ph.D. thesis, Technische Universiteit Delft.
Tax, D. M. J. and R. P. W. Duin (1999). “Data domain description using support vectors,”in Proceedings of the European Symposium on Artificial Neural Networks, 251–256.
Tobin, J. (1958). “Liquidity Preference as Behavior Towards Risk,” The Review of EconomicStudies, 25, 65–86.
Tola, V., F. Lillo, M. Gallegati, and R. N. Mantegna (2008). “Cluster analysisfor portfolio optimization,” Journal of Economic Dynamics and Control, 32, 235 – 258,applications of statistical physics in economics and finance.
Torgerson, W. S. (1952). “Multidimensional Scaling : I Theory and Method,” Psychome-trika, 17, 401–419.
Tsang, I. W., A. Kocsor, and J. T. Kwok (2007). “Simpler Core Vector Machines withEnclosing Balls,” in Proceedings of the 24th International Conference on Machine Learning,New York, NY, USA: ACM, ICML ’07, 911–918.
Tsang, I. W., J. T. Kwok, and P.-m. Cheung (2005). “Core Vector Machines: Fast SVMTraining on Very Large Data Sets,” Journal of Machine Learning Research, 6, 363–392.
Tsiang, S. C. (1972). “The Rationale of the Mean-Standard Deviation Analysis, SkewnessPreference, and the Demand for Money,” The American Economic Review, 62, pp. 354–371.
196
Tutuncu, R. and M. Koenig (2004). “Robust Asset Allocation,” Annals of OperationsResearch, 132, 157?187.
Tversky, A. and D. Kahneman (1992). “Advances in Prospect Theory: Cumulative Rep-resentation of Uncertainty,” Journal of Risk and Uncertainty, 5, pp. 297–323.
Unger, A. (2015). The Use of Risk Budgets in Portfolio Optimization, Wiesbaden: SpringerFachmedien Wiesbaden.
Uppal, R. and T. Wang (2003). “Model Misspecification and Underdiversification,” Journalof Finance, 58, 2465–2486.
van der Hoek, J. and M. Sherris (2001). “A class of non-expected utility risk measuresand implications for asset allocations,” Insurance: Mathematics and Economics, 28, 69 –82.
Vapnik, V. N. (1995). The Nature of Statistical Learning Theory, New York, NY, USA:Springer-Verlag New York, Inc.
——— (1999). “An Overview of Statistical Learning Theory,” Trans. Neur. Netw., 10, 988–999.
VERES, B. (2011). “Goodbye, Old Theory,” Financial Planning, 41, 35–36.
Vermorken, M. A., F. R. Medda, and T. Schroder (2012). “The Diversification Delta: A Heigher-Moment Measure for Portfolio Diversification,” Journal Of Portfolio Manage-ment, 39, 67–74.
Vickson, R. G. (1975). “Stochastic Dominance for Decreasing Absolute Risk Aversion,” TheJournal of Financial and Quantitative Analysis, 10, pp. 799–811.
Vickson, R. G. and M. Altmann (1977). “On the Relative Effectiveness of StochasticDominance Rules: Extension to Decreasingly Risk-Averse Utility Functions,” The Journalof Financial and Quantitative Analysis, 12, pp. 73–84.
Von Neumann, J. and O. Morgenstern (2007). Theory of games and economic behavior,Princeton university press.
Voros, J. (1986). “Portfolio analysis?an analytic derivation of the efficient portfolio frontier,”European Journal of Operational Research, 23, 294 – 300.
——— (1987). “The explicit derivation of the efficient portfolio frontier in the case of degen-eracy and general singularity,” European Journal of Operational Research, 32, 302 – 310,third EURO Summer Institute Special Issue Decision Making in an Uncertain World.
Wagner, W. H. and S. C. Lau (1971). “The Effect of Diversification on Risk,” FinancialAnalysts Journal, 27, 48–53.
197
Wang, G.-J., C. Xie, and S. Chen (2016). “Multiscale correlation networks analysis of theUS stock market: a wavelet analysis,” Journal of Economic Interaction and Coordination,1–34.
Wang, H., R. Pappadà, F. Durante, and E. Foscolo (2017). A Portfolio DiversificationStrategy via Tail Dependence Clustering, Cham: Springer International Publishing, 511–518.
Wang, Q. (2015). “Social Networks, Asset Allocation and Portfolio Diversification,” Master’sthesis, University of Waterloo, Ontario (Canada).
Wang, R., V. Bignozzi, and A. Tsanakas (2015). “How Superadditive Can a Risk MeasureBe?” SIAM Journal on Financial Mathematics, 6, 776–803.
Wang, S. (1995). “Insurance pricing and increased limits ratemaking by proportional hazardstransforms,” Insurance: Mathematics and Economics, 17, 43 – 54.
Wang, S. S., V. R. Young, and H. H. Panjer (1997). “in Honor of Prof. J.A. BeekmanAxiomatic characterization of insurance prices,” Insurance: Mathematics and Economics,21, 173 – 183.
Wang, Y., Y. Zou, S. Zheng, and X. Guo (2008). “Simpler Minimum Enclosing Ball:Fast approximate MEB algorithm for extensive kernel methods,” in Control and DecisionConference, 2008. CCDC 2008. Chinese, 3576–3581.
Wang, Z. (2005). “A Shrinkage Approach to Model Uncertainty and Asset Allocation,” Reviewof Financial Studies, 18, 673–705.
Warner, J. (2010). “Rethinking Modern Portfolio Theory.(cover story),” Bank InvestmentConsultant, 18, 12–17.
Whitmore, G. A. (1970). “Third-Degree Stochastic Dominance,” The American EconomicReview, 60, pp. 457–459.
Willenbrock, S. (2011). “Diversification Return, Portfolio Rebalancing, and the CommodityReturn Puzzle,” Financial Analysts Journal, 67.
Williams, C. K. (2002). “On a Connection between Kernel PCA and Metric Multidimen-sional Scaling,” Machine Learning, 46, 11–19.
Williams, J. B. (1956). The theory of investment value, Amsterdam: North-Holland Pub.Co.
Woerheide, W. and D. Persson (1993). “An Index of Portfolio Diversification,” FinancialServices Review, 2, 73–85.
198
Wright, R. (1987). “Expectation dependence of random variables, with an application inportfolio theory,” Theory and Decision, 22, 111–124.
Xu, S., I. Hwang, and F. In (2016). “The Effect of Diversification on Tail Risk: Evidencefrom US Equity Mutual Fund Portfolios,” International Review of Finance, n/a–n/a.
Yaari, M. E. (1987). “The Dual Theory of Choice under Risk,” Econometrica, 55, 95–115.
Yanagimoto, T. and M. Okamoto (1969). “Partial orderings of permutations and mono-tonicity of a rank correlation statistic,” Annals of the Institute of Statistical Mathematics,21, 489–506.
Yanou, G. (2010). “Mean-Variance Framework and Diversification Objective: Theoreticaland Empirical Implications,” Tech. rep., University of Paris-1, http://ssrn.com/abstract=1722285.
Yildirim, E. A. (2008). “Two Algorithms for the Minimum Enclosing Ball Problem,” SIAMJournal on Optimization, 19, 1368–1391.
Yin, Y. and P. Shang (2014). “Modified multidimensional scaling approach to analyzefinancial markets,” Chaos: An Interdisciplinary Journal of Nonlinear Science, 24, –.
Yitzhaki, S. (1982). “Stochastic Dominance, Mean Variance, and Gini’s Mean Difference,”The American Economic Review, 72, 178–185.
Zhao, Y. and D. N. Naik (2012). “Hypothesis testing with Rao’s quadratic entropy and itsapplication to Dinosaur biodiversity,” Journal of Applied Statistics, 39, 1667–1680.
Zhou, B. and N. Trinajstic (2007). “On the largest eigenvalue of the distance matrix of aconnected graph,” Chemical Physics Letters, 447, 384 – 387.
Zhou, G., K.-C. Tohemail, and J. Sun (2005). “Efficient Algorithms for the SmallestEnclosing Ball Problem,” Computational Optimization and Applications, 30, 147–160.
Zhou, R., R. Cai, and G. Tong (2013). “Applications of Entropy in Finance: A Review,”Entropy, 15, 4909–4931.
Zurcher, S. (2007). “Smallest Enclosing Ball for a Point Set with Strictly Convex LevelSets,” Master’s thesis, ETH Zurich, Zurich.
199
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