Three Essays on Global Minimum Variance Investing

University “Aleksandër Moisiu”, Durrës Business Faculty

Three Essays on Global Minimum Variance Investing

Prepared by: Arben Zibri

Advisors: Prof Dr. Agim Kukeli Prof. Asc. Dr. Orfea Dhuci

Durrës, 2014

Abstrakt Ky disertacion studion performancën jashtë kampionit të portofolit global me variancë minimale. Analiza i referohet gjetjeve të Jagannathan dhe Ma (2003) dhe Behr et al. (2008) në lidhje me reduktimin e riskut dhe analizën e performancës në portofolet e aksioneve në SHBA, të cilat përshtasin kufizime të peshave të portofolit. Të dhënat e përdorura janë indekset javore dhe mujore e aksioneve, obligacioneve, naftës dhe arit për periudhën 1996-2013. Ne vërejme që portofoli me variancë minimale, i cili konsideron aktivet globale performon statistikisht më mirë se dy indekset krahasuese sipas raportit Sharpe, M2, dhe konstantes së regresionit. GMVP javore të aktiveve globale performojnë më dobët se GMVP mujore sipas, duke ju referuar rezultateve të raportit Sharpe, M2 dhe konstantes alfa të regresionit, pavarësisht shpeshtësisë së ribalancimit të peshave apo kostos së transaksionit të aplikuar. Përdorimi i kthimeve javore në optimizim ul variancën relative të portofoleve globale, krahasuar me rastin kur përdoren kthimet mujore. Fjalë kyçe: Optimizim Portofoli; Performanca e Portofolit me Variancë Minimale; Reduktimi i Riskut; Aktivet Globale; Teste Jo-Parametrike; Frequencë e Lartë Kthimesh.

Abstract This dissertation studies the out of sample performance of global minimum variance investing. The analysis follow the discussions of Jagannathan and Ma (2003) and Behr et al. (2008) regarding performance in US stock portfolios using portfolio constraints and performance analysis. The data used are monthly and weekly indices of stocks, bonds, gold oil and spreads from 1996 until 2013. We find that global asset GMVPs significantly outperform the value and equally weighted benchmarks in terms Sharpe Ratio, M2 and risk adjusted excess return. Weekly global asset GMVPs perform worse than monthly GMVPs as measured by Sharpe Ratio, M2 and excess returns, independently from revision frequency and transaction costs. The use of higher frequency weekly returns decreases the tracking error volatility of the global portfolios considered in this research. Keywords: Portfolio Optimization; Minimum Variance Performance; Risk Reduction; Global Assets; Non-Parametric Bootstrap; Higher Return Frequency

Copyright © Arben Zibri 2014

The advisors of Arben Zibri proof that the following is the approved copy of the dissertation:

THREE ESSAYS ON GLOBAL MINIMUM VARIANCE INVESTING

GMVP PERFORMANCE USING MONTHLY RETURNS AS OPTIMIZATION INPUT GMVP PERFORMANCE USING WEEKLY RETURNS AS OPTIMIZATION INPUT MINIMUM TRACKING ERROR VOLATILITY PERFORMANCE: A GLOBAL ASSET APPROACH

Advisors: Prof. Dr. Agim KUKELI, PhD

Prof. Asc. Dr. Orfea DHUCI

THREE ESSAYS ON GLOBAL MINIMUM VARIANCE INVESTING

GMVP PERFORMANCE USING MONTHLY RETURNS AS OPTIMIZATION INPUT GMVP PERFORMANCE USING WEEKLY RETURNS AS OPTIMIZATION INPUT MINIMUM TRACKING ERROR VOLATILITY PERFORMANCE: A GLOBAL ASSET APPROACH

Prepared by: Arben ZIBRI

Dissertation Submitted to the Faculty of Business, University “Aleksandër Moisiu” Durres In Full Compliance With the Requirements For the “Doctor” degree in Economics University “Aleksandër Moisiu”, Durrës, Albania November 2014

ACKNOWLEDGMENTS First of all, I would like to thank Zhaklina and my parents for the care, patients, and sacrifice they have shown for my commitment toward my passion for research. Without their love, encouragement, and support, this would have not been possible. I would like to express my sincere and profound gratitude for the University “Aleksander Mojsiu”, Durres for giving me the amazing opportunity of expanding my knowledge and enhance my productivity on the field that I love most, financial research. The experience have been thrilling and intense, but it was worth to live every second of it. Then I thank Professor Nikolaos Tessaromatis from EDHEC Business School for his unique advises and original ideas. Without his suggestions, my dissertation would have not been possible. I would like to express my deep gratitude for ALBA Graduate Business School, Professor Christos Cabolis, and Athanasios Sakkas. Without their help with the data and academic feedback, it would not be possible to perform the study presented in this dissertation. Professor Orfea Dhuci has been one of the main encouragers of the extension of my knowledge and academic experience since the beginning of my doctoral studies. I thank him for all the academic, professional and human support through all this years. It would have been very difficult to advance in my doctoral program, and not only, without his exceptional unlimited help and availability. Last, but not least. Professor Agim Kukeli is the main reason why I have initiated, continued, and finalized my doctoral studies in economics. His inspiration, encouragement, and academic support gave me the energy and strength to finalize this immense project. It was a great privilege and unique experience to have him as professor and advisor during all these years. I consider myself lucky to have worked with a beautiful mind!

Deklaratë mbi origjinalitetin Arben Zibri Deklaroj se kjo tezë përfaqëson punën time origjinale dhe nuk kam përdorur burime të tjera, përveç atyre të shkruajtura nëpërmjet citimeve. Të gjitha të dhënat. Tabelat, figurat dhe citimet në tekst, të cilat janë riprodhuar prej ndonjë burimi tjetër, duke përfshirë edhe internetin, janë pranuar në mënyre eksplicide si të tilla. Jam i vetëdijshme se në rast të mospërputhjeve, Këshilli i Profesorëve të UAMD-së është i ngarkuar të më revokojë gradën “Doktor”, që më është dhënë mbi bazën e kësaj teze, në përputhje me “Rregulloren e Programeve të Studimit të Ciklit të Tretë (Doktoratë) të UAMD-së, neni 33, miratuar prej Senatit Akademik të UAMD-së me Vendimin nr. , datë ________ Durrës, më _________________

Firma

Përmbledhje Ky disertacion studion performancën jashtë kampionit të portofolit global me variancë minimale. Analiza i referohet gjetjeve të Jagannathan dhe Ma (2003) dhe Behr et al. (2008) në lidhje me reduktimin e riskut dhe analizën e performancës në portofolet e aksioneve në SHBA, të cilat përshtasin kufizime të peshave të portofolit. Së pari, ne vlerësojmë matricën e kovariancës duke përdorur matricën e kovariancës së kampionit të kthimeve mujore, dhe derivojme portofolet optimale me variancë minimale (GMVP) duke aplikuar kufizimet e pozicjoneve të shkurtra dhe të gjata, ose asnjë kufizim, në aktivet e përfshira në këtë punë kërkimore. Rezultatet janë paraqitur në bazë të frekuencave të ndryshme të ribalancimit të peshave dhe kostove të ndryshme të transaksioneve. Të dhënat e përdorura janë indekset e aksioneve, obligacioneve, naftës dhe arit për periudhën 1996-2013. GMVPs të pakufizuar paraqesin variancën më të ulët jashtë kampionit, ndërsa GMVP të pakufizuar të portfolit të obligacioneve globale performojne shfaqin variancën më të ulët nga të gjithë GMVP e konsideruar. Më tej, ne aplikojm teste jo-parametrike për sipas metodologjisë së sugjeruar nga Kossowski et al. (2007) dhe Behr et al. (2008), që masin performancën e GMVP kundrejt dy indekseve krahasuese, sipas raportit Sharpe, CEQ, M2, dhe konstantes së regresionit mes GMVP dhe indeksit krahasues. Ne vërejme që portofoli me variancë minimale, i cili konsideron aktivet globale performon statistikisht më mirë se dy indekset krahasuese sipas raportit Sharpe, M2, dhe konstantes së regresionit. Ribalancimi më i rrallë i peshave nuk përmirëson performancën e GMVP. Më pas, ky punim përdor kthimet javore të indekseve të aksioneve, obligacioneve, arit dhe naftës, për të vlerësuar matricën e kovariancës së kampionit të kthimeve, e cila shërben si input për ndërtimin e portofolit global me variancë minimale (GMVP). GMVP javore të aktiveve globale performojnë më dobët se GMVP mujore sipas, duke ju referuar rezultateve të raportit Sharpe, M2 dhe konstantes alfa të regresionit, pavarësisht shpeshtësisë së ribalancimit të peshave apo kostos së transaksionit të aplikuar. Nëse ribalancimi dymbëdhjetë mujor aplikohet, GMVP javore të kufizuar për pozicjonet e shkurtra dhe të gjata zvogëlojnë riskun më shumë se sa kthimin, krahasuar me GMVP mujore të aktiveve globale. GMVP javore të obligacioneve, të cilat përshtasin kufizime për pozicjone të shkurtra dhe të gjata në peshat e aktiveve, performojnë statistikisht më mirë se sa dy indekset krahasuese nëse ribalancimi i peshave të portofolit bëhet cdo tridhjetegjashtë muaj. Përdorimi i kthimeve javore, në përgjithësi përkeqëson statistikisht performancën e GMVP të aksioneve dhe obligacioneve kundrejt indekseve krahasuese. Pjesa e fundit e kësaj pune kërkimore ka të bëjë me studimin e portofolit me variancë relative minimale kundrejt indeksit krahasues në rastin e portofolit të aktiveve globale, aksioneve globale, dhe obligacioneve globale. Metodologjia aplikohet sipas sugjerimeve te Jagannathan dhe Ma (2003) në lidhje me reduktimin e riskut në portofolet e aksioneve në SHBA, të cilat përshtasin kufizime të peshave. Matrica e kovariancës së diferencës së kthimit mes aktiveve dhe indeksit krahasues, të vlerësuar sipas metodologjise se Chan et al. (1999), përdoret si input për ndërtimin e portofolit me variancë relative minimale. Portofolet me variancë relative minimale optimizohen duke aplikuar kufizimet e pozicjoneve të shkurtra dhe të gjata, ose asnjë kufizim, në peshat e aktiveve. Rezultatet tregojnë që kufizimet e peshave nuk kanë asnjë efekt në reduktimin e variancës relative jashtë kampionit për portofolet me variancë relative minimale të aktiveve globale, obligacioneve globale, aksioneve globale. Përdorimi i 7

kthimeve javore në optimizim ul variancën relative të portofoleve globale, krahasuar me rastin kur përdoren kthimet mujore. Fjalë kyçe: Optimizim Portofoli; Performanca e Portofolit me Variancë Minimale; Reduktimi i Riskut; Aktivet Globale; Teste Jo-Parametrike; Frequencë e Lartë Kthimesh.

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Summary This dissertation studies the out of sample performance of global minimum variance investing. The analysis are drown from the discussions of Jagannathan and Ma (2003) and Behr et al. (2008) regarding the risk reduction in US stock portfolios using portfolio constraints and performance analysis. First, we estimate the covariance matrix using the sample covariance matrix of monthly returns and derive optimal minimum variance portfolios considering upper/lower bounds and no restrictions. Results are shown under different revision frequency and transaction costs. The data used are indices of stocks, bonds, gold oil and spreads from 1996 until 2013. Unconstrained GMVPs result in the lowest out of sample variance, while unconstrained GMVPs of global bond portfolios performs the best in terms of risk reduction. Furthermore, we employ the non-parametric bootstrap approach suggested by Kossowski et al (2007) and Behr et al. (2008) to measure the performance of GMVPs versus the value weighted and equally weighted benchmark in terms of Sharpe Ratio, CEQ, M2 and excess return (alpha). We find that global asset GMVPs significantly outperform the value and equally weighted benchmarks in terms Sharpe Ratio, M2 and risk adjusted excess return. Less frequently portfolio rebalancing policy, however, does not improve significantly performance. Then, this work uses weekly returns for stock, bond, gold, oil indices to estimate the sample covariance matrix of returns, which serves as optimization input for global minimum variance portfolios (GMVPs) construction. Weekly global asset GMVPs perform worse than monthly GMVPs as measured by Sharpe Ratio, M2 and excess returns, independently from revision frequency and transaction costs. If more frequently rebalancing is applied, most restrictive weekly GMVPs reduce risk more than returns vis-à-vis monthly GMVPs of global assets. Weekly upper/lower weight constraint bond GMVPs significantly outperforms the value and equally weighted benchmark if portfolio rebalancing is done every thirty six months. The use of weekly return, based on the findings of this research, generally worsens the performance significance of stock and bond GMVPs toward benchmarks. The last part of this research is the study of the out of sample tracking error of minimum variance portfolios of global assets, equities and bonds. The methodology follows the one presented by Jagannathan and Ma (2003) regarding the risk reduction in US stock portfolios using weight constraints. The sample covariance matrix of excess returns, as per the methodology of Chan et al. (1999), is used as minimum tracking error volatility portfolios optimization input. Optimal minimum tracking error portfolios are derived using upper/lower and no restrictions. Weight constraints have no effect on out of sample relative risk reduction for the global asset, global bond, and global stock minimum tracking error volatility portfolios. The use of higher frequency weekly returns decreases the tracking error volatility of the global portfolios considered in this research. Keywords: Portfolio Optimization; Minimum Variance Performance; Risk Reduction; Global Assets; Non-Parametric Bootstrap; Higher Return Frequency

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Table of Contents ACKNOWLEDGMENTS ......................................................................................................... 5 Deklaratë mbi origjinalitetin...................................................................................................... 6 Përmbledhje ............................................................................................................................... 7 Summary ................................................................................................................................... 9 List of Tables ........................................................................................................................... 13 Introduction ............................................................................................................................. 15 Chapter 1: LITERATURE REVIEW ...................................................................................... 18 1.1

Foundations, Developments and Limitations of the Portfolio Selection Problem .. 19

1.1.1

Markowitz contribution on the Portfolio Selection problem ........................... 19

1.1.2 The Introduction of Risk Factors and Risk Free asset in the asset allocation/security selection problem .............................................................................. 20 1.1.3

Limitations from the Efficient Market Hypothesis and Behavioral Theory .... 22

1.1.4

Roll’s Critique on the definition of the “Market”............................................ 24

1.2 The performance of Minimum Variance Portfolios: Motivations from the Investment Management Profession.................................................................................... 24 1.3 The Definition and Implications of Forecasting Errors of Portfolio Returns and Risk. 26 1.4 The Models Used in Constructing Global Minimum Variance Portfolios and their Implication in Out of Sample Portfolio Performance .......................................................... 28 1.4.1

The Sample Covariance Matrix ....................................................................... 28

1.4.2

The Covariance Matrix Estimated by Common Risk Factors ......................... 29

1.4.3

The Shrinkage Estimators................................................................................ 30

1.4.3.1

Shrinkage Estimators for Portfolio Variance/Covariance ........................... 30

1.4.3.2

Shrinkage Estimators for Portfolio Weights ................................................ 34

1.4.4 The Effect of Weight Constraints on Minimum Variance Portfolio Performance..................................................................................................................... 38 1.4.5

Higher Data Frequency and Minimum Variance Performance ....................... 42

1.5 Models and Implications in Estimating Minimum Tracking Error Volatility Portfolios ............................................................................................................................. 44 Chapter 2: RESEARCH METHODOLOGY .......................................................................... 47 2.1.

The Selection Rationale of the Variance-Covariance Matrix Estimator Model ...... 48

2.2.

Portfolio Optimization ............................................................................................. 49

2.3.

The Consideration of Transaction Costs in Out of Sample Performance ................ 50

2.4.

The Performance Metrics ........................................................................................ 51

2.5.

Statistical Inference of the Performance Metrics Used ........................................... 51

2.5.1.

Brief Summary of Jarque-Bera test for Normal Distribution Of Returns........ 52

2.5.2.

The Observation Bootstrap Approach ............................................................. 52

2.5.3.

The Residual Bootstrap Approach................................................................... 53

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2.6. Optimization and Performance Metrics for Minimum Tracking Error Volatility (TE)Portfolios...................................................................................................................... 54 2.6.1.

Optimization Approach of TE Portfolios ........................................................ 54

2.6.2.

Out of Sample Performance Measure of TE Portfolios ................................... 54

Chapter 3: DATA DESCRIPTION ......................................................................................... 56 3.1.

Global Asset Minimum Variance Portfolios ........................................................... 56

3.1.1.

Global Asset Constituents List ........................................................................ 56

3.1.2.

Benchmark Description ................................................................................... 56

3.2.

Global Bond Minimum Variance Portfolios ........................................................... 57

3.2.1.

Global Bond Constituents List ........................................................................ 57

3.2.2.


3.3.

Global Equity Minimum Variance Portfolios ......................................................... 58

3.3.1.

Global Equity Constituents List ...................................................................... 58

3.3.2.


Chapter 4: GMVP PERFORMANCE USING MONTHLY RETURNS AS OPTIMIZATION INPUT ..................................................................................................................................... 59 4.1.

Introduction ............................................................................................................. 59

4.2.

Short Summary of the Methodology ....................................................................... 61

4.3.

Description of the Data............................................................................................ 63

4.4.

GMVP Performance ................................................................................................ 64

4.4.1.

Risk Reduction Characteristics of Global GMVPs ......................................... 66

4.4.2.

GMVP Performance Using Sharpe Ratio, CEQ, M2 and excess return .......... 68

4.4.2.1.

Specific Markets Performance: The case of Equity and Bond Market .... 70

4.4.2.2. Effects on Performance of Changes in Revision Frequency: The case of 3 years Revision Frequency............................................................................................ 71 Chapter 5: GMVP PERFORMANCE USING WEEKLY RETURNS AS OPTIMIZATION INPUT ..................................................................................................................................... 74 5.1.

Introduction ............................................................................................................. 74

5.2.

Short Summary of the Methodology ....................................................................... 75

5.3.


5.4.

Weekly Versus Monthly GMVP Performance ........................................................ 79

5.4.1.

Risk Reduction Characteristics of Weekly GMVPs ........................................ 79

5.4.2. Weekly Versus Monthly GMVP Performance Using Sharpe Ratio, CEQ, M2 and excess return ............................................................................................................. 83 5.4.2.1. Effects on Performance of Changes in Revision Frequency: The case of 3 years Revision Frequency............................................................................................ 86 5.4.2.2.

Specific Markets Performance: The case of Equity and Bond Market .... 89

Chapter 6: MINIMUM TRACKING ERROR VOLATILITY PERFORMANCE: A GLOBAL ASSET APPROACH .............................................................................................................. 90 6.1.

Introduction ............................................................................................................. 90

6.2.

Short Summary of the Methodology ....................................................................... 91 11

6.3.


6.4.

TE Portfolio Performance........................................................................................ 93

6.4.1. Absolute Risk Reduction Characteristics of Minimum Tracking Error Portfolios 93 6.4.2.

Relative Risk Reduction Characteristics of Minimum Tracking Error Portfolios 95

6.4.2.1. Effects on Performance of Changes in Revision Frequency: The case of 3 years Revision Frequency............................................................................................ 96 6.4.2.2. Effects on Relative Risk Reduction of Higher Frequency Data: The case of Weekly Returns ........................................................................................................ 97 Chapter 7: LIMITATIONS OF THE STUDY ...................................................................... 102 7.1.

The use of more asset classes as optimization input.............................................. 102

7.2.

The use of higher frequency data as daily and intraday returns ............................ 102

7.3.

The use of Shrinkage estimators............................................................................ 103

7.4.

A comparative study regarding transaction costs between different markets/regions 103

7.5.

The Contribution of Forex Volatility to Portfolio Risk ......................................... 104

Chapter 8: Concluding Remarks ........................................................................................... 106 Annex 1 – JARQUE-BERA TEST RESULTS FOR MONTHLY AND WEEKLY GLOBAL MINIMUM VARIANCE PORTFOLIOS AND BENCHMARKS ....................................... 108 BIBLIOGRAPHY ................................................................................................................. 110

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List of Tables Table 1.1: Sharpe over performance of Behr et al. (2008) Global Minimum Variance Portfolios. ................................................................................................................................ 40 Table 1.2: Differences in GMVP characteristics due to changes in data frequency, according to Clarke et al. (2006). ............................................................................................................. 43 Table 3.1: Global Asset Portfolio Constituents ....................................................................... 56 Table 3.2: Global Bond Portfolio Constituents ....................................................................... 57 Table 3.3: Global Equity Portfolio Constituents ..................................................................... 58 Table 4.1: Global Asset Portfolio Constituents ....................................................................... 63 Table 4.2: Summary Statistics of Monthly GMVP Constituent Indices and Spreads ............. 64 Table 4.3: Monthly Return Correlation matrix of the global asset portfolio ........................... 65 Table 4.4: Monthly Return Correlation matrix of the global bond portfolio........................... 65 Table 4.5: Monthly Return Correlation matrix of the global equity portfolio......................... 66 Table 4.6: Summary Statistics of Monthly GMVP and Benchmarks - 1 Year Revision Frequency ................................................................................................................................ 67 Table 4.7: Summary Statistics of Monthly GMVP and Benchmarks - 3 Year Revision Frequency ................................................................................................................................ 68 Table 4.8: Empirical Performance Metrics and Inference of Monthly GMVP - 1 Year Revision Frequency ................................................................................................................. 69 Table 4.9: Empirical Performance Metrics and Inference of GMVP - 3 Years Revision Frequency ................................................................................................................................ 72 Table 5.1: Global Asset Portfolio Constituents ....................................................................... 78 Table 5.2: Summary Statistics of Monthly GMVP and Benchmarks - 1 Year Revision Frequency ................................................................................................................................ 79 Table 5.3: Summary Statistics of Weekly GMVP and Benchmarks - 1 Year Revision Frequency ................................................................................................................................ 80 Table 5.4: Summary Statistics of Monthly GMVP and Benchmarks - 3 Year Revision Frequency ................................................................................................................................ 81 Table 5.5: Summary Statistics of Weekly GMVP and Benchmarks - 3 Year Revision Frequency ................................................................................................................................ 82 Table 5.6: Empirical Performance Metrics and Inference of Monthly GMVP - 1 Year Revision Frequency ................................................................................................................. 84 Table 5.7: Empirical Performance Metrics and Inference of Weekly GMVP - 1 Year Revision Frequency ................................................................................................................................ 85 Table 5.8: Empirical Performance Metrics and Inference of Monthly GMVP - 3 Years Revision Frequency ................................................................................................................. 87 Table 5.9: Empirical Performance Metrics and Inference of Weekly GMVP - 3 Years Revision Frequency ................................................................................................................. 88 Table 6.1: Global Asset Portfolio Constituents ....................................................................... 93 Table 6.2: Summary Statistics of Monthly GMVP and Benchmarks - 1 Year Revision Frequency ................................................................................................................................ 94 Table 6.3: Summary Statistics of Monthly Minimum Tracking Error Volatility Portfolios and Benchmarks - 1 Year Revision Frequency .............................................................................. 95 Table 6.4: Summary Statistics of Monthly GMVP and Benchmarks - 3 Year Revision Frequency ................................................................................................................................ 96 Table 6.5: Summary Statistics of Monthly Minimum Tracking Error Volatility Portfolios and Benchmarks - 3 Year Revision Frequency .............................................................................. 97 Table 6.6: Summary Statistics of Weekly GMVP and Benchmarks - 1 Year Revision Frequency ................................................................................................................................ 98 13

Table 6.7: Summary Statistics of Weekly Minimum Tracking Error Volatility Portfolios and Benchmarks - 1 Year Revision Frequency .............................................................................. 99 Table 6.8: Summary Statistics of Weekly GMVP and Benchmarks - 3 Year Revision Frequency .............................................................................................................................. 100 Table 6.9: Summary Statistics of Weekly Minimum Tracking Error Volatility Portfolios and Benchmarks - 3 Year Revision Frequency ............................................................................ 101 Table A.1: Jarque-Bera Test for Monthly GMVPs and Benchmarks. ................................... 108 Table A.2: Jarque-Bera Test for Weekly GMVPs and Benchmarks. .................................... 109

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Introduction Inspired by the results reported by Nielsen and Aylusubramanian (2008) and Poullaouec (2008), our research is focused on the Global Minimum Variance Portfolio (GMVP) and Global Minimum Tracking Error Volatility performance of global assets. The appealing empirical performance of the GMVP reported so far in the literature, to the best of our knowledge, has not been tested yet for global asset classes. In this paper we take into consideration not only stock markets, but also bond markets, the global oil market and the global market for gold for the portfolio optimization exercise. We also take into consideration not only specific country but regions as well, such as Europe, Asia Pacific, and Developing Markets. As Chopra and Ziemba (1993) suggest, errors in forecasts of first moments of asset returns are ten times greater than the errors in forecasts of variances, while errors in forecasts of variances are two times greater than the errors of forecasts for covariance estimators. The importance of this evidence in our study stands on the fact that GMVP estimation, differently from other portfolios in the efficient front, does not need expected returns as input. As analytically described by Chan et al (1999), predicting variance covariance matrix estimation from history is not necessarily inaccurate. One of the main debates in portfolio optimization, however, is whether allowing security weights to take negative values or not. Green and Hollifield (1992) argue that extreme short or long positions, due to the dominance of the single factor in the covariance structure of returns, is unlikely to be due to sampling error. Extreme short positions, however, due to market regulations and other factors, are not always feasible solutions. Clarke et al (2011), present an analytical solution for optimal portfolio weights when short selling is not allowed. They show empirically that the portion of GMVP risk attributed to the single factor (market) model varies from 80%90%. This result may suggest that there may be imposed some structure to the covariance matrix, which would reduce the number of parameters to be estimated. The main question, in this case, is how much structure to impose? Ledoit and Wolf (2003) addressed this issue by developing a shrinkage estimator model for the variance-covariance matrix, which is a weighted average of the sample covariance matrix (S) and the covariance matrix forecasted by a single factor model or constant correlation model (F). The weight, or shrinkage intensity as called by the authors, is different for different datasets, and depends on the correlation between estimation error of the sample covariance matrix and estimation error of the factor model. The model, according to the authors, reduces the out of sample variance, yields the highest information ratio, and usually yields the highest mean excess return. DeMiguel et al (2009b) and Kempf and Memmel (2006) propose to shrink towards a target weights instead of covariance matrix estimators. Frahm and Memmel (2010) introduce in the finance literature a GMVP shrinkage estimator that targets the reduction of out of sample portfolio variance rather than covariance matrix or GMVP weights. The two shrinkage estimators presented in their work reduce the out of sample portfolio variance as compared to the traditional estimators, especially in the case when the number of assets in the portfolio increases. Jagannnathan and Ma (2003) offer in their paper a different approach in estimating the weights of GMVP. They argue that, in case portfolio weights have no negative (maximum) constraint, they have a shrinkage like effect of reducing (increasing) the covariance of constrained assets with other assets. DeMiguel et al (2009a), prove that the model proposed by Jagannathan and Ma (2003) for estimating GMVPs performs 15

better than other thirteen models, including shrinkage estimators, in terms of Sharpe Ratios. Behr et al (2008) argue that imposing the maximum weight constraint reduces the out of sample standard deviation of GMVP. All papers on GMVP performance so far, however, have shown no consistent outperformance of GMVP strategy, no matter the estimator, towards the naively diversified portfolio (1/N), as argued by DeMiguel et al (2009a). Inspired by the work of Jagannathan and Ma (2003) and Behr et al (2008), we believe that this research is the first attempt for the derivation of global minimum variance and global minimum tracking error volatility portfolios from global assets, including bonds, stocks, energy (oil) and precious metals (gold). The global assets are represented by regional indices, with the main assumption of dividing the world of investment opportunities in four parts, US, Europe, Asia Pacific and Developing Nations. Minimum Variance portfolios are estimated using Markowitz (1952) sample covariance matrix of monthly and weekly indices returns. Since Pantaleo et al (2011) argue that, when the estimation period T is greater than the number of stocks N, other estimators cannot outperform the portfolio obtained by the sample covariance in terms of realized risk when short sale is not allowed, we apply the methodology proposed by Jagannathan and Ma for constructing minimum variance portfolios. Specifically, the unconstrained, short-sale constrained, and upper bound constrained GMVPs are derived for the global asset portfolio, global bond portfolio and global equity portfolio. The performance of unconstrained, short-sale constrained, and upper bound constrained minimum variance portfolios is compared with the value weighted benchmark and equally weighted benchmark of global and distinct asset classes. The performance metrics used for comparison the Sharpe Ratio, CEQ, and Modigliani and Modigliani (M2), where the statistical inference is run as per non-parametric bootstrap approach adapted by Behr et al (2008), and regression alpha, where the statistical inference is run as per the residual bootstrap methodology proposed by Kossowski (2007). Results are presented for one year and three year revision frequency for three global portfolios, where the covariance matrix is estimated using monthly and weekly returns. Transaction costs are included in the model, applying the methodology suggested by DeMiguel et al (2009a). The main research questions that this dissertation addresses are: 1. Does the use of global asset classes as optimization input improve minimum variance risk reduction and performance vis-à-vis benchmarks? 2. Does the application of weight constraints improve GMVP performance in global terms, when different assets and markets are considered? 3. Does the application of different portfolio rebalancing frequencies influence minimum variance strategy performance globally? 4. Does the use of higher return frequency improve global minimum variance performance? 5. Do portfolio constraints and higher return frequency influence the out of sample relative risk reduction of minimum tracking error portfolios in a global framework? This dissertation is organized as follows: Chapter one presents a detailed analyses of the literature review. Next chapter introduces the research methodology of portfolio construction, optimization and benchmark inference tests. Chapter three describes the data and a brief description of their statistical characteristics. Chapter four analyzes 16

the empirical findings of the minimum variance portfolio of global assets, global equity and global bonds derived from monthly returns. Next chapter analyses the performance effect from using weekly instead of monthly returns as optimization input. Chapter six analyses the performance of global minimum tracking error volatility portfolio. Limitations and suggestion for further research are detailed in chapter seven. Chapter eight summarizes the conclusions and recommendations of the dissertation.

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Chapter 1: LITERATURE REVIEW The saga of modern finance, inspired by the seminal work of Markowitz (1952), created the ground for quantifying the assessment of every investment and/or financial choice in the context of a risk-return trade off. Efficient portfolio, with the theoretical hindsight of offering the highest return for a given level of risk, or lowest risk for a given level of return, were difficult to be identified in practice. Portfolios identified using sample moments involve extreme long and short positions among assets, which, as Jagannathan and Ma (2003) argue, may not be possible in practice. Estimation error, in part, was mitigated by the asset pricing models that became famous in academia and through professionals during the last 50 years. The main contribution of the models, from the forecasting point of view, was the significant reduction of the number of parameters to be estimated, by proposing as source of covariation of asset returns common risk factors, and the inclusion of the risk free asset, which, in theory, facilitated the selection of efficient portfolios. However, the utilization of factor models for investment selection requires expected return as input, whose forecasts, as documented by Chopra and Ziemba (1993), are error prone. The finance literature proposes two ways for overcoming the need of expected return as optimization input. One approach is the use of naïve diversification (1/N). DeMiguel et al. (2009a) documents the performance of fourteen different models across seven datasets. None of the strategies, according to the authors, consistently outperforms “primitive” diversification 1/N. The other approach is to invest in the global minimum variance portfolio. This strategy requires only estimates of variance-covariance forecasts for the portfolio optimization exercise. Chopra and Ziemba (1993) document that errors in expected returns are much more important than error in the variance and covariance return estimates from a performance perspective. Inspired by the suggestive performance reports of Nielsen and Aylursubramanian (2008) and Poullaouec (2008) our research is focused on the global minimum variance portfolio performance (GMVP). Despite the findings of Chopra and Ziemba (1993), various studies have focused on the input of implicit assumptions regarding return covariation in ex-ante portfolio optimization and their impact on ex-post portfolio performance. The classical variance-covariance estimator is the sample covariance matrix of returns estimated from equation 1.10, which, according to Ledoit and Wolf (2003), is the maximum likelihood estimator as number of observations approach infinity. Researches so far provide mixed evidence whether the use of sample covariance matrix estimator for the optimization input results in poorer out of sample performance as compared with other models. Other models used for covariance matrix estimates are single/multi risk factor models, which, based on the study of Chan et al. (1999), provide little forecast/out of sample performance incentives as compared with the classical sample estimator to devote more time in determining the structure of return covariation. An innovative step forward in estimation of covariance matrix of returns are shrinkage estimators, highly promoted by Ledoit and Wolf (2003, 2004, 2014). These are weighted average estimators of the structure imposed factor covariance matrix and sample covariance matrix. The shrinkage intensity depends on the correlation between the estimation error of the sample covariance matrix and estimation error of the shrinkage target. These models, in the studies of Ledoit and Wolf (2003, 2004, 2014) show dominance in performance of the shrinkage estimators towards the sample covariance matrix of returns. Application of shrinkage to portfolio weights have been 18

used in existing literature and proved to be at the same or at higher level of efficacy compared with covariance matrix shrinkage estimators in terms of out of sample performance. Jagannathan and Ma (2003) were the first to analyze the shrinkage like effect of short sale and/or maximum weight restrictions in the covariance matrix of return. A series of these studies, further advanced by Behr et al. (2008, 2013) show that imposing weight constraints to the sample covariance matrix optimization may result in better out of sample performance of the global minimum variance portfolios than using other estimators, including shrinkage. The work of Gosier et al. (2005) and Jagannathan and Ma (2003) show that the use of higher frequency return may improve portfolio performance similarly as the use of more sophisticating models in forecasting covariance matrix of returns and/or restricting portfolio weights. A summary of minimum tracking error studies suggests that restrictions in optimization input may improve the performance of minimum tracking error portfolios, as shown by Roll (1992), Jorion (2003) and Bajeux-Besnainou (2011). This chapter continues with a summary of asset pricing theories, their limitations/assumptions, and problems in testing whether theoretical models hold empirically. Then, section 1.2 describes the motivations of studying global minimum variance portfolios from investment management practice. Next section presents an analyses of the current literature regarding the effect of forecasting errors on minimum variance strategy performance. Section 1.4 analytically describes the pros and cons of various models used for global minimum variance portfolio optimization. The last part of this chapter presents a brief summary of the studies, who describe the limits and recommend improvements for minimum tracking error strategies.

1.1 Foundations, Developments and Limitations of the Portfolio Selection Problem 1.1.1 Markowitz contribution on the Portfolio Selection problem As quoted at Reilly and Brown (2012b), Markowitz developed a framework for estimating expected return and risk measures for a portfolio of assets. Under a set of assumptions, Markowitz developed the model for measuring and minimizing portfolio risk. Following the explanatory approach Reilly and Brown (2012b) regarding the Markowitz model, we summarize the assumptions of the model as: 1. Investment alternatives are assessed based on the probability distribution of the expected returns over a predefined holding period. 2. Risk is estimated as variability of expected returns. 3. Investors maximize the expected utility of the predefined holding period. 4. Utility function of investors are characterized by diminishing marginal utility of wealth. 5. Investors’ utility is a function of expected returns, risk, as measured by the variability of expected returns, and the investor’s degree of risk aversion. 6. Investors prefer a higher level of expected return, assuming risk is constant. 7. Investors prefer a lower level of risk, assuming expected return is constant. The set of assumptions allows to define the expected return and risk of a portfolio as: 𝐸𝑅𝑝 = ∑𝑁 (1.1) 𝑘=1 𝑤𝑘 𝑅𝑘 𝑁 𝑁 2 2 𝜎𝑝 = �∑𝑁 𝑘=1 𝑤𝑘 𝜎𝑘 + ∑𝑘=1 ∑𝑙=1 𝑤𝑘 𝑤𝑙 𝐶𝑂𝑉𝑘𝑙

19

, where 𝑘 ≠ 𝑙

(1.2)

where 𝑤𝑘 is the proportion of investment in asset k, 𝑅𝑘 is the expected return of asset k over the predefined holding period, 𝜎𝑘2 is the estimated variance of returns of asset k over the predefined holding period, 𝐶𝑂𝑉𝑘𝑙 is the estimated covariance of returns for asset k and l over the predefined holding period. As explained in Reilly and Brown (2012b), following the above annotation, Markowitz efficient portfolios are those that offer the highest level of 𝐸𝑅𝑝 for a given level of 𝜎𝑝 , and the lowest level of 𝜎𝑝 for a given level of 𝐸𝑅𝑝 . Then, the optimization considers changes in portfolio weights such that they minimize portfolio risk to obtain a specific level of expected portfolio return in the future. The solution to the constraint problem is given by the formula: 𝑚𝑖𝑛𝑊∈𝑅 𝑊` ∑^ 𝑊 (1.3) Subject to: 𝐸𝑅𝑝 = 𝐸𝑅𝑝 ∗ Such that: ∑𝑁 𝑘=1 𝑊𝑘 = 1 where Wk denotes the weight of asset k at period, W denotes the vector of Wk over the predefined holding period, ∑ represents the forecasted variance-covariance matrix of portfolio returns, 𝐸𝑅𝑝 ∗ denotes the desired level of expected return risk minimization is subject to. The higher the level of 𝐸𝑅𝑝 ∗ , the higher the level of 𝜎𝑝 estimated through the portfolio minimization process in equation 1.3. Assuming the input list, expected returns on individual assets, standard deviation of returns on individual assets, covariance estimates between pairs of assets, over the predefined period, is the same, all investors, according to Markowitz, should hold efficient portfolios that offer the highest level of reward (lowest level of risk) for a targeted level of risk (return). The set of efficient portfolios constitute the efficient frontier. The choice of which portfolio to invest from the efficient frontier, according to Markowitz, depends on the risk appetite of the individual investor represented by his/her utility function. Considering the implications summarized at Bodie et al (2014b), however, the number of estimated inputs required for Markowitz efficient portfolios however, is too large to guarantee accurate estimates for each. More specifically, the optimization of a portfolio of N assets requires as input N estimates of expected returns, N estimates of individual asset variances, and (N2 – N)/2 estimates of covariance terms. As indicated by Bodie et al (2014b), for a portfolio comprising all New York Stock Exchange (NYSE) 3000 stocks the optimization input estimates are more than 4.5 million. In the next subsection we discuss how the hypothetical existence of a risk free asset and common risk factors that influence the risk/return performance of all the whole investment universe derive a new efficient frontier called Capital Market Line (CML). We also discuss how such assumptions significantly reduce the number of input list required for estimating efficient portfolios.

1.1.2 The Introduction of Risk Factors and Risk Free asset in the asset allocation/security selection problem As analytically described at Reilly and Brown (2012c), Capital Asset Pricing Model (CAPM), developed by Sharpe (1964) and Lintner (1965), is an asset pricing theory 20

that builds on the foundations of Markowitz (1952) optimal portfolios, and prices expected return of the risky assets based on the underlying risk. The main assumptions considered in formalizing Capital Asset Pricing Model as final product of Capital Market Theory are: 1. Investors choose always Markowitz-efficient portfolios based on their degree of risk aversion, 2. Investors can borrow/lend at the risk free rate, 3. Investors have homogeneous views regarding future asset return scenarios and respective probabilities, 4. Investors have the same predefined holding period, 5. It is possible to buy/sell fractions of investable assets, 6. There are no taxes, 7. There are no transaction costs, 8. Inflation and/or interest rate changes are predictable, 9. All investments are priced based on the underlying risk, 10. Asset returns are described by normal probability distribution. Considering all the above mentioned assumptions, which, in part, are considered with the underlying assumptions of Markowitz efficient portfolios, all investors should hold (1) the market portfolio, (2) the risk free rate, (3) a combination of (1) and (2). Following the explanations of Bodie et al (2014 a,b,c) and Reilly and Brown (2012c), the market portfolio should be the one that maximizes the extra return in excess to the risk free return per unit of risk undertaken, which is formally written: 𝑀𝐴𝑋 𝑆𝐻𝑝 =

𝐸𝑅𝑝 −𝑟𝑓

(1.4)

𝜎𝑝

where 𝑆𝐻𝑝 is the Sharpe Ratio, the slope of the Capital Allocation Line (CAL) from the risk free asset to portfolio p. The higher the slope of the line, the higher the extra reward per unit of risk for the investor. The CAL with the highest slope is the tangent line from the risk free rate to the Markowitz efficient frontier. The line with the highest slope is called the Capital Market Line (CML). As per CAPM, the tangency point between the CML and the efficient frontier of risky assets is the market portfolio M. Capital Asset Pricing Model proposes that individual investors can estimate the optimization input list as: 𝐸𝑅𝑘 = 𝑟𝑓 + 𝛽𝑘 (𝐸𝑅𝑀 − 𝑟𝑓 ) where 𝛽𝑘 = 2 𝜎𝑘2 = 𝛽𝑘2 𝜎𝑀 + 𝜎𝑒2𝑘 2 𝐶𝑂𝑉𝑘𝑙 = 𝛽𝑘 𝛽𝑙 𝜎𝑀

𝜌𝑘𝑀 𝜎𝑘 𝜎𝑀

(1.5) (1.6) (1.7)

where 𝛽𝑘 is the measure systematic risk of asset k, named beta, (𝐸𝑅𝑀 − 𝑟𝑓 ) is the expected market risk premium, and 𝜌𝑘𝑀 is the correlation of returns of asset k with market returns. The number of parameters to be estimated for the input list, as per Capital Market Theory, is dramatically reduced to 3N + 2. The most important contribution of Capital Asset Pricing Model is that asset returns can be forecasted based on their risk characteristics. Assuming all investors hold a well-diversified portfolio that comprises the risk free rate, the market portfolio or both, the risk they should be concern when assessing new investment opportunities is the portion of risk that contributes to total portfolio risk and cannot be diversified away. Un-diversifiable risk comprises all macroeconomic events that influence the 21

performance of all investable assets in the economy. Independently from the source of risk, as per CAPM, macro events do influence the market portfolio risk/return performance. Since the latter is the investment universe portfolio with each asset in proportion to its market value, the result of the impact of the source of risk on the market portfolio is broadly considered by academics and practitioners as a feasible representation for assessing the impact of the source of risk on individual/portfolio of assets. The systematic risk is measured by beta. Following the work of Ross (1976) and Fama and French (1993, 1997), however, the linear model that explains return fluctuations, should include more than one macroeconomic risk representative, to properly identify 1) the source of asset/portfolio fluctuation, 2) better forecast due to more information included in the model, 3) potential hedging strategies on common sources of risk (ex. Inflation, Foreign exchange risk). As explained by Bodie et al (2014d) and Reilly and Brown (2012d), the Arbitrage Pricing Theory (APT) introduced by Ross (1976) explains fluctuations in investment returns with the following model: 𝐸𝑅𝑘 = 𝑟𝑓 + 𝛽𝑘1 𝑅𝐹1 + 𝛽𝑘2 𝑅𝐹2 + ⋯ + 𝛽𝑘𝑛 𝑅𝐹𝑛 (1.8) Where 𝑅𝐹𝑛 represents risk premium of common risk factor n, and 𝛽𝑘𝑛 represents the sensitivity of security k return to fluctuations of common risk factor n. 𝛽𝑘𝑛 is also called the factor loading of asset k on common risk factor n. One of the models highly used in empirical studies and investment industry for pricing returns through common risk factors is the Fama-French three factor model (FF), introduced by Fama and French (1993, 1997), the math representation of which is: 𝐸𝑅𝑘 = 𝑟𝑓 + 𝛽𝑘𝑀 𝐸𝑅𝑀 + 𝛽𝑘𝑆𝑀𝐵 𝑆𝑀𝐵 + 𝛽𝑘𝐻𝑀𝐿 𝐻𝑀𝐿 (1.9) where SMB represents the return on portfolio of small capitalization stocks in excess of the return on portfolio with large capitalization stocks. HML represents the return on portfolio High equity book value-to-market value stocks in excess of Low book value-to-market value stocks. APT is an innovative model with two main drawbacks. It does not suggest the number of common risk factors to include in the model and the identity of the factors. Next section briefly explains the efficient market hypothesis, and the need of fundamental models such as CAPM and APT in order this assumption to hold in the real world. We discuss the source and implications of price/return anomalies in the real world.

1.1.3 Limitations from the Behavioral Theory

Efficient Market

Hypothesis and

The first attempt of formalizing the theory of efficient capital markets is done by Fama (1970). The author, as explained in Reilly and Brown (2012a), describes the theory as a fair game model, where players (investors) may be confident that all information is reflected in the security price. The implication of this hypothesis is that securities are consistently priced based on the underlying systematic risk of the asset, as determined by factor models such as CAPM or APT. Fama (1970) divides efficient market hypothesis in three sub hypothesis. Separation is based on the underlying information required. 22

1. The weak form of efficient market hypothesis states that asset prices currently reflect all available information regarding past prices, trading volumes, rates of return, etc. The weak form implies that nobody can experience higher riskadjusted return without undertaking higher risk, as CAPM and APT infer, just by studying past security prices, rates of return, etc. 2. The semi-strong form of the efficient market hypothesis states that asset prices currently reflect all publicly available information, such as press release, macroeconomic events, company’s performance, etc. The semi-strong form implies that nobody can experience higher risk-adjusted return without undertaking higher risk, as CAPM and APT advocate, just by trying to take advantage of new publicly issued information that affects asset’s return performance. 3. The strong form of the efficient market hypothesis states that asset prices currently reflect all public and private information, where private information includes insider trading, business owner/manager information, etc. The strong form implies that nobody can experience higher risk-adjusted return without undertaking higher risk, as CAPM and APT infer, by trying to take advantage of the privately held information. The work initiated in 1970 by Fama, is later reviewed in 1991 by the same author. Fama (1991) divides empirical tests and subsequent results of the efficient market hypothesis in three groups, consistently with the forms of efficiency he proposed in early studies. One of the main tests used in the weak form of efficiency is the autocorrelation of current and past asset returns. The most popular test of the semi-strong form of the efficient market hypothesis is the event study that examines how fast security prices react to macroeconomic events such as inflation publication, public news for merger and acquisition of public companies, earning announcement, etc. One of the test for efficiency of the strong form considers what happens with assets’ returns after insider trading transactions have been publicly released, as required by the Securities and Exchange Commission (SEC) of the United States. As summarized by Reilly and Brown (2012a), only the tests of the weak form of the efficient market hypothesis supported efficiency. Tests for the semi-strong and the strong form have presented mixed result in support of the efficient market hypothesis. The question mark whether the hypothesis hold or not, imply that the fundamental proposition of every asset pricing theory, asset returns are explained by the underlying risk, may not be always correct. This means that, there may be cases where investors may find opportunities that offer higher/lower return than the one suggested by the asset pricing model they use. These cases are called anomalies. Finance researchers have “borrowed” behavioral sciences to explain anomalies. The subfield of finance that studies anomalies is called behavioral finance. As briefly explained by Reilly and Brown (2012a), some of the behaviors explained by this studies are: 1. Selling winners to soon and holding losers too long, 2. Belief perseverance although new information is available in the capital markets, 3. Overconfidence in forecasts, 4. Escalation bias, which makes investors put more money into a failure rather than success due to the feeling of responsibility.

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Next section presents a summary of studies who criticize the tests used for efficient capital markets.

1.1.4 Roll’s Critique on the definition of the “Market”. The indicative study of Roll (1977), which is extended by Roll and Ross (1994) and Kandel and Stambaugh (1989, 1995), constitute what is known in the modern finance theory as Roll’s critique. As concisely pointed out by Bodie et al (2014g), Roll’s critique states that: 1. There is a single testable hypothesis for CAPM: is the market portfolio meanvariance efficient? 2. The “if and only if” beta-expected return relationship/market portfolio efficiency relation makes them not being independently testable 3. Betas estimated by ex-post mean-variance efficient portfolios satisfy the Security Market Line condition, that all securities return depend on asset’s contribution to total market risk. 4. CAPM is not testable unless the exact composition of the market portfolio is known. 5. Market proxies 1) may be mean-variance efficient when the true market portfolio is not, 2) may be mean-variance inefficient, which does not necessarily make the true market portfolio inefficient as well. This issue is also referred to as benchmark error. 6. Tests that reject the average return-beta relationship of the market proxy point the inefficiency of the proxy and not necessarily the true market expected return-beta relationship. 7. CAPM model cannot be tested meaningfully without an efficient market proxy. 8. It is not possible to determine the level of efficiency of the proxy. The impossibility of observing the true market portfolio, and considering the APT drawbacks mentioned in subsection 1.1.2, makes return forecasts from asset pricing models inaccurate. Eventually, asset allocation/security selection strategies based on such models result in inefficient investment strategies. Since fundamental models like CAPM and APT show ex-post errors in forecasting security returns, researchers came up with two portfolio strategies that do not required expected return in the input list. One strategy is Naive Diversification (1/N), and the other strategy is Global Minimum Variance Strategy. This thesis studies the performance of Global Minimum Variance Strategy in a global asset allocation framework. Next section introduces the motivations of this research from the investment management industry.

1.2 The performance of Minimum Variance Portfolios: Motivations from the Investment Management Profession. Nielsen and Aylursubramanian (2008) present the rationale and performance of MSCI Minimum Variance Equity Index introduced by the MSCIBARRA Research. As the authors cite, minimum variance strategies do not need expected returns as 24

optimization input. Considering the inaccuracy of return forecasts from factor models mentioned in the previous section, minimum variance portfolios may result a more efficient strategy than others, who require a risk/return tradeoff. In the construction of MSCI MV Equity Index the authors use the covariance matrix from the Barra Global Equity Model (GEM). The constraints applied are 1)index apply no hedging on Forex, 2) Investors receive return in USD, 3) The maximum weight on an index constituent is 1.5%, 4) The minimum weight for an index constituent is 0.05%, 5) The sector weights of MSCI MV Index are constraint to +/5% differences from the sector weights of the MSCI World Index, 6) The country weights of the MV Index are constraint to +/-5% differences from the country weights of the MSCI World Index, 7) Barra risk index exposures, Size, Value, Momentum, and Volatility, of MV Index is limited to +/-0.25 standard deviations difference from risk index exposures of MSCI World Index, 8) One-way index turnover is constrained to an upper bound of 10% every portfolio rebalancing. Portfolio rebalancing is done every six months. Considering the above mentioned constraints and revision frequency, Nielsen and Aylursubramanian (2008) construct the MSCI World MV Index for the period June 1995-December 2007. Referring to their findings, for the period under consideration, the MV Index realized higher return than the MSCI World Index resulting in a lower volatility of approximately 30%. During periods of downturn, MV index seemed to experience downside protection relative to the benchmark. It offered a similar risk/return performance during periods of market rise. MV strategy realized risk (9.75% p.a.), according to the authors, is much closer to a Long Duration Fixed Income Index realized risk (8.2% p.a.) than the one realized from the equity benchmark (13.5%). The beta of MV Index, which is measured as a rolling 36 months historical beta, is between 0.5 and 0.8. Simulating the 60/40 equity/fixed income strategy, and replacing the MSCI World Index with MSCI World MV Index, results in a 20% risk reduction as compared with the traditional allocation. The overall excess return of MV strategy above 1 month Treasury Bill was 6.5%, while the excess return of the equity benchmark is 6%. According to Nielsen and Aylursubramanian (2008), the Sharpe Ratio (SH) of the MSCI MV Index for the period June 1995-December 2007 was 0.67, significantly higher as compared with the 0.45 Sharpe Ratio of MSCI World Index. Poullaouec (2008), Vice President of State Street Global Advisors, uses the methodology of MSCI BARRA to construct and analyze the performance of MSCI World Minimum Volatility Index for the period May 1988-May 2008. The author uses monthly data from MSCI BARRA and Ibbotson. The reference currency is USD. Using a five year rolling period, volatility of MV strategy is 25% lower than the volatility of MSCI World Index, according to the author. Looking at extreme top (bottom) six months performance, gains (losses) of MV are reduced by 20% as compared with the equity benchmark. The MV portfolio returns has a higher correlation with the fixed income strategy, represented by Lehman Brothers Global G7 Index, than with MSCI World Index. MV strategy, according to the author, offers a higher Sharpe Ratio than the equity benchmark, because its risk reduction characteristics (30% lower standard deviation of returns) overcome the lower experienced return (20% lower return than MSCI World Index). As per Poullaouec (2008), if MSCI World MV strategy would be considered as an asset to be included in a diversified portfolio, its impact on portfolio performance would be: 25

1. Total risk reduction if MV strategy substitutes the equity index. In a simulation conducted, if the 70/30 equity/fixed income investment strategy would be substituted by the 50/30/20 MSCI World/LM G7/MV strategy, the standard deviation of returns is reduced by 0.7% per annum. 2. The relative risk to the benchmark (tracking error) is increased by 55 basis points if MV strategy substitutes style index strategies. 3. In a portfolio of stocks and bonds, including the MV strategy as additional asset, maximizes returns for the same level of risk, and minimizes risk for the same level of return. The weight of stock and bond indices in the portfolio are reduced due to the inclusion of the MV index. 4. MV strategy may not be necessarily considered as only an equity substitute. As explained above, based on the investment management practice cited in our research, the inclusion of MV strategy as a separate asset in a well-diversified portfolio may reduce absolute risk, but may increase the tracking error volatility vis-àvis the benchmark. Next section presents a summary of research that emphasizes differences in accuracy forecast, and, eventually ex-post performance, of returns and risk from history.

1.3 The Definition and Implications of Forecasting Errors of Portfolio Returns and Risk. Michaud (1989) states mean-variance optimizers have “error maximization” properties, because risk and return estimates are subject to estimation errors. The result of mean-variance optimization, according to the author, is the 1) overweight of securities with large estimated returns, negative correlations, small variances, and 2) underweight of securities with small estimated returns, positive correlations and large variances. Since historical returns are used to estimated expected returns and variance-covariance matrix of returns input for the Markowitz optimization presented in equations 1.1-1.4, estimation error is likely to be present in all input lists. As cited in Michaud (1989), an estimator is “admissible” if no other estimator dominates the earlier for a given risk level. Stein (1956) shows that sample mean of returns is not an admissible estimator for expected returns. Michaud (1989) affirms that sample mean ignores different sources of risk that may affect future returns. Chopra and Ziemba (1993) study the relative impact of errors in means, variance and covariance to performance out of sample. They state that minimum variance optimization is very sensitive to inputs quality. The authors use monthly returns of 10 randomly selected securities from Dow Jones Industrial Average (DJIA) for the period January 1980-December 1989. They compute the historical means, variance and covariance terms as input forecasts. The initial level of risk tolerance was assumed to be 50. Chopra and Ziemba (1993) explain that their result hold independently from the methodology used to estimate optimization input (historical approach or complex forecasting schemes). The assessment of the impact of error in means is done by replacing the true mean �� 𝑅�𝑘� by ��𝑘 (1 + 𝑙𝑧𝑘 ), where 𝑧𝑘 is a parameter with a standard normal an approximation𝑅 distribution, and 𝑙 is varied in the segment 0.05-0.20 by applying increments of 0.05, for examining impact of different error sizes. The procedure is repeated 100 times for each value of k, each iteration having a different set of 𝑧𝑘 .The same procedure is repeated for the set of variances 𝜎𝑘 and covariance terms 𝜎𝑘𝑗 . The risk tolerance is 26

varied in the values 25, 50, 75, to observe the impact of input errors for different risk tolerant investors. According to the findings of Chopra and Ziemba (1993), the error in means is approximately eleven times as important as the error in variance, and the covariance error has half the importance of errors in variance estimates in performance, for an investor with risk tolerance of 50. The authors show that the relative impact of the errors in means, variances and covariance terms depends on the risk aversion of the investor under consideration. At a higher level of risk tolerance, the impact of errors in means increase in importance with respect to errors in variance and covariance. At a lower level of risk tolerance, the impact of errors in means, variance, and covariance is closer. Their finding suggest also that the difference in impact between errors in variance and covariance is relatively stable through the three assumed levels of risk tolerance, as compared with the difference in impact between errors in return and errors in variance/covariance forecasts. Chan et al (1999) focus their study on the second moments of asset returns. The reasoning behind is that 1) first moments are very difficult to forecast, and, as demonstrated by Chopra and Ziemba (1993), optimization process is highly sensible to return estimation errors, and return variances and covariance are more accurate to be predicted from history; 2) they empirically examine the previously published literature regarding sources of return covariation. The authors use monthly stock return data from NYSE and AMEX. They exclude 20% lowest market capitalization stocks from NYSE and AMEX. Securities with prices lower than $5 are also excluded from the input list. On April of each year, from 1968 to 1997, 250 stocks are randomly selected. Pairwise covariance forecast is estimated based on the past 60 months of returns. No short sale and maximum weight constraint of 2% are applied to the optimization process. The forecasting models for covariance matrix of returns are 1) the sample covariance matrix, given by 𝐶𝑂𝑉𝑘𝑙 =

1

59

∑60 �) 𝑙 𝑖=1(𝑟𝑘𝑡−𝑖 − 𝑟�𝑘 ) (𝑟𝑙𝑡−𝑖 − 𝑟

(1.10)

where 𝑟𝑘𝑡 is the return in excess of the monthly T-Bill return of stock k in month t, and 𝑟�𝑘 is the sample mean excess return on stock k during the period under consideration (60 months). 2) The covariance matrix from 1, 2, 3, 4, 8, 10 factor models, given by 𝑉 = 𝛽𝛺𝛽 ′ + 𝐷 (1.11) where 𝛽 is the matrix of factor loadings of stocks, 𝛺, is the covariance matrix of factors, and 𝐷 is a diagonal matrix containing the residual variance of returns. 3) Constant covariance model��, 𝑐𝑜𝑣 where each covariance term is the simple mean of the sample covariance matrix. Model accuracy is measured by the correlation coefficient and mean absolute deviation between true and estimated covariance matrix of returns. In terms of accuracy, according to Chan et al (1999), there is slight difference among model performance. More factors do not necessarily improve prediction. The correlation coefficients for twelve subsequent months varies from 0.16 to 0.2 among various models. The correlation coefficient for thirty six subsequent months vary from 0.34 to 0.36 among various models. As per the authors, variance estimates show higher correlation than covariance estimates. In general, common risk factors help improve 27

forecasting power, but there is little difference between 3 factor and 10 factor models, according to the findings of Chan et al. (1999). The authors compare the out of sample performance of global minimum variance portfolios (GMVPs) optimized by different variance-covariance matrix input estimates. Based on their results, the covariance model used makes little difference in terms of out of sample performance of GMVP portfolios. The out of sample Sharpe Ratio GMVPs derived from covariance estimators varies from 0.64 to 0.69. Portfolio beta with the market index vary from 0.5 to 0.65. The out of sample standard deviation of returns vary from 12.59% to 12.94% among different GMVPs optimized using different estimators of the variance-covariance matrix of returns for the out of sample period 1973-1997. The authors find that the largest part of variation of returns of GMVP portfolios is due to variation of the single factor. More specifically, they find that 75% of the variation in portfolio returns come from variation in the market factor. Section 1.4 presents a detailed analysis of the literature for Global Minimum Variance Portfolio (GMVP) models and respective out of sample risk reduction/performance result.

1.4 The Models Used in Constructing Global Minimum Variance Portfolios and their Implication in Out of Sample Portfolio Performance 1.4.1 The Sample Covariance Matrix As recited by Ledoit and Wolf (2003), the sample covariance matrix estimated by Equation 1.10 has some appealing properties, such as, being the maximum likelihood estimator under the assumption of normal distribution of returns. Maximum likelihood property of this estimator is achieved asymptotically, as the number of return observations approaches infinity. As stressed out by Pantaleo et al (2011), who compare the usefulness of more refined estimators among covariance estimators, if the total number of assets N is the same \ with the total number of return observations T, the total number of parameters is the same order of magnitude with the total size of available data for the sample covariance matrix. This generates large estimation errors, according to the authors. Ledoit and Wolf (2004) state that this implies most extreme coefficient in the matrix tend to take extreme values not because they reflect true values, but because they contain an extreme amount of error. This result in wrong investment choices, as the optimization process weights most extreme coefficients, which, as explained above, are highly unreliable. The phenomenon is called by Michaud (1989) “errormaximization”. Ledoit and Wolf (2003) point out that, for covariance matrix, small sample problems occur unless T is one order of magnitude larger than N at the worst case. In terms of empirical performance, based on the study of Chan et al (1999), the forecasting properties for twelve (thirty-six) subsequent months of the sample covariance matrix, have a correlation coefficient of 0.18 (0.34), while highest correlation coefficients among other estimators are 0.2 (0.36). The out of sample standard deviation of returns is similar to the realized risk of more structured estimators. Similar results show the out of sample Sharpe Ratio of the minimum variance portfolio estimated by the sample covariance matrix of returns. 28

Ledoit and Wolf (2004) find that the sample covariance matrix underperform the shrinkage estimators developed by the authors and Ledoit and Wolf (2003) in terms of out of sample realized risk in five different number of assets applications N, when N = 30, 50, 100, 225, 500. Ledoit and Wolf (2014), show similar findings for their newly developed non-linear shrinkage estimators. DeMiguel et al (2009a), provide evidence that the sample covariance matrix estimators of the global minimum variance strategy is not consistently outperformed from other twelve investment strategies, including factor models and shrinkage estimators. The result is valid for six different datasets of US equity portfolios and two different performance metrics, the Certain Equivalent (CEQ) return and the Sharpe Ratio. Jagannathan and Ma find that sample covariance matrix estimators of minimum variance strategies derived from daily returns outperform shrinkage, factor models, and constrained estimators of the variance covariance matrix in terms of realized risk. This subsection provides mixed evidence, through the studies of other authors, whether using the sample covariance matrix as optimization input is a poor out of sample performance strategy. Next subsections summarize the main findings in the literature referring to the out of sample performance of other covariance estimators.

1.4.2 The Covariance Matrix Estimated by Common Risk Factors Considering the evidenced problems regarding the estimation error of the sample covariance matrix, various studies have attempted to compare the sample covariance performance with more structure models. As highlighted by Ledoit and Wolf (2003), the sample covariance matrix can be interpreted as a N factor model, where N represents the number of assets in the portfolio. As shown by Green and Hollifield (1992), there is a dominant single factor in covariance structure of asset returns, as modeled in equation 1.11. Clarke et al (2011) find that risk attributable to the dominant factor of individual securities accounts for 10%-20%, while risk attributable to the dominant factor of Global Minimum Variance Portfolios account for 80%-90% of the total risk. This suggest that return co-variation source is shifted from the idiosyncratic to systematics influence when investment strategy is changed from individual security selection to minimum variance optimization. Chan et al (1999) confirm that the largest part of performance variation in the minimum variance strategy is due to variations in the single market factor. They confirm that there is little discrimination in terms of risk reduction and Sharpe performance between minimum variance strategies when structure in the covariance matrix of returns is applied through factor models. Moreover, the authors evidence that factors help improving forecasting power of the model. There is little difference, however, between three factors and more factor models in terms of forecasting power according to Chan et al (1999). DeMiguel et al (2009b) find that global minimum variance portfolios derived from single factor estimators of variance-covariance matrix outperform in terms of risk reduction GMVP derived from sample estimators of the variance-covariance matrix in two out of five datasets. Moreover, they outperform the sample counterpart in three out of five datasets in terms of Sharpe Ratio. As per Pantaleo et al (2011), the differences in realized risks of GMVPs computed from sample and factor covariance matrix estimates are statistically significant when 29

revision frequency of the portfolio is one and six months. If one year rebalancing policy is applied, however, the risk reduction empirical characteristics of the single factor estimator seem not to be statistically significant, according to the findings of the authors. Jagannathan and Ma (2003) provide mixed results whether the GMVPs from factor models outperforms the sample GMVPs. They find that factor models seem to outperform the classical sample covariance matrix in terms of risk reduction, there is, however, some evidence that the latter seem to outperform the factor model through the out of sample Sharpe Ratio. The main question arisen from subsections 1.4.1 and 1.4.2 concerns the optimal tradeoff between choosing the so called in sample maximum likelihood estimator (sample covariance matrix in equation 1.10) or the more structured estimators by theoretically sound risk factors (factor covariance matrix in equation 1.11). Next subsection introduces shrinkage estimators of the covariance matrix, which are a compromise between the sample and the factor model approach.

1.4.3 The Shrinkage Estimators 1.4.3.1 Shrinkage Estimators for Portfolio Variance/Covariance As specified by Ledoit and Wolf (2003), the single factor covariance model has a lot of bias due to the heavy structure imposed from a single source of risk, but shows low estimation error. The sample covariance matrix is asymptotically unbiased, but results in high estimation error. The optimal trade-off, as per the authors, is to identify the optimum combination of bias and estimation error. Stein (1956) is the first work that introduces the shrinkage estimator for variable’s moments. He proposed to revise any unbiased maximum likelihood of first moment, second moment or covariance estimate with a weighted average of the prior unbiased estimate with an estimate with a shrinkage target. The weight, κ (between 0 and 1), of the shrinkage target determines the level of structure imposed. The closer to one is the weight, the stronger the structure. Stein (1956) finds that shrinking sample mean towards a constant may improve forecasts under certain conditions. Efron and Morris (1977) further expand the usefulness of shrinkage. Jorion (1986) was the first, to the best of our knowledge, who studied the effect of estimation error on portfolio choice with an application of shrinkage. The author shows through simulation data that, under different scenarios of sample size, the shrinkage estimator outperforms the classical estimator in terms of certain equivalent (CEQ) return. As cited by Behr et al (2008), Jobson and Korkie (1980) confirm, however, that shrinkage estimators suffer the small sample performance problem. Muirhead (1987) confirms, after a review of the large literature on shrinkage estimators, two drawbacks: 1) shrinkage are not valid when N>T, 2) do not take advantage from prior knowledge of cross sectional positive correlation for stock returns. Frost and Savarino (1986) suggest an ex-ante information, according to which the greater is the difference between the sample estimates of a moment of individual securities differs with the mean of the specified moments of all securities, the higher the probability of estimation error on the individual estimate. The shrinkage target is the mean of all securities moments. According to the authors, the smaller the sample, the higher the shrinkage intensity towards grand average of security returns, variances 30

or covariance between return pairs. Frost and Savarino (1986) find that this empirical Bayesian shrinkage strategy selects portfolios with ex-ante superior performance as compared with the investment strategies using as input classical sample estimators or Bayesian shrinkage approaches considered in the previous literature. Frost and Savarino (1986) thus, accommodate the second drawback suggested by Muirhead (1987), but they ignore prior information from correlation of estimation error of the sample estimators and shrinkage targets, as evidenced by Ledoit and Wolf (2003). Ledoit and Wolf (2003, 2004), provide a solution to the breakdown of the shrinkage model when N>T, by minimizing a loss function that does not involve the inverse of the covariance matrix. The authors claim that optimal shrinkage intensity is a function of the correlation between estimation error of the sample covariance model and estimation error of the shrinkage target. The lower the correlation, the higher the benefit in estimating covariance matrix from the combination of sample and highly structured factor model. Their models’ assumptions are: 1. Stock returns are independently identically distributed through time, 2. N is fixed, while T goes to infinity, 3. Stock returns have finite four moments, in order to apply the Central Limit Theorem, 4. The single index covariance matrix Φ ≠ Σ, the covariance matrix of the population. 5. The market model has a positive variance According to Ledoit and Wolf (2003), the equally weighted portfolio explains better stock market variation as compared with the value weighted index. In this model, the choice of how much to weight in the shrinkage estimator 𝑆̂, the sample covariance estimator S, and the single factor covariance matrix estimator F, depends on the tradeoff between estimation error and bias. By applying minimization of the loss function of the difference between the true covariance matrix Σ and the shrinkage estimator 𝑆̂, the authors identify three coefficients to be estimated: π, the sum of asymptotic variances of the entries of the sample covariance matrix 𝑁 scaled by √𝑇: ∑𝑁 𝑘=1 ∑𝑙=1 𝐴𝑠𝑦𝐶𝑜𝑣[ √𝑇𝑠𝑘,𝑙 ], where 𝑠𝑘,𝑙 represents the sample covariance between returns of asset k and l. Each element of π is named π𝑘,𝑙 . ρ, the sum of asymptotic covariances of the entries of the single index covariance matrix with the sample covariance matrix entries scaled by √𝑇: 𝑁 ∑𝑁 𝑘=1 ∑𝑙=1 𝐴𝑠𝑦𝐶𝑜𝑣[√𝑇𝑓𝑘,𝑙 , √𝑇𝑠𝑘,𝑙 ], where 𝑓𝑘,𝑙 represents the single factor covariance estimate between returns of asset k and l. Each element of ρ is named ρ𝑘,𝑙 . 𝑁 2 γ, the misspecification of the single factor model: ∑𝑁 𝑘=1 ∑𝑙=1[𝜙𝑘,𝑙 − 𝜎𝑘,𝑙 ] , where 𝜙𝑘,𝑙 represents the single factor covariance between returns of asset k and l and 𝜎𝑘,𝑙 the true covariance between returns of asset k and l. Each element of γ is named γ𝑘,𝑙 . According to Ledoit and Wolf (2003), a consistent estimator for π𝑘,𝑙 would be p𝑘,𝑙 , given by: p𝑘,𝑙 =

1

𝑇

∑𝑇𝑡=1{( x𝑘𝑡 − m𝑘 ) (x𝑙𝑡 − m𝑙 ) − 𝑠𝑘,𝑙 }2

Where x𝑘𝑡 is return of asset k at month t, and m𝑘 is asset k return mean. 31

(1.12)

A consistent estimator for ρ𝑘,𝑘 is r𝑘,𝑘 , and a consistent estimator for ρ𝑘,𝑙 is r𝑘,𝑙 given by: r𝑘,𝑙 =

Where: r𝑘,𝑙 𝑡 =

1

𝑇

∑𝑇𝑡=1 r𝑘,𝑙 𝑡

(1.13)

𝑠𝑙0 𝑠00 (x𝑘𝑡 − m𝑘 )+ 𝑠𝑘0 𝑠00 (x𝑙𝑡 − m𝑙 )+ 𝑠𝑘0 𝑠𝑙0 (x0𝑡 − m0 ) 𝑠00 2

(x0𝑡 − m0 )(x𝑙𝑡 − m𝑙 ) (x𝑘𝑡 −

m𝑘 ) − 𝑓𝑘,𝑙 𝑠𝑘,𝑙 𝑠00 represents the standard deviation of the single index during the period under

consideration, and 𝑠𝑙0 represents the sample covariance estimate between asset l and the market, during the period under consideration. A consistent estimator for γ𝑘,𝑙 is c𝑘,𝑙 , given by the formula: c𝑘,𝑙 = (𝑓𝑘,𝑙 − 𝑠𝑘,𝑙 ) 2 (1.14) Following the consistent estimators of π𝑘,𝑙 , ρ𝑘,𝑙 and γ𝑘,𝑙 given by equations 1.12, 1.13 and 1.14, a consistent estimator for the optimal shrinkage constant κ is given by: 𝑝−𝑟 𝑘= (1.15) 𝑐 The equation of k indicates that the weight of the index model in shrinkage estimator increases with the error on the sample covariance matrix through p, decreases with the misspecification of the single index model, through c, and decreases with the correlation of errors of the sample covariance matrix with errors of the single index model, through r. By applying the optimal shrinkage constant, Ledoit and Wolf (2003) provide the shrinkage estimator for the variance-covariance matrix of stock return, as: 𝑘 𝑘 𝑆̂ = 𝐹 + �1 − � 𝑆 (1.16) 𝑇

𝑇

The authors use in their study monthly returns of stocks listed in NYSE and AMEX. The shrinkage target is the single factor (market) model suggested by Sharpe (1963). The period under consideration is 1962-1994. The estimating window is ten years. The rebalancing frequency is 12 months. The main findings of Ledoit and Wolf (2003) study are: 1. The t-statistics that the shrinkage estimator yields lower variance out of sample than other estimators varies from 2.73 to 7.39. 2. Shrinkage intensity varies from approximately 0.79 to approximately 0.82 in the period under consideration. 3. Short interest and turnover of the shrinkage estimator is higher than 5 out of 7 minimum variance portfolios. 4. The resulting estimator has the statistical property of being invertible and well-conditioned. Ledoit and Wolf (2004) apply a similar shrinkage estimator. Optimization is applied by constraining the level of information ratio, excess return vis-à-vis the benchmark divided by the standard deviation of excess return vis-à-vis the benchmark during the period under consideration, to 1.5. The authors use the constant correlation model as shrinkage target. Moreover, they use different sample sizes of us stock portfolios, assuming N=30, 50,100,225,500. The benchmarks of the study are the value weighted portfolios for each sample accordingly. The authors highlight the following findings: 1. In all scenarios, shrinkage estimators highest Information Ratio. 2. In most scenarios, shrinkage estimators result in the highest alpha.

32

3. In all scenarios, shrinkage estimators generate the lowest standard deviation of excess returns, alpha. 4. The average information ratio decreases as N increases. 5. Constant correlation shrinkage estimator performs better than the factor model shrinkage estimator of Ledoit and Wolf (2003) if Nk In equation 1.20, 𝑟𝑡,𝑘 represents return on stock k at time t. 𝜖𝑘 is the error term that satisfies the cross correlation assumption of the linear regression model regarding errors. Returns are normal and identical independently distributed (IID). The authors prove that the parameter estimates of the global minimum variance portfolio are: (1.21) 𝛽𝑘 = 𝑤𝑀𝑉,𝑘 𝛼 = 𝐸𝑅𝑀𝑉 (1.22) 2 2 𝜎𝜖 = 𝜎𝑀𝑉 (1.23) Kempf and Memmel (2006) show that weights estimators from the Ordinary Least Squares (OLS) regression are the same as the ones estimated from the traditional approach shown in equation 1.3. As such, distributional results of OLS may be used for inference assessment of the traditional approach. The author conclude that the 2 larger the true 𝜎𝑀𝑉 , the larger N, the shorter T, the higher the unconditional estimation risk, as shown in equation 1.19. One application of the distribution result is to assess the extent of estimation risk. Another application shown in the study of Kempf and Memmel (2006), is to test hypotheses whether there is benefit from international diversification. DeMiguel et al (2009b) provide a general framework for determining the portfolio with superior out of sample performance in the presence of estimation error. They also treat weights rather than moments of asset returns. In the research of weight shrinkage estimators, their study introduces the norm constraint portfolio weights estimates to be smaller than a determined threshold. The author define the norm constrained minimum variance portfolio as: 𝑚𝑖𝑛𝑊∈𝑅 𝑊` ∑^ 𝑊 , s.th. ∑𝑁 (1.24) 𝑘=1 𝑊𝑘 = 1, ‖𝑤‖ ≤ 𝛿 Where ‖𝑤‖ represents the norm of the portfolio weight vector, and 𝛿 is the given threshold. The authors consider the 1-norm and 2-norm p portfolio weight in the following formulation: 1/𝑝

𝑝 ‖𝑤‖𝑝 = �∑𝑁 𝑘=1|𝑤𝑘 | �

(1.25) For portfolio norm p = 1 and 2. The solution of the norm constrained problem, ‖𝑤𝑁𝐶 ‖ for each 𝛿 < ‖𝑤𝑀𝐼𝑁𝑈 ‖ satisfies the inequality: ‖𝑤𝑁𝐶 ‖ < ‖𝑤𝑀𝐼𝑁𝑈 ‖ (1.26) Equation 1.26 shows that ‖𝑤𝑁𝐶 ‖ is the shrinkage estimator of the unconstrained minimum variance portfolio weights, ‖𝑤𝑀𝐼𝑁𝑈 ‖ . The main contribution of the DeMiguel et al (2009b) research, according to the authors, are: 35

1. The provision of a general framework for determining the portfolios in the presence of estimation errors. They suggest to shrink weights rather than variance covariance matrix of returns. 2. The proof that estimators proposed in the studies of Jagannathan and Ma (2003), Ledoit and Wolf (2003, 2004), and DeMiguel et al (2009a), are special cases of the general framework presented in their research, where the first can be replicated by properly calibrating the norm-1 portfolio constraint, and the other three can be replicated by properly calibrating the norm-2 portfolio constraint. 3. Show that these portfolios are for Bayesian investors who have prior beliefs for portfolio weights rather than moments of asset returns. 4. The general framework may be extended to consider portfolio strategies not studied before. 5. Illustrate that norm-constrained portfolio strategies can be calibrated to improve performance. 6. Empirical test for out of sample portfolio variance, Sharpe Ratio, and Turnover. As per the authors, new portfolios can outperform strategies of DeMiguel et al (2009a), Jagannathan and Ma (2003) and Ledoit and Wolf (2004) studies. Frahm and Memmel (2010) develop two shrinkage estimators for GMVP weights that dominate sample covariance estimators with respect to out of sample variance of portfolio returns. The main assumption of their theoretical research analysis is that returns are serially independent and normally distributed. The authors drive the same conclusions even in case more linear constraints are applied to portfolio weights. According to Frahm and Memmel (2010), their approach differs from other shrinkage estimators previously developed for covariance matrix of returns and weights, for: a. Dominance of the strategy analyzed in their research is valid even in small samples. b. The target is not a better covariance matrix, but the reduction of out of sample variance for portfolio returns. As cited by the authors, shrinkage estimators converge towards the optimal portfolio weights as sample size grows to infinity. The main contribution of Frahm and Memmel (2010) paper, according to the authors, is: 1. The derivation of two shrinkage estimators for GMVP that dominate traditional estimators in terms of out of sample variation in returns. 2. Mathematical analysis of both small sample and large sample properties of these estimators. 3. Derivation of a small sample test for deciding to ignore or not time series information vis-à-vis naïve diversification (1/N strategy). As a summary of the results for this research, can be mentioned: 1. The shrinkage estimators developed in the research of Frahm and Memmel (2010) reduce out of sample variation as compared to sample covariance estimators, especially as the number of assets in the portfolio increase. 2. There is mixed evidence whether both estimators dominate naïve diversification in terms of out of sample variance. According to the empirical tests of the study, if the number of assets N is large as compared to the number of observations T, 1/N appears to be the best strategy 36

Clarke et al (2011) suggest an analytical solution to short sale constrained Global Minimum Variance Portfolios under the assumption that return co variation is determined by the single risk factor model. As cited by the authors, the long standing CAPM critique is that high market beta stocks are not rewarded with higher returns necessarily. Low risk/high return strategies are usually associated with idiosyncratic risk, as cited by the authors. Clarke et al (2011) focus on the analytical form and weight parameter values of long only GMVP. The single index model in this research is the value weighted market portfolio. As the authors show in this paper, the weights of individual securities in the long only constrained GMVP are: 𝑤𝑘 =

2 𝜎𝐿𝑀𝑉 2 𝜎𝜖𝑘

∗ �1 −

𝛽𝑘 𝛽𝐿

� for 𝛽𝑘 < 𝛽𝐿 , else 𝑤𝑘 = 0

(1.27)

2 Where 𝜎𝐿𝑀𝑉 is the in sample return variance of long only GMVP, 𝛽𝐿 is the long only threshold beta, 𝑤𝑘 is the weight of security 𝑘 in the minimum variance portfolio, and 2 𝜎𝜖𝑘 is the in sample idiosyncratic variance of asset 𝑘. 𝛽𝐿 is estimated as:

𝛽𝐿 =

1

𝜎2 𝑀

2

𝛽𝑘 + ∑𝑁 𝑘=1 2 𝑓𝑜𝑟𝛽𝑘 N, and constraints are imposed, no improved estimator performs better than the sample covariance matrix of returns in terms of out of sample risk reduction. Second, as the studies of DeMiguel et al. (2009 a,b) and Behr et al. (2013) confirm, no estimator consistently outperforms the unconstrained and short sale constrained Global Minimum Variance Portfolio proposed by Jagannathan and Ma (2003) in terms risk and superior Sharpe Ratio among different datasets. Moreover, Ledoit and Wolf (2014) report small differences in out of sample standard deviation of returns between their improved non-linear shrinkage estimator and the unconstrained GMVP optimized from the sample covariance matrix of returns, when the sample size is relatively small. The extension of this study with the use of shrinkage estimators, however, would contribute to the global GMVP research in two forms. First, it would prove the robustness of the findings presented in this dissertation. Second, it would check the consistency of the findings of existing literature for shrinkage portfolio performance in a global scale. The naïve diversification and constant correlation model proposed by Ledoit and Wolf (2004) may serve as shrinkage targets of the covariance matrix of returns for shrinkage estimators for minimum variance portfolios with global asset input.

7.4. A comparative study regarding transaction costs between different markets/regions One of the limitations of this dissertation research is the assumptions that fractional transaction costs, in the part of the world represented by the indices datasets of our research, are comparatively equal, either 50 basis points or 25 basis points. According to the study of Campbell and Froot (1994), the tax size among 20 developed countries considered in their research varies from 0.33 basis points in US to 50 basis points in UK as of 1991. Matheson (2011) confirms that Brazil applies the highest taxes to securities among G20 countries. Citing previous findings, Matheson (2011) confirms that elasticity of trading volume with respect to transaction costs varies from -0.1 for US Wheat Futures to -2 for S&P 500 Index Futures. To the extent that transaction taxes may be considered a proxy of the relative differences for the cost of financial 103

investment in different markets around the world, previous studies report remarkable difference of transaction costs between different countries/regions, even within the sample of developed economies. We do not employ different relative transaction costs for different markets in terms of asset classes and countries/regions for two main reasons: 1. Segmenting markets in developed/less developed by applying criteria such as the volume of transactions is not feasible, mainly due to the assumption of considering world in four parts, as explained in the previous chapters of our study. Even if we could manage to estimate and rank trading volumes for each market considered in this research, applying 50 bps cost to those markets with lowest volume of transaction, and 25 bps cost to those markets with the highest volume of transactions would be an arbitrary approach. Further investigation of fractional transaction costs for each market considered in this research would have been, to the extent of the difference in legislative/regulative framework and traditional issues of financial exchanges in different regions of the world, to our view, the volume work of a second dissertation. 2. The growing popularity of passive portfolio management through indexing and Exchange Traded Funds (ETFs) around the world makes it easier for a potential investor to presumably invest in different regions of the world by buying/selling ETFs issued in his/her own country or developed financial markets such as US exchanges. Market indices such as Brinson Partner Global Security Market Index (GSMI), which considers performance of US and NonUS stock/bond indices, Barclays Capital international bond indices, MSCI World equity indices, are possibilities for investors to commit their funds internationally through buying ETFs that mimic the above mentioned indices, and facing similar transaction costs like other securities in well-functioning capital markets like US. However, we believe that future research may be devoted to the application of flexible transaction costs in an international optimization context, or global minimum variance investing specifically. This may test the robustness of our findings.

7.5. The Contribution of Forex Volatility to Portfolio Risk In this research we did not quantitatively considered the contribution of foreign exchange volatility to portfolio risk. Oldier and Solnik (1993) find that currency risk contributes to the increase of volatility of approximately 20%-25% of a foreign equity investor and 50%-100% of a foreign bond investor. In the world stock portfolio, the authors find that currency risk contributes to approximately 10%-15% of the volatility. However, they state that part of currency risk is diversified away due to international investment in different currencies. Dumas and Solnik (1995) show that foreign exchange risk premium is a significant component of international portfolios’ rates of return. Solnik (1998) finds that, even for long term investment horizons, the difference between hedged and unhedged foreign portfolio for US dollar investor can be as much as 1.7%. The author suggests a joint optimization approach for market and currency selection. De Santis and Gerard (1998) support the result of Solnik (1998), and suggests an international asset pricing model that includes both market and foreign

104

exchange risk. Wide et al. (2013) find that currency exposure is one of the risk factors that mostly influence total portfolio risk. We do not consider, or quantitatively measure, the impact of currency risk on total portfolio absolute/relative risk due to the fact that world portfolios, optimized in our research, have exposure to many currencies. As such, in line with Oldier and Solnik, and Solnik (1998), we believe that part of the currency risk is diversified away due to diversification among different currencies and long term horizon assumed for the investor holding the proposed strategies in this research. The inclusion of USD/EUR or EUR/GBP exchanged rates, as an additional global investment opportunities in the optimization exercise may produce some highlight, however, regarding the contribution of currency risk in the total portfolio risk. This is a question, which, according to us, is worth to be investigated in future research. Moreover, discovery and application of optimal hedges ratios for major currencies through shorting currency futures/forewords when implementing global asset minimum variance investing, maybe worth the time to be spent for future research.

105

Chapter 8: Concluding Remarks This dissertation studies the performance of global minimum variance portfolios and global minimum tracking error portfolios optimized from global assets, global stocks and global bond returns input list. In performing this study, the world is divided in four regions: US, Europe, Asia – Pacific and Developing Countries. International indices monthly and weekly returns of stocks, corporate/government bonds, spread strategies, oil and gold serve as optimization input. The covariance matrix of returns is estimated according to the Markowitz (1952) sample covariance matrix approach. GMVPs are derived using the unrestricted and upper/lower bound restrictions in the portfolio weights of the sample covariance matrix methodology highly credited by Jagannathan and Ma (2003). Performance significant tests for Sharpe, M2, CEQ and excess return vis-à-vis the benchmarks, based on the factor model proposed by Sharpe (1962), are run according to the methodology proposed by Kossowski et al (2007) and Behr et al (2008). The methodology of Chan et al. (1999) is followed for constructing the minimum tracking error volatility portfolios, with and without weight restrictions considered by Jagannathan and Ma (2003). Unconstrained global minimum variance portfolios provide the greatest out of sample risk reduction, independently from markets or revision frequency, if monthly returns are used as optimization input. Sharpe, M2 and alpha of global asset Monthly GMVPs rebalanced every twelve months show outperformance no matter restrictions or assumed transaction costs towards value and equally weighted benchmarks. Significant outperformance towards benchmarks, as measured by Sharpe Ratio and M2, results only in the unconstrained and short-sale constrained global asset case. Global assets, moreover, deliver e statistically significant alpha versus value and equally weighted benchmark, independently from restrictions or transaction costs, at 5% level of significance. Less frequent rebalancing for monthly GMVPs of global assets improve only when upper/lower bound restriction is imposed. Performance improvement, as compared with both benchmarks, result only when upper/lower bound restrictions are imposed to global bond and stock portfolio. Whether all assets are considered as optimization input or not, highest restrictions improve the out of sample excess returns of GMVPs vis-à-vis value and equally weighted benchmarks. No portfolio, however, outperforms benchmarks based on CEQ performance metric. Less frequent rebalancing improves all performance metrics of constrained Monthly GMVPs of global stocks. Revision frequency seem not to influence, however, the statistical significant outperformance of minimum variance portfolios toward benchmarks. The use of weekly returns result in upper/lower bound constrained GMVPs with higher risk than return reduction, as compared with monthly GMVPs, if rebalanced yearly. However, the use of higher return frequency does not improve GMVP performance, through all performance metrics considered in this dissertation. Weekly data generally worsens the performance of GMVPs that consider specific markets as portfolio input towards benchmark. If highest weight restrictions are imposed, global equity minimum variance portfolios improve performance, although not necessarily significantly, in terms of Sharpe Ratio and risk adjusted excess returns vis-à-vis benchmarks, independently from the transaction costs, return and rebalancing frequency imposed. Three years rebalancing frequency for weekly GMVPs improves significant performance towards benchmarks for global bond portfolios. 106

According to this study, weight constraints have no effect in tracking error volatility minimization, independently from the transaction costs, rebalancing frequency or data frequency applied. The use of weekly return reduces the relative risk towards the benchmark, as measured by the tracking error volatility of the returns of minimum TE portfolio in excess of the benchmark portfolio, according to the results of this study. To the best of our knowledge, this is the first research work that considers: 1. The use of global asset classes in assessing minimum variance investment strategy performance. 2. The effect of weight constraints in the performance of GMVPs of global assets. 3. The effect of transaction cost and rebalancing frequency in the performance of GMVPs of global assets. 4. The performance outcome from using weekly and monthly returns as optimization input for GMVPs of global assets. 5. The analyses of differential performance outcome for specific markets, such as global equity and global bond minimum variance portfolios, from applying alternative weight restrictions, transaction costs, rebalancing frequency, return frequency. 6. The effect of weight constraint, data and revision frequency, and transaction costs in tracking error volatility minimization of a global asset portfolio. One of the extensions of this dissertation research may be the use of more asset classes. Also, the use of shrinkage estimators for GMVPs and daily/weekly/monthly return data may induce the possibility of a more explicit analyses referring the main source of performance enhancement for minimum variance strategies. The application of differential transaction costs in different markets considered as optimization input, would provide more robust results. Moreover, the quantitative assessment of FOREX contribution to portfolio total volatility and, eventually, performance, may prove to be an interesting extension of this research. We believe that future research regarding minimum variance investment strategy or global asset allocation may be inspired by the approach and findings of this dissertation.

107

Annex 1 – JARQUE-BERA TEST RESULTS FOR MONTHLY AND WEEKLY GLOBAL MINIMUM VARIANCE PORTFOLIOS AND BENCHMARKS

Table A.1: Jarque-Bera Test for Monthly GMVPs and Benchmarks. This table reports the Jarque Bera test score for monthly global minimum variance portfolios and Benchmarks. Each portfolio is named (ex. _1YGAMCMVP2) according to: 1) revision frequency (ex. _1Y- one year), 2) asset classes involved (ex. GA - global assets), 3) weight restrictions imposed (ex. MC - upper/lower bound constraints), 4) the portfolio strategy (ex. MVP - minimum variance portfolio, B1- Value weighted benchmark), and the transaction costs applied (ex. 2 - 25 basis points transaction costs). The probabilities highlighted in the table reppresent the probability that the null hypothesis of normal distribution of the portfolio returns is true. If probability is lower than 5%, the null hypothesis may be rejected. Portfolios Jarque-Bera Test Score Probability Portfolios Jarque-Bera Test Score Probability Portfolios Jarque-Bera Test Score Probability Portfolios Jarque-Bera Test Score Probability Portfolios Jarque-Bera Test Score Probability Portfolios Jarque-Bera Test Score Probability

_1YGACM VP1

_1YGACM VP2

_1YGAMC MVP1

_1YGAMC MVP2

_1YGAUM VP1

_1YGAUM VP2

_1YGBCM VP1

0.563

0.834

0.350

0.268

7.521

2.413

1.936

75.46% _1YGBCM VP2

65.90% _1YGBMC MVP1

83.94% _1YGBMC MVP2

87.45% _1YGBUM VP1

2.33% _1YGBUM VP2

29.92% _1YGSCM VP1

37.99% _1YGSCM VP2

2.898

3.516

3.561

202.009

58.105

5.156

5.086

23.48% _1YGSMC MVP1

17.24% _1YGSMC MVP2

16.86% _1YGSUM VP1

0.00% _1YGSUM VP2

0.00% _3YGACM VP1

7.59% _3YGACM VP2

7.86% _3YGAMC MVP1

10.519

10.484

24.297

24.466

1.916

2.218

0.290

0.52% _3YGAMC MVP2

0.53% _3YGAUM VP1

0.00% _3YGAUM VP2

0.00% _3YGBCM VP1

38.36% _3YGBCM VP2

32.99% _3YGBMC MVP1

86.50% _3YGBMC MVP2

0.195

17.568

5.834

2.544

2.959

3.741

3.752

90.73% _3YGBUM VP1

0.02% _3YGBUM VP2

5.41% _3YGSCM VP1

28.02% _3YGSCM VP2

22.78% _3YGSMC MVP1

15.40% _3YGSMC MVP2

15.32% _3YGSUM VP1

211.703

65.486

61.256

61.744

58.070

58.009

82.640

0.00% _3YGSUM VP2

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

GAB1

GAB2

GBB1

GBB2

GSB1

GSB2

82.800

9.234

16.033

36.307

1.505

13.303

13.748

0.00%

0.99%

0.03%

0.00%

47.11%

0.13%

0.10%

108

Table A.2: Jarque-Bera Test for Weekly GMVPs and Benchmarks. This table reports the Jarque Bera test score for weekly global minimum variance portfolios and Benchmarks. Each portfolio is named (ex. _1YGAMCMVP2) according to: 1) revision frequency (ex. _1Y- one year), 2) asset classes involved (ex. GA - global assets), 3) weight restrictions imposed (ex. MC - upper/lower bound constraints), 4) the portfolio strategy (ex. MVP - minimum variance portfolio, B1- Value weighted benchmark), and the transaction costs applied (ex. 2 - 25 basis points transaction costs). The probabilities highlighted in the table reppresent the probability that the null hypothesis of normal distribution of the portfolio returns is true. If probability is lower than 5%, the null hypothesis may be rejected. Portfolios Jarque-Bera Test Score Probability Portfolios Jarque-Bera Test Score Probability Portfolios Jarque-Bera Test Score Probability Portfolios Jarque-Bera Test Score Probability Portfolios Jarque-Bera Test Score Probability Portfolios Jarque-Bera Test Score Probability

_1YGACM VP1

_1YGACM VP2

_1YGAMC MVP1

_1YGAMC MVP2

_1YGAUM VP1

_1YGAUM VP2

_1YGBCM VP1

52.262

54.828

7.245

6.371

876.467

55.170

936.119

0.00% _1YGBCM VP2

0.00% _1YGBMC MVP1

2.67% _1YGBMC MVP2

4.14% _1YGBUM VP1

0.00% _1YGBUM VP2

0.00% _1YGSCM VP1

0.00% _1YGSCM VP2

1047.672

158.432

161.419

4408.936

7294.545

135.731

140.425

0.00% _1YGSMC MVP1

0.00% _1YGSMC MVP2

0.00% _1YGSUM VP1

0.00% _1YGSUM VP2

0.00% _3YGACM VP1

0.00% _3YGACM VP2

0.00% _3YGAMC MVP1

135.579

135.231

60.512

65.719

211.238

211.138

24.029

0.00% _3YGAMC MVP2

0.00% _3YGAUM VP1

0.00% _3YGAUM VP2

0.00% _3YGBCM VP1

0.00% _3YGBCM VP2

0.00% _3YGBMC MVP1

0.00% _3YGBMC MVP2

23.010

6441.548

7640.028

1460.226

1496.164

185.459

185.495

0.00% _3YGBUM VP1

0.00% _3YGBUM VP2

0.00% _3YGSCM VP1

0.00% _3YGSCM VP2

0.00% _3YGSMC MVP1

0.00% _3YGSMC MVP2

0.00% _3YGSUM VP1

8665.162

10568.960

122.149

125.683

163.305

162.331

47.241

0.00% _3YGSUM VP2

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

GAB1

GAB2

GBB1

GBB2

GSB2

GSB1

50.049

95.821

197.901

524.127

37.053

298.089

392.179

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

0.00%

109

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