The effect of increased index futures trading volume on the correlation between index constituents examined through an extended asymmetric DCC-GARCH model

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University of Amsterdam

Faculty of Economics and Business

Supervisor : Dr. Simon Broda

Author : Lodewic van Twillert

Student number : 10216529

Bachelor thesis Econometrics 2014 Lodewic van Twillert

June 27, 2014

The effect of increased index futures trading volume on the

correlation between index constituents examined through an

extended asymmetric DCC-GARCH model

Abstract

Understanding correlations between stocks is vital for investors. Trading index futures affects all index constituents simultaneously, and would intuitively increase the correlation between stocks. It was found in previous research that correlations respond asymmetrically to stock returns. It has also been found that index

futures trading volume responds asymmetrically to stock returns. By implementing an asymmetric DCC-GARCH model, extended with an exogenous variable, the relation between index futures trading volume and stock correlations is researched. Using daily data of the AEX, DAX and EURO STOXX 50 indices it is found that index futures trading volume has no significant effect on the correlation between the index constituents. The correlations are found to respond asymmetrically to returns. However, the correlations are not necessarily responding differently to negative returns than to positive returns, as is often assumed. Correlations rather respond differently to returns less than half their standard deviation. Both the index futures market and the stock correlations are affected similarly by low and negative returns; therefore there is no causal relation

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1. Introduction

Since the introduction of equity derivative contracts in the 1980s there has been a lot of research aimed at whether or not these derivative contracts have caused significant side effects to the equity market. The effect of trading index futures and options has been widely researched regarding changes to trading volume, returns of the underlying index and any volatility effects. Gulen and Mayhew (2000) find that the conditional volatility increases significantly in the US and Japan after introduction of futures contracts whereas in many other countries conditional volatility went significantly down. The authors note that conditional covariances with the world market increased after the introduction of futures contracts. This is explained by the markets being more integrated, although futures are not necessarily the cause of their findings since markets would probably have grown more integrated either way. Another phenomenon related to the index futures trading volume is the expiration day effect. Joensuu (2010) has examined the expiration day effects and has found that trading volume is higher on the underlying index on the expiration day of index futures contracts. Joensuu has also found that the intraday volatility is significantly higher on expiration dates.

While the effects of derivative contracts have been studied widely, there is lacking literature regarding the effect of trading index futures on the correlation between the constituents of the underlying index. Volatility being affected by futures trading suggests that correlations will also be affected in a way, although this research will not be based on the world markets as Gulen and Mayhew (2000) have done but on a much smaller scale, namely the constituents of European stock market indices. In short, this study will be a contribution to the lacking literature of the effects of index futures on stock correlation based on European stock markets.

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1.1 Research motivation

Investors have been constructing portfolios of combinations of stocks, and even indices, to achieve returns higher than the risk-free interest rate with minimal risk. The way to minimize risk in a stock portfolio greatly depends on the correlation of the returns of the selected stocks. While the investor will invest in stocks that he expects to increase in value, it may always go the wrong way. When one of the stocks in a portfolio greatly drops in value then an investor will favor to have other stocks in his portfolio that are uncorrelated, or negatively correlated, with the decreasing stock so that the losses will be contained to a single decreasing stock price. Largely known European indices like the DAX, AEX and the EURO STOXX 50 all consist of a selection of stocks with different weights and are in effect a portfolio of stocks. The way to invest in one of these indices as a whole is mainly through index futures contracts.

Trading volume in index futures has increased dramatically ever since the introduction of futures contracts, although the trading of index futures has been seemingly stable the past decade. However, index futures trading volume on European stock market indices have been quite stable for the past ten years but there are clear periodic spikes of high trading volume on futures expiration dates, where the periodicity differs per index. Trading index futures will intuitively have an impact on the correlation between index constituents since, like any trade on an equity index, all constituents are effectively traded at the same time. To clarify, trading index futures will affect all the stock returns at the same time and when stock returns move in the same direction at the same time then that defines an increase in correlation. Index futures however, are only a small portion of all trades done on an equity market and may have an insignificant impact due to the relative size of the market. Correlation between stocks has been an important factor in the diversification of stock portfolios and it is therefore of interest to study the relationship between futures trading volume and stock correlation.

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3 Correlation amongst euro zone national equity markets have gone up sharply since the early nineties (see, e.g., Kearney and Potì, 2006) due to the introduction of the Euro. The Euro has caused European markets to be more intertwined since its introduction, as is evident from the increased correlation between European indices, but this should have stabilized ever since, just like the futures trading volume seems to be stable over the last 10 years. For the reasons stated this research will be based on data of European markets since 2004 up until 2014.

1.2 Research goal and methods

The goal of this paper is to assess whether increased volume in index futures trades are linked to increased correlation. Specifically, what is the effect of increased index futures trading volume on the correlation between the index constituents? A secondary goal of this paper is to examine the asymmetric response of stock return correlations to stock returns.

Generally, in order to study correlation dynamics of stock markets either the BEKK or DCC-GARCH model, as proposed by Engle and Kroner (1995) and Engle (2002) is often used since both are based on a non-constant covariance factor. Caporin and McAleer (2010) argue that the cDCC model (see Engle, Shepard and Sheppard (2008) and Engle and Kelly (2009)) is preferred over the DCC model by Engle (2002) and that the preference of using either the cDCC model or the BEKK model cannot be judged a priori. However, this research will be based on methods similar to those of Kearney and Potì (2006), namely using an asymmetric DCC-GARCH model. The BEKK and cDCC model may be used to compare performance in future research. Kearney and Potì have argued that the asymmetric effect of historical returns on the covariance between stocks/indices is non-linear, and since the data used for this research will also cover the financial crisis around 2008 it is of considerable interest to aim for a correctly specified asymmetry in the models. The research done by Kearney and Potì is quite similar and

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4 therefore this research will be used to extend their model and apply it to different data. Extending the model in question, by using the index futures trading volume as a exogenous variable for the covariance estimator in the asymmetric DCC-GARCH model, I will be able to examine the relationship between futures trading volume and stock correlation.

1.3 Further structure

The rest of this study will consist of four more sections structured as follows, section 2 will be used to give some background information on correlations and futures contracts. In section 3 the relevant literature will be discussed and it is explained how earlier findings will guide the methods of this research. Section 4 will be used to discuss the relevant data set. Section 5 and 6 will describe the extended asymmetric DCC-GARCH model to be used and will be the analysis of the results of a range of statistical tests to explain the relationship between index futures trading and the correlation of index constituents. The final section will contain a summary of the findings, a discussion of the findings, and proposals for future research.

2 Background information

This study will try to explain the dynamics of correlations through the trading volume of index futures. To understand the concepts of this study it is of importance to grasp the concept of correlations and to understand what futures contracts are. A lot of earlier research has been done to study whether the trading of futures has had a significant effect on the equity markets. Section 2.3 will discuss some of the earlier studies done on the general effect of trading futures.

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2.1 Correlation

When talking about the correlation between stocks it is usually meant that we look at the co-movement of stock returns. Stock returns are often defined as the log-return of daily stock prices, as they will also be in the rest of this paper. The daily log-return of a stock price is defined as,

[

] (1.)

where is the price of the stock at time t. Stock returns are often characterized by a zero mean with a variance that varies for every stock. The conditional correlation between two zero-mean return series is defined as,

( ) √ ( ) ( )

(2.)

When assuming returns to have a mean of zero, it is convenient to write the returns of stock i as a standardized disturbance times the conditional standard deviation,

√ (3.)

where is a disturbance with mean zero and variance of one. To estimate the

unconditional correlation between stock returns it is easiest to use a rolling correlation estimator over a period of k days which is defined as follows for zero mean returns,

̂ ∑

√ ∑ ∑ (4.)

The estimator will always lie in the interval [-1, 1] by probability theory. The correlation estimate is a rough one as recent observations have the same weight as any other observations less than or equal to k days ago while any observation more than k days ago will have zero weight. Another limitation of this estimator is that the researcher has to choose a value for k that is fitting to the current research. This study will also be

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6 based on this estimator because of its simplicity and since the approach to this study is similar to what Kearney and Potì (2006) have done.

Another, arguably better, estimator is the exponential smoother used by RiskMetricsTM which uses declining weights for less recent data points based on a smoothing parameter . The estimator has no termination point in the past and so all available data is considered, ̂ ∑ √ ∑ ∑ (5.)

The default value for used by RiskMetricsTM

is 0.94 for daily data.

When the mean of returns is assumed to be 0, then the returns can be modeled by estimating the values of √ in Equation (3.) When estimating the dynamic variance of

returns it is common practice to use Autoregressive Conditional Heteroskedasticity, or ARCH, models.

2.2 The GARCH model

When modeling a single time series of stock returns, or any other zero mean process, it is often of interest to model the dynamic variance, or volatility, of that time series. Volatility clustering, for example, is often researched (e.g. Klaassen (2002)) through ARCH and GARCH models. While stock returns are generally not auto-correlated, their absolute values are, meaning that the variance of returns is auto-correlated. Periods with highly volatile returns are often followed by periods with highly volatile returns, leading to the phenomenon of volatility clustering. Since it is clear from Equation (2.) that the variance of individual stock returns plays a big role in determining the correlation between two stocks we will need some dynamic model that at least captures the dynamic volatility of returns.

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7 The Generalized Autoregressive Heteroskedasticity model, as proposed by Bollerslev (1986), is a suitable model used to estimate the dynamic variance of some zero mean process. The GARCH model is a generalization of the ARCH model that was proposed by Engle (1982). The ARCH model is based on the lagged values of the squared zero mean returns. The Generalized Autoregressive Conditional Heteroskedasticity, referred to as GARCH, model is an extension to the ARCH model by also including lagged values of the variance. If we were to estimate the time series as in Equation (3.) for all t, then an ARCH(q) model would be defined by,

(6.)

A generalization of the ARCH model also includes lagged values of , leading to a GARCH(p,q) model as follows,

(7.)

Throughout the rest of this research only the GARCH(1, 1) model will be considered since higher orders of lagged values are rarely a better fit. The univariate GARCH model will be used to estimate the dynamic variance of stock returns later in this paper.

So far only univariate models have been shown. The limitation of univariate models is that time series models only depend on themselves and not on other time series, discarding the possibility of sensibly extending the model with any correlation characteristics. To be able to model the correlation dynamics of stock returns it is required to use a multivariate model. The univariate GARCH model is the fundamental basis for the multivariate GARCH model that will be explained in Section 4.1.

2.3 Futures

Stock index futures were first introduced in 1982 on the S&P500 index by CME. Gradually the index futures contract was introduced on different indices around the world. The contract was introduced on the AEX in 1988, on the DAX in 1990 and on the

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8 EURO STOXX 50 in 1998. Quickly after the first introduction of index futures investors realized that they could use index futures to hedge against falling prices, and to this day index futures are still greatly used for that purpose. However, futures have been around a lot longer and older examples of futures may give a clearer idea of what futures contracts are and why one would buy them. When North American farmers would grow crops about 150 years ago they would never know whether it would be a good season for their crops. Farmers and their customers would agree to trade the crops after they had grown at a price agreed upon beforehand. Even if the harvest was bad the buyer would be obligated to buy the crops at the agreed price and would clearly have paid too much. On the other hand, when the harvest is successful the buyer also has the obligation to buy the crops at the agreed price and will most likely have paid less than the present value of the successful harvest. In short, buying a futures contract gives the obligation to buy the underlying asset at a given future expiration date, at a spot price agreed upon today.

Unlike options, futures give the buyer an obligation to buy the entire index at expiration date. Of course, it is impractical to physically buy all of the weighted constituents of the underlying index and therefore the contract is usually settled in cash on expiration date. The settlement rules are different for every exchange, although irrelevant for the rest of this study.

3 Relevant literature

This section will be used to explain the earlier research done that is relevant to this study. This study will examine stock correlations, index futures trading effects and specifically it will use the asymmetric DCC-GARCH model as proposed by Cappiello, Engle and Sheppard (2006).

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9 The asymmetry of the relation between returns and stock correlations is based on the much debated ‘leverage-effect’ which explains the negative correlation between past returns and future volatility. It is often seen that the future volatility is increased after negative past returns (Bouchaud et al. 2008). The leverage-effect was first discussed by Black (1976) and has been the focus of many other studies (see e.g. Christie (1982), Bouchaud et al. (2008)) where the leverage-effect is often confirmed. Research by Andersen et al. (2001) has shown that the leverage-effect is not only applicable to volatility of single stocks. Andersen et al. have found that the correlation between stock returns behaves similar to the stock volatility. Ang and Chen (2002) have found that correlation within a portfolio is higher when both a U.S. equity portfolio and the U.S. market fall. More applicable to this study are the results by Cappiello, Engle and Sheppard (2006), who have used the DCC-GARCH model to study the asymmetric dynamics in correlations. Cappiello, Engle and Sheppard (2006) have found that both bonds and equities exhibit asymmetric behavior of the conditional correlation. There is however one major drawback to the majority of research done on the asymmetric behavior of equity correlations. The asymmetric return-correlation relation is commonly described as correlations reacting differently to negative returns than they do to positive returns.

Research by Kearney and Potì (2006) has uncovered a structural break of correlation between European stock indices a year before the introduction of the Euro. Using data from 1993 to 2002 Kearney and Potì have used a symmetric and asymmetric DCC-GARCH model to examine the correlation between European stocks and stock indices and have argued that the asymmetric relation between European stock returns and correlation is non-linear. Capiello, Engle and Sheppard (2006) have essentially researched the same thing as Kearney and Potì (2006) but have focused on countries outside of Europe as well. It is important to note that the speculated non-linear asymmetric characteristic is argued by Kearney and Potì to be specific to European stock indices and that the linear asymmetric specification is significant when non-European indices were included in their dataset like Cappiello, Engle and Sheppard

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10 (2006) have done. Another, less important, reason for why their linear specification seems incorrectly specified is that the asymmetric correlation coefficient is of less importance with the daily data that Kearney and Potì used than it is with the weekly data that Cappiello, Engle and Sheppard (2006) have used. The non-linear specifications are purely speculation and the next paragraph will be used to propose an alternate linear specification to be tested in this study.

Preis et al. (2012) have examined the correlation between Dow-Jones Industrial

Average stocks over 72 years and have looked into a relationship between correlation of stocks and the index returns. The relationship between stocks and the DJIA (Dow Jones Industrial Average) returns seems to have a quadratic form, or even a near linear relation, which is slightly shifted to the right so that correlations are lowest when returns are around half the standard deviation over a certain period. The former implies that when returns are very small, then the correlation between stocks is probably still higher than when returns are half their standard deviation. Instead of assuming an asymmetric relationship of positive returns versus negative returns, an asymmetric model that distinguishes between returns less than or more than half the standard deviations may be more relevant. In other words, instead of centering the asymmetry on 0 it might be wise to center it on half a standard deviation. Since the main goal of this paper is not to examine the asymmetric effect of returns on covariance, only the shifted linear variant will be discussed and tested.

Some older research by Antoniou, Holmes and Priestley (1998) has been focused on the asymmetric response of volatility to returns. The fact that they have specifically studied the effect that trading index futures has had on the asymmetric response of volatility makes it especially relevant to this study. Antoniou, Holmes and Priestley have found that the introduction of index futures has dampened the asymmetric response of volatility. More importantly, it has been found that the asymmetric volatility characteristic has transferred to the futures market. This means that the volatility of futures markets is found to be higher following negative returns. It should be noted that it is commonly seen that high volatility is paired with higher trading volumes, which has been shown by

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11 studies by for example Chen, Firth and Rui (2001). It is important to understand that this means that index futures trading volume is expected to be higher following negative index returns. So to quickly summarize, both the correlation between stock returns and the index futures trading volume are expected to be higher after periods with negative returns.

Gulen and Mayhew (2000) have found that the introduction of futures has led to a significantly increased conditional volatility in the US and Japan but in many other countries the conditional volatility has either significantly decreased or has not changed. The data that Gulen and Mayhew have used after the introduction of futures contracts dates as far back as 1982, when the United States introduced Index Futures contracts for the first time. It would be of interest to examine the same time-period to examine the rise of futures trading from the start. However, for the same reason that it is not possible to use data from the same period as Kearney and Potì (2006) it is wise to choose a time period, such as the past decade, where the futures market has experienced stable behavior. Gulen and Mayhew also examine the effect of the volume of futures trades and split the volume in an expected and unexpected component. The expected component is supposed to capture the ‘background’ level of trading, while the unexpected component must capture the sudden increases in trading volume caused by unexpected price changes. They find that an unexpected increase of trading activity of futures has a positive effect on volatility, and it is of interest to investigate the effect that unexpected trading of futures may have on the correlation between constituents of an index. Since futures trading volume has distinct periodic peaks of high volume around periods of general expiration dates it is possible to make a distinction between the base-volume and these peaks to see how their impact differs.

4 Data

The data for this research is obtained via DataStream and consists of daily index futures trading volumes and daily closing prices of index constituents expressed in Euros. The

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12 indices used are the European indices AEX (Amsterdam stock exchange), DAX (Frankfurt stock exchange) and the EURO STOXX 50 index. The choice of indices is made in order to capture the different sizes of European indices. All data will be over a period of 10 years, namely 23/42004 to 23/4/2014. Some constituents of the indices stated before do not have continuous data over the period of 2004 to 2014 (e.g. LANXESS on the DAX and OCI on the AEX) and may be left out. The stock returns may still contain NaN values which will be set to 0. There will also be unusual days where the stock market is closed. Days where the index is not traded are characterized by returns of 0 for all stocks at the same time, and these days will be left entirely.

Table 1 shows all the constituents to be used in this research and the constituents that may be left out due to lack of data. Table 2-4 show some standard characteristics of the stock returns. Figures 1 through 3 show the volume of futures traded on the AEX, DAX and Stoxx50 respectively. The graphs show the clear periodic peaks that were discussed earlier. The peaks occur around the dates that futures typically expire, and often these peaks seem to build up before the expiry date and the volume decreases after the expiry dates.

5 Hypotheses and methodology

Section 5 will contain the hypotheses and explanations of the methodology to test those hypotheses. The unconditional correlation of the index constituents’ returns will give a good initial idea on what to look for. Before going on to regressions, the unconditional correlation will be tested for stationarity and a second-order trend.

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5.1 Hypotheses

This study is mostly focused on finding a relation between the trading of index futures and the correlation between index constituents. It is intuitively reasonable that increased index futures trading volume can cause increased correlation between stocks listed on the relevant index. A secondary topic that is linked with correlation is the asymmetric characteristic of correlations. It is of importance to test for asymmetry of correlations since if stock correlations do behave differently in periods with negative returns then that will have had a big impact during the financial crisis starting in 2007. However, the research done by Preis et al. (2012) has pointed out that asymmetry is not necessarily centered around negative returns versus positive returns. Preis et al. have made a convincing point, and although their research was done on an American index, it is likely that European indices behave similarly.

The hypotheses that are deduced based on intuition and earlier research are as follows, - Higher index futures trading volumes increase the correlation between index

constituents.

- Stock correlations respond asymmetrically to stock returns.

- The asymmetric behavior of correlations is not based on negative returns but rather on returns that are smaller than half their standard deviation.

5.2 Unconditional correlation

To get a general idea of the behavior of the correlation between separate constituents’ returns the unconditional monthly correlation estimates will be computed and plotted. The unconditional correlation is favored because of its simplicity but is merely a guideline and not a basis for any definite conclusions. The resulting estimated time series will show whether the correlations exhibit some sort of trend or non-stationarity and will be tested for it using Augmented Dickey-Fuller tests. It is not expected to see

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14 any trend in the unconditional ad hoc correlations. On the other hand, the financial crisis has caused large negative returns which are expected to have increased correlation between stocks, as was researched by Preis et al. (2012), and thus the unconditional correlation may be systematically higher during the financial crisis.

The unconditional correlation can be computed by first computing cross products of the difference between daily log-return and their monthly sample averages for all non-overlapping pairs of stocks i and j. Monthly correlations simply means that the correlation will be based on the last 21 trading days. The sum of cross products described above are then divided by the cross product of monthly standard deviations as follows,

̂

∑ ̅ ̅ √∑ ̅ ∑ ̅

(8.)

Averaging all the computed correlations as in Equation (9.) will yield the unconditional monthly correlation we’re looking for,

_∑

∑ ∑ ̂

(9.)

Here n is the number of constituents on the tested index. Another way to get (9.) it is to simply sum all the elements above the diagonal of the correlation matrix of all stock returns. Unlike Kearney and Potì (2006) these estimates don’t include the correlation between stocks and themselves since they are always 1 by definition and are therefore not dynamic anyway.

The unconditional average correlation of the AEX index constituents is shown in Figure 4 and shows a somewhat similar curvature as the index future trading volume in Figure 1. All unconditional correlations are found in Figure 4-6. The relation between unconditional correlation and index futures trading volume will be discussed in more detail in section 6.

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5.3 Stationarity and trend testing approach

The unconditional correlation as shown in Figure 4-6 must be tested for stationarity if the time series is to be modeled with OLS and GARCH techniques. To test for the presence of a unit root I will use the Augmented Dickey-Fuller (ADF) test allowing up to 12 lags. The test is done using Eviews and the results are shown inTable 5.

The curvature found in the unconditional correlation, as briefly mentioned earlier, do require additional testing. A simple OLS regression including a quadratic trend and the index futures trading volume is used to model the characteristic curvature that is present in the unconditional correlation of all considered indices. The OLS regression described above is defined as follows,

(10.)

Note that this regression will only work for a stationary time series. It is of interest to see if there is still a significant relation between the index futures trading volumes and the unconditional correlations after the second-order time trend has been controlled for. The results are shown in Table 6 and discussed in Section 6.

5.4 The extended asymmetric DCC-GARCH model

Once unconditional correlation estimates have been computed, the next step is to model conditional correlation estimates. The unconditional methods will yield monthly estimates on the correlation between stocks and reveal some interesting characteristics that may be used in the model specification.

As was stated before, this study will be based on the multivariate asymmetric DCC-GARCH model, proposed by Cappiello, Engle and Sheppard (2003). The asymmetric DCC-GARCH model is an extension of the symmetric DCC-GARCH model that was proposed by Engle (2001), Engle and Sheppard (2002) and Engle (2002). The model

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16 has the advantage of being relatively easy to estimate in a two-step procedure. Assuming that there is no deterministic trend in the unconditional correlation estimates, the asymmetric DCC-GARCH model will be extended by including a regressor for the index futures trading volume. The standard asymmetric DCC-GARCH model is as follows,

(11.)

Where the covariance matrix Ht is defined as,

(12.)

The matrix Dt is the n*n diagonal matrix of estimated conditional standard deviations of

all n constituents of the relevant index. The values of the diagonal elements of Dt are the time-varying standard deviations √ of the stock returns. The matrix is estimated as

follows,

̅ (13.)

̅ ̅ (14.)

_√

(15.)

Matrices ̅ and diagonal are n*n non-negative coefficient matrices. The matrices (and therefore ) and are assumed to be positive definite to ensure that the covariance matrix is also positive definite. The parameters and are all scalar coefficients. The matrix is the outer product of all negative return innovations so

that the correlation between two assets that have both had negative returns is allowed to behave differently through the asymmetry coefficient . Equation (15.) is used to make sure that Ct is a correlation matrix with ones on the diagonal. To clarify Equations

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17 conditional correlation to which the matrices mean-revert. The matrix ̅ analogously is the long-run baseline level to which mean-reverts.

As was briefly mentioned earlier, the DCC-GARCH model exhibits a relatively easy method for maximizing the log-likelihood function of (11.). Engle and Sheppard (2002) and Engle (2002) propose a two-step estimation procedure that is computationally feasible. The DCC-GARCH model parameters, and , can be estimated by first estimating all √ through univariate GARCH(1, 1) models to construct the matrix Dt. The log-likelihood function of GARCH(1, 1) models, as in (7.), is maximized for every return series of the individual constituents of the relevant index to obtain the estimates of √ . Ct is easily computed conditional on the estimated matrix Dt by retrieving the values of as follows,

_√

, (16.)

Where is the standardized log-return of asset i at time t. The asset returns transformed by their standard deviations will give the residuals needed in Equation (14.) and are referred to as the standardized returns. Please note that the standardized returns have a standard deviation of 1, which will be of importance later.

Schopen (2012) has proposed ways to include exogenous variables in an asymmetric DCC-GARCH model. The notation by Schopen differs but in order to include the futures trading volume as an exogenous variable to estimate it has to be decided whether the trading volume affects the asset return variance or if it affects all covariances. Since this research’s goal is to examine trading volume effects on correlation it is clear that the index futures trading volume will have to have an effect on the entire covariance matrix. For this case, it is proposed to make the following extension of (14.),

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18 Where is a matrix of ones, is a scalar trading volume coefficient and is the

scalar index futures trading volume at time t-1.1 The index futures trading volume series will be divided by the maximum value in the time series to get all the trading volume variables in a range of 0 to 1.

All models will be estimated using the UCSD GARCH Toolbox by Kevin Sheppard in MATLAB. The UCSD GARCH Toolbox has a successor, which is the MFE Toolbox. The MFE Toolbox has no documentation on the DCC-GARCH model and is much more complicated. The UCSD GARCH Toolbox is expected to give the same results and is easier to extend with the needed specifications.

5.5 Model specifications

There are a number of ways to specify the model described in Section 5.2. The first and simplest model, referred to as Model 1, is to restrict the asymmetry coefficient and the trading volume coefficient to be zero. Model 1 is in effect just the standard symmetric DCC-GARCH model to serve as a baseline for any extensions. Model 2 relieves the restriction off . Model 2 can be tested against model 1 to see if an asymmetric specification performs better than a symmetric specification. For Model 3 no restrictions are imposed so that the full model shown in (17.) can be estimated. Model 4 will be used to test the model with the same restrictions as Model 1 and including a second restriction that the news and persistence parameters, α and β respectively, sum to unity. If Model 1 performs significantly better than Model 4 then that confirms that the correlation process is stationary, similar to the tests done by Kearney and Potì (2006).

Preis et al. (2012) have found a relation between index returns, relative to their historical standard deviation, and the mean correlation of the constituents of the DJIA. The relation can be seen in Figure 7 and may be used to specify the matrix in (17.)

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19 differently. Although the relation found by Preis et al. is found on an American index and is based on the index returns instead of individual stock returns, it may still be interesting to define as the outer product of all returns that are smaller than half their standard deviation over a given time-interval. Since the residuals in (17.) are already standardized to have a standard deviation of one it suffices to define as the outer product of all returns smaller than 0.5. This would imply that correlations increase not only when both returns are negative, but also when returns are small positive values. Kearney and Potì have argued that indices in Europe exhibit a lower tendency of asymmetric behavior than American indices, so using the results of Preis et al. may not necessarily be relevant. A different specification may be to define as the outer

product of all squared negative returns so that when two assets both have extremely negative returns then the correlation will increase relatively more than it would when both assets are only slightly negative. For this research however, only the method of defining as the outer product of all return innovations smaller than 0.5 will be tested. Model 5 will simply be the same model as shown in (17.), except for the different definition of ̅ and . If we blindly take the outer product of all return innovations smaller than 0.5 then it could be that one of the stocks has a negative return while another stock has a return between 0 and 0.5 so that the product of those returns is a negative number. To ensure that all values in the matrix are positive is defined as follows,

(18.)

{ }

{

(19.)

’

Subtracting 0.5 in (18.) ensures that the cross product of all standardized return innovations smaller than 0.5 is a non-negative value.

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20 All parameters are tested for significance with the help of the T-ratio. The T-ratio is simply defined as the parameter value divided by that parameter’s standard error. The T-ratio is asymptotically normally distributed and so the p-value is computed accordingly. The models are tested with LR-ratio tests. The LR-ratio test is simply defined as follows,

(20.) The significance level is then defined as the chance that a random value from the relevant Chi-squared distribution is greater than the LR-statistic. If the significance level is smaller than 0.05 then the unrestricted model performs significantly better than the restricted model at the 5% significance level. The only model that will be hard to test is model 5 since it would make sense to check if model 5 performs significantly better than model 3. However, Model 5 and Model 3 are not nested and therefore cannot be tested with the LR-test. Model 5 will simply be tested against model 2, analogous to model 3, and the LR-statistics will be compared simply by looking at the values.

6. Results and analysis

Section 5 described all the methods used to examine whether an increase in index futures trading volume causes increased correlation between the index constituents. The results of all methods will be described and analyzed in this section, starting with the unconditional correlation.

6.1 Unconditional correlation results

Figures 1-3 show the index futures trading volume for the AEX, DAX and EURO STOXX 50 index. Figures 4-6 show the unconditional correlations for all indices computed as

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21 described in (8.) and (9.). All the data is from 4/23/2004 until 4/23/2014. A second-order trend line is added to the figures to reveal the similar curvature of the trading volumes and the correlations.

By comparing Figure 1 and Figure 4 it can be seen that both the AEX index futures trading volume and the unconditional correlations have started to increase around 2007, which is when the financial crisis started. Both the AEX index futures trading volume and constituents’ unconditional correlation show a similar curvature that is clarified with the help of a second-order trend line. It should be noted that a higher order trend line is effectively the same as the second-order trend line for both time series and is therefore not shown. The graphs show a quite clear relation between AEX index futures trading volume and correlation of index constituents but this relation may not be a causal relation. Correlations are expected to increase in periods with large negative returns either way, which was clearly the case during the financial crisis. The increased trading volume of index futures may have been because of the extra need to hedge portfolios against losses with the help of index futures. It is important to stress that there may have been an exogenous factor that has influenced both the trading volumes and the correlations.

The results of the unconditional correlation of the DAX, shown in Figure 5, are not nearly as curved as was seen for the AEX. The DAX unconditional correlation also seems to have increased steadily since 2007, again corresponding to the financial crisis. The DAX is a larger index than the AEX is and may therefore not have been affected as severely as the smaller AEX index by whatever has caused the increased correlations. By that reasoning, the results for the STOXX50 should be more similar to the DAX than to the AEX, which is indeed the case. Figure 6 shows the unconditional correlation of the STOXX50 constituents which also shows an increase around 2007 but has remained quite stable since 2009.

An asymmetric specification of the DCC-GARCH model will control for higher correlations due to periods with negative returns. If in fact the financial crisis has been

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22 the main cause of increased correlations between index constituents because of larger negative returns than usual then an asymmetric coefficient will be able to capture that relation. The fact that the unconditional correlations show a second-order curvature for all indices, albeit more pronounced for the AEX, is a sign for adding a second-order time trend. A simple OLS regression as defined in (10.) on the unconditional correlations can be used to test if the second-order trend is indeed significant. But to be able to run a regression it must first be confirmed that the unconditional correlations time series has no unit root by doing a series of ADF tests.

6.2 Stationarity and OLS regression results

For the unconditional correlation time series of all three indices an ADF test has been done allowing up to 12 lags. The results are shown in Table 5. For the AEX the null hypothesis is rejected at the 1% level of significance. It is tested without a trend since the graphs clearly showed that the unconditional correlation for the AEX doesn’t exhibit a linear trend. The ADF test including a trend is also rejected at the 1% level simply to test, but the test results for it are not shown since trend-stationarity is hardly relevant in this case. The results for the DAX and STOXX50 index are very similar. For all indices, the null hypothesis of a unit root in the unconditional correlations is rejected at the 1% level. This means that the regression defined in (10.) can be done without any problems.

Table 6 shows the results of regressing the unconditional correlations on a second-order trend and index futures trading volumes. The results are similar across all three tested indices. The regressions show that the parameters of a second-order time trend are significantly differently from 0. The index futures trading volume coefficient is also significantly different from 0, even when controlled for the time trend. A simple time trend does not explain the relation between index futures trading volume and the

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23 unconditional correlations completely, so there may still be a significant relation between stock correlations and the trading of index futures.

6.3 DCC-GARCH model results

In Section 5.3 a total of 5 different models have been described and are clarified in Tables 7-9 alongside the results. The first two models that are tested against each other are Model 1 against Model 4 to control for the stationarity of the conditional correlation series. For the AEX, the restriction of α and β summing up to unity is rejected. This simply confirms that the correlation series does not have a unit root, so that we can continue with the other tests. The second test is Model 2 against Model 1 to test if a symmetric specification is a better fit. With an LR-statistic of 12.78 it is clear that the asymmetric specification performs better at a 0.035% significance level. All parameters are significantly different from 0, judging by the p-values of the parameters. The next extension is to add the index futures trading volume as an exogenous variable. Model 3 relieves the restriction on ϴ that was imposed in Model 2. Looking at the LR-test results of Model 3 versus Model 2 it can be concluded that adding the index futures trading volume is a better fit. Model 3 also shows that the trading volume coefficient is significantly different from 0. However, Model 5 will still reject the fact that index futures trading volumes explain the correlation between index constituents returns. Testing Model 5 against Model 2 we find that the LR-statistic is nearly 6 times as big as it was when testing model 3 against model 2. No actual test is performed to test if Model 5 is performing significantly better than Model 3, but looking at the difference in the log-likelihood it is safe to say that it does. Assuming that Model 5 is indeed a better fit than Model 2 then it turns out the trading volume coefficient is insignificantly different from 0.

For the DAX, the end result is similar to that of the AEX. Model 2 is performing significantly better than Model 1, showing that an asymmetric specification is a better fit. However, using an LR-test to test the restriction of it can be concluded that model

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24 3 is not significantly better than model 2. So far, the results of the DAX reject the hypothesis of a relation between index futures trading volume and the correlation between stocks listed on the DAX. Testing Model 5 against Model 2 however does show a significantly better result when using Model 5. The asymmetry around returns that are smaller than half their standard deviation captures the correlation dynamics better than the traditional asymmetry between positive and negative returns. The trading volume coefficient is insignificantly different from 0, just like the results on the AEX showed.

The third and final index that was tested is the EURO STOXX 50 index. Table 9 shows results similar to the AEX index again. The asymmetric specification in Model 2 is a better fit than the symmetric DCC-GARCH model from Model 1. Testing Model 3 against Model 2 shows that adding the trading volume coefficient results in a significantly better model. Lastly, testing Model 5 against Model 2 gives an LR-statistic that is three times greater than it was when testing Model 3 against Model 2. It is safe to say, although not proven, that Model 5 is a better specification than Model 2. The STOXX50 index is the only index that has a significant trading volume coefficient in the 5th model. Please note that the trading volumes were scaled down to a region between 0 and 1 and that a trading coefficient of 0.00043 is very small. The trading volume may be statistically significant, but it is economically negligible.

7 Conclusion and further research

The goal of this research was to find how the trading of index futures influences the correlation between stocks. Besides simply testing the effect of index futures trading volume, asymmetric behavior of stock return correlations has also been tested. There were three hypotheses that were tested, namely,

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25 - Higher index futures trading volumes increase the correlation between index

constituents.

- Stock correlations respond asymmetrically to stock returns.

- The asymmetric behavior of correlations is based on negative returns but rather on returns that are smaller than half their standard deviation.

By extending the asymmetric DCC-GARCH model proposed by Cappiello, Engle and Sheppard (2006) it has been possible to either support or reject the hypotheses listed above. This study has focused on European indices, namely the AEX, DAX and EURO STOXX 50 index and has found quite conclusive results. However, there is still a lot to be tested on this subject. This research may have stirred interest to further research the unconventional asymmetric specifications used in the DCC-GARCH models.

7.1 Conclusion

The results on both the AEX and the DAX reject the first hypothesis of index futures trading volume influencing the correlation between index constituents. The best fitting DCC-GARCH model for the AEX and the DAX both show that the trading volume coefficient is insignificantly different from 0. On the STOXX50 index however, the best fitting model does have a significantly positive trading volume coefficient. It should be noted that the trading volume used as a variable in the regressions has been scaled down to an interval between 0 and 1. Although the STOXX50 shows a significant relation between index futures trading volume and stock correlations, the trading volume coefficient is only 0.00043 and seems economically negligible. It is safe to say that the first hypothesis is rejected.

For all three tested indices it has turned out that an asymmetric specification of the DCC-GARCH model is a better fit than the symmetric specification. Using LR-tests to compare different models all restrictions on the asymmetric coefficient are rejected. The results support the second hypotheses of asymmetric responses of correlations to stock

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26 returns. The third and last hypothesis was hard to compare since Model 3 and Model 5 had to be compared, as shown in Tables 7-9. Although it is not statistically tested, it seems quite clear that Model 5 is a better fit than Model 3. The LR-statistics of testing Model 5 against Model 2 are 3 to 25 times as large as the LR-statistics of testing Model 3 against model 2. The large increase of the LR-statistics is enough to safely assume that Model 5 is a better specification than Model 3. The finding that Model 5 is the best tested specification supports the third hypothesis. The asymmetric response to stock returns may not be exactly centered on returns being less than half their standard deviation but that specification does perform better than the conventional asymmetry of negative returns against non-negative returns. It is of interest to define the asymmetry in stock correlations based on the results by Preis et al. (2012) more often. To summarize, the index constituents’ correlation is insignificantly explained by index futures trading volumes and the asymmetric response of correlation is based on returns less than half their standard deviation rather than based on negative returns.

It was shown earlier that it may be of interest to use a second-order trend to explain the dynamics of stock correlation between 2004 and 2014 but this was not done due to technical difficulties. The second-order curvature is seen in both the unconditional correlations and in the index futures trading volume which does show some relation.

There are a few explanations for the results found in this research. Antoniou, Holmes and Priestley (1998) had found that the index futures market also has some asymmetry, meaning that the volatility and therefore trading volumes, of index futures are higher following negative returns. The tested models clearly show that stock correlations also respond asymmetrically to past returns. It seems that negative returns are the cause of the similar curvature of unconditional stock correlations and index futures trading volume. Negative returns cause both higher correlations and more activity in the index futures market. When the models are correctly controlled for asymmetric response of correlations, the index futures trading volume effect is insignificant. It was explained earlier that index futures are often traded by investors to hedge their portfolio against losses. Hedging might be the reason that the index futures trading market has

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27 asymmetric characteristics. It is worth looking into the nature of the asymmetry in the futures trading market by regressing index futures trading volume on negative index returns. If the speculated explanations above are true, then that would probably mean that the asymmetry in the index futures market is also not based on a difference response to positive and negative returns but rather some alternative specification. An alternative, less sophisticated, explanation would be that the index futures market is simply too small compared to the trading done on the entire underlying index.

7.2 Suggested further research

All model specifications are clarified in Tables 7-9 along with the results. Below are two additional alternative specifications of the models that may be analyzed. The alternatives are described to motivate future research on the topic of this study but are not tested in this research.

A sixth model would be based on the curvatures that were studied earlier in this research. The regression in (10.) showed that there is in fact a quadratic trend present in the unconditional correlations. Naturally the results from (10.) lead to the model in (17.) to be extended similarly to (10.) with a quadratic time-trend. However, the seventh model was not computed as adding a time-trend to (17.) resulted in errors while computing the log-likelihood of the model. It is still of great interest to test for a quadratic time trend and is therefore left for revision or future research.

Gulen and Mayhew (2000) have also used futures trading volume in their research and, as stated before, have split up the variable into two components through the use of ARIMA models. An expected component of the index futures trading volume, which is meant to capture the base-line levels of futures trading, and an unexpected component that is used to represent sudden movements in the trading volume. A similar approach may be used to separate the period peaks of high trading activity from the ‘base-line’. Since the peaks are consistent with expiration dates of futures, it is not necessary to

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28 estimate ARIMA models like Gulen and Mayhew have done. It may also be sufficient to extend (14.) with a dummy variable for expiry dates, most likely yielding an equivalent effect. This would be more similar to the research done by Joensuu (2010) who has focused his research on the expiry dates of futures. Examining the expiration day effects is worth an entire different study and therefore the methods above are left as a suggestion for future research.

Finally, if we were to say that a different response to positive returns or negative returns is an asymmetric response centered around 0. And that a different response to returns more or less than half the standard deviation of returns is centered around 0.5. Then it is worth finding a way to find the value around which asymmetric behavior is centered. Usually it is assumed that asymmetry is centered around 0, but this research, amongst others, has shown that this is not necessarily the case. The alternative asymmetric specification is still linear, and so finding a correct ‘center’ should not be too challenging. Note that anything can have an asymmetric response to returns, not just correlations or volatility.

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Appendix

0 100000 200000 300000 400000 500000 600000 700000

EUREX-DAX INDEX CONTINUOUS - VOLUME TRADED EUREX-DAX INDEX CONTINUOUS - VOLUME TRADED second-order trendline 0 50000 100000 150000 200000 250000

AEX-AEX INDEX CONTINUOUS -

VOLUME TRADED

AEX-AEX INDEX CONTINUOUS - VOLUME TRADED ETICS00(VM)

second-order trend line

0 1000000 2000000 3000000 4000000 5000000 6000000 7000000

EUREX-EURO STOXX 50 CONTINUOUS - VOLUME TRADED

EUREXEURO STOXX 50 CONTINUOUS -VOLUME TRADED GEXCS00(VM) second-order trend line

Figures 1-3 show the index futures daily trading volume of the AEX, DAX and STOXX50 index respectively. All the data is fitted with a second-order trend line that shows a curvature that is similar across all indices. The similar curvatures lead to believe that there has been a different outside factor that has influenced the indices across Europe, possibly even throughout the world. The periodic peaks in the graphs correspond to expiration days of futures contracts.

Figure 1 Figure 2

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32 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

AEX UNCONDITIONAL CORRELATION

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

DAX UNCONDITIONAL CORRELATION

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 STOXX50 UNCONDITIONAL CORRELATION

Figures 4-6 show the monthly average unconditional correlation of the AEX, DAX and STOXX50 constituents. The blue lines are the monthly unconditional

correlations. The data is fitted with a second-order trend line, shown in red.

The trend line shows the same curvature as was seen in the index futures trading volume although the similarity is less pronounced for the larger DAX and STOXX50 index than it is for the AEX index.

Figure 4 Figure 5

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33 Index Constituents

AEX DAX Stoxx50

AEGON ADIDAS AIR LIQUIDE

AHOLD KON. ALLIANZ AIRBUS GROUP

AKZO NOBEL BASF ALLIANZ (XET)

ARCELORMITTAL BMW ANHEUSER-BUSCH INBEV

ASML HOLDING BAYER ASML HOLDING

BOSKALIS WESTMINSTER BEIERSDORF ASSICURAZIONI GENERALI

CORIO COMMERZBANK AXA

DELTA LLOYD GROUP *** CONTINENTAL BASF (XET)

DSM KONINKLIJKE DAIMLER BAYER (XET)

FUGRO DEUTSCHE BANK BBV.ARGENTARIA

GEMALTO DEUTSCHE BOERSE BANCO SANTANDER

HEINEKEN DEUTSCHE POST BMW (XET)

ING GROEP DEUTSCHE TELEKOM BNP PARIBAS

KPN KON E ON CARREFOUR

OCI *** FRESENIUS MED.CARE CRH (DUB)

PHILIPS ELTN.KONINKLIJKE FRESENIUS DAIMLER (XET)

RANDSTAD HOLDING HEIDELBERGCEMENT DANONE

REED ELSEVIER (AMS) HENKEL PREF. DEUTSCHE BANK (XET)

ROYAL DUTCH SHELL A INFINEON TECHNOLOGIES DEUTSCHE POST (XET)

SBM OFFSHORE K + S DEUTSCHE TELEKOM (XET)

TNT EXPRESS *** LANXESS *** E ON (XET)

UNIBAIL-RODAMCO LINDE ENEL

UNILEVER CERTS. DEUTSCHE LUFTHANSA ENI

WOLTERS KLUWER MERCK KGAA ESSILOR INTL.

ZIGGO *** MUENCHENER RUCK. GDF SUEZ ***

RWE SOCIETE GENERALE

SAP IBERDROLA

SIEMENS ING GROEP

THYSSENKRUPP INTESA SANPAOLO

VOLKSWAGEN PREF. INDITEX

L'OREAL

LVMH

MUENCHENER RUCK. (XET)

ORANGE PHILIPS ELTN.KONINKLIJKE REPSOL YPF RWE (XET) SAINT GOBAIN SANOFI SAP (XET) SCHNEIDER ELECTRIC SIEMENS (XET) TELEFONICA TOTAL UNIBAIL-RODAMCO UNICREDIT UNILEVER CERTS. VINCI VIVENDI

VOLKSWAGEN PREF. (XET)

Table 1

Table 1 shows the constituents of the relevant indices. Stocks that are marked with

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34 Table 2

AEX Stock Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis

Jarque-Bera Prob. AEGON -0.000196 0 0.302154 -0.276836 0.029946 0.241416 17.91965 23805.57 0 AHOLD_KON_ 0.000282 0.000505 0.101762 -0.096419 0.015308 -0.061452 7.825957 2489.747 0 AKZO_NOBEL 0.000207 0 0.183581 -0.114961 0.018752 0.402461 11.89856 8528.742 0 ARCELORMITTAL 9.92E-05 0.000402 0.182928 -0.209882 0.030743 -0.055651 8.498859 3231.692 0 ASML_HOLDING 0.000545 0.00055 0.109628 -0.094378 0.020455 0.168818 4.991455 435.8682 0 BOSKALIS_WESTMINSTER 0.000733 0.000641 0.150654 -0.143606 0.023242 0.027252 8.91135 3733.507 0 CORIO 6.77E-05 0.000657 0.095387 -0.102751 0.018209 -0.268405 6.661015 1462.676 0 DSM_KONINKLIJKE 0.000385 0.000624 0.089373 -0.167019 0.017453 -0.719609 11.75189 8404.256 0 FUGRO 0.000572 0.001157 0.193073 -0.178277 0.02246 -0.584021 13.65348 12270.97 0 GEMALTO 0.000669 0 0.198714 -0.20137 0.022346 -0.104915 11.51495 7750.582 0 HEINEKEN 0.000229 0.000181 0.095864 -0.079444 0.014107 0.092113 7.786472 2451.211 0 ING_GROEP -0.000145 0.000481 0.256527 -0.321361 0.032381 -0.019294 17.255 21709.24 0 KPN_KON -0.00016 0 0.157182 -0.172617 0.017614 -0.176273 18.42713 25439.22 0

PHILIPS_ELTN_KONINKLIJKE 2.92E-05 -8.91E-05 0.117873 -0.1236 0.019143 -0.007177 7.074713 1773.806 0

RANDSTAD_HOLDING 0.000248 0.000397 0.130105 -0.17895 0.024465 -0.336676 7.993778 2712.628 0

REED_ELSEVIER__AMS_ 8.35E-05 4.46E-05 0.100437 -0.13425 0.014749 -0.547825 11.28841 7467.464 0

ROYAL_DUTCH_SHELL_A 0.000107 0.000204 0.131038 -0.09413 0.014304 0.214738 13.62325 12076.2 0

SBM_OFFSHORE 0.000116 0.00036 0.187905 -0.282144 0.023819 -0.58653 17.80575 23565.96 0

UNIBAIL_RODAMCO 0.000399 0.000563 0.088079 -0.183588 0.017662 -0.452033 9.571699 4701.155 0

UNILEVER_CERTS_ 0.000168 0 0.083948 -0.088162 0.013406 -0.053181 8.12763 2810.134 0

WOLTERS_KLUWER 0.000151 0.000468 0.076426 -0.089259 0.014918 -0.177152 6.273256 1158.045 0

Tables 2-4 show some standard characteristics of the returns of the index constituents. The stocks that are marked with ** in Table 1 are no longer present in Tables 2-4.

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35 Table 3

DAX Stocks Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis

Jarque-Bera Prob. ADIDAS 0.000442 0.000246 0.147437 -0.135981 0.017943 0.014369 9.382069 4326.038 0 ALLIANZ 0.000109 0.000509 0.17808 -0.13918 0.020804 0.410273 13.03217 10760.78 0 BASF 0.000508 0.000985 0.126906 -0.129239 0.017828 -0.14852 9.963772 5159.85 0 BAYER 6.07E-04 0.000405 0.108023 -0.104965 0.017368 -0.106076 6.893037 1614.446 0 BEIERSDORF 0.000328 0.000451 0.074247 -0.099668 0.014267 0.055457 6.831418 1560.42 0 BMW 0.00035 0.000519 0.135184 -0.130527 0.019997 0.077537 8.556502 3281.707 0 COMMERZBANK -7.50E-04 -0.000315 0.194573 -0.282482 0.031276 -0.141476 11.23787 7216.069 0 CONTINENTAL 0.000604 0.000776 0.253335 -0.325096 0.028446 -0.32365 18.50064 25563.16 0 DAIMLER 0.000222 0.000649 0.178879 -0.142223 0.021826 0.242705 10.23603 5586.113 0 DEUTSCHE_BANK -0.000282 0.000138 0.22306 -0.180705 0.025485 0.361464 13.19838 11101.91 0 DEUTSCHE_BOERSE 0.000328 0.000189 0.17185 -0.124232 0.02224 0.15673 9.021755 3861.713 0 DEUTSCHE_LUFTHANSA 0.000132 0.00036 0.082238 -0.085876 0.01893 -0.134733 4.787695 347.1382 0 DEUTSCHE_POST 0.000143 0.000495 0.145301 -0.174083 0.018011 -0.353526 11.35628 7469.357 0

DEUTSCHE_TELEKOM -8.49E-05 0.00E+00 0.132554 -0.132554 0.015136 -0.073942 14.03457 12934.44 0

E_ON -0.000134 0 0.158859 -0.107496 0.018139 -0.152384 11.73439 8112.445 0

FRESENIUS 6.31E-04 1.43E-04 0.128833 -0.08948 0.016798 0.00804 6.829066 1557.227 0

FRESENIUS_MED_CARE 0.000373 3.90E-05 0.12229 -0.092021 0.013505 0.075646 9.14773 4016.531 0 HEIDELBERGCEMENT 0.000223 0 0.193769 -0.246684 0.025508 0.176671 13.38789 11474.03 0 HENKEL_PREF_ 0.000461 0.00012 0.106535 -0.074141 0.015592 0.132763 6.691393 1454.723 0 INFINEON_TECHNOLOGIES -7.55E-05 0 0.200781 -0.486434 0.032417 -1.343226 29.48237 75252.11 0 K___S 0.000503 0.000811 0.172112 -0.24961 0.026154 -0.687687 11.04114 7068.328 0 LINDE 0.000467 0.000447 0.128547 -0.082172 0.015959 0.117541 9.204551 4094.514 0 MERCK_KGAA 0.000382 0.000165 0.092487 -0.145506 0.01695 -0.39929 9.0648 3974.265 0 MUENCHENER_RUCK_ 0.000226 0.00047 0.145096 -0.111416 0.016092 0.182872 11.00598 6821.702 0 RWE -0.000123 0.000112 0.142558 -0.115608 0.017441 -0.130621 10.47196 5936.877 0 SAP 0.000217 0.000522 0.134532 -0.165593 0.016167 -0.448328 14.3306 13720.69 0 SIEMENS 0.000169 0.000418 0.165995 -0.16363 0.019122 -0.346962 14.73451 14675.9 0 THYSSENKRUPP 0.000114 0.0007 0.167887 -0.161754 0.023647 -0.112155 8.146186 2818.084 0 VOLKSWAGEN_PREF_ 0.000793 0.000891 0.183426 -0.252441 0.024706 -0.734185 15.99819 18173.2 0