Rao's Quadratic Entropy, Risk Management and Portfolio Theory

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Rao’s Quadratic Entropy, Risk Management and

Portfolio Theory

Thèse

Nettey Boevi Gilles Koumou

Doctorat en économique

Philosophiæ doctor (Ph.D.)

Québec, Canada

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Rao’s Quadratic Entropy, Risk Management and

Portfolio Theory

Thèse

Nettey Boevi Gilles Koumou

Sous la direction de:

Benoît Carmichael, directeur de recherche Kevin Moran, codirecteur de recherche

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Résumé

Cette thèse porte sur le concept de la diversification et sa mesure en théorie des choix de portefeuille. La diversification est un concept clé en finance et en économique, et est au cœur de la théorie des choix de portefeuille. Elle représente l’un des plus importants outils de gestion 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 une approche unifiée et cohérente de mesure et gestion de la diversification.

Cette thèse répond à ce besoin et développe une nouvelle classe de mesures de diversification de portefeuille en adaptant à l’économie financière l’entropie quadratique de Rao, une mesure de diversité proposée par Rao et utilisée en statistique, en biodiversité, en écologie et dans plusieurs autres domaines. La thèse démontre que si l’entropie quadratique de Rao est bien calibrée, elle devient une classe valide de mesures de diversification de portefeuille résumant, de manière simple, les caractéristiques complexes de la diversification de portefeuille, et offrant en même temps une théorie unifiée qui englobe de nombreuses contributions antérieures. Ensuite, la thèse présente deux applications de la classe de mesures proposée. La première application s’est intéressée à la stratégie de diversification de portefeuille maximum diversi-fication (MD) développée par Choueifaty and Coignard (2008). Elle propose de nouvelles formulations de cette dernière en se basant sur la classe de mesures proposée. Ces nouvelles formulations permettent de donner un fondement théorique à la stratégie MD et d’améliorer ses 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, en particulier le processus de rémunération des actifs. Elle offre également de nouvelles possibilités d’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 fundamental importance in finance and insurance domains as risk measurement and management. Conse-quently, 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 limited applications. Developing a coherent measure of portfolio diversification is therefore an active research area in investment management.

In this thesis, a novel, coherent, general and rigorous theoretical framework to manage and quantify portfolio diversification inspiring from Rao (1982a)’s Quadratic Entropy (RQE), a general approach to measuring diversity, is proposed. More precisely, this thesis demonstrates that when RQE is judiciously calibrated it becomes a valid class of portfolio diversification measures summarizing complex features of portfolio diversification in a simple manner and provides at the same time a unified theory that includes many previous contributions. Next, this thesis presents two applications of the proposed class of portfolio diversification measures. In the first application, new formulations of maximum diversification strategy of

Choueifaty and Coignard (2008) is provided based on the proposed class of measures. These new formalizations clarify the investment problem behind the MD strategy, help identify the source of its strong out-of-sample performance relative to other diversified portfolios, and suggest 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 function is provided based on the proposed class of measures. This new formulation significantly improves the mean-variance model understanding, in particular in terms of asset pricing. It also offers new directions along which the mean-variance model can be improved without additional computational costs.

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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. . . 105

5.2 List of Performance Metrics . . . 107

5.3 Characteristics of Portfolios-49Ind . . . 108

5.4 Other Characteristics of Portfolios-49Ind . . . 109

5.5 Annualized Returns During Bear Markets-49Ind. . . 110

5.6 Other Characteristics of Portfolios During Bear Markets-49Ind . . . 111

5.7 Characteristics of Portfolios-World100FF. . . 113

5.8 Other Characteristics of Portfolios-World100FF . . . 114

5.9 Annualized Returns During Bear Markets-World100FF . . . 115

5.10 Other Characteristics of Portfolios During Bear Markets-World100FF . . . 116

5.11 Regression of RQEP_D_ρ on the MDP . . . 116

6.1 List of Portfolios Considered. . . 127

6.2 Scenario I: Performance of Portfolios . . . 130

6.3 Scenario II: Performance of Portfolios . . . 131

6.4 Scenario III: Performance of Portfolios . . . 133

6.5 Portfolio Risks . . . 136

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List of Figures

2.1 Mean-Variance Portfolio Frontier (Without short sales). . . 29

2.2 Mean-Variance Efficient Frontier With Risk-Free Asset . . . 31

2.3 Relationship Between Portfolio Size and Risk . . . 38

3.1 Representation of the Effective Number of Correlated Bets for N = 2, υ = 2

and % = 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 Regression

of ξ2(D)

on ξ_min(S(∞)) . . . 103 5.2 U.S. Market Portfolio Returns. . . 111

6.1 Scenario I: Portfolio Cumulative Returns. . . 129

6.2 Scenario I: Performance Metrics Depending on Transaction Costs Parameters κ 130

6.3 Scenario I: Diversification Level Comparison . . . 131

6.4 Scenario II: Portfolio Cumulative Returns . . . 132

6.5 Scenario II: Performance Metrics Depending on Transaction Costs Parameters κ 133

6.6 Scenario III: Portfolio Cumulative Returns . . . 134

6.7 Scenario III: Performance Metrics Depending on Transaction Costs Parameters κ 135

7.1 Mean-Variance Portfolio Frontier in the Space σ2_{, H} D, µ

(when asset are

risky and short sales are allowed) . . . 144

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À la mémoire de ma mére Anani Kokoe Djiffa Delphine

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An investor who knew future returns with certainty would invest in only one security, namely the one with the highest future return. If several securities had the same, highest, future return then the investor would be indifferent between any of these, or any combination of these. In no case would the investor actually prefer a diversified portfolio. But diversification is a common and reasonable

investment practice. Why? To reduce uncertainty!

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Remerciements

Sans le soutien et l’aide de certaines personnes cette thèse n’aura jamais vu le jour. Je tiens à 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. Kevin Moran 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 de recherche 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 de doctorat en recherche pour étudiants étrangers d’une durée de trois ans;

Tous mes camarades de promotion: Setou Diarra, Ali Yedan, Jean Armand Gnagne, Mbéa Bell, 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 préparation de 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 encouragement et leur sacrifice;

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Acronyms

bcbs Basel Committee on Banking Supervision

bm Bouchaud’s measure

calpers California Public Employees’ Retirement System capm Capital Asset Pricing Model

cd Coefficient of Determination

cebs Committee of European Banking Supervisors

ceiops Committee of European Insurance and Occupational Pension Supervisors cnd Conditional Negative Definite

cr Cumulative Return

crmcr Committee on Risk Management and Capital Requirements csnd Conditional Strictly Negative Definite

cvar Conditional Value-at-Risk cvarr Conditional Value-at-Risk Ratio

dd Delta Diversification

dr Diversification Ratio

eiopa European Insurance and Occupational Pension Authority enb Effective Number of Uncorrelated Bets

enc Effective Number of Constituents encb Effective Number of Correlated Bets

erc Equally Risk Contribution

ercp Equally Risk Contribution Portfolio

eut Expected Utility Theory

mvp Mean-Variance Portfolio

ewp Equally Weighted Portfolio ftse Financial Times Stock Exchange

gs Gini-Simpson

md Maximum Diversification

mdd Maximum Drawdown Risk

mddr Maximum Drawdown Risk Ratio

mdp Most Diversified Portfolio

ml Maximum Loss

mp Market Portfolio

mpt Modern Portfolio Theory

mv Mean-Variance

nc2p 2-Norm Constrained Minimum-Variance Portfolio

pd Positive Definite

pdi Portfolio Diversification Index prqe Portfolio Rao’s Quadratic Entropy

ps portfolio Size psd Positive Semi-Definite pt Portfolio Theory rb Risk Budgeting rc Risk Contribution rp Risk Parity

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rqe Rao’s Quadratic Entropy

rqep Rao’s Quadratic Entropy Portfolio

sr Sharpe Ratio

tobam Think Out of the Box Asset Management

trn Turnover

var Value-at-Risk

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Notation

Basic Sets, Spaces and Operators

|a| _{absolute value of a ∈ R}

Cov(.) covariance operator

E(.) expectation operator

RN, k.k2

Euclidean space

h., .i inner product

h., .iX inner product on space a ∈ X

kAk∞ infinity norm of matrix A ∈ RN. kxk∞= max 1≤j≤N

PN i=1|aij| kAkp Lp norm of matrix A ∈ RN. kAkp =

PN

i=1|aij| p1/p kxk∞ infinity norm of x ∈ RN. kxk∞= max

1≤j≤N PN i=1|xi| kxkp Lp norm of x ∈ RN. kxkp = PN i=1|xi|p 1/p k.k norm

RN N -dimensional real vector space

N set of natural numbers

A universe of N assets. A = {A_i}N

i=1 where Ai is asset i

R(p,q), k.kE

pseudo-Euclidean space

R _{vector space of bounded real-valued random variables. R = L}∞(Ω, F , P ) is the 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 numbers

R− set of negative real numbers R+ set of positive real numbers Var(.) variance operator

W 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)N_i,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 = (r_i, ..., 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, ..., σ2_N)> Σ covariance matrix. Σ = (σij)Ni,j=1

σ vector column of asset volatility. σ = (σ₁, ..., σN)> S similarity matrix. S = (s_ij)N

i,j=1 w portfolio. w = (wi, ..., wN)>

Scalars and Functions

σ2_(w) _{portfolio w weighted average variance. σ}2_{(w) = w}>_σ2 βi covariance between asset i and the market. βi= σ_σim2

m

ρij correlation between asset i and j. ρij = σij

σiσj

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 ∈ R rit observation of Ri at period t. rit∈ R

µ(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−1_σ2_.

σi asset volatility or standard deviation. σi=pVar(Ri) σ_i2 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 her consumption and how to allocate it among goods and services. Second, how to invest his or her saving among various assets. The first problem is known as the consumption-saving decision and the second as the portfolio selection decision. The two problems can be analyzed separately (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 is devoted to the study of the concept of diversification and its measurement. It develops a novel, coherent, flexible, unified, computational efficient and rigorous approach for diversification measurement 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 primary concern is portfolio selection problem and asset pricing theory. Its Achilles’ heel is portfolio diversification.

Portfolio diversification consists in investing in variety assets and “do not put all your eggs in 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. The key to the success of this multilateral insurance lies in the interdependence or more generally in the dissimilarity between assets. More assets are dissimilar, more the probability that they do poorly at the same time in the same proportion is low and the better is the protection offered by this multilateral insurance which is the diversification.

Diversification is a well-established risk management technique in economic and finance long before the birth of portfolio theory. Mention of it can be found in Babylonian Talmud: Tractate Baba Mezia, folio 42a 1 (seeDeMiguel 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.

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inBernoulli (1738/1954)

It is advisable to divide goods which are exposed to some small danger into several small portions rather than to risk them all together.

and more recently in Leavens (see1945, 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 probable results 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 by diversification, in narrowing the spread between probable losses and gains, varies as the square root of the number of items. For example, it would take 40 issues to give results twice as good (that is, with half the spread) as 10 issues.

Thus, after a reasonable diversification, which naturally can (and must, because of inadequate supplies) be larger for a portfolio measured in millions of dollars than for one measured in thousands of dollars, the advantage of any further diver-sification hardly balances the difficulty of choosing (and watching) a great variety of 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 among industries is needed for that purpose. Nor can diversification among industries protect against cyclical factors that may depress all industries at the same time. Diversification is only one principle of investment management; it is primarily to offset lack of full knowledge. The investor must still use all the information and judgment that he can muster in choosing specific securities and in timing his purchases and sales.

What was missing, as mentioned inMarkowitz(1999), “was an adequate theory of investment that covered the effects of diversification when risks are correlated, distinguished between efficient and inefficient portfolios, and analyzed risk-return trade-off on the portfolio as a whole”. Portfolio theory contribution therefore has been to provide a set of adequate theories of investment 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 portfolio theory or efficient diversification. This model marks the beginning of portfolio theory and constitutes 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 portfolio return for a given level of portfolio risk, where return is measured by expected return, risk by variance and the level of risk is determined by the investor’s ability to assume the risk or the investor’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 on academic research and the financial industry as a whole. It is the foundation of investment education for chartered financial planner (CFP), for chartered financial analysts (CFA) and for master of business administration (MBA) (Kitces,2012). The funds under its management is estimated to around $7 trillion (Solin,2012). The number of articles in Google Scholar citing Markowitz’s original paper Portfolio Selection is about 25,857. The paper is also the most cited articles of all time in The Journal of Finance. When you search for modern portfolio theory, you obtain about 4,230,000 results. Not surprisingly, Markowitzwon a Nobel Prize in Economics 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 diversification fails 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, CIO and managing partner at Quoniam, a quantitative asset management firm based in Frankfurt, inFabozzi et al.(see 2014, pp. 28-35) says

The financial crisis has clearly shown that when you need diversification most, it may not work. Historical correlations may simply be wrong. Different liquidity of different asset classes may mean that some less risky assets may still be punished because 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 real world...I began to watch it closely starting in the 1980s, and with the passage of each year I’ve appreciated more and more that it’s really nothing more than a marvelous academic concept that has been heavily popularized within the financial planning community. It’s nothing more than an abstraction from reality that doesn’t have a lot of client-based relevance.

As a result, a new portfolio diversification strategy was proposed. This new diversification strategy, 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 over

Markowitz’s diversification remains an open question (see Maxey,2015;Laise,2010;Corkery et al.,2010).

For the defenders, diversification does not fail during the 2008-2009 financial crisis. It is misunderstood. 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, other than high-quality sovereign debt, moved in the same direction (down). We argue that the attacks are undeserved. Instead, we believe that the problem is "user error"; most investors were never as diversified as they thought they were. There is ample room for improvement by shifting the focus from asset class diversification to factor diversification.

Statman(2013, pp. 11) says

Even those who believe that Markowitz is wrong continue to use the language of mean-variance portfolio theory. We have gained Markowitz’s mathematical formulation of diversification and adopted its language of correlations, but in the process 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 crash of 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 “As predicted by portfolio theory” and followed by the phrase, “which is why one should 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 into account 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 the regulators, in particular in insurance and banking industries, to take into account diversifica-tion in risk management reguladiversifica-tion (see CEBS,2010;Laas and Siegel,2016;CEIOPS,2010a;

CRMCR,2016;BCBS,2010,2013;EIOPA,2014;CEIOPS,2010b). However, less is done for the 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 and Shay,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 proven totally 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 managers have usually based their choice on convenience, familiarity, empirical properties, or on vague methodological grounds. This can have major implications in terms of risk and consequently social welfare if they are mistaken. Thus, as in the case of inequality (seeSen,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 coherent framework to manage and quantify portfolio diversification. This is the goal of this thesis. The thesis develops a novel, coherent, flexible, unified, computational efficient and rigorous approach to portfolio diversification measurement. The suggested diversification statistics inspiring from Rao’s Quadratic Entropy (RQE) summarizes complex features of portfolio diversification in a simple manner and provides at the same time a unified theory that includes many previous contributions.

RQE is a general approach to measuring diversity introduced by Rao (1982a,b) and 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 energy policy (see

Stirling, 2010) and in income inequality analysis (see Nayak and Gastwirth, 1989). This thesis adapts and extends its use to portfolio theory as a new class of portfolio diversification measures.

Contributions

Principal Contribution

Based on RQE, the thesis develops a novel, coherent, flexible, unified, computational efficient and 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 multivariate expected utility;

2. It has low computational cost i.e. computational efficient; 3. It is easy to implement;

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asset dissimilarity matrix is homogeneous, translate-invariant and conditionally negative definite;

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 Choueifaty and Coignard(2008),Goetzmann and Kumar(2008)’s measure andFrahm 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 utility

and Bouchaud et al. (1997)’s general free utility offering therefore a novel and useful interpretation of these utility functions.

The thesis also provides some theoretical properties of the optimal portfolio of RQE called RQE portfolio (RQEP) (Chapter 5, Section 5.1), which is obtained by maximization. The 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 5.1.10and 5.1.11).

It also provides some empirical properties of RQEP examining the performance of twenty RQEP out-of-sample, using different performance metrics, on two different empirical datasets, against four standard diversified portfolios which are: the equally weighted portfolio, the equally 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 returns and 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 in portfolio 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 the risk contribution diversification strategies. The main portfolio theory models covered these four diversification strategies are presented. These models are the mean-variance model, its

Tobin(1958) andSharpe(1964)’s extension, the expected utility theory and the risk budgeting approach. The portfolio diversification measures covered these four diversification strategies are 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 diversification measure (Chapter 3). This definition is established in two steps. First, nine desirable proper-ties are postulated formalizing some intuitions of diversification and adapting some properproper-ties of risk measures in portfolio theory (Section 3.1). The measures satisfying these properties are 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 (reviewed inChapter 2) is examined against these properties (Section 3.3,Propositions 3.3.1and3.3.2).

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 billion dollars 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 of portfolio RQE to portfolio variance or, alternatively, minimizes portfolio variance subject to diversification constraint, where the diversification is measured by RQE. These new formal-izations clarify the investment problem behind the MD strategy and help identify the source of 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 the criticism from Lee (2011) and Taliaferro (2012). Moreover, these new formulations suggest new directions along which its out-of-sample performance can be improved, and it shows that these 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 to

Fernholz (2010), it shows that there is a specific measure of portfolio diversification in the MV model, which is the diversification return (Chapter 7, Section 3.2). As a result, since the 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 utility function based on RQE (Chapter 7,Section 7.1). This new formulation significantly improves the 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 without additional computational costs (Section 7.3).

Outline

The remainder of this thesis is organized as follows. Chapter 1 presents a brief review of some of mathematics notions and results that are used in this thesis. Chapter 2 reviews the concept of diversification and its measurement in portfolio selection. Chapter 3presents the minimum desirable properties for a measure of diversification to be considered as coherent.

Chapter 4 adapts and extends the use ofRao’s Quadratic Entropy (RQE) to portfolio theory as measure of portfolio diversification. The resulting new class of portfolio diversification measures 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 6

deeply analyzes the relationship between PRQE and diversification ratio of Choueifaty and Coignard (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 are provided. Chapters 1and2 constitute the first part of the thesis,Chapters 3to 5the second part and Chapters 6 and7 the third part.

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Part I

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Chapter 1

Mathematics Material

This first part reviews some background of mathematics and portfolio diversification. The first chapter presents a brief review of some of mathematics notions and results that are used in this 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 number sets are denoted R, R+ and N, respectively. The derivatives of univariate function f (.) are denoted in the usual fashion by f0, f00 and so forth. Higher-order derivatives are denoted by f(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 with a, 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 all elements except the ith, which is one. Transpose operator is denoted by >. Vectors are, unless otherwise specified, column vectors, transposed vectors are row vectors. The inverse of a square matrix A is denoted by A−1. In the case where the spectral decomposition of A exists, the eigenvectors matrix is denoted by EA and the eigenvalues diagonal matrix by ΞA with the diagonal consisting of eigenvalues of A ranked in descending order. The i-th eigenvalue of a square matrix A is denoted by ξ_i(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 of positive eigenvalues of A is denoted by ξ₊(A) and that of negative eigenvalues by ξ−(A). The determinant 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)N_i,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 , a_ij = 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 , a_ij ≤ a_ik + 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, I2such 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 of B 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 = (a_ij)N_i,j=1.

1) A is said to be positive semi-definite (positive definite) if and only if x>Ax ≥ 0 for all x ∈ RN (x>_{Ax > 0 for all x ∈ R}N 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 if

x>A x ≤ (≥)0 for all x ∈ RN such that PN

i=1xi= 0;

4) A is said to be conditionally strictly (positive (CSPD)) negative definite (CSND) if and only if x>_{A x < (>)0 for all x ∈ R}N such that PN

i=1xi= 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 definite matrix 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 any α ∈ RN+ such that PN i=1αi = 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 G_A(α) is the N × N Gram matrix associated to A.

Proposition 1.1.3 (Pekalska and Duin (2005_{): Remark 3.6). If there is α ∈ R}N₊ such thatPN

i=1αi = 1 and GA(α) is PSD, then GA(α) is PSD for all α ∈ RN+ such that PN

i=1αi = 1.

Proposition 1.1.4 (Lau (1985): Theorem 2.1). Let A be a symmetric matrix. If A is CND, 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 matrix of rank such that 1 ≤ rank(B) = r ≤ N , and A + B has eigenvalues ξ1(A + B) ≥ ξ2(A + B) ≥ ... ≥ ξN(A + B), then

1) ξ_i(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))N_i,j=1 is called a dissimilarity matrix.

This definition is a special case of that proposed in Orozco(2004), and it is equivalent to the definition of quasimetric in Pekalska and Duin (2005). In this thesis a dissimilarity matrix will be denoted D = (dij)N_i,j=1 with dij ≡ d(i, j).

Definition 1.1.5. Let D = (d_ij)N_i,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)N_i,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 individuals of X .

Definition 1.1.7. D = (d_ij)N_i,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 matrix S = (sij)Ni,j=1 verifying the following properties

1) 0 ≤ s_ij ≤ s_max; 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, k.k2) i.e. there is N points x1, ..., xN in space (RM, k.k2) such that dij = kxi− xjk22. The points x1, ..., xN are obtained through the spectral decomposition of the Gram matrix G_D(α). The point xi is the i-th column of the matrix X such that

X = EGDΞ 1 2

GD. (1.3)

Proposition 1.1.7 (Pekalska and Duin (2005): Theorem 3.13). A dissimilarity matrix D 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) = Z ∞

0

1 − exp(−α x)

α g(α)dα, (1.4)

where g(α)dα is a non-negative measure on R+ and R∞

1 g(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 derivatives φ(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) = _α(α+x)x , α > 0 g(α) = _Γ(1−α)α λ−α φ(x) = xα_{, 0 < α < 1} g(α) = δ(λ − α) φ(x) = 1−exp(−α x)_α , α ≥ 0 g(α) = exp(−λα) φ(x) = ln 1 + x_α , α > 0

Berg et al.(see 2008) φ(x) = _1+xxαα, 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, then

φ(D) = (φ(dij))N_i,j=1 is an Euclidean dissimilarity matrix.

Readers are referred toSchoenberg(1938a) andBavaud(2011) for more details on Schoenberg transformations.

Proposition 1.1.10 (From Cailliez (1983)). If D is a dissimilarity matrix, then there is constant d∗ such that for all d≥ d∗ _{the matrix with elements d}

ij + d, i 6= j is Euclidean. d∗ is the largest eigenvalue of the matrix

0N ×N −J _N1 D2J _N1 −I_{N ×N} 2 J _N1 D J 1 N !

Proposition 1.1.11 (Gower and Legendre (1986): Theorem 6, pp. 10). If S is a PSD similarity matrix with elements sij such that 0 ≤ sij ≤ 1 and sii = 1, then the dissimilarity matrix D =√11>− S with elements dij =p1 − sij is Euclidean.

Definition 1.1.11 (Circum-Euclidean matrix (Tarazaga et al., 1996)). A Circum-Euclidean matrix is an Euclidean distance matrices generated by points lying on a hypersphere. Formally, consider D = (d_ij)N_i,j=1an Euclidean distance matrix. FromDefinition 1.1.9, there is N points x1, ..., xN in space (RM, k.k2) such that dij = kxi− xjk22. If kx1k2 =, ..., = kxNk2, then D is Circum-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-EuEu-clidean 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-EuEu-clidean if and only if rank(D) = rank(G_D(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 h., .iE. E admits a direct orthogonal decomposition E = E+⊕ E−, where E+ = Rp and E− = Rq and the inner product is positive definite on E+ and negative definite on 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 hx, yiE = x>Ip qy = P

p i=1x + i y + i − Pq i=1x − i y − i , where Ip q = [Ip×p 0; 0 Iq×q]. As a result, norm of x ∈ E denoted by k.k2E is defined as kxk2E = x>Ip qx =Ppi=1x + i x + i − Pp i=1x − i x − i .

Definition 1.1.13 (Pseudo-Euclidean Embedding (Pekalska and Duin, 2005)). An N × N dissimilarity matrix D is said to be Euclidean if it can be embedded in a Pseudo-Euclidean space

R(p,q), k.kE

i.e. there is N points x1, ..., xN in space

R(p,q), k.kE

such that dij = kxi− xjk2E. The points x1, ..., xN are obtained through the spectral decomposition of the Gram matrix G_D(α). The point xi is the i-th column of the matrix X such that

X = EGD|ΞGD| 1

2. (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 functions and some of their characteristics. Next, it reviews the definition of Lipschitz continuous function 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, x}₂ _{∈ X and α ∈ [0, 1]}

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(ii) f is (strictly) quasi-concave if f (α x₁ + (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 ∇f is the gradient of f .

The definitions of (strictly) convex, (strictly) quasi-convex and (strictly) pseudo-convex can be obtained replacing f by −f .

Proposition 1.2.1 (Avriel et al. (2010_{): Corollary 6.4). Let X be an open convex set in}

RN and let X denote the closure of X. Then F (x) = 1₂x>A x + a>x is quasi-concave on X if and 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 are satisfied

(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 following statements 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 set X ⊆ 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 ⊆ R}N → R be a pseudo-convex (strictly pseudo-convex) function on an open convex set X and let φ : R → R be a differentiable function such that φ0(x) > 0, ∀ x ∈ R. Then, the composite function 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 ⊂ R}N _{→ R is}

said to be Lipschitzian on X if there is a constant γ1 such that

|g(x1) − g(x2)| ≤ γ1kx1− x2k, ∀ x1, x2∈ X. (1.6) where γ1 is a Lipschitz constant for the function g and k.k is a norm defined on X and |.| is absolute value operator.

Definition 1.2.4 (Bonnans and Shapiro (1998)). Consider the following optimization prob-lem

min

x∈X⊂RNf (x). (1.7)

Denote X∗ the set of optimal solutions. f satisfies the growth condition of order κ > 0 if there exists a constant γ₂ > 0 such that

f (x) − f (x∗) ≥ γ2(dist(x, X∗))κ, (1.8) where dist(x, X∗) = inf

x∗_∈X∗kx − x

∗_k.

Definition 1.2.5 (Guigues (2011): Definition 3.1, pp. 561). For any symmetric matrix A, let γ(A) be such that the quadratic function x>A x is γ(A)-strongly convex with respect to k.k₁ i.e. γ(A) = inf x6=0 x>A x kxk2 1 (1.9)

Proposition 1.2.6 (Guigues (2011): Proposition 3.1, pp. 561). Consider the two op-timization problems

min

x∈Xf1(x) (1.10)

min

x∈Xf2(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 a solution of problem (1.11). If (i) f₁ _{satisfies a second-order growth condition on X and (ii) the} function f2− f1 is Lipschitz continuous with modulus γ1 on X, then there is a constant γ2 such that

dist (x∗₂, X∗1) ≤ γ1 γ2

(1.12) Proposition 1.2.7 (Guigues (2011): Lemma 3.1, pp. 561). Let A be a real symmetric matrix, then sup

x∈RN +, x>1=1

kAxk2 = max

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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 measuring diversity introduced by Rao (1982a,b). Given a population of individuals P, it is defined as 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 by P the probability distribution function of X. RQE of P is defined as

HDP(P ) =

Z

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 population P2. Then

HD_P1×P2(P1, P2) = Z

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| X i,j=1 dP,ijpipj, (1.15) HD_P1×P2(p1, p2) = |P1| X i=1 |P2| X j=1 dP1×P2,ijp1ip2j, (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 individual

j from P2. The interpretation of RQE is straightforward: the higher HDP(p) is, the higher

the diversity of P is; the higher H_D_P1×P2(p1, p2) is, the higher the difference between P1 and P2 is.

RQE has been used extensively in fields such as statistics (seeRao,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). 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 and to extend its use to portfolio theory as a new class of portfolio diversification measures. The

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following results will be necessary for this achievement. When it is not necessary to distinguish between population of individuals, the notation H_D_P(p) will be replaced by HD(p).

Proposition 1.3.1 (Rao (1982a): pp. 7). H_D(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 1.3.1is equivalent to the following one.

Proposition 1.3.2 (Rao (2010): Lemma 3.1, pp. 76). HD(p) is concave if and only if D is CND.

Pavoine (2012) establishes the condition under which HD(p) is strictly concave.

Proposition 1.3.3 (Pavoine (2012): pp. 515). H_D(p) is strictly concave if and only if rank GD _N1 = N − 1.

Proposition 1.3.4 (Rao and Nayak (1985)). If P1, ..., PM are the distributions of X in populations P1, ..., PM with a priori probabilities α1, ..., αM, then the distribution in the mixture population P₀≡PM

i=1αiPi is P0=PMi=1αiPi. If RQE is concave, then

HD_P0(P0) = M X i=1 αiHD_Pi(Pi) + M X i,j=1 αiαjDH_DP i×Pj (Pi, Pj), (1.19) where PM i,j=1αiαjDH_DP i×Pj

(Pi, Pj) is called the Jensen difference and DH_DP

i×Pj

(Pi, Pj) is called the directed divergence or cross entropy (Rao and Nayak,1985) defined as follows

DH_DP

i×Pj

(Pi, Pj) = 2HD_Pi×Pj(Pi, Pj) − HD_Pi(Pi) − HD_Pj(Pj) (1.20)

Rao and Nayak(1985) also show thatqDH_DP

i×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). pD_H(., .) is a met-ric on the space of multinomial distributions when D is CND.

Shimatani (2001, pp. 140) also shows that H_D(p) can be decomposed as follows.

Proposition 1.3.6 (Shimatani (2001): pp. 140). For any D and p, H_D(p) can be de-composed as follows HD(p) = 1 2G(p) × A (D) + 1 2B (p, D) , (1.21)

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where G(p) = |P| X i6=j=1 pipj (1.22) A (D) = 2 |P|(|P| − 1)DT (D) (1.23) B (p, D) = |P| X i6=j=1 (d(xi, xj) − A (D)) pipj − ˜G(p) (1.24) with ˜G(p) = _|P|(|P|−1)2 G(p) and DT (D) =P|P| i=1d(xi, xj).

The optimal distribution p is obtained maximizing H_D(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

max

p 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, D_H(p∗₁, p∗₂) = 0. 4) If √D is ultrametric, then

a) 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) D_H DP×P(p, p ∗_{) = (H} D(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 selection rules 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 (capital asset pricing model) and the risk contribution strategies. The pre-modern rules are a list of advice and prescriptions. The modern rules are the mean-variance models (includingTobin’s extension and the CAPM), the expected utility theory and the risk budgeting approaches.

Section 2.2focuses 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 measures are grouped in four categories according to the diversification strategies. The advantages and shortcomings of each considered measure are also discussed.

2.1 Portfolio Selection Rules

Consider an investor with a given initial wealth denoted W . The problem faces by this investor is how to allocate his wealth W among N alternative investment opportunities. This problem is known under the name of portfolio selection problem. More formally, the portfolio selection problem is to find the optimal allocation or portfolio w = (w1, ..., wN)> which best suits the needs of the investor, where w_i is the share of W invested in asset i such that PN

i=1wi = 1. In the case where the investment opportunities or assets are risk-free, the solution of this problem is trivial. The optimal allocation is to put the total wealth W on the asset with the highest future return. However, in the case where some assets are risky, it is no longer optimal to concentrate. Diversification becomes recommended. Therefore, several rules based

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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 w_i = 0 for at least one i and wi > 0 for at least two i. A portfolio w will be said to be completely diversified if wi > 0 for all i. A portfolio w will be said to be well-diversified if it is an optimal portfolio of a measure of portfolio diversification or of a diversification strategy. A portfolio w will be said to be concentrated if wi = 1 for some i. The concentrated portfolio is also called a single asset portfolio. It is assumed that all asset are not risk-free. and the initial wealth W is normalized to 1, unless otherwise specified.

There are two types of portfolio selection problems: the one-period and the multi-period problems. In the one-period specification, the investor constructs his or her portfolio at the beginning of the period (time 0) and get the pay off at the end (time T ). In a multi-period problem, 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 the final period, at time T . This thesis concerns the one-period problem. For the multi-period problem readers are referred to Samuelson (1969), Merton (1969), Steinbach (2001), Li and Ng (2000) andMarkowitz (1959).

2.1.1 Pre-Modern Rules

The situation prior to Markowitz (1952)’s seminal work Portfolio selection published in The Journal of Finance, is described byBernstein(2005) and Markowitz(1999) itself.

Bernstein (2005, pp. 55)

Before Markowitz, we had no genuine theory of portfolio construction, only rules of thumb and folklore.

Markowitz (1999, pp. 5)

What was lacking prior to 1952 was an adequate theory of investment that covered the effects of diversification when risks are correlated, distinguished between effi-cient and ineffieffi-cient portfolios, and analyzed risk-return trade-offs on the portfolio as a whole.

The following reports some of these rules of thumb and folklore.

Shakespeare and Phelps (c1923, Act I, Scene 1, pp.) (see alsoMarkowitz, 1999; Rubinstein,

2002)

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estate Upon the fortune of this present year; Therefore, my merchandise makes me 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 alsoSullivan,2011, pp. 74)

It is advisable to divide goods which are exposed to some small danger into several small portions rather than to risk them all together.

Keynes (1934)2 (seeKeynes and (Grande-Bretagne),1971)

As time goes on I get more and more convinced that the right method in investment is to put fairly large sums into enterprises which one thinks one knows something about and in the management of which one thoroughly believes. It is a mistake to think that one limits one’s risk by spreading too much between enterprises about which one knows little and has no reason for special confidence. [...] One’s knowledge and experience are definitely limited and there are seldom more than two or three enterprises at any given time in which I personally feel myself entitled to put full confidence.

Loeb (1965)

When an investment is made, its prospect must be so good that placing a rather large 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 only a moderate portion of total capital need invested to get the desired percentage appreciation on total funds. Expressing the matter in different way, this means that diversification is undesirable. One or two, or at most three or four securities should be bought. And the should be so well selected, their purchase so expertly timed and their profit possibilities so large that it will never be necessary to risk in any of them a large proportion of available capital. Under this policy, only the best is bought at the best possible time. Risk are reduced in two ways-first by the 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 time will be given to the choice of each so that every important detail about them

1_{http://juchre.org/talmud/babametzia/babametzia.htm}_.

2_{From Keynes’ letter to F. C. Scott, 15 August, 1934 (see}_{Keynes and (Grande-Bretagne)}_, ₁₉₇₁_{). This}