A deep dive into the Mean/Variance Efficiency of the ......A deep dive into the Mean/Variance Efficiency of the Market Portfolio Master Thesis Supervised by Prof. Dr. Karl Schmedders

A deep dive into the Mean/Variance Efficiency of the

Market Portfolio

Master Thesis

Supervised by Prof. Dr. Karl Schmedders

Chair for Quantitative Business Administration at UZH Zurich

Co-supervised by Prof. Dr. Didier Sornette

Chair of Entrepreneurial Risks at ETH Zurich

Theodoros Giannakopoulos

Zurich

2013

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Abstract

Ever since the appearance of the Capital Asset Pricing Model (CAPM), the mean/variance

efficiency of various market proxies has been theoretically and empirically tested, by a

significant number of previous studies. The findings of almost all of these studies have

shown that the CAPM cannot work in practice; a motion generally accepted amongst the

economists worldwide. The mean/variance efficiency of the Market Portfolio was first

suggested and supported quantitatively by Levy and Roll (2010),1 and afterwards by Ni,

Malevergne, Sornette and Woehrmann, (2011)2. The above mentioned authors attempted

to calculate the return parameters and standard deviations of a sample portfolio by using an

approach different (rather reverse) from the previous researchers, so that they would satisfy

the mean/variance efficiency constraints; by comparing (statistically) their values with the

actual ones for each specific portfolio, they concluded that the difference is within the

statistical error margins, and therefore the CAPM model could hold (at least it cannot be

rejected). In this Thesis, we attempt to stress the robustness of the results supporting the

CAPM, by examining whether the calculated parameters are indeed so close to the actual

ones in a variety of portfolios, in different model implementations and market conditions.

After detailed presentation of the results from analyzing this method, we reached the

conclusion that it has limited power, and by no means can be universally used across

different markets and/or portfolios. Subsequently, an alternative theory was investigated,

which was first presented by Malevergne, Santa-Clara and Sornette (2009)3. In this paper,

the authors employ the idea discovered by Professor George Kingsley Zipf in 19494, in order

to manifest that the heavy-tailed distribution of firm sizes could be used in the form of an

additional risk factor in a time series regression model. The new factor is named the Zipf

factor, and together with the market factor they form a two-factor model that performs

equally well, and sometimes even better, than the three-factor Fama French model5.

1 Levy, M. and Roll, R. (2010). The Market Portfolio May Be Mean/Variance Efficient After All. Oxford

Journals, Review of Financial Studies, 23(6), 2464-2491. 2 Ni, X., Malevergne, Y., Sornette, D. and Woehrmann, P. (2011). Robust Reverse Engineering of Cross

Sectional Returns and Improved Portfolio Allocation Performance using the CAPM. Journal of Portfolio Management, 37 (4), 76-85. 3 Malevergne, Y., Santa-Clara, P. and Sornette, D. (2009). Professor Zipf goes to Wall Street. NBER

Working Paper No. 15295. 4 Zipf, G. K. (1949). Human behavior and the principle of least effort. Oxford, England: Addison-Wesley

Press. xi 573 pp. 5 Fama, E. F. and French, K. R. (1993). Common risk factors in the returns on stocks and bonds.

Journal of Financial Economics. 33 (1), 3–56.

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Statement regarding plagiarism when submitting written work at the University of Zurich By signing this statement, I affirm that I have read the information notice on plagiarism, independently produced this Thesis, and adhered to the general practice of source citation in this subject-area. Der Verfasser erklärt an Eides statt, dass er die vorliegende Arbeit selbständig, ohne fremde Hilfe und ohne Benutzung anderer als die angegebenen Hilfsmiffel angefertigt hat. Die aus fremden Quellen (einschliesslich elektronischer Quellen) direkt oder indirekt übernommenen Gedanken sind ausnahmslos als solche kenntlich gemacht. Die Arbeit ist in gleicher oder ähnlicher Form oder auszugsweise im Rahmen einer anderen Prüfung noch nicht vorgelegt worden. Matriculation number: 11-746-823 _______________________ ___________________________________ place and date signature

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Acknowledgements

Every project is a learning voyage, and this Thesis proved to be the most fascinating research

endeavor I have undergone in my entire academic career. During the time required, working

on this Thesis, a significant number of people contributed to the final success of my work.

Regardless of whether that was directly or indirectly, the final result would have not been

the same without the valuable input from my Professors, friends and colleagues.

From the ETH Zurich and specifically the Chair of Entrepreneurial Risks, I need to primarily

thank Professor Dr. Didier Sornette and Dr. Peter Cauwels for their pioneering ideas that

initially brought this project into life, and subsequently guided me through the research

process with their key remarks. I also need to thank them for inspiring me initially to engage

this project, even though I could foresee that working with the financial markets in such

depth, would pose a serious challenge to my engineering background. I also feel compelled

to thank Dr. Qunzhi Zhang from the same Chair, with whose valuable suggestions I managed

to optimize significantly my Matlab code, a fact that allowed me to perform all the

optimization tests required for my analysis within the deadlines.

In the University of Zurich, I am grateful to Professor Dr. Karl Schmedders from the Chair of

Quantitative Business Administration, for making the cooperation with ETH Zurich feasible

and for his assistance, whenever that was required. I would also like to thank Professor Dr.

Uschi Backes-Gellner and Simone Balestra, as the Management and Economics Program

Director and Coordinator, respectively, for all their valuable efforts that resulted in my

successful graduation. Furthermore, the guidance provided by Dr. Ioannis Akkizidis proved

to be valuable in my early steps in Financial Risk and Portfolio Management.

In personal level, I am deeply grateful to my family for supporting me for all this time.

Finally, I want to thank all my friends and colleagues in Portfolio Investments (Dow Chemical

Europe), for the insightful discussions and comments that were exchanged.

Thank you all.

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Table of contents

Abstract ..................................................................................................................................... 3

Acknowledgements ................................................................................................................... 7

Table of contents ....................................................................................................................... 9

Table of graphs and figures ..................................................................................................... 13

Introduction ............................................................................................................................. 15

1 Financial Markets ................................................................................................................. 17

2 Capital Asset Pricing Model (CAPM) Theory ........................................................................ 22

3 The Levy/ Roll approach ....................................................................................................... 27

3.1 Implementation in Matlab ............................................................................................ 30

3.2 The “fmincon” Function ................................................................................................ 31

3.3 Nonlinear Optimization ................................................................................................. 32

4 Optimization Problem 1 ....................................................................................................... 40

4.1 Methodology ................................................................................................................. 40

4.2 Covariance Matrix Shrinkage Methods ......................................................................... 43

4.3 Definition of the Dependent Variable in all Matlab Functions...................................... 45

4.4 Flowchart of the Matlab code ....................................................................................... 46

4.5 Construction of the 25 Fama French Portfolios ............................................................ 46

4.6 Statistical Analysis of the test results ............................................................................ 53

4.7 Investigation of correlations and other varying attributes ........................................... 56

4.8 Definition of Market Proxy – Indices ............................................................................. 59

4.9 Optimization Problem 1 Robustness Evaluation ........................................................... 62

4.9.1 Portfolio: first 100 Stocks from S&P 1200 Global Index ......................................... 64

4.9.2 Portfolio: first 100 stocks from the S&P 500 Index ................................................ 77

4.9.3 The 25 Fama French NYSE Portfolios...................................................................... 84

4.9.4 The 25 Fama French NASDAQ/NYSE/AMEX Portfolios........................................... 86

4.9.5 Portfolios of variable size from the top Market Capitalization Firms of S&P 500

Index ................................................................................................................................ 88

4.10 Conclusions for the Optimization Problem 1 .............................................................. 88

5 Optimization Problem 2 ....................................................................................................... 90

5.1 Mathematical Problem Description .............................................................................. 90

5.2 Methodology ................................................................................................................. 92

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5.2.1 Constraints Function funCeq2 ................................................................................ 94

5.2.2 Constraints Function funCeq2b .............................................................................. 95

5.2.3 Constraints Function funCeq (+linear equality)...................................................... 95

5.2.4 Constraints Function funCeq2 (+linear equality).................................................... 96

5.2.5 Constraints Function funCeq3 ................................................................................ 97

5.2.6 Constraints Function funCeq2 (+linear INequality) ................................................ 97

5.3 Test results: Portfolio of the 100 first stocks from S&P 500 Index ............................... 98

5.4 Conclusions for the Optimization Problem 2 .............................................................. 109

6 Zipf Approach ..................................................................................................................... 110

6.1 Introduction ................................................................................................................. 110

6.2 Matlab Implementation .............................................................................................. 111

6.3 Data ............................................................................................................................. 112

6.4 Time Series Regression Models ................................................................................... 112

6.4.1 Market Model ....................................................................................................... 112

6.4.2 Zipf Model ............................................................................................................ 114

6.4.3 Three-Factor Fama French Model ........................................................................ 115

6.5 Test results .................................................................................................................. 116

6.5.1 Results for the 25 Fama French portfolios (January 1926 to November 2013) ... 116

6.5.2 Results for the 25 Fama French portfolios (January 2003 to November 2012) ... 118

6.5.3 Results for the 25 Fama French Portfolios (January 1991 to November 2008) ... 120

6.5.4 Results for the extra Portfolios (January 1991 to December 2012) ..................... 122

6.5.5 Results for the 10 Deciles Portfolios (January 1991 to December 2012) ............. 124

7 Conclusion .......................................................................................................................... 126

8 Suggestions for continuing this work ................................................................................. 128

9 References .......................................................................................................................... 130

9.1 Books ........................................................................................................................... 130

9.2 Papers .......................................................................................................................... 130

9.3 Websites ...................................................................................................................... 131

Appendix ................................................................................................................................ 145

1 Acquiring Bloomberg Data ................................................................................................. 145

1.1 Introduction ................................................................................................................. 145

1.2 Bloomberg Formulas ................................................................................................... 145

1.3 Bloomberg Fields ......................................................................................................... 147

2 Matlab Code ....................................................................................................................... 151

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2.1 Matlab script v2.4.m .................................................................................................... 151

2.2 Optimization Problem 1 .............................................................................................. 155

2.2.1 Command line script ................................................................................................. 155

2.2.2 Function funInitial .................................................................................................... 156

2.2.3 Function funD ........................................................................................................... 157

2.2.4 Function funCeq ....................................................................................................... 159

2.2.5 Function runfmincon ................................................................................................ 163

2.3 Optimization Problem 2 .............................................................................................. 171

2.3.1 Command line script ................................................................................................. 171

2.3.2 Function funInitial .................................................................................................... 172

2.3.3 Function funD ........................................................................................................... 172

2.3.4 Function funCeq2 ..................................................................................................... 172

3 Portfolios Stock Constituents ............................................................................................. 177

3.1 The 100 first stocks from S&P 1200 Global Index (1047 stocks with full data during the

period 2003-2013) ............................................................................................................ 177

3.2 The 100 first stocks from S&P 500 (423 stocks with full data during the period 2003-

2013) .................................................................................................................................. 179

3.2 Company lists of the 25 Fama French portfolios ......................................................... 181

3.2.1 Portfolio 1 ............................................................................................................. 182

3.2.2 Portfolio 2 ............................................................................................................. 183

3.2.3 Portfolio 3 ............................................................................................................. 183

3.2.4 Portfolio 4 ............................................................................................................. 184

3.2.5 Portfolio 5 ............................................................................................................. 185

3.2.6 Portfolio 6 ............................................................................................................. 187

3.2.7 Portfolio 7 ............................................................................................................. 188

3.2.8 Portfolio 8 ............................................................................................................. 189

3.2.9 Portfolio 9 ............................................................................................................. 190

3.2.10 Portfolio 10 ......................................................................................................... 192

3.2.11 Portfolio 11 ......................................................................................................... 193

3.2.12 Portfolio 12 ......................................................................................................... 195

3.2.13 Portfolio 13 ......................................................................................................... 196

3.2.14 Portfolio 14 ......................................................................................................... 197

3.2.15 Portfolio 15 ......................................................................................................... 198

3.2.16 Portfolio 16 ......................................................................................................... 199

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3.2.17 Portfolio 17 ......................................................................................................... 201

3.2.18 Portfolio 18 ......................................................................................................... 203

3.2.19 Portfolio 19 ......................................................................................................... 203

3.2.20 Portfolio 20 ......................................................................................................... 204

3.2.21 Portfolio 21 ......................................................................................................... 205

3.2.22 Portfolio 22 ......................................................................................................... 208

3.2.23 Portfolio 23 ......................................................................................................... 209

3.2.24 Portfolio 24 ......................................................................................................... 209

3.2.25 Portfolio 25 ......................................................................................................... 210

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Table of graphs and figures

Picture 1: S&P 500 Index crashes, 1928 to 2008. .................................................................... 18

Picture 2: The rally of S&P 500 Index in 2013. ........................................................................ 19

Picture 3: The effect of the Quantitative Easing (QE) program on the S&P 500 Index. .......... 19

Picture 4: Increase of the U.S. 30 Year Mortgage Rate, as an effect of the FED Chairman’s

statement. ............................................................................................................................... 21

Picture 5: The Efficient Frontier. ............................................................................................. 23

Picture 6: Risk-Return graphs for varying correlation between two assets. ........................... 24

Picture 7: The value of the Objective Function being minimized, during the optimization

process..................................................................................................................................... 36

Picture 8: The Maximum Constraint Violation being minimized, during the optimization

process..................................................................................................................................... 37

Picture 9: The plot of the Matlab function Peaks. ................................................................... 38

Picture 10: Different starting points resulting in different end points. .................................. 39

Picture 11: Tolfun, Tolx. ........................................................................................................... 41

Picture 12: Matlab code flowchart. ......................................................................................... 46

Picture 13: Data acquisition and analysis flowchart. .............................................................. 48

Picture 14: Correlations are significantly higher when in crisis. ............................................. 56

Picture 15: S&P Global 1200 Index first 100 stocks all tests. .................................................. 68

Picture 16: Testing the performance of the model for weights based on the market

capitalization of the firms in different points in time. ............................................................ 69

Picture 17: Testing a number of parameter combinations: none of them works, including the

ones with correct market values. ............................................................................................ 70

Picture 18: Testing the Rf: the model should not work for 0.12%, but for 1.5%, which is the

historical average. The results indicate abnormal behavior. .................................................. 71

Picture 19: Alpha’s effect on the results: insignificant. ........................................................... 72

Picture 20: The model was also tested with the parameters (return and standard deviation)

of the Wilshire 5000, as market parameters. As we can see, it failed the test....................... 73

Picture 21: The model was unable to produce Levy/Roll complied results, with this market

proxy. ....................................................................................................................................... 74

Picture 22: Standard deviations SP1200GL MKT VL. ............................................................... 76

Picture 23: S&P 500 Index first 100 stocks all tests. ................................................................ 79

Picture 24: Optimization Problem 1 tested for the set of stocks with wrong market values

(overall S&P 500 Index average for 90 years). ........................................................................ 80

Picture 25: The model is fully in line with the Levy/ Roll theory, for values close to the correct

market ones. ............................................................................................................................ 81

Picture 26: Model tested for deviations of market μ. ............................................................. 82

Picture 27: Testing variations in the risk free ratio, as well as to the market standard

deviation (by keeping all others constant). The model works only for the correct Rf ............ 83

Picture 28: Optimization Problem 2, all subproblems. ......................................................... 100

Picture 29: Optimization Problem 1. As we can see, the results demonstrate almost perfect

compliance with the Levy/Roll model. .................................................................................. 101

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Picture 30: From the plot of tests for Ceq2, we can see large rejection rates –especially in

standard deviations. .............................................................................................................. 102

Picture 31: In this test, we can see the massive effect of the choice of the extra constraint

(μ0) on the estimated standard deviations............................................................................ 103

Picture 32: In this plot we can see perhaps the most important test: Ceq2with linear

inequality. The estimated parameters are statistically close to the sample ones. ............... 104

Picture 33: Objective Function in the case of a model break-down. .................................... 105

Picture 34: Maximum Constraint Violation in the case of a model break-down. ................. 106

Picture 35: The constraint violations resulting by feeding the estimated (from the

Optimization problem ceq2 and inequality) vector xIN into the subproblems constraints

functions. ............................................................................................................................... 108

Picture 36: The official Wilshire calculator. ........................................................................... 113

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Introduction

Levy and Roll, in their 2010 paper6 rejuvenated in full power one of the most debated issues

in modern finance; their results imply that there is still a possibility that the original Capital

Asset Pricing Model (CAPM) presented by Sharpe (1964)7, could describe real market

conditions, a notion which is widely believed to be incorrect, amongst economists.

The motivation of this Thesis is divided in two main parts; first, to fully challenge the Levy

and Roll approach of reverse engineering, by reconstructing the entire code in Matlab and

applying it in a variety of different portfolios, under different market conditions. The goal of

this process is to acquire a deep understanding of the behavior of this model, used by Levy

and Roll, and ultimately to conclude whether it indeed stands true or not. Second, a

different model is tested; the one proposed by the authors of the paper “Professor Zipf goes

to Wall Street”8. Based on the original idea discovered by Zipf (1949)9, we evaluate and test

further the financial use that was first proposed in this paper. As we will see in Chapter 6,

the results are remarkable.

In Chapter 1, an introduction to the financial markets as of the second half of 2013 is given.

Right after, in Chapter 2, a second introductory chapter follows with an insight to notions

such as the CAPM, the efficient frontier and the work of Levy and Roll. Hopefully, these two

chapters would provide enough justification in regard to whether portfolio management is

of such importance, given the special nature of the financial markets.

In Chapter 3, a short introduction to the Levy and Roll model is provided, along with the

fmincon function used (in Matlab) and the general concept of nonlinear optimization. The

purpose of this chapter is to prepare the ground for a more detailed analysis that will follow

in Chapter 4. In this chapter, the entire analysis of the Optimization Problem 1 is analyzed,

starting from its definition, continuing with its implementation and the data preparation,

and completing with a presentation of the results and conclusions. Similarly, Chapter 5 is

devoted to the Optimization Problem 2, by following the same structure as in Chapter 4.

In Chapter 6, the two-factor Zipf model is analyzed; again, from the perspective of the

Matlab process, as well as the data preparation. Finally, in Chapter 7 we draw the final

conclusions of the entire Thesis for both methods (Levy/Roll and Zipf). Chapter 9 includes

some suggestions of the author for further research on some issues that were not

exhausted, scientifically, in this Thesis. That could be either because they deviated

significantly from the main goals of this Thesis, or because there was simply not enough time

to explore all different ideas for testing the different models.

6 Levy, M. and Roll, R. (2010). The Market Portfolio May Be Mean/Variance Efficient After All. Oxford

Journals, Review of Financial Studies, 23(6), 2464-2491. 7 Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk.

The Journal of Finance, 19 (3), 425-442. 8 Malevergne, Y., Santa-Clara, P. and Sornette, D. (2009). Professor Zipf goes to Wall Street. NBER

Working Paper No. 15295. 9 Zipf, G. K. (1949). Human behavior and the principle of least effort. Oxford, England: Addison-Wesley

Press. xi 573 pp.

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In Chapter 9, the references on which this work is based on are provided, in order to

facilitate future research. Due to the nature of this project, the largest number of references

provided refers to website pages, especially from Mathworks.

Finally, a complete Appendix is included, divided in three sectors: first, the full description of

the data acquisition procedure from Bloomberg is provided. Second (sector B), the main

modules of the Matlab code are documented, along with the necessary comments for the

understanding of their basic functionality. In the third part of the Appendix, the lists with the

firm names are provided, which are the constituents of the portfolios used throughout the

tests of the optimization problems.

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Chapter 1

1 Financial Markets

It has been well more than a century, since the first time that an economist tried to explain

the market moves and predict the next one, in order to consequently draft a –hopefully-

lucrative strategy. Such analysis was initially carried out by simply studying the

fundamentals, as well as basic economic notions such as supply and demand, in almost

isolated (from one another) markets. In today’s globalized financial system, the situation is

completely different. Enormous volumes of data are exchanged and analyzed every second

of each day by using the most sophisticated computer systems, in an attempt to evaluate

the current market position, and as such forecast the next move of the highly intercorrelated

markets; a zero-sum game, in theory, which sometimes creates a tremendous amount of

social turbulence.

In a system extremely volatile, which is often characterized as a “random walk” due to its

unpredictable nature, the procedure is usually the same: a new model is tested by using

(existing) historical data. If the model’s explanatory power is consistently high enough, in a

series of different time periods under different macroeconomic conditions, then it is

(considered to be) a good candidate as a predictor of future market moves. For that

purpose, the model is usually fed with a large number of data from one of the known data

providers (Bloomberg, Thomson Reuters, Wharton), analyzes them, and subsequently

produces some quant signals that the analysts of portfolio managers need to evaluate.

But the market “state” belongs usually in one of two different modes: the distressed or the

normal one. While there is no official definition for either state, it is broadly accepted that a

market is in distress (crisis) when we have “a situation in which the value of financial

institutions or assets drops rapidly“10. Usually, the subsequent financial event that takes

place, often called as an “extreme event”, has (or used to have) a possibility to happen so

small, that it was considered almost impossible. Regarding the cause of such a market

downturn, one could argue by studying historical data that sometimes a market crash is

based on an actual event such as a war, while other times it can be based on mere financial

conditions and practices, such as the subprime crisis of 2008.

In the graph below, we can see the most market crashes -by using as market proxy the S&P

500 Index- for the last approximately 100 years, where they are explained as the reaction of

the market to extreme macroeconomic conditions.

10

Investopedia: Financial Crisis. http://www.investopedia.com/terms/f/financial-crisis.asp, 04.08.2013.

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Picture 1: S&P 500 Index crashes, 1928 to 2008.11

As we can imagine, in general it is very difficult for any quantitative model, to be able to

predict accurately the outcome of the markets or a specific market move in the occurrence

of such extreme events. In addition, it is not always desirable for a model to include modules

able to measure the effect of a crisis on the assets, since that would make the model

severely biased and probably not performing well in “normal” market conditions. On top of

that, there are many factors affecting the performance of an asset or portfolio that cannot

be modeled in a quantitative way.

For all the above reasons, in most cases portfolio managers prefer to have their models

reflecting pure fundamental qualities of the firms they analyze, so that they will be able to

add an overlay analysis regarding for example possible political risks or sustainability - social

responsibility targets that need to be taken into account. By using both approaches

together, one is able to have the overall picture, which is the valuation of the assets based

on their own value, but co-calculating the macroeconomic conditions as well.

Last summer, the S&P 500 Index managed to surpass the peak levels of 2007 for the first

time since the occurrence of the subprime crisis. It took several years for the system to

recover from a crash, that for the first time it was widely accepted as a purely systemic one,

based only on incorrect valuations of assets and as such, without any reflection to shift of

real value.

11

185.Bond vigilantes. Index GPL. http://www.bondvigilantes.com/blog_files/UserFiles/Image/stock_market_crashes2.jpg, 19.04.2013.

19

Picture 2: The rally of S&P 500 Index in 2013.12

In the same spirit, the rally of most equity indices in year-to-date 2013, which broke the

upper trend line several times (even with the correction in May 2013), is considered to be

the result of the third Quantitative Easing (QE3) program of the United States Federal

Reserve Bank (FED) and does not necessarily proves that the macroeconomic conditions are

considerably more favorable than when the subprime crisis emerged.

Picture 3: The effect of the Quantitative Easing (QE) program on the S&P 500 Index.13

12

Source: Russell Investment Group, Standard & Poor’s, FactSet, J.P. Morgan Asset Management.

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At the same time the economic conditions for most of Europe remain depressed, with the

Eurozone’s first-quarter GDP retreating for the sixth consecutive quarter. In Japan, another

heavy-weighted pole of the global financial system, the effects of a first time in history (with

regards to its size) Quantitative Easing program has been launched (Bank of Japan, BoJ) on

the country’s monetary policy are still not clear. In the meanwhile, “Abenomics” (the

measures introduced by the Japanese Prime Minister Shinzo Abe) have already created a

tsunami of consequences to all the peripheral economies, varying from drops in FX rates and

shifts in the imports/exports trade balance, to the capacity of core industries and actual

people’s jobs. Finally, in China, the first quarter growth rate of 2013 did not reach the

expectations for the first time in the last 10 years, and decelerated to approximately 7.5%,

instead of the double-digit numbers that were promised to the investors and the “Western

World”. This resulted in a sizable decrease in the demand of commodities, which was –by

many economists- the main driver of the drop of the prices of many basic commodities in

stock markets all over the world from the second quarter of 2013 onwards.

These developments resulted in the end of the broadly accepted perception of the financial

world during the recent years, that emerging markets are the main contributors of growth in

the global system, bringing back the economy of the United States of America under the

spotlight. Finally, in order to demonstrate the influence of the FED’s strategy in the domestic

market, as well as the influence of the U.S. economy to the global market, we will refer to

the effect of one statement made by FED's chairman, Ben S. Bernanke (May 2013): the

chairman announced that the FED intends to reduce its asset-buying program (of

approximately $80 billion per month) by the end of the year, putting an end in abundant

liquidity and therefore extremely low interest rates. The fear created by this “tapering of

QE” statement pushed the markets into a very nervous reaction, with an example being

provided below:

13

Financial Sense. Clues to watch for the End of QE “Infinity”. http://www.financialsense.com/contributors/lance-roberts/clues-watch-end-qe-infinity, 23.08.2013.

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Picture 4: Increase of the U.S. 30 Year Mortgage Rate, as an effect of the FED Chairman’s statement.

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The yield on U.S. 10-year Treasuries increased 1% during the first 7 weeks after the

announcement by Mr. Bernanke, the sharpest increase in such a short time period for the

last 50 years.

The above stated events and their consequences in asset pricing all over the world should

make clear that active portfolio management is key to whoever wishes to achieve the

highest possible performance of their portfolio of assets. Naturally, there are a lot more

factors that affect the assets’ prices, varying from regulatory changes to behavioral finance.

The focus of this Thesis is concentrated into two competitive theories; first, to add new

evidence in one of the most fundamental issues affiliated with the Modern Portfolio

Management Theory, the Mean/Variance Efficiency of the Market Portfolio. Second, to

demonstrate the explanatory power of the two-factor Zipf Model.

14

The Motley Fool. What “Tapering” Means for Middle America. http://www.fool.com/investing/general/2013/08/17/what-tapering-means-for-middle-america.aspx, 02.09.2013.

22

Chapter 2

2 Capital Asset Pricing Model (CAPM) Theory

In 1952, Markowitz presented the mean/variance approach15. Sharpe16 (1964) and Lintner17

(1964) presented in their work what would afterwards be known as the Capital Asset Pricing

Model (CAPM).

The whole idea can be described by the following equality:

ri,t = Rf,t + βi ( rm,t – Rf,t)

Where ri,t corresponds to the portfolio i’s (expected) returns, Rf,t is the risk free ratio for

each time period (usually monthly), and rm,t is the (expected) market return for the same

period t.

In words, that equality means that the return of a certain portfolio (which could be

constituted by only one security) can be explained by the risk free return plus a risk premium

(market excess return) multiplied by a factor called beta, which captures the sensitivity of

the return of the particular portfolio towards the market returns.

Therefore, somebody could apply this formula in known past returns of a certain portfolio,

obtain its beta, and under the assumption that its sensitivity to the market returns will not

change, use the above formula to calculate the expected returns of that portfolio. The idea

of a linear relationship between risk and returns is also introduced, meaning that the riskier

an asset is considered to be, the higher its returns will be.

The value of this theory, as it is used today, lies basically at the notion that a certain portfolio

is mean/variance efficient under two interchangeable conditions:

It provides with the maximum possible returns, given a certain amount of risk taken.

It bears the minimum possible amount of risk, given a certain return.

This theory turned to be the corner stone of Modern Portfolio Theory, which brings us to the

Efficient Frontier: the set of optimal portfolios that provide with the highest expected return

given a certain level of risk, or the minimum possible risk for a given level of expected

return. The portfolios that reside below the efficient frontier are called “sub-optimal”, since

15

Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7, 77-91. 16

Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. The Journal of Finance, 19 (3), 425-442. 17

Lintner, L. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics, 47, 13-37.

23

they are characterized by lower return for any given level of risk; portfolios that reside on

the right of the efficient frontier are also called sub-optimal, since they bear more risk for

any given level of return.

The curved shape of the efficient frontier, in the space of the returns-risk graph of all

attainable portfolio combinations, pointed out the precious benefit of the diversification

that can be achieved, by choosing the right combination of portfolio constituents.

Picture 5: The Efficient Frontier.18

In order to achieve this most desirable diversification property for a portfolio, and therefore

reduce (ideally eliminate) the systematic risk19, the portfolio constituents need to be assets

with negative correlation, or at least very small (correlation). This can be observed in the

following simplified graph, where we can see that in the case of negative correlation, it is

possible to increase the portfolio (constituted by these two assets) returns while reducing its

risk (standard deviation of return) at the same time. In real market conditions it is rather

18

Mathworks. Newsletters. Developing Portfolio Optimization Models. http://www.mathworks.com/company/newsletters/articles/developing-portfolio-optimization-models.html, 23.04.2013. 19

Systematic risk or undiversifiable risk is called the amount of risk that is considered to be “systemic”, i.e. cannot be eliminated by further diversification.

24

rare to obtain two assets with such a perfectly desired correlation, but it is still possible to

minimize the systematic risk of a portfolio.

Picture 6: Risk-Return graphs for varying correlation between two assets.20

Ever since, there has been an extensive research activity testing whether these theories

(which basically constitute the mean/variance efficiency of a portfolio) actually apply in real

market data. The efficiency of the market portfolio, except from being the corner stone of

modern finance, is a key subject regarding the debate of active versus passive investing and

whether the riskiness of various assets can be represented by their “beta” (β). Especially

since the last financial crisis, where many professionals in the industry underperformed the

various benchmark indices, there has been a tremendous amount of discussion regarding

whether they “deserve” their management fees, and if they actually create some “alpha”

(α).

Most of the studies have tested various market proxies, which have turned out to be far

from the efficient frontier; in addition, it has been proven that the portfolios on the efficient

frontier usually include short positions. That verifies the opinion than the market portfolio

(in which there cannot be such thing as a “short position”) is not efficient by definition.

To be noted, in the previous studies an extensive collection of sample/modified parameters

has been employed, by using various shrinkage techniques (we will come back to this at the

implementation of the model).

All these could ultimately mean that the portfolio managers and investment consultants

could be dismissed, and the investors could simply invest in an index like the S&P 500, or

even a blend between an index and some risk free assets (like T-bills or German Bunds).

20

IndUS Business Journal. Asset Allocation and the Efficient Frontier. http://www.indusbusinessjournal.com/ME2/dirmod.asp?sid=CA41EB12BCD14CBBB683703875E204A5&nm=&type=Blog&mod=BlogTopics&mid=1AE0C859CE3C43E0B50B7615AAC20015&tier=7&id=958C2D62F6014598A76C3D623A997EE6, 22.08.2013.

25

Amongst the professionals in the industry, it has been accepted for years now that the

CAPM does not really work, therefore there is a series of alternative models, whose base is

the CAPM.

Ross’s (1976) Arbitrage Pricing Theory (APT) is one of them. In general, this model supports

the theory that “the expected return of a financial asset can be modeled as a linear

function“21 of a chosen number of factors, which can be macroeconomic ones and/or market

indices. The coefficient of each factor represents the sensitivity, or “beta” (β), of the asset’s

return to the changes of the value of this factor. After these betas have been calculated, the

model can then be used for pricing this asset.

APT is considered to be one of the basic after-CAPM models, but the most prominent one is

the three-factor Fama-French model (1993). This model expands the CAPM, by having two

additional factors, the “SMB- small (capitalization) minus big” and the “HML- high (book-to-

market) minus low”. More on this model will be provided in Chapter 6.

More recently, in 1993, another model was presented by Cahart, which introduces the

momentum effect; the inclusion of the momentum effect that gains more and more ground

against competitive models, especially in the explanation of returns of hedge funds and

funds of funds.

The first core area of interest of this Thesis lies in the examination of the explaining power of

CAPM, or better, in the determination of whether the market portfolio is mean/variance

efficient; a notion which is considered by the majority of the economists worldwide as not

true. In that case, CAPM would be just a mere pedagogical exercise for finance students,

that does not reflect any actual economic conditions.

In order to accomplish that, a rather reverse approach is being employed. This approach was

first introduced by Levy and Roll in 201022, and was further expanded by Ni, Malevergne,

Sornette and Woehrmann (2011)23.

Their approach tackles the problem from the opposite perspective; most researchers prior

to Levy and Roll implemented the “usual” technique of gathering returns from different

portfolios, market proxies, etc. and feeding them into the CAPM model, in order to

determine whether it holds true. The brilliant idea that basically constitutes the paper of

Levy and Roll is the calculation of the return parameters and standard deviations of a sample

portfolio, so that they would satisfy the mean/variance efficiency constraints.

More precisely put, that means that for a certain portfolio of equities, the authors

determined what the return and standard deviation of each individual stock should be, so

that the entire portfolio would be mean/variant efficient. In order to estimate these returns

21

Arbitrage pricing theory - Wikipedia, the free encyclopedia. http://en.wikipedia.org/wiki/Arbitrage_pricing_theory, 21.08.2012. 22

Levy, M. and Roll, R. (2010). The Market Portfolio May Be Mean/Variance Efficient After All. Oxford Journals, Review of Financial Studies, 23(6), 2464-2491. 23

Ni, X., Malevergne, Y., Sornette, D. and Woehrmann, P. (2011). Robust Reverse Engineering of Cross Sectional Returns and Improved Portfolio Allocation Performance using the CAPM. Journal of Portfolio Management, 37 (4), 76-85.

26

and standard deviations, they used a quantitative model in Matlab (more on this model

under the “implementation” Chapters 4 and 5). That way, they acquired a second set of

returns and standard deviations for each portfolio (by “first set” we mean the actual values,

or sample values of the portfolio), which they compared statistically with the first set of

values. The results were astonishing for the scientific community: the difference between

the values of each set, the sample and the calculated (mean/variance efficient) one, was

lying within the statistical error margins; that would mean that the CAPM theory in terms of

explaining the returns of the particular portfolio (and therefore, market proxy) cannot be

rejected. Details of how this second set of values is calculated, and how the statistical

comparison is being made, will be presented in the respective chapter.

To be noted, a great debate regarding the validity of any testing of the CAPM theory has as

its main subject the validity of any market proxy, from the market proxies that are available

to the researchers. That is because, as many economists support, for a definite test of the

respective theory, one should include all assets, such as human capital, real estate, etc. So

far, there has not been a satisfying approach capable of incorporating equivalent factors in a

model.

27

Chapter 3

3 The Levy/ Roll approach

As stated in the previous chapter, Levy and Roll applied a reverse-engineering model in

order to calculate the values of a pair of parameters (return and standard deviation) of each

stock for a given portfolio, and afterwards compare it with the actual (sample) values of this

stock. Two optimization problems were used by the authors for this procedure, where the

second one is significantly more constrained than the first.

In their approach, the main issue is the calculation of the parameters (returns and standard

deviations) which, given a sample portfolio, will satisfy the mean/variance efficiency

conditions. In this chapter, we will attempt to provide a short introduction of the two

Optimization Problems (Optimization Problem 1 and Optimization Problem 2), so that it will

be possible later in our analysis to elaborate further on them, and build on their

characteristics. The mathematical and programming details for these two Problems will be

analyzed in detail in the two chapters devoted to them.

Regarding Optimization Problem 1, its sole objective is to find a set of μ , σ vectors that

satisfy the nonlinear equation constraint and at the same time make sure that the objective

function f(x) (which is the function D((μ,σ),(μ,σ)sample) in our case) has a value as low as

possible.

We can see the mathematical description of the Optimization Problem 1 right below (all the

parameters are explained after the presentation of Optimization Problem 1):

Minimize (objective function):

Subject to (nonlinear mean/variance constraint):

28

As we will see in Chapter 5, the Optimization Problem 2 is a more constraint version of the

Optimization Problem 1. The objective function and the first constraint remain the same as

in Optimization Problem 1; the additional constraints correspond to the notion that there

might be a predefined (mean) return μ0 and standard deviation σ0 of the estimated

portfolio’s values, which we impose as extra constraints to the initial optimization problem.

For that purpose, the Optimization Problem 1 is equipped with two more constraints, a

linear equation for the returns, and a set of nonlinear equations for the standard deviations.

The Optimization Problem 2 can be formulated as:


Subject to:

a) Nonlinear mean/variance constraint:

b) Liner constraint for estimated portfolio returns (μ):

c) Nonlinear constraint for estimated portfolio standard deviation (σ):

29

For the rest of this Thesis, we will try to use a coherent naming of the various basic

parameters. There are constituted by two main sets, the sample parameters and the

adjusted (calculated from Matlab) ones.

For the sample parameters, we have μsample and σsample which are vectors constituted

from the return and standard deviation, respectively, of each stock of the sample

portfolio.

For the adjusted parameters, we have μ and σ which are vectors constituted from

the return and standard deviation, respectively, of each stock that have been

calculated via the Matlab procedure.

Furthermore, the variable N corresponds to the number of stocks in the portfolio being

used, rf (or Rf) the risk free ratio, q the constant of proportionality and α (alpha) a parameter

defining the relative weight given to the deviations of the mean returns (versus the weight

given to the deviations of the standard deviations).

By following the most popular approach, the risk free ratio used is the 3-month T-bill rate.

Given the fact that the Levy and Roll Models do not have a time dimension (the naïve

averages are used for the returns of each stock), the same naïve average was calculated and

used for the risk free ratio. As we will see later on, for the period 2003 to 2013 for example,

that was calculated as approximately 1.5% (precise calculations are following, right before

the presentation of the tests results).

The formula for calculating q, reads:

Where σmarket and Rmarket are the market standard deviation and (average) return,

respectively.

Regarding the selection of α (for which holds 0 ≤ α ≤ 1), it has been proven that the model

returns the most robust results when α varies between 0.5 and 0.75. Indeed, the value of α

used in this Thesis is within this range, quantified by 0.6 (its effect is examined in several

optimization tests).

In addition to the previous defined parameters, we also have the vector xmi which represents

the weight of stock i, for each i from 1 until N, of the portfolio. The weights of the stocks,

following the previous work by Levy/Roll24 (as well as almost all available relevant studies),

are calculated based on the market capitalization of each firm. There are several different

approaches as to the market capitalization of which date should be used. In the first tests

(several portfolios constituted by different number of top market capitalization S&P 500

INDEX stocks) the simple average of the market capitalization of each stock was used; later

on, in the first 100 from S&P 500 Index and the S&P 1200 Index, as well as in all the Fama

24

Levy, M. and Roll, R. (2010). The Market Portfolio May Be Mean/Variance Efficient After All. Oxford Journals, Review of Financial Studies, 23(6), 2464-2491.

30

French sets, the market capitalization of each firm as of the last month of the time period

under examination was used.

The formula that gives us the weight xi for every stock i is simply:

The intuition behind D((μ,σ),(μ,σ)sample), where “D” stands for “Distance”, is a mathematical

way to represent how close the sample values are, in relation to the ones calculated by the

Matlab procedure. As expected, the closer the estimated values are to the sample ones, the

lower is the value of D. As we know from statistics, the largest the standard deviation (of a

stock’s returns), the largest the statistical error will be from calculating the estimated

parameter for the same stock, and the larger the confidence interval as well. This is why

they are divided by the sample standard deviation, as an attempt for normalization.

The set of nonlinear equations represents the constraints necessary so that the set of stocks

will satisfy the mean/variance efficiency conditions. For this optimization problem, the

overall number of nonlinear constraint equations is equal to the number of stocks in our

stock set.

Both optimization problems and their constraints will be discussed in detail in the respective

chapters (Chapters 4 and 5 for Optimization Problem 1 and 2, respectively).

Having somewhat described the parameters used, we will see in paragraph 4.6 what

“reasonable close” parameters means, when it comes to comparing the sample and adjusted

parameters, for the stocks of each portfolio. After the calculation of the mean/variance

efficient parameters for each set of stocks, we would accept them as “reasonably close” to

the sample set’s, if at least 95% of the parameters calculated are within the 95% confidence

intervals of the sample parameters. Analytical tables with the statistical results of these

comparison tests and their thorough explanations will be provided at the respective sections

with the results of the optimization tests.

3.1 Implementation in Matlab

For the needs of this Thesis, the entire Matlab code was hardcoded from scratch25, for two

main reasons. First, we needed to make sure that previous coding attempts of the same –or

similar- problems will not influence our approach, which tries to be as accurate to the

original mathematical problem description as possible. Second, and as we will see in the

chapters to come, there are a lot of calibration issues and assumptions that need to be

applied when executing the several functions of the Matlab code, for each specific portfolio

under examination. Hence, only by reconstructing the entire model (in terms of

programming) it would be possible to gain a deep understanding of all these subtle issues

25

With the exception of the Ledoit/Wolf shrinkage technique, as we will see in the respective paragraph.

31

that affect the performance of the model, and result in a stronger conclusion regarding its

validity.

The code is viewed as a tool which would be used to calculate the estimated parameters, so

that we would test the model fit. As such, not much effort was spent in the optimization of

the code itself, except from one very important aspect: the speed of the optimization for

each portfolio, and more specifically, the execution time required for each run. Initially, as

the code was not optimized at all in terms of execution speed, even an execution over a

small portfolio required a tremendous amount of time. To put things into perspective, the

(continuous) execution time in a personal pc for each optimization test with just 10 stocks

was approximately 2 hours, with 20 stocks it was more than 5 hours, and with 50 stocks it

was almost 30 hours; all these, even when the initial guess (x0) was very close to the actual

result. Since it was not possible to conduct all required tests within the time schedule of the

Thesis given these execution times, with the significant help of Dr. Qunzhi Zhang (ETH Zurich,

Chair of Entrepreneurial Risks), the code was optimized (time-wise) and the time frame of 5

hours became approximately 10 seconds; mainly because of eliminating the import-export

process of excel spreadsheets into Matlab, for each iteration. That was the main action

undertaken regarding the performance of the code. Of course, it could be further enhanced,

but since such actions would have absolutely no impact on its functionality, and almost none

on its execution speed, no further action was taken.

We will continue with a short description of the main Matlab function that was used (called

fmincon) throughout the evaluation of the entire Levy/Roll approach, before we come back

to the specifics of the code.

3.2 The “fmincon” Function

For the calculation of the adjusted parameters sets, all the work has been implemented in

Matlab environment, and more specifically by using the fmincon function as the key module.

The descriptive definition of this function, as given by the Matlab documentation, is to find

the minimum value of a certain function f(x) provided by the user, under certain linear

and/or nonlinear constraints (also provided by the user):

c(x) ≤ 0

ceq(x) = 0

A·x ≤ 0

Aeq·x = beq

lb ≤ x ≤ ub

32

Where x, b, beq, lb, and ub are vectors, A and Aeq are matrices, c(x) and ceq(x) are functions

that return vectors, and f(x) is a function that returns a scalar. The functions f(x), c(x), and

ceq(x) can be nonlinear.26

In our case, f(x) is our objective “Distance” function and the vector x corresponds to the

returns and standard deviations of each individual stock for each portfolio. As we will see in

detail further down this Thesis, for the Optimization Problem 1’s constraints, we have only

one set of nonlinear equations, with the actual number of equations being equal to the

number of the stocks of the portfolio; for the Optimization Problem 2, we employ an

additional set of nonlinear equation constraints (with the number of equations again equal

to the number of stocks in the portfolio), as well as a linear one (a single equation,

regardless of the number of stocks).

3.3 Nonlinear Optimization

In order to gain a basic understanding of how the function fmincon produces the results, we

will attempt to describe shortly the basic framework around it, as well as the principles of

nonlinear optimization in similar problems in general.

Based on the documentation available by Matlab, fmincon begins the optimization

procedure by calculating the f(x) function (which must be minimized) for the values of an

initial vector x0 provided by the user. For these initial values, fmincon calculates the value of

f(x), as well as the maximum constraint violation. The maximum constraint violation is

basically the maximum violation that has occurred amongst the constraint functions, given

the values of x0 (at the first run of the function).

For example, an execution of the code implemented for the Optimization Problem 1, for the

BtM1 (first Book-to-market quintile), Market Cap 2 (second Market Capitalization quintile)

portfolio from the 25 Fama-French portfolios (of the entire U.S. equities market-

NYSE/AMEX/NASDAQ), is demonstrated below:

26

Description of the variables taken from the official Mathworks page for the function fmincon: http://www.mathworks.com/help/optim/ug/fmincon.html, 21.08.2012.

33

clear all

global xexcel;

global weights;

global x0;

global x0i;

global lb;

funInitial

options=optimset('Display','iter-detailed','Diagnostics','on','FunValCheck','on');

[x,fval,exitflag,output] = fmincon(@funD,x0,[],[],[],[],lb,[],@funCeq,options);

global variables xexcel,weights,x0,x0i and lb have been created and stored to the workspace :)

Warning: The default trust-region-reflective algorithm does not solve problems with the constraints you have

specified. FMINCON will use the active-set algorithm instead. For information on applicable algorithms, see

Choosing the Algorithm in the documentation.

> In fmincon at 486

____________________________________________________________

Diagnostic Information

Number of variables: 44

Functions

Objective: optimfcnchk/checkfun

Gradient: finite-differencing

Hessian: finite-differencing (or Quasi-Newton)

Nonlinear constraints: optimfcnchk/checkfun

Nonlinear constraints gradient: finite-differencing

Constraints

Number of nonlinear inequality constraints: 0

Number of nonlinear equality constraints: 22

34

Number of linear inequality constraints: 0

Number of linear equality constraints: 0

Number of lower bound constraints: 22

Number of upper bound constraints: 0

Algorithm selected

medium-scale: SQP, Quasi-Newton, line-search

____________________________________________________________

End diagnostic information

Max Line search Directional First-order

Iter F-count f(x) constraint steplength derivative optimality Procedure

0 45 0.00617755 0.004689 Infeasible start point

1 91 0.239494 0.00229 0.5 -0.741 0.949

2 136 0.276686 0.003016 1 -0.774 0.941

3 182 0.23385 0.00148 0.5 -1.16 1.19

4 229 0.163491 0.001178 0.25 -0.876 0.647

5 275 0.150393 0.001333 0.5 -0.606 0.577

6 320 0.229915 0.001213 1 -0.779 0.871

7 365 0.221513 0.001088 1 -0.889 0.8

8 410 0.205579 0.001145 1 -0.931 1.01

9 455 0.242629 0.001049 1 -0.785 1.07

10 500 0.204216 0.001316 1 -0.919 0.78

11 546 0.13408 0.0006917 0.5 -0.798 1.01

12 593 0.0954456 0.0006253 0.25 -0.747 0.589

13 639 0.163942 0.0005746 0.5 -0.587 1.08

14 684 0.108503 0.0005996 1 -0.823 0.39

15 730 0.159044 0.0005533 0.5 -0.648 0.733

16 775 0.137833 0.000849 1 -0.733 0.408

17 821 0.0938025 0.000512 0.5 -0.709 0.461

18 868 0.0898399 0.0004677 0.25 -0.524 0.695

19 913 0.136677 0.0002424 1 -0.652 0.594

20 959 0.105435 0.0001595 0.5 -0.644 0.629

35

21 1005 0.0906033 0.0002612 0.5 -0.729 0.413

22 1051 0.0847407 0.0001485 0.5 -0.488 0.437

23 1098 0.06863 0.0001534 0.25 -0.542 0.285

24 1144 0.0813658 0.0001529 0.5 -0.431 0.43

25 1189 0.0707969 0.0002185 1 -0.527 0.287

26 1235 0.0709694 0.0001992 0.5 -0.411 0.317

27 1280 0.0723218 8.924e-05 1 -0.424 0.316

28 1326 0.0621664 6.905e-05 0.5 -0.423 0.205

29 1372 0.0627063 5.411e-05 0.5 -0.293 0.238

30 1418 0.0579527 4.812e-05 0.5 -0.314 0.154

31 1465 0.0564062 3.589e-05 0.25 -0.18 0.0843

32 1510 0.057422 1.719e-05 1 -0.145 0.111

33 1555 0.0568897 1.575e-05 1 -0.148 0.127

34 1601 0.0559108 1.622e-05 0.5 -0.147 0.0622

35 1646 0.0557735 1.513e-06 1 -0.0997 0.0755

36 1691 0.0558733 1.39e-06 1 -0.0794 0.0636

37 1736 0.0556121 1.205e-06 1 -0.0811 0.0573

38 1782 0.0555657 1.033e-06 0.5 -0.0352 0.0131

39 1827 0.0555815 9.625e-07 1 -0.0198 0.0376 Hessian modified








Optimization stopped because the predicted change in the objective function,

4.834686e-07, is less than options.TolFun = 1.000000e-06, and the maximum constraint

violation, 2.486700e-09, is less than options.TolCon = 1.000000e-06.

Optimization Metric Options

abs(steplength*directional derivative) = 4.83e-07 TolFun = 1e-06 (default)

max(constraint violation) = 2.49e-09 TolCon = 1e-06 (default)

No active inequalities

As we can see, there are 22 stocks in this particular portfolio, and this equals the number of

nonlinear equation constraints as well.

The warning message produced simply means that given the nonlinearity of the problem,

Matlab changed the solver from the default trust-region-reflective algorithm to the active-

set algorithm.

We can see that for every step (iteration) fmincon calculates f(x) several times, each time

changing one parameter towards a different direction. In principle, both f(x) and “Max

constraint” should get continuously lower values in each step, although the procedure is not

monotonous, especially in highly nonlinear problems with a large number of variables (like

this one). The optimization will conclude successfully if the tolerance of the constraint

violation (TolCon) is below the prespecified one (the default is set at 10-6) and/or if the

termination tolerance on the function value is also below the given level, where the default

is again 10-6.

For the above provided execution of fmincon, we can see how the function handles the

minimization of the objective function. Since it literally “tries” different values of x, and

afterwards calculates the value of f(x), the convergence is not monotonous. We also see that

it doesn’t stop once it reaches the minimum value of the objective function, since it needs to

ensure that the maximum constraint violation is also minimized.

0

0.05

0.1

0.15

0.2

0.25

0.3

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47

Ob

ject

ive

Fu

nct

ion

Val

ue

Fmincon Iteration Number

Picture 7: The value of the Objective Function being minimized, during the optimization process.

37

After having reviewed the graph for f(x), we now provide the one for the maximum

constraint violation, for the exact same optimization problem, portfolio and parameters as

before (they could not be combined in one representative graph, since their values differ

significantly).

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

0.0045

0.005

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47

Max

imu

m C

on

stra

int

Vio

lati

on

Val

ue


Picture 8: The Maximum Constraint Violation being minimized, during the optimization process.

We observe a similar pattern as with f(x), although with less fluctuations (more

monotonous). What is particularly interesting, from the intuitive point of view, is that in

several cases that one of the two graphs has a steep jump (upwards), the other one has the

opposite first derivative: it plunges. That reveals the way fmincon works, since it is not

possible for the function to minimize simultaneously the objective function and the

maximum constraint violation. Therefore, it takes turns in minimizing the one or the other,

and then comes back and evaluates them to determine in which direction it should continue

(in terms of configuring the x vector, i.e. the stocks’ returns and standard deviations).

Now, let us assume that we want to find the (global) minimum of the Peaks27 function:

z = 3*(1-x).^2.*exp(-(x.^2) - (y+1).^2) ...

- 10*(x/5 - x.^3 - y.^5).*exp(-x.^2-y.^2) ...

- 1/3*exp(-(x+1).^2 - y.^2)

27

Mathworks. Peaks Minimization with MultiStart. http://www.mathworks.com/matlabcentral/fileexchange/27178-global-optimization-with-matlab/content/html/msPeaksExample.html, 25.10.2012.

38

The plot of this function is displayed in the following graph:

Picture 9: The plot of the Matlab function Peaks.

We can see multiple local minimums in this plot and only one global minimum. In order to

have a more clear visualization of the problem that fmincon is called to resolve, we can see

the different mountain-valley tops in the following picture.

39

Picture 10: Different starting points resulting in different end points.28

From the different isoheight lines, we can see that if fmincon does not start from the

appropriate initial point, it is possible that it will not return the global minimum, but a local

one. Similarly, if the first derivative of f(x) gets a very small value (below the given TolFun),

then fmincon considers that point as a minimum, even if the function continues further (but

slowly) to a lower value point.

As a last attempt of making absolutely clear how the fmincon function works, even to

somebody that has no previous experience on the whole subject, I find it of use, to cite here

a creative description of how the function works. Even though it is not a pure scientific

explanation, but it feels like the beginning of a novel, after studying its behavior through

hundreds of experiments with it, I believe that the following passage captures the

fundamental workflow of the algorithm with great success:

“Think of fmincon as a blind man trying to find the bottom of a valley. He can stop at any

point and determine the local gradient. From there, he will choose where to search next, as

he also has some idea of the local shape (curvature) of the surface given his previous

meanderings. This is an iteration.

Within each iteration, he will try a few new points before updating his estimate of the local

gradient, which will start a new iteration.”29

At this point, we will continue our analysis with Chapter 4, where the detailed investigation

of Optimization Problem 1 begins.

28

Mathworks. Peaks Minimization with MultiStart. http://www.mathworks.com/matlabcentral/fileexchange/27178-global-optimization-with-matlab/content/html/msPeaksExample.html, 25.10.2012. 29

Mathworks. Thread Subject: Questions about fmincon options. http://www.mathworks.com/matlabcentral/newsreader/view_thread/237130, 26.10.2012.

40

Chapter 4

4 Optimization Problem 1

4.1 Methodology

As already mentioned in this Thesis, while previous researchers tried to determine if the

Levy/Roll model works, which translates to whether the market portfolio can be

mean/variance efficient, the goal is to further examine:

a) If the model indeed works, but more specifically;

b) Under which exact conditions the model works, as well as;

c) Why it works.

For that purpose, while going through the presentation of the results for each optimization

test, we will abandon the previous practice, in which the Levy/Roll process was viewed as a

“black box” which just produced results (but there was no effort spent in describing the

specifics of the Matlab models themselves).

The formulas for Optimization Problem 1 are repeated here, for the kind convenience of the

reader.


Subject to (nonlinear mean/variance constraint):

The objective function D is constructed in Matlab by basically using the “sum” function.

Initially, a script (script v2.4m in the Appendix) was created with all necessary commands for

the full implantation of Optimization Problem 1, but by using the Matlab Symbolic

(variables) Toolbox. The purpose of this exercise was to fully understand the code and verify

that it functions exactly as expected, without having to use as inputs actually stocks’ returns.

41

Every time the fmincon function is called, in every iteration it evaluates (in the sense of

calculating the value) the D, and for the same values of the vector x, it also evaluates each of

the constraint equations. The nonlinear constraints have been set in a form (all the right

hand side to the left one) so that they will result in a right hand side equation, which must

be equal to zero, and this is what the fmincon checks (in each iteration). At the “max

constraint violation” field, it returns the largest value of these equations (largest distance

from 0), which is named maximum constraint violation.

As the program is running, in each new iteration, if that maximum violation has a value less

than the default (10-6 or defined by the user), for three or more consecutive iterations, and

the predicted change in the objective function D is less than the respective threshold (10-6 or

defined by the user), then the fmincon stops the optimization and returns the result (the

adjusted parameters that it estimated, the value of the objective function, the stopping

criteria, etc.).

The criterion by which fmincon decides if D can be minimized even more, is the TolFun

threshold. Also 10-6 by default, if by each iteration (and therefore change of the objective

function’s input parameters) the difference (towards a smaller value) of the objective

function is less than this threshold, then this value of D is considered to be the minimum

one. After fmincon reaches the condition of the minimization of the objective function, it

would try to minimize even more the maximum constraint violation (without increasing D),

and once this is satisfied as well, it will stop.

We can see a simple representation of this in the following graph:

Picture 11: Tolfun, Tolx.30

If the maximum violation has a value above the TolCon threshold, then fmincon will return

the values that correspond to the minimum constraint violation, unless if the steps of

changing the value of the objective function are very small, but not small enough so that the

program will terminate. In this case, we can observe a very large number of iterations (when

30

Mathworks. Matlab Central. Fmincon, Tolx and Tolfun: How they work. http://www.mathworks.com/matlabcentral/newsreader/view_thread/33662, 21.03.2013.

42

the relevant ‘MaxIter’ setting is set to “infinite”), where fmincon basically goes back and

forward with the values of the estimated parameters, without being able to terminate.

At this point, we understand how important is the selection of the initial point, from which

fmincon begins the optimization, and therefore the evaluation of the objective function.

In similar problems faced in the industry, a linearization of the entire optimization problem

is often attempted, so that the whole procedure will give more robust results. In the context

of this Thesis, a lot of effort was consumed in order to make sure that the best possible

approach is being used, in order to avoid the local-global minimum problem.

Matlab provides the built-in options of Multistart and Global Search31 in order to address

such problems, but in our case we have a great advantage: based on our approach, we know

that the values for the returns and standard deviations that Matlab will return, should be

very close to their sample counterparts. That means, that if we start the optimization with

initial points the vectors of the sample returns and standard deviations, the optimization will

most likely begin from a point in this 2N-D dimensions space, which is really close to the

actual global minimum for our objective function. In other words, it is impossible for our

given problem, to identify a vector x’ of 2N variables that is “far away” from x0, and at the

same time if it is given as input to the objective function, D(x’) will result in a value lower

than the D(x0).

Therefore, in the data sets used to run the optimization problems, the following tests were

conducted, in order to make sure that the minimum acquired is actually (or better, most

likely) the global minimum one.

In a number of stock sets, first we ran the optimization with the sample vectors indeed as x0.

From doing so, we got a vector x*, with the estimated parameters for the same stocks,

returns and standard deviations. We fed these estimated parameters as input in the

objective D, and calculated the “distance” (D(x*)), overall, between these estimated

parameters and the sample ones. That value was saved, and we ran again the optimization

for x0 larger, than the sample values. Then again, we fed the new set of estimated

parameters as input in the objective D, and calculated the “distance”. This procedure was

repeated several times, also for smaller values than the sample ones, as well as for mixed

(some smaller, some larger) ones.

By comparing all the different D(x*), with x* the set of returns and standard deviations from

each run of the optimization mentioned above, it appears that the estimated parameters x*

that we estimated from the first time we ran the model, give us the lowest value for the

objective function. In other words, it always holds that:

D(x0) ≤ D(x*), ∀ x* ≠ x0

With this simple exercise, we consider that the area around the values of x0 was scanned for

resulting in possible lower values of the objective function, and therefore we concluded that

31

Mathworks. Global Optimization Toolbox. Global Search and Multistart Solvers. http://www.mathworks.com/products/global-optimization/description3.html, 20.03.2013.

43

our D(x0) with x0 the sample parameters, was indeed the minimum one. Of course, all the

x*’s that were returned satisfy all constraints. In fact, although some different local

minimums were found, in some occasions, a general multi-dimensional monotonicity was

observed. Most of the times, the problem would converge to the exact same set of values as

the first results (with the sample parameters as x0, and as expected, the further away the

starting vector from it, the more time and iterations would take for fmincon, to “return” to

the global minimum (or to stop to another, local minimum, which would give a larger

distance). Therefore, from now on, for each market proxy we use, we will consider the

parameters that are returned by fmincon with x0 as initial point, the ones that lead to a

global minimum for the objective (distance) function.

The conclusion is that, since the final estimated parameters should be “close” to their

sample counterparts, there should not be a vector with values “far away” from the x0’s ones,

which will return a smaller distance function value.

4.2 Covariance Matrix Shrinkage Methods

There is one particular problem that needs to be addressed, by anybody that is doing

research that includes simulating portfolios with large numbers of stocks; if the number of

observations (in our case, monthly returns) is smaller than the number of free variables (in

our case, the returns and standard deviations for each stock) then the covariance matrix

faces the issue of singularity.

Across literature, several methods have been implemented in order to deal with this, with

most prominent being the following ones32:

a) Shrinkage approach of Jagannathan and Ma (2003)

b) 1/N Portfolio by DeMiguel, Garlappi and Uppal (2007)

c) Shrinkage method of Ledoit and Wolf (2003, 2004)

d) Factor Portfolios (several authors)

In this Thesis, following the originally mentioned paper33 we will use the shrinkage method

presented by Ledoit and Wolf (2003)34. Their code was modified slightly, only for the

purposes of fitting in the original model hardcoded for the Levy/Roll approach; its

functionality was not changed:

(The code is provided here as implemented in all the funCeq.m functions; no separate

function for the shrinking of the covariance matrix was necessary.)

32

The papers in which details of all methods mentioned here can be found, are provided in the Chapter 9 (References). 33

Ni, X., Malevergne, Y., Sornette, D. and Woehrmann, P. (2011). Robust Reverse Engineering of Cross Sectional Returns and Improved Portfolio Allocation Performance using the CAPM. Journal of Portfolio Management, 37 (4), 76-85. 34

Ledoit, O. and Wolf, M. (2003). Honey, I Shrunk the Sample Covariance Matrix. UPF Economics and Business, Working Paper No. 691.

44

% de-mean returns [t,n]=size(xexcel);

%n columns' number m=s_m(1:n); s=s_m(n+1:n+n);

meanx=mean(xexcel); xexcel2=xexcel-meanx(ones(t,1),:);

% compute sample covariance matrix sample=(1/t).*(xexcel2'*xexcel2);

% compute prior var=diag(sample); sqrtvar=sqrt(var);

rho=(sum(sum(sample./(sqrtvar(:,ones(n,1)).*sqrtvar(:,ones(n,1))')))-

n)/(n*(n-1)); prior=rho*sqrtvar(:,ones(n,1)).*sqrtvar(:,ones(n,1))'; prior(logical(eye(n)))=var; % initianilization of shrink variable shrink = -1; if (nargin < 2 | shrink == -1) % compute shrinkage parameters

c2=norm(sample-prior,'fro')^2; y=xexcel2.^2; p=1/t*sum(sum(y'*y))-sum(sum(sample.^2)); rdiag=1/t*(sum(sum(y.^2)))-sum(var.^2); v=((xexcel2.^3)'*xexcel2)/t-(var(:,ones(1,n)).*sample); v(logical(eye(n)))=zeros(n,1); roff=sum(sum(v.*(sqrtvar(:,ones(n,1))'./sqrtvar(:,ones(n,1))))); r=rdiag+rho*roff;

% compute shrinkage constant k=(p-r)/c2; shrinkage=max(0,min(1,k/t));

else % use specified number shrinkage = shrink;

end % compute the estimator sigma=shrinkage*prior+(1-shrinkage)*sample;

The effect of this piece of code onto the covariance matrix is demonstrated below, with a

simple example:

By running the Matlab script v2.4.m for a portfolio constituted only by the top 5 (by market

capitalization) S&P 500 Index stocks from Bloomberg, we get the following two matrices:

45

% compute sample covariance matrix (before Ledoit-Wolf)

sample =

0.0112 0.0015 0.0070 0.0008 0.0044 0.0015 0.0025 0.0020 0.0005 0.0020 0.0070 0.0020 0.0098 0.0011 0.0031 0.0008 0.0005 0.0011 0.0034 0.0029 0.0044 0.0020 0.0031 0.0029 0.0108

% sample covariance matrix from Ledoit-Wolf

sigma =

0.0112 0.0017 0.0052 0.0015 0.0041 0.0017 0.0025 0.0018 0.0008 0.0019 0.0052 0.0018 0.0098 0.0015 0.0033 0.0015 0.0008 0.0015 0.0034 0.0025 0.0041 0.0019 0.0033 0.0025 0.0108

As we can see, we could simplify the explanation of the effect of the Ledoit/Wolf procedure

on the covariance matrix, by stating that while it leaves the variances unchanged, it

increases the lower covariances and decreases the higher ones, in an attempt to reduce

their difference from the mean covariance in a non-proportional way. As mentioned by the

authors of the original paper, the biggest statistical challenge of the entire process is to

identify the optimal shrinkage intensity; details of this method can be found in the original

paper35.

To highlight the importance of such a procedure, based on the 1/N method by DeMiguel,

Garlappi and Uppal36, a portfolio of 25 assets would require (without shrinkage) at least

3000 months of returns for any statistical use of the stocks’ covariance matrix, and a

portfolio of 50 stocks would require 6000 months of returns. As we will see in the relevant

chapter in the Appendix regarding the data acquisition procedure from Bloomberg, such a

case would make any research attempt in past returns of stocks ex ante completely

infeasible.

4.3 Definition of the Dependent Variable in all Matlab Functions

The first issue that had to be solved is the fact that fmincon accepts only a vector x as

variable, while in the optimization problems we have at least37 two sets of vector-variables,

one for the returns and one for the standard deviations. As such the x used be fmincon as

free variable, and of course the very important x0, is a vector of 2·N values, where the first

35

Ledoit, O. and Wolf, M. (2003). Honey, I Shrunk the Sample Covariance Matrix. UPF Economics and Business, Working Paper No. 691. 36

DeMiguel, V., Garlappi, L., Nogales, J. and Uppal, R. (2007). Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy? Rev. Financ. Stud, 22(5), 1915-1953. 37

Across literature, one can see different approaches. In the first part of this Thesis, our x used for fmincon is constituted only be returns and standard deviations.

46

N (1, 2, .., N) values are the returns of the proxy’s stocks, and the second N (N+1, N+2, .., 2N)

represent the standard deviations of these stocks. A very important notice given the nature

of the algorithm, is that the two parameters (return and standard deviation) of each stock,

have the same index in both sets. That means that if the return of a specific stock has the

index 45, for example, then its standard deviation will also have the index 45. Therefore, at

the vector x of fmincon, the same stock’s return and standard deviation will be located at

the places 45 and N+45, respectively.

4.4 Flowchart of the Matlab code

In this paragraph we will provide a simple diagram of the Matlab code, along with a short

description of the main functions.

Picture 12: Matlab code flowchart.

As we can see in the simplified diagram above, for every optimization test (Optimization

Problems 1 & 2) we need to define an objective function to be minimized (named funD.m in

all our tests) and one function that includes all the nonlinear constraints. Depending on the

implementation, that can be one of the funCeq, funCeq2 or funCeq3. In addition, some

variations of funCeq2 were used for Optimization Problem 2, all of which are explained in

detail in Chapter 5. Finally, the entire Matlab code can be found in the relevant Appendix

chapter.

4.5 Construction of the 25 Fama French Portfolios

The ultimate challenge in regard to the power of the Levy/Roll theory is to test it on the 25

Fama French portfolios structure. These portfolios are known throughout the literature, and

have the name of the researchers that first formulated them, because of their ability to

capture, or even better reveal, crucial inner characteristics of any given model that is being

tested on them. In this Thesis, they are considered to be a very important test for the

validity of the model (although, as we will see further on, the objectivity of the results of

fmincon

funCeq.m funCeq2.m funCeq3.m

funD.m

47

such a test can be challenged), and as such, great weight is given to the performance of the

model when testing it on these portfolios.

These 25 Fama French portfolios were formed from data downloaded from

Bloomberg.38There are a lot of issues with missing data from this database, especially when

one tries to download several thousands of stocks for which continuous monthly returns and

other Bloomberg fields are required, over a large number of years (for example, from 1991

until 2009). For that reason, the processing of the raw data downloaded from Bloomberg,

until they were in appropriate format and ready to be used as input to the Matlab

optimization problem, is being described as follows:

38

The entire process of data acquisition from Bloomberg is described in detail in the Appendix.

48

Picture 13: Data acquisition and analysis flowchart.

Downloading of raw data from Bloomberg.

Preparation in excel, with returns, Market Capitalization and Market to Book

Bloomberg fields.

Execution of a small VBA script to get only the stocks with full continuous values for

the given time period.

Execution of a C# script to sort the remaining stocks in two dimensions,

Market Capitalization and Book to Market.

Splitting the whole set of stocks left into the 25 Fama French portfolios, based on

the output of the previous step.

Running the MATLAB optimization procedures for every one of the 25 sets, in

order to get the estimated parameters.

Analyzing the "statistical equality" of the estimated parameters with the sample

ones, for each of the 25 sets.

49

After the acquisition of the raw data from Bloomberg, the form of the data is shown below.

Here the monthly returns (in percentage points, %) of a random selection of alphabetically

concequitive stocks from the NASDAQ Index; the columns correspond to the time series of

the stocks’ monthly returns. The empty cells correspond to missing values, as downloaded

from Bloomberg:

OBAF US EQUITY

OBCI US EQUITY

OPTT US EQUITY

ORIG US EQUITY

OSHC US EQUITY

OCFC US EQUITY

OCLR US EQUITY

OFED US EQUITY

OCLS US EQUITY

OCZ US EQUITY

-9.09 3.42 -10.91 13.04 -23.23 0.39

-15.00 6.75 0.93 4.95 7.62 -22.73

-23.53 -6.83 -2.11 2.09 -9.87 3.03

0.00 -15.25 -6.20 5.64 -26.24 19.90

-15.38 25.76 5.03 -0.97 -40.06 10.95

18.18 12.15 1.98 11.76 49.07 -3.01

0.00 -17.70 8.32 6.58 -5.96 -12.01

7.69 11.85 0.78 -8.85 -6.67 7.76

-21.43 -17.68 -4.11 6.77 13.21 1.34

9.09 -6.59 -2.14 8.08 0.32 -34.21

0.00 -2.08 2.19 11.02 31.60 -18.00

37.50 -20.38 2.86 -0.14 16.61 5.85

39.39 -6.27 -0.17 -3.93 5.53 30.41

4.35 -0.66 1.30 5.58 -8.74 -10.60

-8.33 -2.87 5.41 6.20 21.70 -17.39

-4.55 -4.19 -0.65 -1.00 16.26 -4.55

-14.29 -0.96 3.20 0.34 4.96 -39.60

22.22 -34.25 0.08 -4.83 36.68 19.50

9.09 10.38 3.17 0.00 -36.16 -0.69

12.50 10.40 -0.23 4.96 -36.78 -33.57

-18.52 -2.45 7.41 -12.73 -47.37

24.09 0.77 -2.77 -11.31 -54.00

-15.38 3.53 -0.41 6.71 210.87

0.00 4.17 -2.31 1.26

0.00 -1.82 -25.25 -2.80

-9.09 -0.07 8.11 28.75

15.00 -2.89 -2.53 0.99

8.70 -1.15 9.05 -28.26

-8.00 1.85 5.14 -10.27

-8.70 -2.96 -6.02 -13.36

-4.76 -14.06 -4.03 0.88

20.00 1.82 -3.36 -6.11

2.81 1.25 10.61 4.65

10.64 -3.44 12.30 16.89

Table 4.1: Step one, raw data downloaded from Bloomberg (before deleting the columns with

incomplete data).

50

Each row represents one month; not the full time period is depicted here. In our analysis we

can use the stocks for which we have returns (for example) for all months within the time

period under consideration, therefore by using the following VBA script we delete all the

incomplete ones (parts of the code were found in the forum39 cited):

The sole purpose of this small script is to delete the columns (each column represents a

different stock) for which there is even one value missing (for each row we have the value

for one month, where the oldest date is on top).

Now that we are left only with the stocks for which we have continuous monthly returns for

the desired time period, in order to populate the 25 Fama French portfolios we need to

perform a “two dimensional sorting”, so that we will realize which stock belongs to which

quintile, based on the methodology given by Fama and French.40 The break values for

market capitalization and book-to-market used in this Thesis are the very popular ones in

the literature and previous research work, taken from the database41 maintained by

Professor Kenneth French42.

The procedure followed for their calculation, as described by Professor Kenneth French,

started with the gathering of data from NYSE stocks, for the end of June of every year T.

These stocks were initially sorted by market equity (ME-size, the number of stocks

outstanding times the current stock price). Afterwards, they were also sorted by book-to-

market equity (BE/ME, with BE being book value and ME the market value); the market

39

MrExcel.com. Forum. http://www.mrexcel.com/forum/excel-questions/52132-delete-row-where-specific-cell-blank-meets-condition.html, 02.03.2013. 40

The exact same procedure was followed in the paper on which we have been referring before: Ni, X., Malevergne, Y., Sornette, D. and Woehrmann, P. (2011). Robust Reverse Engineering of Cross Sectional Returns and Improved Portfolio Allocation Performance using the CAPM. Journal of Portfolio Management, 37 (4), 76-85. 41

Dartmouth College. Current Research Returns. http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html, 13.03.2013. 42

Kenneth R. French is the Roth Family Distinguished Professor of Finance at the Tuck School of Business at Dartmouth College. http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/curriculum_vitae.html, 20.09.2012.

Sub MaxRows()

For i = 1 To 6000

If Sheets(1).Columns(i).End(xlDown).Row <

Sheets(1).Cells.SpecialCells(xlCellTypeLastCell).Row Then

Columns(i).EntireColumn.Delete

i = i - 1

End If

Next i

End Sub

51

value of a stock is defined as the ME at the end of December of the year T-1, and the book

value is the BE for the fiscal year ending in calendar year T-1.

All the possible values for market value and book-to-market are then split into five quintiles

each, the intersection of which provides the 25 Fama French portfolios.

The new script, which will sort the stocks into these 25 Fame French portfolios, was

implemented in C# and it is presented as follows:

using System;

using System.Collections.Generic;

using System.Linq;

using System.Text;

using Excel = Microsoft.Office.Interop.Excel;

namespace Excel_Ted

{

class Program

{

static void Main(string[] args)

{

Excel.Application excelApp = new Excel.Application();

excelApp.Visible = true;

string workbookPath = "c:/NYSE only BE ME to be

sorted.xlsx";

Excel.Workbook excelWorkbook =

excelApp.Workbooks.Open(workbookPath,

0, false, 5, "", "", false,

Excel.XlPlatform.xlWindows, "",

true, false, 0, true, false, false);

Excel.Sheets excelSheets = excelWorkbook.Worksheets;

string currentSheet = "Sheet1";

Excel.Worksheet excelWorksheet =

(Excel.Worksheet)excelSheets.get_Item(currentSheet);

string workbookPath2 = "c:/MEB&BEME_LAST.xlsx";

Excel.Workbook excelWorkbook2 =

excelApp.Workbooks.Open(workbookPath2,

0, false, 5, "", "", false,

Excel.XlPlatform.xlWindows, "",

true, false, 0, true, false, false);

Excel.Sheets excelSheets2 = excelWorkbook2.Worksheets;

string currentSheet2 = "Sheet1";

Excel.Worksheet excelWorksheet2 =

(Excel.Worksheet)excelSheets2.get_Item(currentSheet2);

int i_TEDs = 2;

string i_str_TEDs, j_str_TEDs;

for (i_TEDs = 2; i_TEDs <= 560; i_TEDs++)

{

i_str_TEDs = 'B' + i_TEDs.ToString();

Excel.Range excelCell =

(Excel.Range)excelWorksheet.get_Range(i_str_TEDs); // BtoM

Console.WriteLine("The BtoM " + i_str_TEDs + " value is " + fl);

if (fl <= 0.358)

excelWorksheet.get_Range('E' +

i_TEDs.ToString()).Value = 1;

52

else if (fl <= 0.562)



else if (fl <= 0.798)



else if (fl <= 1.102)



else if (fl <= 6.905)



j_str_TEDs = 'C' + i_TEDs.ToString();

Excel.Range excelCell2 =

(Excel.Range)excelWorksheet.get_Range(j_str_TEDs); // MKT CAP

fl = excelCell2.Cells.Value;

Console.WriteLine("The MKT CAP " + j_str_TEDs + "

value is " + fl);

if (fl <= 568.09)

excelWorksheet.get_Range('D' +


else if (fl <= 1480.28)



else if (fl <= 3172.82)



else if (fl <= 8557.61)



else if (fl <= 394611.12)



}

// string ans = Console.ReadLine();

excelWorkbook.SaveAs(workbookPath, true);

excelApp.Quit();

}

}

}

With each command like the one following, we compare the book-to-market or market capitalization value of the stock with the break points (here, the book-to-market value of the stock (BtoM) is compared to the lowest breakpoint (0.358): Console.WriteLine("The BtoM " + i_str_TEDs + " value is " + fl);

if (fl <= 0.358)

The output of this script is two additional columns in the spreadsheet which we give as

input, where in the first one it returns the classification of the particular stock in one of the

five quintiles in respect to its market capitalization, and the second the classification of the

stock in respect to its book-to-market ratio. The structure of the output for the first 20

stocks can be seen in the following table:

53

Firm Name Book-to -market

Market Cap.

Book-to -market Rank

Market Cap. Rank

Mean Return

DDD US EQUITY 0.57 177.58 3 1 1.84

MMM US EQUITY 0.27 39906.48 1 5 0.87

AIR US EQUITY 0.90 656.20 4 2 1.26

ABT US EQUITY 0.20 82853.35 1 5 1.04

ABM US EQUITY 0.85 759.51 4 2 1.23

ATU US EQUITY 0.61 1011.49 3 2 2.23

AMD US EQUITY 0.09 1327.62 1 2 1.45

AFL US EQUITY 0.65 21389.63 3 5 1.66

GAS US EQUITY 0.64 2410.82 3 3 0.86

APD US EQUITY 0.41 10539.27 2 5 0.97

ARG US EQUITY 0.48 3162.83 2 3 1.97

ALK US EQUITY 0.63 1061.04 3 2 0.68

AIN US EQUITY 1.34 384.67 5 1 0.41

AA US EQUITY 1.74 9011.57 5 5 0.79

ALX US EQUITY 0.18 1297.85 1 2 1.52

Y US EQUITY 0.97 2332.92 4 3 1.02

AGN US EQUITY 0.32 12260.83 1 5 1.40

ALE US EQUITY 0.80 1052.00 4 2 0.93

AB US EQUITY 1.05 1877.83 4 3 1.46

LNT US EQUITY 0.88 3222.90 4 4 0.67

Table 4.2: Data from Bloomberg sorted. (Market capitalization in USD Million, throughout the

whole Thesis.)

Afterwards, it comes down to a simple exercise in excel, to create the 25 different portfolios

that we need from this original spreadsheet.

4.6 Statistical Analysis of the test results

Before we proceed with the various tests of the Optimization Problems 1 & 2, it is necessary

to provide with the procedure with which the sample values are compared with the

generated from Matlab ones. Hence, for every portfolio, the table below is constructed for

each one of the tests:

54

1 2 3 4 5 6 7 8 9

μ sample

μ Matlab

σ sample

σ Matlab

t statistic |t| < 1.96 satisfied

lower bound satisfied

variances ratio

upper bound satisfied

0.014 0.017 0.109 0.108 0.29 yes yes 0.982 yes

0.001 0.017 0.110 0.110 1.56 yes yes 0.991 yes

0.041 0.017 0.115 0.107 -2.37 no yes 0.871 yes

0.012 0.016 0.062 0.060 0.61 yes yes 0.956 yes

0.009 0.016 0.050 0.049 1.42 yes yes 0.959 yes

0.012 0.016 0.059 0.057 0.71 yes yes 0.952 yes

0.016 0.016 0.066 0.064 0.08 yes yes 0.941 yes

0.011 0.016 0.073 0.072 0.71 yes yes 0.974 yes

0.014 0.017 0.097 0.096 0.34 yes yes 0.977 yes

0.013 0.016 0.088 0.087 0.33 yes yes 0.976 yes

0.021 0.017 0.122 0.121 -0.32 yes yes 0.975 yes

0.005 0.016 0.052 0.052 2.34 no yes 0.982 yes

0.011 0.016 0.050 0.049 1.09 yes yes 0.949 yes

0.016 0.017 0.115 0.114 0.08 yes yes 0.980 yes

0.017 0.016 0.097 0.095 -0.09 yes yes 0.968 yes

0.011 0.017 0.100 0.099 0.69 yes yes 0.984 yes

0.013 0.016 0.065 0.064 0.52 yes yes 0.957 yes

0.009 0.020 0.301 0.299 0.40 yes yes 0.988 yes

0.013 0.017 0.113 0.112 0.37 yes yes 0.984 yes

0.042 0.017 0.189 0.185 -1.50 yes yes 0.954 yes

0.008 0.016 0.083 0.083 1.06 yes yes 0.983 yes

0.015 0.016 0.096 0.094 0.22 yes yes 0.975 yes

0.033 0.016 0.114 0.109 -1.72 yes yes 0.919 yes

0.006 0.017 0.090 0.089 1.25 yes yes 0.988 yes

0.010 0.018 0.182 0.182 0.48 yes yes 0.995 yes

0.008 0.016 0.072 0.071 1.28 yes yes 0.982 yes

0.026 0.016 0.078 0.074 -1.39 yes yes 0.897 yes

0.028 0.017 0.125 0.122 -1.01 yes yes 0.948 yes

0.016 0.016 0.101 0.099 0.06 yes yes 0.974 yes

0.015 0.017 0.136 0.135 0.18 yes yes 0.987 yes

0.014 0.016 0.073 0.071 0.33 yes yes 0.962 yes

0.012 0.016 0.090 0.089 0.54 yes yes 0.978 yes

0.016 0.017 0.098 0.097 0.10 yes yes 0.974 yes

0.010 0.016 0.065 0.064 1.04 yes yes 0.970 yes

0.020 0.016 0.083 0.080 -0.42 yes yes 0.945 yes

-0.001 0.016 0.120 0.120 1.54 yes yes 0.996 yes

0.019 0.016 0.091 0.089 -0.36 yes yes 0.956 yes

0.031 0.018 0.176 0.174 -0.84 yes yes 0.972 yes

0.017 0.016 0.078 0.076 -0.06 yes yes 0.953 yes

0.004 0.016 0.089 0.089 1.48 yes yes 0.990 yes

55

Table 4.3: Statistical tests carried out for each optimization sets’ results.

At the table above we can see a sample of the results of the statistical test. We have picked

40 stocks out of a sample portfolio constituted by 100 stocks of the S&P 500 Index, sorted by

alphabetical order.

Each row corresponds to one specific stock, for which as has already explained there are two

sets of parameters: the estimated ones μ and σ, and the sample ones μsample and σsample.

More specifically (explanation of the variables):

μ sample: the (simple) average return of each stock of the portfolio.

μ Matlab: the return of each stock estimated by fmincon.

σ sample: the standard deviation of each stock of the portfolio.

σ Matlab: the standard deviation of each stock estimated by fmincon.

The objective of this statistical exercise is to determine how “close” the sample and the

estimated parameters are; so we compare the μ with the μsample, and the σ with the σsample.

Two simple statistical tests are employed in order to test how “close” the estimated

parameters are to the real ones:

a) T-test for the returns. The t-values for the estimated returns are given in column 5

of the above table. As we can see, for each and every one of the stocks of this

sample, the t-value states that the difference between the adjusted and the sample

parameter is not significant at the 95% level of confidence level. This can be seen at

column 6 of our table, where an if-function in excel informs us whether the null

hypothesis (H0: μisample – μi = 0, i = 1, 2, .., 121 at the 5% significance level) holds true.

b) Confidence interval for the standard deviations. In order to test the standard

deviations, we test whether the confidence interval of the ratio (σi)2/(σi

sample)2 for

each stock is within range; the range differs depending on the number of stocks, for

the above sample it is [0.7873-1.27016]. This is calculated as follows: We use 121

monthly return observations, therefore n=121. Since we want are results to be in

line with the 95% confidence interval for (σi)2/(σi

sample)2, and under the null

hypothesis that the estimated standard deviations returned by Matlab follow the

same distribution as the sample parameters, we are looking for the critical values c1

and c2, such that P(x2120-1 > c1 ) = 0.025 and P(x2

120-1 < c2 ) = 0.025. For the first

equation we have (2·c1)^(0.5)-(2·121-1)^(0.5) = 1.96, and for the second

(2·c2)^(0.5)-(2·121-1)^(0.5) = 1.96. From these two equations, we get c1 = 152.42

and c2 = 90.58. In order to calculate the confidence interval, we employ the standard

formula c1 < (121-1)·(σi)2/(σi

sample)2 < c2, which gives us the above numbers for the

specific stock set.

Another candidate for testing the null hypothesis is the Bonferroni test (1945), which states

that a multiple comparison null hypothesis like this one should be rejected at the 5%

significance level if any one of the estimated return parameters is significantly different from

its sample counterpart at the 2.5% significance level. In this particular case two of the

56

estimated parameters are significant at the 5% significance level, therefore even by

Bonferroni’s test, the multiple null hypothesis cannot (marginally) be rejected.

In the chapters devoted to the tests results for the optimization problems under the

Levy/Roll procedure (Chapter 4 for Optimization Problem 1 and Chapter 5 for the

Optimization Problem 2), a large number of tests will be presented. They were picked out of

a much larger pool of all the relevant optimization tests conducted, for the purposes of this

Thesis, in an attempt to familiarize with the behavior of these models under real market

conditions.

4.7 Investigation of correlations and other varying attributes

Picture 14: Correlations are significantly higher when in crisis.43

The rapid change of correlations, even within a short period of only few years, poses a

challenge to the statistical analysis of stock attributes. We believe that using the same (or

“average”) correlation for the entire period, like in the implementation by Levy and Roll

(2010), corresponds to a serious over-generalization and represents a very strong

assumption for projections of future returns. In addition, a lot of questions have been raised

regarding the structural robustness of the model -that is, whether it is robust because of its

mathematical nature- and it does not necessarily reflect any characteristics of the actual

stocks.

For that reason, two distinct series of tests with different correlation matrices were

implemented. In order to accomplish that, we will replace the matrix “sample” at line 52 of

the constraint functions (funCeq.m), which represents the covariance matrix of the sample

set of stocks (the “real one”) with the artificial covariance matrix, for which we want to

stress the model. This artificial covariance matrix is constructed by using the original

43 Empirical Research Partners LLC, Standard & Poor’s, J.P. Morgan Asset Management. Capitalization

weighted correlation of top 750 stocks by market capitalization, daily returns, 30.06.2013.

57

standard deviations of the stocks, but a different correlation matrix, for which we will create

for different versions (two sets). The portfolio chosen for these tests is the most robust one

from all the optimizations ran so far, the first 100 stocks from the S&P 500 Index. The set-up

that we will use for testing the correlation matrices is the “proper” one, based on market

conditions and risk free rate; the one that, as we will see in the subsequent chapters, is

proven to be robust. This way we believe that the necessary conditions are met, in order to

achieve a reliable conclusion regarding the use of different correlation matrices in a known

problem.

1. For the first test we will create a random correlation matrix. The following simple

Matlab script is being used for that purpose:

% generate a pseudo random definite correlation matrix % positive definite NxN matrix SPD = randn(N); % N the number of stocks in our portfolio %covariance matrix: SPD = SPD'*SPD; % Convert S into a correlation matrix: CMATR = sqrt(diag(SPD)); C = diag(1./ CMATR)* SPD *diag(1./ CMATR);

2. For the second test we will create a random positive correlation matrix, given that

most stocks have positive correlations. The following simple Matlab script is being

used for that purpose:

% generate a pseudo random definite correlation matrix % positive definite NxN matrix constituted only by positive

values SPD = randn(N); % N the number of stocks in our portfolio %covariance matrix: SPD = SPD'*SPD; % Convert S into a correlation matrix: CMATR = sqrt(diag(SPD)); C = diag(1./ CMATR)* SPD *diag(1./ CMATR);

3. For the third test, we employ the Cholesky factor for sampling, for the generation of

a mixed random set of correlations:

CL = chol(SPD) %generate positive pseudo-random deviates XCL = randn(10,N)*CL; corr(XCL);

4. Finally, for the fourth test we employ the Cholesky factor for sampling, for the

generation of only positive correlations:

CL = chol(SPD) %generate mixed pseudo-random deviates XCL = rand(10,N)*CL; corr(XCL);

58

To be noted, a very explanatory description of the generation of the above correlation

matrices was found in the stackoverflow forum (see footnote for reference).44

As already mentioned, we do not expect the standard deviations of the stocks’ returns to

change. Therefore, in perhaps a little more detail, the procedure followed for the replacing

of the current correlation matrices with the new ones, is:


% Compute the standard deviations from the original sample covariance

matrix var=diag(sample); sqrtvar=sqrt(var);

% calculate the artificial correlation matrix (procedure demonstrated % above)

ArtCOR; % Use the new correlation matrix in the Final expression for non the

linear restriction ceqshrunkinv = S* ArtCOR*S*xma - q*mr2n;

The reason why we refer to the “original” sample covariance matrix is that as we know, the covariance matrix used in all tests is the shrunk one (based on the cited paper45), although the standard deviations remain unchanged after shrinking; there is no apparent reason why the random covariance matrices should be shrunk, therefore we substitute the shrunk ones with them, at the final formulas in the Matlab code. The results from running the model with these correlation matrices will indicate whether the

mathematical model is robust by itself, disregarding the actual relationships between the

stocks, or if it actually reveals some intuition behind these relationships. In the following

table we have summarized the execution trials of Optimization Problem 1:

Model name

Rejected μ's

Rejected σ's

MaxFunEvals Levy/Roll complied

1 44% 87% 40000 N

2 43% 91% 40000 N

3 85% 82% 40000 N

4 86% 52% 84690 N

Table 4.4: Results of the optimizations conducted by using the generated random correlation

matrices (the explanation of the parameters is provided in page 66).

As we can see from the results, as well from the behavior of the model while running in

Matlab, the different correlation matrices caused the complete breakdown of the whole

optimization procedure (even though the maximum –objective- function evaluations were

several times above the usual number).

44

Stackoverflow. How to generate pseudo-random positive definite matrix with constraints on the off-diagonal elements? http://stackoverflow.com/questions/1037340/how-to-generate-pseudo-random-positive-definite-matrix-with-constraints-on-the-o, 20.08.2013. 45

Ledoit, O. and Wolf, M. (2003). Honey, I Shrunk the Sample Covariance Matrix. UPF Economics and Business, Working Paper No. 691.

59

Our results indicate that there is a significant amount of information included in the

correlation matrices. Therefore, in the final chapter we will propose an idea of repeating the

tests with random correlation matrices constrained by certain conditions, so that the largest

part of information will not be lost.

For that reason, it is considered reasonable to proceed with tests in actual market data, right

after we analyze how to deal with the final calibration of the input parameters.

4.8 Definition of Market Proxy – Indices

Extensive research was carried out in order to determine the best set of market parameters

that need to be fed into the model for each different set of tests. Therefore, it was

considered absolutely necessary to devote a subchapter of this Thesis on this issue, in order

for the reader to gain a better insight of the historical return and standard deviation values,

for different indices which are considered to be market proxies, throughout the literature.

For the needs of the entire Thesis, the following Indices were used:

1. S&P 500

2. S&P 500 Equally Weighted

3. S&P Global 1200

4. Willishire 5000

5. Willishire 5000 Equally Weighted

6. Russell 3000

The equally weighted indices were used at the third main part of the Thesis, in order to

calculate the Zipf factor. For that particular part, the Equally Weighted Market Returns from

CRSP Database were used as well, but as it is not officially considered an Index, it is not

stated here.

From the analysis of the below provided graphs, as well as the tables downloaded from

Bloomberg, we are trying to demonstrate two major characteristics: the returns and the

standard deviation of each index, and of course the degree of correlation between the

different indices for the same market, as well as for overlapping markets. The reason for

undertaking so much effort to analyze the behavior of the main (U.S.) market Indices, is that,

as we will see in the respective chapters, we need to calculate accurately (sometimes up to

10-4) the proxy market’s returns and standard deviations, in order to feed them into the

models. Since the backbone of the stress testing of the Levy/Roll approach (Optimization

Problems 1 and 2) is based on whether they perform well under the specific market

conditions for each, we attempted to specify for each optimization test, based on the

constituents of the portfolio used, and the specific time period, the most appropriate mean

market returns and standard deviation.

60

Zephyr StyleADVISORZephyr StyleADVISOR: Dow Chemical Company

Trailing Market ReturnsJanuary 2003 - February 2013 (not annualized if less than 1 year)

Retu

rn

0

5

10

15

20

YTD 1 quarter 1 year 3 years 5 years 10 years

Citigroup 3-month T-bill

Russell 3000

MSCI EAFE Index

S&P 500

S&P 500 Equal Weight

S&P 1000 Pure Growth

Wilshire 5000 (Full Cap)

Wilshire 5000 - Equal Weight

Trailing Market ReturnsJanuary 2003 - February 2013 (not annualized if less than 1 year)


Russell 3000

MSCI EAFE Index

S&P 500

S&P 500 Equal Weight

S&P 1000 Pure Growth

YTD 1 quarter 1 year 3 years 5 years 10 years

0.01% 0.02% 0.08% 0.09% 0.35% 1.67%

6.89% 8.19% 13.65% 13.83% 5.38% 8.85%

4.32% 7.67% 10.37% 7.35% -0.77% 9.88%

6.61% 7.58% 13.46% 13.50% 4.94% 8.24%

7.80% 10.27% 15.29% 15.22% 8.08% 11.60%

6.11% 8.76% 10.83% 17.99% 11.76% 13.93%

Wilshire 5000 (Full Cap) 7.02% 8.28% 13.45% 13.81% 5.58% 9.19%

Wilshire 5000 - Equal Weight 8.57% 11.46% 13.88% 10.16% 8.53% 15.13%

61

Zephyr StyleADVISORZephyr StyleADVISOR: Dow Chemical Company

Risk / Return TableAnnualized Summary Statistics: January 2003 - February 2013


Russell 3000

Return(%)

Std Dev(%)

Downside Risk(%)

Betavs.

Market

Alphavs. Market

(%)

R-Squaredvs. Market

(%)

R-Squaredvs. Style

(%)

SharpeRatio

Tracking Errorvs. Market

(%)Observs.

1.66 0.51 0.30 -0.0017 1.68 0.25 100.00 0.0000 14.7529 122

8.26 15.31 11.80 1.0376 0.33 99.45 100.00 0.4306 1.2599 122

MSCI EAFE Index 9.01 18.34 14.14 1.1249 0.81 81.50 83.07 0.4004 8.0983 122

S&P 500 7.66 14.72 11.30 1.0000 0.00 100.00 99.82 0.4072 0.0000 122

S&P 500 Equal Weight 10.86 17.90 13.34 1.1834 1.93 94.70 97.29 0.5140 4.9274 122

S&P 1000 Pure Growth 13.03 19.32 13.97 1.1960 4.08 83.06 95.15 0.5886 8.4568 122

Wilshire 5000 (Full Cap) 8.57 15.28 11.75 1.0341 0.65 99.23 99.95 0.4523 1.4330 122

Wilshire 5000 - Equal Weight 14.76 21.77 15.68 1.2910 5.36 76.18 85.49 0.6016 11.4559 122

S&P 500 7.66 14.72 11.30 1.0000 0.00 100.00 99.82 0.4072 0.0000 122

62

Tables 4.5 and 4.6: Returns and risk metrics for the main market Indices used46

.

4.9 Optimization Problem 1 Robustness Evaluation

Now that we have gained a solid understanding of the whole optimization procedure, we are

called to investigate the Levy/Roll approach for Optimization Problem 1 with real data.

In order to do that, with leaving zero -or at least the minimum amount of- doubts regarding

the robustness of our conclusions, a very large amount of optimizations was conducted.

At this point, it is worth mentioning again that the overall methodology followed in this

Thesis, is significantly differentiated from the previous research attempts. The main

differentiating point is that so far, the parameters which we call “calibration parameters” for

this model (market returns and standard deviation, risk free ratio and the choice of alpha),

were considered to be free parameters of the optimization procedure; some, or even all of

them. We believe that this is a wrong approach, since except from the choice of alpha

(which by based on previous research, should produce robust results when varying between

0.6 and 0.75) the values of the other three calibration parameters is known, for each set.

The idea is that for every portfolio constructed and tested with our optimization model, its

market proxy needs to be defined, as well as the risk free ratio.

Afterwards, the portfolio is tested according to the optimization procedure described above,

whether it meets the Levy/Roll conditions, based on these calibration parameters, initially.

In order to consider that the model is robust for a certain stock selection, two general

conditions should be met:

a) The Matlab-generated parameters should be close enough (within 5% statistical

significance) to their sample counterparts, when the calibration parameters are

exactly or very close to the real ones, which are being dictated by the respective

market proxy.

b) The Matlab-generated parameters should not be close enough to their sample

counterparts, when the calibration parameters are - on purpose- given significantly

different than the real ones.

If any of these two conditions does not hold true for a certain portfolio, we consider that the

Levy/Roll procedure failed for the specific portfolio.

In order to verify the validity of these two conditions, except from running the model with

the true calibration parameters, a number of other, secondary optimizations tests is

conducted for each portfolio under examination, so that we will stress the sensitivity of the

model to each of these parameters. Usually, three out of the four remain the same, and a

different value is given to the fourth one, in order to document the behavior of the model.

For the new value, usually an increased one and a decreased one are given, always within

46

Specialized software provided by Dow Chemical GmbH.

63

the range of what is expected from this parameter. In addition, the values given to this

“test” parameter are not necessarily arbitrary; they can be the mean return and standard

deviation of another market proxy who could also be expected to perform well in the

particular set up.

Having explained the choice of the market proxies, we will continue with the definition of

the portfolios chosen as benchmarks for the whole Levy/Roll procedure.

In the next section, we will start the presentation of results by employing the first 100 stocks

from the S&P Global 1200 Index. The entire amount of data of each stock for the Index is

downloaded from Bloomberg for that purpose, from 2003 until 2013.

As mentioned earlier, each time that we run the Optimization Problem 1, we need to specify

certain parameters.

For each optimization test presented, the following parameters used will be provided, in

order to get a better grip of the importance of the different calibration parameters47 of the

model:

Parameter Value

Market μ 0.052

Market σ 0.104

Rf 0.0012

α 0.75

Table 4.7: The calibration parameters with example values. In this case, Rf is set as the 3 month T-

Bill rate for 2013, and a is set at 0.75.

In addition, we need to provide the model with all the stocks weights, for the specific stock

composition of the portfolio; throughout the literature, no standard way seems to exist for

calculating weights over a period of time, based on the market capitalization of the firms

(which obviously vary). In the previous literature tests, for the calculation of the stocks’

weights of a portfolio for a specific period of time, the market capitalization of the last

month of that time period was used. This is the main approach being followed in this Thesis

as well, although for the last tests for different number of stocks (from 5 until 150) the

weights of the stocks were calculated by using the average market capitalization over the

entire time period. In addition, as we will see in the next paragraph, we ran the tests for the

same portfolio, under the same market conditions, but with weights calculated from the

market capitalization of the firms in three different points in time.

47 "According to Standard & Poor's, trailing five-year periods for the S&P 500 averaged 5.2% with a

standard deviation of 10.4% between 1928 and 2007 (on average). "Investopedia. Stock Market Risk:

Wagging the Tails.

http://www.investopedia.com/articles/financial-theory/09/bell-curve-wag-tails.asp#axzz2LvXpyznj,

07.10.2012.

64

4.9.1 Portfolio: first 100 Stocks from S&P 1200 Global Index

The first benchmark portfolio is constituted by the first 100 stocks from the S&P 1200 Global

Index, based on the data drawn from Bloomberg. In almost all the previous research

attempts, the sample used has always been the top stocks of the market proxy (index),

sorted by market capitalization. But in a real-life portfolio, that is not always the case,

therefore for the first two sets, the random first 100 stocks were used (in an attempt to

avoid possible size-effects, as much as possible).

For the sake of being consistent with the previous discussion about the calculation of the

weights by the market capitalization of the firms, and before we get into the main analysis

we will make an exploratory parenthesis: In order to detect possible differences in the

model’s performance, because of calculating the weights by using the market capitalization

of the firms in different months, the first runs of the Optimization Problem 1 code for this

portfolio were conducted with three different values for the weights:

a) Market capitalization as of 2003

b) Market capitalization as of 2008

c) Market capitalization as of 2013

As we can see in the respective picture (Picture 15), the number of μ’s (returns) that are not

“close” enough to their sample counterparts vary from 3, in the first two tests, to 2, at the

last one. The set appears to follow the Levy/Roll theory, since the number of rejected null

hypothesis remained almost the same by using the first versus the last market capitalization

of the firms.

Having resolved this, we are moving towards the main structural robustness tests for the

Optimization Problem 1, with the set of the 100 first stocks from the S&P 1200 Global Index.

The table with all the different tests and their corresponding parameters will be presented

initially, to be followed by a 3D graph, for better overview of the results for the returns of all

individual stocks. Afterwards, for each cluster of tests (where different test results are

grouped together in order to make a specific point in the analysis) we will be presented

again with the table of the particular tests, the graph of the returns, and a short

commentary.

In the table below the calibration parameter sets that were used in the tests are presented,

based on the respective market proxies:

Index μ σ Time period

S&P 500 0.087 0.1807 2003-2013

W5000 0.079 0.19 2003-2013

S&P 1200 Global 0.084 0.047 2003-2013

S&P 500 0.052 0.104 1928-2007

Table 4.8: Market proxy values (return and standard deviation).

65

Equivalently, for the determination of what is considered to be risk free ratio (Rf or Rf in the

code) we employed the following benchmarks:

Time period 2003-2013 2003-2013 Rf as of 02.2013

Mean Rf 1.66% 1.57% 0.12%

Source Professor Kenneth French

(CRSP Database) Bloomberg 3m T-Bill

monthly Bloomberg

Table 4.10: Risk free ratio references.

For each of the following returns (μ) graphs presented, it holds that:

Vertical axis: absolute monthly return.

Horizontal axis: stock number, therefore 1 to 100 (the specific names are given in

the end of the Appendix).

After the presentation of all tests for this portfolio, we will continue with a series of tests for

other portfolios, which were constructed in a completely different way (details will be given

in the respective paragraphs). In the end of this chapter, an overall conclusion will be drawn

regarding the robustness of the entire Optimization Problem 1 for this portfolio.

Out of the hundreds of different optimization tests conducted for this and each portfolios

included in this Thesis, we have chosen to group and present the results together, of the

ones that reveal the effect of a certain parameter on the entire model.

A list of the tests conducted with the first 100 stocks of the S&P 1200 Global Index is

provided, with the short description of each one following in the following paragraph:

1. Three weights

2. S&P 500 Index MKT VL (market values)

3. Rf test

4. Alpha (α) test

5. Alternative market proxies

6. SP1200GL MKT VL (market values)

The “3 weights” test aims to reveal the effect on using different months of market

capitalization for the calculation of the portfolio weights, the “S&P 500 INDEX MKT VL” and

“SP1200GL MKT VL” are the two test clusters that show how the model performs when

using the respective Index’s values for the given period, the “Rf test” and the “Alpha (α) test”

demonstrate the behavior of the model when Rf or alpha varies, and the “Market proxies”

compares in one graph the output of the model when using as calibration parameters the

ones from the Indices S&P 500, Wilshire 5000 and S&P 1200 Global.

Below we give the explanation of columns fields for the next and each subsequent table

(only for the “new” parameters; for parameters such as the risk free ratio that have been

explained before, we do not repeat the definitions):

66

Market μ: average return of the market proxy.

Market σ: standard deviation of the market proxy.

Rejected μ's (#): the number of stocks in the portfolio, whose estimated returns are

not statistically identical with the sample returns (for the same stocks).

Rejected μ's (%): the percentage of stocks in the portfolio, whose estimated returns

are not statistically identical with the sample returns. It is calculated by dividing the

number of stocks with “rejected returns”, over the overall number of stocks in the

particular portfolio.

Rejected σ's (#): the number of stocks in the portfolio, whose estimated standard

deviations are not statistically identical with the sample standard deviations (for the

same stocks).

Rejected σ's (%): the percentage of stocks in the portfolio, whose estimated

standard deviations are not statistically identical with the sample standard

deviations. It is calculated by dividing the number of stocks with “rejected standard

deviations”, over the overall number of stocks in the particular portfolio.

Levy/Roll complied: it is set at “Y” (yes) if at least 95% of the stocks of the given

portfolio are not rejected by the statistical tests for the stock returns and standard

deviation; it is set at “N” (no) otherwise.

Finally, in the entire Levy/Roll results sector, when we say that a certain set-up for a

portfolio “works”, we mean that the estimated (by Matlab) vector of returns and standard

deviations is statistically “close enough” to the sample values, based on the statistical

comparison we have established.

67

Test name - S&P 1200 Global Index first 100 stocks market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

BL TOP 100 first market cap 0.052 0.104 0.0012 0.75 3 3% 0 0% Y

BL TOP 100 MIDDLE market cap 0.052 0.104 0.0012 0.75 3 3% 0 0% Y

BL TOP 100 orig altered market values 0.1 0.05 0.0012 0.75 4 4% 15 15% N

BL TOP 100 orig altered market values low bounded 0.1 0.05 0.0012 0.75 22 22% 45 45% N

BL TOP 100 orig altered only market m 0.104 0.07 0.0012 0.75 3 3% 1 1% Y

S&P 500 MKT VL BL TOP 100 orig 0.087 0.1807 0.015 0.75 12 12% 4 4% N

S&P 500 MKT VL BL TOP 100 orig s=0.1 0.087 0.1 0.015 0.75 13 13% 9 9% N

S&P 500 MKT VL BL TOP 100 orig m=0.05 0.05 0.1807 0.015 0.75 8 8% 1 1% N

S&P 500 MKT VL, Rf = 0.0012 shouldn’t work 0.087 0.1807 0.0012 0.75 1 1% 8 8% N

S&P 500 MKT VL, Rf = 0 shouldn’t work 0.087 0.1807 0 0.75 3 3% 8 8% N

S&P 500 MKT VL, a = 0.7 0.087 0.1807 0.0012 0.7 4 4% 8 8% N

S&P 500 MKT VL, a = 0.6 0.087 0.1807 0.0012 0.6 4 4% 8 8% N

BL TOP 100 original 0.052 0.104 0.0012 0.75 2 2% 0 0% Y

BL TOP 100 original 0.02 Rf 0.052 0.104 0.02 0.75 44 44% 8 8% N


Wilshire 5000 MKT VL 0.079 0.19 0.015 0.75 12 12% 1 1% N

Wilshire 5000 MKT VL a=0.6 0.079 0.19 0.015 0.6 12 12% 1 1% N

SP1200GL MKT VL BL TOP 100 original 0.084 0.047 0.015 0.6 22 22% 10 10% N

SP1200GL MKT VL Rf =0.001 0.084 0.047 0.001 0.6 8 8% 9 9% N

SP1200GL MKT VL Rf =0.001_s=0.1 0.084 0.1 0.001 0.6 7 7% 2 2% N

SP1200GL MKT VL Rf =0.001_s=0.1_a=0.75_m=0.05 0.05 0.1 0.001 0.75 3 3% 0 0% N

SP1200GL MKT VL s=0.1 0.084 0.1 0.015 0.6 29 29% 8 8% N

x0 (sample parameters)

Table 4.10: S&P 1200 Global Index first 100 stocks all tests.

68

Picture 15: S&P Global 1200 Index first 100 stocks all tests.

-1%

0%

1%

2%

3%

4%

5%

6%

7%

1 15 29 43 57 71 85 99

Return

Stock #

BL TOP 100 first market cap BL TOP 100 MIDDLE market cap BL TOP 100 orig altered market valuesBL TOP 100 orig altered market values low bounded BL TOP 100 orig altered only market m S&P 500 MKT VL BL TOP 100 origS&P 500 MKT VL BL TOP 100 orig s=0.1 S&P 500 MKT VL BL TOP 100 orig m=0.05 S&P 500 MKT VL, Rf = 0.0012S&P 500 MKT VL, Rf = 0 S&P 500 MKT VL, a = 0.7 S&P 500 MKT VL, a = 0.6BL TOP 100 original BL TOP 100 original 0.02 RF BL TOP 100 original 0.015RFWilshire 5000 MKT VL Wilshire 5000 MKT VL a=0.6 SP1200GL MKT VL BL TOP 100 originalSP1200GL MKT VL Rf=0.001 SP1200GL MKT VL Rf=0.001_s=0.1 SP1200GL MKT VL Rf=0.001_s=0.1_a=0.75_m=0.05SP1200GL MKT VL s=0.1 x0 (sample parameters)

69

Three weights market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

BL TOP 100 first market cap 0.052 0.104 0.0012 0.75 3 3% 0 0% Y

BL TOP 100 MIDDLE market cap 0.052 0.104 0.0012 0.75 3 3% 0 0% Y

BL TOP 100 original 0.052 0.104 0.0012 0.75 2 2% 0 0% Y


Table 4.11: Testing the model for weights based on the market capitalization of the firms in different points in time; small variations.

-1.0%

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0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

1 15 29 43 57 71 85 99

Return

Stock #

BL TOP 100 first market cap BL TOP 100 MIDDLE market cap BL TOP 100 original x0 (sample parameters)

Picture 16: Testing the performance of the model for weights based on the market capitalization of the firms in different points in time.

70

S&P 500 MKT VL market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied


S&P 500 MKT VL BL TOP 100 orig s=0.1 0.087 0.1 0.015 0.75 13 13% 9 9% N

S&P 500 MKT VL BL TOP 100 orig m=0.05 0.05 0.1807 0.015 0.75 8 8% 1 1% N

S&P 500 MKT VL, Rf = 0.0012 0.087 0.1807 0.0012 0.75 1 1% 8 8% N

S&P 500 MKT VL, Rf = 0 0.087 0.1807 0 0.75 3 3% 8 8% N

S&P 500 MKT VL, a = 0.7 0.087 0.1807 0.0012 0.7 4 4% 8 8% N

S&P 500 MKT VL, a = 0.6 0.087 0.1807 0.0012 0.6 4 4% 8 8% N


Table 4.12: Full cluster of sets for market values based on the S&P 1200 Global Index.

Picture 17: Testing a number of parameter combinations: none of them works, including the ones with correct market values.

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2%

3%

4%

1 15 29 43 57 71 85 99

Return

Stock #S&P 500 MKT VL BL TOP 100 orig shouldn’t work S&P 500 MKT VL BL TOP 100 orig s=0.1 shouldn’t work S&P 500 MKT VL BL TOP 100 orig m=0.05

S&P 500 MKT VL, Rf = 0.0012 shouldn’t work S&P 500 MKT VL, Rf = 0 shouldn’t work S&P 500 MKT VL, a = 0.7

S&P 500 MKT VL, a = 0.6 x0 (sample parameters)

71

Rf test market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

BL TOP 100 original 0.0012 Rf 0.052 0.104 0.0012 0.75 2 2% 0 0% Y




Table 4.13: Testing the model for different values of Rf.

Picture 18: Testing the Rf: the model should not work for 0.12%, but for 1.5%, which is the historical average. The results indicate abnormal behavior.

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-0,5%

0,0%

0,5%

1,0%

1,5%

2,0%

2,5%

3,0%

3,5%

4,0%

1 15 29 43 57 71 85 99

Return

Stock #

BL TOP 100 original 0.0012 Rf BL TOP 100 original 0.02 Rf BL TOP 100 original 0.015Rf x0 (sample parameters)

72

Alpha (α) test market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

S&P 500 MKT VL, a = 0.75 0.087 0.1807 0.0012 0.75 1 1% 8 8% N

S&P 500 MKT VL, a = 0.7 0.087 0.1807 0.0012 0.7 4 4% 8 8% N

S&P 500 MKT VL, a = 0.6 0.087 0.1807 0.0012 0.6 4 4% 8 8% N


Table 4.14: Testing the model for different values of α.

Picture 19: Alpha’s effect on the results: insignificant.

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0,5%

1,0%

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1 15 29 43 57 71 85 99

Return

Stock #

S&P 500 MKT VL, a = 0.75 S&P 500 MKT VL, a = 0.7 S&P 500 MKT VL, a = 0.6 x0 (sample parameters)

73

Alternative market proxies market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied


Wilshire 5000 MKT VL 0.079 0.19 0.015 0.75 12 12% 1 1% N

Wilshire 5000 MKT VL a=0.6 0.079 0.19 0.015 0.6 12 12% 1 1% N

S&P 1200GL MKT VL BL TOP 100 original 0.084 0.047 0.015 0.6 22 22% 10 10% N


Table 4.15: Testing the model output when using as calibration parameters the ones from the Indices S&P 500, Wilshire 5000 and S&P 1200 Global.

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1%

2%

3%

4%

5%

6%

7%

1 15 29 43 57 71 85 99

Return

Stock #S&P 500 MKT VL BL TOP 100 orig Wilshire 5000 MKT VL Wilshire 5000 MKT VL a=0.6 S&P 1200GL MKT VL BL TOP 100 original x0 (sample parameters)

Picture 20: The model was also tested with the parameters (return and standard deviation) of the Wilshire 5000, as market parameters. As we can see, it failed the test.

74

Table 4.16: Finally, the model was tested for the actual market parameters, for this market proxy: the returns and standard deviation of S&P 1200 Global Index.

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2%

3%

4%

5%

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7%

1 15 29 43 57 71 85 99

Return

Stock #SP1200GL MKT VL BL TOP 100 original SP1200GL MKT VL Rf=0.001 SP1200GL MKT VL Rf=0.001_s=0.1

SP1200GL MKT VL Rf=0.001_s=0.1_a=0.75_m=0.05 SP1200GL MKT VL s=0.1 x0 (sample parameters)

Picture 21: The model was unable to produce Levy/Roll complied results, with this market proxy.

SP1200GL MKT VL market

μ market

σ Rf α Rejected

μ's (#) Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

SP1200GL MKT VL BL TOP 100 original 0.084 0.047 0.015 0.6 22 22% 10 10% N

SP1200GL MKT VL Rf =0.001 0.084 0.047 0.001 0.6 8 8% 9 9% N

SP1200GL MKT VL Rf =0.001_s=0.1 0.084 0.1 0.001 0.6 7 7% 2 2% N

SP1200GL MKT VL Rf =0.001_s=0.1_a=0.75_m=0.05 0.05 0.1 0.001 0.75 3 3% 0 0% Y

SP1200GL MKT VL s=0.1 0.084 0.1 0.015 0.6 29 29% 8 8% N


75

After having conducted these tests, two issues are under the spotlight: first, when the model

indeed performs well (less that 5% statistically significant), should it actually work? And

second, when the model does not perform well (more that 5% statistically significant, and

therefore rejected), should that also be the case?

The problem is that the whole model is extremely sensitive to a number of input factors, for

which most of the times there is not just one “right” value. For the Optimization Problem 1,

these are the market returns and standard deviation, the risk free ratio and the choice of

alpha. For the Optimization Problem 2, we have in addition the ex-ante return and standard

deviation.

If the results of a portfolio works better (less statistically different estimated parameters)

with market parameters other than the actual ones, or at least the most descriptive given

the specific portfolio, that will consist evidence that the Optimization Problem 1 or 2 that

was used for the particular test series, is a mere mathematical exercise that can work for

certain sets of numbers, but it does not reveal any idiosyncratic characteristics of the whole

model.

In the table below we summarize the above tests, and whether the model actually managed

to depict actual financial values or not. Once again, by “should work” we indicate whether

the model was expected to work in line with the Levy/Roll theory, given the calibrated

parameters that it was fed with.

Test set Works Should work Levy/Roll complied

3 weights Y (not applicable) -

S&P 500 MKT VL

Y (when market values are correct)

N N

Rf test Y (when market

values are incorrect) N for correct market values,

Y for the wrong ones N

Alpha test N N (a makes difference) N

Market N/Y (not applicable) -

SP1200GL MKT

N Y (market proxy consistent) N

Table 4.17: Overview of tests for Optimization Problem 1, first 100 stocks from the S&P 1200 Global

Index.

To be noted, throughout the whole procedure we do not present the results for the

standard deviations explicitly, because they are not that sensitive in the various parameters:

their behavior is more or less in line with the behavior of the returns between the different

tests. In fact, one could state that if the returns are in order (Levy-Roll complied), then it is

highly unlikely that the standard deviations will not be; their mathematical restrictions are

very strict. Therefore, even when the statistical tests for the returns fail massively, the

statistical tests for the respective standard deviations (same portfolio-calibration

parameters) are usually proper. Another issue that was revealed is that the standard

deviations of the first stocks were the ones with the largest standard deviation amongst the

different optimization tests. In order to illustrate that, their values for a random test are

briefly below:

76

Set title market

μ market

σ Rf α

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

SP1200GL MKT VL BL TOP 100 orig. 0.084 0.047 0.015 0.6 10 10% N

SP1200GL MKT VL Rf =0.001 0.084 0.047 0.001 0.6 9 9% N

SP1200GL MKT VL Rf =0.001_s=0.1 0.084 0.1 0.001 0.6 2 2% N

SP1200GL MKT VL s=0.1 0.084 0.1 0.015 0.6 8 8% N


Table 4.18: Demonstration of standard deviations for the SP1200GL MKT VL tests.

As we can see in the radar graph below, the estimated standard deviations follow a similar

pattern throughout the different implementations of the table. An issue that is addressed in

the last chapter, regarding ideas to continue this work, is that it seems like the same stocks

always fail the standard deviation test. That might be an inspiration for investigating the

behavior of the returns of these stocks, since they seem to be not that much correlated with

the respective market proxy in which they belong.

Graph

Picture 22: Standard deviations of the stocks for the tests SP1200GL MKT VL.

Overall, we can see that the model failed almost all tests, either but not working when it

should have, or by working when it should not have. This serves as a preliminary indication

-10%

-5%

0%

5%

10%

15%

1 15 29 43 57 71 85 99

Standard deviation

Stock Number

BL TOP 100 original with SP1200GL MKT VL Rf=0.001 Rf=0.001_s=0.1 s=0.1 X0

77

that the model does not work at a whole, but without having 100% concrete evidence yet. It

has been brought to our attention that in all the previous research attempts, only U.S. stocks

were used. As we know, S&P 1200 Global Index (Ticker: SPGLOB Index) is constituted by

approximately 50% non-U.S. stocks. Therefore, before we come into discussing what would

intuitively mean, in case that the model works for U.S. or non-U.S. stocks, we will continue

with the second large cluster of tests, for an equivalent portfolio but with constituents

drawn only by U.S. stocks.

For that purpose, the results of the first 100 stocks (alphabetically) from the S&P 500 Index

(Ticker: SPX Index) will be presented in the next paragraph. The names of these stocks are

given in tables that can be found in the Appendix.

4.9.2 Portfolio: first 100 stocks from the S&P 500 Index

As mentioned in the previous paragraph, it makes sense to perform a similar procedure for a

set which is formed only by U.S. stocks (since that has always been the case in previous

research attempts). For this reason, we created a portfolio of also 100 stocks but this time

from the first 100 stocks of S&P 500 Index (similarly to the previous set of 100 stocks, but

not from the S&P 1200 Global Index).

The full list of companies for both portfolios can be found in the Appendix. All data have

been acquired from Bloomberg; the first 100 stocks for which full data were available over

the entire 2003-2013 period were used.

To be noted, the effect of alpha was not tested that thoroughly in this portfolio; it was

exhausted in the previous set (since it is more of a model structural issue, the results of the

tests regarding alpha for this set are deliberately not reported: they are consistent with the

previous tests’ results). Varying the value of alpha between 0.6 and 0.75 affects the value of

the objective function, but not the performance of the entire model regarding whether the

estimated values are mean/variance efficient.

The tests conducted with the first 100 stocks of the S&P 500 Index provided are the same in

methodology as the ones conducted with the previous portfolio. A commentary of each one

is presented along with the test results.

78

S&P 500 first 100 stocks market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

TOP100FROM421 (a=0.75) 0.052 0.104 0.0012 0.75 16 16% 0 0% N

TOP100FROM421_a=0.6 0.052 0.104 0.0012 0.6 16 16% 0 0% N

S&P 500 MKT VL mrktV,RfBL 0.052 0.104 0.0012 0.6 17 17% 0 0% N

S&P 500 MKT VL m=0.1 0.1 0.18 0.015 0.6 3 3% 0 0% Y

S&P 500 MKT VL m=0.2 0.2 0.18 0.015 0.6 8 8% 0 0% N

S&P 500 MKT VL m=0.05 0.05 0.18 0.015 0.6 3 3% 0 0% Y

S&P 500 MKT VL m=0.07 0.07 0.18 0.015 0.6 3 3% 0 0% Y

S&P 500 MKT VL m=0.15 0.15 0.18 0.015 0.6 4 4% 0 0% Y

S&P 500 MKT VL m=0.97 0.97 0.18 0.015 0.6 53 53% 14 14% N

S&P 500 MKT VL Rf =0.001 0.087 0.18 0.001 0.6 24 24% 0 0% N

S&P 500 MKT VL Rf =0.02 0.087 0.18 0.02 0.6 21 21% 0 0% N

S&P 500 MKT VL s=0.1 0.087 0.1 0.015 0.6 12 12% 1 1% N

S&P 500 MKT VL s=0.25 0.087 0.25 0.015 0.6 3 3% 0 0% Y


Table 4.19: All the tests that were conducted with the first 100 stocks from the S&P 500.

79

Picture 23: S&P 500 Index first 100 stocks all tests.

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1 15 29 43 57 71 85 99

Returns

Stock #

TOP100FROM421 (a=0.75) TOP100FROM421_a=0.6 S&P 500 MKT VL mrktV,RfBL S&P 500 MKT VL m=0.1 S&P 500 MKT VL m=0.2

S&P 500 MKT VL m=0.05 S&P 500 MKT VL m=0.07 S&P 500 MKT VL m=0.15 S&P 500 MKT VL m=0.97 S&P 500 MKT VL RF=0.001

S&P 500 MKT VL RF=0.02 S&P 500 MKT VL s=0.1 S&P 500 MKT VL s=0.25 x0

80

S&P 500 Index 1920-2000 market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

TOP 100 FROM 421 a=0.75 0.052 0.104 0.0012 0.75 16 16% 0 0% N

TOP 100 FROM 421 a=0.6 0.052 0.104 0.0012 0.6 16 16% 0 0% N


Table 4.20: Results not compatible with Levy/Roll. In addition, no effect from alpha.

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1 15 29 43 57 71 85 99

Return

Stock #TOP100FROM421 a=0.75 TOP100FROM421 a=0.6 x0

Picture 24: Optimization Problem 1 tested for the set of stocks with wrong market values (overall S&P 500 Index average for 90 years).

81

S&P 500 MKT VL test # 1 market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

S&P 500 MKT VL mrktV,RfBL 0.052 0.104 0.0012 0.6 17 17% 0 0% N

S&P 500 MKT VL m=0.1 0.1 0.18 0.015 0.6 3 3% 0 0% Y

S&P 500 MKT VL m=0.2 0.2 0.18 0.015 0.6 8 8% 0 0% N

S&P 500 MKT VL m=0.05 0.05 0.18 0.015 0.6 3 3% 0 0% Y

S&P 500 MKT VL m=0.07 0.07 0.18 0.015 0.6 3 3% 0 0% Y


Table 4.21: Small model sensitivity to market μ. (Actual values for S&P 500 Index are 8.7% for mean returns and 18.07% standard deviation. The average risk free ratio

at 1.5%.)

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1%

2%

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4%

5%

1 15 29 43 57 71 85 99

Return

Stock #

S&P 500 MKT VL mrktV,RfBL S&P 500 MKT VL m=0.1 S&P 500 MKT VL m=0.2 S&P 500 MKT VL m=0.05 S&P 500 MKT VL m=0.07 x0

Picture 25: The model is fully in line with the Levy/ Roll theory, for values close to the correct market ones.

82

S&P 500 MKT VL test # 2 PASSED market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

S&P 500 MKT VL m=0.07 0.07 0.18 0.015 0.6 3 3% 0 0% Y

S&P 500 MKT VL m=0.15 0.15 0.18 0.015 0.6 4 4% 0 0% Y

S&P 500 MKT VL m=0.97 0.97 0.18 0.015 0.6 53 53% 14 14% N


Table 4.22: We notice a small but sufficient sensitivity to market μ (the middle test perhaps should have been rejected, but on the other hand, 0.15 can be considered

not that “far” from 0.087, which is the correct value).

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3%

4%

5%

6%

1 15 29 43 57 71 85 99

Return

Stock #S&P 500 MKT VL m=0.07 S&P 500 MKT VL m=0.15 S&P 500 MKT VL m=0.97 x0

Picture 26: Model tested for deviations of market μ.

83

Rf and σ test market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

S&P 500 MKT VL m=0.07 0.07 0.18 0.015 0.6 3 3% 0 0% Y

S&P 500 MKT VL Rf =0.001 0.087 0.18 0.001 0.6 24 24% 0 0% N

S&P 500 MKT VL Rf =0.02 0.087 0.18 0.02 0.6 21 21% 0 0% N

S&P 500 MKT VL s=0.1 0.087 0.1 0.015 0.6 12 12% 1 1% N

S&P 500 MKT VL s=0.25 0.087 0.25 0.015 0.6 3 3% 0 0% Y


Table 4.23: The model exhibits sufficiently sensitivity, where tests with wrong value fail. Rf = 0.001: model too sensitive to the value of Rf .

-1%

0%

1%

2%

3%

4%

5%

1 15 29 43 57 71 85 99

Return

Stock #S&P 500 MKT VL m=0.07 S&P 500 MKT VL RF=0.001 S&P 500 MKT VL RF=0.02 S&P 500 MKT VL s=0.1 S&P 500 MKT VL s=0.25 x0

Picture 27: Testing variations in the risk free ratio, as well as to the market standard deviation (by keeping all others constant). The model works only for the correct Rf.

84

Test Works Should work Levy/Roll complied

S&P 500 1920-2000 MKT VL N N Y

Correct market values # 1 Y Y Y

Correct market values #2 N N Y

Rf and σ test Rf =0.001 N N Y

Rf and σ test b Rf =0.002 N N Y

Rf and σ test b σ=0.1 N N Y

Rf and σ test b σ=0.25 Y Y Y

Table 4.24: Overview of tests for Optimization Problem 1, first 100 stocks from the S&P 500

Index.

Finally, just like with the previous set of 100 stocks (from the S&P 1200 Index), we aggregate

all the results in the above table. Surprisingly, by following the exact same methodology as

before, but only for U.S. stocks this time, all the results comply with the Levy/Roll theory.

Therefore, the final conclusion of the cluster of tests on the second portfolio in our analysis

proved to be 100% supportive to the Levy/Roll theory, and the overall approach.

In order to further investigate the validity of the proposed approach, we will continue by

testing it on the 25 Fama French portfolios, the construction of which was described in

previous paragraphs.

4.9.3 The 25 Fama French NYSE Portfolios

In this paragraph, the performance of the Levy/Roll approach on the 25 Fama French NYSE

portfolios will be presented.

The number of firms in each Fama French portfolio is presented in the next table:

Book to market

Quantile low 2 3 4 high

small 15 16 19 14 63

Size 2 15 20 30 29 32

3 10 19 40 20 20

4 25 14 19 19 14

big 36 24 26 15 10

Table 4.25: Stocks constituting the 25 Fama French NYSE portfolios. Total number of stocks: 564.

For the next table, we have that:

Time period: 1.1991-12.2008.

BtMx, x=1, 2, 3, 4, and 5: Book-to-Market quintile; low (1) to high (2).

85

Second column x=1, 2, 3, 4, and 5: market capitalization quintile; small (1) to large

(2).

The entire Levy/Roll analysis’ results are given in the table:

Book to Market

Size # stocks in

the Portfolio Rejected


Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

NYSE BtM1 small 15 1 6.67% 0 0.00% N

NYSE BtM1 2 15 0 0.00% 0 0.00% Y

NYSE BtM1 3 10 0 0.00% 0 0.00% Y

NYSE BtM1 4 25 0 0.00% 0 0.00% Y

NYSE BtM1 large 36 0 0.00% 0 0.00% Y

NYSE BtM2 small 16 0 0.00% 0 0.00% Y

NYSE BtM2 2 20 0 0.00% 0 0.00% Y

NYSE BtM2 3 19 1 5.26% 0 0.00% N

NYSE BtM2 4 14 0 0.00% 0 0.00% Y

NYSE BtM2 large 24 2 8.33% 0 0.00% N

NYSE BtM3 small 19 0 0.00% 0 0.00% Y

NYSE BtM3 2 30 0 0.00% 0 0.00% Y

NYSE BtM3 3 40 1 2.50% 0 0.00% Y

NYSE BtM3 4 19 0 0.00% 0 0.00% Y

NYSE BtM3 large 26 1 3.85% 0 0.00% Y

NYSE BtM4 small 14 0 0.00% 0 0.00% Y

NYSE BtM4 2 29 0 0.00% 0 0.00% Y

NYSE BtM4 3 20 1 5.00% 0 0.00% Y

NYSE BtM4 4 19 0 0.00% 0 0.00% Y

NYSE BtM4 large 15 0 0.00% 0 0.00% Y

NYSE BtM5 small 63 3 4.76% 0 0.00% Y

NYSE BtM5 2 32 1 3.13% 0 0.00% Y

NYSE BtM5 3 20 0 0.00% 0 0.00% Y

NYSE BtM5 4 14 2 14.29% 0 0.00% N

NYSE BtM5 large 10 0 0.00% 1 10.00% N

Table 4.26: Performance of the 25 Fama French NYSE portfolios.

Overall, 20 out of the 25 portfolios’ estimated parameters are fully in line with the Levy/Roll

method, while from the five that are not, only two have rejected values (either in returns or

standard deviations) more than 10%. The last column is set at “N” (No) if the particular

portfolio’s returns or standard deviations tests are rejected at more than a 5% significance

level. Overall, we consider that the behavior of these 25 portfolios is consistent with the

approach of Levy and Roll.

86

4.9.4 The 25 Fama French NASDAQ/NYSE/AMEX Portfolios

Now the results of the Levy/Roll approach on the 25 Fama French NYSE/AMEX/NASDAQ

(entire U.S. Equity Market) portfolios will be presented.

The number of firms in each Fama French portfolio is presented in the next table:

Book to market

Quantile low 2 3 4 high

small 35 46 60 89 118

Size 2 22 36 55 45 25

3 17 36 39 31 20

4 48 41 27 28 19

big 74 60 35 24 4

Table 4.27: Stocks constituting the 25 Fama French NYSE/AMEX/NASDAQ portfolios. Total number

of stocks: 1034.

For the next table, we have that:

Time period: 1.1991-12.2008.

BtMx, x=1, 2, 3, 4, and 5: Book-to-Market quintile; low (1) to high (2).

Second column x=1, 2, 3, 4, and 5: market capitalization quintile; small (1) to large

(2).

The entire Levy/Roll analysis’ results are given in the table:

87

Book to Market Size

# stocks in the Portfolio

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

U.S. BtM1 small 35 0 0.00% 0 0.00% Y

U.S. BtM1 2 22 0 0.00% 0 0.00% Y

U.S. BtM1 3 17 1 5.88% 1 5.88% N

U.S. BtM1 4 48 0 0.00% 0 0.00% Y

U.S. BtM1 large 74 0 0.00% 0 0.00% Y

U.S. BtM2 small 47 2 4.26% 0 0.00% Y

U.S. BtM2 2 36 0 0.00% 0 0.00% Y

U.S. BtM2 3 36 1 2.78% 0 0.00% Y

U.S. BtM2 4 41 0 0.00% 0 0.00% Y

U.S. BtM2 large 60 3 5.00% 0 0.00% Y

U.S. BtM3 small 60 0 0.00% 0 0.00% Y

U.S. BtM3 2 55 4 7.27% 0 0.00% N

U.S. BtM3 3 39 1 2.56% 0 0.00% Y

U.S. BtM3 4 27 0 0.00% 0 0.00% Y

U.S. BtM3 large 35 1 2.86% 0 0.00% Y

U.S. BtM4 small 89 0 0.00% 0 0.00% Y

U.S. BtM4 2 45 0 0.00% 0 0.00% Y

U.S. BtM4 3 31 0 0.00% 0 0.00% Y

U.S. BtM4 4 28 0 0.00% 0 0.00% Y

U.S. BtM4 large 24 1 4.17% 0 0.00% Y

U.S. BtM5 small 118 2 1.69% 0 0.00% Y

U.S. BtM5 2 25 3 12.00% 0 0.00% N

U.S. BtM5 3 20 1 5.00% 0 0.00% Y

U.S. BtM5 4 19 1 5.26% 0 0.00% N

U.S. BtM5 large 4 0 0.00% 2 50.00% N

Table 4.28: 26 Performance of the 25 Fama French NYSE/AMEX/NASDAQ portfolios.

Even though the NYSE stocks represent only approximately half of the U.S. Equity Market,

and that was the case in the portfolio used after the data cleaning (after the downloading of

the raw data from Bloomberg) as well, the results of the same procedure onto the entire

U.S. market are almost identical with the ones for the NYSE stocks only.

The results produced for the 25 Fama French portfolios for the purposes of this Thesis are

slightly different than the ones in previous papers. That could very well be because of the

data preparation process, where it is impossible to guarantee that every researcher uses the

exact same set of stocks (after the “cleaning” of raw data).

As a conclusion of the 25 Fama French portfolios tests with our code, it could be stated that

their performance is in line with the Levy/Roll theory- a significantly strong result.

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4.9.5 Portfolios of variable size from the top Market Capitalization Firms of S&P

500 Index

Finally, we chose to present in this chapter the results of the top market capitalization (as of

February 2013) stocks from the S&P 500 Index, with a variable number of stocks (5, 10, 20,

50, 100, 146 and 200), in an attempt to manifest the behavior of the Levy/Roll model onto

portfolios constituted by top market capitalization stocks. This practice (constructing

portfolios by the top market capitalization firms of a market proxy) holds a dominant

position in the literature, like for example in the original paper by Levy and Roll (there, the

authors did all their tests by using the first 100 top capitalization stocks).

Portfolio Stocks

# Rejected


Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

top 5 S&P 500 5 0 0.00% 0 0.00% Y

top 10 S&P 500 10 0 0.00% 0 0.00% Y

top 20 S&P 500 20 2 10.00% 0 0.00% N

top 50 S&P 500 50 3 6.00% 0 0.00% N

top 146 S&P 500 146 4 2.74% 0 0.00% Y

top 200 S&P 500 200 5 2.50% 0 0.00% Y

Table 4.29: Performance summary for the variable size top market capitalization portfolios.

The results of these tests are mixed. But there is an important bias: the very top market

capitalization stocks have completely different attributes from the rest of the stocks, even in

a portfolio formulated by the largest market capitalization firms. Therefore, every time the

number of stocks changes, so does the distribution (although not that much), but the same

huge market capitalization stocks remain in the portfolio. That results in the phenomenon

that, after the portfolio size of 20, it is the exact same stocks that fail to meet the Levy/Roll

statistical tests. This is recognized as a possible idea for further research, and is discussed in

more detail in the final chapter.

4.10 Conclusions for the Optimization Problem 1

In this chapter, our goal was to determine whether the model performs as it is supposed to,

given the inputs. That means that there were tests when the inputs were correct, therefore

the model should work, and there are tests were the inputs were partially or completely

wrong, in which case the model should fail.

As a conclusion for this first major cluster of tests, the model did not perform in line with the

theory at the portfolio constructed by the S&P 1200 Index, but it performed perfectly well

when tested on the portfolio constructed by the S&P 500 Index. With regard to the 25 Fama

French portfolios, the results inclined towards the validity of the model, while the tests on

the various size portfolios were rather ambiguous.

89

Overall, it seems that the model could possibly reveal some financial attributes of a certain

portfolio, but not under all conditions. Given that the results are not conclusive, we will test

the portfolio that performed fully in line with the Levy/Roll approach under the much more

constrained Optimization Problem 2 in the next Chapter.

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Chapter 5

5 Optimization Problem 2

The analysis so far, of Optimization Problem 1, has provided evidence that the model can

work under certain circumstances, but the results are not robust enough in order for us to

be absolutely positive towards whether it worked due to the fact that it is mathematically

correct, or because it reflects some meaningful, intuitive financial attributes.

In order to strengthen our methodology, and by following the original Levy and Roll

approach, we apply two extra constraints to the Optimization Problem 1, which is now

named Optimization Problem 2. The idea is that the estimated parameters should, on top of

the mean/variance constraints of Optimization Problem 1, satisfy two additional constraints:

the returns and standard deviation of the estimated portfolio should be ex ante identical to

the sample portfolio’s ones (or to another, also prespecified pair of returns and standard

deviation values.

5.1 Mathematical Problem Description

The formulas for Optimization Problem 2 are repeated here, for the kind convenience of the

reader.


Subject to:


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c) Nonlinear constraint for estimated portfolio standard deviation (σ):

The first extra constraint (b) corresponds to a single equation (for the portfolio return, μ). It

states that the overall average (value-weighted) return, calculated from the estimated

parameters for our portfolio, should be equal to the actual average (value-weighted) return

of the sample. This is a very strong condition, given that the constraint violation tolerance is

just 10-6. This equation is linear, whereas already described, xm is the value-weighted vector

of the returns, μ the estimated returns and μ 0 the actual sample portfolio return.

The second extra constraint (c) is constituted by a set of nonlinear equations, similarly with

the first constraint (and only one, for Optimization Problem 1). It states that the estimated

portfolio’s standard deviation (σ) is equal to the sample portfolio’s one (σ0). More

specifically, by assuming that the correlations between the stocks should remain the same,

only the standard deviations are allowed to change, and that can be seen in the respective

mathematical formula above.

The σ0 and μ0 variables are being calculated as follows, from the sample stocks:

σ0 = sqrt(xm*sigma*xm'); μ0 = [xm,zeros(1,100)]*x0';

The variable xm (xm) corresponds to the weights derived from the market capitalization of the firms, sigma (σ) to the shrunk covariance matrix of the sample portfolio, and the variable x0 (x0) to the vector comprised by the N returns and N standard deviations, as it has been explained before. The variable x0 is multiplied with the vector [xm, zeros (1,100)], in order to get μ 0 because as we have stated in the Matlab variables paragraph, only the first half of the elements of every x0 corresponds to stocks’ returns – and these are the ones we need here; the second half elements of x0 correspond to the respective standard deviations (which are multiplied by 0, here). For our particular set of stocks, by using the above formulas, we have that

σ0 = sqrt(xm*sigma*xm'); σ0 = 0.0137

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μ0 = [xm,zeros(1,100)]*x0'; μ0 = 0.0044

The mean return might appear particularly low (0.44%), but we should co-calculate the fact that this portfolio also bears very little risk (1.37%) and it is basically a random portfolio. Therefore, no techniques have been applied in terms of stock picking that would possibly maximize returns and/or minimize the standard deviation; the goal was to have a portfolio as random as possible. In the following paragraphs of this chapter, we will always use the correct calibration parameters, that were proven to work (with this specific portfolio) in the previous chapter. These would be the following values:

market μ market σ Rf α

0,087 0,18 0,015 0,6

Table 5.1: Calibration parameters and their values for this portfolio.

According to the theory and the empirical data provided by Levy and Roll, for the above values of parameters, the Optimization Problem 2 procedure should provide us with a range of σ0 and μ0 values, in the two-dimensional σ0 - μ0 space, for which the estimated parameters would be close enough to their sample counterparts. Since the actual values of σ0 and μ0 are equal to 1.37% and 0.44%, respectively, we expect the Optimization Problem 2 to work perfectly well for these exact values, as well as for small deviations of either one or both of them, around the original values. But in contradiction of the expected outcome, as we will see in the following paragraphs, the Optimization Problem 2 as provided with all its constraints, failed systematically to provide with proper convergence vectors (returns and standard deviations) in at least twenty different portfolios tested, whereas the Optimization Problem 1 worked for all of them. Based on these unexpected results, we decided to break down the different sets of extra constraints of the Optimization Problem 2 (one set for the returns and another for the standard deviations) and examine the behavior of the model in each of these individual subproblems. In the following paragraph, the specifics of these subproblems are presented, by presenting their mathematical formulas, as well the details of adjusting the Matlab implementation to each case. It is of high importance to state the exact specifications of the Matlab code used, when calling the fmincon function, since, as we will see, there are a lot of different combinations and misconceptions that can alter the final result. The detailed construction of the subproblems could also serve as a reference for future research.

5.2 Methodology

As stated in the previous paragraph, in order to dive deeper into the analysis, we impose the

new sets of constraints to the problem one by one; the reason for this careful approach was

that based on existing research, the model gets extremely sensitive to minor changes of the

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μ0 and σ0 extra constraints, once all of them are in place. Therefore, we continue with the

construction of the Optimization subproblems ceq2, ceq2b, ceq and linear equality, ceq2

and linear equality, ceq3 and finally, ceq2 with linear inequality for the m0 constraint. Below

we explain the formation and the reasoning behind each different combination of the

constraints.

In the fmincon function of Matlab there is a possibility, structural, to define a linear equation

either as linear or nonlinear. More specifically, from the definition48 of the function:

c(x) ≤ 0

ceq(x) = 0

A·x ≤ 0

Aeq·x = beq

lb ≤ x ≤ ub

We have the possibility of only two sets (it can be more than just one equation in each set)

equations: the nonlinear set, which is defined by the ceq(x) =0 equation, and the linear one,

which is the Aeq·x = beq. Equivalently, we can impose nonlinear and linear inequality

constraints, by using c(x) ≤ 0 and A·x ≤ b. The linear (in)equalities are implemented

differently from the nonlinear (in)equalities, when calling the fmincon function; the

parameters of the linear ones are given as input parameters when calling the fmincon, while

the nonlinear ones are implemented in the ceq.m function. In this Thesis, the fmincon is

always called with the formation:49

[x,fval,exitflag,output] = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

The parameters A and b correspond to the linear inequality, while Aeq and beq to the linear

equality (their positions as inputs are fixed by the function). The most strict constraint,

proved to be (although it was quite obvious) the linear equality for the μ0. Since fmincon

treats the linear equations in a different way than the nonlinear ones, internally when

evaluating the algorithm, it was considered that this constraint should be tested with both

implementations.

The different optimization sub problems implemented are being described as follows.

Minimize the objective function D:

48

Mathworks. Fmincon. http://www.mathworks.com/help/optim/ug/fmincon.html, 20.08.2012. 49

Mathworks. Fmincon. http://www.mathworks.com/help/optim/ug/fmincon.html, 20.08.2012.

94

Under the sets of constraints presented in the following paragraphs.

5.2.1 Constraints Function funCeq2

Calling the fmincon function by: [x,fval,exitflag,output] = fmincon(@funD,x0,[],[],[],[],[],[],@funCeq2,options)

Linear constraints: none

ceq (nonlinear constraints) formulation used: % Final expression for nonlinear restriction with LW

% S: estimated standard deviations matrix

% RSIGMA: correlation matrix

% xma: weights of the stocks in the portfolio

% q: constant of proportionality

% mr2n: vector ceqshrunkinv = S*RSIGMA*S*xma - q*mr2n;

% Final expression for the σ squared nonlinear restriction of OPT 2:

% prsq: variance

ceq2= xm*S*RSIGMA*S*xma-prsq;

% Final nonlinear constraints used:

ceq = [ceqshrunkinv; ceq2];


b) Nonlinear constraint for estimated portfolio standard deviation (σ):

This problem corresponds to the Optimization Problem 2 of Levy and Roll, but without the

linear equation for the estimated portfolio returns (which, as we will see in the following

pages, is the strongest constraint).

95

5.2.2 Constraints Function funCeq2b

Calling the fmincon function by: [x,fval,exitflag,output] = fmincon(@funD,x0,[],[],[],[],[],[],@funCeq2b,options)


ceq (nonlinear constraints) formulation used: % Final expression for nonlinear restriction with LW ceqshrunkinv = S*RSIGMA*S*xma - q*mr2n;

% Final expression for the μ linear restriction of OPT 2: % m: estimated returns vector

ceq3 = xm*m'-0.0137;





The set of nonlinear equations from Optimization Problem 1, as well as the linear equation

for the estimated portfolio returns, are implemented as nonlinear equation (in the ceq.m

function, not with the Aeq, beq parameters when calling fmincon).

5.2.3 Constraints Function funCeq (+linear equality)

Calling the fmincon function by: [x,fval,exitflag,output] = fmincon(@funD,x0,[],[],Aeq,beq,[],[],@funCeq,options);

Linear constraints:

Aeq=[xm,zeros(1,100)];

beq=[xm,zeros(1,100)]*x0';


ceqshrunkinv = S*RSIGMA*S*xma - q*mr2n;

96


ceq = ceqshrunkinv.';

The set of nonlinear equations from Optimization Problem 1, as well as the linear equation

for the estimated returns, but with the linear equation for the estimated portfolio returns is

implemented as a linear equation by fmincon.

5.2.4 Constraints Function funCeq2 (+linear equality)

Calling the fmincon function by: [x,fval,exitflag,output] = fmincon(@funD,x0,[],[],Aeq,beq,[],[],@funCeq2,options);

Linear constraints:

Aeq=[xm,zeros(1,100)];

beq=[xm,zeros(1,100)]*x0';








b) Nonlinear constraint for estimated portfolio standard deviation (σ):

c) Liner constraint for estimated portfolio returns (μ):

97

In this subproblem all the sets of constraints that describe the Optimization Problem 2 have

been included. The linear equation for the estimated portfolio returns is implemented as a

linear equation and the subproblem corresponds to the “original” Optimization Problem 2.

5.2.5 Constraints Function funCeq3

Calling the fmincon function by: [x,fval,exitflag,output] = fmincon(@funD,x0,[],[],[],[],[],[],@funCeq3,options);





% Final expression for the μ linear restriction of OPT 2: ceq3 = xm*m'-0.016;


ceq = [ceqshrunkinv; ceq2; ceq3];

In this subproblem all the sets of constraints that describe the Optimization Problem 2 have

been included, but with the linear equation for the estimated portfolio returns is

implemented as a nonlinear equation. In fact, all the constraints are given to the program by

a single Matlab file called funCeq3.m.

5.2.6 Constraints Function funCeq2 (+linear INequality)

Calling the fmincon function by: [x,fval,exitflag,output] = fmincon(@funD,x0,A,b,[],[],[],[],@funCeq2,options);

Linear constraints: none A=-[xm,zeros(1,100)];

b=-[xm,zeros(1,100)]*x0';






98

The final test mimics a practice of previous research work:50 transforming the linear equality

for μ0, into a linear inequality. The reason for changing the problem by replacing the equality

with an inequality, is that we are giving fmincon an much less constraint problem to resolve,

so that we will collect the estimated values (xIN) and feed them as inputs to the sets of

constraints (ceq, ceq2, ceq2b and ceq3) and validate whether they hold true.

Because Matlab accepts the inequality constraint only in the form A·x ≤ b, in our case this

translates into:

But since we want the overall returns of the estimated portfolio to be larger (ideally) or at

least equal to the sample ones, we make the following transformation:

Overall, as mentioned before, the model “ceq2 and linear equality” is the one described as

Optimization Problem 2 in the literature. Nevertheless, we find that by imposing each

additional set of constraints to Optimization Problem 1 separately, we could gain a better

insight of the behavior of the Levy/Roll approach; the results are demonstrated below (in all

the following graphs, the first one hundred points correspond to the returns for the 100

stocks, while the last 100 points correspond to the standard deviations of the same stocks).

5.3 Test results: Portfolio of the 100 first stocks from S&P 500 Index

50


99

Constraints Function Short description μ0 σ0 market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

ceq non liner eq. OP1 - - 0.087 0.18 0.015 0.6 3 3% 0 0% Y

ceq2 ceq + nonlinear eq. OP2 (σ) - 0.0589 0.087 0.18 0.015 0.6 3 3% 20 20% N

ceq2 ceq + nonlinear eq. OP2 (σ) - 0.0137 0.087 0.18 0.015 0.6 7 7 % 7 7% N

ceq2 ceq + nonlinear eq. OP2 (σ) - 0.005 0.087 0.18 0.015 0.6 10 10 % 46 46% N

ceq2b ceq + m0 eq. as nonlinear eq. 0.044 - 0.087 0.18 0.015 0.6 18 18 % 63 63% N

ceq2b ceq + m0 eq. as nonlinear eq. 0.0252 - 0.087 0.18 0.015 0.6 13 13% 46 46% N

ceq2b ceq + m0 eq. as nonlinear eq. 0.01 - 0.087 0.18 0.015 0.6 4 4% 20 20% N

ceq2b ceq + m0 eq. as nonlinear eq. 0.0044 - 0.087 0.18 0.015 0.6 2 2% 1 1% Y

ceq+linear eq ceq + m0 eq. as linear eq. 0.0044 - 0.087 0.18 0.015 0.6 2 2% 1 1% Y

ceq2+linear eq ceq2 + m0 eq. as linear eq. 0.0044 0.0137 0.087 0.18 0.015 0.6 2 2% 8 8% N

ceq3 ceq2 + m0 eq. as nonlinear eq. 0.0044 0.0137 0.087 0.18 0.015 0.6 2 2% 15 15% N

ceq2 with linear INeq ceq2 + m0 eq. as nonlinear INeq.

0.0044 0.0137 0.087 0.18 0.015 0.6 3 3% 0 0% Y


Table 5.2: Overview of the tests with all subproblems of Optimization Problem 2 for the first 100 stocks of S&P 500 Index.

100

.

Picture 28: Optimization Problem 2, all subproblems.

-50%

0%

50%

100%

150%

200%

250%

300%

1 15 29 43 57 71 85 99 113 127 141 155 169 183 197

Returns/st.deviation

Number of N+N stocks

x0 ceq ceq2 s0=0.0589 ceq2 s0=0.0137 ceq2 s0=0.005 ceq2b m0=0.044

ceq2b m0=0.0252 ceq2b m0=0.01 ceq2b m0=0.0044 ceq+linear eq ceq2 with linear INeq

101

ceq μ0 σ0 market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

ceq - - 0.087 0.18 0.015 0.6 3 3% 0 0% Y


Table 5.3: For illustration purposes, before we move on with the Optimization Problem 2, we picture once again the results for Optimization Problem 1, for the same

portfolio.

-5%

0%

5%

10%

15%

20%

25%

30%

35%

1 15 29 43 57 71 85 99 113 127 141 155 169 183 197


Number of N+N stocksx0 ceq (OPT1)

Picture 29: Optimization Problem 1. As we can see, the results demonstrate almost perfect compliance with the Levy/Roll model.

102

Table 5.4: By imposing only the second set of constraints, for σ0, the model is incapable of producing parameters close enough to their sample counterparts.

-20%

0%

20%

40%

60%

80%

100%

120%

140%

1 15 29 43 57 71 85 99 113 127 141 155 169 183 197


Number of N+N stocksx0 ceq2 s0=0.0589 ceq2 s0=0.0137 ceq2 s0=0.005

Picture 30: From the plot of tests for Ceq2, we can see large rejection rates –especially in standard deviations.

ceq2 μ0 σ0 market

μ market

σ Rf α

Rejected μ's (#)

Rejected μ's (%)

Rejected σ's (#)

Rejected σ's (%)

Levy/Roll complied

ceq2 s0=0.0589 - 0.0589 0.087 0.18 0.015 0.6 3 3% 20 20% N

ceq2 s0=0.0137 - 0.0137 0.087 0.18 0.015 0.6 7 7% 7 7% N

ceq2 s0=0.005 - 0.005 0.087 0.18 0.015 0.6 10 10% 46 46% N

x0

103

Table 5.5: The model seems to work well with the extra constraint for μ0 (0.0044 is the correct value). It is extremely sensitive to this constraint, since we can see that

for the “wrong” values (0.044, 0.0252 and 0.01), the rejection rates are high.

-50%

0%

50%

100%

150%

200%

250%

300%

1 15 29 43 57 71 85 99 113 127 141 155 169 183 197


Number of N+N stocksx0 ceq2b m0=0.044 ceq2b m0=0.0252 ceq2b m0=0.01 ceq2b m0=0.0044

Picture 31: In this test, we can see the massive effect of the choice of the extra constraint (μ0) on the estimated standard deviations.

ceq2b μ0 σ0 market μ market σ Rf α Rejected

μ's (#) Rejected μ's

(%) Rejected

σ's (#) Rejected σ's

(%) Levy/Roll complied

ceq2b m0=0.044 0.044 - 0.087 0.18 0.015 0.6 18 18 % 63 63% N

ceq2b m0=0.0252 0.0252 - 0.087 0.18 0.015 0.6 13 13% 46 46% N

ceq2b m0=0.01 0.01 - 0.087 0.18 0.015 0.6 4 4 % 20 20% N

ceq2b m0=0.0044 0.0044 - 0.087 0.18 0.015 0.6 2 2% 1 1% Y

x0

104

ceq2 with linear INeq μ0 σ0 market μ market σ Rf α Rejected

μ's (#) Rejected μ's

(%) Rejected

σ's (#) Rejected σ's

(%) Levy/Roll complied

ceq+linear eq 0.0044 - 0.087 0.18 0.015 0.6 2 2% 1 0% Y

ceq2 with linear INeq 0.0044 0.0137 0.087 0.18 0.015 0.6 3 3% 0 0% Y

x0

Table 5.6: When the linear inequality is imposed, it has a minor effect on the results, in comparison to the exact same problem but without the linear inequality. In both

cases, the model seems to work in line with the Levy/Roll theory (2% and 3% rejection rate, respectively).

-5%

0%

5%

10%

15%

20%

25%

30%

35%

1 15 29 43 57 71 85 99 113 127 141 155 169 183 197


Number of N+N stocks

x0 ceq ceq+linear eq ceq2 with linear INeq

Picture 32: In this plot we can see perhaps the most important test: Ceq2with linear inequality. The estimated parameters are statistically close to the sample ones.

105

After a series of tests with all the different subproblems presented in paragraph 5.2, we

came to the conclusion that, while the model can work (most of the times) for different

combinations of two out of the three sets of constraints, it is impossible for fmincon to solve

the Optimization Problem 2 with all three sets of constraints for this portfolio.

This is a very strong result, since in the previous chapter, this particular portfolio, with the

exact same market parameters, managed to pass all tests with almost 100% success rate.

Regarding the inability of the model to return a “proper” vector of estimated returns and

standard deviations, we could state that the biggest difficulty seems to lie when including

the linear equality constraint for the mean portfolio returns (μ0).

As we did in paragraph 3.3, we present here the graph of the objective function and the

maximum constraint violation. We can see that as fmincon tries to satisfy all the constraints

(the particular graphs correspond to the ceq3 model which is exactly equivalent to the

Optimization Problem 2), the value of the objective function increases, up to an extreme

value. For the record, the value of D for this set should be approximately 0.07, when

optimized; in this situation it goes well above 10.

Picture 33: Objective Function in the case of a model break-down.

The behavior of the objective function in combination with the behavior of the maximum

constraint violation, shown in the graph right below, depicts clearly the inability of the

model to return a “proper” vector of returns and standard deviations. Of course, by

“proper” we mean a vector of estimated parameters, which will be close enough,

statistically, with the sample parameters. The largest accepted value of a constraint

violation, with which we would assume that the constraint is satisfied, is 10-6. As we can see

below, in this case the maximum constraint violation could not go below approximately 10-2,

0

2

4

6

8

10

12

14

1

15

29

43

57

71

85

99

11

3

12

7

14

1

15

5

16

9

18

3

19

7

21

1

22

5

23

9

25

3

26

7

28

1

29

5

30

9

32

3

33

7

35

1

36

5

37

9

39

3

Ob

ject

ive

Fun

ctio

n V

alu

e


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even though the number of iterations was almost eight times more than the number

required in the Optimization Problem 1, with the exact same setting (without the two extra

sets of constraints for Optimization Problem 2).

Picture 34: Maximum Constraint Violation in the case of a model break-down.

As a way around this problem, we followed a practice that has been used before in the

literature;51 we employed a linear inequality instead of the linear equality (for μ0), in which

as shown in paragraph 5.2.6 the returns of the estimated parameters can be larger or equal

than the returns of the sample ones. Indeed, Matlab returned a vector xIN (by executing the

subproblem named ceq2 with linear inequality) that is in line with the Levy Ross approach:

the estimated parameters (xIN) are not statistically different from the sample parameters

(xsample), at the 5% significance level.

Before we accept this solution, as a valid solution for the Optimization Problem 2, we need

to verify that the estimated vector of 200 elements (the 100 returns and 100 standard

deviations of the 100 stocks) satisfies all the constraints of the problem, but as they are

defined originally; with an equality for the estimated portfolio returns equation. Based on

the generally accepted standards, which are in line with Matlab’s default thresholds, that

would mean that all constraints should be satisfied with a maximum deviation of 10-6.

In order to accomplish that, we used the vector xIN that was produced by the last test (ceq2

with linear inequality set) as input to all the constraint functions demonstrated above. If xIN

is indeed a solution to the original problem, the following inequalities should hold:

51


0.009

0.0095

0.01

0.0105

0.011

0.0115

1

16

31

46

61

76

91

10

6

12

1

13

6

15

1

16

6

18

1

19

6

21

1

22

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24

1

25

6

27

1

28

6

30

1

31

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33

1

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6

36

1

37

6

39

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Max

imu

m C

on

stra

int

Vio

lati

on

Val

ue


107

ceq(xIN) ≤ 10-6

ceq2(xIN) ≤ 10-6

ceq2b(xIN) ≤ 10-6

ceq3(xIN) ≤ 10-6

That would mean that xIN, the solution found by employing an inequality instead of an

equality in the original Optimization Problem 2 can serve as a solution to the whole problem.

As it has been mentioned before, the nonlinear constraint that corresponds to the

mean/variance efficiency condition for the Optimization Problem 1, is a set of N equalities,

where N=100 in this case of a 100-stock portfolio. That means that ceq(xIN) will be a vector

of also 100 values, which will correspond to the constraint violation for each stock of the

portfolio.

On the other hand, the extra constraints of Optimization 2, i.e. the linear equality for μ0 and

the non linear equation for σ0 correspond to one equation each; that means that ceq2(xIN)

and ceq2b(xIN) will include N+1 (101 in this case) values, where the 100 first will be the

constraints violations of ceq, and the 101st the constraint violation for σ0 and μ0,

respectively.

Finally, the vector resulting from ceq3(xIN) will include N+2 (102 in this case) values, where

the first 100 will be the constraints violations of ceq, the 101st the constraint violation for μ0

and the 102nd the constraint violation for μ0.

Overall, the constraint violations resulting by feeding the estimated (from the Optimization

problem ceq2 and inequality) vector xIN, to the actual constraints of the full Optimization

problem 2 (and its subproblems), are depicted in the following graph:

108

Picture 35: The constraint violations resulting by feeding the estimated (from the Optimization problem ceq2 and inequality) vector x IN into the subproblems constraints functions.

-0.00014

-0.00012

-0.0001

-0.00008

-0.00006

-0.00004

-0.00002

0

0.00002

13 5

79

11

13

15

17

19

21

23

25

27

29

31

33

35

37

39

41

4345

4749515355

5759

61

63

65

67

69

71

73

75

77

79

81

83

85

87

89

91

9395

9799 101

ceq ceq2 ceq2b ceq3

109

As we can see, the equality constraint for the returns is satisfied with a deviation of 1.4·10-4,

which is approximately 100 times larger than the maximum allowed threshold (10-6). This is

observed for both ceq2b and ceq3 subproblems, which identifies that the divergence comes

from the linear constraint for estimated portfolio returns (μ0).

5.4 Conclusions for the Optimization Problem 2

After the step-by-step analysis of the Optimization Problem 2 that we demonstrated in this

chapter, we concluded that it is not possible to produce results in accordance with the

Levy/Roll approach, when the two additional constraints are applied simultaneously. More

specifically, the addition of the linear equality constraint for the estimated portfolio returns

(μ0) which corresponds to the first extra constraint imposed on the Optimization Problem 1

by Levy and Roll for the Optimization Problem 2, cannot be satisfied for a given portfolio,

having the rest of the constraints implemented in the Matlab code. Even by replacing it by

an inequality so that we will relax the constraint and produce smoother results, backtesting

of the Matlab function revealed that not all constraints are satisfied.

110

Chapter 6

6 Zipf Approach

All the details regarding the Zipf distribution can be found in the relevant pages in the

References sector of this Thesis. In the Introduction of this Chapter, a brief description of the

Zipf Theory will be presented; the following work is based mainly on the paper “Professor

Zipf goes to Wallstreet”52.

6.1 Introduction

The key idea of this approach, an idea first stated by George Kingsley Zipf (1949)53, is that in

many systems in nature (including human-related activities) the size of an object or network

is dependent to its ranking amongst its fellow objects or networks of the same type. In fact,

the size is found to be inversely proportional to the rank of the object studied. This

remarkable relationship has been tested and proved to be true in a lot of different contexts,

such as English words, cities, distribution of wealth and others.

Zipf’s law, in its original form, it is written as

, where x is the size of the object, i is

its rank and xs is the maximum size in the specific system (or set) of objects.

In our case we employ the Zipf law in the context of firm sizes, stating the idea from the

paper “Professor Zipf goes to Wallstreet” cited previously, that because the distribution of

the firms’ sizes are heavy tailed, the market portfolio is not diversified enough; since the top

few firms in reality represent a huge portion of the total market capitalization. That

relationship has been proven to be valid in real life when finding that approximately 20% of

the firms in a given market correspond to the 80% of the market capitalization.

The authors of the paper54 mentioned in the previous paragraph were inspired by this fact,

and supported the opinion that a factor representing this effect could have significant

explanatory power in a time series regression model. Indeed, as we will see in the rest of this

chapter, by adding the “Zipf factor” to the plain market model, we were able to accomplish

results even better than the three-factor Fama French model.

52

Malevergne, Y., Santa-Clara, P. and Sornette, D. (2009). Professor Zipf goes to Wall Street. NBER Working Paper No. 15295. 53

Zipf, G. K. (1949). Human behavior and the principle of least effort. Oxford, England: Addison-Wesley Press. xi 573 pp. 54

Malevergne, Y., Santa-Clara, P. and Sornette, D. (2009). Professor Zipf goes to Wall Street. NBER Working Paper No. 15295.

111

6.2 Matlab Implementation

The third and last Matlab test set of this Thesis (as the first and second Matlab set tests we

consider the Optimization Problem 1, and the Optimization Problem 2, respectively) is solely

a number of linear regressions of portfolios’ returns on different factors. Having as reference

the examples of Mathworks55, the code used is the minimum one, and it is presented below

as such:

Dataset_1= dataset('XLSFile','1.xlsx');

With the above command, we import an excel file (the file '1.xlsx') into Matlab which is

being converted to a data set (named Dataset_1 in our example) with all the variables’

names.

Finally, with the following command we run a linear regression on the previously created

data set:

M_MARKET_MODEL_1= LinearModel.fit(Dataset_1)

The most important part of this procedure is that the excel spreadsheet imported should be

constructed such that the first row has the variables names, and the factors are one by one

ordered in a different column (starting from the first factor), with the last column being the

regressand (portfolio returns).

An example of the Matlab output for the three-factor Fama French model is given as follows:

Kenneth_mkt_factor_MODEL =

Linear regression model:

smallLow_Rf ~ 1 + Mkt_RF + SMB + HML + RF

Estimated Coefficients:

Estimate SE tStat pValue

(Intercept) -1.0238 0.29969 -3.4161 0.00073691

Mkt_RF 1.112 0.038951 28.547 4.1672e-82

SMB 1.3527 0.052288 25.871 7.1059e-74

HML -0.34482 0.05453 -6.3235 1.1088e-09

RF 1.1994 0.95988 1.2495 0.2126

Number of observations: 265, Error degrees of freedom: 260

Root Mean Squared Error: 2.66

R-squared: 0.901, Adjusted R-Squared 0.899

F-statistic vs. constant model: 590, p-value = 4.02e-129

As we can see, from such an output we can gather all the information that we need in order

to evaluate the performance of each model (each model’s evaluation will be analyzed in the

subsequent paragraphs).

55

MathWorks. Documentation Center. Time Series Regression I: Linear Models. http://www.mathworks.com/help/econ/examples/time-series-regression-i-linear-models.html, 20.12.2012.

112

6.3 Data

The data used for this chapter can be divided in two separate data sets: the portfolios’

returns which we will regress on a number of factors, and the factors themselves.

The portfolios returns used, are the ones that Professor Kenneth French has calculated and

published at his website56; the first set that has been tested is the 25 Fama French portfolios,

always by size and book-to-market. Afterwards, data downloaded directly from the CRSP

Database (Wharton interface) are used, namely the entire universe of NYSE/AMEX/NASDAQ

stocks divided in 10 deciles, then 4 quadrilles, and also just 2 sets (halves), by market

capitalization (with the largest companies in portfolio 1 and the smallest in portfolio 10).

For the factors, we initially applied 5 different tests, overall. First, the only factor was the

market one, but there were two different versions: one with the market factor Rm from the

database of Professor French, and a second one by using the returns of the Wilshire 5000

Index. This index is constituted by all the stocks of NYSE, NASDAQ and AMEX Indices,

therefore it could be considered as the stocks “universe”, being a representative proxy of

the market.

6.4 Time Series Regression Models

While in the literature it is always the case that very long time periods are used for such

kinds of tests, initially we wanted to test this model for the exact same period employed for

testing the 100 stocks (for the S&P500 Index and S&P 1200 Index) the Optimization Problem

1. Having already seen how the previous models cope with this specific data set, it is

interesting to examine the performance of this new approach as well.

In this subchapter, we will present the different time series regression models used in the

rest of the chapter. For reasons of completeness, we will demonstrate a table of results for

the R2 statistical parameter for each different model, so that the reader will acquire a

“feeling” of the different models’ fit and as such have certain expectations from their

performance. Thereafter, we will go on with the presentation of the main results.

6.4.1 Market Model

The first time series regression model employed is the “Market Model”: only one

explanatory factor, the Market Factor. The first set of regressions is described by the

following simple formula:

ri,t – Rf,t = αi + βi (rm,t – Rf,t) + εi,t

56

Dartmouth College. Research Returns. http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html, 13.03.2013.

113

Where ri,t corresponds to each different portfolio’s returns, for i = [1,25], Rf,t is the risk free

ratio for each month, and rm,t is the market return time series. As in all regressions, εi,t

represents the residual (or error term).

At the first set, as mentioned above, the market return rm,t is the one given at the website of

Professor Kenneth French, with the following results:

Book-to-market

low 2 3 4 high

small 0.56 0.55 0.61 0.58 0.59

2 0.69 0.71 0.69 0.66 0.63

Size 3 0.72 0.79 0.74 0.68 0.63

4 0.79 0.81 0.75 0.72 0.64

big 0.89 0.82 0.75 0.62 0.57

Table 6.1: R

2 for the Market model, CRSP Data.

The average R2 equals to 68.7%.

At the second set, as mentioned above, the market return rm,t is calculated by the official

Wilshire website,57 and it was retrieved from the official website of the Wilshire 5000 Index:

Picture 36: The official Wilshire calculator.

The results were the following:

57

Wilshire. Wilshire Index Calculator. http://www.wilshire.com/Indexes/calculator, 31.07.2013.

114

Book-to-market

low 2 3 4 high

small 0.55 0.54 0.54 0.58 0.59

2 0.68 0.71 0.70 0.66 0.63

Size 3 0.71 0.79 0.75 0.69 0.65

4 0.78 0.82 0.77 0.74 0.66

big 0.89 0.83 0.77 0.63 0.58

Table 6.2: R

2 for the Market model, Wilshire 5000 Data.

The average R2 equals to 68.8%: just 0.1% higher in comparison to the market data from the

website of Professor Kenneth French; that indicates that the Wilshire 5000 Index could be

used as a market proxy for such experiments, with a performance almost identical to the

one of the widely accepted database of Professor Kenneth French.

6.4.2 Zipf Model

For the second round of regressions, we will add another factor to the market model, the

Zipf Factor.

Based on the theory presented in the beginning of this chapter, the addition of this factor to

the market model is expected to provide with considerable better fitting (or explanation

power) to the portfolios’ returns, and not much worse than the fitting by the three-factor

Fama French model.

The time series for the Zipf factor is calculated as the difference between the returns of the

equally weighted market portfolio (rm_EQ.W,t), and the value (market capitalization) market

portfolio (rm_V.W,t):

rz,t = rm_EQ.W,t – rm_V.W,t

Given the Zipf factor, the new regression model will be:

ri,t – Rf,t = αi + βi (rm,t – Rf,t) + γi (rz,t – Rf,t) + εi,t

At the first set, the market return rm,t is the one given at the website of Professor Kenneth

French, with the following results:

115

Book-to-market

low 2 3 4 high

small 0.82 0.79 0.82 0.80 0.84

2 0.81 0.81 0.77 0.74 0.72

Size 3 0.79 0.82 0.75 0.69 0.66

4 0.82 0.82 0.76 0.73 0.65

big 0.91 0.84 0.76 0.63 0.57

Table 6.3: R

2 for the Zipf model, CRSP Data.


At the second set, the market return rm,t is calculated by the official Wilshire website and it is

the one of the Wilshire 5000 Index:

Book-to-market

low 2 3 4 high

small 0.81 0.79 0.81 0.80 0.84

2 0.81 0.81 0.78 0.74 0.72

Size 3 0.78 0.82 0.76 0.70 0.67

4 0.81 0.83 0.77 0.74 0.66

big 0.90 0.85 0.77 0.65 0.58

Table 6.4: R

2 for the Zipf model, Wilshire 5000 Data.

The average R2 equals to 76.8%. Once again, the use of the Wilshire 5000 Index

outperformed slightly (by 0.4%) the market proxy returns provided by the website of

Professor Kenneth French.

6.4.3 Three-Factor Fama French Model

Finally, the last test employed in this cluster is the regression of the portfolios’ returns with

the use of the three-factor Fama French model. The values for all three factors (market, SMB

and HML) are taken from the database of Professor Kenneth French. The formula is:

ri,t – Rf,t = αi + βi (rm,t – Rf,t) + bi,S SMBt + bi,H HMLt + εi,t

The results for the 25 Fama French portfolios can be seen in the following table:

116

Book-to-market

low 2 3 4 high

small 0.90 0.93 0.94 0.92 0.93

2 0.95 0.93 0.93 0.92 0.94

Size 3 0.94 0.87 0.86 0.86 0.85

4 0.93 0.87 0.86 0.86 0.86

big 0.95 0.89 0.86 0.88 0.78

Table 6.5: R

2 for the three-factor Fama French model, CRSP Data.


6.5 Test results

From previous research attempts, the most robust (and complete) data set to be used is

considered the entire 1925 to 2012 CRSP data for U.S. market (NYSE/AMEX/NASDAQ). For

the purposes of this Thesis, this database was downloaded from the Wharton Interface and

was split into tests as follows. For each different test section, three basic parameters will be

presented, indicating the model fit:

a) R2

b) The value of intercept a

c) The t-statistic of the intercept a

All the values are being returned by the Matlab procedure (time series regression) described

previously.

6.5.1 Results for the 25 Fama French portfolios (January 1926 to November 2013)

At first, we will examine the performance of the three regression models onto the data from

the whole time period, formulated into the 25 Fama French portfolios (standard procedure

from the website of Professor Kenneth French, as described before). As we know, the most

characteristic parameter of the entire model fit is R2.

For the plain market model, that value reads 0.75; for the Zipf model 0.84 and for the three-

factor Fama French model 0.88 (mean absolute values, in all cases). Even from the first test,

it has been established that the Zipf model has significantly more explanatory power (9%)

from the market model, and only slightly less (ΔR2=4%) from the unbeatable Fama French

one. But we have to bear in mind that the Fama French model employs three factors, while

the Zipf model only two, therefore the question is, can the Zipf model outperform the three-

factor Fama French one? In order to answer that, we will conduct a series of tests, presented

in the next paragraphs.

117

Table 6.6: Statistical results for the 25 Fama French portfolios January 1926 to November 2013.

Parameter: R2 a t(a)

Market low 2 3 4 high low 2 3 4 high low 2 3 4 high

small 0.53 0.57 0.67 0.67 0.63 -0.59 -0.12 0.14 0.34 0.51 -2.24 -0.54 0.85 2.19 2.79

2 0.72 0.77 0.77 0.77 0.72 -0.21 0.14 0.28 0.30 0.33 -1.59 1.18 2.58 2.59 2.30

3 0.82 0.86 0.85 0.81 0.76 -0.12 0.16 0.25 0.28 0.26 -1.23 2.15 3.06 3.05 1.98

4 0.87 0.90 0.87 0.82 0.76 0.02 0.05 0.15 0.20 0.14 0.25 0.77 2.02 2.21 1.01

high 0.92 0.91 0.86 0.79 0.26 -0.02 0.02 0.04 -0.02 -1.00 -0.32 0.45 0.52 -0.16 -2.82

Mean abs: 0.75 0.23 1.63

Zipf

small 0.66 0.78 0.85 0.90 0.90 -0.46 0.02 0.25 0.46 0.65 -2.10 0.11 2.27 5.35 6.70

2 0.81 0.90 0.90 0.90 0.88 -0.14 0.22 0.35 0.37 0.43 -1.32 2.75 4.99 5.02 4.43

3 0.88 0.91 0.90 0.88 0.85 -0.07 0.20 0.29 0.33 0.33 -0.88 3.24 4.32 4.53 3.12

4 0.87 0.92 0.89 0.86 0.82 0.02 0.07 0.17 0.24 0.20 0.34 1.25 2.64 2.91 1.67

high 0.94 0.92 0.86 0.80 0.26 -0.03 0.01 0.03 0.00 -0.97 -0.79 0.18 0.50 0.02 -2.75

Mean abs: 0.84 0.25 2.57

Fama French

small 0.65 0.81 0.86 0.93 0.94 -0.85 -0.39 -0.13 0.03 0.09 -3.75 -2.71 -1.19 0.42 1.22

2 0.90 0.93 0.94 0.95 0.95 -0.25 -0.03 0.08 0.04 -0.02 -3.12 -0.41 1.41 0.76 -0.42

3 0.93 0.93 0.92 0.93 0.93 -0.15 0.08 0.10 0.08 -0.07 -2.44 1.49 1.70 1.39 -0.94

4 0.93 0.92 0.91 0.92 0.92 0.09 -0.02 0.03 0.00 -0.19 1.79 -0.37 0.52 0.07 -2.30

high 0.96 0.93 0.91 0.92 0.31 0.07 0.04 -0.03 -0.21 -1.25 2.05 1.00 -0.63 -3.52 -3.64

Mean abs: 0.88 0.17 1.57

118

6.5.2 Results for the 25 Fama French portfolios (January 2003 to November 2012)

As with the testing methodology of the Optimization Problems 1 and 2, it is in our best

interest not to just demonstrate some test results that might or might not comply with our

proposed theory, but to try to examine the models’ behavior in variations of the original

model. That is because there are several factors that may strengthen the model

performance, for which we will try to intuitively control, as well as because we want to see

the performance of the Zipf Model during the same time period that we tested the

Optimization Problems 1 and 2.

Therefore, in this section we present the results for the exact same procedure but for the

time period January 2003 to November 2012.

As we can see in the table below, all models pose enhanced performance for this time

period; the market model bears an R2 of 0.86, the Zipf model 0.89 and the three-factor Fama

French model 0.94. Once again, the performance of the Zipf Model lies in the middle

(between the Market and the three-factor Fama French one), but now the spread from the

Market model is smaller (three percentage points) while it is significantly larger from the

three-factor Fama French model (five percentage points). That boosts less support for the

use of the Zipf model than in the previous test. Such fluctuations in performance are exactly

the reason why, while each time we are evaluating a model proposal, a series of tests with

slightly different specifications are carried out, so that in end we will have a broader and

more robust picture.

119



small 0.79 0.84 0.85 0.80 0.79 -0.42 0.13 0.06 0.09 0.40 -1.43 0.57 0.30 0.37 1.39

2 0.84 0.86 0.85 0.85 0.80 0.14 0.25 0.35 0.11 0.14 0.62 1.29 1.71 0.51 0.49

3 0.87 0.89 0.89 0.89 0.81 0.13 0.27 0.35 0.27 0.48 0.70 1.61 2.09 1.56 2.00

4 0.89 0.92 0.91 0.88 0.84 0.29 0.11 -0.04 0.22 -0.03 1.87 0.76 -0.25 1.23 -0.12

high 0.91 0.91 0.91 0.88 0.80 -0.01 0.14 -0.16 -0.19 0.01 -0.07 1.23 -1.22 -1.30 0.05

Mean abs: 0.86 0.19 0.99

Zipf

small 0.90 0.92 0.91 0.85 0.88 -0.59 0.00 -0.04 -0.02 0.25 -2.88 0.01 -0.25 -0.08 1.12

2 0.88 0.90 0.88 0.87 0.83 0.05 0.17 0.28 0.04 0.05 0.24 1.01 1.49 0.23 0.17

3 0.89 0.91 0.90 0.90 0.82 0.07 0.22 0.32 0.23 0.43 0.38 1.39 1.93 1.36 1.83

4 0.90 0.93 0.92 0.89 0.85 0.25 0.06 -0.10 0.16 -0.08 1.67 0.48 -0.67 0.98 -0.39

high 0.92 0.92 0.91 0.89 0.80 0.01 0.17 -0.15 -0.17 0.00 0.10 1.59 -1.15 -1.16 0.00

Mean abs: 0.89 0.16 0.90

Fama French

small 0.92 0.95 0.97 0.96 0.96 -0.62 -0.06 -0.13 -0.13 0.15 -3.34 -0.49 -1.23 -1.22 1.14

2 0.96 0.96 0.97 0.96 0.96 -0.02 0.09 0.16 -0.08 -0.10 -0.20 0.85 1.60 -0.68 -0.71

3 0.95 0.95 0.93 0.92 0.92 0.03 0.15 0.24 0.18 0.32 0.25 1.31 1.81 1.19 1.96

4 0.95 0.94 0.92 0.91 0.92 0.23 0.04 -0.10 0.13 -0.10 2.08 0.34 -0.67 0.82 -0.62

high 0.96 0.92 0.93 0.95 0.85 0.05 0.17 -0.13 -0.17 -0.01 0.64 1.58 -1.11 -1.68 -0.07

Mean abs: 0.94 0.14 1.10

Table 6.6: Statistical results for the 25 Fama French portfolios January 2003 to November 2012.

120

6.5.3 Results for the 25 Fama French Portfolios (January 1991 to November 2008)

In this paragraph we present the third test with the 25 Fama French portfolios, for another

period that was used in the Optimization Problem 1. Several major financial crashes

(extreme events) occurred in this period, therefore it could be that the differences in the

models’ performance are even larger.

Indeed, the market model bears an R2 of 0.64, the Zipf model 0.74 and the three-factor Fama

French model 0.88. This is the largest spread noticed between all three models, with the Zipf

model being almost exactly in the middle, in terms of explanatory power.

Given these numbers, the Zipf model is much better from the plain market one, but it is

questionable whether the easiness of using one factor less would compensate sufficiently

for a difference in performance of 14 percentage points.

Since this section has left with mixed results for the tradeoff between the number of factors

and the explanatory power of the model, we will continue with a different set up of tests, in

terms of formulating the portfolios on which we will test the same (three) models.

121



small 0.53 0.49 0.53 0.49 0.52 -0.63 0.34 0.49 0.76 0.76 -1.55 0.95 1.87 3.08 2.91

2 0.67 0.66 0.63 0.59 0.55 -0.31 0.12 0.50 0.49 0.47 -1.07 0.56 2.54 2.29 1.89

3 0.70 0.76 0.68 0.61 0.58 -0.26 0.19 0.44 0.41 0.72 -1.01 1.14 2.55 2.09 3.32

4 0.79 0.78 0.70 0.67 0.58 0.00 0.17 0.21 0.41 0.20 0.01 1.15 1.13 2.31 0.95

high 0.89 0.79 0.70 0.53 0.50 -0.04 0.23 0.08 0.16 0.21 -0.43 1.75 0.48 0.83 0.85

Mean abs: 0.64 0.34 1.55

Zipf

small 0.83 0.79 0.82 0.82 0.86 -0.65 0.32 0.47 0.75 0.74 -2.62 1.41 2.93 5.05 5.28

2 0.82 0.80 0.75 0.70 0.70 -0.32 0.11 0.49 0.49 0.46 -1.49 0.67 3.04 2.67 2.25

3 0.79 0.80 0.70 0.62 0.62 -0.27 0.19 0.44 0.40 0.71 -1.23 1.23 2.61 2.13 3.47

4 0.82 0.79 0.70 0.67 0.58 0.00 0.17 0.20 0.40 0.20 -0.02 1.16 1.12 2.31 0.95

high 0.91 0.80 0.70 0.55 0.51 -0.04 0.23 0.08 0.16 0.85 -0.45 1.85 0.49 0.21 0.86

Mean abs: 0.74 0.37 1.90

Fama French

small 0.90 0.92 0.93 0.90 0.91 -0.66 0.09 0.17 0.34 0.19 -3.40 0.65 1.58 3.10 1.69

2 0.95 0.93 0.91 0.91 0.93 -0.32 -0.17 0.09 -0.03 -0.16 -2.74 -1.59 0.95 -0.25 -1.56

3 0.94 0.86 0.84 0.85 0.82 -0.14 -0.07 0.05 -0.08 0.19 -1.10 -0.52 0.39 -0.64 1.33

4 0.93 0.86 0.84 0.84 0.82 0.13 -0.12 -0.18 0.00 -0.27 1.13 -1.00 -1.35 0.01 -1.87

high 0.95 0.87 0.84 0.85 0.76 0.18 0.12 -0.14 -0.19 -0.20 2.34 1.20 -1.19 -1.63 -1.12

Mean abs: 0.88 0.17 1.37

Table 6.7: Statistical results for the 25 Fama French Portfolios (January 1991 to November 2008).

122

6.5.4 Results for the extra Portfolios (January 1991 to December 2012)

In this section, the same NYSE/AMEX/NASDAQ stocks from the CRSP database were used,

but divided based on market capitalization, only. More particularly, the universe of U.S.

stocks was divided in 10 deciles, which were afterwards used as such for the portfolio

construction. A short table is submitted for the detail construction of these “extra”

portfolios:

Name Deciles included

Portfolio 1_5 1, 2, 3, 4, 5

Portfolio 6_10 6, 7, 8, 9, 10

Portfolio 1_2 1, 2

Portfolio 3_5 3, 4, 5

Portfolio 6_8 6, 7, 8

Portfolio 9_10 9, 10

Table 6.8: Formulation of the extra Portfolios (based on the deciles in which the U.S. Equities

Market is divided).

Given the definition (and as such, the construction) of the Zipf factor, the Zipf model is

expected to perform better than in the 25 Fama French portfolios tests. The time period

used is the last 20 years; the reason is not only the extreme volatility of the equity indices

during this time period, but also because these are the most common data used today when

it comes to testing equities models. Older data can no longer be considered to be influenced

by similar macroeconomic and other, market related factors.

Indeed, by dividing the entire U.S. stocks universe in just two portfolios by market

capitalization, we can achieve the following results: the market model’s R2 is 0.88, the Zipf

model’s is 0.96 and the three-factor Fama French model demonstrates an impressive R2,

equal to 0.99. Therefore all models perform even better, with the Zipf model further closing

the performance gap to the three-factor Fama French one (only three percentage points).

123

R2

Portf.1_5 Portf.6_10 Average

Market 0.99 0.77 0.88

Zipf 1.00 0.93 0.96

Fama French 1.00 0.97 0.99

a


Market 0.0025 0.0045 0.0035

Zipf 0.0026 0.0036 0.0031

Fama French 0.0027 0.0023 0.0025

t(a)


Market 12.30 2.51 7.40

Zipf 18.39 3.75 11.07

Fama French 20.55 3.69 12.12

Table 6.9: Statistical results: U.S. Equities Market divided in two portfolios (January 1991 to

December 2012).

By testing the models onto the second set of portfolios, where the U.S. stocks universe is

divided in four portfolios, based on the market capitalization of the stocks, we can see that

the market model’s R2 is 0.83, the Zipf model’s 0.95 and the three-factor Fama French model

0.95. Quite impressively, the Zipf model performs (on average) exactly as well as the three-

factor Fama French one: a high R2 of 95% is achieved.

124

R2

Portf.1_2 Portf.3_5 Portf.6_8 Portf.9_10 Average

Market 0.98 0.90 0.79 0.65 0.83

Zipf 0.99 0.93 0.91 0.96 0.95

Fama French 0.99 0.96 0.97 0.89 0.95

a


Market 0.0024 0.0036 0.0043 0.0056 0.0040

Zipf 0.0025 0.0033 0.0035 0.0043 0.0034

Fama French 0.0028 0.0026 0.0023 0.0028 0.0026

t(a)


Market 6.24 3.69 2.50 2.40 3.71

Zipf 9.66 4.05 3.15 5.42 5.57

Fama French 13.10 3.92 3.79 2.14 5.74

Table 6.10: Statistical results: U.S. Equities Market divided in four portfolios (January 1991 to

December 2012).

6.5.5 Results for the 10 Deciles Portfolios (January 1991 to December 2012)

The final series of tests for the Zipf model was chosen to be the U.S. stocks universe divided

in 10 deciles by market capitalization (decile 1 contains the largest market capitalization

stocks), always for the same time period.

This time the results are reversed, indicating the strength of the Zipf Model in portfolios

composed of evenly shorted stocks by market capitalization.

The R2 of the Zipf Model is 0.95, while the three-factor Fama French one is just 0.83 (and the

plain market model follows with only 0.65). It is an extraordinary result, having a two-factor

model demonstrating better explanatory power than the three-factor Fama French model.

125

R2

Decile 1 Decile 2 Decile 3 Decile 4 Decile 5 Decile 6 Decile 7 Decile 8 Decile 9 Decile 10 Average

Market 0.27 0.40 0.53 0.58 0.66 0.70 0.74 0.81 0.87 0.99 0.65

Zipf 0.81 0.88 0.92 0.93 0.93 0.93 0.90 0.91 0.92 0.99 0.91

Fama French 0.47 0.60 0.74 0.82 0.90 0.93 0.96 0.96 0.96 1.00 0.83

a


Market 0.0139 0.0085 0.0067 0.0066 0.0058 0.0046 0.0042 0.0038 0.0039 0.0023 0.0060

Zipf 0.0119 0.0070 0.0055 0.0054 0.0047 0.0036 0.0034 0.0032 0.0035 0.0024 0.0051

Fama French 0.0119 0.0066 0.0049 0.0044 0.0035 0.0025 0.0023 0.0020 0.0027 0.0025 0.0043

t(a)


Market 3.39 2.80 2.73 2.85 2.79 2.34 2.33 2.55 3.38 7.67 3.28

Zipf 5.70 5.03 5.26 5.89 5.10 3.79 3.00 3.11 3.89 9.86 5.06

Fama French 3.36 2.61 2.67 2.89 3.00 2.48 3.06 2.70 4.01 12.40 3.92

Table 6.11: Statistical results: U.S. Equities Market divided in ten portfolios (January 1991 - December 2012).

126

Chapter 7

7 Conclusion

The process of evaluating quantitative models by applying them on real-market financial

data is known to be very data dependent. For that reason, to make our case as robust as

possible, we ran the models constructed for the purpose of this Thesis numerous times, with

their parameters varying in each execution in order to make sure that we have a

representative pool of results for each distinct case.

Regarding the Levy/Roll approach, the results for the optimizations are very sensitive to the

choice of the portfolio used, the market returns and standard deviation, as well as to the

choice of the risk free ratio. In the personal opinion of the author of this Thesis, it is possible

to manipulate these results, up to a certain point, by demonstrating specific cases/

combinations of the above stated parameters, in order to accomplish a better outcome and

improve the robustness of the model. Of course, such results would be severely biased,

since they would basically demonstrate the performance of a model for a certain subcase

and not for the universe of stocks, corresponding to the real market conditions. Our own

analysis showed that if we control for the “calibration parameters” and feed the models

with their real market values, the performance of the models is not robust enough in order

to justify global acceptance.

More specifically, having already drawn the detailed conclusions regarding each

optimization problem in the respective chapters (Chapters 4 and 5), we will attempt here to

sum up the main points and conclude about the entire approach as proposed by Levy and

Roll in 2011.

Regarding the Optimization Problem 1, we showed that the Levy/Roll approach can work for

certain portfolios, but not universally. It appears that it has increased explanatory power

over portfolios comprised exclusively by U.S. equities, versus portfolios that include stocks

from other markets. Previous research work has focused its efforts solely in U.S. equities.

For that reason, we proceeded with the step-by-step evaluation of the Optimization

Problem 2. In order to make our case more robust, we used only the portfolio for which the

Optimization Problem 1 produced results fully in line with the Levy/Roll approach (we refer

to the portfolio composed of the first 100 stocks from the S&P 500 Index). But even after

relaxing the most difficult constraint for the model to meet, the inability of the process to

produce results in line with the Levy/Roll approach is considered to be the final hit to the

economists’ theory (at least as it has been manifested). Therefore, the value of the whole

approach diminishes significantly, with the Levy/Roll approach being dismissed, conclusively.

127

In our quest for better “fitting” the returns of a portfolio, an idea coming from the paper

“Professor Zipf goes to Wall Street”58 seemed that it was worth exploring. Chapter 6 of this

Thesis is devoted to a series of tests, aiming to prove whether this new theory holds.

After conducting numerous time series regressions with three different regression models

(the market model, the Zipf model and the three-factor Fama French model), the results

verify at full power the initial motion introduced in the paper.

Two very large data sets were used, from which six different test sets were produced. Three

of them were constructed in the known format of the 25 Fama French portfolios, and the

other three were based on portfolios constructed by deciles, in which the stocks were

ranked by market capitalization. As we saw in the respective chapter, Zipf’s model

performance in all three 25 Fama French portfolios test sets was fully competitive, by always

overtaking the simple market model, and always lacking slightly (with some fluctuation)

from the three-factor Fama French model. But the results of the last three portfolio settings,

that were constituted by stocks based on (decreasing) market capitalization were even more

impressive. When the market was divided in only two parts, the Zipf model lagged slightly

over the Fama French one (but still has an R2 of 96%) and when the market was divided in

four parts (by market capitalization), both models had the exact same performance. But in

our last test, when the market was divided in 10 deciles based on the firms’ market

capitalization, the Zipf model surpassed the Fama French one (in terms of performance), by

boasting an R2 of 91% versus 83% (of the three-factor Fama French one). That is because of

the theory behind the Zipf model; in this case it can deploy its full potential.

As a final comment on the two approaches, we believe that the Levy Roll procedure could be

employed in order to achieve superior returns, in certain portfolios that their constituents

belong to well-defined markets. Our tests have shown that when we used representative

(for the market) values for the calibration parameters, and most importantly it was possible

to calculate the risk free ratio accurately, the model had a high probability to work well. On

the other hand, the enormous explanatory power of the Zipf factor in portfolios composed

of equities sorted by market capitalization, could be the foundation for further research on

achieving higher portfolio returns by using the Zipf model.

58

Malevergne, Y., Santa-Clara, P. and Sornette, D. (2009). Professor Zipf goes to Wall Street. NBER Working Paper No. 15295.

128

Chapter 8

8 Suggestions for continuing this work

As with every research process, several issues emerged along the way of this Thesis, which

were not “visible” or were not considered as important enough from the beginning. Most of

them were addressed until they were resolved, or at least until their mechanism was fully

understood by the author, so that later they would be presented in a decent way in this

Thesis. But not each and every idea that comes along while working on a certain research

topic can always be explored in its full detail, given the several constraints (with time being

the most important one). Therefore, in this section we will shortly present some issues that

were considered of secondary importance while working on this Thesis, but at the same

time are worth mentioned, for future research. These points are:

1) During the various tests conducted while examining the behavior of Optimization

Problem 1, it became evident that the estimated by Matlab parameters for certain

stocks were always not “close enough” to their sample counterparts. After a brief

look, it appeared that these stocks did not move along with the others in the same

portfolio, meaning that the correlation of their returns to the portfolio’s returns was

low. Therefore, further research could attempt to determine under which exact

conditions, a specific stock in a given portfolio will most probably not comply with

the Levy/Roll theory.

2) It has also been documented that the behavior of a portfolio (again, in the

optimization problems by Levy and Roll) changes, with regard to the order of the

stocks that it is being constituted. That could pose, for example, an idea for

investigating what the effects of market capitalization sorting versus sorting by

volatility are, in the same stocks of a given portfolio.

3) In the same spirit, it could be possible to find a critical number of stocks per

portfolio under examination with the Levy/Roll procedure, which would achieve the

maximum compliance to the theory. Again, it is difficult to control for such an effect

while keeping everything else constant (ceteris paribus), therefore such an approach

would require a tremendous amount of tests, covering all different cases (depending

on given markets/indices, sorting method of stocks in the portfolio, time-periods

used, etc.).

4) From a different perspective, it is possible that the number of monthly returns used

in any test (optimization for the Levy/Roll approach or time series regression

analysis for the Zipf factor) also has its implications on the results. Extensive

research has been already conducted on a similar subject, which is how many

monthly returns should be used in a model so that it would be able to forecast

market moves with a certain accuracy rate. This point was taken under

consideration in this Thesis, up to a certain extend.

129

5) As an extension of the previous point, but from the perspective of macroeconomics,

it is almost certain that even if the number of months is the same, it matters which

exact time period is covered. Even intuitively, we can understand that if the period

that is being used to test a model contains an “extreme event”, or it is generally

considered as a distressed period for the financial sector, it is expected to have

completely different characteristics versus a “normal” period. Examples of attributes

that are key to a financial analysis and vary significantly in such a distressed period

are the correlations between assets and the liquidity shortages caused after such

events (that affect asset pricing). Therefore, all tests could demonstrate significantly

different results, depending on which crisis they coincide with.

6) Referring to Paragraph 4.7, a different –and perhaps more intuitive- way of testing

the model with random correlation matrices could be applied, by using the view of

Malevergne and Sornette in their 2004 paper.59 In their work, they discovered that

the largest part of the information included in the correlation matrix of (in our case)

a portfolio, is located in the highest eigenvalues, while the rest of the correlation

matrix is white noise. Therefore, we could repeat the random correlation model

tests by imposing the necessary constraints on the (average) correlations of the

random correlation matrices, so that the information of the original correlation

matrix will not be lost (as it is the case with completely random correlation

matrices).

7) Regarding the optimization algorithm (solver) itself, there is a popular alternative

module that gains ground, Tomlab.60 It is possible that we would have different

results, by implementing Tomlab in the existing optimization problems for the

Levy/Roll procedure. That is because, in nonlinear optimization problems, the choice

of the solver employed is very important, and can –in certain cases of high

nonlinearity as in our case- lead into different results.

8) Finally, an issue that is more of a statistical nature: while the paper61 on which this

Thesis is based for the Levy/Roll approach part employs “naïve” averages for the

stocks’ returns, there was recently a paper62 in which the authors had similar results

but by using annualized mean returns. Since in this Thesis in none of the tests

annualized mean returns were used, it is unknown how all the different models

would behave in that case; but it is certainly an issue that could be addressed in

future work.

59

Malevergne, Y. and Sornette, D. (2004). Collective origin of the coexistence of apparent random matrix theory noise and of factors in large sample correlation matrices. Physica A: Statistical Mechanics and its Applications. 331(3–4), 660–668. 60

Tomlab Optimization. http://tomopt.com/tomlab/, 02.09.2012. 61

Levy, M. and Roll, R. (2010). The Market Portfolio May Be Mean/Variance Efficient After All. Oxford Journals, Review of Financial Studies, 23(6), 2464-2491. Ni, X., Malevergne, Y., Sornette, D. and Woehrmann, P. (2011). Robust Reverse Engineering of Cross Sectional Returns and Improved Portfolio Allocation Performance using the CAPM. Journal of Portfolio Management, 37(4), 76-85. 62

Brière, M., Drut, B., Mignon, V., Oosterlinck, K. and Szafarz, A. (2011). Is the Market Portfolio Efficient? A New Test to Revisit the Roll (1977) versus Levy and Roll (2010) Controversy. EconomiX Working Papers, 2011-20.

130

Chapter 9

9 References

General note on the references for this entire Thesis: each time there is need for a direct

reference (for a graph, for example) throughout the text, the source is given in a footnote in

the same page. All the sources used for this work, including the ones mentioned in the

footnotes, are provided in this Chapter, organized in three categories: Books, Papers and

Websites.

9.1 Books

1. Brammertz, W., Akkizidis, I., Breymann, W., Entin, R., Rüstmann, M. (2011). Unified

Financial Analysis: The Missing Links of Finance. John Wiley & Sons, The Wiley

Finance Series.

2. Brealey, R. A., Myers, S. T., Franklin, A. (2008). Principles of Corporate Finance.

Mcgraw-Hill, Irwin, 9.

3. Hull, J. C. (2011). Options, Futures and Other Derivatives. Prentice Hall, 9.

9.2 Papers

1. Brière, M., Drut, B., Mignon, V., Oosterlinck, K. and Szafarz, A. (2011). Is the Market

Portfolio Efficient? A New Test to Revisit the Roll (1977) versus Levy and Roll (2010)

Controversy. EconomiX Working Papers, 2011-20.

2. Chesnay, F. and Jondeau, E. (2000). Does Correlation between Stock Returns Really

Increase during Turbulent Period? Economic Notes, 30(1), 53-80.

3. Cristelli, M., Batty, M. and Pietronero, L. (2012). There is More than a Power Law in

Zipf. Scientific Reports 2, 812.

4. DeMiguel, V., Garlappi, L., Nogales, J. and Uppal, R. (2007). Optimal versus Naive

Diversification: How Inefficient is the 1/N Portfolio Strategy? Rev. Financ. Stud,

22(5), 1915-1953.

5. DeMiguel, V., Garlappi, L., Nogales, J. and Uppal, R. (2009). A Generalized Approach

to Portfolio Optimization: Improving Performance by Constraining Portfolio Norms.

Management Science, 55(5), 798-812.

6. Disatnik, D. and Benninga, S. (2006). Estimating the Covariance Matrix for Portfolio

Optimization. Available at SSRN: http://ssrn.com/abstract=873125 or

http://dx.doi.org/10.2139/ssrn.873125.

131

7. Fama, E. F. and French, K. R. (1993). Common risk factors in the returns on stocks

and bonds. Journal of Financial Economics. 33 (1), 3–56.

8. Flavin, T. J., Hurley, M. J., Rousseau, F. (2001). Explaining Stock Market Correlation: A

Gravity Model Approach. The Manchester School, 70(S1), 87-106.

9. Forbes, K. J. and Rigobon, R. (2002). No Contagion, Only Interdependence:

Measuring Stock Market Co-Movements. The Journal of Finance, 57(5), 2223-2261.

10. Jagannathan, R. and Ma, T. (2003). Risk Reduction in Large Portfolios: Why Imposing

the Wrong Constraints Helps. The Journal of Finance, 58)4), 1651–1684.

11. Ledoit, O. and Wolf, M. (2003). Honey, I Shrunk the Sample Covariance Matrix. UPF

Economics and Business, Working Paper No. 691.

12. Levy, M. and Roll, R. (2010). The Market Portfolio May Be Mean/Variance Efficient

After All. Oxford Journals, Review of Financial Studies, 23(6), 2464-2491.

13. Lintner, L. (1965). The valuation of risk assets and the selection of risky investments

in stock portfolios and capital budgets. Review of Economics and Statistics, 47, 13-

37.

14. Malevergne, Y. and Sornette, D. (2004). Collective origin of the coexistence of

apparent random matrix theory noise and of factors in large sample correlation

matrices. Physica A: Statistical Mechanics and its Applications. 331(3–4), 660–668.

15. Malevergne, Y., Santa-Clara, P., and Sornette, D. (2009). Professor Zipf goes to Wall

Street. NBER Working Paper No. 15295.

16. Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7, 77-91.

17. Ni, X., Malevergne, Y., Sornette, D. and Woehrmann, P. (2011). Robust Reverse

Engineering of Cross Sectional Returns and Improved Portfolio Allocation

Performance using the CAPM. Journal of Portfolio Management, 37 (4), 76-85.

18. Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under

Conditions of Risk. The Journal of Finance, 19 (3), 425-442.

19. Zipf, G. K. (1949). Human behavior and the principle of least effort. Oxford, England:

Addison-Wesley Press. xi 573 pp.

9.3 Websites

Note: all websites cited in this section were visited from August 2012 until September 2013,

for the purposes of this Thesis. For each website, the date is stated individually at the end of

each link, of which the general website origin/Institution is also provided as a title.

1.4 Guys from Rolla. Sorting a Two-Dimensional Array using the Bubble Sort.

http://www.4guysfromrolla.com/webtech/011601-1.shtml, 27.06.2013.

2.All Business. S&P global 1200 index.

http://www.allbusiness.com/glossaries/s-p-global-1200-index/4955586-1.html

3.Alpha Hive. Asset Pricing.

http://alphahive.wordpress.com/2013/05/20/asset-pricing-part-3a-time-series-

regression-capm/, 12.06.2013.

132

4.Attain Capital Management. Gold and Stocks Decoupling?

http://managed-futures-blog.attaincapital.com/2012/12/20/gold-and-stocks-

decoupling/, 13.07.2013.

5.Bespoke Investment Group. Asset Class Correlations in 2012.

http://www.bespokeinvest.com/thinkbig/2012/8/14/asset-class-correlations-in-

2012.html, 12.08.2013.

6.Bond vigilantes. Index GPL.

http://www.bondvigilantes.com/blog_files/UserFiles/Image/stock_market_crashe

s2.jpg, 19.04.2013.

7.Bowgett Investments. General Trading Rules.

http://www.bowgett.com/Resources/GeneralRules.aspx, 11.06.2013

8.Calculated Risks. Comparing Stock Market Crashes.

http://www.calculatedriskblog.com/2008/10/comparing-stock-market-

crashes.html, 14.07.2013.

9.Carnegie Mellon University. Matlab in Chemical Engineering. Constrained

optimization.

http://matlab.cheme.cmu.edu/2011/12/24/constrained-optimization/,

12.11.2012.

10.Center of Research in Security Prices. Indexes Databases.

http://www.crsp.com/products/indices.htm, 12.01.2013.

11.Chart your Trade. S&P 500 Index.

http://chartyourtrade.com/wp-content/uploads/2013/06/SP-500-Daily-06-21-

2013-2.jpg, 15.04.2013.

12.Chart Your Trade. State of the Market of 09.13.2012.

http://chartyourtrade.com/tag/nyse/, 13.02.2013.

13.Chicago Booth. Center for Research in Security Prices. Indexes Databases.


14.Chron. Small Business. How to Input Excel Files into Matlab.

http://smallbusiness.chron.com/input-excel-files-matlab-40068.html, 19.04.2013.

15.Codeforge. Fmincon.

http://www.codeforge.com/read/45196/fmincon.m__html, 25.11.2012.

16.Colorado State University. Constrained Optimization using fmincon.

http://www.math.colostate.edu/~gerhard/classes/331/lab/fmincon.html,

25.11.2012.

17.Columbia University. Business Guides Bloomberg.

http://library.columbia.edu/indiv/business/guides/bloomberg.html, 05.12.2012.

18.Computer Groups. Too many output arguments error using fmincon.

http://compgroups.net/comp.soft-sys.matlab/too-many-output-arguments-error-

using-fm/1900928, 20.02.2013.

19.Czech Technical University in Prague. Symbolic Math Toolbox.

http://radio.feld.cvut.cz/matlab/toolbox/symbolic/ch115.html, 28.10.2012.

20.Daily Markets. Claymore Launches Three Wilshire ETFs (WFVK, WXSP, WREI).

http://www.dailymarkets.com/stock/2010/03/09/claymore-launches-three-

wilshire-etfs-wfvk-wxsp-wrei/, 13.01.2013.

133

21.Dartmouth College. Current Research Returns.

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html,

13.03.2013.

22.Dartmouth College. Developed Market Factors and Returns.

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html#Deve

loped, 26.03.2013.

23.Data mining in Matlab BlogSpot. Getting data into and out of excel.

http://matlabdatamining.blogspot.ch/2009/02/getting-data-into-and-out-of-

excel.html, 19.04.2013.

24.Department of Chemical Engineering, Bogazici University. Nonlinear Programming

Examples.

http://www.che.boun.edu.tr/courses/che477/matlab477/html/nlp477.html,

10.07.2013.

25.Department of Mathematics, University of Washington. Markowitz Mean-Variance

Portfolio Theory.

http://www.math.washington.edu/~burke/crs/408/fin-proj/mark1.pdf,

12.07.2013.

26.Department of Radio Engineering, Czech Technical University in Prague. Output

Headings: Medium-Scale Algorithms.

http://radio.feld.cvut.cz/matlab/toolbox/optim/opttut33.html, 14.07.2013.

27.Dimensional. Fama/French Forum.

http://www.dimensional.com/famafrench/, 11.07.2013.

28.Edaboard. Matlab define Matrix.

http://www.edaboard.com/thread128612.html, 19.11.2012.

29.Efficient Solutions Inc. Mean-Variance Optimization. Modern Portfolio Theory,

Markowitz Portfolio Selection.

http://www.effisols.com/basics/MVO.htm, 26.06.2013.

30.Elite Trader. Wilshire 5000 Equal Weight.

http://www.elitetrader.com/vb/printthread.php?threadid=92694, 24.08.2013.

31.Empirical Finance Blog. How to use the Fama French Model.

http://blog.empiricalfinancellc.com/2011/08/how-to-use-the-fama-french-

model/, 18.09.2012.

32.ETF Database. RSP-S&P Equal Weight ETF Fundamentals.

http://etfdb.com/etf/RSP/fundamentals/, 15.12.2012.

33.ETF Database. S&P Equal Weight Index ETF Returns.

http://etfdb.com/index/sp-equal-weight-index/returns/, 15.12.2012.

34.Federal Reserve Bank of St. Louis. Financial Indicators.

http://research.stlouisfed.org/fred2/categories/46/downloaddata, 14.04.2013.

35.Federal Reserve. Selected Interest Rates.

http://www.federalreserve.gov/releases/h15/update/, 11.06.2013.

36.Financial Sense. Clues to watch for the End of QE “Infinity”.

http://www.financialsense.com/contributors/lance-roberts/clues-watch-end-qe-

infinity, 23.08.2013.

37.Financial Web Ring. Equal weighted Indices.

http://www.financialwebring.org/gummy-stuff/equal-weights.htm, 11.08.2013.

134

38.Florida State University. Fmincon.

http://people.sc.fsu.edu/~jburkardt/m_src/fmincon/fmincon.html, 24.11.2012.

39.Fool. The Power Law Distribution of Market Capitalization.

fool.com/The_Power-Law_Distribution_of_Market_Capitalization, 29.06.2013.

40.Forex Live. Guest trader: the Schubes Files.

http://www.forexlive.com/blog/2013/06/17/forexlive-new-feature-guest-trader-

the-schubes-files-day-1/, 30.08.2013.

41.Generation X Finance. How Correlation between asset classes affects your

portfolio.

http://genxfinance.com/how-correlation-between-asset-classes-affects-your-

portfolio/, 07.04.2013.

42.Google Groups. Fmincon and TolCon.

https://groups.google.com/forum/?fromgroups=#!topic/comp.soft-

sys.matlab/HP0zp6CIevY, 25.10.2012.

43.Gus Hart’s Homepage, Brigham Young University. Loops.

http://msg.byu.edu/matlab/node42.html, 16.06.2013.

44.Harvard Business School, Baker Library Bloomberg Center. Wharton Research Data

Service.

http://www.library.hbs.edu/go/wrds.html, 05.12.2012.

45.HP Labs. Zipf, Power-laws, and Pareto – a ranking Tutorial.

http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html, 23.05.2013.

46.Index Universe. Claymore Filling: Equal-Weighting the Wilshire 5000.

http://www.indexuniverse.com/sections/news/7052-claymore-filing-equal-

weighting-the-wilshire-5000.html, 17.08.2013.

47.Intmath. Zipf Distributions log log Graphs and Site Statistics.

http://www.intmath.com/blog/zipf-distributions-log-log-graphs-and-site-

statistics/702, 12.09.2012.

48.Investopedia. Efficient Frontier.

http://www.investopedia.com/terms/e/efficientfrontier.asp, 02.05.2013.

49.Investopedia. S&P 500: Market weight Vs Equal Weight.

http://www.investopedia.com/articles/exchangetradedfunds/08/market-equal-

weight.asp, 05.08.2013.

50.Investopedia. Stock Market Risk: Wagging the Tail.

http://www.investopedia.com/articles/financial-theory/09/bell-curve-wag-

tails.asp#axzz2LvXpyznj, 07.10.2012.

51.Investopedia. Stock Market Risk: Wagging the Tails.

http://www.investopedia.com/articles/financial-theory/09/bell-curve-wag-

tails.asp#axzz2LvXpyznj, 07.10.2012.

52.Investor Place. Until the S&P 500 breaks this level, don’t expect much.

http://investorplace.com/2013/01/daily-stock-market-news-until-the-sp-500-

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53.Issuu. Bloomberg Tutorial.

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145

Appendix

1 Acquiring Bloomberg Data

1.1 Introduction

There was a large amount of peculiarities that had to be dealt with, when attempting to

download data with the Bloomberg excel add-in. The most important one is the way that the

add-in uses, in order to fill in the columns (it is always the case, that each column represents

the field of a different stock), when it comes to large numbers of columns (above several

hundred). In several cases the add-in would fill in every second column the intermediates

were completely blank) and we would have to click “enter” in the formula of each separate

cell, in order to be filled (when the number was small). In other cases, especially above 1500

columns, after a certain point (the last 3-5 hundred, for example) were also completely

blank.

For these reasons and a few more, in order to acquire all the available data for each major

index, a massive amount of time was consumed. This would include the break-up of the

large sets in several smaller excel files (so that the add-in would not jam) and then checking

that the columns that were still blank are because there are actually no data available for

the particular stock for the particular period and not because of any other bug. After the

check was concluded, all the partial excel sheets would be accumulated back into a master

file, in which we would run the VBA script, in order to keep only the stocks for which we

have full data over the desire period (even if one out of 215 months of returns was missing,

the entire column would be deleted).

1.2 Bloomberg Formulas

Below we can see all the different Bloomberg formulas used for the data acquisition, as well

as the explanation of the Bloomberg fields. For reasons of space, every different formula is

presented only for a particular set/time period.

1. Index Members

=BDS("Security ID","Field Mnemonic") In order to download the members of the Wilshire 5000 Index, we would use the previous formula as:

=BDS("W5000 Index","INDX_MEMBERS")

146

2. Bloomberg Security Names

After the previous command, we need to make the index members names compatible with the Bloomberg security names. For that reason, we employ the following excel formula:

=A$1&" EQUITY"

3. General formula (historical fields)

For downloading any historical field from the Bloomberg database, we use the formula: =BDH("Security ID","Field mnemonic","Start Date","End Date")

4. Monthly returns (using day-to-day field with the option “per=cm”):

Field mnemonic: "DAY_TO_DAY_TOT_RETURN_GROSS_DVDS"

The formula used for this field is the following:

=BDH(B3,"DAY_TO_DAY_TOT_RETURN_GROSS_DVDS","01/01/1991","03/01/2009","per=cm

","dts=h")

* dts=h is the option for hide the dates column, particularly handy when downloading

historical returns or other fields for 5000 stocks with their names in a single row, one next to

each other.

5. Market capitalization

=BDH(B3,"CUR_MKT_CAP","01/01/2008","03/01/2009","per=cm","dts=h")

*The dates of the last year under examination were set in this formula, and not for the

entire period. The reason for that is that in previous research attempts, the weight of each

stock was always calculated based on the last month of data. In addition, the Book-to-

Market and Market capitalization for the formulation of the 25 Fama French portfolios used

the T-1 years last month (where T again, the most recent year of data), and since this

approach was used in this Thesis too, no previous data for these particular fields were

necessary. Only for certain cases, like the tests for weights for different periods (first month

of data, middle one and last) the time series for the entire period were downloaded.

6. Market-to-Book

In Bloomberg, no historical field for Book-to-Market exists; therefore the simple inverse 1/x

of this one was used instead.

=BDH(L7,"MARKET_CAPITALIZATION_TO_BV ","01/01/2008","01/01/2009","dts=h")

7. Indices Formulas

Below we can see the formulas used in order to get the monthly/yearly returns of the

various benchmark indices:

147

=BDH($A$1;$B$2:$C$2;"1/1/1990";"";"Dir=V";"Dts=S";"Sort=A";"Quote=C";"QtTyp=Y";"Days

=T";"Per=cm";"DtFmt=D";"UseDPDF=Y";"CshAdjNormal=N";"CshAdjAbnormal=N";"CapChg=

N";"cols=3;rows=283")

Used for the Indices: W5000 Index, SPW Index, RAY Index (RUSSELL 3000) and SPOGLOB

Index (S&P 1200).

1.3 Bloomberg Fields

In this paragraph we provide with the descriptions for the fields used from Bloomberg. All

definitions were acquired from the Bloomberg Terminal system, and as such are property of

Bloomberg. They are given in this sector for the kind convenience of the reader, so that it

will be clear, which exact returns and other parameters were used.

A) DAY_TO_DAY_TOT_RETURN_GROSS_DVDS

=BDH(B3,"DAY_TO_DAY_TOT_RETURN_GROSS_DVDS","01/01/1991","03/01/2009","per=cm

","dts=h")

Day to Day Total Return (Gross Dividends)

One day total return as of today. The start date is one day prior to the end date (as of date).

Historically, this is a series of day to day total return values for daily periodicity. Applicable

periodicity values are daily, weekly, monthly, quarterly, semi-annually and annually. Gross

dividends are used.

There is a limitation of 5000 price observations between the start and end dates.

B) MARKET_CAPITALIZATION_TO_BV

=BDH(C2;"MARKET_CAPITALIZATION_TO_BV";"01/01/2003";"03/15/2013";"per=cm";"dts=h

")

Also available as Historical field

Measure of the relative value of a company compared to its market value. Calculated as:

Market Capitalization/Book Value

Where:

Market Capitalization is RR902 (CUR_MKT_CAP) for current or daily ratio

Market Capitalization is RR250 (HISTORICAL_MARKET_CAP) for historical fundamental

period ratio

Book Value is RR010, TOT_COMMON_EQY.

148

Unit: Actual.

C) CUR_MKT_CAP

=BDH(C2;"CUR_MKT_CAP";"01/01/2003";"03/15/2013";"per=cm";"dts=h")


Total current market value of all of a company's outstanding shares stated in the pricing

currency. Capitalization is a measure of corporate size. For the historical market value, use

Historical Market Cap (RR250, HISTORICAL_MARKET_CAP), returned in the fundamental

currency. Market capitalization will be returned in the pricing currency of the security

except for the cases:

If Pricing Currency then Market Currency

GBp (BRITISH PENCE) GBP (BRITISH POUND)

ZAr (S. AFR. CENTS) ZAR (SOUTH AFRICAN RAND)

IEp (Irish Pence) IEP (IRISH PUNT)

ILs (Israeli Agorot) ILS (ISRAELI SHEKEL)

ZWd (Zimbabwe Cents) ZWD (ZIMBABWE DOLLAR)

BWp (Botswana Thebe) BWP (BOTSWANA PULA)

KWd (KUWAIT FILS) KWD (KUWAITI DINAR)

SZl (Swaziland cents) SZL (SWAZILAND LILANGENI)

MWk (MALAWI TAMBALA) MWK (MALAWI KWACHA)

NON MULITIPLE-SHARE COMPANIES:

Current market capitalization is calculated as:

Current Shares Outstanding * Last Price

Where:

Current Shares Outstanding is DS124, EQY_SH_OUT

Last Price is PR005, PX_LAST

For certain countries, DS124 excludes treasury shares. Please see DS124, EQY_SH_OUT

definition for details.

If the last price available is past more than 50 days, refer to Market Cap - Last Trade (RX066,

MKT_CAP_LAST_TRD) (available to set as Market Cap Default on FPDF settings).

149

For companies which trade on multiple regional exchanges, the Composite Ticker is used in

the end-of-day calculation of the market cap. Refer to Composite Exchange Code (DS291,

COMPOSITE_EXCH_CODE) for the exchange code of the composite ticker. The intraday value

of market cap is calculated separately for each local exchange ticker.

MULTIPLE-SHARE COMPANIES:

Current market cap is the sum of the market capitalization of all classes of common stock, in

millions. If only one class is listed, the price of the listed-class is applied to any unlisted

shares to determine the total market value. If there are two or more listed classes and one

or more unlisted classes, the average price of the listed classes is applied to the unlisted

shares to compute the total market value. 'Company Has Multiple Shares' (DS738,

MULTIPLE_SHARE) indicates if the company has multiple shares.

If a class of shares has not traded for more than 60 days, 'Last Price' will take what is

available in the order of:

1. Ask price

2. Bid price

3. Weighted average of all related priced securities.

For a single class of market capitalization, please refer to Current Market Capitalization of a

Share Class (RR233, CURRENT_MARKET_CAP_SHARE_CLASS).

Figure is reported in million; the Scaling Format Override (DY339, SCALING_FORMAT) can be

used to change the display units for the field.

Equity index market capitalization values are stated in pricing currency.

D) CURR_BOOK_VAL

=BDH(C2;"CURR_BOOK_VAL";"01/01/2003";"03/15/2013";"per=cm";"dts=h")

Current book value of properties held by the trust, calculated as:

Net Fixed Assets + Disclosed Intangibles + Net Purchase Amounts (from last period end

date).

Where:

'Net Fixed Assets' is BS032, BS_NET_FIX_ASSET,

'Disclosed Intangibles' is BS138, BS_DISCLOSED_INTANGIBLES, and

'Net Purchase Amounts' is derived from contributed data.

E) TOT_COMMON_EQUITY

=BDH(C2;"TOT_COMMON_EQUITY";"01/01/2003";"03/15/2013";"per=cm";"dts=h")

150


INDUSTRIALS

Total common equity is calculated using the following formula:

Share Capital & APIC + Retained Earnings

BANKS



FINANCIALS



INSURANCES



UTILITIES



151

2 Matlab Code

Initially, a script was created with all necessary commands for the full implantation of

Optimization Problem 1, but with symbolic variables. The purpose of this exercise was to

fully understand the code and verify that it functions exactly as expected.

2.1 Matlab script v2.4.m

# Copyright (c) 2013. All rights reserved.

###########################################################################

# File: “Matlab script v2.4.m”

# Project: Master Thesis

# Description: Symbolic full Optimization Problem 1 script

# Author: Theodoros Giannakopoulos (TG)

###########################################################################

% excel handling

fileName = 'returns div 100.xlsx';

% a = xlsread(fileName);

x = xlsread(fileName);

% r = x(:, 1);

% plot (r);

% get five rows

% rr = x(:, 1:5);

% % % % % % % % % %

% Ledoit and Wolf %

% % % % % % % % % %

% de-mean returns

[t,n]=size(x);

%n columns' number

meanx=mean(x);

%already limiting the outliers:

x=x-meanx(ones(t,1),:);

% compute sample covariance matrix

sample=(1/t).*(x'*x);

152

% compute prior

var=diag(sample);

sqrtvar=sqrt(var);


n)/(n*(n-1));

prior=rho*sqrtvar(:,ones(n,1)).*sqrtvar(:,ones(n,1))';

prior(logical(eye(n)))=var;

%create sample standard deviations matrix:

stdeviations = diag(sqrtvar);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% initianilization of shrink variable

shrink = -1;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

if (nargin < 2 | shrink == -1) % compute shrinkage parameters

c2=norm(sample-prior,'fro')^2;

y=x.^2;

p=1/t*sum(sum(y'*y))-sum(sum(sample.^2));

rdiag=1/t*(sum(sum(y.^2)))-sum(var.^2);

v=((x.^3)'*x)/t-(var(:,ones(1,n)).*sample);

v(logical(eye(n)))=zeros(n,1);

roff=sum(sum(v.*(sqrtvar(:,ones(n,1))'./sqrtvar(:,ones(n,1)))));

r=rdiag+rho*roff;

% compute shrinkage constant

k=(p-r)/c2;

shrinkage=max(0,min(1,k/t));

else % use specified number

shrinkage = shrink;

end

% compute the estimator

sigma=shrinkage*prior+(1-shrinkage)*sample;

%-------------------------------------------------------------------

% construction of the objective (Distance) function

% first sum, m for returns μ

i=1:n;

m = sym(zeros(1, n));

for k=1:n

m(k) = sym(sprintf('m%d', k));

end

153

% sample return vector

% msam = x(1, 1:n);

msam = meanx;

% sample standard deviation matrix

% sqrtvar estimated after the shrinkage of x=x-meanx(ones(t,1),:);

ssam = sqrtvar;

ssamrev = sqrtvar.';

firstsum(i) = ((m(i) - msam(i))./ssamrev(i)).^2;

myy = sum(firstsum);

%=================================================================

% second sum, s for standard deviations σ i=1:n;

s = sym(zeros(1, n));

for k=1:n

s(k) = sym(sprintf('s%d', k));

end

secondsum(i) = ((s(i) - ssamrev(i))./ssamrev(i)).^2;

g = sum(secondsum);

%==================================================================

% first sum:

myy;

% second sum:

g;

% a parameter

a=0.75;

% final expression for the distance D:

D = ((1/n)*a*myy + (1/n)*(1-a)*g)^(1/2);

%====================================================================

%

% construnction of nonlinear constraint

%

%====================================================================

% standard deviations matrix:

% S = diag(sym('s',[1 n]));

S = diag(s);

% returns matrix:

%M = diag(m);

m;

154

%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

% sample covariance matrix

sample;

% sample CORRELATION matrix

R = corrcoef(x);

RCOV = corrcov(sample);

% The above two methods return the exact same result

% Correlation matrix based on the shrunk covariance matrix:

RSIGMA = corrcov(sigma);

%++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

%====================================================================

% weights

% value-weighted market proxy portfolio weights

% endweight = n/10 - 0.1;

% xm = 0:0.1:endweight;

% xma = xm';

% Actual weights from Bloomberg:

xm = [0.00122638067003372 0.00201676814145104 0.0166343578669329

0.000796167105894909 0.00763137808267098 …….

0.000493801758618225 0.00141373789183443];

xma = xm';

% proper weights (change the row/column field)

% for k=1:n

% marketcap(k) = x(k)/sum(x);

% end

%====================================================================

% [μ - rf] term

%risk free rate

rf = 0.0012;

mr2 = m - rf;

mr2n = mr2.';

% Final expression for nonlinear restriction

% q is missing

% q = (market standard deviaton)^2/(market expected returns - risk

free rate)

% setting market parameters equal to sample parameters:

% marketsqrtvar = mean(sqrtvar);

% marketmean = mean(meanx);

%

% q = (marketsqrtvar)^2/(marketmean-rf);

marketsqrtvar = 0.104;

155

marketmean = 0.052;

q = (marketsqrtvar)^2/(marketmean-rf);

%q=2.2906;

% and,finally:

%ceqorig = S*R*S*xma - q*mr2n;


%R = corrcoef(x);

%====================================================================

%

%construnction of nonlinear constraint with shrunk correlation matrix

%

%====================================================================

% compute sample covariance matrix (before LW)

sample;


sigma;

% sample CORRELATION matrix from Ledoit-Wolf

RSIGMA;


ceq = S*RSIGMA*S*xma - q*mr2n;

%x0=[meanx';sqrtvar];

2.2 Optimization Problem 1

Note: for several of the functions presented in the rest of this chapter, most of the values of

the “weights” variable had to be deleted, so that we will not have an excess amount of plain

numbers throughout several pages, each time. As with all code files used for this Thesis,

these can be found in the accompanying digital disk.

(Non-automatic history approach.)

2.2.1 Command line script


###########################################################################

# File: “command line script”


156

# Description: commands for running the Matlab scripts, Optimization Problem 1 (OPT 1)

# Author: Theodoros Giannakopoulos(TG)

###########################################################################

clear all global xexcel; global weights; global x0; global x0i; global lb; funInitial options=optimset('Display','iter-

detailed','Diagnostics','on','FunValCheck','on'); [x,fval,exitflag,output] =

fmincon(@funD,x0,[],[],[],[],lb,[],@funCeq,options);

###########################################################################

2.2.2 Function funInitial


###########################################################################

# File: “funInitial.m”


# Description: variable xexcel has been created and stored to the workspace, OPT 1


###########################################################################

function [] = funInitial()

% global n; % global x0; global xexcel; global weights; global x0; global x0i; global lb;

fileName = 'r.xlsx'; xexcelR = xlsread(fileName); xexcel=xexcelR/100;

fileName = 'w.xlsx'; mktcaps = xlsread(fileName);

[t,n]=size(xexcel);

meanx=mean(xexcel);

157

% compute sample covariance matrix sample=(1/t).*(xexcel'*xexcel);

% compute prior

var=diag(sample); sqrtvar=sqrt(var);

mktcapsSum=sum(mktcaps);

i=1:n;

weights(i)=mktcaps(i)/mktcapsSum;

% first sum, m for returns ì

i=1:n; % m = sym(zeros(1, n)); % for k=1:n % m(k) = sym(sprintf('m%d', k)); % end

x0i=[meanx';sqrtvar]; x0=x0i';

lb=[-Inf*ones(1,n);zeros(1,n)];

disp('global variables xexcel,weights,x0,x0i and lb have been created

and stored to the workspace :)');

end

###########################################################################

2.2.3 Function funD


###########################################################################

# File: “funD.m”


# Description: objective function (to be minimized), OPT 1


###########################################################################

function [D] = funD(s_m)

% s_m is a 2n*1 vector % funInitial(1); % global n;

158

global xexcel; global weights; global x0; global x0i;

% m=varm; % s=vars; % S = struct('m','s'); % C = struct2cell(S); % objecfunD(C{:}); % d = m,s;

% excel handling

%------------------------------------------------ % fileName = 'Bloomberg top 50.xlsx'; % x = xlsread(fileName); %------------------------------------------------

%r = x(:, 1); %plot (r);

%get five rows %rr = x(:, 1:5);

% % % % % % % % % % % Ledoit and Wolf % % % % % % % % % % %

% de-mean returns

[t,n]=size(xexcel);

%n columns' number

i=1:n;

m=s_m(1:n); s=s_m(n+1:end); %s>= 0;

meanx=mean(xexcel); xexcel2=xexcel-meanx(ones(t,1),:);


% compute prior


% first sum, m for returns ì

i=1:n;

159

% m = sym(zeros(1, n)); % for k=1:n % m(k) = sym(sprintf('m%d', k)); % end

% sample return vector msam = meanx; % msam = x(1, 1:n);

% sample standard deviation matrix % ssam = sqrtvar; ssamrev = sqrtvar.';



%=================================================

% second sum, s for standard deviations ó

i=1:n;

% s = sym(zeros(1, n)); % for k=1:n % s(k) = sym(sprintf('s%d', k)); % end


g = sum(secondsum);

%=================================================

% myy; % % g; % a parameter

a=0.75;

% final expression for the distance D: D = ((1/n)*a*myy + (1/n)*(1-a)*g)^(1/2);

%DISTANCE = D;

%x0=[meanx';sqrtvar]; %vpa(D)

end

###########################################################################

2.2.4 Function funCeq

160


###########################################################################

# File: “funCeq.m”


# Description: nonlinear equality constraints, OPT 1


###########################################################################

function [c,ceq] = funCeq(s_m)

%x0=[meanx';sqrtvar]; %m=varm;

global xexcel; global weights; global x0; global x0i;

%s=vars; %nonlinear inequality constraint c c=[];

% excel handling

%fileName = 'bloom day 2 fav 2.xlsx';

%------------------------------------------------ % fileName = 'Bloomberg top 50.xlsx'; % x = xlsread(fileName); %------------------------------------------------

%r = x(:, 1); %plot (r); %get five rows %rr = x(:, 1:5);

% % % % % % % % % % % Ledoit and Wolf % % % % % % % % % % %

% de-mean returns [t,n]=size(xexcel);

%n columns' number

m=s_m(1:n); s=s_m(n+1:n+n); %s>= 0;

meanx=mean(xexcel);

161

xexcel2=xexcel-meanx(ones(t,1),:);


% compute prior



n)/(n*(n-1)); prior=rho*sqrtvar(:,ones(n,1)).*sqrtvar(:,ones(n,1))';



%stdeviations = diag(sqrtvar);

% initianilization of shrink variable:

shrink = -1;


c2=norm(sample-prior,'fro')^2; y=xexcel2.^2; p=1/t*sum(sum(y'*y))-sum(sum(sample.^2)); rdiag=1/t*(sum(sum(y.^2)))-sum(var.^2); v=((xexcel2.^3)'*xexcel2)/t-(var(:,ones(1,n)).*sample); v(logical(eye(n)))=zeros(n,1); roff=sum(sum(v.*(sqrtvar(:,ones(n,1))'./sqrtvar(:,ones(n,1))))); r=rdiag+rho*roff;

% compute shrinkage constant k=(p-r)/c2; shrinkage=max(0,min(1,k/t));

else % use specified number shrinkage = shrink;

end

% compute the estimator sigma=shrinkage*prior+(1-shrinkage)*sample;

%=================================================

% standard deviations matrix: %S = diag(sym('s',[1 n])); S = diag(s);

% returns matrix: %M = diag(m); %m;

162

% sample covariance matrix %sample;


% R = corrcoef(x); % RCOV = corrcov(sample);

%The above two methods return the exact same result

% Correlation matrix based on the shrunk covariance matrix: RSIGMA = corrcov(sigma);

% value-weighted market proxy portfolio weights % example values %endweight = n/10 - 0.1; %xm = 0:0.1:endweight; %xma = xm';

%Actual weights from Bloomberg:

%xm = [0.0280910250000000 0.0382039180000000 0.00273670100000000

0.0246119830000000 0.0107766570000000 0.0144955560000000

0.00309649000000000 0.0269766410000000 0.00580240200000000

0.0179166430000000 0.0178648600000000 0.0364889520000000

0.0210662310000000 0.0124436950000000 0.0135646230000000

0.0233340910000000 0.0188631610000000 0.0198748980000000

0.00581223200000000 0.00868560200000000 0.0225968770000000

0.0229924130000000 0.0101871690000000 0.0216415520000000

0.00529777100000000 0.00848838600000000 0.0102632990000000

0.0145289290000000 0.0368832980000000 0.00406042400000000

0.0111329630000000 0.00579704200000000 0.00778162300000000

0.00454872600000000 0.00763275800000000 0.0188112170000000

0.0114758390000000 0.0206147130000000 0.0226579860000000

0.0107347910000000 0.00640195400000000 0.00589182400000000

0.0240276140000000 0.0199382420000000 0.0110246710000000

0.00579213500000000 0.0196435420000000 0.00838663000000000

0.0326277380000000 0.0240396930000000 0.00505543900000000

0.00612670100000000 0.0261632590000000 0.00120394500000000

0.00469026000000000 0.0130799570000000 0.0311204910000000

0.0204878650000000 0.0378937720000000 0.00681394300000000

0.0116468760000000 0.0366378620000000 0.0124714520000000];

xma = weights'; % [ì - rf] term

%risk free rate rf = 0.0012;

mr2 = m - rf; mr2n = mr2.';


% q is missing % q = (market standard deviaton)^2/(market expected returns - risk

free rate) % setting market parameters equal to sample parameters (FOR NOW):

CHANGED

163

% marketsqrtvar = mean(sqrtvar); % marketmean = mean(meanx); % q = (marketsqrtvar)^2/(marketmean-rf);

marketsqrtvar = 0.104; marketmean = 0.052;


%q=2.2906;

% and,finally: %ceqorig = S*R*S*xma - q*mr2n;

% sample CORRELATION matrix %R = corrcoef(x);

%==================================================================== % %construnction of nonlinear constraint with shrunk correlation matrix % %====================================================================

% sample covariance matrix from Ledoit-Wolf %sigma;

% sample CORRELATION matrix from Ledoit-Wolf %RSIGMA;

% Final expression for nonlinear restriction %ceqinv = S*R*S*xma - q*mr2n;

%ceqinv.';

% Final expression for nonlinear restriction with LW


%ceqshrunk = ceqshrunkinv.';



End

###########################################################################

2.2.5 Function runfmincon # Copyright (c) 2013. All rights reserved.

###########################################################################

# File: “runfmincon.m”

164


# Description: script for the entire OPT 1, saving all intermediate values


###########################################################################

function [history,searchdir] = runfmincon

% Set up shared variables with OUTFUN

history.x = [];

history.fval = [];

searchdir = [];

%clear all

%reset(symengine)

% call optimization

x0 = [0.0140511013386440 0.0165456219521299 0.0305610291613639

0.0724360224951741 0.124277940596655 0.0396964114859922];

options = optimset('outputfcn',@outfun,'display','iter',...

'Algorithm','active-set');

xsol = fmincon(@funD,x0,[],[],[],[],[-Inf -Inf -Inf -Inf -Inf -Inf -

Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf

-Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf












-Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf -Inf 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0],[],@funCeq,options);

% Following function “stop” implemented as found in the cited link.63

function stop = outfun(x,optimValues,state)

stop = false;

switch state

case 'init'

hold on

case 'iter'

% Concatenate current point and objective function

63

Mathworks. Output Functions. http://www.mathworks.com/help/optim/ug/output-functions.html, 20.08.2012.

165

% value with history. x must be a row vector.

history.fval = [history.fval; optimValues.fval];

history.x = [history.x; x];

xlswrite('Matlaboutput.xlsx', x);

%x

% Concatenate current search direction with

% searchdir.

searchdir = [searchdir;...

optimValues.searchdirection'];

plot(x(1),x(2),'o');

% Label points with iteration number and add title.

% Add .15 to x(1) to separate label from plotted 'o'

text(x(1)+.15,x(2),...

num2str(optimValues.iteration));

title('Sequence of Points Computed by fmincon');

case 'done'

hold off

otherwise

end

end

function [D] = funD(s_m)

% s_m is a 2n*1 vector

funInitial(1);

global n;

% m=varm;

% s=vars;

% S = struct('m','s');

% C = struct2cell(S);

% objecfunD(C{:});

% d = m,s;

% excel handling

fileName = 'top 200 div 100.xlsx';


% r = x(:, 1);

% plot (r);

% get five rows

% rr = x(:, 1:5);

% % % % % % % % % %

% Ledoit and Wolf %

% % % % % % % % % %

% de-mean returns

[t,n]=size(x);

%n columns' number

i=1:n;

m=s_m(1:n);

s=s_m(n+1:end);

%s>= 0;

166

meanx=mean(x);




% compute prior

var=diag(sample);

sqrtvar=sqrt(var);


n)/(n*(n-1));





shrink = -1;



y=x.^2;






r=rdiag+rho*roff;


k=(p-r)/c2;



shrinkage = shrink;

end



%-------------------------------------------------


i=1:n;

% m = sym(zeros(1, n));

% for k=1:n

% m(k) = sym(sprintf('m%d', k));

% end


msam = meanx;

167

% msam = x(1, 1:n);


ssam = sqrtvar;




%=================================================

% second sum, s for standard deviations σ

i=1:n;

% s = sym(zeros(1, n));

% for k=1:n

% s(k) = sym(sprintf('s%d', k));

% end


g = sum(secondsum);

%=================================================

% myy;

%

% g;

% a parameter

a=0.75;

% final expression for the distance D:

D = ((1/n)*a*myy + (1/n)*(1-a)*g)^(1/2);

% DISTANCE = D;

% x0=[meanx';sqrtvar];

% vpa(D)

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% function [c, ceq] = confun(x)

% % Nonlinear inequality constraints

% c = [1.5 + x(1)*x(2) - x(1) - x(2);

% -x(1)*x(2) - 10];

% % Nonlinear equality constraints

% ceq = [];

% end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [c,ceq] = funCeq(s_m)


% m=varm;

% s=vars;

% nonlinear inequality constraint c

168

c=[];

% excel handling

% fileName = 'bloom day 2 fav 2.xlsx';

fileName = 'top 200 div 100.xlsx';


% r = x(:, 1);

% plot (r);

% get five rows

% rr = x(:, 1:5);

% % % % % % % % % %

% Ledoit and Wolf %

% % % % % % % % % %

% de-mean returns

[t,n]=size(x);

%n columns' number

m=s_m(1:n);

s=s_m(n+1:n+n);

% s>= 0;

meanx=mean(x);




% compute prior

var=diag(sample);

sqrtvar=sqrt(var);


n)/(n*(n-1));



% create sample standard deviations matrix:

stdeviations = diag(sqrtvar);

% initianilization of shrink variable

shrink = -1;



y=x.^2;

169






r=rdiag+rho*roff;


k=(p-r)/c2;



shrinkage = shrink;

end



%------------------------------------------------


i=1:n;

% m = sym(zeros(1, n));

% for k=1:n

% m(k) = sym(sprintf('m%d', k));

% end


msam = meanx;

% msam = x(1, 1:n);


% ssam = sqrtvar;




%=================================================

% second sum, s for standard deviations σ i=1:n;

% s = sym(zeros(1, n));

% for k=1:n

% s(k) = sym(sprintf('s%d', k));

% end


g = sum(secondsum);

%=================================================

%

%construnction of nonlinear constraint

%

%=================================================

170


%S = diag(sym('s',[1 n]));

S = diag(s);

% returns matrix:

%M = diag(m);

m;


sample;


% R = corrcoef(x);

% RCOV = corrcov(sample);





% example values

%endweight = n/10 - 0.1;

%xm = 0:0.1:endweight;

%xma = xm';

%Actual weights from Yahoo Finance:

xm = [0.00275758188502065 0.00453480997299087 0.0374032346102973

0.00179022390217937 0.0171595577724969 0.00390140750942289

0.00657224287469942 0.00107182696301482 0.00364033457124486

0.00404380187640757 0.00157630025144305 ......

0149586349522 0.00215126703163037 0.000826926752136789

0.0184719696330407 0.00148913297472914 0.00118499689906005

0.00115714079690051 0.00180626078806991 0.00256774675530488];

xma = xm';

% [μ - rf] term

%risk free rate

rf = 0.0012;

mr2 = m - rf;

mr2n = mr2.';


% q is missing


free rate)



%



marketmean = 0.052;

171


% q=2.2906;

% and,finally:

% ceqorig = S*R*S*xma - q*mr2n;


% R = corrcoef(x);

%====================================================================

%

% construnction of nonlinear constraint with shrunk correl. matrix

%

%====================================================================


sigma;


RSIGMA;


% ceqinv = S*R*S*xma - q*mr2n;

% ceqinv.';



% mceqshrunk = ceqshrunkinv.';



end

end

2.3 Optimization Problem 2

In this chapter we will present only the additional functions created for the implementation

of Optimization Problem 2. The ones that are re-used, exactly as stated in the Optimization

Problem 1, will not be repeated.

2.3.1 Command line script

172


###########################################################################

# File: “command line script”


# Description: commands for running the Matlab scripts, OPT 2

# Author: Theodoros Giannakopoulos(TG)

###########################################################################

clear all

global xexcel;

funInitial

options=optimset('Display','iter-

detailed','Diagnostics','on','FunValCheck','on');

x0 = [0.0403569838238442 0.00940221891604133 0.00874400668113271

0.00319408770902294 0.00347398775072959 0.00431993692215445

0.00935585243541237 0.0125356698027965 .......

0.00987978702834312 0.00739523378757699 0.00986021319292649

0.0128226500244603 0.00987621936963414 0.00866795369270682

0.0204022179328837 0.0146942593300487];

Aeq= -[xm,zeros(1,50)];

beq= -0.0092;

[x,fval,exitflag,output] = fmincon(@funD,x0,Aeq,beq,[],[],[-Inf -Inf




-Inf -Inf -Inf -Inf -Inf -Inf 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0],[],@funCeq2,options);

###########################################################################

2.3.2 Function funInitial

Same as in Optimization Problem 1.

2.3.3 Function funD

Same as in Optimization Problem 1.

2.3.4 Function funCeq2

173


###########################################################################

# File: “funCeq2.m”


# Description: nonlinear equality constraints, OPT 2


###########################################################################

function [c,ceq] = funCeq2(s_m)


% m=varm;

global xexcel;

% s=vars;

% nonlinear inequality constraint c

c=[];

% excel handling

% fileName = 'bloom day 2 fav 2.xlsx';

%------------------------------------------------

% fileName = 'top 20 YF FINAL.xlsx';

% x = xlsread(fileName);

%------------------------------------------------

%r = x(:, 1);

%plot (r);

%get five rows

%rr = x(:, 1:5);

% % % % % % % % % %

% Ledoit and Wolf %

% % % % % % % % % %

% de-mean returns

[t,n]=size(xexcel);

%n columns' number

m=s_m(1:n);

s=s_m(n+1:n+n);

%s>= 0;

meanx=mean(xexcel);

xexcel2=xexcel-meanx(ones(t,1),:);


174

sample=(1/t).*(xexcel2'*xexcel2);

% compute prior

var=diag(sample);

sqrtvar=sqrt(var);


n)/(n*(n-1));





% initianilization of shrink variable:

shrink = -1;



y=xexcel2.^2;



v=((xexcel2.^3)'*xexcel2)/t-(var(:,ones(1,n)).*sample);



r=rdiag+rho*roff;


k=(p-r)/c2;



shrinkage = shrink;

end



%=================================================

%

% construnction of nonlinear constraint

%

%=================================================


%S = diag(sym('s',[1 n]));

S = diag(s);

% returns matrix:

%M = diag(m);

%m;


% sample;


175

% R = corrcoef(x);

% RCOV = corrcov(sample);





% example values

% endweight = n/10 - 0.1;

% xm = 0:0.1:endweight;

% xma = xm';

% Actual weights from Yahoo Finance:

xm = [0.0273949642352741 0.0603798052558709 0.0271359894286672

0.0447680466363324 0.0417075297237040 0.0342492713609203

0.0265834454272898 0.0250328047309234 0.0287739772360470

0.0279067663775509 0.0285331386379770 0.0244773717179720

0.0225206753319110 0.0198011681107040 0.0204479974885029

0.0167583498028455 0.0241459171740712 0.0266961790746409

0.0269708102846110 0.0227470998891522 0.000819543951002549

0.0166886772603584 0.0157440031158128 0.0231230397426509

0.0290511355404517 0.0155067643713051 0.00704917098699307

0.0227298612324019 0.0115327880152489 0.0159559912101363

0.0102784179373393 0.0228563049279067 0.0108961513921709

0.0206162449639539 0.0199735504373512 0.0107637602483512

0.0210847952024665 0.0150518181546959 0.0110448044600769

0.00643198634301398 0.00999571535939178 0.0121756328633725

0.00987978702834312 0.00739523378757699 0.00986021319292649

0.0128226500244603 0.00987621936963414 0.00866795369270682

0.0204022179328837 0.0146942593300487];

xma = xm';

% [μ - rf] term

% risk free rate

rf = 0.0012;

mr2 = m - rf;

mr2n = mr2.';


% q is missing


free rate)





marketmean = 0.052;


% q=2.2906;

% and,finally:

% ceqorig = S*R*S*xma - q*mr2n;

176


% R = corrcoef(x);

%====================================================================

%

% construnction of nonlinear constraint with shrunk correl. matrix

%

%====================================================================


% sigma;


% RSIGMA;


%ceqinv = S*R*S*xma - q*mr2n;

%ceqinv.';



%ceqshrunk = ceqshrunkinv.';

%ceqshrunkinv = ceqshrunkinvbef.';

%===========================%

% SECOND NONLINEAR EQUALITY

%===========================%

%Nonlinear Equality constraint #2

SigmaMiden = 0.042;

prsq = SigmaMiden^2;

%opt2 = prsq*ones(1,5);

%opt2i=opt2.';

%ceqtest = S*RSIGMA*S*xma - opt2i;


%ceq2MUL = ceq2*ones(10,1);

%ceq3= xm*S*RSIGMA*S*xma-5;


%ceq = [ceqshrunkinv.'];


end

177

3 Portfolios Stock Constituents

3.1 The 100 first stocks from S&P 1200 Global Index (1047 stocks

with full data during the period 2003-2013)

SECURITY NUMBER INDX_MEMBERS SECURITY NAME

1 000270 KP Kia Motors Corp

2 000830 KP Samsung C&T Corp

3 005380 KP Hyundai Motor Co

4 005490 KP POSCO

5 005930 KP Samsung Electronics Co Ltd

6 009540 KP Hyundai Heavy Industries Co Lt

7 012330 KP Hyundai Mobis

8 033780 KP KT&G Corp

9 055550 KP Shinhan Financial Group Co Ltd

10 066570 KP LG Electronics Inc

11 1 HK Cheung Kong Holdings Ltd

12 11 HK Hang Seng Bank Ltd

13 13 HK Hutchison Whampoa Ltd

14 1301 TT Formosa Plastics Corp

15 1303 TT Nan Ya Plastics Corp

16 1326 TT Formosa Chemicals & Fibre Corp

17 16 HK Sun Hung Kai Properties Ltd

18 1801 JT Taisei Corp

19 1802 JT Obayashi Corp

20 1803 JT Shimizu Corp

21 1812 JT Kajima Corp

22 19 HK Swire Pacific Ltd

23 1925 JT Daiwa House Industry Co Ltd

24 1928 JT Sekisui House Ltd

25 1963 JT JGC Corp

26 2 HK CLP Holdings Ltd

27 2002 TT China Steel Corp

28 2267 JT Yakult Honsha Co Ltd

29 2282 JT Nippon Meat Packers Inc

30 2317 TT Hon Hai Precision Industry Co

31 2330 TT Taiwan Semiconductor Manufactu

32 2382 TT Quanta Computer Inc

33 2388 HK BOC Hong Kong Holdings Ltd

34 2412 TT Chunghwa Telecom Co Ltd

35 2454 TT MediaTek Inc

178

36 2502 JT Asahi Group Holdings Ltd

37 2503 JT Kirin Holdings Co Ltd

38 2651 JT Lawson Inc

39 2802 JT Ajinomoto Co Inc

40 2882 TT Cathay Financial Holding Co Lt

41 2897 JT Nissin Foods Holdings Co Ltd

42 2914 JT Japan Tobacco Inc

43 3 HK Hong Kong & China Gas Co Ltd

44 3401 JT Teijin Ltd

45 3402 JT Toray Industries Inc

46 3405 JT Kuraray Co Ltd

47 3407 JT Asahi Kasei Corp

48 386 HK China Petroleum & Chemical Cor

49 3861 JT Oji Holdings Corp

50 388 HK Hong Kong Exchanges and Cleari

51 4005 JT Sumitomo Chemical Co Ltd

52 4063 JT Shin-Etsu Chemical Co Ltd

53 4183 JT Mitsui Chemicals Inc

54 4185 JT JSR Corp

55 4324 JT Dentsu Inc

56 4452 JT Kao Corp

57 4502 JT Takeda Pharmaceutical Co Ltd

58 4503 JT Astellas Pharma Inc

59 4507 JT Shionogi & Co Ltd

60 4519 JT Chugai Pharmaceutical Co Ltd

61 4523 JT Eisai Co Ltd

62 4528 JO Ono Pharmaceutical Co Ltd

63 4543 JT Terumo Corp

64 4661 JT Oriental Land Co Ltd/Japan

65 4901 JT FUJIFILM Holdings Corp

66 4902 JT Konica Minolta Inc

67 4911 JT Shiseido Co Ltd

68 494 HK Li & Fung Ltd

69 5012 JT TonenGeneral Sekiyu KK

70 5108 JT Bridgestone Corp

71 5201 JT Asahi Glass Co Ltd

72 5332 JT TOTO Ltd

73 5333 JT NGK Insulators Ltd

74 5401 JT Nippon Steel & Sumitomo Metal

75 5406 JT Kobe Steel Ltd

76 5411 JT JFE Holdings Inc

77 5711 JT Mitsubishi Materials Corp

78 5713 JT Sumitomo Metal Mining Co Ltd

79 5802 JT Sumitomo Electric Industries L

80 5901 JT Toyo Seikan Group Holdings Ltd

179

81 5938 JT LIXIL Group Corp

82 6 HK Power Assets Holdings Ltd

83 6201 JT Toyota Industries Corp

84 6273 JT SMC Corp/Japan

85 6301 JT Komatsu Ltd

86 6326 JT Kubota Corp

87 6367 JT Daikin Industries Ltd

88 6471 JT NSK Ltd

89 6501 JT Hitachi Ltd

90 6502 JT Toshiba Corp

91 6503 JT Mitsubishi Electric Corp

92 6594 JO Nidec Corp

93 6701 JT NEC Corp

94 6702 JT Fujitsu Ltd

95 6752 JT Panasonic Corp

96 6753 JT Sharp Corp/Japan

97 6758 JT Sony Corp

98 6762 JT TDK Corp

99 6806 JT Hirose Electric Co Ltd

100 6857 JT Advantest Corp

3.2 The 100 first stocks from S&P 500 (423 stocks with full data

during the period 2003-2013)

SECURITY NUMBER INDX_MEMBERS SECURITY NAME

1 A UN Agilent Technologies Inc

2 AA UN Alcoa Inc

3 AAPL UW Apple Inc

4 ABC UN AmerisourceBergen Corp

5 ABT UN Abbott Laboratories

6 ACE UN ACE Ltd

7 ACN UN Accenture PLC

8 ACT UN Actavis Inc

9 ADBE UW Adobe Systems Inc

10 ADM UN Archer-Daniels-Midland Co

11 ADSK UW Autodesk Inc

12 AEE UN Ameren Corp

13 AEP UN American Electric Power Co Inc

14 AES UN AES Corp/VA

15 AET UN Aetna Inc

16 AFL UN Aflac Inc

17 AGN UN Allergan Inc/United States

18 AIG UN American International Group I

180

19 AIV UN Apartment Investment & Managem

20 AKAM UW Akamai Technologies Inc

21 ALL UN Allstate Corp/The

22 ALTR UW Altera Corp

23 ALXN UW Alexion Pharmaceuticals Inc

24 AMAT UW Applied Materials Inc

25 AMD UN Advanced Micro Devices Inc

26 AMGN UW Amgen Inc

27 AMT UN American Tower Corp

28 AMZN UW Amazon.com Inc

29 AN UN AutoNation Inc

30 ANF UN Abercrombie & Fitch Co

31 AON UN Aon PLC

32 APA UN Apache Corp

33 APC UN Anadarko Petroleum Corp

34 APD UN Air Products & Chemicals Inc

35 APH UN Amphenol Corp

36 APOL UW Apollo Group Inc

37 ARG UN Airgas Inc

38 ATI UN Allegheny Technologies Inc

39 AVB UN AvalonBay Communities Inc

40 AVP UN Avon Products Inc

41 AVY UN Avery Dennison Corp

42 AXP UN American Express Co

43 AZO UN AutoZone Inc

44 BA UN Boeing Co/The

45 BAC UN Bank of America Corp

46 BAX UN Baxter International Inc

47 BBBY UW Bed Bath & Beyond Inc

48 BBT UN BB&T Corp

49 BBY UN Best Buy Co Inc

50 BCR UN CR Bard Inc

51 BDX UN Becton Dickinson and Co

52 BEAM UN Beam Inc

53 BEN UN Franklin Resources Inc

54 BF/B UN Brown-Forman Corp

55 BHI UN Baker Hughes Inc

56 BIIB UW Biogen Idec Inc

57 BK UN Bank of New York Mellon Corp/T

58 BLK UN BlackRock Inc

59 BLL UN Ball Corp

60 BMS UN Bemis Co Inc

61 BMY UN Bristol-Myers Squibb Co

62 BRCM UW Broadcom Corp

63 BRK/B UN Berkshire Hathaway Inc

181

64 BSX UN Boston Scientific Corp

65 BTU UN Peabody Energy Corp

66 BWA UN BorgWarner Inc

67 BXP UN Boston Properties Inc

68 C UN Citigroup Inc

69 CAG UN ConAgra Foods Inc

70 CAH UN Cardinal Health Inc

71 CAM UN Cameron International Corp

72 CAT UN Caterpillar Inc

73 CB UN Chubb Corp/The

74 CBS UN CBS Corp

75 CCE UN Coca-Cola Enterprises Inc

76 CCI UN Crown Castle International Cor

77 CCL UN Carnival Corp

78 CELG UW Celgene Corp

79 CERN UW Cerner Corp

80 CHK UN Chesapeake Energy Corp

81 CHRW UW CH Robinson Worldwide Inc

82 CI UN Cigna Corp

83 CINF UW Cincinnati Financial Corp

84 CL UN Colgate-Palmolive Co

85 CLF UN Cliffs Natural Resources Inc

86 CLX UN Clorox Co/The

87 CMA UN Comerica Inc

88 CMCSA UW Comcast Corp

89 CMI UN Cummins Inc

90 CMS UN CMS Energy Corp

91 CNP UN CenterPoint Energy Inc

92 CNX UN CONSOL Energy Inc

93 COF UN Capital One Financial Corp

94 COG UN Cabot Oil & Gas Corp

95 COH UN Coach Inc

96 COL UN Rockwell Collins Inc

97 COP UN ConocoPhillips

98 COST UW Costco Wholesale Corp

99 CPB UN Campbell Soup Co

100 CSC UN Computer Sciences Corp

3.2 Company lists of the 25 Fama French portfolios

182

In this sector, the Security ID’s for the 25 portfolios created by the Fama French procedure

(US Equities) are provided. Only the stocks for which full data could be retrieved constitute

these portfolios; for that reason, about 5 out of every 6 stocks from the original downloaded

file had to be discarded. It is believed that the next tables, as trivial as they might appear,

could save a tremendous amount of time and effort to somebody that would attempt to

repeat the experiments.

3.2.1 Portfolio 1

SECURITY NUMBER SECURITY ID SECURITY NAME

1 BMI US EQUITY Badger Meter Inc

2 FRM US EQUITY Furmanite Corp

3 GNI US EQUITY Great Northern Iron Ore Proper

4 LDR US EQUITY Landauer Inc

5 LXU US EQUITY LSB Industries Inc

6 TISI US EQUITY Team Inc

7 TPL US EQUITY Texas Pacific Land Trust

8 TYL US EQUITY Tyler Technologies Inc

9 ABMD US EQUITY ABIOMED Inc

10 ACUR US EQUITY Acura Pharmaceuticals Inc

11 AEPI US EQUITY AEP Industries Inc

12 AKRX US EQUITY Akorn Inc

13 ARDNA US EQUITY Arden Group Inc

14 ATRO US EQUITY Astronics Corp

15 BCPC US EQUITY Balchem Corp

16 CLDX US EQUITY Celldex Therapeutics Inc

17 CHYR US EQUITY ChyronHego Corp

18 COKE US EQUITY Coca-Cola Bottling Co Consolid

19 CBRX US EQUITY Columbia Laboratories Inc

20 EBIX US EQUITY Ebix Inc

21 ENZN US EQUITY Enzon Pharmaceuticals Inc

22 FLOW US EQUITY Flow International Corp

23 IMMU US EQUITY Immunomedics Inc

24 MTRX US EQUITY Matrix Service Co

25 PMFG US EQUITY PMFG Inc

26 RAVN US EQUITY Raven Industries Inc

27 INTG US EQUITY InterGroup Corp/The

28 TRNS US EQUITY Transcat Inc

29 UG US EQUITY United-Guardian Inc

30 URRE US EQUITY Uranium Resources Inc

183

31 ECOL US EQUITY US Ecology Inc

32 VALU US EQUITY Value Line Inc

33 WLB US EQUITY Westmoreland Coal Co

34 XOMA US EQUITY XOMA Corp

35 ZIXI US EQUITY Zix Corp

3.2.2 Portfolio 2


1 ARB US EQUITY Arbitron Inc

2 FUN US EQUITY Cedar Fair LP

3 CBB US EQUITY Cincinnati Bell Inc

4 CLH US EQUITY Clean Harbors Inc

5 DLX US EQUITY Deluxe Corp

6 GY US EQUITY GenCorp Inc

7 HLS US EQUITY HealthSouth Corp

8 LNN US EQUITY Lindsay Corp

9 PII US EQUITY Polaris Industries Inc

10 SBR US EQUITY Sabine Royalty Trust

11 TRC US EQUITY Tejon Ranch Co

12 UIS US EQUITY Unisys Corp

13 VGR US EQUITY Vector Group Ltd

14 CBRL US EQUITY Cracker Barrel Old Country Sto

15 HCSG US EQUITY Healthcare Services Group Inc

16 HTLD US EQUITY Heartland Express Inc

17 IIVI US EQUITY II-VI Inc

18 IDCC US EQUITY InterDigital Inc/PA

19 TILE US EQUITY Interface Inc

20 VIVO US EQUITY Meridian Bioscience Inc

21 SVNT US EQUITY Savient Pharmaceuticals Inc

22 MIDD US EQUITY Middleby Corp

3.2.3 Portfolio 3


1 ALX US EQUITY Alexander's Inc

2 ATW US EQUITY Atwood Oceanics Inc

3 BYI US EQUITY Bally Technologies Inc

4 BRY US EQUITY Berry Petroleum Co

184

5 GGG US EQUITY Graco Inc

6 HXL US EQUITY Hexcel Corp

7 HFC US EQUITY HollyFrontier Corp

8 ROL US EQUITY Rollins Inc

9 BID US EQUITY Sotheby's

10 THC US EQUITY Tenet Healthcare Corp

11 TTC US EQUITY Toro Co/The

12 UDR US EQUITY UDR Inc

13 VHI US EQUITY Valhi Inc

14 VMI US EQUITY Valmont Industries Inc

15 WMS US EQUITY WMS Industries Inc

16 MLHR US EQUITY Herman Miller Inc

17 MCRS US EQUITY MICROS Systems Inc

3.2.4 Portfolio 4


1 ATK US EQUITY Alliant Techsystems Inc

2 BF/A US EQUITY Brown-Forman Corp

3 BF/B US EQUITY Brown-Forman Corp

4 CVC US EQUITY Cablevision Systems Corp

5 COG US EQUITY Cabot Oil & Gas Corp

6 CLF US EQUITY Cliffs Natural Resources Inc

7 CLX US EQUITY Clorox Co/The

8 CVA US EQUITY Covanta Holding Corp

9 CCK US EQUITY Crown Holdings Inc

10 DCI US EQUITY Donaldson Co Inc

11 EV US EQUITY Eaton Vance Corp

12 EQT US EQUITY EQT Corp

13 FRT US EQUITY Federal Realty Investment Trus

14 FLS US EQUITY Flowserve Corp

15 FMC US EQUITY FMC Corp

16 FCE/A US EQUITY Forest City Enterprises Inc

17 HRB US EQUITY H&R Block Inc

18 HOG US EQUITY Harley-Davidson Inc

19 HSY US EQUITY Hershey Co/The

20 IFF US EQUITY International Flavors & Fragra

21 MTW US EQUITY Manitowoc Co Inc/The

22 MKC US EQUITY McCormick & Co Inc/MD

23 NAV US EQUITY Navistar International Corp

185

24 JWN US EQUITY Nordstrom Inc

25 OII US EQUITY Oceaneering International Inc

26 PLL US EQUITY Pall Corp

27 PBI US EQUITY Pitney Bowes Inc

28 PCL US EQUITY Plum Creek Timber Co Inc

29 RHI US EQUITY Robert Half International Inc

30 SHW US EQUITY Sherwin-Williams Co/The

31 JOE US EQUITY St Joe Co/The

32 HOT US EQUITY Starwood Hotels & Resorts Worl

33 TSS US EQUITY Total System Services Inc

34 VAR US EQUITY Varian Medical Systems Inc

35 ALTR US EQUITY Altera Corp

36 ADSK US EQUITY Autodesk Inc

37 BMC US EQUITY BMC Software Inc

38 XRAY US EQUITY DENTSPLY International Inc

39 EXPD US EQUITY Expeditors International of Wa

40 FAST US EQUITY Fastenal Co

41 FISV US EQUITY Fiserv Inc

42 FTR US EQUITY Frontier Communications Corp

43 JBHT US EQUITY JB Hunt Transport Services Inc

44 LLTC US EQUITY Linear Technology Corp

45 ROST US EQUITY Ross Stores Inc

46 SEIC US EQUITY SEI Investments Co

47 SIAL US EQUITY Sigma-Aldrich Corp

48 XLNX US EQUITY Xilinx Inc

3.2.5 Portfolio 5


1 MMM US EQUITY 3M Co

2 ABT US EQUITY Abbott Laboratories

3 AFL US EQUITY Aflac Inc

4 APD US EQUITY Air Products & Chemicals Inc

5 AGN US EQUITY Allergan Inc/United States

6 MO US EQUITY Altria Group Inc

7 AXP US EQUITY American Express Co

8 AVP US EQUITY Avon Products Inc

9 BAX US EQUITY Baxter International Inc

10 BDX US EQUITY Becton Dickinson and Co

11 BBY US EQUITY Best Buy Co Inc

186

12 BA US EQUITY Boeing Co/The

13 BMY US EQUITY Bristol-Myers Squibb Co

14 BCR US EQUITY CR Bard Inc

15 CPB US EQUITY Campbell Soup Co

16 CAT US EQUITY Caterpillar Inc

17 KO US EQUITY Coca-Cola Co/The

18 CL US EQUITY Colgate-Palmolive Co

19 GLW US EQUITY Corning Inc

20 DE US EQUITY Deere & Co

21 DD US EQUITY EI du Pont de Nemours & Co

22 ECL US EQUITY Ecolab Inc

23 ELN US EQUITY Elan Corp PLC

24 LLY US EQUITY Eli Lilly & Co

25 EMR US EQUITY Emerson Electric Co

26 EOG US EQUITY EOG Resources Inc

27 EXC US EQUITY Exelon Corp

28 XOM US EQUITY Exxon Mobil Corp

29 F US EQUITY Ford Motor Co

30 FRX US EQUITY Forest Laboratories Inc

31 GPS US EQUITY Gap Inc/The

32 HAL US EQUITY Halliburton Co

33 HON US EQUITY Honeywell International Inc

34 IBM US EQUITY International Business Machine

35 IGT US EQUITY International Game Technology

36 JEC US EQUITY Jacobs Engineering Group Inc

37 JNJ US EQUITY Johnson & Johnson

38 K US EQUITY Kellogg Co

39 KMB US EQUITY Kimberly-Clark Corp

40 MDR US EQUITY McDermott International Inc

41 MCD US EQUITY McDonald's Corp

42 MHFI US EQUITY McGraw Hill Financial Inc

43 MDT US EQUITY Medtronic Inc

44 MRK US EQUITY Merck & Co Inc

45 NKE US EQUITY NIKE Inc

46 NUE US EQUITY Nucor Corp

47 OMC US EQUITY Omnicom Group Inc

48 ORCL US EQUITY Oracle Corp

49 PEP US EQUITY PepsiCo Inc

50 POT US EQUITY Potash Corp of Saskatchewan In

51 PCP US EQUITY Precision Castparts Corp

52 STR US EQUITY Questar Corp

187

53 SLB US EQUITY Schlumberger Ltd

54 SWN US EQUITY Southwestern Energy Co

55 STJ US EQUITY St Jude Medical Inc

56 SYK US EQUITY Stryker Corp

57 SYY US EQUITY Sysco Corp

58 TXT US EQUITY Textron Inc

59 SCHW US EQUITY Charles Schwab Corp/The

60 TJX US EQUITY TJX Cos Inc

61 ADBE US EQUITY Adobe Systems Inc

62 AAPL US EQUITY Apple Inc

63 AMAT US EQUITY Applied Materials Inc

64 ADP US EQUITY Automatic Data Processing Inc

65 CELG US EQUITY Celgene Corp

66 CSCO US EQUITY Cisco Systems Inc

67 DELL US EQUITY Dell Inc

68 DTV US EQUITY DIRECTV

69 EA US EQUITY Electronic Arts Inc

70 FWLT US EQUITY Foster Wheeler AG

71 MSFT US EQUITY Microsoft Corp

72 PAYX US EQUITY Paychex Inc

73 TROW US EQUITY T Rowe Price Group Inc

74 TXN US EQUITY Texas Instruments Inc

3.2.6 Portfolio 6


1 DDD US EQUITY 3D Systems Corp

2 AVD US EQUITY American Vanguard Corp

3 ARSD US EQUITY Arabian American Development C

4 AZZ US EQUITY AZZ Inc

5 CBM US EQUITY Cambrex Corp

6 ENZ US EQUITY Enzo Biochem Inc

7 SJW US EQUITY SJW Corp

8 SR US EQUITY Standard Register Co/The

9 UHT US EQUITY Universal Health Realty Income

10 ASEI US EQUITY American Science & Engineering

11 AMSWA US

EQUITY

American Software Inc/Georgia

12 ARKR US EQUITY Ark Restaurants Corp

13 ARTW US EQUITY Art's-Way Manufacturing Co Inc

14 ATRI US EQUITY Atrion Corp

188

15 DJCO US EQUITY Daily Journal Corp

16 DAIO US EQUITY Data I/O Corp

17 DWSN US EQUITY Dawson Geophysical Co

18 EXPO US EQUITY Exponent Inc

19 HURC US EQUITY Hurco Cos Inc

20 IMGN US EQUITY ImmunoGen Inc

21 KOSS US EQUITY Koss Corp

22 KLIC US EQUITY Kulicke & Soffa Industries Inc

23 MXWL US EQUITY Maxwell Technologies Inc

24 MDCI US EQUITY Medical Action Industries Inc

25 MOCO US EQUITY MOCON Inc

26 MTSC US EQUITY MTS Systems Corp

27 LABL US EQUITY Multi-Color Corp

28 OSUR US EQUITY OraSure Technologies Inc

29 POWL US EQUITY Powell Industries Inc

30 RADA US EQUITY Rada Electronic Industries Ltd

31 RGEN US EQUITY Repligen Corp

32 SPAN US EQUITY Span-America Medical Systems I

33 SUBK US EQUITY Suffolk Bancorp

34 USLM US EQUITY United States Lime & Minerals

35 UTMD US EQUITY Utah Medical Products Inc

36 VICR US EQUITY Vicor Corp

37 VIDE US EQUITY Video Display Corp

38 VSEC US EQUITY VSE Corp

39 WDFC US EQUITY WD-40 Co

40 WSCI US EQUITY WSI Industries Inc

41 BZC US EQUITY Breeze-Eastern Corp

42 COVR US EQUITY Cover-All Technologies Inc

43 GRC US EQUITY Gorman-Rupp Co/The

44 BKR US EQUITY Michael Baker Corp

45 PW US EQUITY Power REIT

46 TOF US EQUITY Tofutti Brands Inc

47 TMP US EQUITY Tompkins Financial Corp

3.2.7 Portfolio 7


1 ATU US EQUITY Actuant Corp

2 ACO US EQUITY AMCOL International Corp

3 AIT US EQUITY Applied Industrial Technologie

189

4 CCC US EQUITY Calgon Carbon Corp

5 CEC US EQUITY CEC Entertainment Inc

6 CHE US EQUITY Chemed Corp

7 CUZ US EQUITY Cousins Properties Inc

8 DW US EQUITY Drew Industries Inc

9 EGP US EQUITY EastGroup Properties Inc

10 GTY US EQUITY Getty Realty Corp

11 GDP US EQUITY Goodrich Petroleum Corp

12 HL US EQUITY Hecla Mining Co

13 HEI US EQUITY HEICO Corp

14 HNI US EQUITY HNI Corp

15 KDN US EQUITY Kaydon Corp

16 MSA US EQUITY Mine Safety Appliances Co

17 NEU US EQUITY NewMarket Corp

18 ORB US EQUITY Orbital Sciences Corp

19 RES US EQUITY RPC Inc

20 RTI US EQUITY RTI International Metals Inc

21 TNC US EQUITY Tennant Co

22 TPC US EQUITY Tutor Perini Corp

23 WRE US EQUITY Washington Real Estate Investm

24 WST US EQUITY West Pharmaceutical Services I

25 WGO US EQUITY Winnebago Industries Inc

26 WWW US EQUITY Wolverine World Wide Inc

27 AMAG US EQUITY AMAG Pharmaceuticals Inc

28 ASTE US EQUITY Astec Industries Inc

29 DIOD US EQUITY Diodes Inc

30 LANC US EQUITY Lancaster Colony Corp

31 NPBC US EQUITY National Penn Bancshares Inc

32 PDCE US EQUITY PDC Energy Inc

33 SMTC US EQUITY Semtech Corp

34 SIGM US EQUITY Sigma Designs Inc

35 TRST US EQUITY TrustCo Bank Corp NY

36 WABC US EQUITY Westamerica Bancorporation

3.2.8 Portfolio 8


1 AXE US EQUITY Anixter International Inc

2 WTR US EQUITY Aqua America Inc

3 AJG US EQUITY Arthur J Gallagher & Co

190

4 BOH US EQUITY Bank of Hawaii Corp

5 BIG US EQUITY Big Lots Inc

6 BIO US EQUITY Bio-Rad Laboratories Inc

7 BRE US EQUITY BRE Properties Inc

8 EAT US EQUITY Brinker International Inc

9 BRO US EQUITY Brown & Brown Inc

10 CRS US EQUITY Carpenter Technology Corp

11 CLC US EQUITY CLARCOR Inc

12 CNW US EQUITY Con-way Inc

13 CR US EQUITY Crane Co

14 FDO US EQUITY Family Dollar Stores Inc

15 HUB/A US EQUITY Hubbell Inc

16 HUB/B US EQUITY Hubbell Inc

17 ITG US EQUITY Investment Technology Group In

18 JW/A US EQUITY John Wiley & Sons Inc

19 KEX US EQUITY Kirby Corp

20 MDP US EQUITY Meredith Corp

21 NYT US EQUITY New York Times Co/The

22 NVR US EQUITY NVR Inc

23 OMI US EQUITY Owens & Minor Inc

24 PVA US EQUITY Penn Virginia Corp

25 RSH US EQUITY RadioShack Corp

26 SNA US EQUITY Snap-on Inc

27 THO US EQUITY Thor Industries Inc

28 VLY US EQUITY Valley National Bancorp

29 WRI US EQUITY Weingarten Realty Investors

30 GNTX US EQUITY Gentex Corp/MI

31 JKHY US EQUITY Jack Henry & Associates Inc

32 MSCC US EQUITY Microsemi Corp

33 NDSN US EQUITY Nordson Corp

34 PCH US EQUITY Potlatch Corp

35 PMTC US EQUITY PTC Inc

36 SGMS US EQUITY Scientific Games Corp

37 AXE US EQUITY Anixter International Inc

38 WTR US EQUITY Aqua America Inc

3.2.9 Portfolio 9


1 ARG US EQUITY Airgas Inc

191

2 AB US EQUITY AllianceBernstein Holding LP

3 AVY US EQUITY Avery Dennison Corp

4 BLL US EQUITY Ball Corp

5 BCO US EQUITY Brink's Co/The

6 CNP US EQUITY CenterPoint Energy Inc

7 CHD US EQUITY Church & Dwight Co Inc

8 CMC US EQUITY Commercial Metals Co

9 SSP US EQUITY EW Scripps Co

10 EGN US EQUITY Energen Corp

11 ESV US EQUITY Ensco PLC

12 EFX US EQUITY Equifax Inc

13 GPC US EQUITY Genuine Parts Co

14 HRS US EQUITY Harris Corp

15 HSC US EQUITY Harsco Corp

16 HP US EQUITY Helmerich & Payne Inc

17 HRC US EQUITY Hill-Rom Holdings Inc

18 HRL US EQUITY Hormel Foods Corp

19 IPG US EQUITY Interpublic Group of Cos Inc/T

20 LTD US EQUITY L Brands Inc

21 NFG US EQUITY National Fuel Gas Co

22 NWL US EQUITY Newell Rubbermaid Inc

23 OKE US EQUITY ONEOK Inc

24 SPW US EQUITY SPX Corp

25 SWK US EQUITY Stanley Black & Decker Inc

26 TEX US EQUITY Terex Corp

27 TIF US EQUITY Tiffany & Co

28 VFC US EQUITY VF Corp

29 VTR US EQUITY Ventas Inc

30 GWW US EQUITY WW Grainger Inc

31 ADI US EQUITY Analog Devices Inc

32 BEAV US EQUITY B/E Aerospace Inc

33 CERN US EQUITY Cerner Corp

34 HAS US EQUITY Hasbro Inc

35 IEP US EQUITY Icahn Enterprises LP

36 KLAC US EQUITY KLA-Tencor Corp

37 LRCX US EQUITY Lam Research Corp

38 MAT US EQUITY Mattel Inc

39 GT US EQUITY Goodyear Tire & Rubber Co/The

40 TRMB US EQUITY Trimble Navigation Ltd

41 WDC US EQUITY Western Digital Corp

192

3.2.10 Portfolio 10


1 APA US EQUITY Apache Corp

2 ADM US EQUITY Archer-Daniels-Midland Co

3 BHI US EQUITY Baker Hughes Inc

4 ABX US EQUITY Barrick Gold Corp

5 CAH US EQUITY Cardinal Health Inc

6 CVX US EQUITY Chevron Corp

7 CI US EQUITY Cigna Corp

8 CSX US EQUITY CSX Corp

9 CMI US EQUITY Cummins Inc

10 DHR US EQUITY Danaher Corp

11 D US EQUITY Dominion Resources Inc/VA

12 ETN US EQUITY Eaton Corp PLC

13 EMC US EQUITY EMC Corp/MA

14 ETR US EQUITY Entergy Corp

15 BEN US EQUITY Franklin Resources Inc

16 GD US EQUITY General Dynamics Corp

17 GE US EQUITY General Electric Co

18 GIS US EQUITY General Mills Inc

19 HES US EQUITY Hess Corp

20 HPQ US EQUITY Hewlett-Packard Co

21 HSH US EQUITY Hillshire Brands Co

22 HD US EQUITY Home Depot Inc/The

23 HUM US EQUITY Humana Inc

24 ITW US EQUITY Illinois Tool Works Inc

25 ITT US EQUITY ITT Corp

26 KR US EQUITY Kroger Co/The

27 MGM US EQUITY MGM Resorts International

28 MUR US EQUITY Murphy Oil Corp

29 NEM US EQUITY Newmont Mining Corp

30 NEE US EQUITY NextEra Energy Inc

31 NE US EQUITY Noble Corp

32 NBL US EQUITY Noble Energy Inc

33 OXY US EQUITY Occidental Petroleum Corp

34 PH US EQUITY Parker Hannifin Corp

35 PFE US EQUITY Pfizer Inc

36 PPG US EQUITY PPG Industries Inc

37 PPL US EQUITY PPL Corp

193

38 PG US EQUITY Procter & Gamble Co/The

39 PGR US EQUITY Progressive Corp/The

40 PEG US EQUITY Public Service Enterprise Grou

41 PSA US EQUITY Public Storage

42 RTN US EQUITY Raytheon Co

43 STT US EQUITY State Street Corp

44 TGT US EQUITY Target Corp

45 USB US EQUITY US Bancorp/MN

46 UTX US EQUITY United Technologies Corp

47 UNH US EQUITY UnitedHealth Group Inc

48 VNO US EQUITY Vornado Realty Trust

49 WAG US EQUITY Walgreen Co

50 WMT US EQUITY Wal-Mart Stores Inc

51 WFT US EQUITY Weatherford International Ltd/

52 WMB US EQUITY Williams Cos Inc/The

53 AMGN US EQUITY Amgen Inc

54 CA US EQUITY CA Inc

55 COST US EQUITY Costco Wholesale Corp

56 INTC US EQUITY Intel Corp

57 NTRS US EQUITY Northern Trust Corp

58 PCAR US EQUITY PACCAR Inc

59 SLM US EQUITY SLM Corp

60 SPLS US EQUITY Staples Inc

3.2.11 Portfolio 11


1 AWR US EQUITY American States Water Co

2 AP US EQUITY Ampco-Pittsburgh Corp

3 CATO US EQUITY Cato Corp/The

4 CPK US EQUITY Chesapeake Utilities Corp

5 GPX US EQUITY GP Strategies Corp

6 MTRN US EQUITY Materion Corp

7 MPR US EQUITY Met-Pro Corp

8 NL US EQUITY NL Industries Inc

9 OFG US EQUITY OFG Bancorp

10 ODC US EQUITY Oil-Dri Corp of America

11 PKE US EQUITY Park Electrochemical Corp

12 PIR US EQUITY Pier 1 Imports Inc

13 KWR US EQUITY Quaker Chemical Corp

194

14 ROG US EQUITY Rogers Corp

15 SCL US EQUITY Stepan Co

16 STL US EQUITY Sterling Bancorp/NY

17 RGR US EQUITY Sturm Ruger & Co Inc

18 UBP US EQUITY Urstadt Biddle Properties Inc

19 APFC US EQUITY American Pacific Corp

20 APOG US EQUITY Apogee Enterprises Inc

21 AROW US EQUITY Arrow Financial Corp

22 BMTC US EQUITY Bryn Mawr Bank Corp

23 BTUI US EQUITY BTU International Inc

24 CTWS US EQUITY Connecticut Water Service Inc

25 CRRC US EQUITY Courier Corp

26 ELSE US EQUITY Electro-Sensors Inc

27 FFBC US EQUITY First Financial Bancorp

28 GLDC US EQUITY Golden Enterprises Inc

29 HWKN US EQUITY Hawkins Inc

30 ICAD US EQUITY Icad Inc

31 INDB US EQUITY Independent Bank Corp/Rockland

32 IIN US EQUITY IntriCon Corp

33 JJSF US EQUITY J&J Snack Foods Corp

34 KEQU US EQUITY Kewaunee Scientific Corp

35 FSTR US EQUITY LB Foster Co

36 LYTS US EQUITY LSI Industries Inc

37 LTXC US EQUITY LTX-Credence Corp

38 MAG US EQUITY Magnetek Inc

39 MGRC US EQUITY McGrath RentCorp

40 MBVT US EQUITY Merchants Bancshares Inc

41 MFRI US EQUITY Mfri Inc

42 MSEX US EQUITY Middlesex Water Co

43 NSSC US EQUITY NAPCO Security Technologies In

44 NATR US EQUITY Nature's Sunshine Products Inc

45 PATR US EQUITY Patriot Transportation Holding

46 DFZ US EQUITY RG Barry Corp

47 SEV US EQUITY Sevcon Inc

48 LNCE US EQUITY Snyders-Lance Inc

49 SPAR US EQUITY Spartan Motors Inc

50 SUPX US EQUITY Supertex Inc

51 TCCO US EQUITY Technical Communications Corp

52 WTSL US EQUITY Wet Seal Inc/The

53 TWIN US EQUITY Twin Disc Inc

54 VLGEA US EQUITY Village Super Market Inc

195

55 WEYS US EQUITY Weyco Group Inc

56 CCF US EQUITY Chase Corp

57 CTO US EQUITY Consolidated-Tomoka Land Co

58 DXR US EQUITY Daxor Corp

59 ESP US EQUITY Espey Manufacturing & Electron

60 GHM US EQUITY Graham Corp

3.2.12 Portfolio 12


1 AIR US EQUITY AAR Corp

2 ABM US EQUITY ABM Industries Inc

3 AIN US EQUITY Albany International Corp

4 B US EQUITY Barnes Group Inc

5 BRS US EQUITY Bristow Group Inc

6 CWT US EQUITY California Water Service Group

7 CMO US EQUITY Capstead Mortgage Corp

8 CSH US EQUITY Cash America International Inc

9 CUB US EQUITY Cubic Corp

10 ELX US EQUITY Emulex Corp

11 ESE US EQUITY ESCO Technologies Inc

12 FICO US EQUITY Fair Isaac Corp

13 GCO US EQUITY Genesco Inc

14 GVA US EQUITY Granite Construction Inc

15 FUL US EQUITY HB Fuller Co

16 LG US EQUITY Laclede Group Inc/The

17 MTZ US EQUITY MasTec Inc

18 MNI US EQUITY McClatchy Co/The

19 NJR US EQUITY New Jersey Resources Corp

20 NWN US EQUITY Northwest Natural Gas Co

21 ONB US EQUITY Old National Bancorp/IN

22 OLN US EQUITY Olin Corp

23 PULS US EQUITY Pulse Electronics Corp

24 SXT US EQUITY Sensient Technologies Corp

25 SJI US EQUITY South Jersey Industries Inc

26 SF US EQUITY Stifel Financial Corp

27 SFY US EQUITY Swift Energy Co

28 TR US EQUITY Tootsie Roll Industries Inc

29 UIL US EQUITY UIL Holdings Corp

30 WSO US EQUITY Watsco Inc

196

31 WSO/B US EQUITY Watsco Inc

32 INT US EQUITY World Fuel Services Corp

33 WOR US EQUITY Worthington Industries Inc

34 ACXM US EQUITY Acxiom Corp

35 ALOG US EQUITY Analogic Corp

36 ASNA US EQUITY Ascena Retail Group Inc

37 BOBE US EQUITY Bob Evans Farms Inc/DE

38 CASY US EQUITY Casey's General Stores Inc

39 CHCO US EQUITY City Holding Co

40 CGNX US EQUITY Cognex Corp

41 CVBF US EQUITY CVB Financial Corp

42 FMBI US EQUITY First Midwest Bancorp Inc/IL

43 FMER US EQUITY FirstMerit Corp

44 FELE US EQUITY Franklin Electric Co Inc

45 GBCI US EQUITY Glacier Bancorp Inc

46 IMKTA US EQUITY Ingles Markets Inc

47 MGEE US EQUITY MGE Energy Inc

48 OTTR US EQUITY Otter Tail Corp

49 PLXS US EQUITY Plexus Corp

50 SAFM US EQUITY Sanderson Farms Inc

51 SWKS US EQUITY Skyworks Solutions Inc

52 SIVB US EQUITY SVB Financial Group

53 USTR US EQUITY United Stationers Inc

54 WEN US EQUITY Wendy's Co/The

55 PRK US EQUITY Park National Corp

3.2.13 Portfolio 13


1 GAS US EQUITY AGL Resources Inc

2 BMS US EQUITY Bemis Co Inc

3 BRC US EQUITY Brady Corp

4 BPL US EQUITY Buckeye Partners LP

5 CSL US EQUITY Carlisle Cos Inc

6 CRK US EQUITY Comstock Resources Inc

7 CFR US EQUITY Cullen/Frost Bankers Inc

8 CW US EQUITY Curtiss-Wright Corp

9 HAR US EQUITY Harman International Industrie

10 HTSI US EQUITY Harris Teeter Supermarkets Inc

11 IEX US EQUITY IDEX Corp

197

12 SJM US EQUITY JM Smucker Co/The

13 KSU US EQUITY Kansas City Southern

14 KEG US EQUITY Key Energy Services Inc

15 MATX US EQUITY Matson Inc

16 MOG/A US EQUITY Moog Inc

17 OGE US EQUITY OGE Energy Corp

18 OSK US EQUITY Oshkosh Corp

19 PKI US EQUITY PerkinElmer Inc

20 PNY US EQUITY Piedmont Natural Gas Co Inc

21 PVH US EQUITY PVH Corp

22 RAD US EQUITY Rite Aid Corp

23 RPM US EQUITY RPM International Inc

24 SKS US EQUITY Saks Inc

25 SCI US EQUITY Service Corp International/US

26 SON US EQUITY Sonoco Products Co

27 TCB US EQUITY TCF Financial Corp

28 TFX US EQUITY Teleflex Inc

29 TER US EQUITY Teradyne Inc

30 TXI US EQUITY Texas Industries Inc

31 TDW US EQUITY Tidewater Inc

32 UGI US EQUITY UGI Corp

33 UNT US EQUITY Unit Corp

34 UVV US EQUITY Universal Corp/VA

35 UHS US EQUITY Universal Health Services Inc

36 VAL US EQUITY Valspar Corp/The

37 WSM US EQUITY Williams-Sonoma Inc

38 CBSH US EQUITY Commerce Bancshares Inc/MO

39 UMBF US EQUITY UMB Financial Corp

3.2.14 Portfolio 14


1 DOV US EQUITY Dover Corp

2 DRE US EQUITY Duke Realty Corp

3 FST US EQUITY Forest Oil Corp

4 HCP US EQUITY HCP Inc

5 HCN US EQUITY Health Care REIT Inc

6 HST US EQUITY Host Hotels & Resorts Inc

7 MAN US EQUITY Manpowergroup Inc

8 MKL US EQUITY Markel Corp

198

9 MAS US EQUITY Masco Corp

10 MDU US EQUITY MDU Resources Group Inc

11 NBR US EQUITY Nabors Industries Ltd

12 PNR US EQUITY Pentair Ltd

13 RDC US EQUITY Rowan Cos Plc

14 R US EQUITY Ryder System Inc

15 TE US EQUITY TECO Energy Inc

16 TSO US EQUITY Tesoro Corp

17 TMK US EQUITY Torchmark Corp

18 USM US EQUITY United States Cellular Corp

19 VMC US EQUITY Vulcan Materials Co

20 WPO US EQUITY Washington Post Co/The

21 WHR US EQUITY Whirlpool Corp

22 WEC US EQUITY Wisconsin Energy Corp

23 CTAS US EQUITY Cintas Corp

24 CY US EQUITY Cypress Semiconductor Corp

25 HOLX US EQUITY Hologic Inc

26 MXIM US EQUITY Maxim Integrated Products Inc

27 RRD US EQUITY RR Donnelley & Sons Co

3.2.15 Portfolio 15


1 AA US EQUITY Alcoa Inc

2 AEP US EQUITY American Electric Power Co Inc

3 APC US EQUITY Anadarko Petroleum Corp

4 AON US EQUITY Aon PLC

5 T US EQUITY AT&T Inc

6 BK US EQUITY Bank of New York Mellon Corp/T

7 BEAM US EQUITY Beam Inc

8 BRK/A US EQUITY Berkshire Hathaway Inc

9 CCL US EQUITY Carnival Corp

10 CCE US EQUITY Coca-Cola Enterprises Inc

11 CAG US EQUITY ConAgra Foods Inc

12 CVS US EQUITY CVS Caremark Corp

13 DVN US EQUITY Devon Energy Corp

14 DOW US EQUITY Dow Chemical Co/The

15 EIX US EQUITY Edison International

16 FDX US EQUITY FedEx Corp

17 IP US EQUITY International Paper Co

199

18 JCP US EQUITY JC Penney Co Inc

19 JCI US EQUITY Johnson Controls Inc

20 LUK US EQUITY Leucadia National Corp

21 LOW US EQUITY Lowe's Cos Inc

22 MRO US EQUITY Marathon Oil Corp

23 MMC US EQUITY Marsh & McLennan Cos Inc

24 NSC US EQUITY Norfolk Southern Corp

25 PCG US EQUITY PG&E Corp

26 SWY US EQUITY Safeway Inc

27 SO US EQUITY Southern Co/The

28 TMO US EQUITY Thermo Fisher Scientific Inc

29 UNP US EQUITY Union Pacific Corp

30 VZ US EQUITY Verizon Communications Inc

31 DIS US EQUITY Walt Disney Co/The

32 WFC US EQUITY Wells Fargo & Co

33 WY US EQUITY Weyerhaeuser Co

34 XRX US EQUITY Xerox Corp

35 FOX US EQUITY Twenty-First Century Fox Inc

3.2.16 Portfolio 16


1 AXR US EQUITY AMREP Corp

2 BXMT US EQUITY Blackstone Mortgage Trust Inc

3 CAS US EQUITY AM Castle & Co

4 CDI US EQUITY CDI Corp

5 CIA US EQUITY Citizens Inc/TX

6 CSS US EQUITY CSS Industries Inc

7 DCO US EQUITY Ducommun Inc

8 DX US EQUITY Dynex Capital Inc

9 EBF US EQUITY Ennis Inc

10 HNR US EQUITY Harvest Natural Resources Inc

11 KID US EQUITY Kid Brands Inc

12 LUB US EQUITY Luby's Inc

13 MCS US EQUITY Marcus Corp

14 MEI US EQUITY Methode Electronics Inc

15 MNR US EQUITY Monmouth Real Estate Investmen

16 MYE US EQUITY Myers Industries Inc

17 NPK US EQUITY National Presto Industries Inc

18 PAR US EQUITY PAR Technology Corp

200

19 PKY US EQUITY Parkway Properties Inc/Md

20 RPT US EQUITY Ramco-Gershenson Properties Tr

21 SFE US EQUITY Safeguard Scientifics Inc

22 SGK US EQUITY Schawk Inc

23 SKY US EQUITY Skyline Corp

24 TRR US EQUITY TRC Cos Inc

25 UTL US EQUITY Unitil Corp

26 WPP US EQUITY Wausau Paper Corp

27 FUR US EQUITY Winthrop Realty Trust

28 SHLM US EQUITY A Schulman Inc

29 ACET US EQUITY Aceto Corp

30 AEGN US EQUITY Aegion Corp

31 ATAX US EQUITY America First Tax Exempt Inves

32 AMWD US EQUITY American Woodmark Corp

33 ANEN US EQUITY Anaren Inc

34 ALOT US EQUITY Astro-Med Inc

35 BELFA US EQUITY Bel Fuse Inc

36 BOTA US EQUITY Biota Pharmaceuticals Inc

37 BRID US EQUITY Bridgford Foods Corp

38 CACH US EQUITY Cache Inc

39 CRUS US EQUITY Cirrus Logic Inc

40 COHU US EQUITY Cohu Inc

41 JCS US EQUITY Communications Systems Inc

42 CTG US EQUITY Computer Task Group Inc

43 ATX US EQUITY AT Cross Co

44 DRAM US EQUITY Dataram Corp

45 DGAS US EQUITY Delta Natural Gas Co Inc

46 DGII US EQUITY Digi International Inc

47 DGICB US EQUITY Donegal Group Inc

48 EML US EQUITY Eastern Co/The

49 EEI US EQUITY Ecology and Environment Inc

50 ELRC US EQUITY Electro Rent Corp

51 ESIO US EQUITY Electro Scientific Industries

52 RDEN US EQUITY Elizabeth Arden Inc

53 ESCA US EQUITY Escalade Inc

54 FARM US EQUITY Farmer Bros Co

55 FBNC US EQUITY First Bancorp/Troy NC

56 FFCH US EQUITY First Financial Holdings Inc/O

57 FFKY US EQUITY First Financial Service Corp

58 FRME US EQUITY First Merchants Corp

59 FFEX US EQUITY Frozen Food Express Industries

201

60 GIII US EQUITY G-III Apparel Group Ltd

61 GSBC US EQUITY Great Southern Bancorp Inc

62 TINY US EQUITY Harris & Harris Group Inc

63 HIFS US EQUITY Hingham Institution for Saving

64 IIIN US EQUITY Insteel Industries Inc

65 INPH US EQUITY Interphase Corp

66 ITIC US EQUITY Investors Title Co

67 LAWS US EQUITY Lawson Products Inc/DE

68 MRTN US EQUITY Marten Transport Ltd

69 MFLR US EQUITY Mayflower Bancorp Inc

70 NAFC US EQUITY Nash Finch Co

71 NTSC US EQUITY National Technical Systems Inc

72 ORBT US EQUITY Orbit International Corp

73 PKOH US EQUITY Park-Ohio Holdings Corp

74 PENX US EQUITY Penford Corp

75 PWX US EQUITY Providence and Worcester Railr

76 REXI US EQUITY Resource America Inc

77 RBPAA US EQUITY Royal Bancshares of Pennsylvan

78 SYNL US EQUITY Synalloy Corp

79 TSRI US EQUITY TSR Inc

80 VIRC US EQUITY Virco Manufacturing Corp

81 WSFS US EQUITY WSFS Financial Corp

82 ZAZA US EQUITY ZaZa Energy Corp

83 ZIGO US EQUITY Zygo Corp

84 ACU US EQUITY Acme United Corp

85 BWL/A US EQUITY Bowl America Inc

86 FRS US EQUITY Frisch's Restaurants Inc

87 EGAS US EQUITY Gas Natural Inc

88 HWG US EQUITY Hallwood Group Inc/The

89 VSR US EQUITY Versar Inc

3.2.17 Portfolio 17


1 ALE US EQUITY ALLETE Inc

2 BLC US EQUITY Belo Corp

3 BKH US EQUITY Black Hills Corp

4 BGG US EQUITY Briggs & Stratton Corp

5 BWS US EQUITY Brown Shoe Co Inc

6 CACI US EQUITY CACI International Inc

202

7 CNL US EQUITY Cleco Corp

8 CBU US EQUITY Community Bank System Inc

9 CTB US EQUITY Cooper Tire & Rubber Co

10 DRL US EQUITY Doral Financial Corp

11 EDE US EQUITY Empire District Electric Co

12 FSS US EQUITY Federal Signal Corp

13 FOE US EQUITY Ferro Corp

14 GLT US EQUITY PH Glatfelter Co

15 IDA US EQUITY IDACORP Inc

16 IVC US EQUITY Invacare Corp

17 KAMN US EQUITY Kaman Corp

18 NNN US EQUITY National Retail Properties Inc

19 PKD US EQUITY Parker Drilling Co

20 PEI US EQUITY Pennsylvania Real Estate Inves

21 PBY US EQUITY Pep Boys-Manny Moe & Jack/The

22 ZQK US EQUITY Quiksilver Inc

23 RBC US EQUITY Regal-Beloit Corp

24 RLI US EQUITY RLI Corp

25 AOS US EQUITY AO Smith Corp

26 SWX US EQUITY Southwest Gas Corp

27 TG US EQUITY Tredegar Corp

28 UNF US EQUITY UniFirst Corp/MA

29 UNS US EQUITY UNS Energy Corp

30 VVI US EQUITY Viad Corp

31 WTS US EQUITY Watts Water Technologies Inc

32 WMK US EQUITY Weis Markets Inc

33 CATY US EQUITY Cathay General Bancorp

34 COHR US EQUITY Coherent Inc

35 CNMD US EQUITY CONMED Corp

36 GK US EQUITY G&K Services Inc

37 MENT US EQUITY Mentor Graphics Corp

38 ORBK US EQUITY Orbotech Ltd

39 SIGI US EQUITY Selective Insurance Group Inc

40 SKYW US EQUITY SkyWest Inc

41 NAVG US EQUITY Navigators Group Inc/The

42 TRMK US EQUITY Trustmark Corp

43 UBSI US EQUITY United Bankshares Inc/WV

44 UFCS US EQUITY United Fire Group Inc

45 WERN US EQUITY Werner Enterprises Inc

203

3.2.18 Portfolio 18


1 Y US EQUITY Alleghany Corp

2 ATO US EQUITY Atmos Energy Corp

3 BXS US EQUITY BancorpSouth Inc

4 CBT US EQUITY Cabot Corp

5 CYN US EQUITY City National Corp/CA

6 CDE US EQUITY Coeur Mining Inc

7 COO US EQUITY Cooper Cos Inc/The

8 DBD US EQUITY Diebold Inc

9 ESL US EQUITY Esterline Technologies Corp

10 FNP US EQUITY Fifth & Pacific Cos Inc

11 GMT US EQUITY GATX Corp

12 GXP US EQUITY Great Plains Energy Inc

13 HE US EQUITY Hawaiian Electric Industries I

14 IRF US EQUITY International Rectifier Corp

15 SFI US EQUITY iStar Financial Inc

16 KBH US EQUITY KB Home

17 KMT US EQUITY Kennametal Inc

18 LEG US EQUITY Leggett & Platt Inc

19 MDC US EQUITY MDC Holdings Inc

20 MCY US EQUITY Mercury General Corp

21 PL US EQUITY Protective Life Corp

22 RJF US EQUITY Raymond James Financial Inc

23 TKR US EQUITY Timken Co

24 TRN US EQUITY Trinity Industries Inc

25 WR US EQUITY Westar Energy Inc

26 WGL US EQUITY WGL Holdings Inc

27 CDNS US EQUITY Cadence Design Systems Inc

28 FCNCA US EQUITY First Citizens BancShares Inc/

29 FULT US EQUITY Fulton Financial Corp

30 PPC US EQUITY Pilgrim's Pride Corp

31 WAFD US EQUITY Washington Federal Inc

3.2.19 Portfolio 19


1 AMD US EQUITY Advanced Micro Devices Inc

204

2 LNT US EQUITY Alliant Energy Corp

3 ARW US EQUITY Arrow Electronics Inc

4 AVT US EQUITY Avnet Inc

5 CTL US EQUITY CenturyLink Inc

6 CMS US EQUITY CMS Energy Corp

7 CSC US EQUITY Computer Sciences Corp

8 STZ US EQUITY Constellation Brands Inc

9 STZ/B US EQUITY Constellation Brands Inc

10 DTE US EQUITY DTE Energy Co

11 TEG US EQUITY Integrys Energy Group Inc

12 KEY US EQUITY KeyCorp

13 LM US EQUITY Legg Mason Inc

14 NU US EQUITY Northeast Utilities

15 SCG US EQUITY SCANA Corp

16 SFD US EQUITY Smithfield Foods Inc

17 SNV US EQUITY Synovus Financial Corp

18 TDS US EQUITY Telephone & Data Systems Inc

19 TSN US EQUITY Tyson Foods Inc

20 USG US EQUITY USG Corp

21 WRB US EQUITY WR Berkley Corp

22 WTM US EQUITY White Mountains Insurance Grou

23 XEL US EQUITY Xcel Energy Inc

24 ASBC US EQUITY Associated Banc-Corp

25 LSI US EQUITY LSI Corp

26 MOLX US EQUITY Molex Inc

27 MOLXA US EQUITY Molex Inc

28 PBCT US EQUITY People's United Financial Inc

3.2.20 Portfolio 20


1 AIG US EQUITY American International Group I

2 BAC US EQUITY Bank of America Corp

3 BBT US EQUITY BB&T Corp

4 CB US EQUITY Chubb Corp/The

5 C US EQUITY Citigroup Inc

6 COP US EQUITY ConocoPhillips

7 ED US EQUITY Consolidated Edison Inc

8 IR US EQUITY Ingersoll-Rand PLC

9 JPM US EQUITY JPMorgan Chase & Co

205

10 LNC US EQUITY Lincoln National Corp

11 L US EQUITY Loews Corp

12 MTB US EQUITY M&T Bank Corp

13 TAP US EQUITY Molson Coors Brewing Co

14 MSI US EQUITY Motorola Solutions Inc

15 NOC US EQUITY Northrop Grumman Corp

16 PNC US EQUITY PNC Financial Services Group I

17 LUV US EQUITY Southwest Airlines Co

18 STI US EQUITY SunTrust Banks Inc

19 TRV US EQUITY Travelers Cos Inc/The

20 TYC US EQUITY Tyco International Ltd

21 CMCSA US EQUITY Comcast Corp

22 CMCSK US EQUITY Comcast Corp

23 FITB US EQUITY Fifth Third Bancorp

24 SYMC US EQUITY Symantec Corp

3.2.21 Portfolio 21


1 AXLL US EQUITY Axiall Corp

2 BH US EQUITY Biglari Holdings Inc

3 BRT US EQUITY BRT Realty Trust

4 CPF US EQUITY Central Pacific Financial Corp

5 CRD/A US EQUITY Crawford & Co

6 CRD/B US EQUITY Crawford & Co

7 CTS US EQUITY CTS Corp

8 DY US EQUITY Dycom Industries Inc

9 FAC US EQUITY First Acceptance Corp

10 GFF US EQUITY Griffon Corp

11 HRG US EQUITY Harbinger Group Inc

12 HVT US EQUITY Haverty Furniture Cos Inc

13 IHC US EQUITY Independence Holding Co

14 ISH US EQUITY International Shipholding Corp

15 SCX US EQUITY LS Starrett Co/The

16 LZB US EQUITY La-Z-Boy Inc

17 LEE US EQUITY Lee Enterprises Inc/IA

18 LDL US EQUITY Lydall Inc

19 MEG US EQUITY Media General Inc

20 MTH US EQUITY Meritage Homes Corp

21 MOD US EQUITY Modine Manufacturing Co

206

22 OLP US EQUITY One Liberty Properties Inc

23 OXM US EQUITY Oxford Industries Inc

24 REX US EQUITY REX American Resources Corp

25 RT US EQUITY Ruby Tuesday Inc

26 SPA US EQUITY Sparton Corp

27 SMP US EQUITY Standard Motor Products Inc

28 SPF US EQUITY Standard Pacific Corp

29 SXI US EQUITY Standex International Corp

30 STC US EQUITY Stewart Information Services C

31 SUP US EQUITY Superior Industries Internatio

32 TCI US EQUITY Transcontinental Realty Invest

33 UFI US EQUITY Unifi Inc

34 SRCE US EQUITY 1st Source Corp

35 AVHI US EQUITY AV Homes Inc

36 AMOT US EQUITY Allied Motion Technologies Inc

37 ASBI US EQUITY Ameriana Bancorp

38 AMIC US EQUITY American Independence Corp

39 ASRV US EQUITY AmeriServ Financial Inc

40 ANLY US EQUITY Analysts International Corp

41 ACAT US EQUITY Arctic Cat Inc

42 AAME US EQUITY Atlantic American Corp

43 BWINB US EQUITY Baldwin & Lyons Inc

44 BSET US EQUITY Bassett Furniture Industries I

45 BERK US EQUITY Berkshire Bancorp Inc/NY

46 CAMP US EQUITY CalAmp Corp

47 CSWC US EQUITY Capital Southwest Corp

48 CNBKA US EQUITY Century Bancorp Inc/MA

49 CHFC US EQUITY Chemical Financial Corp

50 COBR US EQUITY Cobra Electronics Corp

51 CRWS US EQUITY Crown Crafts Inc

52 CSPI US EQUITY CSP Inc

53 DXLG US EQUITY Destination XL Group Inc

54 DRCO US EQUITY Dynamics Research Corp

55 EMCI US EQUITY EMC Insurance Group Inc

56 ESBF US EQUITY ESB Financial Corp

57 EXAR US EQUITY Exar Corp

58 FLXS US EQUITY Flexsteel Industries Inc

59 FEIM US EQUITY Frequency Electronics Inc

60 GIGA US EQUITY Giga-tronics Inc

61 GLCH US EQUITY Gleacher & Co Inc

62 HELE US EQUITY Helen of Troy Ltd

207

63 HTCH US EQUITY Hutchinson Technology Inc

64 IBCP US EQUITY Independent Bank Corp/MI

65 JOUT US EQUITY Johnson Outdoors Inc

66 KTCC US EQUITY Key Tronic Corp

67 KBALB US EQUITY Kimball International Inc

68 LAKE US EQUITY Lakeland Industries Inc

69 LSCC US EQUITY Lattice Semiconductor Corp

70 MASC US EQUITY Material Sciences Corp

71 MERC US EQUITY Mercer International Inc

72 MGPI US EQUITY MGP Ingredients Inc

73 NANO US EQUITY Nanometrics Inc

74 NHTB US EQUITY New Hampshire Thrift Bancshare

75 NBBC US EQUITY NewBridge Bancorp

76 NEWP US EQUITY Newport Corp

77 PFIN US EQUITY P&F Industries Inc

78 PATK US EQUITY Patrick Industries Inc

79 PLAB US EQUITY Photronics Inc

80 RELL US EQUITY Richardson Electronics Ltd/Uni

81 SBCF US EQUITY Seacoast Banking Corp of Flori

82 SNFCA US EQUITY Security National Financial Co

83 SENEB US EQUITY Seneca Foods Corp

84 SPEX US EQUITY Spherix Inc

85 SGC US EQUITY Superior Uniform Group Inc

86 SYMM US EQUITY Symmetricom Inc

87 TBAC US EQUITY Tandy Brands Accessories Inc

88 TECUB US EQUITY Tecumseh Products Co

89 DXYN US EQUITY Dixie Group Inc/The

90 TORM US

EQUITY

TOR Minerals International Inc

91 TWMC US EQUITY Trans World Entertainment Corp

92 USEG US EQUITY US Energy Corp Wyoming

93 UNAM US EQUITY Unico American Corp

94 VOXX US EQUITY VOXX International Corp

95 AMS US EQUITY American Shared Hospital Servi

96 BRN US EQUITY Barnwell Industries Inc

97 CAW US EQUITY CCA Industries Inc

98 CVR US EQUITY Chicago Rivet & Machine Co

99 CRV US EQUITY Coast Distribution System Inc/

100 CUO US EQUITY Continental Materials Corp

101 CMT US EQUITY Core Molding Technologies Inc

102 BDL US EQUITY Flanigan's Enterprises Inc

103 FRD US EQUITY Friedman Industries Inc

208

104 JOB US EQUITY General Employment Enterprises

105 GV US EQUITY Goldfield Corp/The

106 HH US EQUITY Hooper Holmes Inc

107 IOT US EQUITY Income Opportunity Realty Inve

108 IHT US EQUITY InnSuites Hospitality Trust

109 ITI US EQUITY Iteris Inc

110 LGL US EQUITY LGL Group Inc/The

111 RWC US EQUITY Relm Wireless Corp

112 SIF US EQUITY SIFCO Industries Inc

113 SLI US EQUITY SL Industries Inc

114 SSY US EQUITY SunLink Health Systems Inc

115 STS US EQUITY Supreme Industries Inc

116 VII US EQUITY Vicon Industries Inc

117 VGZ US EQUITY Vista Gold Corp

118 WGA US EQUITY Wells-Gardner Electronics Corp

3.2.22 Portfolio 22


1 ALK US EQUITY Alaska Air Group Inc

2 AM US EQUITY American Greetings Corp

3 AVA US EQUITY Avista Corp

4 BHE US EQUITY Benchmark Electronics Inc

5 BDN US EQUITY Brandywine Realty Trust

6 BC US EQUITY Brunswick Corp/DE

7 DDS US EQUITY Dillard's Inc

8 FBP US EQUITY First BanCorp/Puerto Rico

9 HOV US EQUITY Hovnanian Enterprises Inc

10 LPX US EQUITY Louisiana-Pacific Corp

11 NC US EQUITY NACCO Industries Inc

12 PNK US EQUITY Pinnacle Entertainment Inc

13 PNM US EQUITY PNM Resources Inc

14 POL US EQUITY PolyOne Corp

15 RYL US EQUITY Ryland Group Inc/The

16 UAM US EQUITY Universal American Corp/NY

17 WBS US EQUITY Webster Financial Corp

18 AGII US EQUITY Argo Group International Holdi

19 CAR US EQUITY Avis Budget Group Inc

20 KELYA US EQUITY Kelly Services Inc

21 NWLI US EQUITY National Western Life Insuranc

209

22 WSBC US EQUITY WesBanco Inc

23 YRCW US EQUITY YRC Worldwide Inc

24 SEB US EQUITY Seaboard Corp

3.2.23 Portfolio 23


1 AFG US EQUITY American Financial Group Inc/O

2 ASH US EQUITY Ashland Inc

3 AN US EQUITY AutoNation Inc

4 CWH US EQUITY CommonWealth REIT

5 FHN US EQUITY First Horizon National Corp

6 FL US EQUITY Foot Locker Inc

7 KMPR US EQUITY Kemper Corp

8 LEN US EQUITY Lennar Corp

9 MBI US EQUITY MBIA Inc

10 ODP US EQUITY Office Depot Inc

11 OMX US EQUITY OfficeMax Inc

12 OCR US EQUITY Omnicare Inc

13 SIG US EQUITY Signet Jewelers Ltd

14 VSH US EQUITY Vishay Intertechnology Inc

15 ANAT US EQUITY American National Insurance Co

16 IDTI US EQUITY Integrated Device Technology I

17 BPOP US EQUITY Popular Inc

18 SUSQ US EQUITY Susquehanna Bancshares Inc

19 TECD US EQUITY Tech Data Corp

20 TLAB US EQUITY Tellabs Inc

3.2.24 Portfolio 24


1 CNA US EQUITY CNA Financial Corp

2 CMA US EQUITY Comerica Inc

3 CLGX US EQUITY CoreLogic Inc/United States

4 UFS US EQUITY Domtar Corp

5 GCI US EQUITY Gannett Co Inc

6 MGA US EQUITY Magna International Inc

210

7 NI US EQUITY NiSource Inc

8 ORI US EQUITY Old Republic International Cor

9 PNW US EQUITY Pinnacle West Capital Corp

10 PHM US EQUITY PulteGroup Inc

11 SVU US EQUITY SUPERVALU Inc

12 TOL US EQUITY Toll Brothers Inc

13 UNM US EQUITY Unum Group

14 URS US EQUITY URS Corp

15 CINF US EQUITY Cincinnati Financial Corp

16 HBAN US EQUITY Huntington Bancshares Inc/OH

17 MU US EQUITY Micron Technology Inc

18 MYL US EQUITY Mylan Inc/PA

19 ZION US EQUITY Zions Bancorporation

3.2.25 Portfolio 25


1 CBS US EQUITY CBS Corp

2 CBS/A US EQUITY CBS Corp

3 DUK US EQUITY Duke Energy Corp

4 RF US EQUITY Regions Financial Corp

A deep dive into the Mean/Variance Efficiency of the ......A deep dive into the Mean/Variance Efficiency of the Market Portfolio Master Thesis Supervised by Prof. Dr. Karl Schmedders

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