UNIVERSITY OF GHENT

FACULTY OF ECONOMICS AND BUSINESS

ADMINISTRATION

ACADEMIC YEAR 2014 – 2015

HOW MANY STOCKS DOES AN INVESTOR NEED TO DIVERSIFY WITHIN EUROPE?

Master thesis recited for obtaining the degree of

Master of Science in Business Administration

Olivier De Keyzer Michiel De Schaepmeester

Headed by

Prof. K. Inghelbrecht

UNIVERSITY OF GHENT

FACULTY OF ECONOMICS AND BUSINESS

ADMINISTRATION

ACADEMIC YEAR 2014 – 2015

HOW MANY STOCKS DOES AN INVESTOR NEED TO DIVERSIFY WITHIN EUROPE?

Master thesis recited for obtaining the degree of

Master of Science in Business Administration

Olivier De Keyzer Michiel De Schaepmeester

Headed by

Prof. K. Inghelbrecht

I

I. Confidentiality clause

PERMISSION I declare that the contents of this master thesis may be consulted and / or reproduced, provided that the source is acknowledged Name student: Olivier De Keyzer PERMISSION I declare that the contents of this master thesis may be consulted and / or reproduced, provided that the source is acknowledged Name student: Michiel De Schaepmeester

II

II. Dutch summary

Sinds het baanbrekende werk van Markowitz (1952), heeft asset allocatie veel aandacht

gekregen van onderzoekers. Deze studie richt zich op een enkel onderdeel van de asset

allocatie, namelijk aandelen binnen een portfolio. Veel onderzoek is uitgevoerd in de afgelopen

decennia over het optimaal aantal aandelen binnenin een portefeuille. Het doel van dit

onderzoek is om na te gaan of het optimaal aantal aandelen is veranderd door de jaren heen.

Dit onderzoek wordt uitgevoerd voor verschillende perioden op de Europese beurs; de

volledige periode (2000-2014), voor de crisis (2004-2006), tijdens de crisis (2007-2009) en na de

crisis (2010-2012). Hierbij is het mogelijk om de financiële crisis van 2008 te onderzoeken.

Naast dit, vergelijkt het onderzoek op basis van het optimaal aantal aandelen 5 beter

presterende landen (Denemarken, Frankrijk, Duitsland, Verenigd Koninkrijk en Zweden) met de

PIIGS-landen (Portugal, Italië, Ierland, Griekenland en Spanje), die een zwakkere economie

hebben. Verder zijn er 5 verschillende sectoren (Consumptie goederen, Financiële sector,

Olie&Gas, Technologie en Nutsvoorzieningen) onderzocht om het onderlinge verschil te

onderzoeken. Resultaten tonen aan dat optimale diversificatie wordt bereikt met een

portefeuille bestaande uit 14 aandelen voor de S&P Europe 350. Bovendien werd er een

gemiddelde vastgesteld van 14,8 aandelen voor de PIIGS-landen, een gemiddelde van 16,4 voor

de beter presterende landen en een gemiddelde van 31,6 voor de sectoren. Deze resultaten

kunnen een impact hebben op de beslissing van individuele stock-pickers en beheerders van

beleggingsfondsen.

Sleutelwoorden: Optimale diversificatie, beurs, optimaal aantal aandelen

III

III. Abstract Since the groundbreaking work of Markowitz (1952), asset allocation has been given a lot of

attention by researchers. This study focuses on a single part of asset allocation, namely the

stocks within a portfolio. A lot of research has been documented in the last few decades on the

optimal required amount of stocks. The purpose of this research is to find whether the optimal

required amount of stocks has changed throughout the years. This research is conducted for

different time periods on the European stock market; the full time period (2000-2014), before

the crisis (2004-2006), during the crisis (2007-2009) and after the crisis (2010-2012) hereby it’s

possible to examine the financial crisis of 2008. Moreover, 5 better performing countries

(Denmark, France, Germany, Sweden and the United Kingdom) are compared with the PIIGS

countries (Portugal, Italy, Ireland, Greece and Spain) that have a weaker economy. Next to

these countries, 5 different sectors (Consumer goods, Financials, Oil&Gas, Technology and

Utilities) are selected to examine whether the required amount of stocks differs between them.

Our results show that for the S&P Europe 350 an optimal diversified portfolio consists out of 14

stocks. Moreover, an average of 14,8 stocks is required for the PIIGS countries, an average of

16,4 is required for the better performing countries and an average of 31,6 is required for the

sectors. This could have an impact on the decision of the individual stock-picker and mutual

fund manager.

Keywords: Optimal diversified portfolio, stock market, required amount of stocks

IV

IV. Acknowledgements

We want to thank one person in particular, Koen Inghelbrecht. This could not have been done

without the feedback, time and support throughout the entire process of executing this

research. Moreover, we want to thank the University of Ghent for the use of their resources.

V

Table of Contents I. List of used abbreviations ............................................................................................ VII

II. List of figures .............................................................................................................. VIII

III. List of tables .................................................................................................................. IX

1. Introduction ....................................................................................................................... 1

2. Literature review ............................................................................................................... 4 2.1. Concepts .............................................................................................................................................................................. 4

2.1.1. Diversification .................................................................................................................................................................... 4 2.1.2. Idiosyncratic risk .............................................................................................................................................................. 4 2.1.3. Systematic risk ................................................................................................................................................................... 6 2.1.4. Correlation ........................................................................................................................................................................... 6

2.2. Timeline ............................................................................................................................................................................... 7 2.2.1. 1960-1969 ............................................................................................................................................................................ 7 2.2.2. 1970-1979 ............................................................................................................................................................................ 9 2.2.3. 1980-1989 ......................................................................................................................................................................... 10 2.2.4. 2000+ .................................................................................................................................................................................. 10 2.2.5. Conclusion ......................................................................................................................................................................... 11

3. Hypothesis ....................................................................................................................... 14

4. Data ................................................................................................................................. 16 4.1. S&P Europe 350 Index ................................................................................................................................................ 16 4.2. Market indices Europe ............................................................................................................................................... 17

4.2.1. PIIGS (Portugal, Italy, Ireland, Greece and Spain) ......................................................................................... 18 4.2.2. Denmark, France, Germany, Sweden and the United Kingdom ............................................................... 20 4.2.3. Different sectors ............................................................................................................................................................. 21

5. Methodology ................................................................................................................... 23 5.1. Constructing a portfolio ............................................................................................................................................. 23 5.2. Applying benchmark ................................................................................................................................................... 26

6. Results ............................................................................................................................. 28 6.1. Europe ............................................................................................................................................................................... 29 6.2. PIIGS ................................................................................................................................................................................... 30

6.2.1. Portugal ............................................................................................................................................................................. 31 6.2.2. Italy ...................................................................................................................................................................................... 31 6.2.3. Ireland ................................................................................................................................................................................ 32 6.2.4. Greece .................................................................................................................................................................................. 32 6.2.5. Spain .................................................................................................................................................................................... 33 6.2.6. Overview ............................................................................................................................................................................ 34

6.3. Better performing countries .................................................................................................................................... 35 6.3.1. Denmark ............................................................................................................................................................................ 35 6.3.2. France ................................................................................................................................................................................. 36 6.3.3. Germany ............................................................................................................................................................................. 36 6.3.4. Sweden ................................................................................................................................................................................ 37 6.3.5. United Kingdom.............................................................................................................................................................. 38 6.3.6. Overview ............................................................................................................................................................................ 39

6.4. Different sectors ............................................................................................................................................................ 41 6.4.1. Consumer goods ............................................................................................................................................................. 41 6.4.2. Financial sector .............................................................................................................................................................. 41

VI

6.4.3. Oil & Gas ............................................................................................................................................................................. 42 6.4.4. Technology ....................................................................................................................................................................... 42 6.4.5. Utilities ............................................................................................................................................................................... 43 6.4.6. Overview ............................................................................................................................................................................ 44

6.5. Financial crisis of 2008 .............................................................................................................................................. 45 6.5.1. Before the crisis (2004-2006) .................................................................................................................................. 46 6.5.2. During the crisis (2007-2009) ................................................................................................................................. 46 6.5.3. After the crisis (2010-2012) ..................................................................................................................................... 47 6.5.4. Conclusion crisis ............................................................................................................................................................. 48

7. Conclusion ....................................................................................................................... 49

References ........................................................................................................................... 54

Appendices .......................................................................................................................... 58 Appendix 1: Exploration of idiosyncratic risk........................................................................................................... 59 Appendix 2: Unemployment rate Europe different countries. ........................................................................... 60 Appendix 3: GDP to debt ratio Europe for different countries .......................................................................... 61 Appendix 4: GDP Euro-countries 2005 - 2014 ......................................................................................................... 62 Appendix 5: S&P Europe 350 – Benchmark 4%, 6% and 8% ............................................................................. 63 Appendix 6: PIIGS countries – Benchmark 4% ......................................................................................................... 64 Appendix 7: PIIGS countries – Benchmark 6% ......................................................................................................... 65 Appendix 8: PIIGS countries – Benchmark 8% ......................................................................................................... 66 Appendix 9: Better performing countries – Benchmark 4% .............................................................................. 67 Appendix 10: Better performing countries – Benchmark 6% ............................................................................ 68 Appendix 11: Better performing countries – Benchmark 8% ............................................................................ 69 Appendix 12: Sectors – Benchmark 4%....................................................................................................................... 70 Appendix 13: Sectors – Benchmark 6%....................................................................................................................... 71 Appendix 14: Sectors – Benchmark 8%....................................................................................................................... 72 Appendix 15: Financial crisis – Benchmark 4% ....................................................................................................... 73 Appendix 16: Financial crisis – Benchmark 6% ....................................................................................................... 74 Appendix 17: Financial crisis – Benchmark 8% ....................................................................................................... 75 Appendix 18: Average Transaction Costs MSCI World Index by Sector in 2009 ....................................... 76

VII

I. List of used abbreviations E&A Evans & Archer E&G Elton & Gruber F&L Fisher & Lorie Return portfolio Standard deviation portfolio STDEV Standard deviation

VIII

II. List of figures Figure 1 - Firm-specific risk and market risk ................................................................................... 5 Figure 2 - Asymptotic function portfolios ....................................................................................... 8 Figure 3 - Overview literature ....................................................................................................... 12 Figure 4 - Standard deviation S&P Europe 350 ............................................................................. 30 Figure 5 - Standard deviation of Portugal ..................................................................................... 31 Figure 6 - Standard deviation Italy ................................................................................................ 31 Figure 7 - Standard deviation Ireland ............................................................................................ 32 Figure 8 - Standard deviation Greece ........................................................................................... 33 Figure 9 - Standard deviation Spain .............................................................................................. 33 Figure 10 - Standard deviation Denmark ...................................................................................... 35 Figure 11- Standard deviation France ........................................................................................... 36 Figure 12 - Standard deviation Germany ...................................................................................... 37 Figure 13 - Standard deviation Sweden ........................................................................................ 37 Figure 14 – Standard deviation United Kingdom .......................................................................... 38 Figure 15 - Standard deviation consumer goods .......................................................................... 41 Figure 16 - Standard deviation financial sector ............................................................................ 41 Figure 17 - Standard deviation Oil & Gas ...................................................................................... 42 Figure 18 - Standard deviation technology ................................................................................... 43 Figure 19 - Standard deviation utilities ......................................................................................... 43 Figure 20 - Standard deviation before the crisis ........................................................................... 46 Figure 21 - Standard deviation during the crisis ........................................................................... 46 Figure 22 - Standard deviation after the crisis .............................................................................. 47

IX

III. List of tables Table 1- Literature review ............................................................................................................. 13 Table 2- Overview Rank within European Union according to GDP, Debt-to-GDP ratio, unemployment rate and global competitiveness ranking for the PIIGS countries. ...................... 19 Table 3 - Overview Rank within European Union according to GDP, Debt-to-GDP ratio, unemployment rate and global competitiveness ranking for the better performing countries. . 21 Table 4 - January 2009: Average transaction costs ....................................................................... 28 Table 5 - Overview S&P Europe 350 .............................................................................................. 29 Table 6 - Overview PIIGS countries ............................................................................................... 34 Table 7 - Overview Denmark, France, Germany, Sweden and United Kingdom .......................... 39 Table 8 - Overview different sectors ............................................................................................. 44 Table 9 - Overview different time periods S&P Europe 350 ......................................................... 48 Table 10 - Overview S&P Europe 350, different countries and sectors ........................................ 51

1

1. Introduction In the last decades we’ve seen that a lot of researches focus on the diversification effects

through asset allocation, which is the most important step in creating an optimal

portfolio. Previous studies have shown that asset allocation is responsible for 91.5% of

the performance. (Ibbotson&Kaplan, 2000) The groundbreaking work of Markowitz

(1952), which is a milestone in developing theories about asset allocation and optimal

portfolio selection, has proven the necessity of diversification for mean-variance portfolio

optimization. Interestingly, there are not many studies conducted in the past about the

required number of stocks in a portfolio to obtain optimal diversification. The key to

successfully invest your money into different alternatives (such as commodities, bonds,

stocks, etc.) requires specific knowledge about those markets. With this research we want

to focus on one aspect of asset allocation and provide investors knowledge on investing

stocks. In particular, this paper discusses the relationship between the number of stocks

and the reduction of the risk of the portfolio. Moreover, the transaction costs are taken

into account. Many investors tend to forget that diversification is one of the crucial

aspects of investing so we want to provide a clear view about this topic.

Moreover, due to recent developments in the world markets we’ve seen that risk-free

investments yield less. This results in people searching for alternatives to invest their

money, such as investing in stock markets. A lot of starting investors and sometimes

professional investors tend to forget diversifying their portfolios because mainly they are

not educated about this topic. They prefer high risk to low risk when there is a great

upward potential, this can go hand in hand with unnecessary risk and can cause huge

losses. (Statman, 2002) Despite all the research since the 1960’s, many studies have

concluded that you need at least more than 10 stocks to diversify. However, Goetzmann

& Kumar (2001) concluded that people hold on average 4 stocks. This is far below the 120

stocks concluded by Statman (2002) to reach optimal diversification. This research will

indicate the effects on the reduction of risk when adding more stocks to the portfolio.

2

Diversification can lead to a decline or an elimination of the idiosyncratic risk or the

diversifiable risk. Research showed that to eliminate idiosyncratic risk in today’s market, a

portfolio must hold many more stocks than that was necessary in the 1960’s. Which is an

indication that the idiosyncratic risk has raised during the past decades. (Malkiel, 2002)

On the other hand there is systematic risk, which is not diversifiable. Results show that

systematic risk has increased due to the globalization and integration of the world

market. (Bodie, Kane, &Marcus, 2011)

In the past decades research has been done about portfolio selection. First we have Evans

& Archer (1968); They concluded that 10 stocks are adequate to have a well-diversified

portfolio. On the other hand, Fisher & Lorie (1970) concluded that only 8 stocks are

required to have a well-diversified portfolio. Upson, Jessup & Matsumoto (1975) looked

at the standard deviation and concluded that more than 16 stocks are required to

diversify. In the period 1980 to 1989 four studies have been conducted. Elton & Gruber

concluded in 1977 that adding stocks beyond 15 is still beneficial in terms of risk

reduction. In 1983, Gup wrote that the diversifiable risk is almost entirely eliminated with

8 to 9 stocks. Reilly (1985) argues that a portfolio of 12 to 18 stocks is required to reach

adequate diversification. Francis (1986) wrote that investors only need 10 to 15 stocks

within a portfolio to have a well-diversified portfolio. If investors hold more than 15

stocks, it won’t have a significant impact on the variance of the portfolio. Statman (1987)

wrote that an adequately diversified portfolio consists of 30 to 40 stocks. In the period

1990 to 1999 no research has been done about this subject. Statman concluded in 2002

that more than 120 stocks are required in a portfolio by the rules of the mean-variance

theory. The most recent research has been done by Zhou (2014), who concluded that only

10 stocks are required for a well-diversified portfolio for the American market. This is an

affirmation of the results of Evans & Archer (1968). These different researches are mainly

situated in 1960 to 1990, with an exception in 2002 and 2014. Throughout the years there

is a rising trend of stocks required to have a well-diversified portfolio, except for the

research of Zhou (2014). With our research we want to examine whether this trend is

constant throughout the years.

3

Recent studies show that there is an increase in the correlations between international

stock markets since the 1970’s. (Longin&Solnik, 1995; Goetzmann et al., 2005) This might

be an explanation for the rising trend in the number of stocks needed to have a well-

diversified portfolio. With this paper we want to research this hypothesis and possibly

confirm the rising trend in the required amount of stocks.

“We show that high volatility of markets is directly linked with strong

correlations between them. This means that markets tend to behave as

one during great crashes.“ (Sandoval&De Paula, 2012)

This research will be executed on the stock market of Europe. 2 databases are retrieved

from DataStream; the S&P Europe 350 and a database consisting out of 2608 stocks

representing ±90% of the market capitalization of Europe. Moreover, we will take into

account five better performing countries (Denmark, France, Germany, Sweden and the

United Kingdom) and five countries with a weaker economy (PIIGS; Portugal, Italy,

Ireland, Greece and Spain). Besides those countries, five sectors will be researched as

well. This will be done for the time period 2000-2014.

This study will also be conducted for the financial crisis of 2008. We will do this by

examining three different time periods; before the crisis (2004-2006), during the crisis

(2007-2009) and after the crisis (2010-2012).

4

2. Literature review This literature review gives an overview of the existing literature that’s relevant for this

research. First, we will define some important concepts. Thereafter, an overview of the

previous studies can be found about the required amount of stocks to make a well-

diversified portfolio.

2.1. Concepts

2.1.1. Diversification ‘Don’t put all your eggs in one bucket’. This is what diversification is all about. It’s a

strategy used to reduce the total risk of the portfolio. Moreover, diversification leads to

more stable returns as it reduces lower and upper performance. Sharpe (1966), who

implemented variation as risk, introduced that the total variation of a portfolio return can

be divided into two aspects. Namely; (1) idiosyncratic or non-systematic variation (risk)

and (2) systematic variation (risk).

In this research we will search for the optimal diversified portfolio; in other words to

reach optimal diversification. The number of stocks in this portfolio depends on the

marginal benefits versus the marginal costs. As long as the marginal benefits exceed the

marginal costs an additional stock should be add to the portfolio to reach the optimal

amount of stocks.

2.1.2. Idiosyncratic risk Idiosyncratic risk or firm-specific risk is the part of risk that can be eliminated by

diversification. It’s the risk that is peculiar to the firm. This risk can take various forms

such as sales growth, R&D, relative performance, etc. of the specific firm.

“Recent studies by Campbell, Lettau, Malkiel & Xu (CLMX) (2001), Xu & Malkiel (2003),

Fama & French (2004), Wei & Zhang (2006), and Jin & Myers (2006) document that, over

a 4-decade period from the early 1960s to the end of the 1990s, U.S. public firms exhibit

5

increasing firm-specific return volatility, more volatile income and earnings, lower

profitability, and lower survival rates. As seen in appendix 1, the recurring theme in these

studies is that firm-specific risk, however defined, has been increasing.” (Fink, Fink,

Grullon, &Weston, 2010)

The key of diversification is to eliminate idiosyncratic risk, which can be done by

increasing the number of stocks in your portfolio. As shown in figure 1.A; higher amounts

of stocks decrease the variance of the portfolio.

However, Bennett & Sias identified in 2004 that three factors could be responsible for the

rise of idiosyncratic risk between 1962 to 1997. These factors are: (1) The growth of

industries containing above-average levels of firm specific risk, (2) The increased role of

small companies in the market and (3) changes in estimated firm-specific risk induced by

changing within-industry concentration.

Fink, Fink, Grullon & Weston (2010) document that an increase in idiosyncratic risk is

caused by the maturity of listed firms and the increase of IPO’s. The last 40 years the age

Figure 1 - Firm-specific risk and market risk

6

of the firms issuing public equity has decreased from 40 years in 1960 to 5 years in 1990.

The result of Fink et al. (2010) contrasts with the work of Brandt, Brav, Graham & Kumar

(2010), which says that irrational “noise” traders drive the high levels of idiosyncratic risk

during the Internet boom.

2.1.3. Systematic risk Systematic risk or market risk is the part of risk that affects all firms in the economy and

can’t be eliminated by diversification. As shown in figure 1.B the market risk stays

constant, even with a higher number of stocks. Recent studies have shown that there is

an increase in systematic risk due to globalization and integration of the world market.

(Bodie, Kane, &Marcus, 2011)

2.1.4. Correlation Correlation measures the relation between 2 or more variables. The range of this

measure lies between -1 and +1. Between the interval 0 and +1, this measure indicates a

positive correlation. If one variable goes up, the other one will go up too depending on

how strong their relationship is. In terms of diversification it’s better to invest into assets

that are negatively correlated. Moreover, a correlation between -1 and 0 means that if

one variable goes up, the other one will go down.

Previous studies have concluded that during times of high volatility, which indicates

possible financial crashes, correlations between assets tend to move into the same

direction and therefore stronger relationships occur. (Solnik, Boucrelle, &Le Fur, 1996;

Meric&Meric, 1997; Longin&Solnik, 1999; Cizeau, Potters, &Bouchaud, 2001; Hartmann,

Straetmans, &De Vries, 2001; Lillo, Bonanno, &Mantegna, 2001; Longin&Solnik, 2001;

Ang&Chen, 2002; Marshal&Zeevi, 2002; Bartram&Wang, 2005; Malevergne&Sornette,

2006; Knif, Kolari, &Pynnönen, 2007; Meric, Kim, Kim, &Meric, 2008; Maskawa&Souma,

2010; Reigneron, Allez, &Bouchaud, 2011; Sandoval&De Paula, 2012)

7

2.2. Timeline

This topic has caught the attention of many professional investors during the last

decades. Since the beginning of diversification strategies developed by Markowitz (1952),

different studies have been conducted about the relationship between the number of

stocks in a portfolio and the reduction of variance in a portfolio. One of the fundamental

papers is the one from Evans & Archer (1968). The methodology developed by E&A has

been implemented by most of the follow-up studies, while some researchers developed a

new approach to take into account new developments throughout the years. Below, we

provide the most relevant literature to this topic together with their findings.

2.2.1. 1960-1969 One of the first papers conducted about this subject is the research of Evans & Archer

(1968). We will base our research on this fundamental paper. E&A were the first to give

attention to the relationship between the decrease in variance (risk) and the amount of

stocks in a portfolio. Based on the research of Sharpe (1966), E&A made a distinction

between systematic variation and unsystematic variation. Their goal was to eliminate the

unsystematic variation by extending the portfolio. If enough securities are implemented

in a portfolio, E&A expected the total variation to drop until the level of systematic

variation and thus equal to the market portfolio.

The methodology developed by E&A is used by almost all the following studies. E&A

selected 60 portfolios consisting out of 1-40 stocks. Each of the portfolios were randomly

selected (each portfolio with 1,2,3…40 stocks), which results in a total of 2400 portfolios

(60 x 40). For every portfolio the mean portfolio return ( ) and standard deviation ( )

were computed. To examine whether the standard deviation of a portfolio decreases

when the diversification increases, they used a regression where Y= mean portfolio

standard deviation and X= the number of stocks in the portfolio.

Y = B1(1/X) +A2

1 B is a parameter used in the regression of E&A, the regression coefficient. 2 A is a parameter used in the regression of E&A, the asymptote.

8

Figure 2 shows that there is an asymptote. This asymptote indicates that if you decrease

the variation significantly; you end up close to the systematic risk, which is illustrated by

the asymptote (A= 0,1191). Two different tests are conducted, T-test and F-test, to

indicate whether it’s significant to add another stock in terms of decreasing variation.

As seen in figure 2 most of the unsystematic variation is diversified when reaching 8

stocks in the portfolio. Evans & Archer (1968) concluded that there exists a negative

relationship between the number of stocks included and the standard deviation of the

portfolio. A portfolio size beyond 10 stocks lacks economic justification as the variation

doesn’t decrease significantly.

E&A concluded that a ten-security portfolio provided adequate diversification.

Source: Evans, John, and Stephen Archer. 1968. “Diversification and the Reduction of Dispersion: An Empirical Analysis,” Journal of Finance, XXIV: 761-769.

Figure 2 - Asymptotic function portfolios

9

2.2.2. 1970-1979 Fisher & Lorie (1970) used a different approach to measure the risk. Compared to

Markowitz’ (1952) rate of returns, F&L used wealth ratios. Wealth ratios have 2

advantages; (1) annual rates of return are difficult to distinguish. Some rates are

compounded annually while others have a constant rate over the years. (2) Rates of

return are frequently misinterpreted.

F&L used wealth ratios to measure the dispersion of various portfolio sizes namely

1,2,8,16,32,128. The stocks are randomly selected and equal investments were made in

each stock in the different portfolios.

“The findings indicate that roughly, 40 percent of achievable reduction is obtained by

holding two stocks; 80 percent, by holding eight stocks; 90 percent, by holding sixteen

stocks; 95 percent, by holding thirty-two stocks; and 99 percent, by holding 128 stocks.”

(Fisher&Lorie, 1970)

Upson, Jessup & Matsumoto (1975) based their research on Fischer & Lorie (1970).

Instead of using wealth ratios, they used the standard deviation. They concluded that

increasing the stocks in a portfolio result in a higher possibility reaching the market

return. 8 to 16 stocks are enough to reach a significant diversified portfolio, but many

mutual funds hold 30 to 100 stocks to provide the investors with confidence that the

market return will be reached.

However, the research of Elton & Gruber (1977) distinguishes from the standard

literature of E&A in 2 ways: (1) they created an analytical expression to examine the

relationship between risk and portfolio size (2) risk is defined differently. E&G defined

that risk from holding a single stock isn’t only caused by the variability of the stock’s

return, but also caused by the uncertainty of the average return. They compared 2

portfolios; (1) one consisting out of 150 stocks and (2) one consisting out of 3290 stocks.

The results showed that there is only a minor effect on the variation when the portfolio is

expanded from 150 stocks to 3290 stocks.

10

E&G concluded that a 15 stock portfolio has 32% more risk than a 100 stock portfolio. In

contradiction to previous studies, which have shown that a well-diversified portfolio

shouldn’t consist out of more than 10-20 stocks, adding stocks beyond 15 is still beneficial

in terms of risk reduction.

2.2.3. 1980-1989 In the beginning of the 1980’s Gup (1983) says that a well-diversified portfolio doesn’t

require large numbers. The idiosyncratic risk is almost entirely eliminated when reaching

8 or 9 stocks.

Reilly (1985) wrote that 200 stocks aren’t required to make a diversified portfolio. A

portfolio with 12 to 18 stocks is sufficient for most of the diversification.

Francis (1986) concluded that portfolio managers shouldn’t hold too many assets. 10 or

15 assets are adequate to reach the maximum benefits from diversification. A higher

number of stocks are redundant to attain more diversification.

Statman (1987) contradicts previous studies that 10 to 15 stocks are enough to have a

well-diversified portfolio. Instead 30 to 40 stocks are required. The benefits of

diversification result in a decline in variation. A portfolio benefits from diversification as

long as the marginal costs are lower than the marginal benefits. To measure the marginal

costs and benefits Statman (1987) used returns. 2 portfolios are used to compare risk

reduction; (1) a benchmark portfolio consisting out of 500 stocks and (2) smaller

portfolios. By including the marginal costs into the model, a minimum of 30 to 40 stocks

are required to have an optimal diversified portfolio.

2.2.4. 2000+ Statman (2002) concluded by the rules of the mean-variance portfolio theory that a

portfolio consisting out of more than 120 stocks is the optimal portfolio. Statman (2002)

also implied a behavioral portfolio theory in his research, which has different rules

compared to the mean-variance theory. The mean-variance portfolio theory doesn’t take

11

into account the behavior of investors, which can give biased results. For both theories

investors act the same in the downside protection layers, where they prefer low risk to

high risk. On the other hand in the upside potential layers, investors in behavioral

portfolio theory tend to prefer a high risk over low risk as those investors only see the

upward potential return and forget the downside risk. Due to this desire for upward

potential investors tend to forget that diversification is needed to reduce risk.

Zhou reconstructed the study of Evans & Archer (1968) in 2014 by using the data of the

S&P 500 until 2013. Similar to E&A, he used portfolios ranging from 1 to 40 stocks and

repeated this 60 times which gives a total of 2400 portfolios. By extending the range of

the portfolio to 100, instead of the 40 used by E&A, he wanted to research further

decrease in the unsystematic variation. Zhou (2014) confirmed the conclusion of E&A that

increasing your portfolio beyond 10 stocks doesn’t result in a significant reduction of the

risk.

2.2.5. Conclusion Most research about this topic situates in the period 1960 to 1989. As seen in figure 3, we

can conclude that there is a rising trend in the required amount of stocks to have a well-

diversified portfolio.

12

Zhou (2014) is the only exception for this rising trend throughout the years. A possible

explanation for this is that he used the exact same methodology as Evans & Archer

(1968), which doesn’t take into account the recent developments.

Some of the recent studies have implemented other theories besides the mean-variance

portfolio theory; such as the use of wealth ratios (Upson, Jessup, &Matsumoto, 1975) and

the behavioral portfolio theory (Statman, 2002). Both theories implement an innovative

approach that results in different conclusions.

Figure 3 - Overview literature

13

Number of stocks Period Country/index

Evans & Archer (1968) 10 1958 - 1967 S&P500 Fisher & Lorie (1970) 8 1926 - 1965 NYSE Upson, Jessup and Matsumoto (1975) 16 1926 - 1965 NYSE Elton & Gruber (1977) 15 1971 - 1974 NYSE Gup (1983) 9 1983 USA Reilly (1985) 18 1985 USA Francis (1986) 15 1986 USA Statman (1987) 40 1979 - 1984 Vanguard index Statman (2002) 120 1926 - 2001 Vanguard index Zhou (2014) 10 2008 - 2013 S&P500

Table 1- Literature review Previous studies had different conclusions about the required number of stocks. The

dependent variable in our study is the portfolio variation. Moreover, the required amount

of stocks is reached when the difference in the reduction of the portfolio variation is no

longer significant.

As seen in figure 3 and concluded in our literature review, there is a possible rising trend

throughout the years. We want to research whether the required number of stocks in a

portfolio has risen.

14

3. Hypothesis As seen in the literature review there is a rising trend in the required number of stocks to

have an optimal diversified portfolio throughout the last decades. Most of the researches

regarding this subject are executed between 1960-1990. Moreover, all the researches

that have been conducted in the past are based on the American stock market. This paper

focuses on the European stock market, which could lead to different results compared to

the American stock market. Therefore we want to examine whether this trend can be

confirmed with recent data and for different countries and sectors within Europe.

Hypothesis 1: Is there a rising trend in the required amount of stocks to reach an

optimal diversified portfolio?

A lot of factors can be responsible for this rising trend such as the globalization and

internationalization. This leads to higher correlation between international markets.

Moreover, correlation is higher during times of high volatility. (Sandoval&De Paula, 2012)

This could be one of the explanations why investors require more stocks in a portfolio

during times of high volatility to meet the same level of diversification as in stable times.

Investors need to take in mind that a higher amount of stocks in their portfolio is

associated with higher transaction costs. These costs exceed the marginal benefits at a

certain amount of stocks. Therefore, in some countries you need to invest in a higher

amount of stocks to reach the same level of diversification. Investors should take into

account that in some situations it’s better to diversify in more than one country.

Moreover, in some countries it’s better to be exposed at a higher level of risk if you

compare reducing the risk with the associated transaction costs.

Another explanation for this rising trend could be the increase of idiosyncratic risk. More

idiosyncratic risk could lead to a higher required number of stocks in a diversified

portfolio. (Campbell et al, 2001)

If this trend is confirmed, this could have a major impact on investors who still maintain a

strategy that is conducted in past researches or based on the American stock market.

15

Investors should take in mind that strategies based on the American stock market might

have a different outcome on the European stock market.

Besides the rising trend we want to research whether there is a difference in the required

amount of stocks to diversify within different countries and sectors. Moreover, we want

to compare five better performing countries with five countries that were more affected

by political and economical instabilities in the last decade. Next to the countries, five

sectors were chosen to compare within Europe. Therefore, the next hypotheses are

defined:

Hypothesis 2: Is there a difference in the required amount of stocks to reach

optimal diversification in better performing countries compared to countries with a

weaker economy within Europe?

Hypothesis 3: Is there a difference in the required amount of stocks to reach

optimal diversification between sectors within Europe?

The financial markets encountered several crises in the past few decades such as in 1987

(Black Monday), 1998 (Russian Crisis), 2001 (Dot-Com bubble) and in 2008 (Subprime

Mortgage crisis). This indicates that the last decades the frequency of financial crises

augments, hence we want to examine whether the required amount of stocks to reach

optimal diversification is affected during times of crises3.

As mentioned above, during times of high volatility the correlation among stocks tend to

increase. (Sandoval&De Paula, 2012) We’ve chosen the financial crisis of 2008 because

it’s the most recent global economic recession. This crisis caused a major crash on the

worldwide stock markets. Therefore, we want to inform investors how to change their

strategy during times of crises. The required amount of stocks to have an optimal

3 http://www.theguardian.com/business/2001/jul/10/globalrecession

16

diversified portfolio might differ during these times. Having said this, the next hypothesis

is defined:

Hypothesis 4: Does a financial crisis affect the required number of stocks to reach

optimal diversification?

We think that the rising trend of required stocks to reach optimal diversification will be

confirmed, as well as the second, third and fourth hypothesis; whereas we think that

respectively the state of the economy influences the required amount of stocks, the

sectors will differ with each other and the financial crisis of 2008 will have an impact on

the required amount of stocks.

4. Data Most studies concerning our subject are based on the stock market of the USA.

(Evans&Archer, 1968; Fisher&Lorie, 1970; Upson, Jessup&Matsumoto, 1975;

Elton&Gruber, 1977; Gup, 1983; Reilly, 1985; Francis, 1986; Statman, 1987; Statman,

2002; Zhou, 2014)

On the other hand, this study will provide research based on the European stock market,

which might give different results. We will take into account an overview of the European

market and compare 2 groups of countries retrieved from DataStream. Namely PIIGS

countries (Portugal, Italy, Ireland, Greece and Spain) and better performing countries

(Denmark, France, Germany, Sweden and the United Kingdom). Furthermore, this

research will compare different sectors and their characteristics around diversification.

4.1. S&P Europe 350 Index We shall use the S&P Europe 350 index retrieved from DataStream. This gives an

overview of the European stock market; therefore this index is used to calculate the

average required amount of stocks in an optimal diversified portfolio. The index is made

up from 350 individual stocks originating from 17 countries that represents 70% of the

market capitalization in Europe. Besides this, there is a good balance between different

17

sectors. Just like the S&P 500 Index in the USA, the S&P Europe 350 Index is used as a

benchmark for the European stock market. Investors can invest in an index fund so their

portfolio gets automatically regionally diversified.

The data that is collected consists out of monthly data, starting from 01/01/1990 until the

end of 2014. A problem that occurs is that for some stocks there is only data available for

a few months or years. If we would take into account those stocks then there is a

possibility for biased results. For our chosen time period of ‘2000 – 2014’ we resolved this

by erasing stocks that don’t have data before 31/12/2007. In other words we don’t take

into account stocks that have less than 50% data for the full time period.

Moreover, this index will be used to research the effects on diversification during the

financial crisis of 2008. We will use three different time periods; one preceding the

financial crisis, 2004-2006; one during the financial crisis, 2007-2009 and one after the

financial crisis, 2010-2012.

4.2. Market indices Europe We retrieved an index from DataStream consisting out of 2608 stocks representing ±90%

of the market capitalization of Europe. It’s an index that can be divided into different

individual countries, sectors and sub-sectors. By using this index it’s possible to select and

research particular countries and sectors. As mentioned above, we selected 10 countries:

PIIGS (Portugal, Italy, Ireland, Greece and Spain) and five better performing countries

(Denmark, France, Germany, Sweden and the United Kingdom). Besides those 10

countries we selected 5 sectors as well: Consumer goods, Financial sector, Oil&Gas,

Technology and Utilities.

We used monthly data to compare the different countries and sectors for the period 2000

–2014. To be consistent with our research based on the S&P Europe 350, we have erased

those stocks that don’t have data before 31/12/2007. This will prevent possible biased

results, as we don’t take into account stocks that have less than 50% data available for

the full time period.

18

4.2.1. PIIGS (Portugal, Italy, Ireland, Greece and Spain) The term PIIGS is a non-official term, but used by many investors around the globe. We’ve

selected the PIIGS countries because they have an unstable economy, which was a real

problem during the global economic crisis of 2008. Some countries, like the United States,

have larger external debts than the PIIGS countries. The risk for these countries is lower

than for the PIIGS because they show signs of a solid and vibrant economy.4 By choosing

these countries we want to examine whether their unstable economy has an effect on

the required number of stock to have an optimal diversified portfolio.

The first country is Portugal, which ranks 16th in the European Union according to the GDP

in 2014. The debt to GDP ratio exceeded 128% in 2013 and the unemployment rate has

reached an alarming rate of 12,5% in 2014. Portugal exports over 75% of its agriculture

based on products such as grain, cattle, cork, wheat and olive oil.5

The second country is Italy, the economy of Italy is ranked 4th within the European Union

in 2014. In 2009, its GDP went down by about 4%. The debt to GDP ratio of the country

stands at 127,9% and its unemployment rate exceeds 10,6%.6 Moreover, Italy is a popular

holiday destination for people all over the world and depends on the income of tourism.

However, tourism has been negatively affected since the global economic crisis in 2008.

Since two-thirds of the working population works in the service sector, this could be a

possible explanation of the current high unemployment rate.7

The third country is Ireland; this is the most recent member of the group. Similar to

Portugal, Ireland is also one of the smaller countries within the European Union. It’s

ranked 14th according to GDP within Europe. Further, Ireland has a debt to GDP ratio of

4 http://www.investopedia.com/articles/economics/12/countries-in-piigs.asp 5 http://ec.europa.eu/eurostat 6 http://ec.europa.eu/eurostat 7 http://www.investopedia.com/articles/economics/12/countries-in-piigs.asp

19

123,3% in 2013 and its unemployment rate exceeds 10,0% in 2014.8 Ireland was like many

other countries heavily affected by the housing bubble. Moreover, Ireland was the first

euro zone country to fall into recession in 2008. Today, Ireland is recovering from this

recession but carries heavy debts and high unemployment.

Greece is the most controversial of all. Considering the GDP in 2014, Greece ranks 15th

within the European Union. It has a debt to GDP ratio of 174,9% in 2013 and an

unemployment rate of 24,8% in 2014.9 Greece differs in its economic structure compared

to other European nations since the public sector accounts for about half of its GDP. This

sector is known for reacting very slow to changes in the market. This could be one of

many explanations why Greece is facing a slow recovery from the global economic crisis.

Moreover, the corruption and political unrest has a major impact on the recovery of

Greece.10

Finally, according to the GDP, Spain is the 5th largest economy in the European Union,

with the lowest debt-to-GDP ratio of all PIIGS countries, namely 92,1% in 2013. However,

the unemployment rate is 22,3% in 2014.11

Rank according to GDP in 2014

Debt-to-GDP ratio in 2013 (%)

Unemployment rate in 2014

Global comp. ranking

Portugal 16 128 12,5 36 Italy 4 127,9 10,6 49 Ireland 14 123,3 10 25 Greece 15 174,9 24,8 81 Spain 5 92,1 22,3 35

Average 129,24 16,04 45 Table 2- Overview Rank within European Union according to GDP, Debt-to-GDP ratio, unemployment rate and global competitiveness ranking for the PIIGS countries.

8 http://ec.europa.eu/eurostat 9 http://ec.europa.eu/eurostat 10 http://www.investopedia.com/articles/economics/12/countries-in-piigs.asp 11 http://ec.europa.eu/eurostat

20

4.2.2. Denmark, France, Germany, Sweden and the United Kingdom By examining Denmark, France, Germany, Sweden and the United Kingdom we can

compare the results of these 5 countries with the results of the PIIGS. In general, these 5

countries performed better in the last decade compared to the PIIGS. By doing this, we

can see if there are any differences between those 2 groups in the required number of

stocks to have an optimal diversified portfolio. These countries are selected based on the

global competitiveness report 2014-2015 and other factors. This competitiveness report

ranks 144 economies according to twelve key measures that influence competitiveness,

including infrastructure, education and innovation12. Table 2 and 3 show that the PIIGS

countries and the better performing countries have respectively an average global

competitiveness ranking of 45 and 12. This difference indicates that the better

performing countries have a stronger economy compared to the PIIGS countries.

The first country is Denmark. In the European Union, Denmark is ranked 12th according to

the GDP in 2014 with a debt-to-GDP ratio of 45% in 2013 and an unemployment rate of

5,5% in 2014.13

The second country we’ve chosen is France. France was the third biggest economy within

Europe in 2014 according to the GDP. It has a debt-to-GDP ratio of 92,2% and an

unemployment rate of 8,8% in 2014.14

Next, Germany has the biggest economy according to GDP in 2014. Germany has a debt-

to-GDP ratio of 76,9% in 2013 and an unemployment rate of 4,7% in 2014.15 Further,

Germany is considered as the most stable country of the European Union. Its long-term

interest rates are often used as a benchmark in many calculations.

12 http://reports.weforum.org/global-competitiveness-report-2014-2015/rankings/ 13 http://ec.europa.eu/eurostat 14 http://ec.europa.eu/eurostat 15 http://ec.europa.eu/eurostat

21

The fourth country we’ve selected is Sweden. Sweden is ranked 7th in the European Union

according to the GDP of 2014. Sweden has a debt-to-GDP ratio of 38,6% in 2013, which is

the lowest ratio for these 5 countries. Moreover, the unemployment rate for Sweden was

5,7% in 2014.16

Finally, the United Kingdom is ranked 2th within the European Union according to the GDP

of 2014. UK has a debt-to-ratio of 87,2% in 2013 and an unemployment rate of 4,4% in

2014.17 Moreover, the United Kingdom is the only country with an own currency. Besides

this, UK has a stable economy and many headquarters are based in London.

Rank according to GDP in 2014

Debt-to-GDP ratio in 2013 (%)

Unemployment rate in 2014

Global comp. ranking

Denmark 12 45 5,5 13 France 3 92,2 8,8 23 Germany 1 76,9 4,7 5 Sweden 7 38,6 5,7 10 United Kingdom 2 87,2 4,4 9

Average 67,98 5,82 12 Table 3 - Overview Rank within European Union according to GDP, Debt-to-GDP ratio, unemployment rate and global competitiveness ranking for the better performing countries.

4.2.3. Different sectors

We also selected 5 different sectors to compare the impact on the diversification effect of

a portfolio on sectoral level. These sectors cross borders, which gives an indication of the

impact on 1 particular sector within Europe. The 5 sectors we’ve chosen are: Consumer

goods, Financial sector, Oil&Gas, Technology and Utilities. Some of these sectors are

heavily affected by the economic crisis, while others are less volatile. With this research

we want to examine whether the effects in a particular sector are reflected on the

required amount of stocks to have an optimal diversified portfolio.

16 http://ec.europa.eu/eurostat 17 http://ec.europa.eu/eurostat

22

The first sector is consumer goods. These products are bought for consumption and not

for further manufacturing. Cars, food but also electronic devices are examples of

consumer goods. Further, this sector depends on the demand for a particular product and

this can change for a lot of various reasons. Therefore, we think that this sector will be

volatile and have a high standard deviation. Due to this high standard deviation the

required amount of stocks to have optimal diversification will be higher.

The second sector we’ve chosen is the financial sector. This sector reacts fast when

economical events occur (e.g. stock market crash) since they have a lot of knowledge

about the economy. Due to this fast reaction we think that the standard deviation will be

rather neutral. This sector contains a lot of different organizations such as banks, credit

card companies, insurance companies, accountancy companies, etc.

.

The third sector we wanted to examine is Oil&Gas. This sector includes the oil and gas

extraction as well as the petroleum refining. It’s a sector that requires a lot of capital

investments and therefore is associated with high debts. Moreover, the political impact

on this sector is enormous which makes this sector volatile. Therefore we think the

standard deviation will be above neutral.

The fourth sector we’ve chosen is the technology sector. This sector develops and

distributes technologically based goods and services. Goods such as software, hardware

and telecommunications equipment but also services such as IT consultants, provider of

Internet, etc. This sector had a hard time because of the Dot-Com bubble but the recent

years were very promising. This resulted in a consistent growth over the last years.

The final sector we’ve chosen is utilities. It’s a sector that requires a lot of investments

because of the necessary infrastructure. Even during crises the demand remains stable

because their services or products are a basic need. Therefore, we think the standard

deviation will be low.

23

5. Methodology The principle of our methodology is similar with the research of Statman (1987).

Diversification of a portfolio should be increased as long as the marginal benefits exceed

the marginal costs. People should take in mind that the marginal costs increase faster

than the marginal benefits and the pace of the increase varies through different sectors,

regions and countries. Moreover the pace of the increase of marginal costs and benefits

depends on the correlation between stocks in a country, the level of idiosyncratic risk,

systematic risk, etc. By applying our method on different countries, sectors and Europe as

a whole we want to examine whether we obtain different results. Our results are

applicable for various (individual) investors and mutual fund managers.

Although, it might be more useful for individual investors or smaller mutual funds

because large mutual fund managers feature funds that are worth billions of dollars

which results in transactions with a large amount of stocks. Due to those large amounts,

transaction costs are relatively lower and therefore those managers could argue with this

research because the margin we use for our research may not be applicable for those

large mutual funds. Nevertheless, it’s useful for every investor that picks stocks himself

instead of investing in an ETF, index, tracker, mutual fund, etc. as we can point out the

optimal number of stocks.

To reach the optimal number of stocks we applied the following formulas:

5.1. Constructing a portfolio Having said this we apply the following, consistent method through our research so our

results are non-biased. The returns retrieved from DataStream are expressed in monthly

data. To calculate the % change of the total return index, we used the following

formula :

Where: = % change of the total return index = The monthly return of stock X in year t = The monthly return of stock X in year t-1

24

To investigate the volatility of the returns, we calculate the standard deviation for each

portfolio and this for a given time period. Since monthly data is used, the standard

deviation is multiplied by the square root of 12 to obtain yearly data.

Where: = Standard deviation = The total amount of stocks = The return of stock x µ = The average return of the total amount of stocks

Next, correlation is calculated for Europe (2000-2014; 2004-2006; 2007-2009; 2010-

2012), the different countries and sectors (2000-2014). The following formula is used:

Where: = Correlation = The total amount of stocks = The return of stock x y = The return of stock y = The average return of stock x = The average return of stock y = Standard deviation of stock x = Standard deviation of stock y

This formula is used to calculate the correlation matrix for n stocks as seen in the table

below:

1 2 3 4 5 … n Stocks selected 1 1

2 1

3 1

4 1

5 1

…

…

n

1

25

Investors should take in mind that correlations are important when choosing stocks.

Stocks that have a high positive correlation tend to move into the same direction. On the

other hand, stocks with a negative correlation tend to move in the opposite direction. The

level of correlation can influence the diversification benefits when adding one additional

stock to the portfolio and therefore it influences the required amount of stocks to have

an optimal diversified portfolio. These correlations are supportive for our findings and

might affirm our conclusions.

Similar to the study of Evans & Archer (1968) we have 3 assumptions:

First, we don’t claim that a portfolio with the maximum usable amount of stocks is a

proxy for the market portfolio. The market portfolio contains every asset available in a

country with each asset in proportion to its weight in the total market. The portfolios

constructed in our research are equally weighted. By doing this we assume that equal

amounts are invested in each stock.

The second assumption is that there is no capitalization. Every dividend received from a

stock will not be reinvested.

Third, we will look at the specific relation between the size of a portfolio and the standard

deviation (risk). In order to research this relationship we randomly select stocks from size

1,2,…,20,25,30,40,50,75,100 in a portfolio. In case that the country doesn’t contain 100 or

more stocks we select the maximum usable amount of stocks.

Robustness test: For countries or sectors with a higher amount of

stocks than 100 we select portfolios consisting out of 100 stocks. On

the other hand, we select the maximum usable amount of stocks for

countries or sectors that doesn’t contain 100 stocks or more. To

examine whether those both methods are consistent we used for a

country with more than 100 stocks the same amount of stocks that

26

are used for the country with the lowest amount of usable stocks.

These results were consistent. Therefore, we can conclude that even

with fewer stocks available the results can be compared with

countries that have 100 or more stocks available.

To construct a portfolio consisting out of 1 stock; we select 1 stock at random from the

dataset. For a portfolio with size 2; 2 stocks are selected at random and divided by 2 to

have the average return. This process is repeated for portfolios sizing from 1 to 100/N

(i.e. N is the amount of stocks).

Each portfolio is calculated 200 times. Therefore if we take the mean of all the standard

deviations, it will be a good representative of a portfolio consisting out of N stocks. As this

representative will be a close estimate of the real mean, we can conclude that our results

are non-biased and applicable for real life situations.

5.2. Applying benchmark

Since we don’t take into account the transaction costs, a benchmark is used. The

benchmark is the CAGR of the S&P Europe 350 over the last 10 years.

Where: = Compounding Annual Growth Rate = The total amount of years = The value of the S&P Europe 350 on 31/12/2014 = The value of the S&P Europe 350 on 31/12/2004

If we use this formula to calculate the CAGR for the past 10 years, the average annual

return is 6.78%. In our model we use 6% as a benchmark. One benchmark is applied for all

the countries and sectors; the 0,78% is an extra safety margin to take into account the

additional taxes and various costs in the different countries and sectors. Besides this

benchmark, other benchmarks are used to give a comparison and therefore this could be

27

of use for investors with a different profile – defensive neutral aggressive. This can be

found in appendix 5 to 17.

We use the standard deviation of the portfolio with 100 stocks or the portfolio with the

maximum stocks available in that country and compare it with other, less diversified

portfolios. By multiplying this with our benchmark it’s possible to compare the different

countries and sectors with each other. We can compare these by leveraging the

benchmark to make the risk equal to the risk of the portfolio as seen in the formula

below:

=

Where: = Standard deviation of a portfolio with 100 or the maximum amount of stocks available in a country or sector.

= Standard deviation of a portfolio with x amount of stocks = Expected average return of N stocks

By calculating the leverage we examine the benefits relative to the benchmark. After

calculating the leverage, we subtract the leverage by the benchmark to receive the

difference between the returns. This difference is compared with the chosen transaction

cost. Therefore, if the transaction cost is higher than the possible return, an additional

stock will not be beneficial because the reduction in risk doesn’t compensate the

additional transaction cost.

This formula is applied in the following example:

28

With this given information applied in our formula we obtain a leverage of 8% (= ). Further we subtract this with the benchmark: . This 2% is

compared with the transaction cost, which is 0,5% in this example. The transaction costs

are relatively lower than the increase in return; therefore we can conclude that an

additional stock is beneficial for this portfolio.

Table 4 - January 2009: Average transaction costs

We will maintain 0,5% transaction costs through our research. As seen in table 4 and

appendix 16 the average transaction cost for those European countries is ±0,5%.18 Two

assumptions are made considering the transaction costs; (1) transaction costs are equal

over countries and sectors, (2) transaction costs are equal for each portfolio size. This

makes it possible to compare countries and sectors with each other.

Moreover, as seen in appendix 6-17 different transaction costs are applied in our

research. This makes it possible to use higher/lower transaction costs to compare

countries and sectors as this can be useful for the PIIGS countries, because transaction

costs might be higher. Whereas within the appendix 5-17; the yellow indicator is used to

implement 0,50% transaction costs. Further the dark grey indicator represents 0,25% and

the light grey indicator represents 0,75%.

6. Results The results are based on a benchmark of 6% and transaction costs of 0,5%. To compare

different investor profiles we use three different benchmarks and three different

18 http://www.itg.com

Average transaction cost

Great Britain 0,56% France 0,44% Germany 0,40% Switzerland 0,38%

29

transaction costs. By doing this we take into consideration benchmarks that have a lower

or higher performance. With this comparison we want to make sure that every investor

can retrieve his investor profile within our research. Therefore, the investor is able to find

the required amount of stocks to have a diversified portfolio within his investor profile.

On the other hand we take into account that transaction costs can be more or less

expensive than our chosen transaction cost of 0,5%. Therefore it’s possible for investors

to select a transaction cost that suits best for its country or sector, as transaction costs

can be higher or lower in various countries.

The chosen transaction cost or benchmark can vary through investor profiles and certain

countries and sectors. Furthermore, the different benchmarks or transaction costs could

significantly affect the required amount of stocks to have an optimal diversified portfolio.

6.1. Europe Our main research is based on the S&P Europe 350 since we focus on the European

market. As seen in appendix 5 we’ve come to the conclusion that 14 stocks with a

correlation of 0,3077 and a standard deviation of 0,1644 are enough to have an optimal

diversified portfolio. The difference in return for adding 1 more stock to a portfolio with

14 stocks, gives an extra return of 0,49%. Considering the transaction cost of 0,5%, this

extra return won’t cover the additional transaction cost associated with adding one more

stock to the portfolio.

Period Number of

stocks Average return

Standard deviation Correlation

S&P Europe 350 2000 - 2014 14 6,78% 0,1644 0,3077 Table 5 - Overview S&P Europe 350

Results show that 14 stocks are enough to diversify. If we compare this result with the

research of Evans & Archer (1968), who concluded that 10 stocks are enough to diversify,

we can say that throughout the years an additional 4 stocks are required to have an

optimal diversified portfolio. On the other hand, if the result is compared with the follow-

up studies Upson, Jessup & Matsumoto (1975) Elton & Gruber (1977) Reilly (1985) Francis

30

(1986), who concluded that 16-18 stocks are required to reach optimal diversification, we

can reject the first hypothesis because the required amount of stocks to diversify is

remains unchanged.

Figure 4 - Standard deviation S&P Europe 350

Further, if the result is compared with the research of Statman (1987, 2002), who

concluded that respectively 40 and 120 stocks are required to reach optimal

diversification, the first hypothesis isn’t ratified. This is a first indication that contradicts

the rising trend in the required amount of stocks to have optimal diversification and

therefore deny the first hypothesis.

For following calculations we will use this result as a benchmark to compare it with

individual countries and sectors. By doing this we want to see if countries or sectors show

different results. These results can be affected by the countries’ political instability,

economic instability, fraud, corruption and other risk factors.

6.2. PIIGS We selected the PIIGS countries (Portugal, Italy, Ireland, Greece and Spain) because these

5 countries were heavily affected by the economic crisis in 2008-2009. By doing this we

want to examine whether there is an impact on the diversification effect of a portfolio in

these countries.

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

25

30

40

50

75

100

S&P Europe 350

31

6.2.1. Portugal

The first country is Portugal. The stock market of Portugal consists out of 40 usable stocks

for our chosen time period. Figure 5 shows that most of the diversification effects are

reached by having a portfolio of 15-20 stocks.

Figure 5 - Standard deviation of Portugal The results in appendix 7 show that 20 stocks are enough to diversify. The difference in

return between the leverage and the benchmark for a portfolio with 20 stocks is equal to

0,62%. The additional return of 0,62% outweighs the transaction costs of 0,5%.

6.2.2. Italy The second country we examined is Italy. The stock market of Italy consists out of 158

usable stocks for our chosen time period.

Figure 6 - Standard deviation Italy

0,00

0,10

0,20

0,30

0,40

0,50

0,60

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40

Portugal

STDEV

0,00

0,10

0,20

0,30

0,40

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

25

30

40

50

75

100

Italy

STDEV

32

Appendix 7 shows that having a portfolio with 16 stocks is enough to diversify. The

difference in return between a portfolio with 16 stocks and 17 stocks is 0,46%. This is not

enough to cover the additional transaction cost of 0,5%.

6.2.3. Ireland The third country is Ireland. The stock market of Ireland consists out of 32 usable stocks

for our chosen time period.

Figure 7 - Standard deviation Ireland

Figure 7 shows that most of the diversification effects are reached with a portfolio of 12-

16 stocks.

As you can see in appendix 7 a portfolio with 14 stocks is enough to have an optimal

diversified portfolio. The difference between the leverage and the benchmark for a

portfolio with 14 stocks is 0,5%. This extra stock will cause a break-even between the

additional return and additional transaction cost. Since an extra stock reduces the

standard deviation (risk), this will be beneficial for the portfolio.

6.2.4. Greece The next country we’ve chosen is Greece. The stock market of Greece consists out of 50

usable stocks for our chosen time period.

0,00

0,10

0,20

0,30

0,40

0,50

0,60

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 32

Ireland

STDEV

33

Figure 8 - Standard deviation Greece As seen in the figure above, most of the diversification effects are reached with a

portfolio consisting out of 8 stocks. Appendix 7 shows that more than 8 stocks doesn’t

give enough additional return (0,46%) to cover the transaction cost of 0,5%. Notice that

the standard deviation is higher than in other countries.

6.2.5. Spain The final country of the PIIGS is Spain. The stock market of Spain consists out of 93 usable

stocks for our time period.

Figure 9 - Standard deviation Spain

We can derive from figure 9 that most of the diversification benefits are reached by

having a portfolio of 12-16 stocks.

0,00

0,10

0,20

0,30

0,40

0,50

0,60

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50

Greece

STDEV

0,00

0,10

0,20

0,30

0,40

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50 75 93

Spain

34

On the other hand, appendix 7 confirms this observation. 16 stocks are required within a

portfolio to reach optimal diversification. The difference in return between the leverage

and the benchmark for a portfolio with 16 stocks is equal to 0,57%. Moreover, adding one

more stock to the portfolio will give an extra 0,49% return. This is not enough to cover the

additional transaction cost.

6.2.6. Overview We can conclude for PIIGS (Portugal, Italy, Ireland, Greece and Spain) that most of the

diversification benefits are reached by having a portfolio between 14 and 16 stocks, as

you can see below in table 6. Adding 1 additional stock to one of the portfolios in each

country won’t result in higher marginal benefits compared to the marginal costs. This

result is similar to the research based on the S&P Europe 350. Therefore, these results are

another indication to reject our first hypothesis: “Is there a rising trend in the required

amount of stocks to reach an optimal diversified portfolio?”

Fund managers should take in mind that adding more stocks beyond 14-16 stocks,

depending on the country, doesn’t give a sufficient reduction of risk compared to the

associated transaction costs. If fund managers tend to do this anyway, it might lead to

higher transaction costs and therefore to lower profits.

Period Number of stocks

Average return

Standard deviation Correlation

Portugal 2000 - 2014 20 -4,27% 0,2289 0,1450 Italy 2000 - 2014 16 1,66% 0,1743 0,3197 Ireland 2000 - 2014 14 3,21% 0,2459 0,1780 Greece 2000 - 2014 8 -10,63% 0,3197 0,4081 Spain 2000 - 2014 16 4,63% 0,1813 0,2360 Average: 14,8 -1,08% 0,23 0,2574

Table 6 - Overview PIIGS countries As mentioned above, the number of stocks in your portfolio that is necessary to reach

diversification ranges from 8 to 20 with an average of 14,8. The level of standard

deviation ranges from 0,1743 to 0,3197 with an average of 0,2300.

35

Greece has the highest level of standard deviation, namely 0,3197. This could be

explained by the fact that Greece is a country with political and economical problems

whereby all stocks react the same on events. Moreover Greece was heavily affected by

the crisis in the previous decade. As you can see in table 6 the correlation of Greece is

0,4081. This is higher than the average of the PIIGS countries (0,2855) and the S&P

Europe 350 (0,3077).

Sandoval & De Paula (2012) concluded that correlations of financial markets are higher

during times of crises. The political and economical instability causes a high correlation on

the stock market in Greece, which makes it difficult or nearly impossible to obtain the

same level of standard deviation compared to other countries.

6.3. Better performing countries We’ve chosen Denmark, France, Germany, Sweden and the United Kingdom because they

were less affected by the economic crisis in 2008. By doing this we can compare 5 better

performing countries with 5 countries that were heavily affected by the economic crisis.

6.3.1. Denmark Denmark is the first country we’ve examined. The stock market of Denmark consists out

of 222 usable stocks for our time period.

Figure 10 - Standard deviation Denmark

0

0,1

0,2

0,3

0,4

0,5

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

25

30

40

50

75

100

Denmark

STDEV

36

The results in appendix 10 show that 18 stocks are enough to diversify. The difference in

return between the leverage and the benchmark for a portfolio with 18 stocks is equal to

0,55%. Adding one more stock to the portfolio will give an extra 0,49% return. This is not

enough to cover the additional transaction cost.

6.3.2. France The second country we discussed is France. The stock market of France consists out of

226 usable stocks for our chosen time period.

Figure 11- Standard deviation France Figure 11 shows that most of the diversification benefits are reached by having a portfolio

of 17-20.

On the other hand, appendix 10 shows that having a portfolio with 17 stocks is enough to

diversify. The difference in return between a portfolio with 17 stocks and 18 stocks is

0,47%. This is not enough to cover the transaction cost of 0,5%.

6.3.3. Germany The third country we discussed is Germany. The stock market of Germany consists out of

218 usable stocks for our chosen time period.

0 0,05

0,1 0,15

0,2 0,25

0,3 0,35

0,4

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

25

30

40

50

75

100

France

STDEV

37

Figure 12 - Standard deviation Germany Figure 12 shows that most of the diversification benefits are reached by having a portfolio

ranging from 15-19 stocks. As you can see in appendix 10, 19 stocks are necessary to

diversify. Adding one more stock to a portfolio with 19 stocks causes an additional return

of 0,47% which is lower than the transaction costs of 0,5% and therefore it’s not

beneficial to add more stocks.

6.3.4. Sweden The fourth country we’ve chosen is Sweden. The stock market of Sweden consists out of

66 usable stocks for our chosen time period.

Figure 13 - Standard deviation Sweden Figure 13 shows that most of the diversification effects are reached with a portfolio of 8-

11 stocks.

0

0,1

0,2

0,3

0,4

0,5 1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

25

30

40

50

75

10

0

Germany

STDEV

0 0,05

0,1 0,15

0,2 0,25

0,3 0,35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50 66

Sweden

STDEV

38

As you can see in appendix 10 a portfolio with 11 stocks is enough to have an optimal

diversified portfolio. The difference between the leverage and the benchmark for a

portfolio with 11 stocks is 0,5%. This extra stock will cause a break-even between the

additional return and additional transaction cost. Since an extra stock reduces the

standard deviation (risk), this will be beneficial for the portfolio.

6.3.5. United Kingdom The last country that we’ve examined is the United Kingdom. The stock market of the

United Kingdom consists out of 432 usable stocks for our chosen time period.

Figure 14 – Standard deviation United Kingdom We can derive from figure 14 that most of the diversification benefits are reached by

having a portfolio of 17-20 stocks.

On the other hand, appendix 10 confirms this observation. 17 stocks are necessary within

a portfolio to reach optimal diversification. The difference in return between a portfolio

with 17 stocks and 18 stocks is 0,48%. This is not enough to cover the transaction cost of

0,5%.

0

0,1

0,2

0,3

0,4

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

25

30

40

50

75

100

United Kingdom

STDEV

39

6.3.6. Overview We can conclude for Denmark, France, Germany, Sweden and the United Kingdom that

most of the diversification benefits are reached by having a portfolio between 17 and 19

stocks, as you can see below in table 7. Adding more stocks to one of the portfolios won’t

reduce the risk significantly.

Besides the 14 required stocks within Europe and the 14,8 required stocks within the

PIIGS-countries to reach optimal diversification, results show this is another indication to

reject our hypothesis: “Is there a rising trend in the required amount of stocks to reach an

optimal diversified portfolio?”

Period Number of stocks

Average return

Standard deviation Correlation

Denmark 2000 - 2014 18 7,79% 0,1899 0,1854 France 2000 - 2014 17 7,04% 0,1871 0,2571 Germany 2000 - 2014 19 7,86% 0,1918 0,2088 Sweden 2000 - 2014 11 12,84% 0,2043 0,3820 United Kingdom 2000 - 2014 17 10,21% 0,1725 0,2959 Average: 16,4 9,49% 0,1889 0,2859

Table 7 - Overview Denmark, France, Germany, Sweden and United Kingdom We can conclude for these countries that the required number of stocks in a portfolio is

quite similar, with an exception for Sweden. Furthermore, the level of standard deviation

in those countries is similar. This could be explained because all these countries have a

strong, solid economy that maintained their strong position even during the crisis of

2008-2009.

If you compare these 5 countries with the PIIGS countries we can say that the average

optimal number of stocks in a portfolio is slightly different. For the PIIGS countries you’ll

need 14,8 stocks and for the better performing countries 16,4 stocks. It indicates that

there is a difference between the required amount of stocks to have an optimal

40

diversified portfolio in better performing countries and countries with a weaker economy.

This is a strong indication to accept our second hypothesis:

Hypothesis 2: Is there a difference in the required amount of stocks to reach

optimal diversification in better performing countries compared to countries with a

weaker economy within Europe?

This difference can be explained due to the fact that large instability comes with a higher

correlation in the stock market of a particular country or sector. (Maskawa&Souma, 2010;

Sandoval&De Paula, 2012) Stocks with a higher correlation tend to move in the same

direction, which causes lower diversification possibilities. Due to this higher correlation

it’s hard to obtain a low level of standard deviation in particular countries, such as the

PIIGS-countries. Moreover, most of the diversification possibilities are reached at a faster

pace compared to stock markets with a lower correlation. This could be a reason for the

lower required amount of stocks to reach optimal diversification in Sweden.

The level of standard deviation differs more than the required amount of stocks; with an

average STDEV of 0,1891 for Denmark, France, Germany, Sweden and the United

Kingdom compared to an average STDEV of 0,2300 for the PIIGS countries. This difference

can be explained by the higher debts for the PIIGS countries, higher unemployment rate,

unstable politics, corruption, etc. Combined, all these factors lead to unstable, volatile

countries.

So if investors would like to reach the same level of standard deviation for the PIIGS

countries as for Denmark, France, Germany, Sweden and the United Kingdom they’ll have

to add more stocks (average STDEV = 0,1891). In some countries it will be hard or even

impossible to reach such levels of standard deviation.

Furthermore, investors should know that adding more stocks might lead to more

marginal transaction costs compared to the marginal benefits. This could have a possible

downward pressure on the profit margin.

41

6.4. Different sectors

6.4.1. Consumer goods To examine this sector we used 273 usable stocks to compare the standard deviation of

portfolios consisting out of 1-100 stocks.

Figure 15 - Standard deviation consumer goods

The results of appendix 13 show that a portfolio with 50 stocks is enough to have a

diversified portfolio. The difference between the leverage and the benchmark consisting

out of 50 stocks is 0,56%, which is higher than the 0,50% additional transaction cost.

6.4.2. Financial sector

To examine this sector we used 605 usable stocks that are based in different countries

within Europe.

Figure 16 - Standard deviation financial sector

0 0,1 0,2 0,3 0,4 0,5

1 3 5 7 9 11 13 15 17 19 25 40 75

Consumer goods

Consumer goods

0

0,1

0,2

0,3

0,4

0,5

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

25

30

40

50

75

100

Financial sector

STDEV

42

Appendix 13 shows that a portfolio with 50 stocks is enough to reach optimal

diversification. The difference between the leverage and the benchmark consisting out of

50 stocks is 0,54%, which is higher than the 0,50% additional transaction cost.

6.4.3. Oil & Gas We used 94 usable stocks from the different countries to examine the diversification

effect on a portfolio.

Figure 17 - Standard deviation Oil & Gas

The results in appendix 13 show that 18 stocks are enough to diversify. The difference in

the return between the leverage and the benchmark for a portfolio with 18 stocks is

equal to 0,53%. Adding one more stock to the portfolio will give an additional return of

0,48% . This is not enough to cover the additional transaction cost.

6.4.4. Technology To examine this sector we used 130 usable stocks within the technology sector.

0,00

0,10

0,20

0,30

0,40

0,50

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 40 50 75 94

Oil & Gas

STDEV

43

Figure 18 - Standard deviation technology The results in appendix 13 show that 20 stocks are enough to reach optimal

diversification. The difference in return between the leverage and the benchmark for a

portfolio with 20 stocks is equal to 0,64%. The additional return of 0,64% outweighs the

transaction cost of 0,50%.

6.4.5. Utilities To examine this sector we’ve used 69 usable stocks.

Figure 19 - Standard deviation utilities

As you can see in appendix 13, 20 stocks are required to reach optimal diversification. The

difference between the leverage and the benchmark consisting out of 20 stocks is 0,59%,

which is higher than the 0,50% additional transaction cost.

0

0,1

0,2

0,3

0,4

0,5

0,6 1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

25

30

40

50

75

10

0

Technology

STDEV

0 0,05

0,1 0,15

0,2 0,25

0,3 0,35

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

25

30

40

50

75

100

Utilities

STDEV

44

6.4.6. Overview

As seen below in table 8 we can conclude that an average of 31,6 stocks is required to

have a diversified portfolio. In contradiction to the results of the countries, we can

conclude that sectors have a higher average. Therefore we can conclude that there is a

rising trend compared to the following studies; Upson, Jessup & Matsumoto (1975) Elton

& Gruber (1977), Reilly (1985) and Francis (1986), who concluded that 16-18 stocks are

required to reach optimal diversification. Nevertheless, if we compare this result with the

conclusion of Statman (1987, 2002) this is another indication next to the results of

Europe, the PIIGS- and better performing countries to reject our first hypothesis: “Is there

a rising trend in the required amount of stocks to reach an optimal diversified portfolio?”

Period Number of stocks

Average return

Standard deviation Correlation

Consumer goods 2000 - 2014 50 9,97% 0,1481 0,1770 Financials 2000 - 2014 50 6,70% 0,1701 0,3256 Oil&Gas 2000 - 2014 18 10,03% 0,2025 0,2611 Technology 2000 - 2014 20 -4,49% 0,2569 0,2853 Utilities 2000 - 2014 20 9,13% 0,1308 0,1934 Average: 31,6 6,27% 0,1817 0,2485

Table 8 - Overview different sectors

Most sectors have a standard deviation between 0,17 and 0,20 when reaching an optimal

diversified portfolio. Only the technology sector has a higher standard deviation. One of

the factors could be the Dot-Com bubble that happened in 2001. Since our researched

data begins in the year 2000, the results of the technology sector can be affected by this

crisis.

Results in table 8 show that in particular sectors, such as Consumer goods and Financials,

more stocks are required compared to the other sectors. This is a confirmation for the

third hypothesis:

45

Hypothesis 3: Is there a difference in the required amount of stocks to reach

optimal diversification between sectors within Europe?

Compared to the countries; sectors have a remarkable higher average of stocks needed to

diversify; in particular Consumer goods and Financials. This high amount of stocks is due

to the fact that those sectors are widespread in Europe; therefore there are more

diversification possibilities, which makes it more interesting to have a portfolio with a

higher amount of stocks.

6.5. Financial crisis of 2008 The financial crisis of 2008 had a major impact on the world economy. People lost their

confidence in the financial system. A lot of companies had a difficult time and some went

bankrupt. Banks suffered from the same problems, such as the run on the bank in Ireland

that caused a domino effect because banks are interrelated with each other. All these

problems combined created a higher unemployment rate and high volatility of the stock

market. We want to research whether this volatility had an impact on the required

number of stocks to have a well-diversified portfolio. Therefore we selected 3 equal

periods to investigate the impact of the financial crisis; one preceding the financial crisis,

2004-2006; one during the financial crisis, 2007-2009 and one after the financial crisis,

2010-2012.

The last decades, crises happen to appear more frequently than before. Investors don’t

know how to react during these times. Therefore, our research could be of great value for

individual investors and managers of mutual funds to react correctly in times of crisis. The

proper approach during those times could minimize the downward risk. The results of this

research could be implemented in risk management of individual investors and mutual

funds.

46

6.5.1. Before the crisis (2004-2006)

Figure 20 - Standard deviation before the crisis Figure 20 shows that most diversification benefits are reached by having a portfolio

ranging from 16 to 20 stocks. As you can see in appendix 16, 18 stocks are required to

diversify. The difference between the leverage and the benchmark for a portfolio with 18

stocks is 0,5%. This extra stock will cause a break-even between the additional return and

additional transaction cost. Since an extra stock reduces the standard deviation (risk), this

will be beneficial for the portfolio.

6.5.2. During the crisis (2007-2009)

Figure 21 - Standard deviation during the crisis We can derive from figure 21 that most of the diversification benefits are reached by

having a portfolio of 6-9 stocks.

0

0,05

0,1

0,15

0,2

0,25

1 3 5 7 9 11 13 15 17 19 25 40 75

Before the crisis

Before the crisis

0

0,1

0,2

0,3

0,4

0,5

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

25

30

40

50

75

100

During the crisis

During the crisis

47

Appendix 16 confirms this observation. 7 stocks are necessary within a portfolio to

diversify. The difference in return between the leverage and the benchmark for a

portfolio with 7 stocks is equal to 0,54%. Adding one more stock to the portfolio will give

an extra 0,49% return. This is not enough to cover the additional transaction cost of

0,50%.

6.5.3. After the crisis (2010-2012)

Figure 22 - Standard deviation after the crisis

Figure 22 shows that most of the diversification benefits are reached by having a portfolio

of 12-14. On the other hand, appendix 16 shows that having a portfolio with 12 stocks is

enough to diversify. The difference in return between a portfolio with 12 stocks and 13

stocks is 0,39%. This is not enough to cover the transaction cost of 0,5%.

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

After the crisis

After the crisis

48

6.5.4. Conclusion crisis The number of stocks required varies for the different time periods. As you can see in

table 9; in 2004-2006, 18 stocks are needed to have a diversified portfolio, which is above

the average of the S&P Europe 350 for the full time period (2000-2014).

Table 9 - Overview different time periods S&P Europe 350

Between 2007-2009, only 7 stocks are required to reach optimal diversification that is

similar to the result of Greece. Due to the recession during this time period, there was a

high volatility (i.e. STDEV= 0,2655) in the market and this is directly linked with strong

correlations between them. This means that markets tend to behave as one during

economical crises. (Sandoval&De Paula, 2012) Therefore, it’s hard to reach the same level

of diversification as in 2004-2006. Investors should take in mind that if they want to reach

the same level of standard deviation as in the period of 2004-2006. The associated costs

by adding more stocks to a portfolio will exceed the potential benefits of diversification at

a certain point.

Furthermore, the period of 2010-2012 contains a higher correlation than before the crisis.

This could be explained by the joint recovery after the crisis. Moreover, it takes multiple

years to restore the confidence in the market.

Table 9 shows that the financial crisis of 2008 has a major impact on the required amount

of stocks in an optimal diversified portfolio. Therefore, we can conclude that the number

of stocks is significantly affected by the financial crisis. This confirms our fourth and final

hypothesis: ”Does a financial crisis affect the required number of stocks to reach optimal

diversification?”

Period Number of stocks Average return Standard

deviation Correlation

S&P Europe 350 2000 - 2014 14 6,78% 0,1644 0,3077 2004 - 2006 18 23,58% 0,1045 0,2513 2007 - 2009 7 -5,23% 0,2655 0,4052 2010 - 2012 12 6,33% 0,1721 0,3506

49

7. Conclusion Since the groundbreaking work of Markowitz (1952), asset allocation has been given a lot

of attention by researchers. Subsequently, a lot of studies have been conducted between

1960-1990 about the relationship between the number of stocks in a portfolio and the

reduction of variance in a portfolio. One of the studies that has been used by many

follow-up studies, as well as this research, is the research of Evans & Archer (1968). This

study concluded that 10 stocks are required to reach optimal diversification. Fisher &

Lorie (1970), Gup (1983) & Zhou (2014) confirms the conclusion of E&A, with respectively

8,9 and 10 stocks.

The studies in the past are mainly conducted on the American stock market. In

contradiction to these studies, this research focuses on the European stock market with

recent data. Next to the European stock market as a whole, we executed our research on

five better performing countries, five countries with a weaker economy (PIIGS) and five

sectors. This could be of great interest for investors that want to invest in the European

stock market or individual European countries. As it is possible that investors need to

apply a different method when investing in the European stock market compared to the

American stock market.

Previous studies have shown that asset allocation is responsible for 91.5% of the

performance. (Ibbotson & Kaplan, 2000) With this study we want to focus on a single

part of asset allocation, namely the stocks within a portfolio. This is an important factor

within asset allocation and contributes to the performance, as many funds consist out of

more than 50% stocks.

By examining the previous studies we have observed that there is a rising trend in the

required amount of stocks to diversify. Therefore we’ve set up the following hypothesis:

“Is there a rising trend in the required amount of stocks to reach an optimal diversified

portfolio?”

50

Supplementary to the previous hypothesis, we compare five better performing countries

with five countries that were more affected by political and economical instabilities in the

last decade. Next to the countries, five sectors were chosen to compare within Europe.

Therefore, the next hypotheses are defined: “Is there a difference in the required amount

of stocks to reach optimal diversification in better performing countries compared to

countries with a weaker economy within Europe?” and “Is there a difference in the

required amount of stocks to reach optimal diversification between sectors within

Europe?”

Furthermore, the financial markets encountered several crises in the past few decades

such as in; 1987 (Black Monday), 1998 (Russian Crisis), 2001 (Dot-Com bubble) and in

2008 (Subprime Mortgage crisis). This indicates that the last decades the frequency of

financial crises augments, hence we want to examine whether the required amount of

stocks to reach optimal diversification is affected during times of crises. This paper

focuses on the financial crisis of 2008 because it’s the most recent global economic

recession. Having said this, the next hypothesis is set up: “Does a financial crisis affect the

required number of stocks to reach optimal diversification?”

As seen in table 10, we can conclude for the S&P Europe 350 that the required amount of

stocks to have an optimal portfolio is 14. This result is supported by the similar results of

the PIIGS- and five better performing countries. This is a confirmation of the following

studies Upson, Jessup & Matsumoto (1975) Elton & Gruber (1977) Reilly (1985) and

Francis (1986), who concluded that 16-18 stocks are required to reach optimal

diversification. On the other hand, the required amount of stocks to reach optimal

diversification for sectors is higher. Nevertheless this result, we can conclude that there is

no rising trend throughout the years and therefore it leads to the rejection of the first

hypothesis. This is in contradiction with our expectations about the rising trend.

51

Table 10 - Overview S&P Europe 350, different countries and sectors Further, table 10 shows that there is a difference in the required amount of stocks within

an optimal diversified portfolio between the different countries. If you compare the five

better performing countries with the PIIGS countries we can say that the average optimal

required number of stocks in a portfolio is different. For the PIIGS countries you’ll need

14,8 stocks and for the better performing countries 16,4 stocks. Moreover, the average

standard deviation differs between the PIIGS-and better performing countries. These

indications confirm the second hypothesis as we expected.

Results in table 10 show that the sectors have an average of 31,6 stocks when reaching

optimal diversification, which exceeds the required amount in the countries and the S&P

52

Europe 350 as a whole. Moreover, to investigate our second hypothesis we compared the

individual sectors with each other. The results show that there is a difference between

the sectors; in particular Consumer goods and Financial sector that require 50 stocks to

have optimal diversification. This amount of stocks is in contradiction to the 18-20 stocks

required in the Oil&Gas-, technology- and utilities sector. Based on this evidence we

conclude that the third hypothesis can be accepted, as we expected.

Table 10 shows that the financial crisis of 2008 has a major impact on the required

amount of stocks in an optimal diversified portfolio. The difference in the required

amount of stocks between; before, during and after the financial crisis is notable.

Therefore, we can conclude that the number of stocks is significantly affected by the

financial crisis. This confirms the fourth hypothesis, as we expected.

Investors should take in mind that correlations are important when choosing stocks.

Stocks that have a high correlation tend to move into the same direction, which can

influence the required amount of stocks to have an optimal diversified portfolio. These

correlations are supportive for our findings and can be found for the S&P Europe 350,

each individual country and sector in table 10. The correlations are discussed in the

sections 6.1.; 6.2.6.; 6.3.6.; 6.4.6.

The results of the above study should be taken into account by individual investors as well

as fund managers. Large mutual fund managers feature funds that are worth billions of

dollars, which results in transactions with a large amount of stocks. Due to those large

amounts, transaction costs are relatively lower and therefore those managers could argue

with this research because the margin we use for our research may not be applicable for

those large mutual funds. Therefore, individual stock pickers and smaller mutual funds

might benefit more from this research.

To answer different investor profiles –defensive, neutral and aggressive- we included

appendix 5-17 with respectively a benchmark of 4%, 6% and 8% applied on the study

53

above, as well as different transaction costs of 0,25% and 0,75%. By doing this, this study

can be applied on countries with a higher/lower transaction cost and with other

benchmarks.

Further research can be done by calculating the idiosyncratic risk throughout the years

and compare it with the required amount of stocks in the corresponding time periods.

Moreover, this can be done for the different countries and sectors within Europe or other

stock markets. This could be an additional factor explaining the required number of stocks

to reach optimal diversification.

54

References

1. Alexander, G J., Francis, JC. (1986). Portfolio Analysis, (3th ed.). Prentice-Hall.

2. Ang, A., & Chen, J. (2002). Asymmetric correlations of equity portfolios. Journal of Financial Economics, 63, 443-494.

3. Bartram, S.M., & Wang, Y-H. (2005). Another look at the relationship between

cross-market correlation and volatility Finance Research Letters, 2, 75-88.

4. Bennett, J., & Sias, R. (2004). Why has firm-specific risk increased over time? Washington State working paper.

5. Bodie, Z., Kane, A., Marcus, A. J. (2011). Investments and Portfolio Management.

6. Bodie, Z., Kane, A., Marcus, A. J. (2013). Essentials of Investments, (9th ed.). New

York: McGraw-Hill Education.

7. Brandt, M., Brav, A., Graham, J., &Kumar, A. (2010) .The Idiosyncratic Volatility Puzzle: Time Trend or Speculative Episodes? Review of Financial Studies, 23, 863-899.

8. Campbell, J., Lettau, M., Malkiel, B., & Xu, Y. (2000). Have individual stocks

become more volatile? An empirical exploration of idiosyncratic risk. Journal of Finance.

9. Cizeau, P., Potters, M., & Bouchaud, J-P. (2001). Correlation structure of extreme

stock returns. Quantitative Finance, 1, 217-222.

10. Elliot, L., & Milner, M. (2001). Age of anxiety. The Guardian. Consulted on 16 April 2015, via http://www.theguardian.com/business/2001/jul/10/globalrecession

11. Elton, E. J., & Gruber, M. J. (1977). Risk Reduction and Portfolio Size: An Analytical

Solution. Journal of Business, 50, 415-437.

12. Eurostat. (2013-2014). Consulted on 4 May 2015 via http://ec.europa.eu/eurostat

13. Evans, J., & Archer, S. (1968). Diversification and the Reduction of Dispersion: An Empirical Analysis. Journal of Finance XXIV, 761-769.

14. Fama, E., & French. (2004). New Lists: Fundamentals and Survival Rates. Journal

of Financial Economics, 73, 229-269.

55

15. Fink, J., Fink, K. E., Grullon, G., & Weston, J. P. (2010). What Drove the Increase in Idiosyncratic Volatility during the Internet Boom? Journal of Financial and Quantitative Analysis, 45, 1253–1278.

16. Fisher, Lawrence, & Lorie, J.H. (1970). Some Studies of Variability of Returns on

Investments in Common Stocks. Journal of Business, 43, 99-l 17.

17. Francis, J. C (1986). Investments: Analysis and Management (4th ed.). New York: McGraw-Hill.

18. Global competitiveness report, world economic forum. (2014-2015). Consulted on

6 May 2015 via http://reports.weforum.org/global-competitiveness-report-2014-2015/rankings/

19. Gup, B. E. (1983). The Basics of Investing (2nd ed.). New York: John Wiley & Sons.

20. Goetzmann, W. & Alok, K. (2001). Equity Portfolio Diversification. National Bureau

of Economic Research working paper series.

21. Goetzmann, W., Li, L., & Rouwenhorst, G., (2005). Long-term global market correlations. Journal of Business, 78 (1), 1–38.

22. Hartmann, P., Straetmans, S., & de Vries, C.G. (2001). Asset market linkages in

crisis periods. Tinbergen Institute Discussion Paper, TI 2001-71/2.

23. Investment Technology Group. (2009). 2009 Global index transaction costs. (New York, USA). Consulted on 10 May 2015 via http://www.itg.com.

24. Jin, L., & S. Myers. (2006). Around the World: New Theory and New Tests. Journal

of Financial Economics, 79 ,257-292.

25. Knif, J., Kolari, J., & Pynnönen, S. (2007). What drives correlation between stock market returns? IMFWorking Paper WP/07/157.

26. Lillo, F., Bonanno, G., & Mantegna, R.N. (2001). Variety of stock returns in normal

and extreme market days: the August 1998 crisis, Proceedings of Empirical Science of Financial Fluctuations. Econophysics on the Horizon, Edited by H. Takayasu.

27. Longin, F., & Solnik, B. (1995). Is the correlation in international equity returns

constant: 1960–1990? Journal of International Money and Finance, 14 (1), 3–26.

28. Longin, F., & Solnik, B. (1999). Correlation structure of international equity markets during extremely volatile periods. Le Cashiers de Reserche, HEC Paris, 646.

56

29. Longin, F., & Solnik, B. (2001). Extreme correlation of international equity

markets. The Journal of Finance, 56, 649-675.

30. Markowitz, H. (1952). Portfolio Selection. Journal of Finance, 7, 77–91.

31. Malevergne, Y., & Sornette, D. (2006). Investigating extreme dependences: concepts and tools, Extreme Financial Risks (From dependence to risk management). Springer, Heidelberg.

32. Malkiel, B. G. (2002). How Much Diversification Is Enough? In Equity Portfolio

Construction.

33. Marshal, R., & Zeevi, A. (2002). Beyond correlation: extreme co-movements between financial assets. Working Paper, Columbia Business School.

34. Maskawa, J., & Souma, W. (2010). Large correlations as a signal of instability in

stock market.

35. Meric, I., &Meric, G. (1997). Co-movements of European equity markets before and after the 1987 crash. Multinational Finance Journal, 1, 137-152.

36. Meric, I., Kim, S., Kim, J.H., & Meric, G. (2008). Co-movements of U.S., U.K., and

Asian stock markets before and after September 11, 2001. Journal of Money, Investment and Banking, 3, 47-57.

37. Reigneron, P. A., Allez, R., & Bouchaud, J.P. (2011). Principal regression analysis

and the index leverage effect. arXiv:1011.5810.

38. Reilly. F. K. (1985). Investment Analysis and Portfolio Management. (2nd ed.). San Francisco: Dryden Press.

39. Sandoval L., & De Paula Franca I. Jr. (2012). Correlations of financial markets in

times of crisis. Physica A, 391, 187-208.

40. Schmidt M., (2011). Consulted on 16 april 2015, via http://www.investopedia.com/articles/economics/12/countries-in-piigs.asp

41. Sharpe, W. (1966). Mutual Fund Performance. Journal of Business, 39, (2), 119-

138.

42. Solnik, B., Boucrelle, C., & Le Fur,Y. (1996). International market correlation and volatility. Financial Analysts Journal, 52, 17-34.

57

43. Statman, Meir. (1987). How Many Stocks Make a Diversified Portfolio? Journal of Financial and Quantitative Analysis, 22, 353-363.

44. Statman, Meir. (2002). How much diversification is enough? Working Paper,

Santa Clara University.

45. Stevenson, R. A., & Jennings, E. H. (1984). Fundamentals of Investments (3rd ed.). San Francisco: West Publ. Co.

46. Upson, Roger, B., Paul, F., Jessup, & Matsumoto, K. (1975). Portfolio

Diversification Strategies. Financial Analysts Journal, 3 (1), 86-88.

47. Wei, S. X., & Zhang, C. (2006). Why Did Individual Stocks Become More Volatile? Journal of Business, 79, 259-292.

48. Xu, Y, & Malkiel, B. (2003). Investigating the Behavior of Idiosyncratic Volatility.

Journal of Business, 76, 613-644.

49. Zhou, Y. (2014). Number of Stocks in Portfolio and risk reduction. University of California, Los Angeles, USA.

58

Appendices

59

Appendix 1: Exploration of idiosyncratic risk

60

Appendix 2: Unemployment rate Europe different countries. S_ADJ Not seasonally adjusted data

AGE Total SEX Total

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Belgium 8,5 8,3 7,5 7,0 7,9 8,3 7,2 7,6 8,4 8,5 Bulgaria 10,1 9,0 6,9 5,6 6,8 10,3 11,3 12,3 13,0 11,4 Czech Republic 7,9 7,1 5,3 4,4 6,7 7,3 6,7 7,0 7,0 6,1 Denmark 4,8 3,9 3,8 3,4 6,0 7,5 7,6 7,5 7,0 6,6 Germany 11,2 10,1 8,5 7,4 7,6 7,0 5,8 5,4 5,2 5,0 Estonia 8,0 5,9 4,6 5,5 13,5 16,7 12,3 10,0 8,6 7,4 Ireland 4,4 4,5 4,7 6,4 12,0 13,9 14,7 14,7 13,1 11,3 Greece 10,0 9,0 8,4 7,8 9,6 12,7 17,9 24,5 27,5 26,5 Spain 9,2 8,5 8,2 11,3 17,9 19,9 21,4 24,8 26,1 24,5 France 8,9 8,8 8,0 7,4 9,1 9,3 9,2 9,8 10,3 10,3 Croatia 13,0 11,6 9,9 8,6 9,2 11,7 13,7 16,0 17,3 17,3 Italy 7,7 6,8 6,1 6,7 7,7 8,4 8,4 10,7 12,1 12,7 Cyprus 5,3 4,6 3,9 3,7 5,4 6,3 7,9 11,9 15,9 16,1 Latvia 10,0 7,0 6,1 7,7 17,5 19,5 16,2 15,0 11,9 10,8 Lithuania 8,3 5,8 4,3 5,8 13,8 17,8 15,4 13,4 11,8 10,7 Luxembourg 4,6 4,6 4,2 4,9 5,1 4,6 4,8 5,1 5,9 5,9 Hungary 7,2 7,5 7,4 7,8 10,0 11,2 11,0 11,0 10,2 7,7 Malta 6,9 6,8 6,5 6,0 6,9 6,9 6,4 6,3 6,4 5,9 Netherlands 5,9 5,0 4,2 3,7 4,4 5,0 5,0 5,8 7,3 7,4 Austria 5,6 5,3 4,9 4,1 5,3 4,8 4,6 4,9 5,4 5,6 Poland 17,9 13,9 9,6 7,1 8,1 9,7 9,7 10,1 10,3 9,0 Portugal 8,8 8,8 9,2 8,7 10,7 12,0 12,9 15,8 16,4 14,1 Romania 7,1 7,2 6,4 5,6 6,5 7,0 7,2 6,8 7,1 6,8 Slovenia 6,5 6,0 4,9 4,4 5,9 7,3 8,2 8,9 10,1 9,7 Slovakia 16,4 13,5 11,2 9,6 12,1 14,5 13,7 14,0 14,2 13,2 Finland 8,4 7,7 6,9 6,4 8,2 8,4 7,8 7,7 8,2 8,7 Sweden 7,7 7,1 6,1 6,2 8,3 8,6 7,8 8,0 8,0 7,9 United Kingdom 4,8 5,4 5,3 5,6 7,6 7,8 8,1 7,9 7,6 6,1 Iceland 2,6 2,9 2,3 3,0 7,2 7,6 7,1 6,0 5,4 5,0 Norway 4,5 3,4 2,5 2,5 3,2 3,6 3,3 3,2 3,5 3,5 Turkey 9,5 9,0 9,1 10,0 13,0 11,1 9,1 8,4 9,0 9,9 United States 5,1 4,6 4,6 5,8 9,3 9,6 8,9 8,1 7,4 6,2 Japan 4,4 4,1 3,8 4,0 5,1 5,0 4,6 4,3 4,0 3,6

Source: Eurostat

61

Appendix 3: GDP to debt ratio Europe for different countries

2005 2006 2007 2008 2009 2010 2011 2012 2013 Belgium 94,8 90,8 86,9 92,2 99,3 99,6 102,1 104 104,5 Bulgaria 27,1 21,3 16,6 13,3 14,2 15,9 15,7 18 18,3 Czech Republic 28 27,9 27,8 28,7 34,1 38,2 41 45,5 45,7 Denmark 37,4 31,5 27,3 33,4 40,4 42,9 46,4 45,6 45 Germany 66,8 66,3 63,5 64,9 72,4 80,3 77,6 79 76,9 Estonia NA NA NA NA NA 6,5 6 9,7 10,1 Ireland 26,2 23,8 24 42,6 62,2 87,4 111,1 121,7 123,3 Greece : 103,4 103,1 109,3 126,8 146 171,3 156,9 174,9 Spain 42,3 38,9 35,5 39,4 52,7 60,1 69,2 84,4 92,1 France 67 64,2 64,2 67,8 78,8 81,5 85 89,2 92,2 Croatia 38,6 36,1 34,4 36 44,5 52,8 59,9 64,4 75,7 Italy 101,9 102,5 99,7 102,3 112,5 115,3 116,4 122,2 127,9 Cyprus 63,3 58,9 53,7 44,7 53,5 56,5 66 79,5 102,2 Latvia 11,7 9,9 8,4 18,6 36,4 46,8 42,7 40,9 38,2 Lithuania 18,3 18 16,7 15,4 29 36,3 37,3 39,9 39 Luxembourg 6,3 7 7,2 14,4 15,5 19,6 18,5 21,4 23,6 Hungary 60,8 65 65,9 71,9 78,2 80,9 81 78,5 77,3 Malta 70,1 64,6 62,4 62,7 67,8 67,6 69,8 67,9 69,8 Netherlands 49,4 44,9 42,7 54,8 56,5 59 61,3 66,5 68,6 Austria 68,3 67 64,8 68,5 79,7 82,4 82,1 81,7 81,2 Poland 46,7 47,1 44,2 46,6 49,8 53,6 54,8 54,4 55,7 Portugal 67,4 69,2 68,4 71,7 83,6 96,2 111,1 124,8 128 Romania 15,7 12,3 12,7 13,2 23,2 29,9 34,2 37,3 37,9 Slovenia 26,3 26 22,7 21,6 34,5 37,9 46,2 53,4 70,4 Slovakia 33,8 30,7 29,8 28,2 36 41,1 43,5 52,1 54,6 Finland 40 38,2 34 32,7 41,7 47,1 48,5 53 56 Sweden 48,2 43,2 38,2 36,8 40,3 36,7 36,1 36,4 38,6 United Kingdom 41,5 42,5 43,6 51,6 65,9 76,4 81,9 85,8 87,2 Iceland NA NA NA NA NA NA NA NA NA Norway NA NA NA NA NA 41,7 27,5 29,2 29,3 Switzerland NA NA NA NA NA NA NA NA NA

Source: Eurostat

62

Appendix 4: GDP Euro-countries 2005 - 2014

Source: Eurostat

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Belgium 311.150 327.368 345.069 355.066 349.703 365.747 379.991 388.254 395.262 402.270 Bulgaria 23.582 26.827 31.883 36.450 36.078,40 36.764 40.103 40.926 41.047 42.010 Czech Republic 109.394 123.743 138.004 160.961 148.357,40 156.369 163.579 160.947 157.284 154.929 Denmark 212.906 225.592 233.439 241.087 230.213,30 241.516 246.074 250.786 252.938 256.937 Germany 2.297.820 2.390.200 2.510.110 2.558.020 2.456.660,00 2.576.220 2.699.100 2.749.900 2.809.480 2.903.790 Estonia 11.260 13.518 16.241 16.511 14.138,20 14.709 16.403 17.636 18.738 19.526 Ireland 169.152 183.759 196.748 186.870 168.114,00 164.928 171.042 172.754, 174.791 185.411 Greece 199.152 217.830 232.831 242.096 237.431,00 226.209 207.751 194.203 182.438 179.080 Spain 930.566 1.007.974 1.080.807 1.116.207 1.079.034,00 1.080.913 1.075.147 1.055.158 1.049.181 1.058.469 France 1.771.978 1.853.267 1.945.670 1.995.850 1.939.017,00 1.998.481 2.059.284 2.091.059 2.113.687 2.142.022 Croatia 36.508 40.197 43.925 48.129 45.090,70 45.004 44.708 43.933 43.561 43.084 Italy 1.490.409 1.549.188 1.610.304 1.632.933 1.573.655,10 1.605.694 1.638.857 1.615.131 1.609.462 1.616.047 Cyprus 14.906 16.097 17.406 18.768 18.423,10 19.062 19.486 19.411 18.118 17.506 Latvia 13.733 17.239 22.623 24.403 18.816,10 18.015 20.197 22.217 23.265 24.059 Lithuania 21.002 24.079 29.040 32.696 26.934,80 28.001 31.247 33.314 34.955 36.287 Luxembourg 29.771 33.303 35.953 37.522 36.093,90 39.370 42.410 43.812 45.288 NA Hungary 90.027 90.950 101.240 107.150 93.371,70 97.814 100.350 98.699 100.536 103.302 Malta 5.142 5.386 5.757 6.128 6.138,60 6.599 6.902 7.226 7.571 7.961 Netherlands 540.656 573.444 608.729 635.794 617.650,00 631.512 642.929 640.644 642.851 655.375 Austria 253.009 266.478 282.346 291.930 286.188,40 294.207 308.675 317.213 322.594 328.996 Poland 244.822 273.417 313.654 363.691 314.689,40 359.816 377.028 386.143 395.962 412.189 Portugal 158.652 166.248 175.467 178.872 175.448,20 179.929 176.166 169.668 171.211 174.384 Romania 80.225 98.418 125.403 142.396 120.409,20 126.746 133.305 133.806 144.282 150.664 Slovenia 29.235 31.561 35.152 37.951 36.166 36.219 36.868 36.006 36.144 37.246 Slovakia 39.335 45.435 56.063 65.679 63.798 67.204 70.159 72.184 73.593 75.214 Finland 164.387 172.614 186.584 193.711 181.029 187.100 196.869 199.793 201.995 204.015 Sweden 313.218 334.876 356.434 352.317 309.678 369.076 404.945 423.340 436.342 429.468 United Kingdom 1.940.128 2.059.064 2.164.064 1.907.212 1.663.573 1.816.615 1.863.940 2.041.491 2.017.193 2.222.361 Iceland 13.524 13.675 15.676 10.761 9.182 10.013 10.551 11.076 11.583 12.853 Norway 248.332 275.289 293.128 316.813 278.386 323.587 358.248 396.678 393.098 377.224 Switzerland 327.755 342.123 348.864 376.326 388.781 439.140 501.642 518.204 516.068 NA Macedonia 5.032 5.472 6.095 6.744 6.765 7.108 7.550 7.585 8.122 NA Serbia 21.103 24.434 29.451 33.704 30.654 29.766 33.423 31.683 34.262 NA

63

Appendix 5: S&P Europe 350 – Benchmark 4%, 6% and 8%

64

Appendix 6: PIIGS countries – Benchmark 4%

65

Appendix 7: PIIGS countries – Benchmark 6%

66

Appendix 8: PIIGS countries – Benchmark 8%

67

Appendix 9: Better performing countries – Benchmark 4%

68

Appendix 10: Better performing countries – Benchmark 6%

69

Appendix 11: Better performing countries – Benchmark 8%

70

Appendix 12: Sectors – Benchmark 4%

71

Appendix 13: Sectors – Benchmark 6%

72

Appendix 14: Sectors – Benchmark 8%

73

Appendix 15: Financial crisis – Benchmark 4%

74

Appendix 16: Financial crisis – Benchmark 6%

75

Appendix 17: Financial crisis – Benchmark 8%

76

Appendix 18: Average Transaction Costs MSCI World Index by Sector in 2009