masterthesis_venetia_christodoulopoulou.pdf

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Morphological Factor Analysis inPortfolio Design

Master Thesis

Venetia Christodoulopoulou

MSc. Computational Science and Engineering

Field of Specialization: Financial Engineering

in the Department of Mathematics and Physics

Supervisor: Prof. Dr. Diethelm Wrtz

ETH Zrich, June 2013


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Abstract

The Mean-Variance (MV) theory of Markowitz (1952) is widely regarded

as one of the major theories in financial industry and it has been the de-facto

standard for portfolio optimization in the past fifty years. In this work the

feasible set of Markowitz portfolio is approximated by an ellipse character-

ized by the basic geometric parameters (area, center of mass, eccentricity

and orientation) of the hull. Similar to a Principal Component Analysis, a

dimensionality reduction for the first few geometric moments of the feasible

set is performed to identify and to extract hidden features. This method

simplifies the evolution analysis of the feasible set, thus making it possible

to describe the variability over time.

Using the introduced geometric shape factor analysis the interplay ofthe Eurozone bond market is explored. Barclays all maturities bond price

index is investigated by analyzing the feasible set of a Markowitz portfolio.

The components consist of the peer group of the major 11 members of the

European Monetary Unit (EMU). The dynamic behavior of the hulls over a

rolling time window, expressed by the shape factors, illustrates the progres-

sion of the Eurozones debt crisis and shows that the geometric shape factor

analysis successfully describes the evolution of complex structures, such as

the feasible sets.

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Acknowledgments

I would specially like to thank my supervisor, Prof. Dr. Diethelm Wuertz (ETH Zurich, In-

stitute for Theoretical Physics, Econophysics Group), for his invaluable mentoring and guid-

ance during the process of writing my master thesis.

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Contents

Abstract ii

Acknowledgments iv

Table of Contents vi

List of Figures viii

List of Tables ix

1 Introduction 1

2 European Sovereign Debt Crisis 3

2.1 The Eurozone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Timeline of key events in the European debt crisis . . . . . . . . . . . . . . . . . . 42.3 Eurozone Financial and Economic Indicators . . . . . . . . . . . . . . . . . . . . . 8

3 Principal Component Analysis and Clustering 14

3.1 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 How many principal components? . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Rolling Time Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4.1 Hierarchical Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 The Mean-Variance Markowitzs Model 22

4.1 Assumptions of the MV model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Settings of the Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Efficient Frontier and Minimum Variance Locus . . . . . . . . . . . . . . . . . . . 24

4.4 Special Portfolios of the Feasible set . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.4.1 Minimum Variance Portfolio (MVP) . . . . . . . . . . . . . . . . . . . . . . 26

4.4.2 Equal Weights Portfolio (1/n) . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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4.4.3 Most Diversified Portfolio (MDP) / Tangency Portfolio (TGP) . . . . . . . 26

4.5 Optimal Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.6 Illustration of the Mean Variance model . . . . . . . . . . . . . . . . . . . . . . . . 28

4.7 Limitations of Mean-Variance Optimization . . . . . . . . . . . . . . . . . . . . . . 30

5 Statistical Shape Analysis 31

5.1 Portfolio Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Morphological Characterization of the Feasible Set . . . . . . . . . . . . . . . . . 34

5.2.1 Area and Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2.2 Centroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2.3 Eccentricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2.4 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.3 Time Evolution of Morphological Factors . . . . . . . . . . . . . . . . . . . . . . . 38

5.4 Moments invariants to translation, rotation and scaling . . . . . . . . . . . . . . . 40

5.5 Empirical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Robust Estimation 43

6.1 Minimum Volume Ellipsoid Estimator . . . . . . . . . . . . . . . . . . . . . . . . 43

6.2 Minimum Covariance Determinant Estimator . . . . . . . . . . . . . . . . . . . . 44

6.3 Orthogonalized Gnanadesikan-Kettenring Estimator . . . . . . . . . . . . . . . . 45

6.4 Empirical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7 Conclusions 49

Bibliography 51

A Timeline of European debt crisis 55

B Datasets 63

B.1 Barclays European Bond Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

C Tables 64

D Conferences and Proceedings 70

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List of Figures

2.1 Eurozone map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 General Government gross debt (%GDP) . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Real GDP growth rate, percentage change on previous year . . . . . . . . . . . . 10

2.4 Unemployment Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Long-term interest rates, annual average (%) . . . . . . . . . . . . . . . . . . . . . 112.6 Annual mean of the bond returns calculated at the end of the year for the

period [1998-2012]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.7 Annual standard deviation of the bond returns calculated at the end of the year

for the period [1998-2012]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Screen graph and Eigenvalue Ratio of nine time windows. . . . . . . . . . . . . . 17

3.2 Number of PCs according to the Eigenvalue Ratio estimator. . . . . . . . . . . . . 18

3.3 Number of PCs needed for at least 90% of the total variation. . . . . . . . . . . . 19

3.4 Evolution of Cluster Dendrograms over 3-year rolling time windows. . . . . . . 21

4.1 Different views of Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Feasible Set of Barclays European Bond Indices dataset. . . . . . . . . . . . . . . 29

4.3 Efficient Frontier of European Bond portfolio. . . . . . . . . . . . . . . . . . . . . 29

5.1 Feasible set of the Eurozone bond portfolio for the entire period [1996-01-01 -

2013-04-17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Evolution of feasible sets of the Eurozone bond portfolio. . . . . . . . . . . . . . 33

5.3 Time evolution of the returns versus covariance risk (sd) of Greece, Ireland,Portugal, Italy and Spain bond prices. . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4 Morphological Characterization of the Eurozone bond portfolio. . . . . . . . . . 37

5.5 Evolution of the morphological factors of the dataset. . . . . . . . . . . . . . . . . 39

5.6 Evolution of the seven Hu moments of the dataset. . . . . . . . . . . . . . . . . . 42

6.1 Feasible set of the Eurozone bond dataset for the entire period [1999-01-01 -

2013-04-30] using several robust methods for the estimate of the covariance

matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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6.2 Evolution of the hull using sample and robust estimate covariance matrix and

evolution of the morphological factors of sample and robust feasible sets. . . . . 47

6.3 Difference between the sample and the robust morphological factors. . . . . . . 48

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List of Tables

3.1 Monthly Rolling Window initial and final dates for the bond returns dataset. . . 16

C.1 Data of General Government gross debt % GDP. . . . . . . . . . . . . . . . . . . . 65

C.2 Data of Real GDP growth rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

C.3 Data of unemployment rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

C.4 Data of Long-term interest rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

C.5 Mean of the returns of the bond prices. . . . . . . . . . . . . . . . . . . . . . . . . 69

C.6 Standard Deviation of the returns of the bond prices. . . . . . . . . . . . . . . . . 69

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Chapter 1

Introduction

The recent financial crisis of 2008, triggered by a subprime mortgage default, initiated a

cascading process that eventually led to a dramatic worldwide economic collapse. In particu-lar, after the Lehman collapse in September 2008 the world went through the largest recession

since the 1930s. Fiscal deficits increased in all countries. Europe faced the biggest challenge

since 1990, the European sovereign-debt crisis. These events were a direct consequence of

the financial crisis and provide a clear motivation for the continuous efforts to predict eco-

nomic patterns and market behaviors as well as to improve frameworks for financial stability

analysis and policy.

In this work the focus is on the Eurozone financial crisis and the study of its evolution

in a more comprehensible way. Initially, in order to gain a primary intuition on the evolu-

tion of the Eurozone bond market, a principal component analysis and hierarchical clustering

analysis are performed. Thereafter, treating the Eurozone bond market as a portfolio with

components the peer group of the major 11 members of the European Monetary Unit (EMU)

and using the Barclays European Bond Indices, the Mean-Variance (MV) Markowitzs portfo-

lio optimization model is applied. Due to the extreme events occurred during the past four

years in the Eurozone bond market, MV model results in rather complex feasible sets. The

complexity of these curves renders the analysis impractical. In an attempt to obtain a more

abstract characterization, a geometric shape factor analysis is proposed, which makes it eas-

ier to investigate the interplays within Eurozone bond market. Finally, due to the inevitableestimation errors the MV model is subject to, an analysis of the hulls by means of a robust

estimator of covariance matrix is carried out.

The thesis is organized as follows:

Chapter 1 contains the motivation to pursue a more abstract characterization of Markowitzs

Portfolio feasible set and includes an outline of the organization of the thesis.

Chapter 2 gives an general description of the Eurozone, presents a timeline of the European

sovereign-debt crisis with the most important events and the most significant Eurozone

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financial and economic indicators.

In Chapter 3 an analysis of Barclays European Bond Indices is undertaken, using Principal

Component Analysis and Hierarchical Clustering Analysis over a rolling time window.

These analyses give an insight into the bond market over the last 14 years.

Chapter 4 describes the model assumptions of Mean-Variance framework and the general

setting of Markowitzs portfolio. It also defines the hull of all possible portfolios (feasi-

ble set), the efficient frontier and possible investment strategies based on Markowitzs

Portfolio Optimization. Finally, it presents the limitations of Mean-Variance model.

Chapter 5 deals with the evolution of the feasible sets of the dataset using rolling analysis

and introduces a morphological characterization of the portfolios hull. Moreover the

feasible set is approximated by an ellipse characterized by the shape factors of hull and

the evolution of the geometrical parameters is displayed.

Chapter 6 presents three robust estimators as well as the evolution of feasible sets produced

by using a robust estimation of covariance matrix on the rolling time windows.

Chapter 7 addresses the purpose for using geometrical shape factor analysis in the feasible

sets resulted from Markowitzs Portfolio Optimization, summarizes the methods em-

ployed in the previous chapters and discusses the effectiveness of the proposed method-

ologies.

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Chapter 2

European Sovereign Debt Crisis

2.1 The Eurozone

The Eurozone, officially called the euro area, is an economic and monetary union (EMU) of

17 European Union (EU) member states that have adopted the euro (e) as their common cur-

rency and sole legal tender. The currency of euro officially came into existence on January 1,

1999. The Eurozone currently consists of Austria, Belgium, Cyprus, Estonia, Finland, France,

Germany, Greece, Ireland, Italy, Luxembourg, Malta, the Netherlands, Portugal, Slovakia,

Slovenia, and Spain. All Member States of the European Union, except Denmark and the

United Kingdom, are required to adopt the euro and join the euro area. To do this they must

meet certain conditions known as convergence criteria" [1]. Of the Member States outside

the euro area, Denmark and the United Kingdom have opt-outs from joining for reasons of

economic sovereignty. These two countries can join in the future if they so wish. Bulgaria,

the Czech Republic, Denmark, Hungary, Latvia, Lithuania, Poland, Romania, Sweden, and

the United Kingdom are EU Member States but do not currently use the single European

currency.

The euro is also used in countries outside the EU [2], [3]. Three states, Monaco, San

Marino, and Vatican City have signed formal agreements with the EU to use the euro and

issue their own coins. Nevertheless, they are not considered part of the Eurozone by the ECB

and do not have a seat in the ECB or Euro Group. Andorras monetary agreement with theEU to use the euro came into force in April 2012 and will permit it to issue its own euro coins

as early as 1 July 2013, provided that Andorra implements relevant EU legislation. Kosovo

and Montenegro officially adopted the euro as their sole currency without an agreement and,

therefore, have no issuing rights. These states are not considered part of the Eurozone by the

ECB. Figure 2.1 shows the current situation of the Eurozone.

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Figure 2.1: Eurozone map. Source: http://en.wikipedia.org/wiki/Eurozone

Eurozone (17)EU states obliged to join the Eurozone (8)

EU state with an opt-out on Eurozone participation (2)

Areas outside the EU using the euro with an agreement (4)

Areas outside the EU using the euro without an agreement (2)

2.2 Timeline of key events in the European debt crisis

The European sovereign debt crisis is an ongoing financial crisis which has made it difficult

or impossible for some countries in the euro area to repay of re-finance their government debt

without the assistance of third parties. In the this section it will be presented a timeline with

the most important events of the Eurozone crisis. A more detailed timeline can be found in

Appendix A.

The following timeline is borrowed from [4], [5], [6], [7] and [8].

2009 October 4: George Papandreouss Pan-Hellenic Socialist Movement (PASOK) wins na-

tional elections in Greece.

October 20: Greek Finance Minister George Papakonstantinou announces a revised deficit

figure of 12.5% of GDP for the year, double the previous estimate. The statement shocks

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bond markets and is widely considered to be the beginning of the European debt crisis.

December: Agencies cut their ratings of Greek debt.

2010 February 12: Euro falls as EU fails to agree Greece rescue plan.

March 26: Euro leaders agree, together with the IMF, a rescue package for debt-stricken

Greece and the euro bounces back from a 10-month low against the dollar.

April 23: Greece officially requests for a e45 billion bailout package financed by the EU

and the International Monetary Fund (IMF).

April 27: Standard & Poors cuts Greek debt to junk status and downgrades Por-

tuguese debt to A-.

April 28: Standard & Poors downgrades Spanish bonds from AA to AA- because of

poor growth prospects.

May 2: The EU agrees to a e110 billion bailout package for Greece.

In exchange, Greece vows to slim down its budget by e30 billion over the next three

years. It is the first rescue of a member of the 16-nation Eurozone.

May 3: ECB supports Greeces efforts for fiscal consolidation.

May 5: Greek Austerity Plan Approved.

May 9-10: EU ministers agree e500 billion fund, European Financial Stability Facility

(EFSF), to save euro from disaster.

May 28: Ratings agency Fitch downgrades Spains credit rating from AAA to AA+, say-

ing austerity measures will affect growth.June 10: As part of the restructuring of Spains struggling financial sector, Caja Madrid

takes over Bancaja. The previous month Caja madrid announced its merger with five

regional savings and loan banks.

June 30: The ECBs Governing Council ends its covered bond purchase program as

planned, after one year. A nominal amount ofe60 billion is purchased on the pri-

mary and secondary markets.

July 19: Moodys cuts Irelands sovereign bond rating by one notch as IMF pulls e20

billion finance deal for Hungary.

July 23: Stress Tests of banks fail to reassure. 7/91 largest banks in Europe failed the

tests.

September 23: Irish economy faces double dip recession as figures reveal national out-

put dropped by 1.2% in the second quarter of 2010.

November 21: Ireland seeks financial support.

November 22: IMF and EU bail out Ireland amid fears of Eurozone contagion.

November 28: Ireland applies for and gets an e85 billion bailout.

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2011 March 21: EU finance ministers agree to replace the ESFF with the permanent e500 bil-

lion European Stability Mechanism (ESM), which could start operating in 2012.

March 31: Ireland forced into new e21 billion bailout by debt crisis.

April 6: Portugal Seeks Bailout

May 16: EU finance ministers approve e78 billion bailout for Portugal.

June 13: Standard & Poor cuts Greek debt to CCC, the lowest rating of any country it

reviews.

June 23: European Financial Stability Facility is boosted.

July 5: Despite the rescue package that Portugal received in May, Moodys Investors Ser-

vice lowers its rating to junk status and suggest that further downgrades are likely.

July 21: An EU summit agrees to a second Greek bailout ofe120 billion.

August 8: The ECB launches its e22 billion operation to buy Spanish and Italian bonds.

September 3: Ireland getse

1.2 billion IMF payout.September 20: Standard & Poors downgrades Italys sovereign credit rating, with the

ratings agency keeping the countrys outlook on negative.

October 7: Credit ratings agency Fitch cut Italys credit rating by one notch to A+ from

AA- and cut Spains rating to AA- from AA+.

October 18: France and Germany reach agreement to boost Eurozones rescue fund to

e2 trillions.

October 26: European leaders obtain an agreement from banks to take a 50% loss on the

face value of their Greek debt, a plan that will bring greek debt down by 2020 to 120%

of that nations GDP.

November 6: Greek Prime Minister George Papandreou resigns. A government of na-

tional unity, led by former ECB vice president Lucas Papademos, takes over.

November 12: Italys Prime Minister Silvio Berlusconi resigns after parliament approves

a package of debt reduction measures. Seasoned technocrat Mario Monti takes over.

November 23: Sovereign debt crisis spreads to Germany. Shock as e6 billion German

bond sale ends in failure.

December 9: Euro area leaders agree on a new fiscal compact", compact to co-ordinate

economic policies with a ceiling on debts.December 21: Eurozone banks borrow e500 billion from ECB.

2012 January 15: Standard & Poors downgrades French and Austrian debt to AA+. The

ESFS is also downgraded from AAA to AA+. Moreover, S& P lowers Spain, Italy and

five other euro members further, and maintains the top credit rating for Finland, Ger-

many, Luxembourg, and the Netherlands

January 30: EU countries, except the UK and the Czech Republic, agree to a German-led

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fiscal compact that caps the annual structural deficits of EU member states to 0.5% of

their respective GDP.

February 15: Eurozone economy shrinks for first time since 2009.

February 21: Eurogroup agrees on second financial aid package for Greece.

February 29: ECB hands e529 billion in emergency loans to 800 European banks.

March 13: Fitch upgrades Greeces credit rating from junk" to B.

March 16: Eurozone firewall increased to e700 billion.

March 30: Eurozone ministers agree e500 billion in new bailout funds.

April 27: Standard & Poors downgrades Spains credit rating from A to BBB+ and

warns that Spains recession is likely to deepen by the end of the year.

May 6: Socialist Francois Hollande is elected president of France on an anti-austerity

platform. His victory weakens the German-led consensus for austerity as the way to

steer Europe out of its debt crisis.May 6: Greece holds parliamentary elections. Anti-bailout party Syriza finishes a sur-

prising second behind center-right and pro-bailout party New Democracy.

May 14: Greece euro exit fears rattle stock markets.

May 15: Eurozone avoids double-dip recession as Germany makes up for losses.

May 19 G8 leaders end summit with pledge to keep Greece in Eurozone.

May 23: Greece euro exit fears hit stock markets.

June 7: Spains credit rating downgraded by Fitch as international bailout looms.

June 9: Eurozone finance ministers agree to a e100 billion bailout of Spanish banks.

With this latest rescue, the EU has committed e500 billion to finance European bailouts.

June 20: Greece: New Democracy leader Antonis Samaras is sworn in as a prime minis-

ter after his party forms a coalition government with the socialist PASOK and the Demo-

cratic Left. Samaras and his team are expected to renegotiate the terms of Greeces e130

billion bailout.

June 21: Banks downgraded as size of Spanish crisis revealed.

June 27: Cyprus requests financial support. Cyprus becomes fifth Eurozone country to

ask for outside financial help after it is caught in backwash of Greek crisis. | Spain

seeks financial support.June 29: Euro area leaders agree to establish a new banking supervisory agent run by

the ECB.

July 5: Ireland returns to the debt markets with e500m sale of treasury bills.

July 10: Eurozone finance ministers agree to a e30 billion bailout of Spains troubled

banks. Minister also extend by one year -until 2014- the countrys deadline, to achieve

a budget deficit of 3% of GDP.

July 17: Portugal gets IMF approval for next bailout payment.

July 20: Eurogroup grants financial assistance to Spains banking sector.

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July 23: Spain in crisis talks with Germany over e300bn bailout.

July 24: Panic selling of shares in Spain as countrys bond yields go above 7.5%. Big

falls on world financial markets as fears grow of second successive Eurozone summer

crisis.

September 4: Moodys turns negative on EUs Aaa rating.

October 11: Downgrade raises pressure on Madrid to accept bailout. Standard & Poors

says it has lowered the rating on Spains debts two notches, to BBB- following slump in

countrys fortunes.

2013 February 28: Spain falls further into recession as GDP plunges by 0.8%. Gloomy figures

for final quarter of 2012 offset by news that deficit has fallen to 6.7%, close to EU target.

March 20: Cyprus rejects bailout deal leaving Eurozone facing fresh crisis.

March 25: Eurogroup reaches agreement on future macroeconomic adjustment pro-

gramme for Cyprus.

2.3 Eurozone Financial and Economic Indicators

In the following graphs we will show some of the most important indicators of the financial

crisis in the Eurozone. We will focus on the following countries: Belgium, Germany, Ireland,

Greece, Spain, France, Italy, the Netherlands, Austria, Portugal, Finland and UK. The dataused for the extraction of these graphs are given in tables C.1, C.2, C.3 and C.4 in Appendix

C.

Figure 2.2 shows the general government gross debt percentage of GDP. This indicator

shows how large a countrys outstanding sovereign debt compares to its annual GDP. This

number allows you to easily compare how sustainable debt is across very different countries

and economies. The higher the amount of debt compared to the economy, the more of a

countrys resources must go to paying down debt rather than other things such as investment

or consumption. Greece has the biggest general government gross debt % of GDP for all years

(2008-2012) followed by Italy, Portugal and Ireland.

Moreover, in Figure 2.3 it is depicted the growth rate of the GDP 1. The change in Gross

Domestic Product is a commonly used indicator of economic activity in a country. Positive

GDP growth can make it easier for a country to pay its debts, as it has more revenue to draw

from. Negative GDP growth has the opposite effect. It is observed that for the year 2009 all

1Gross domestic product (GDP) is a measure of the economic activity, defined as the value of all goods andservices produced less the value of any goods or services used in their creation. The calculation of the annualgrowth rate of GDP volume is intended to allow comparisons of the dynamics of economic development bothover time and between economies of different sizes.

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Figure 2.2: General Government gross debt (%GDP). Source of Data: Eurostat, [9]

countries had negative GDP growth rate. It must be noted that the U.S. subprime mortgage

crisis which led to a financial crisis and subsequent recession began in 2008 2. In addition,

in 2009 was the beginning of the European sovereign-debt crisis. Greece is the only country

among the Eurozone and UK with negative GDP growth rate for the period 2009-2013. The

forecast for 2013 indicates Greece as the country with the smallest GDP growth rate, followed

by Portugal, Spain, Italy and the Netherlands.Figure 2.4 shows the annual Unemployment Rate 3 for the period [2008-2012]. Higher un-

employment can reduce government revenues from payroll taxes. Rising unemployment can

also encourage political instability. In the EU-27 the unemployment rate in 2012 was 10.5%,

up from 9.7% in 2011 and 2010. Among the Member States, the lowest unemployment rates

in 2012 were recorded in Austria (4.3%), the Netherlands (5.3%) and Germany (5.5%), and

the highest rates in Spain (25%), Greece (24.3%) and Portugal (15.9%). Compared with a year

ago, the unemployment rate increased in sixteen Member States, fell in eight and remained

the same in three. The highest increases were registered in Greece (17.7% to 24.3%), Cyprus

(7.9% to 11.9% ), Spain (21.7% to 25%) and Portugal (12.9% to 15.9%). The largest decreases

among the countries we are focusing on were observed in Germany (5.9% to 5.5%), Finland

(7.8% to 7.7%) and Ireland remained in the same levels (14.7%).

2Several major financial institutions collapsed in September 2008, with significant disruption in the flow ofcredit to businesses and consumers and the onset of a severe global recession.

3Unemployment rates represent unemployed persons as a percentage of the labor force. The labor force is thetotal number of people employed and unemployed. Unemployed persons comprise persons aged 15 to 74 whowere: a. without work during the reference week, b. currently available for work, c. actively seeking work, orwho found a job to start later.

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Figure 2.3: Real GDP growth rate, percentage change on previous year. Source of Data: Eurostat, [9]

Figure 2.4: Unemployment Rate. Source of Data: Eurostat, [9]

Bar plot 2.5 shows the Long-term interest rates, 10-year government bond yields, annual

average (%). An interest rate is the cost or price of borrowing, or the gain from lending,

normally expressed as an annual percentage amount. Ten year government bond yields are

often used as a measure for long-term interest rates. Yields vary according to the price of the

bond. Bond Yields show how much interest a government needs to pay annually in order to

borrow money. Here we show the yield as a percentage. When investors become fearful that

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Figure 2.5: Long-term interest rates, 10-year government bond yields. Annual average (%). Source ofData: OECD.Stat, [10]

a government may not pay them back, they often demand a higher interest rate, which can

be seen in rising bond yields. Higher bond yields make it more expensive for governments

to borrow and their debt harder to bear. A 10-year yield above 7% has triggered bailouts for

Greece, Portugal, and Ireland. In 2010 only Greece had a substantial increase in the long-term

interest rates from 5.17% to 9.09%. In 2011 the ten-year bond yields increased dramaticallyfor Greece 73% (9.09% to 15.75% ), Portugal 89% (5.4% to 10.24%) and Ireland 60% (5.99%

to 9.58%). In 2012 the highest interest rates were recorded in Greece (22.5%) and in Portugal

(10.55%) while there was a decrease for Ireland (9.58% to 5.99%).

In the current master thesis we will try to analyze the European bond market since the

existence of the Eurozone using factor analysis. The dataset we will use is the Barclays

European Bond Indices which is introduced in Appendix B.1. This dataset holds bond prices

of Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Netherlands, Portugal,

Spain, Sweden, UK and US, together with an aggregate Euro Government index for the time

period [1997-12-31 - 2013-04-30]. In Figures 2.6 and 2.7 we can see the mean and the standard

deviation, respectively, of the returns for every country for the period [1998-2012]. The data

used for the extraction of these graphs are given in tables C.5 and C.6, in Appendix C.

The line chart 2.6 depicts the average of the bond returns of each year for the Eurozone

countries between 1998 and 2012. It may be seen clearly that for the period 1998-2009 the

returns for the Eurozone countries move together. In 2010 it is observed a substantial decrease

for Greece, Ireland, Portugal and Spain. In 2011 bond returns dropped dramatically for

Greece. Portugal and Italy had a steady decline, in contrast to Ireland and Spain, which had

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a significant recovery. Finally, in 2012 there was rapid raise for Greece, Portugal and Italy.

Ireland had a moderate progression and Spain remained stable.

0.3

0.2

0.1

0.0

0.1

0.2

Time

Mean

AT

BE

DE

ES

FI

FR

GR

IE

IT

NL

PT

EU

GB

US

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

Figure 2.6: Annual mean of the bond returns calculated at the end of the year for the period [1998-2012].

Figure 2.7 displays the standard deviation of the bond returns for the same period. In 2010

it is recorded a vast jump for Greece and a considerable increase for Portugal and Ireland,

while for Italy and Spain there was a little raise. In 2011 standard deviation stayed con-

stant for Greece and increased substantially for Portugal, Ireland, Italy and Spain. Lastly, in

2012 Greece shows a dramatic increase, Portugal and Spain raised slightly, Ireland decreased

quickly and Italy had a slight reduction.

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0.0

0.5

1.0

1.5

2.0

2.5

Time

StandardDeviation

AT

BE

DE

ES

FI

FR

GR

IE

IT

NL

PT

EU

GB

US

1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

Figure 2.7: Annual standard deviation of the bond returns calculated at the end of the year for theperiod [1998-2012].

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Chapter 3

Principal Component Analysis and

Clustering

In the current chapter we will analyze Barclays European Bond Indices, introduced in

Appendix B.1. To do so, we will consider the time series of the bond returns in the Eurozone

as a portfolio with the countries forming the peer group.

In order to analyze the complexity and the evolution of our portfolio over time we will

apply the Principal Component Analysis (PCA) and the Hierarchical Clustering Analysis on

a rolling time window. In the following sections it will be presented the theory of PCA and

hierarchical clustering as well as the application to our dataset.

3.1 Principal Component Analysis

Principal component analysis (PCA) is a multivariate technique which analyzes a data

set in which observations are described by several inter-correlated quantitative dependent

variables. The central idea of PCA is to reduce the dimensionality of the data set, while

retaining as much as possible of the variation present in the data. This is achieved by a

transformation to a new set of orthogonal variables, the principal components (PCs), which

are linearly uncorrelated, and the number of those is less than or equal to the number of

original variables [12]. This transformation is defined in such a way that the first principal

component has the largest possible variance and each succeeding component in turn has the

highest variance possible under the constraint of being orthogonal to (i.e., uncorrelated with)

the preceding components.

The goals of PCA are to [13]:

extract the most important information from the data set,

compress the size of the data set by keeping only this important information,

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simplify the description of the data set, and

analyze the structure of the observations and the variables.

In order to achieve these goals, PCA computes principal components which are obtained

as linear combinations of the original variables.

PCA is mathematically defined as an orthogonal linear transformation that transforms

the data to a new coordinate system such that the greatest variance by any projection of the

data comes to lie on the first coordinate (called the first principal component), the second

greatest variance on the second coordinate, and so on. PCA can be done by either eigenvalue

decomposition of a data covariance (or correlation) matrix or singular value decomposition of a

data matrix, usually after mean centering the data matrix for each attribute. Details on the

derivation of the PCs of both methods can be found in Jolliffe and Abdi. H., & Williams, L.J,

[12] and [13].

3.1.1 How many principal components?

In this section we present a number of rules for deciding how many PCs should be retained

in order to account for most of the variation in the original data set.

Cumulative Percentage of Total Variation

Perhaps the most obvious criterion for choosing the number of PCs is to select a (cumu-

lative) percentage of total variation which one desires that the selected PCs contribute, e.g.

90%. The required number of PCs is then the smallest value of m, the number of PCs, for

which this chosen percentage is exceeded. So, basically, we keep enough PCs to account for

90% of the variation, since principal components are successively chosen to have the largest

possible variance.

Screen Graph

The first rule described above involves a degree of subjectivity in the choice of the cut-

off levels, i.e. the percentage of total variation. The screen graph, which was discussed

and named by Cattell [15], is even more subjective, as it involves looking at a plot of the

eigenvalues against the number of PCs and deciding at which principal component the slope

of the plotted points is steep enough. This value of principal components is then taken to

be the number of components to be retained.

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Eigenvalue Ratio

Ahn and Horenstein in [14] proposed two new estimators for determining the number of

factors (r) in static approximate factor models, the Eigenvalue Ratio and Growth Ratio.

We will focus on the former one. The Eigenvalue Ratio estimator is obtained simply by

maximizing the ratio of two adjacent eigenvalues arranged in descending order and it is

defined as follows:

rER = max1kkmax

ER(k) (3.1)

ER(k) =lk

lk+1(3.2)

where lk is the k th largest eigenvalue.

3.2 Rolling Time Windows

A rolling analysis of time series data is often used to track the evolution of a a given

property over time. When analyzing financial time series data using a statistical model, a

common technique to assess the constancy of a models parameters is to compute parameter

estimates over a rolling window of a fixed size through the sample. If the parameters are

constant over the entire sample, then the estimates over the rolling windows should not be

too different. If the parameters change at some point during the sample, then the rolling

estimates should capture this instability, [16].

3.3 Empirical Results

Taking the returns of the Barclays European Bond Indices dataset and as a starting point

the date 1996-01-01 (3 years before the start of the economic and monetary union (EMU)) we

apply a 3-year rolling window with monthly shift, which results in 173 windows. Table 3.1

presents a part of the initial and final dates of the time windows.

Start Date End Date Start Date End Date

1996-01-01 1998-12-31...

...1996-02-01 1999-01-31 2009-11-01 2012-10-311996-03-01 1999-02-28 2009-12-01 2012-11-301996-04-01 1999-03-31 2010-01-01 2012-12-311996-05-01 1999-04-30 2010-02-01 2013-01-311996-06-01 1999-05-31 2010-03-01 2013-02-281996-07-01 1999-06-30 2010-04-01 2013-03-31

...... 2010-05-01 2013-04-30

Table 3.1: Monthly Rolling Window initial and final dates for the bond returns dataset.

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For each of these time windows, a Principal Component Analysis is performed, using the

singular value decomposition. This is generally the preferred method for numerical accuracy.

Moreover we plot the eigenvalues against the number of the PCs (screen graph), calculate

the eigenvalue ratio (ER) and the number of PCs needed for cumulative percentage of total

variation 90%. The aforementioned are displayed in the following figures.

2 4 6 8 10

0

2

4

6

8

10

19960201 19990131

i i l

Eigenvalue q

qq q q q q q q q q

2 4 6 8 10

2

4

6

8

10

12

14

Eige

nvalueRatio

Principal Component number

2 4 6 8 10

0

2

4

6

8

10

19990101 20011231

i i l

Eigenvalue

q

q q q q q q q q q q

2 4 6 8 10

0

10

20

30

40

Eige

nvalueRatio


2 4 6 8 10

0

2

4

6

8

10

20061101 20091031

i i l

Eigenvalue

q

q q q q q q q q q q

2 4 6 8 10

5

10

15

20

Eige

nvalueRatio


2 4 6 8 10

0

2

4

6

8

10

20070501 20100430

i i l

Eigenvalue

q

qq q q q q q q q q

2 4 6 8 10

1

2

3

4

5

6

7

8

EigenvalueRatio


2 4 6 8 10

0

2

4

6

8

10

20070601 20100531

i i l

Eigenvalue q

q

q q q q q q q q q

2 4 6 8 10

2

4

6

8

10

12

EigenvalueRatio


2 4 6 8 10

0

2

4

6

8

10

20080601 20110531

i i l

Eigenvalue

q

q

q q q q q q q q q

2 4 6 8 10

1

2

3

4

5

6

EigenvalueRatio


2 4 6 8 10

0

2

4

6

8

10

20081101 20111031

i i l

Eigenvalue

q

q

q q q q q q q q q

2 4 6 8 10

1.5

2.0

2.5

3.0

3.5

4.0

EigenvalueRatio


2 4 6 8 10

0

2

4

6

8

10

20090301 20120229

i i l

Eigenvalue

q

q

q q q q q q q q q

2 4 6 8 10

1.5

2.0

2.5

3.0

EigenvalueRatio


2 4 6 8 10

0

2

4

6

8

10

20090501 20120430

i i l

Eigenvalue

q

q

q q q q q q q q q

2 4 6 8 10

1.0

1.5

2.0

2.5

EigenvalueRatio


Figure 3.1: Screen graph and Eigenvalue Ratio of nine time windows.

Figure 3.1 displays the screen graph and the eigenvalue ratio of nine critical time windows

which show the evolution of our data. The top-left graph is for the time window before the

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start of EMU [1996-02-01 - 1999-01-31], which is the reason why the eigenvalue ratio estimator

indicates 10 PCs, as it is shown in Figure 3.2. After the time window [1996-04-01 - 1999-03-31]

the eigenvalue ratio estimator indicates one PC and the estimate stays constant till [2007-

06-01 - 2010-05-31]. Between the time windows [1996-04-01 - 1999-03-31] and [2006-01-01 -

2008-12-31] the eigenvalue ratio increases gradually as the first eigenvalue becomes bigger.

In [2006-11-01 - 2009-02-31] the situation starts changing as the second eigenvalue increases

and the eigenvalue ratio changes. In the fourth and fifth graphs of Figure 3.1 are shown

two consecutively time windows, [2007-05-01 - 2010-04-30] and [2007-06-01 - 2010-05-31], in

which it can be observed the change of the ER estimator from one to two PCs as well as the

decrease of the first eigenvalue and the increase of second one. The reason for this change

is the beginning of the Eurozone debt crisis at the end of 2009. The status does not change

till the end of our data. The first eigenvalue decreases considerably and the second and third

eigenvalue increase substantially. Even though the ER estimator stays constant for the rest ofthe time windows it is observed that the ER for higher number of PCs, becomes larger.

qqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq2

4

6

8

10

Time

NumberofPCs

1 99 8 12 3 1 2 00 0 12 3 1 2 00 2 12 3 1 2 00 4 12 3 1 2 00 6 12 3 1 2 00 8 12 3 1 2 01 0 12 3 1 2 01 2 12 3 1

Figure 3.2: Number of PCs according to the Eigenvalue Ratio estimator.

Figure 3.3 depicts the number of principal components needed for at least 90% of the total

variation, which means that the first m PCs give us 90% of the information the original data

have. In the x-axis of the plot we see the final date of each time window. The blue dashed

line indicates the time window [1999-01-01 - 2001-12-31] which has as starting date the start

of the EMU. It may be seen clearly the reduction of PCs needed as the Eurozone countries are

trying to fulfill certain economic and legal conditions known as convergence criteria during

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the first years. From 2000-11-30 1 to 2009-11-30 one PC is needed for the 90% variation of our

dataset. Basically, that means that all the countries of our portfolio behave similar during that

period. In 2009-12-31 the number of PCs needed starts increasing gradually as the Eurozone

crisis begins and some euro area members as Greece, Ireland, Spain, Portugal and Italy de-

viate from the rest of the countries in the Eurozone. Consequently, the more outlying values

the portfolio has the more PCs are needed.

qqqqqq

qqqqqq

qqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqq

qqqqqqqq

qqqq

qqqqqqqqqqqqqqqq

1

2

3

4

5

Time

NumberofPCs

1 99 8 12 3 1 2 00 0 12 3 1 2 00 2 12 3 1 2 00 4 12 3 1 2 00 6 12 3 1 2 00 8 12 3 1 2 01 0 12 3 1 2 01 2 12 3 1

Figure 3.3: Number of PCs needed for at least 90% of the total variation.

3.4 Clustering

When we are dealing with portfolio selection, we want to select in a data pre-processing

step the most dissimilar assets in a large data set, in order to reduce the number of assets

in portfolio design [24]. This can be done by using statistical approaches which sort outthe assets in groups with similar behavior and similar properties. To those approaches be-

long cluster algorithms, like hierarchical clustering or k-means clustering, [17], which group

similar assets together and separate dissimilar ones. In this section, we apply hierarchical

clustering to our dataset.

1This date indicates the final date of the 3-year time window.

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3.4.1 Hierarchical Clustering

In a hierarchical classification the data are not partitioned into a particular number of

classes or clusters at a single step. Instead the classification consists of a series of partitions,

which may run from a single cluster containing all individuals, to n clusters each containing

a single individual. Hierarchical clustering techniques may be subdivided into agglomerative

methods, which proceed by a series of successive fusions of the n individuals into groups, and

divisive methods, which separate the n individuals successively into finer groupings.

Hierarchical classifications produced by either the agglomerative or divisive route may

be represented by a two-dimensional diagram known as a dendrogram, which illustrates

the fusions or divisions made at each stage of the analysis. Further information on cluster

analysis can be found in Everitt [17] and Hartigan [18].

The function which used for clustering performs a hierarchical cluster analysis using a

set of dissimilarities for the n objects being clustered. Initially, each object is assigned to its

own cluster and then the algorithm proceeds iteratively, at each stage joining the two most

similar clusters, continuing until there is just a single cluster. At each stage distances between

clusters are recomputed by the LanceWilliams dissimilarity update formula, [19] according

to the particular clustering method being used. The agglomeration method which was used

is the complete linkage method, [20], which finds similar clusters. The following figures display

the results of clustering in our dataset.

Figure 3.4 depicts the cluster dendrograms of nine time windows. The y-axis shows the

clustering height, which is the value of the criterion associated with the clustering methodfor the particular agglomeration. In addition to each dendrogram a plot with the heights

of the clusters is displayed. For the first years of Eurozone the clusters are quite close as it

can be observed from the height plot in the two first graphs. The countries do not have big

dissimilarities. In 2009-04-302 (third dendrogram) the height of the clusters starts increasing

slightly and in 2009-11-30 (fourth dendrogram) it can already be noticed the separation of

Ireland, Greece and Italy from the rest of the countries. In 2010-04-30 (fifth dendrogram)

the height of the clusters is doubled and after one month (sixth dendrogram) almost tripled,

separating clearly Greece from the rest of the EMU members countries. Finally, in the last

three dendrograms it can be seen the rapid rise of the heights and the partition of Greece,

Ireland, Portugal, Spain and Italy.

2This date indicates the final date of the 3-year time window.

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GR

AT IE

FI

IT

PT

ES

BE

FR

DE

NL

0.5

1.5

2.5

3.5

19990101 20011231

*

Height

q q q q q q q q qq

2 4 6 8 10

0

10

20

30

40

50

60

Height

IE

GR

IT FI

AT

PT

DE

FR

NL B

EES

0.5

1.0

1.5

2.0

2.5

20000501 20030430

*

Height

q q q q q q q q q q

2 4 6 8 10

0

10

20

30

40

50

60

Height

IE

GR IT

AT

DE

FR

NL

FI

PT

BE

ES0

1

2

3

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5

6

20060501 20090430

*

Height

q q q q q q qq q

q

2 4 6 8 10

0

10

20

30

40

50

60

Height

FI

DE

FR

NL

AT

PT

BE

ES

IE

GR IT

0

1

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3

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6

20061201 20091130

*

Height

q q q q q qq q

q q

2 4 6 8 10

0

10

20

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50

60

H

eight

GR

IT

FI

DE

FR

NL

AT

BE

ES

IE PT

0

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20070501 20100430

*

Height

q q q q qq q

q q

q

2 4 6 8 10

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10

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60

H

eight

GR

FI

DE

FR

NL A

TBE

ES IT

IE PT

0

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20070601 20100531

*

Height

q q q qq q

q q

q

q

2 4 6 8 10

0

10

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40

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60

H

eight

GR

ES IT B

E

AT

FR F

I

DE

NL

IE PT

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10

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20081101 20111031

*

Height

q q q qq q

q q

q

q

2 4 6 8 10

0

10

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60

Height

GR

PT

IE

ES IT B

E

DE F

I

NL

AT

FR

0

10

20

30

40

50

20090601 20120531

*

Height

q q qq q

q

q

q

q

q

2 4 6 8 10

0

10

20

30

40

50

60

Height

GR

PT

IE

ES IT

DE F

I

NL B

E

AT

FR

0

10

30

50

20100401 20130331

*

Height

q q qq

q q

qq

q

q

2 4 6 8 10

0

10

20

30

40

50

60

Height

Figure 3.4: Evolution of Cluster Dendrograms over 3-year rolling time windows.

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Chapter 4

The Mean-Variance Markowitzs Model

In 1952, Harry Markowitz presented his model on Portfolio Selection in the Journal of Fi-

nance [21], which became a foundation of modern finance theory and revolutionized invest-ment practice. He won the Nobel prize in Economics for his work in 1990. Markowitz

formulated the problem of optimal portfolio construction, the process of efficiently allocating

wealth among asset classes, in a Mean-Variance (MV) framework where one assumes that the

rational, risk averse investor either seeks to maximize the expected return for a given risk

level (volatility level) or minimize the risk for a given expected return.

4.1 Assumptions of the MV model

The assumptions of the classical Mean-Variance model of Markowitz are the following: [ 22]

1. There are N risky assets, no risk-free assets.

2. Prices of all assets are known and exogenous given. Investors are price takers, i.e. their

actions do not influence prices.

3. There is a single time period.

4. There is a probability space (, F, P). Returns are following a stochastic process and are

elliptically distributed.

5. There are no transaction costs.

6. Markets are liquid for all assets.

7. Assets are infinitely divisible.

8. Full investment is considered.

9. Portfolios are selected according to the mean-variance criterion.

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4.2 Settings of the Portfolio

Consider a portfolio of n different assets with weights wi, . . . , wn. Let Pit be the price of

asset i at time t. The one-period linear return1 will be:

Rit+1 =Pit+1 Pit

Pit=

Pit+1Pit

1

The expected return of the portfolio is the weighted average of the expected return of each

asset class in the portfolio:

= E[Rp] =n

i=1

wii, (4.1)

where E[Rp] is the expected return of the portfolio, wi the weight of the ith asset and i is the

expected return of asset i, i = E[Ri].

The risk of the portfolio is measured by the standard deviation of its return, . In order

to calculate it we compute the variance of portfolios return:

var(Rp) =n

i=1

n

j=1

wiwjij = wTw. (4.2)

where ij = cov(Ri, Rj) is the covariance between the returns of assets i and j. Covariance

measures the extent to which the returns move together. Moreover, we require full investment,

meaning the sum of the weights should be equal to one, and when short selling is forbidden

we have the additional long-only constraint which forces the weights to be positive.

n

i=1

wi = 1 (4.3)

wi 0 (4.4)

Equations 4.1, 4.2 and 4.3 contains the mathematical framework of MV Markowitz Model.

For different choices ofw1, . . . , wn the investor will get different combinations of and . The

set of all possible (, ) combinations is called feasible set. The portfolios with minimum risk

for a given expected return , or with maximum return for a given level of risk , are

called efficient portfolios forming the efficient frontier.

1It should be noted that in the analysis performed in Chapter 5 logarithmic returns are used, which are defined

as: Rit+1 = ln

Pit+1

Pit

.

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4.3 Efficient Frontier and Minimum Variance Locus

The efficient frontier is the curve which shows all the efficient portfolios in a risk-return

framework. In order to calculate the efficient frontier we have to either minimize the risk

(standard deviation) given some expected return, or maximize the return for given level ofrisk.

The Minimum Risk Portfolio

In the minimum risk portfolios the objective function which has to be minimized is the

standard deviation. However, we take the variance (ww) as objective function, which isallowed since the standard deviation can only be positive.

There are two constraints which must hold for minimizing the variance. First, the expected

return must be fixed, since we are minimizing the risk for given return and secondly, we havethe full investment constraint. As a result, the MV Markowitz model is expressed as the

following convex optimization problem:

w = minw

{ww} Covariance Risk (4.5)

subject to:

w = r Target Return

i

wi = 1 Full Investment

The Maximum Return Portfolio

The objective function which has to be maximized is the portfolio return ( w), subjectto the constraints of given level of covariance risk and full investment. The mathematical

formulation of the optimization problem is the following:

w = maxw

{w} (4.6)

subject to:

ww i

wi = 1

In the minimum risk portfolios, the optimization problem (Equation 4.5) is a quadratic

programming problem with linear constraints (QP1), which can be easily solved by the

Goldfarb-Idnani quadratic solver [30], while in the maximum return portfolios, the opti-

mization problem (Equation 4.6) is a linear programming problem with quadratic constraints

(QP2), which can be solved by second order cone programming solvers. In Figure 4.1 we can

see the different ways of portfolio optimization.

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When short selling is forbidden we have the Long-only portfolio optimization problem.

This requires the additional long-only constraint which forces the weights of the portfolio to

be positive.

wi 0 Long only constraint

Set A gives all the possible combinations of return (mean) and risk (standard deviation)

for a given portfolio:

A := {(E[Rp],

var[Rp])|w Rn, wT1 = 1} (4.7)

and sets +A , A are the efficient and inefficient frontiers of the Markowitz model respec-tively:

+A := {(r, (r))|r r}

A := {(r, (r))|r < r}

where r is the return of the portfolio with the lowest overall risk, the Minimum VariancePortfolio. The A set is called minimum variance locus.

Figure 4.1: Different views of Portfolio Optimization. QP1 is the efficient portfolio resulting from min-

imizing the risk for given level of return and QP2 is the efficient portfolio resulting from maximizingthe return for given level of risk.

4.4 Special Portfolios of the Feasible set

As it has already been mentioned, the solution of the MV Markowitz problem leads to

a attainable set of portfolios. Portfolios with special characteristics will be presented in the

following sections.

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4.4.1 Minimum Variance Portfolio (MVP)

The Minimum-Variance Portfolio is a specific portfolio on the Mean-Variance efficient fron-

tier and the only objective is to lower the risk (Figure 4.2). MVP is the only portfolio located

on the efficient frontier which is not dependent on the expected returns hypothesis. More-

over, it is easy to compute since the solution is unique. The Minimum Variance portfolio is

defined by the following quadratic programming problem:

w = arg min{ww} (4.8)

s. t. : i

wi = 1, and wi 0

The main disadvantage is that the portfolio is concentrated in relatively few assets. All

portfolios which are computed through optimization based on the covariance matrix share

this drawback.

4.4.2 Equal Weights Portfolio (1/n)

In the equally-weighted portfolio we attribute an equal amount to all of the portfolio

weights. The idea of the 1/n portfolio is to define a portfolio independently from estimated

statistics and properties of assets (Figure 4.2). The structure of the portfolio depends only on

the number n of assets because the weights are equal and uniform:

wi = 1n

(4.9)

The 1/n portfolio is the least concentrated portfolio in terms of weights. The main weak-

ness is that it does not consider individual risks and correlations between these risks.

4.4.3 Most Diversified Portfolio (MDP) / Tangency Portfolio (TGP)

The MDP defined by Choueifaty and Coignard [23], is a long-only portfolio which maxi-

mizes the Diversification Ratio. The diversification ratio is the ratio of the weighted average

of volatilities divided by the portfolio volatility, DP = www . If the expected excess returns ofassets are proportional to their risks (volatilities), then ER[] = sw, where s is constant, andmaximizing DP is equivalent to maximizing ER

[]ww

, which is the Sharpe ratio 2 of the port-

folio. This is the reason why MDP is synonymous with the Maximum Sharpe Ratio (MSR)

portfolio. In this case, the MDP is also the Tangency Portfolio (TGP) (Figure 4.2).

2Sharpe ratio is defined as the return-risk ratio and it represents the expected return per unit of risk. So theportfolio with maximum Sharpe Ratio gives the highest expected return per unit of risk, and is thus the most"risk-efficient" portfolio.

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The traditional optimization problem is defined as follows:

w = arg max sh(w) (4.10)

s. t. :

n

i wi = 1, and wi 0

where

sh(w) =w rF

ww

In the case of MDP/MSR portfolio we assume that i = r + si, and the objective function

becomes

sh(w) = swww

where = (1, . . . , n) is the vector of volatilities, the vector of expected return and rF is

the risk free rate.

This portfolio presents appealing properties, such as better diversification and less sensi-

tivity to inputs, than the MVP, and it is not dependent upon any expected return hypothesis.

In addition, MDP is the optimal portfolio if all Sharpe ratios are equal (tangency portfolio).

Finally, it may still be concentrated in few assets.

4.5 Optimal Portfolio

The efficient frontier is the set of the efficient portfolios, but which is the optimal portfolio?For the selection of the best portfolio, the investors risk-return preferences are analyzed.

According to Markowitzs theory the investors goal is to maximize his utility function:

u = w 2

ww. (4.11)

Parameter is the parameter of absolute risk aversion and it measures the investors risk

averseness. The greater the the more risk averse the investor is. The parameter of risk

aversion is assumed to be positive, because all investors are assumed to be risk averse. An

investor who is highly risk averse will hold a portfolio on the lower left hand of the frontier

and an investor who is not too risk averse will choose a portfolio on the upper portion of the

frontier.

The portfolio optimization problem which trades risk against return in the objective func-

tion using the absolute risk aversion index is [26]:

w = maxw

{w 2

ww} (4.12)

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subject to:

i

wi = 1

wi 0

By using the Critical Line Algorithm introduced by Markowitz we calculate the specificrisk and return values along the efficient frontier. Further information on the critical line

algorithm can be found in Markowitzs books [27], [28].

In practice it is common to model the risk-return trade-off using a parameter , 0 1,with the following optimization problem:

w = minw

{ww (1 )w} (4.13)

subject to:

i

wi = 1

wi 0

where = +2 .

4.6 Illustration of the Mean Variance model

To illustrate Mean-Variance Analysis we will use Barclays European Bond Indices dataset

which is introduced in Appendix B.1. In the interest of clarifying the MV model as clear as

possible, Greece and Portugal are excluded, which results in a more suitable feasible set.

In Figure 4.2 we can see the Feasible Set of European Bond portfolio. As it is expected the

bonds of Spain (ES), Italy (IT) and Ireland (IE) are the most risky components and the bonds

of Finland (FI), the Netherlands (NL), Germany (DE), France (FR), Belgium (BE) and Austria

(AT) are the less risky ones with higher return. Moreover, we can see the Minimum Variance

Portfolio (MVP), the Equal Weights Portfolio (EWP) and the Tangency Portfolio (TGP), which

maximizes the Sharpe Ratio. The magenta line is the Capital Market Line (CML). The CML

is the tangent line drawn from the point of the risk-free asset to the feasible region for risky

assets. The point where the CML touches the efficient frontier is the Tangency Portfolio.Figure 4.3 depicts the Efficient Frontier of European Bond portfolio (black points) and the

Minimum-Variance Locus (gray points). In addition to the efficient frontier and the minimum-

variance locus, Figure 4.3 also shows the available bond indices. The graph makes it apparent

that investing in only one asset class or a subset of assets classes is highly inefficient.

In order to compute the feasible set, the efficient frontier and the weights of the MVP and

TGP we used the fPortfolio package from the R/Rmetrics statistical software environment

[24].

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0.20 0.25 0.30 0.35 0.40

0.0

17

0.0

18

0.0

19

0

.020

Target Risk (sd)

TargetReturn(mean)

qq

q

q

q q

q

q

q

AT

BE

DE

ES

FI

FR

IE

IT

NLEWPq

MVP

qTGP

Figure 4.2: Feasible Set of Barclays European Bond Indices dataset.

0.20 0.25 0.30 0.35 0.40

0.0

17

0.0

18

0.0

19

0.0

20

Target Risk (sd)

TargetReturn(mean)

q

qqqqqqqqqq

qqqq

q qq q

q qq q

q

qqqqqqqqqqq

qqqq

q qq q

q qq q

q

qq

q

q

q q

q

q

q

AT

BE

DE

ES

FI

FR

IE

IT

NL

qMVPq

TGP qEWP

Figure 4.3: Efficient Frontier of European Bond portfolio.

The optimization routine which is used to solve the Quadratic Programming Problem is

the solve.QP() from R package quadprog. This routine implements the dual method of Goldfarb

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and Idnani [30] for solving quadratic programming problems of the form:

minx

cTx + 12

xTDx (4.14)

subject to:ATx b

4.7 Limitations of Mean-Variance Optimization

Mean-variance analysis provides a powerful framework for asset allocation, therefore it

has gained wide spread acceptance among practitioners. However, there are certain caveats

which must be considered when applying it.

The inputs needed for MV model are asset expected returns, expected standard deviations,

and correlations of returns of assets. If the inputs are free of estimation error, MV optimization

is guaranteed to find the optimal or efficient portfolio weights. However, because the inputs

are statistical estimates (typically created by analyzing historical data), are inevitable subject

to estimation error. This inaccuracy will lead to over-investment in some assets and under-

investment in others. The estimation errors introduced can distort the optimization results.

It can be shown that even small estimation errors can result in large deviations from optimal

allocations in an optimizers results. Estimation error can also cause an efficient portfolio to

appear inefficient. An approach to limit the impact of estimation error is the use of robust

statistics. In chapter 6 several methodologies will be presented.Another limitation of Mean-Variance Optimization is that the results can sometimes be

unstable. Small changes in the input assumptions can lead to large changes in the solutions.

One important reason for this behavior is ill-conditioning of the covariance matrix. Input

assumptions that do not reflect financially meaningful estimates or the use of parameter

estimates based on insufficient historical data are often associated with ill-conditioning and

instability.

In order to minimize dramatic changes in recommended portfolio composition sensitivity

analysis can be used. This technique involves selecting an efficient portfolio and then altering

the Mean-Variance Optimization inputs and seeing how close to efficient the portfolio is under

the new set of inputs. The goal is to identify a set of asset class weights that will be close to

efficient under several different sets of plausible inputs.

Further details on the limitations of Mean-Variance Optimization and possible solutions

to those, can be found in Michaud and Lummer et al., [31], [32], [33] .

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Chapter 5

Statistical Shape Analysis

As it has already been mentioned, a way to analyze the European bond market is to treat

the bonds of Eurozone countries as assets of a portfolio. A primary analysis was undertakenin Chapter 3 using the Principal Component Analysis and Hierarchical Clustering algorithm.

This analysis gave us an insight into the evolution of the bond market over the last 14 years.

Moreover, in Chapter 4, it was presented the Mean-Variance Markowitz Model with the

mathematical framework for portfolio optimization as well as the most relevant investment

strategies for a given set of assets.

In this chapter we will present the evolution of the Eurozone bond portfolio over time

using the Mean-Variance Framework. Furthermore, in a attempt to understand the portfolio

as a physical entity, a statistical shape analysis 1 of the Hull of the feasible set will be carried

out.

5.1 Portfolio Evolution

In order to track the changes of the feasible set in the Eurozone bond portfolio, the method

of rolling time windows is implemented, using the same time windows as in Chapter 3,

3-year rolling window with monthly frequency (Table 3.1) starting at 1996-01-01. For each

of these windows, a long-only minimum risk portfolio optimization is performed, which is

formulated mathematically in section 4.3.In Figure 5.1 the feasible set of Barclays European Bond Indices , the components of

the portfolio (Eurozone countries) as well as two investment strategies, the Equal Weights

Portfolio (EWP) and the Minimum Variance Portfolio (MVP), are displayed.

Figure 5.2 shows the evolution of the portfolio. The first graph depicts the feasible set of

the time window [1999-01-01 - 2001-12-31], it is observed that all the countries are grouped

together and have similar returns and risks. The countries "move" together as a single group

1Statistical shape analysis is a geometrical analysis from a set of shapes in which statistics are measured todescribe geometrical properties from similar shapes or different groups.

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for the first ten years of the Eurozone (1999-2009). At the end of 2009 Ireland and Greece start

diverging from the group. The first noticeable separation of these two countries is occurring

in the time window [2007-02-01 - 2010-01-31] (second feasible set). The rest of the hulls in

Figure 5.2 show the deterioration of the situation in Greece, Ireland Portugal, Italy and Spain.

It is worth mentioning the dramatic decrease of return and increase of risk of Greek bonds

in the periods [2008-10-01 - 2011-09-30] and [2009-06-01 - 2012-05-31]. Greece reached a low

point in the second period.

0.2 0.4 0.6 0.8 1.0

0.0

05

0.0

10

0.0

15

0.0

20

Risk (sd)

Return

qq

qq q

q

q

qq

q

ATBE

DE

ES

FI FR

GR

IE

IT

NL

PTqEWPqMVP

Figure 5.1: Feasible set of the Eurozone bond portfolio for the entire period [1996-01-01 - 2013-04-17].

The line graphs in Figure 5.3 depict the positions of Greece, Ireland, Portugal, Italy and

Spain for every time window. It can be seen clearly the huge drop of Greece as well as the

upswing of its position after the time window [2009-07-01 - 2012-06-30]. This change can be

explained by the e110 billion bailout package Greece received by the European Union and

International Monetary Fund (IMF) in May, 2010.

Using the methodology of rolling time window one can track the evolution of the feasibleset of a portfolio. However, due to the high complexity of the hulls, it becomes difficult to

derive the information from its shape. In order to provide an objective characterization of the

hull, shape representation and description techniques can be implemented. The geometrical

parametrization of the feasible set will be presented in the next section.

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0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.10

0.0

5

0.0

0

0.0

5

19990101 20011231

Risk

R

eturn

qqqqATBEDEESFIFRGR IEITNLPTqEWPqMVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.10

0.0

5

0.0

0

0.0

5

20070201 20100131

Risk

R

eturn

qqqqq

ATBEDEES

FI FR

GR

IEITNLPTqEWPMVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.10

0.0

5

0.0

0

0.0

5

20070501 20100430

Risk

R

eturn

qqqq

qqATBEDEESFI FRGR

IE

ITNL

PTEWP

MVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

20070601 20100531

Risk

Return

qqqq

qqATBEDEESFI FRGR

IEIT

NL

PTEWP

MVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

20071201 20101130

Risk

Return

qqqqq

qq q

AT

BE

DE

ES

FI FR

GR

IE

IT

NL

PT

qEWPqMVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

20080901 20110831

Risk

Return

qqqqqq

q

qqq

q

AT

BE

DE

ES

FI FR

GR

IE

IT

NL

PT

qEWPqMVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

20081001 20110930

Risk

Return

qqqqqq

q

qq

q

AT

BE

DE

ES

FI FR

GR

IEIT

NL

PT

qEWPqMVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

20081101 20111031

Risk

Return

qqqqqq

q

qq

q

AT

BE

DE

ES

FIFR

GR

IEIT

NL

PT

qEWPqMVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

20081201 20111130

Risk

Return

qqqqq

q

qq

q

AT

BE

DE

ES

FIFR

GR

IEIT

NL

PT

qEWPqMVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

20090201 20120131

Risk

Return

qqqq

q

qqq

q

ATBE

DE

ES

FIFR

GR

IE

IT

NL

PT

qEWPqMVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

20090501 20120430

Risk

Return

qq qqq

q

qqqq

ATBE

DE

ES

FIFR

GR

IE

IT

NL

PT

qEWPqMVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

20090601 20120531

Risk

Return

qqq

qq

q

qqq

q

AT

BE

DE

ES

FI FR

GR

IEIT

NL

PT

qEWPqMVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

20090901 20120831

Risk

Return

qqq

qq

q

qqq

qAT

BEDE

ES

FI FR

GR

IE

IT

NL

PTqEWPqMVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

20091201 20121130

Risk

Return

qqqqq

q

qq qATBEDE

ES

FI FR

GR

IE

IT

NL

PTEWP

qMVP

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

20100201 20130131

Risk

Return

qq qq

q

qq qATBEDE ESFI FR

GR

IE

ITNL

PTqEWPMVP

Figure 5.2: Evolution of feasible sets of the Eurozone bond portfolio.

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0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

Greece

Risk

eturn

qqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q

qq q

qqqqqq

qq q

q

q

q

qqqqqqqqq

qqqq

q

qq

q

qq

qqq

q

q q

q

q

q

q

q

qq

q

q

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

Ireland

Risk

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqq

qqqqqqqq

qqqqq

qqqqq qq

q

qqq

qq

qqqqqqqqqqq

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

Portugal

Risk

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqq q q qq

qqqqqq

q qqq

qq

qqqqqqqq

qqqq

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

Italy

Risk

qqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqq

q qqq

qqqqqqq q qq

qqqq

0.0 0.5 1.0 1.5 2.0

0.2

0

0.1

5

0.1

0

0.0

5

0.0

0

0.0

5

Spain

Risk

qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqqq

qqqqqqq qqq

qqqqqqq q qq

qqqq

Figure 5.3: Time evolution of the returns versus covariance risk (sd) of Greece, Ireland, Portugal, Italyand Spain bond prices.

5.2 Morphological Characterization of the Feasible Set

The problem of characterizing the shape and the main properties of a particle has been

studied in many application areas such as Mechanics, Biology, Material Science, Pattern

Recognition among others. Shape is an important visual feature and it is one of the basic

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features used to describe image content. Shape representation and description techniques

generally look for effective and perceptually shape features based on either shape boundary

information or boundary plus interior content [34].

The main idea, for extracting information from the hull of the feasible set, is to estab-

lish global morphological properties which can represent and uniquely describe a contour,

making it distinguishable from others. This can be achieved by using global contour-based

shape representation techniques, which usually compute a multi-dimensional numeric fea-

ture vector from the shape boundary information. Common simple global descriptors are

area, perimeter, centroid, eccentricity and orientation. These simple global descriptors usu-

ally can discriminate shapes with differences. This method was first implemented by Saaibi

[35] and is extended in this work.

In the following sections the aforementioned geometrical features of the hull will be cal-

culated by considering the hull as a binary image.

5.2.1 Area and Perimeter

The area A of the feasible set indicates the initial dispersion of the portfolio componentsand particularly the size of the feasible set, i.e. the amount of the portfolios constructed by

different possible combinations of the assets. In a binary image B(x,y) the pixels at position

(x,y) of the object, the feasible set, have value 1 and the background pixels have value 0.

The area A is the sum of all the object pixels in the image and can be described by the 0-thmoment,

00.

A = xy

B(x,y) = 00. (5.1)

The perimeter Pof the feasible set, which is the length of the hull, in combination with thearea can give us the information of the portfolios complexity. The perimeter can be derived

from a modified image matrix C(x,y) in which all the internal pixels of the object are also set

to 0. As a result, only the pixels which form the contour of the feasible set have value 1. So,

the perimeter of the object, analogous to the area, is the sum of all pixels:

P=x y C(x,y). (5.2)

The area A and the perimeter Pare invariant with respect to translation and rotation butthey are scale variant.

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5.2.2 Centroid

The coordinates of the centroid Cx,y of a figure are given by the following equations, andcan be obtained by using the corresponding first moments (10, 01):

x = 1Ai

xi = 10

00, (5.3)

y =1

Ai

yi =0100

,

Centroid is rotation invariant but translation variant.

The location of the centroid determines a certain combination of risk and return which lies

in the geometrical center of the feasible set. Its distance to the EWP could be used to estimate

the risk diversification of the investments at a given point in time. In order to provide a

comparison to the Sharpe Ratio (SR) of risk free investments the Centroid Ratio is defined as:

Cy/x = yx =CyCx (5.4)

5.2.3 Eccentricity

The eccentricity e of an image, or how elongated it is, is given by the ratio between the

major and the minor axes of the image intensity, which correspond to the eigenvectors of the

ima

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