Correlations across idiosyncratic risks in the Dutch stock market

Faculty of Economics and Business

Correlations across idiosyncratic risks in the Dutch

stock market

Menno Aardema

11033088

Economics and Business Economics

Supervisor:

Dhr. dr. M.I. Dröes

Study year 2017-2018

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Abstract

In this research paper, the correlations across idiosyncratic risks in Sharpe’s single-index model are investigated. It is the first research that analyses these correlations and provides new insights in the implications of using this model in portfolio selection. The research question is therefore: Are there significant correlations of idiosyncratic risks in Sharpe’s single-index model? The most important stocks on the Dutch stock market are used in the research, which yields 300 correlations. More than half (151) of them turned out to be statistically significant, which suggests that the assumption of uncorrelated residuals does not hold in practice. Also, the correlations are used in a portfolio construction procedure to illustrate the consequences of the interrelationships, with regard to portfolio weights, return and risk. The result was that idiosyncratic risk of the portfolio decreased, where overall risk and return increased slightly. However, these increases were not proportional, which implies that with this sample, potential diversification value is missed when the original single-index model is used. Finally, the robustness of the results on the single-index model is assessed by performing the same correlation analysis on several asset pricing models. This led to the same 151 significant correlations for the three-factor model of Fama and French, 150 of which had the same sign as in the single-index model. The analysis on the four-factor model of Carhart resulted in 147 significant correlations, and the five-factor model of Fama and French produced 142 significant correlations. The findings of the research suggest that idiosyncratic shocks might be more important than literature presumes, and demonstrate an important drawback of using the single-index model in the portfolio construction process.

Statement of originality

This document is written by Student Menno Aardema who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Content

1. Introduction p. 6

2. Theory and Literature review p. 8

2.1 Theoretical background p. 8

2.2 Idiosyncratic risk in portfolio theory p. 10

2.3 Potential factors affecting correlation p. 12

2.3.1 Competitive relationships p. 12

2.3.2 Supplier networks p. 13

2.3.3 Omitted variables p. 13

2.3.4 Other factors p. 15

3. Data and Methodology p. 17

3.1 Datasets p. 17

3.1.1 Descriptive statistics single-index model p. 17

3.1.2 Descriptive statistics extended models p. 20

3.2 Research method p. 21

3.3 The analytical example p. 24

3.4 Hypothesis and expectations p. 25

4. Results p. 27

4.1 Analysis of correlations p. 27

4.2 Analytical example with sampled correlations p. 31

4.3 Robustness analysis p. 35

4.3.1 The three-factor model of Fama and French p. 35

4.3.2 The four-factor model of Carhart p. 36

4.3.3 The five-factor model of Fama and French p. 36

5. Conclusion and Discussion p. 39

6. References p. 41

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1. Introduction

Since the 1950’s, research in finance is concerned with portfolio selection. Several models, like the selection model of Markowitz (1952), have addressed the issue of diversification and the construction process of a stock portfolio. Just like the single-index model of Sharpe, that was developed in 1963 to be a simplification of the then existing procedures. This model is based on the assumption that the idiosyncratic parts that explain stock return are not correlated across different stocks. More recent literature demonstrates that these idiosyncrasies are getting bigger and more important in explaining stock return (Campbell et al., 2001). Also, there are studies that form an indication that the interrelationships between these idiosyncratic shocks might be getting more important over the last years. So, it is possible that the assumption of uncorrelated idiosyncratic volatilities was valid at the time when Sharpe developed his model, but it may be that this is no longer the case. However, as the literature section of this thesis suggests, the possible existence of idiosyncratic risk correlation has never been tested, neither in the 1960’s nor in the last years. These potential correlations could have serious implications on portfolio selection, and could suggest that the single-index model is more attractive in theory than in practice. Persistent idiosyncratic correlation would be a drawback for the single-index model, as it could lead to the underestimation of risk or the missing out on diversification potential.

The aim of this research is to investigate the presence of correlations across idiosyncratic risks and to illustrate the consequences thereof in an analytical example. This leads to the following research question:

Are there significant correlations of idiosyncratic risks in Sharpe’s single-index model?

In order to answer this research question, data on stock prices of the companies in the Amsterdam Exchange Index (AEX) is used. Companies from the Netherlands are used because I expect drivers of idiosyncratic risk correlation to be present here and the computations are manageable with several hundreds of correlations. If, for example, the stocks in the S&P500 were researched, more than 120.000 correlations should be studied. In total, 29.867 data points on stock prices are retrieved from the network Datastream. To analyse the correlations across the stocks, the residuals are collected by performing regressions on stock returns. The obtained correlations are used in a portfolio selection

7 procedure to analyse the consequences of idiosyncratic risk correlation. This will be done by using the procedure developed by Treynor and Black (1973).

The consideration whether to use Markowitz’s model or the single-index model is as old as the models themselves, because both have their own benefits and drawbacks. The model of Markowitz is sophisticated, but faces a lot of estimation risk when dealing with large input lists. The single-index model offers a solution by providing a less complicated computation of portfolio characteristics, but does this by simplifying reality even more. This paper contributes by analysing this disadvantage of the single-index model as it investigates the correlations of idiosyncratic risks across different companies and gives new insight in the implications of using the single-index model in portfolio construction.

Of all the 300 investigated correlations, 151 turned out to be statistically significant. There were 78 negative correlations and 73 positive correlations. These interrelations could be the result of the factors mentioned in the literature section of this thesis, such as competitive relationships or supplier networks. The obtained results can have implications in practice, as these could form an argument in deciding whether to make use of the single-index model or not. To check the robustness of the research, the idiosyncratic correlations are also analysed for the three-factor model of Fama and French, the four-factor model of Carhart, and the five-factor model of Fama and French. These models add systematic five-factors to the CAPM. The same 151 statistically significant correlations were found for the three-factor model, of which 150 had the same sign as the corresponding correlations in the single-index model. The four-factor model led to 147 statistically significant correlations, and the five-four-factor model resulted in 142 significant correlations.

This paper is structured in the following way. In Section 2, a brief theoretical background of the single-index model is provided, together with an overview of existing literature. This section provides possible explanations for the results we might find. After that, the research method is presented. Here, the data source will be described and the used formulas are indicated. In Section 4, the results of the research are given and discussed, together with an analysis on the idiosyncratic risk correlation in other asset pricing models. In addition, the results following from the Treynor-Black procedure are presented here. The paper ends with some summarizing statements. Also, the limitations of this research and suggestions for further research are mentioned.

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2. Theory and Literature review

In this section, a theoretical overview is given to illustrate the fundamental elements of the single-index model. After this, there will be some subsections about the existing research on idiosyncratic volatility, followed by possible explanations for idiosyncratic risk correlation.

2.1 Theoretical background

The development of theories on portfolio selection started with research from Markowitz (1952). However, in the following decades, several problems and disadvantages came forward on his method of portfolio selection. The idea behind the model of Markowitz is to include all covariances of stocks in a portfolio, to get a representative figure of the portfolio variance and therefore the perception of risk in portfolio management. The main problem here is that for portfolios with large input lists, the construction of the optimal portfolio gets computationally intense. On top of that, all the estimates can lead to serious estimation risk, leading to wrong results. Namely, the estimated moments of stock returns are used and not the true population values. This could even lead to a negative portfolio variance. So, the advantage of including all estimates of the covariances can be illusory (Bodie, Kane, & Marcus, 2014). Finally, Konno and Yamazaki (1991) indicate that the use of the Markowitz model is accompanied by severe management costs.

In order to simplify the portfolio construction process, Sharpe (1963) came up with his diagonal model, or the single-index model. His main idea was to decompose risk into a systematic and an idiosyncratic part, which greatly simplifies the computation of the covariances. On top of that, far less parameters need to be estimated compared to the model of Markowitz, which reduces the estimation risk and leads to the fact that less computational power is needed to optimise portfolios. Sharpe introduced the single-index model in a theoretical equation, presented in equation (1):

Ri,t = αi,t + βi,tIt + εi,t , (1)

where Ri is the excess return, αi and βi are parameters, I is an index variable and εi is the residual. The residuals are assumed to have an expected mean of zero and to be independent and identically distributed. The subscript i indicates that the variables concern the individual security i.

9 The single-index model is based on some assumptions, like a uniform holding period for every stock and homogenous expectations. However, the assumption of the uncorrelated unsystematic shocks is the most essential one. This is crucial to simplify the notion of risk, as illustrated in equation (2) and (3).

𝜎

= 𝛽

𝜎

+ 𝜎

(𝑒

)

with

𝜎

(𝑒

)

∑

𝑥

𝜎

(𝑒

)

As one can see in the equations above, the covariances are neglected in the idiosyncratic part of the portfolio risk and they are simplified in the systematic part of the portfolio risk. It is also implied that the last term of the portfolio risk is diversifiable, as the number of stocks in the portfolio increases. The estimation of the covariance of two stock returns is simplified to a multiplication of the companies’ betas and the market risk, as illustrated in equation (4).

𝐶𝑜𝑣(𝑟_𝑖, 𝑟_𝑗) = 𝐶𝑜𝑣(𝛽_𝑖𝑟_𝑚+ 𝑒_𝑖, 𝛽_𝑗𝑟_𝑚+ 𝑒_𝑗) = 𝛽_𝑖𝛽_𝑗𝜎_𝑚2 ₍₄₎

In this way, not 2n + (n2-n)/2 estimates are needed to construct the optimal portfolio, as in the Markowitz model, but just 3n +2 estimates. In Section 3, I propose a different formula for the variance of the residuals, illustrated in equation (14).

It is not the case that the simplification provided by Sharpe leads to the fact that this model is the good one, and Markowitz’ model is the bad one. Sharpe’s modification leads to less estimation risk, but part of the covariance is ignored. However, it is often suggested that the benefits of the single-index model are most prevalent when the amount of stocks in the portfolio (n) is large, which is the case with the portfolio sizes that portfolio theory suggests.

Although the single-index model is developed many years ago, it still holds an important role in finance, which means that research on it is still relevant. Due to its simplicity, it is still commonly used in portfolio construction, but with caution (Bodie et al., 2014). Portfolio managers often use the single-index model to optimise portfolios and to obtain a convenient benchmark in portfolio analysis, but they do take into account that the single-index model might not provide a perfect representation of return. This is why multi-factor models that, for example, take into account the interest rate and the unanticipated growth of GDP are also used in practice, especially to construct hedging portfolios. On top of that, the single-index model is used by private investors whose main goal is to diversify risk.

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The simplicity of the model offers great help to the increasing amount of private investors, who often lack experience and professional knowledge to invest in the equity market (Nandan and Strivastava, 2017). Also, the related CAPM is used in reality, despite the criticism the model gets (Grinold, 1993). This model has applications in project finance and performance evaluation, among others.

Besides its practical relevance, the single-index model has great theoretical importance. This started with the development of the CAPM and subsequently led to many other extensions, like the model of Bilbao et al. (2006) that expands the original single-index model by introducing expert betas. The single-index model also created new insights in portfolio theory, which then led to the development of construction processes, like the mentioned procedure of Treynor and Black (1973). But also more recently, the single-index model still forms the basis of numerous studies, which primarily consist of extensions of the single-index model and empirical studies looking at the validity of the model under certain conditions. These studies aim to get a better insight in the debate whether to use the single-index model in portfolio theory, or to use another model.

2.2 Idiosyncratic risk in portfolio theory

In portfolio theory, there is still room for research on idiosyncratic volatility. The only attention it got is in studies that focus on the role of unsystematic risk in relation to stock prices. However, the results and conclusions of these studies are conflicting. Goyal and Clara (2003) find that stock variance is positively correlated with returns. They study the relationship between stock variance and returns, where this variance is almost exclusively idiosyncratic risk in their research. Namely, they find that the average volatility consists largely out of idiosyncratic risk. Just like Jiang and Lee (2006), who find that more than 85% of the volatility in stock return is unsystematic. Studies on this risk-return relationship were also conducted in the early days of portfolio theory, for example by Douglas (1969). He found that the residual variance is very significant in explaining stock returns. More recently, such a significant relation between idiosyncratic risk and stock returns was also found (Storesletten, Telmer, & Yaron, 2010). These studies indicate that idiosyncratic risks and asset prices are positively related, but there are also studies that find a negative relation of these variables, like Fama and MacBeth (1973), and Ang et al. (2006).

Although the evidence of a positive relationship between idiosyncratic risk and stock returns seems more repelling than a negative relationship, clear-cut evidence has not been

11 provided yet. What all these studies have in common is that the correlation across idiosyncratic risks of companies is not researched. In almost all of the studies mentioned above, the assumption of no correlation across idiosyncratic risks is only briefly stated. Even in the paper where the single-index model was introduced, Sharpe (1963) only mentioned the assumption of uncorrelated idiosyncratic risks, without going into detail about the validity thereof. Only a few studies give some remarks on the assumption of uncorrelated residuals. Komo and Yamazaki (1991) call it unrealistic and Malkiel and Xu (2002) state that the assumption will probably not hold in practice. Though, these statements are not backed-up by evidence.

An analysis on idiosyncratic risk is only given by Campbell et al. (2001). They acknowledge the importance of idiosyncratic risk, and analyse this unsystematic risk over-time and on an industrial level, but did not research correlations across stocks. Although the research on portfolio theory has developed in the last decades, they remark that there is “surprisingly little empirical research” (Campbell et al., 2001, p.2) on idiosyncratic volatility. This might be considered strange, because of the importance this kind of risk could have in practice. Namely, individual investors do not diversify as much as portfolio theory suggests. This leads to a greater exposure to idiosyncratic volatility. More than 75% of the investors do not hold more than three stocks (Fu, 2009), and even if investors hold more than twenty stocks to fully diversify the idiosyncratic risks, it is not certain that all unsystematic risk is avoided, because of the differing levels of idiosyncratic risks of the stocks in the portfolio. For this reason, not the conventional twenty random stocks must be acquired for complete diversification, but rather fifty stocks (Campbell et al., 2001). On top of that, high unsystematic volatility can lead to considerable pricing errors of assets (Shleifer and Vishny, 1997).

So, these two arguments imply that large firm-specific risks could have serious implications in reality, where theory on diversification and idiosyncratic risk would ignore this. And indeed, we see that the idiosyncratic shocks to firms are increasing greatly over the last years. This was the case in a sample period of 1962 to 1997, but also more recent research validated that the idiosyncratic parts of risk are increasing and getting more volatile (Zhang et al., 2016). Noteworthy is that the increase in unsystematic volatility is not accompanied by the same increase in market volatility. This is in line with earlier research that found that the market volatility remained unchanged or only increased by a small amount, where the yearly increase in idiosyncratic volatility turned out to be more than six percent (Irvine and Pontiff, 2008). So, the complete market has not incurred substantially more risk, but the firm-specific

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volatility did increase in the recent years. Not only is there evidence for an upward trend in unsystematic shocks over time, the volatility of the shocks is also increasing (Fu, 2009). The residuals are getting larger and more volatile, which both contributes to the increasing idiosyncratic volatility.

2.3 Potential factors affecting correlation

The previous subsection suggested that idiosyncratic volatilities are getting more and more important over time, which makes research on them more relevant and raises the question whether idiosyncrasies might be correlated. However, research on portfolio theory ignores these possible correlations, so in the following subsections, research on risk management and strategic management is studied in trying to identify potential factors affecting the correlation of idiosyncratic volatilities across companies. These factors have not been tested to see whether they indeed have a significant impact on the correlation.

2.3.1 Competitive relationships

The first factor that could influence correlations across idiosyncratic volatilities of firms is competition. If there are two companies and one of them is hit by an idiosyncratic shock, it will be likely that the other company is affected as well when the two companies have a competitive relationship. The stock returns will probably be negatively related, which would have an impact on the covariance term in the variance of a portfolio. Gaspar and Massa (2006) researched interfirm-relationships and found that competition has increased in almost every industry. They claim that firms obtain less market power in general, due to a more competitive environment. These findings hold for developed countries, including the Netherlands. The increased competition can be seen as a cause for the increased volatility in idiosyncratic shocks (Fu, 2009). On top of that, Irvine and Pontiff (2008) argue that the amount of idiosyncratic risk of firms increases due to this increased competitive environment. They also confirm that the covariance will be negatively affected by this development. Though, this claim is of theoretical nature and is not tested by them. When negative correlations are found across competitive firms in this paper, this could form the confirmation of this claim.

So, these findings imply a negative relationship between firms’ returns. Because of this, diversification potential is missed out when the single-index model is used to decompose risk. The covariance term is limited to the form in equation (4), and the correlation of the

13 idiosyncratic shocks is ignored. In our sample, some competitive relationships might be expected, like between ABN and ING.

2.3.2 Supplier networks

At the same time, we see the increase of another interfirm-relationship, which could also affect correlation across idiosyncratic volatilities. In the last couple of years, firms engage more and more in supplier networks (Hallikas and Lintukangas, 2016). The awareness of firms about the importance of acquiring goods they cannot produce efficiently themselves has increased in the recent years. Therefore, the traditional role of acquiring resources has changed. The increasing trend in purchasing would cause a positive relationship in stock returns. Like in the previous subsection, an adjustment to the covariance term would be in place, but now it is a positive one. Namely, if the supplier is idiosyncratically hit, the purchasing firm is hit as well. The other way around is also possible. This increases the risks of the companies. Hallikas and Lintukangas (2016) find that a positive correlation in these risks is the highest for companies that intensively purchase from suppliers. In other research, it was already found that these dependencies across firms are deepening (Hallikas et al., 2014): companies are getting more and more reliant on a small number of suppliers. Organizations may not use an increasingly amount of suppliers, but the ties with three or four suppliers are getting more substantial.

So, a supplier network might be a success factor for companies, but the deepening relationship across firms leads to more exposure to the risks of other organizations. This could be an indication for idiosyncratic risk correlation. In addition to the supplier network, the trend of increasing strategic alliances could also impact idiosyncratic risk correlation.

Blome and Schoenherr (2011) indicate that the increasing correlation of supply risk is probably less for service firms, because they generally depend less on suppliers. So, also in the sample of this research, we would probably only observe positive correlations across manufacturing firms with a buyer-supplier relationship, like between Altice and Gemalto.

2.3.3 Omitted variables

When a single-index model is used for regressions, it is possible that residuals of this regression are correlated across firms, because of omitted variables. After all, only one independent variable is used. This means that any factor affecting the prices of stocks, other than the excess market return, is represented by the residuals. This could lead to correlation if certain stocks are affected by the same omitted factor. However, this problem is not only present in the single-index model. The more general problem is that the systematic factors

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included in asset pricing models are not the only factors affecting stock prices (Chen, Roll, & Ross, 1986). It is simply unknown how much, and which systematic factors are priced in the market. On top of that, the relation between the macro-economy and financial markets is not clear-cut, which means that we often do not exactly know to what extent the systematic factors indeed have an impact (Chen et al., 1986).

So, if we apply this remark on the single-index model, we could say that if we use the residuals of the CAPM to measure unsystematic volatility, we can only state that idiosyncratic risk is measured in the context of this CAPM (Malkiel and Xu, 2002). Omitted variable bias would be a major drawback in research on idiosyncrasies, including this research, but we still see a lot of studies using the CAPM in determining idiosyncratic risk. This concerns both recent and older research: Zhang et al. (2016), Huang et al. (2009), and Gaspar and Massa (2006) all base their studies on the residuals from the CAPM. Malkiel and Xu (2002) justified the use of these CAPM residuals by showing that the relationship between idiosyncratic volatility and stock return did not change when there was controlled for more factors. So, when residuals of models with more variables were used, the results did not change significantly. This suggests that the omitted factors that may be present do not significantly affect idiosyncratic volatility, when this volatility is measured using the CAPM residuals. This is probably because the macro-factor captures much of the systematic influences, so that indeed the true idiosyncratic parts remain, represented by the residuals. This is favourable for this research, since the correlation of just these idiosyncratic parts is studied.

On top of that, Jiang and Lee (2006) show that the use of CAPM residuals as proxy for idiosyncratic risk achieves the same results as using the three-factor model of Fama and French and the four-factor model of Carhart. Related to that, Spiegel and Wang (2007) also show that the definition of idiosyncratic volatility yields no different results in researching idiosyncratic volatility and stock returns: CAPM residuals indicate the same relationship as other definitions of idiosyncratic volatility.

Based on this subsection, the outcomes of this research are not expected to be affected by the omission of variables, if we work with the CAPM residuals to compute idiosyncratic risk correlation.

2.3.4 Other factors

The literature also provides other factors that could be an indication for idiosyncratic risk correlation. For example, the location of the companies (Emmons, 2004). When an idiosyncratic event occurs in a particular geographical area and a lot of companies would be

15 situated in the same place, for example in an industrial area, there will probably be correlation of idiosyncratic risk across the companies. The companies in the sample of this research are not located at the same location, however they are all established in the Netherlands, which is a relatively small area. Because all companies are situated in the same country, no idiosyncratic risk correlation will be generated due to correlation across countries. Namely, Bekeart et al. (2012) find that idiosyncratic risks are very much correlated across countries. This could be a substantial factor if stocks from around the world are analysed.

Bennett and Sias (2006) show that increases in unsystematic risk are related to the total market value of firms: companies with a low market capitalization have higher idiosyncratic risk, than companies with a high market value. In this research, the biggest companies on the Dutch stock market are analysed, which could mean that the idiosyncratic risk of those companies is increasing, which could cause some correlation.

In addition, Campbell et al. (2001) illustrate that the volatility of idiosyncratic shocks could increase due to the division of a conglomerate into more specialising entities. In our sample, we see such a break-up, with Philips and Philips Lightning. So if we observe a correlation in idiosyncratic risks between these organizations, this could be the result of increased volatilities in unsystematic shocks for both firms, after the split in 2015.

On top of that, due to the way of defining return by only using the stock prices, as illustrated in equation (5), there could arise correlation across idiosyncratic risks. Namely, if companies would distribute dividend, this could lead to a decline in stock price, which lowers their returns. So, we would probably see correlation across idiosyncratic risks then as well. This effect on the correlation of two companies will be most prevalent when the distributions of dividend take place at the same date. However, not all stocks from the sample pay dividend, and the twenty-two companies that do pay dividend, all showed different ex-dividend dates and pay-out dates, except for Philips and Philips Lightning.

Furthermore, correlation across idiosyncratic risks could be the result of neglecting a relatively new systematic factor, namely low-volatility. The market risk factor assumes higher return with higher risk, where low-volatility assumes the opposite: high return comes with low risk. Blitz and Van Vliet (2007) presented evidence that low-volatility stocks generate too high returns, after adjusting for risk. Likewise, Ang et al. (2006) find that stocks with high idiosyncratic volatility earn exceptionally low risk-adjusted returns. Jiang and Lee (2006) already showed that the volatility in stock return is mainly attributable to idiosyncratic risk. This means that the low-volatility factor can lead to idiosyncratic risk correlation. Namely, if two stocks experience low volatility, their exposure to this omitted systematic factor will

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probably be quite similar, which means that the residuals are affected in the same way. This can lead to correlation of their idiosyncratic risks. The correlations are analysed, also with regard to the low-volatility factor, and presented in Section 4.

Finally, the idiosyncratic risk correlation can be affected through the liquidity effect (Malkiel and Xu, 2001). The liquidity of a share measures how accessible trading in it is. The liquidity of companies is not included in the CAPM, so the idiosyncratic volatility could capture the liquidity effect. If several firms are more easy to trade, or difficult to trade, this could lead to correlation. However, research is not unanimous about the impact of liquidity on stock returns, and therefore on the residuals in this research, if the liquidity effect is left out (Spiegel and Wang, 2005).

The fact that the factors above are discussed less extensively, does not mean that they are not important. The impact on the correlation of idiosyncratic risk depends on the characteristics of the individual stocks.

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3. Data and Methodology

In this section, a description of the complete dataset for the single-index model will be given and the table with descriptive statistics is analysed. After that, this section provides a description of other datasets that were used in this research. Finally, the approach of the research is provided, including the used formulas.

3.1 Datasets

Four datasets will be used in this research, one with only stock prices and market index levels to perform the regressions in the single-index model, and the other three with additional factors, namely the SMB, HML, UMD, RMW and CMA factors. The datasets with these control variables will be used in the robustness analysis.

3.1.1 Descriptive statistics single-index model

In the research on correlations between residuals in the single-index model, data on stock prices will be used, which will be obtained from Thomson Reuters Datastream. This network is well-known in providing long-term data on stock prices. The daily index levels will also be collected from Datastream. The sample for my research will consist out of the twenty-five most prominent stocks included in the Dutch stock market, all represented in the AEX. A list of all the companies in the sample, and their industries, can be found in the Appendix, as Table A1.

The returns of the stocks are computed on a daily basis with a time horizon going from May 7th, 2013 to May 7th 2018. The selected time window is five years, which is determined after studying research from Ang and Kristensen (2012). They state that the typical time window with CAPM-regressions is five years, and they find that the optimal sample size lies between 1.5 and 8.5 years. This means that we obtain enough data points to perform a proper regression with a sample size of five years. So, the dataset of this research consists of 1310 data points of stock prices per company, and the daily levels of the market index over this five-year period. Five companies in the sample went public during the selected time window, so these have less data points available. The returns are calculated in the following way:

𝐴𝑑𝑗.𝐶𝑙𝑜𝑠𝑒 𝑥_𝑡−𝐴𝑑𝑗.𝐶𝑙𝑜𝑠𝑒 𝑥_𝑡−1

18 0 500 1000 1500 2000 2500 3000 3500 4000 0 20 40 60 80 ₁00 ₁20 ₁40 0₁6 ₁80 ₂00 0₂2 ₂40 ₂60 Fre q u ency Stock price Frequency

where x stands for a company and t for time. Important to note is that the returns are solely calculated on the basis of stock prices, and no dividends or other distributions are involved. The descriptive statistics of the dataset can be found in Table 1. The data on the level of the market index is included in the table below, but in a separate row as it is not a stock price.

There are twenty-five more observations on stock prices than on daily returns, which is the result of the way of calculating the returns, illustrated in equation (5). In Table 1, we see that the maximum and minimum stock prices are very divergent. The minimum is less than one standard deviation from the mean, but the maximum stock price lays almost five standard deviations from the mean. This alone does not suggest a positive skewness in the distribution of the sampled data, but it could be an indication. Indeed, if we look at the distribution of stock prices, we see such a pattern in Figure 1.

Figure 1. Distribution of sampled stock prices.

Note: This figure shows the distribution of the stock prices of the sample, going from 2013 to

2018.

Table 1. Descriptive statistics of base dataset.

Observations Mean (Mean annualized return) Standard

deviation Maximum Minimum

Stock price 29867 € 43.08 € 44.47 € 260.45 € 1.39 Daily stock return 29842 0.0549% (10.436%) 1.6959% 34.5677% -22.5866% Market index 1310 453.84 56.47 570.82 332.25 Daily market return 1309 0.0391% (9.370%) 0.9871% 4.0511% -5.7041%

Note: This table contains descriptive statistics on the dataset for the single-index model, going from 2013 to 2018. This dataset includes stock prices and market index levels, which are transformed to returns. The returns are annualized by assuming 252 trading days per year.

19 -8.00% -6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00% 7-5-2013 7-5-2014 7-5-2015 7-5-2016 7-5-2017 7-5-2018 Ret urn Date

Average daily return

When looking at the descriptive statistics of the stock returns, the mean is more in the middle of the minimum and the maximum of daily returns, with a slight tendency towards the minimum. The annualized mean return of all the stocks was 10.436%. Over the years, the average daily returns have been stable, with some exceptions. This can be seen in Figure 2, where the average return of all stocks in the sample is computed per day.

Figure 2. The average daily stock returns over the sample period.

Note: This graph contains the development of the daily stock returns, going from May 7th 2013, to May 7th 2018.

The highest average daily return can be found on August 25, 2015. The lowest average daily return was on June 24, 2016, when the twenty-five stocks lost 5.976% of their stock value on average. This is not surprising since the results of the voting on the Brexit were presented that day. In the days after this, the return continued to be very low.

Finally, the maximum and minimum stock prices and daily returns are mentioned in Table 1. The minimum stock price in the sample was €1.39 of KPN. This seems low, but the maximum share price of KPN in the sample period was just €3.69. The table illustrates that firms could lose more than 20% of their stock value in just one day. On the other hand, the maximum daily return in the sample was 34.568%. Compared to the minimum and maximum daily stock return, the range of daily market return is limited. Also, the average daily market return is lower than the average daily stock return. This can be explained by the fact that the market return has a lower standard deviation.

The table with descriptive statistics of all the individual stocks, including the market index, can be found in the Appendix as Table A2. The number of observations is given, which are 1310 data points for most of the companies. The number of observations of the five companies that went public during the selected time window varies from 1112 to 497. Also, the average stock prices and average annualized returns are given. The market index has no stock price, so the descriptive statistics are given on the index level. The range of mean stock

20

prices is very diverse, going from €2.82 to €215.33. The annualized returns vary across the companies as well. For example, we see that over this five-year period, firms like ASML and Galapagos experience annualized returns of 22.746% and 31.896%, respectively. This could be the result of the booming high-tech market. On the other hand, we also see negative returns. For example, Gemalto (-4.059%) and VOPAK (-1.230%). At Gemalto, there was a big scandal on a hack of customer data in 2015. This could be an explanation for the low return over the sample period. The negative return of VOPAK could be explained by the growing interest in environmental sustainability, which lowers the business potential of an oil storage company.

Finally, the standard deviations of the stock returns are provided. There is not very much spread in these annualized standard deviations: most of them are in the range of 20%-30%, with a few exceptions. The standard deviation of stock return measures the total risk, which consists of both systematic and idiosyncratic risk.

3.1.2 Descriptive statistics of extended models

Besides the single-index model, the correlations of idiosyncratic risk are also analysed for three other models, namely the three-factor model of Fama and French (1993), the four-factor model of Carhart (1997) and the five-factor model of Fama and French (2015). This can be seen as tests on the robustness of the obtained results from the single-index model. As the three added models build on each other, their data is described in this joint section.

The three used models all add control variables to the original single-index model. This means that for the excess market return factor, the same data is used as in the single-index model, again obtained from Datastream. The three-factor model of Fama and French subsequently adds the SMB and HML factors. The data on these factors is obtained from the personal website of Kenneth French1. However, this data is updated until April 30th 2018, so this does not yield 1310 observations like in the research on the single-index model, but 1303 data points. The time window of the sample is then May 7th 2013 to April 30th 2018. The table with descriptive statistics on the SMB and HML factors can be found below, as Table 2. Due to the small decrease in sample size, the mean and standard deviations of the stock return and market return changed slightly.

The four-factor model of Carhart not only includes the excess market return, SML and HML factors, but also incorporates a UMD factor. Descriptive statistics on this factor can also

1

The data on the SMB and HML factors is obtained from:

21 be found in Table 2. Again, the data on this factor is provided by Kenneth French2. Just like the data on the SMB and HML factors, the UMD factors are provided until April 30th 2018, which yields 1303 observations and the same time window as the three-factor model.

The last model that is used to check the robustness of the results from the single-index model is the five-factor model of Fama and French. Compared to the single-index model, the SML, HML, RMW and CMA factors are added. This means that the UMD factor, provided in the four-factor model, is now neglected. The data on the RMW and CMA factors are again obtained from Kenneth French’s website3

. Once more, French provided data on these factors until April 30th 2018, which leads to 1303 data points.

Table 2 provides the descriptive statistics on the dataset that is used to estimate the three extensions of the single-index model. The added factors show quite similar characteristics. The mean of all factors is close to zero, with relatively large standard errors.

3.2 Research method

Idiosyncratic risk is defined as the risk that only applies to one company, and is given by the volatility of the regression residuals. It is also known as firm-specific risk and unsystematic risk, and these concepts are used interchangeably throughout this paper. There are two main

2 _{The data on the UMD factor is obtained from:}

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html#Developed

3 _{The data on the RMW and CMA factors is obtained from:}

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html#Research

Table 2. Descriptive statistics of extended dataset.

Observations Mean (Mean annualized return) Standard

deviation Maximum Minimum Daily stock return 29.667 0.0549% (10.438%) 1.6978% 34.5677% -22.5866% Daily market return 1303 0.0392% (9.371%) 0.9873% 4.0511% -5.7041% SMB 1303 0.01693 0.4187 1.87 -1.62 HML 1303 -0.00039 0.3972 1.75 -2.07 UMD 1303 0.03489 0.5288 4.51 -2.41 RMW 1303 0.01359 0.2762 1.61 -1.51 CMA 1303 0.00401 0.2498 1.50 -1.47

Note: This table contains descriptive statistics on the dataset for the Fama-French models and the four-factor model of Carhart, going from 2013 to 2018. The stock prices and market index levels are transformed to returns. The returns are annualized by assuming 252 trading days per year.

22

ways of computing firm-specific risk, one using the R2of a regression and the other using the residuals. Zhang et al. (2016) show that these concepts do not provide the same results and that the use of the residuals is more appropriate. This is why the residuals are used in this research and not the R2.

To perform the regressions to obtain the residuals, a complete asset pricing model is used, namely the CAPM. This is also a single-index model, so the regression model is based on the original diagonal model of Sharpe:

Ri,t = αi,t + βi,t Rm,t + εi,t

,

where αi is the intercept of the regression, βi measures the sensitivity to the macro-factor, and the residual is given by εi. Again, the residuals are assumed to have an expected mean of zero and to be independent and identically distributed. The capital characters R are used in the regression equation, to indicate that excess stock return is regressed on excess market return (Bodie et al., 2014). Every observation for stock and market return and the corresponding residual takes place at the same time, denoted by the subscript t. The risk-free rate is derived from the yield on Dutch short-term treasury bills, during the same time as the analysed time horizon for the stocks, and turned out to be slightly negative. Data on the treasury bills is collected from the Ministerie van Financiën4. The regressions will be OLS-regressions, with excess market return as proxy for the macro-index. The market return is given by the return on the AEX. This index variable is chosen, because it is thought to be the most relevant general driver of stock returns, and it is according to the application that Sharpe (1963) provided in his research.

After collecting the residuals, the correlation across the idiosyncratic shocks is calculated and tested for significance. The (Pearson) correlation coefficients will be calculated with the following formula:

𝜌

=

,

The data on the yield on short-term treasury bills is obtained from the official website: (https://www.dsta.nl/actueel/statistieken/rentereeksen-nederlandse-staatsobligaties).

23 where x and y are residuals of two different companies. The correlations of the companies with less data available are calculated with the data points that are available. All correlations are presented in a correlation matrix and will be tested on significance using the p-value. To check whether the results from the single-index model are robust, the same analysis on idiosyncratic risk correlations is performed in other asset pricing models.

First, the idiosyncratic correlations are analysed for the three-factor model of Fama and French, which includes more variables than the single-index model. Fama and French (1993) extended the CAPM by adding small-minus-big and high-minus-low factors. The regression equation will be:

Ri,t = αi,t + β1Rm,t + β2SMBt + β3HMLt + εi,t

,

where Ri,t, αi,t, Rm,t and the betas have their now usual meaning. The residuals also have the same characteristics as the residuals mentioned before. The SMB is the small-minus-big factor and HML forms the high-minus-low factor. These factors are included because of the pricing anomalies that exist with these asset classes. Empirical evidence suggests that companies with small market capitalisations have higher returns than companies with high market capitalisations, after controlling for the CAPM. This is a pattern that does not disappear and could be the result of behavioural biases (Goetzmann & Massa, 2003).

After performing the regressions on all stocks, the residuals are obtained, that are used to compute the correlations of idiosyncratic risks. These results will be compared to the results of the single-index model.

In addition, the correlation of idiosyncratic risk is analysed in the four-factor model of Carhart (1997). This model is developed as expansion on the three-factor model of Fama and French. It not only includes the SMB and HML factors, but also a UMD factor. This factor is also called the momentum factor and is added because of the momentum anomaly, first described by Jegadeesh and Titman (1993). This anomaly suggests that the stock price of companies with good past performance keeps rising, and the stock price of companies that do not perform well will continue to fall. This factor is included in the following regression equation:

24

where the added UMD factor is the up-minus-down factor, and β4 measures the exposure on this factor for the individual stock. The other variables and parameters have their now usual meaning. The residuals are again assumed to have an expected mean of zero and to be independent and identically distributed.

Finally, the five-factor model of Fama and French (2015) is compared to the single-index model. This model forms an expansion on the previous three-factor model, by adding a profitability factor and an investment strategy factor, RMW and CMA respectively. The profitability factor suggests that high profitability of a company increases its stock price and returns. The investment strategy factor distinguishes aggressive and conservative investment strategies, and their impact on stock returns. This leads to an extension of the regression equation in the following way:

Ri,t = αi,t + β1Rm,t + β2SMBt + β3HMLt + β4RMWt + β5CMAt + εi,t

,

where the variables and parameters have their now usual meaning and implications. Equation (10) is different from Carhart’s four-factor model, because it ignores the momentum factor, but adds the RMW and CMA factors. Compared to equation (8), the RMW and CMA factors are both new. It is important to note that this five-factor model is the most recent model of Fama and French, and not the five-factor model that was earlier developed by them.

In Section 4, the idiosyncratic risk correlations of all three extensions are compared to the correlations in the single-index model.

3.3 The analytical example

To analyse the consequences of the possible correlations in the single-index model, the correlations are used in a portfolio construction process that uses this model, namely the procedure developed by Treynor and Black (1973). The effects on portfolio weights, risk and return are analysed and the results will be presented in a statistical and a graphical way. The Treynor-Black procedure helps in constructing the optimal portfolio. It first identifies the optimal risky portfolio, by assigning weights to stocks and the market index, which leads to the portfolio return and its risk. This procedure uses the single-index model to distinguish idiosyncratic and systematic risk. The idiosyncratic risk of the active portfolio is given by:

σ

_(e

_{) =}

_∑

_𝑤

𝑖2

𝜎

(𝑒

)

25 where wi is the weight of each individual stock, based on its alpha. This idiosyncratic risk is crucial in computing the weight of the active portfolio:

𝑤

=

,

𝑤

=

First, a portfolio with all the companies in the sample and the market index is constructed via the original single-index model in the analytical example. After that, the portfolio construction process is adjusted in this research, by allowing covariances of the residuals to enter the formula for idiosyncratic risk of the active portfolio. Equation (11) is therefore modified to the following equation:

𝜎

(𝑒

)

∑

𝑤

𝜎

(𝑒

)

+ 2𝑤

𝑤

𝐶𝑜𝑣(𝑒

, 𝑒

)

This equation is used to analyse whether the correlations across idiosyncratic risks affect the construction of the optimal risky and optimal complete portfolio. The optimal risky portfolio consists of an active portfolio of stocks and a passive market index, which are securities where the risk-return relationship is important. The optimal complete portfolio regards the optimal risky portfolio as one security, and the risk aversion of the investor then determines the weight that will be invested in this risky portfolio, versus the weight that will be invested in a risk-free asset.

3.4 Hypothesis and expectations

The consulted literature from the previous section could be an indication of correlation of idiosyncratic shocks, so I expect the correlation across the residuals of the researched individual stocks to be different from zero. I do not expect the correlation of two competitive firms to be negative per se, because there could be other factors affecting the correlation positively. Similarly, the correlation between two firms in a buyer-supplier relationship does not necessarily have to be positive. So, I expect that the assumption of uncorrelated residuals in Sharpe’s single-index model will not hold in reality. The hypothesis of this research will therefore be:

H0 : ρ_ei,ej = 0 H1 : ρ_ei,ej ≠ 0

26

The single-index model contains several assumptions, but only the assumption of uncorrelated residuals is analysed in this paper.

When using the adjustment of the single-index model to construct a portfolio of the stocks from the sample, I expect to find different stock weights, portfolio return and risk. If the covariances turn out to be negative, I predict that the overall weight on stocks will increase. Likewise, when the idiosyncratic risk is positively hit, I expect the overall weight on stocks to decrease.

Further, I expect that the analysis of the three other asset pricing models will generate fairly similar outcomes as those of the single-index model. Of course, the values will be different with different inputs, but I expect that the amount of statistically significant correlations will not change by much. Especially in the three-factor model, the exposures of the companies to the SMB and HML factors are probably quite similar, which will most likely lead to the same significant correlations as in the single-index model. As more variables are added, like in the four-factor model and in the five-factor model, the residuals will probably be more and more affected, which also affects the correlations of idiosyncratic risk.

27

4. Results

In this section, the correlations of idiosyncratic risk in the single-index model are presented and analysed. After that, I will use these correlations in the Treynor-Black procedure. Finally, the correlations in the three other asset pricing models are analysed.

4.1 Analysis of residuals

After performing the twenty-five regressions of excess stock return on excess market return, the residuals were obtained. The correlations across all stocks were tested for significance using the p-value and are presented in the correlation matrix on the next page, in Table 3. The p-values are not included in the table. An asterisk (*) indicates that the correlation is significant with a significance level of 5%. The correlation matrix presents 300 correlations, from which 151 turned out to be statistically significant. This forms evidence that Sharpe’s assumption of uncorrelated residuals does not hold in practice. There were more negative than positive correlations, but the difference was not substantial: 78 negative correlations and 73 positive correlations. Every company in the sample carried some of the correlations, so there were no companies without any correlation. The maximum correlation was 0.6320 and the minimum was -0.4756. With a mean of -0.0066, it could be stated that the correlations tend to cancel out as this is very close to zero.

Some interesting results come from the correlation matrix. A lot of the correlations can be explained by the indications provided in the literature section, such as a competitive relationship or a buyer-supplier relation. For example, the companies Wolters Kluwer and RELX turned out to have a substantial negative correlation, which could be explained by the competition between the firms, since both are publishing companies. If one of the companies was idiosyncratically hit in a negative way, the other would probably have gained an advantage, and vice versa. The same holds for NN and Aegon, both insurance companies. These companies produce the most negative correlation. On the other hand, there are also positive correlations. This could be an indication that a buyer-supplier relation is present. Indeed, we see significant correlations between, for example, Royal Dutch Shell and VOPAK and between NN and ING. NN forms the insurance department of the ING Group, which continued after the initial public offering of NN. This means that ING would refer clients in need of an insurance to NN. The two companies are highly dependent on each other, which explains the positive correlation of idiosyncratic risks. Another example is formed by Royal

28 Tab le 3 . Cor re la tion ma trix of r esidua ls ( S IM) . A A L B A B N AGN AD A K Z A A L T AM ASML A S R N L DSM G L P G G T O H E IA IN G A KPN NN P H IA L IG H T R A N D R E N R D S A UL U N A VPK W K L A A L B 1. 0000 A B N 0. 0972 * 1. 0000 AGN 0. 1372 * 0. 3631 * 1. 0000 AD -0. 054 7* -0. 088 9* -0. 115 7* 1. 0000 A K Z A 0. 1422 * 0. 1434 -0. 010 1 0. 0359 1. 0000 A L T 0. 0264 0. 0751 0. 0464 -0. 085 1 -0. 003 8 1. 0000 AM 0. 0513 0. 0715 0. 1587 * -0. 13 6 5* -0. 038 3 0. 0190 1. 0000 ASML 0. 1155 * -0. 219 6* -0. 117 2* -0. 067 8* -0. 018 6 -0. 037 6 -0. 070 6* 1. 0000 A S R N L 0. 0837 0. 1838 * 0. 2532 * -0. 031 6 0. 0535 0. 0156 0. 0117 -0. 058 1. 0000 DSM 0. 0785 * -0. 022 9 -0. 010 6 0. 00 70 0. 1087 * -0. 017 2 0. 0191 0. 1223 * 0. 0337 1. 0000 G L P G 0. 0780 * -0. 046 5 -0. 020 7 -0. 123 2* -0. 056 8* 0. 0343 0. 0010 0. 0901 * 0. 0308 0. 0195 1. 0000 G T O 0. 0065 -0. 010 3 0. 0491 -0. 050 3 0. 0093 0. 1073 * -0. 028 0 0. 0211 0. 0155 -0. 017 1 0. 0170 1 .0000 H E IA -0. 018 9 -0. 210 4* -0. 203 2* 0. 0577 * 0. 0087 -0. 055 2 -0. 259 8* -0. 102 5* -0. 126 2* 0. 0190 0. 0265 0. 1175 * 1. 0000 IN G A -0. 036 2 0. 6320 * 0. 2693 * -0. 156 5* 0. 1255 * 0. 0190 0. 1139 * -0. 207 9* 0. 1425 * -0. 068 8* 0. 0109 -0. 004 1 -0. 280 9* 1. 0000 KPN -0. 001 5 -0. 017 2 -0. 068 4* 0. 0160 -0. 018 4 0. 0470 -0. 079 3* -0. 129 9* -0. 030 5 -0. 047 8 -0. 015 4 0. 0195 0. 1401 * -0. 001 6 1. 0000 NN 0. 1379 * 0. 2867 * -0. 475 6* -0. 083 6 0. 0625 0. 0301 0. 0573 -0. 138 8* 0. 2421 * -0. 031 9 -0. 013 8 0. 0553 -0. 225 1* 0. 3046 * -0. 022 8 1. 0000 P H IA 0. 1214 * 0. 0530 -0. 017 6 -0. 004 1 0. 0271 -0. 019 0 -0. 047 7 -0. 035 9 0. 0442 0. 1031 * -0. 057 8* -0. 009 2 0. 1231 * -0. 042 9 0. 0119 0. 0541 1. 0000 L IG H T 0. 0963 * -0. 014 7 -0. 015 4 -0. 047 7 -0. 035 1 -0. 022 9 0. 0060 0. 0848 0. 1586 * 0 .0837 0. 0086 0. 0280 -0. 031 1 -0. 024 2 -0. 126 3* -0. 051 3 0. 0531 1. 0000 R A N D 0. 2196 * 0. 2994 * 0. 1731 * -0. 062 6* 0. 0974 * 0. 0742 0. 0970 * -0. 133 5* 0. 0789 0. 0063 0. 0445 0. 0260 -0. 063 5* 0. 1193 * 0. 0158 0. 3049 * 0. 0646 * -0. 031 8 1. 0000 R E N 0. 0958 * -0. 265 9* -0. 189 6* 0. 0600 * -0. 011 2 -0. 055 6 -0. 268 1* 0. 0089 -0. 045 7 -0. 001 5 0. 0338 -0. 004 1 0. 1491 * -0. 219 9* 0. 0760 * -0. 154 7* -0. 017 7 0. 0952 * 0. 0654 * 1. 0000 R D S A -0. 145 8* -0. 149 6* -0. 017 4 -0. 137 6* -0. 200 7* -0. 084 9 0. 2469 * -0. 232 5* -0. 052 9 -0. 099 1* -0. 040 3 -0. 089 1* -0. 211 7* -0. 186 4* -0. 109 4* -0. 111 2* -0. 212 6* -0. 019 3 -0. 186 2* -0. 182 4* 1. 0000 UL 0. 0191 -0. 174 4* -0. 170 8* 0. 0044 -0. 026 8 -0. 068 2 -0. 163 7* 0. 1366 * -0. 118 1* 0. 0021 0. 0214 -0. 079 3* 0. 1864 * -0. 135 9* 0. 0630 * -0. 097 0* -0. 046 8 -0. 078 6 0. 0171 0. 1176 * -0. 24 2 9* 1. 0000 U N A -0. 161 7* -0. 326 8* -0. 311 1* 0. 0620 * -0. 054 4* -0. 125 3* -0. 323 5* -0. 102 3* -0. 142 9* -0. 104 9* -0. 018 6 -0. 047 3 0. 3576 * -0. 403 3* -0. 038 5 -0. 273 4* -0. 106 5* -0. 050 1 -0. 166 1* 0. 1311 * -0. 226 4* 0. 1442 * 1. 0000 V P K 0. 1342 * 0. 0206 0. 0526 0. 0055 0. 0554 * -0. 022 9 0. 0628 * -0. 077 9* 0. 0293 0. 0159 0. 0397 0. 0525 0. 1258 * -0. 119 7* 0. 0029 0. 0970 * -0. 006 1 0. 0311 -0. 011 6 0. 1159 * 0. 3 209 * -0. 019 9 -0. 062 1* 1. 0000 W K L 0. 1165 * -0. 272 5* -0. 125 1* 0. 0928 * -0. 000 9 -0. 045 8 -0. 205 9* 0. 0132 0. 0256 0. 0143 0. 0878 * 0. 0155 0 .1680 * -0. 174 1* 0. 0585 * -0. 183 4* -0. 016 9 0. 0554 -0. 138 6* -0. 450 1* -0. 178 2* 0. 0832 * 0. 0807 * 0. 0800 * 1. 0000 N o te : T h is ta bl e c on ta in s the c o rr el a ti on s a cr o ss t h e 25 com pa ni es fr om t he A EX , w it hi n t he t im ef ra m e of 2013 -2018. T h e re g re ss ion s a re ba se d on t he s ing le -i n de x m ode l. * r epr es en ts si gn if ic a n c e a t 5% si gn if ic a n c e le ve l.

29 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 20 -11 -20 15 20 -1-201 6 20 -3-201 6 20 -5-201 6 20 -7-201 6 20 -9-201 6 20 -11 -20 16 20 -1-201 7 20 -3-201 7 20 -5-201 7 20 -7-201 7 20 -9-201 7 20 -11 -20 17 20 -1-201 8 20 -3-201 8 Da ily re sidu a ls Date ABN residuals ING residuals

Dutch Shell and VOPAK, which have a correlation of 0.3209. This could be explained by the fact that VOPAK is a worldwide oil storage company, so if Royal Dutch Shell is idiosyncratically hit, VOPAK would probably experience part of this effect as well. These results support the statements made by, among others, Hallikas and Lintukangas (2016).

In the following figures, the lowest, highest, and most insignificant correlations across two companies are presented, to illustrate how the bilateral movements in residuals can differ across firms. In Figure 3, the residuals with the highest correlation are presented. The lines of ABN and ING are very much alike: if the ABN-residuals increase, so do the ING-residuals and vice versa. It can almost be seen as one line. However, the magnitude of the increases and decreases seems somewhat higher with ABN, which can be the result of the fact that the correlation is not perfect, but 0.6320. Figure 4 illustrates the residuals of the companies with the lowest correlation. Here, a clear negative relationship is visible: if the residuals of NN are high, the residuals of Aegon tend to be low, and vice versa. The two lines can be distinguished clearly. The last figure, Figure 5, presents the residuals with the most insignificant correlation. In the beginning of the sample, the relationship looks positive, but after one and a half year, it looks negative. Onwards, no distinguishable pattern can be observed and in the end, it again looks like a positive relationship. All in all, an insignificant correlation seems appropriate if you look at this graph.

𝜌̂ = 0.6320

Figure 3. The movements in residuals of ABN and ING.

Note: This graph contains the residuals of the performed regressions, with the timeframe of November 20th 2015 until May 7th 2018.

30 -0.1 -0.05 0 0.05 0.1 0.15 7-5-2013 7-5-2014 7-5-2015 7-5-2016 7-5-2017 7-5-2018 Da ily re sidu a ls Date Wolters residuals AKZA residuals -0.1 -0.05 0 0.05 0.1 0.15 2-7-2014 2-7-2015 2-7-2016 2-7-2017 Da ily re sidu a ls Date NN residuals Aegon residuals 𝜌̂ = -0.4756

Figure 4. The movements in residuals of Aegon and NN.

Note: This graph contains the residuals of the performed regressions, with the timeframe of July 2nd 2014 until May 7th 2018.

𝜌̂ = -0.0009

Figure 5. The movements in residuals of AKZA and Wolters.

Note: This graph contains the residuals of the performed regressions, with the timeframe of May 7th 2013 until May 7th 2018.

In Section 2 of this paper, another possible explanation for idiosyncratic risk correlation was provided, namely the omission of the low-volatility factor. In the sample, there were four companies with remarkably low volatility, namely Aalberts Industries, RELX, Royal Dutch Shell and Wolters Kluwer. With an average idiosyncratic volatility of 22.89% for the complete sample, these companies exhibited 13.95%, 14.59%, 12.68% and 13.63%,

31 respectively. All correlations across these companies turned out to be statistically significant, according to Table 3. Of course, this could be coincidence, but when we look at the company with the highest idiosyncratic volatility, which is Altice by far with 41.06%, we see that none of the low-volatility stocks is correlated with Altice. This is noteworthy, because this is unique in the sample: every stock is correlated with at least one of the low-volatility stocks, except for Altice. If the correlations across the low-volatility stocks would be coincidence, we would also expect at least one significant correlation with Altice, as every other stock exhibits this. The fact that this is not the case, forms an indication that the low-volatility factor could certainly be a determinant of idiosyncratic risk correlation. The exposure to this factor is probably quite similar for the low-volatility stocks, but will be different for Altice. This could be the reason that the correlations are significant across the low-volatility stocks, but not between the low-volatility stocks and Altice.

4.2 Analytical example with sampled correlations

Here, the effects of using the original single-index model in portfolio selection are analysed. This original model decomposes the notion of risk, but does this by greatly simplifying reality. Equation (14) offered an alternative computation of the idiosyncratic part of the variance, which now includes the computed correlations across the idiosyncratic risks. This will form the ‘modified single-index model’ in the following subsection. It is analysed whether the inclusion of these correlations affects portfolio characteristics. The Treynor-Black procedure is used for portfolio construction, because this procedure uses the single-index model. So, the Treynor-Black procedure generates the portfolio weights of the individual stocks in the active portfolio. As indicated in Section 3, the active portfolio and the passive market index together form the optimal risky portfolio. The optimal complete portfolio subsequently consists of the optimal risky portfolio and a risk-free asset. So, the results of changing portfolio weights hold for the active portfolio, as well as for the optimal risky portfolio. However, the portfolio characteristics of the active portfolio are easier to interpret, so those are used in the analysis. The weights on stocks increased from 95.8365% to 97.1798%, which means that the invested amount in the market decreased from 4.1635% to 2.8202%.One can notice that the overall weight on stocks is very high, which is the result of the set-up of the example. All the stocks are also part of the market index, so the overall stock weight should indeed be close to 100%. The fact that it is not exactly 100% can be explained