The four-factor model supplemented by an endogenous factor Dutch mutual funds

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The four-factor model supplemented by an endogenous factor

Dutch mutual funds

Author: Sander van Laarhoven

Master thesis

Master of Science in Business Administration, specialization Finance

August 2012

Supervisor: dr. L. (Lammertjan) Dam

2

nd

Supervisor: dr. A. (Auke) Plantinga

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The four-factor model supplemented by an endogenous factor

Dutch mutual funds

Author: Sander van Laarhoven

*

Abstract

This research adds the endogenous factor to the four-factor model for 137 Dutch equity mutual funds in the period January 1993 to December 2010. The endogenous factor is based on funds classified in the same group. These groups are based on fund style and magnitude. The resulting models with the aid of this endogenous factor are the omitted factor model, the correlated errors model and the single-factor model and doing very well compared to the four-factor model in explaining fund returns. These models shows that the endogenous factor is very significant and thus increases the ability to explain the Dutch mutual equity fund returns.

JEL codes: G11, G23

Keywords: endogenous factor, Dutch equity mutual funds, alpha, four-factor model, omitted factors model, correlated errors model, single factor model

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Introduction

The endogenous factor integrated into the four-factor model. There is much said and written about the Capital Asset Pricing Model (CAPM) introduced by Sharpe (1964) and Lintner (1965). Over the years, this model is even transformed into other models, like the three-factor model of Fama and French (1993) and the extended version of this, the four-factor model of Carhart (1997). The aim of these models is to explain the expected returns of assets. Many researchers have given their advantages and disadvantages about these models. While nobody can predict the future, why should we still build more and new extended models? Do we want to beat the market? In this case, researchers try over and over to explain the returns in the best possible way. In this research, I made just like the paper by Hunter et al. (2011) an adjustment on the four-factor model. This adjustment is to add an endogenous factor to the four-factor model. The endogenous factor is obtained by group returns which compose of funds with the same style and magnitude. With this newly obtained variable, the omitted factors model and the correlated errors model are created. The relevant question here is whether this endogenous factor is better in explaining the Dutch equity fund returns in the period January 1993 to December 2010 than the four-factor model. In order to answer this question, I conducted several regressions. Four-factor regressions are used on group returns and individual funds and shows that fund returns are not very well explained by this model. In addition, the omitted factors model, the correlated errors model and the single-factor model are regressed and shows evidence that the endogenous factor is important to increase the ability in explaining the Dutch equity fund returns in the period January 1993 to December 2010. Further, I will briefly discuss that the endogenous factor affects the alpha’s in its size and accuracy, but also the standard error is influenced by the endogenous factor.

Hunter et al. (2011) focuses on funds in the United States. Also focused on funds in the United States is the paper by Garyn-Tal and Lauterbach (2011). To see whether this methodology of the endogenous factor is also applicable to other markets, I take the Dutch fund market. Fixed income funds are excluded from this research, I base my research on equity funds only.

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I. Literature review

Assets can be priced in various ways. A way to describe stock returns in the asset pricing and portfolio management, Fama and French (1993) designed a three-factor model. This is an adjustment of the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965). They use only one factor to describe the returns of stocks. The equation of the CAPM is shown in equation (1):

( ) (1)

Where, ( ) is the only factor and is called the market factor. Jensen (1968) used the CAPM model to create the alpha, the well-known Jensen’s alpha. When the alpha is positive, it suggest that there is an abnormal return relative to the benchmarks. Today the CAPM model is much criticized model. Fama and French (1993) added two factors in their model, this are the size and the book-to-market ratio. This model is known as the three-factor model and is shown in equation (2):

( ) (2)

Where stands for Small Minus Big. Put differently, the small market capitalization minus the big one. This is the portfolio that mimics the factors of risk that are related to size. It represents the portfolio that buys small stocks and sells big stocks. stands for High Minus Low. Put differently, this is the high book-to-market ratio minus the low one. This is the portfolio that mimics the factors of risk that are related to the book-to-market value ratio. This factor represents the return on a portfolio that buys high book-to-market stocks and sells low book-to-market stocks. These factors are also known as the growth and value stocks.

An adjustment to this model leads to the four-factor model. This is the strategy of buying winners and selling losers, as documented by Jegadeesh and Titman (1993). Carhart (1997) incorporated this factor to create a four-factor model, this model is shown in equation (3):

( ) (3)

Where, stands for Winners Minus Losers. This factor relates to the fact that funds that performed well during the last twelve months, tend to earn higher than expected returns in the next twelve months.

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models as mentioned above. This decreases the power of these models to select good performing funds. That is why the paper of Hunter et al. (2011) uses another additional factor. This additional factor is based on an equal investment in all same category funds. They call their additional factor an endogenous benchmark, because each fund chooses a group with which it intends to compete. Hunter et al. (2011) show in their paper that the endogenous benchmark improves the selection of funds with future outperformance. They wanted a simple and an easy implementable way to control for fluctuations in markets that affect fund returns in an economic, and asset group wide area. Thus they have chosen for the additional benchmark, in other words the group return. The model of Hunter et al. (2011) is shown in equation (4):

( )

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In this equation, stands for the group return. Groups are formed by their category and style of the funds. Thus a group contains funds with the same category. Hunter et al. (2011) have formed five categories of equity funds.

Hunter et al. (2011) gives four reasons to use this endogenous benchmark instead of multiple exogenous factors which individually funds are dealing with.

Firstly, from the perspective of an investor with an already selected asset allocation, but has yet to choose which funds to invest in. Thus he needs help in choosing the best funds within a specific group. To ensure that the individual investor invest his money in the fund instead of the entire group, the fund manager can persuade the investor that the fund has superior performance compared to the other funds. By controlling for the risks of the individual fund compared to the entire group, the superior performance is calculated. This allows us to use the investment in the group as an endogenously-established benchmark for each individual fund that belongs to a specific group. The fund that claims to have superior performance will also be considered to be part of the growth funds. Hunter et al. (2011) argues that the managers of these individual funds that claim to have superior performance relative to other funds, implicitly choose to constitute a benchmark of all growth funds. By using the entire group for benchmarking the risks of the individual funds has an alternative investment interpretation. This means that the aim is on identifying the best strategies within a certain group that has its focus on the same area of stocks.

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group are exposed to these factors, then the entire group is exposed to these factors. Hunter et al. (2011) shows that the group returns can be used to control for the average exposure of these unperceived factors of risk. It gives a better estimate of the alpha of the fund with a simple control variable.

Thirdly, also when there are no hidden priced factors, it can be possible that many managers of the funds in a specific group undertake the same security bets. This is because they use the same models and they have the same behavioral biases. They may also be geographically in the same area. This leads to correlated error terms of individual fund returns in a same group. In that case, the group return captures these similarities.

Finally, the last reason described by Hunter et al. (2011) is that when the exposure of the funds to the known factors of risk varies during the time period, and in this variation there is a similar component between the funds, the group return will capture this co-movement.

There consists a very large list of papers which examines the performance of (equity) mutual funds. A few examples are the studies of Lehmann and Modest (1987), Grinblatt and Titman (1992), Hendricks, Patel and Zeckhauser (1993), Grinblatt and Titman (1994), Malkiel (1995), Gruber (1996) and Shukla and Singh (1997). Papers which are more in line with this paper by means of the models to be used and or sample time periods are for example Otten and Bams (2002), Bauer, Koedijk and Otten (2005), Fama and French (2010), Garyn-Tal and Lauterbach (2011).

The paper of Otten and Bams (2002) investigates mutual funds, using the four-factor model. Their data is based on the five most important mutual fund countries in Europe including the Netherlands. They use in their paper equity funds, as I do in this paper. A result of their paper is that they show that the group of small cap mutual funds outperform their benchmark. A difference between large and small mutual fund sectors according to Otten and Bams (2002) is that the larger the fund sector will be, the more difficult it is to outperform the market as a group. The ultimate question of Otten and Bams (2002) is whether managers of European funds have ‘hot hands’, which means that they are persistence in performance. They find weak evidence for this. In addition, Wermers (2003) find in his paper that returns of mutual funds strongly persist over periods with many years. He relates this result to the fact that the persistence is partially attributable to the behavior of the fund manager and also the consumer, because they actively chase for funds with high historical returns.

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investments. Finally, Bauer, Koedijk and Otten (2005) find that the ethical indices compared to the standard ones perform weaker in explaining returns.

For the years 1984 to 2006, Fama and French (2010) evaluate equity fund performance in the United States. There benchmarks they use are their own three-factor model and the four-factor model of Carhart (1997). Fama and French (2010) find that the value weight portfolio with active funds and the market portfolio are close to each other. The alpha’s they investigated before the expenses are close to zero within common benchmarks. They conclude that the investors of the funds have net returns that not only underperform the CAPM, but also the three-factor model and the four-factor model. This underperforming can be related to the costs in expense ratios.

Garyn-Tal and Lauterbach (2011) compares in their paper several benchmark models, like the three-factor model, the four-factor model and the five-factor model where the fifth factor is the endogenous factor according to Hunter et al. (2011). They evaluate the influence of the alpha’s of non-specialized equity mutual funds in the United States in the period 2001 to 2009 on the just mentioned various models. Garyn-Tal and Lauterbach (2011) find in their paper that funds underperform their benchmarks, thus they do not beat their benchmark. The alpha’s vary in magnitude when other models are used.

Wei Wei Shi and Seiler (2002) used the same category classification as I will do in this paper (see the subsection data in the second section for the classification). The aim of the paper by Wei Wei Shi and Seiler (2002) is to investigate whether there are significant differences between the returns and the risks of growth and value funds. They conclude that the size of the funds affects the results. They show that the large growth and large value funds do not outperform the benchmarks. But when the size for the growth and value funds is small, then they outperform the benchmarks. Wei Wei Shi and Seiler (2002) conclude the paper by saying that the performance in the past is not a guarantee of performance in the future.

II. Data and methodology

In the first part of this section I will explain the gathered data and some descriptive statistics. The second part of this section I devote to the methodology.

A. Data

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classification is also used in the paper of Wei Wei Shi and Seiler (2002). Equity funds can be divided into a style, these are: Growth, Mixed or Value. Also the equity funds will be divided according to their magnitude, these are: Large, Middle or Small. An overview of the number of funds classified by style and magnitude is presented in Appendix A. After this subdivision, I have the following groups: Growth Large, Growth Middle, Growth Small, Mixed Large, Mixed Middle, Mixed Small, Value Large, Value Middle and Value Small.

I have obtained monthly returns from Datastream for the Dutch mutual funds. First I removed the fixed income funds from the list. To distinguish between fixed income and equity funds, I used the website of Morningstar to search for their category. For several equity funds their classification into style and or magnitude were not available. Also for some equity funds their returns were not available. And there were some equity funds which has not a sufficient amount of months for which returns were available in the given time period. Here I have set the limit that for at least 30 months returns must be available in time periods of 36 months. See the following subsection for this time classification. I left out these funds with no complete data. After all these omissions I have 137 equity funds left. These funds are included in Appendix B. Table I presents the sample size of the equity funds in the different groups to the related periods.

Table I: Sample size Dutch equity funds

This table presents the sample size of the Dutch equity funds in the period January 1993 to December 2010. The first column shows the group names. The following six columns shows the time periods of each three years with their amount of funds in that specific time period.

Jan 1993 - Dec 1995 Jan 1996 - Dec 1998 Jan 1999 - Dec 2001 Jan 2002 - Dec 2004 Jan 2005 - Dec 2007 Jan 2008 - Dec 2010 Growth Large 4 4 7 9 11 17 Growth Middle 0 1 2 3 4 5 Growth Small 1 1 1 2 2 2 Mixed Large 10 15 22 40 51 66 Mixed Middle 0 1 2 3 3 5 Mixed Small 0 1 1 1 1 2 Value Large 1 3 8 16 23 28 Value Middle 0 2 2 4 4 4 Value Small 1 3 3 4 4 8

The groups with the most funds are: Growth Large, Mixed Large and Value Large. It may be concluded that most of the equity funds have a large magnitude.

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European factors. Another important reason to choose for the European factors is that 125 funds out of the 137 total funds invest in European countries with a mean of 54.9%. And 81 funds out of the 137 total funds invest in the United States with a mean of 39.7%. Therefore, I conclude that fewer Dutch funds invest in the United States as compared to Europe. And when they invest in the United States, the average investment is less than when the Dutch funds invest in Europe.

The descriptive statistics from monthly group returns over the time period January 1993 to December 2010 are presented in Table II. In this table the average, standard deviation, minimum and maximum of the group returns are presented. In the same table, these descriptive statistics are also shown for the factors of the four-factor model. Which are the market return, SMB, HML and WML.

Table II: Descriptive statistics

This table presents the descriptive statistics of the monthly group returns and the factors; market return, SMB, HML and WML of the four-factor model. These descriptive statistics are based on the entire time period of January 1993 to December 2010. The first column shows the group name. The averages are listed in de second column. The third column contains the standard deviation. And in the last two columns the minimum and maximum are presented respectively.

Average

Standard

deviation Minimum Maximum Growth Large 0.007 0.051 -0.168 0.140 Growth Middle 0.005 0.042 -0.175 0.150 Growth Small 0.012 0.076 -0.206 0.307 Mixed Large 0.006 0.040 -0.114 0.096 Mixed Middle 0.011 0.051 -0.196 0.265 Mixed Small 0.013 0.066 -0.221 0.310 Value Large 0.006 0.042 -0.127 0.104 Value Middle 0.008 0.049 -0.241 0.260 Value Small 0.010 0.047 -0.166 0.166 Market return 0.006 0.050 -0.221 0.138 SMB 0.001 0.024 -0.069 0.093 HML 0.006 0.025 -0.096 0.110 WML 0.009 0.044 -0.260 0.138

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2000s which I also include in my data. Causes of these are the ‘Internet bubble’ and the financial crisis. It can be seen that, also for the factors of the four-factor model, the averages are all positive.

B. Methodology

I divide the entire time period January 1993 to December 2010 in parts of three years, which results in six periods of three years. This division of time periods is in line with the paper of Hunter et al. (2011) and Garyn-Tal and Lauterbach (2011). In these periods of three years or 36 months, I remove the periods of funds which have less than 30 month return observations in a period. This therefore results in a number of 137 funds remaining.

I will use the four-factor model of Carhart (1997) first. I run this regression for all the funds separately and also for every time period of three years with more than 30 month return observations. The four-factor regression on funds is shown in equation (5):

( ) (5)

Where is the return of fund and stands for the corresponding time period. The factor loadings for the risk variables of the market, SMB, HML and WML are respectively , , and

. The error term is denoted by .

I use all the fund returns which belong to the same group and equal weight them to create the group returns. The four-factor regression is also made for these equal weighted group returns, see equation (6). Again this regression is made for every time period of three years with more than 30 month return observations. These group returns are denoted by .

( ) (6)

With the aid of this last regression, I obtain estimates for every time period of the group alpha ( ) and the standard error of the group ( ). Adding these two together, it yields the following variable (7):

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et al. (2011) calls this model the omitted factors model. Further in this paper, if I mention this model, I will also refer to the omitted factors model. This model is shown in equation (8):

( )

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The additional factor here is of equation (7) and the corresponding factor loading is . This regression, the omitted factor model, in equation (8) is made for every fund and for every time period within group .

A modification to this omitted factors model helps to control for dynamic factor loadings. This leads to the correlated errors model and is shown in equation (9):

( )

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Again the variable comes from the four-factor model on the group returns. So be aware that the group alpha is excluded from this model compared with the previous model. Also this regression is made for every fund within group and for every time period. Further in this paper, if I mention this model, I will refer to the correlated errors model.

The differences between the omitted factors model (equation (8)) and the correlated errors model (equation (9)) leads to a distinction between the group alpha and the alpha of an individual fund. The alpha of the group is related to co-movement and the alpha of the individual fund is unrelated to co-movement. In the following equation (10) the difference between the alpha’s of the omitted factors model and the correlated errors model is shown.

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the fund’s alpha. The co-movement related effects, which I just mentioned, is the part that is acquired from equation (10). Hunter et al. (2011) states, by using a common strategy, that this correlated alpha appears if a group of funds is able to outperform the average fund in that specific group.

In addition to the omitted factors model and the correlated errors model, I also apply the single-factor model for every fund. This model consists of one factor, namely the endogenous benchmark, see equation (11):

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In contrast to the other two models, the single-factor model does not contain any other factors. The only factor included in this model is the endogenous benchmark ( ). Hunter et al. (2011) appoints an advantage and a disadvantage of this model in their paper. The advantage of this model is that the degrees of freedom are maintained. And since there are no other factors, this model is a simple investment strategy. But the disadvantage is that the outcomes of alpha in this single-factor model are less accurate than the outcomes of alpha in the omitted factors model. So what value adds this single-factor model to this paper? The added value of this single-factor model is to show how the endogenous factor performs in a particular case.

III. Results

This section contains the results of this research. Subsection A is about the results of the four-factor regression on the composed groups. In subsection B, I will explain the results of the regressions of the four-factor model, omitted factors model, correlated errors model and the single-factor model on the equity funds. The answer on how accurate the alpha’s are is given in subsection C. And the last subsection (subsection D) shows the impact of the endogenous factor.

A. Group results

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Table III: Four-factor model regression on group returns for January 1993 to December 2010 This table presents the outcomes of the four-factor model regression on group returns ( ( ) ). Group returns are equally weighted and grouped by their category. For every group and time period of three years the regressions is made. This table shows for every group the alpha and the F-statistic. This is shown in the first and second column. The column three to eight gives the outcomes of the regression. One asterisk (*) behind the alpha and F-statistic indicates that it is significant at the 10% significance level. Two asterisks (**) indicates a significance level of 5%. Three asterisks (***) indicates a significance level of 1%.

Group Jan 1993 - Dec 1995 Jan 1996 - Dec 1998 Jan 1999 - Dec 2001 Jan 2002 - Dec 2004 Jan 2005 - Dec 2007 Jan 2008 - Dec 2010 Growth Large alpha 0.002 -0.013* 0.020*** 0.002 -0.011* 0.002 F-statistic 6.230*** 12.430*** 28.401*** 10.038*** 7.511*** 12.194*** Growth Middle alpha -0.003 0.000 -0.001 -0.009** 0.002 F-statistic 3.305** 3.133** 12.654*** 25.015*** 19.653*** Growth Small alpha -0.003 -0.006 0.036** -0.007 -0.005 0.000 F-statistic 5.003*** 8.187*** 9.558*** 7.088*** 13.261*** 15.093*** Mixed Large alpha -0.002 -0.002 0.010* -0.004 -0.010** 0.000 F-statistic 7.479*** 13.049*** 10.329*** 18.793*** 13.554*** 14.279*** Mixed Middle alpha 0.003 0.021*** 0.003 -0.012* 0.003 F-statistic 2.292* 4.865*** 10.400*** 10.626*** 24.162*** Mixed Small alpha 0.003 0.002 0.007 0.011 0.009 F-statistic 7.620*** 3.925** 25.053*** 3.846** 8.075*** Value Large alpha 0.000 -0.001 0.008 -0.004 -0.009** -0.002 F-statistic 7.023*** 17.010*** 14.907*** 26.303*** 16.024*** 20.318*** Value Middle alpha -0.001 0.000 0.005 0.007 -0.012* -0.002 F-statistic 0.6230 16.190*** 6.745*** 13.891*** 8.367*** 43.857*** Value Small alpha 0.004 -0.007 0.009 0.001 -0.003 -0.005 F-statistic 6.073*** 11.420*** 5.977*** 16.605*** 44.665*** 23.926***

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Within the alpha’s, they are most of the time not significant. Only five, four and two alpha’s of all the groups are significant respectively at the 10% level, 5% level and the 1% level. There are two groups, ‘Mixed Small’ and ‘Value Small’, which does not exhibit significant alpha’s at all. Furthermore, there is no clear difference between positive and negative alpha’s. Although the group ‘Mixed Small’ is the only group with only positive alpha’s for every time period.

A noticeable point is that all the alpha’s are very low. The average of the positive alpha’s is 0.63% and the average for the negative alpha’s is -0.56%. Thus in both cases, this is approximately slightly more than a half percentage point.

B. Regressions of the various models on equity funds

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Table IV: Regressions on equity funds for three year periods from January 1993 to December 2010

This table presents what percentages of the coefficients are positive or negative and whether they are significant or not significant at the 5% significance level. The first three models for which this is done are the four-factor model ( ( )

), the omitted factors model ( ( ) _{) and the correlated errors model (}

( ) ). Next to these columns of the models are the columns that presents the percentage of the RM-Rf (market), SMB, HML and WML coefficients obtained either from the omitted factors model or the correlated errors model. The penultimate column presents what percentage of the endogenous factor of either the omitted factors model or the correlated errors model is positive or negative and whether this is significant or not. And the last column presents what percentage of the alpha’s of the single-factor model ( ) are positive or negative and whether they are significant or not. Further the adjusted R2 is shown for every model in the last row. The regressions of the four models are run for every fund and over 36 month periods from January 1993 to December 2010.

Four-factor alpha Omitted factors model alpha Correlated errors model alpha RM-Rf SMB HML WML Endogenous factor Single-factor alpha Group Gro w th L arg

e Negative, not significant 37% 57% 33% 0% 41% 25% 22% 0% 51%

Negative, significant 8% 0% 14% 0% 18% 37% 43% 0% 12%

Positive, significant 4% 2% 10% 73% 18% 20% 14% 94% 0%

Positive, not significant 51% 41% 43% 27% 24% 18% 22% 6% 37%

Adjusted R2 37% 70% 70% 67% Gro w th M id d

le Negative, not significant 47% 47% 27% 0% 20% 40% 40% 0% 47%

Adjusted R2 45% 76% 76% 72%

Gro

w

th Sm

all Negative, not significant 70% 50% 50% 0% 0% 40% 30% 0% 40%

Adjusted R2 46% 88% 88% 82%

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Continued Table IV Group Four-factor alpha Omitted factors model alpha Correlated errors model alpha RM-Rf SMB HML WML Endogenous factor Single-factor alpha M ixed Larg

Adjusted R2 46% 77% 77% 73% M ixed M id d

Adjusted R2 46% 82% 82% 72%

M

ixed S

m

all Negative, not significant _{Negative, significant} 0% _0% 67% _17% 0% _0% 0% _0% 17% 33% 17% _{0% 33% 33%} _0%0% 17% _67%

Adjusted R2 40% 92% 92% 85%

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Continued Table IV Group Four-factor alpha Omitted factors model alpha Correlated errors model alpha RM-Rf SMB HML WML Endogenous factor Single-factor alpha Val u e Larg

Adjusted R2 34% 83% 83% 79% Val u e M id d

Adjusted R2 42% 78% 78% 74%

Val

u

e

Small

Negative, not significant 43% 48% 26% 0% 4% 39% 35% 0% 35%

Adjusted R2 50% 75% 75% 73%

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A remarkable point is that for the four-factor model the alpha’s are generally not significant at the 5% level. The adjusted R2 for this four-factor model vary between 37% and 55%, with an average of 45%. Thus the returns are not very weak or strong explained by the four-factor model. This means that the returns are not very well explained by the four-factor model. Another remarkable point is that almost all the endogenous factors for every group are positive and significant at the 5% level, see the second last column of Table IV. Further, the coefficients of the market return (RM-Rf) of either the omitted factors model or the correlated errors model are for the most part positive and the largest part is also significant. The other factors of these models are SMB, HML and WML where the coefficients are widely distributed. Hunter et al. (2011) has the same results in their paper. This is true for the alpha’s of the four-factor model, the coefficients; market return, SMB, HML and WML and for the endogenous factor. They conclude that the regression submitted with a group return is equally or even more important than the SMB, HML and WML factors. I can also draw this conclusion that the group return (see column of the endogenous factor) is more important.

The adjusted R2 for the single-factor model ranches between 67% and 85% with an average of 75%. This is not bad for a model with just one factor. Indeed, this is very high for a single-factor model. Just like the four-factor model, the alpha’s of the single-factor model are generally not significant at the 5% level. But there is a difference between these models, namely the percentages of significant alpha’s have increased a little bit.

A result which is recognized from this table is consistent with an argument made by Hunter et al. (2011). They claim in their paper that it is helpful for all equity funds and in particular for groups of funds that are less explained by the four-factor model, to include the return of the group as a control for unobserved commonalities. A result of this table, which I just appointed above, is that for almost every fund, the endogenous factor is positive and significant and thereby the fund returns are not very well explained by the four-factor model. Thus the claim of Hunter et al. (2011) holds also for the results in Table IV.

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C. Accuracy of the alpha’s

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Table V: Accuracy of significant alpha's

This table presents percentages of significant alpha's classified by their t-statistic of the four-factor model ( ( ) ) versus the omitted factors model ( ( ) ) in the first matrix and the four-factor model versus the correlated errors model ( ( )

) in the second matrix. This table presents the entire period of January 1993 to December 2010. The first column of both matrices indicates the classification of the t-statistics of all significant alpha's of the four-factor model. The head-row of both matrices indicates the classification of the t-statistics of all significant alpha's of the omitted factors model and the correlated errors model. When the rows and columns will be crossed, the intersection shows what percentage of significant alpha's of both models falls into the corresponding classification.

Four-factor significant

alpha’s

Omitted factors model significant alpha’s

t < -2 -2 < t < -1.645 -1.645 < t < 0 0 < t < 1.645 1.645 < t < 2 t > 2 t < -2 8.7% 0% 0% 0% 0% 0% -2 < t < -1.645 0% 0% 0% 0% 0% 0% -1.645 < t < 0 0% 0% 0% 0% 0% 0% 0 < t < 1.645 0% 0% 0% 0% 0% 0% 1.645 < t < 2 30.4% 0% 0% 0% 0% 0% t > 2 0% 0% 0% 0% 43.5% 17.4% Four-factor significant alpha’s

Correlated errors model significant alpha’s

t < -2 -2 < t < -1.645 -1.645 < t < 0 0 < t < 1.645 1.645 < t < 2 t > 2 t < -2 52.4% 0% 0% 0% 0% 0% -2 < t < -1.645 0% 0% 0% 0% 0% 0% -1.645 < t < 0 0% 0% 0% 0% 0% 0% 0 < t < 1.645 0% 0% 0% 0% 0% 0% 1.645 < t < 2 0% 0% 0% 0% 0% 0% t > 2 0% 0% 0% 0% 34.5% 13.1%

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But there is one deviation. There is a group of significant alphas which are not at or near the diagonal line. According to this first matrix, there are more than 30% of the funds with significant alpha’s which have a positive statistic in the four-factor model between 1.645 and 2 and a negative t-statistic in the omitted factors model that is lower than -2. Therefore, I conclude that the omitted factors model is worse in increasing the accuracy of the alpha’s then the correlated errors model.

D. Impact of the endogenous factor

I also have the single-factor model in this study for which is something to say about. For this, I have calculated the rank correlation (Spearman’s rank correlation) of all the alpha’s between the four-factor model and the single-factor model. For an explanation of the Spearman rank correlation I refer to Appendix C. The results of this correlation are presented in Table VI. The reason why I get all funds together is that some groups have in the earliest periods a few funds in which alpha’s are available. This means that the coincidence that the ranking is the same is larger, this is due the very few available alpha’s. Thus to avoid the problem that the outcomes rather rely on coincidence than empirical evidence, I only calculate the rank correlations of all funds together.

Table VI: Alpha rank correlation

This table presents the rank correlations of the alpha's between the four-factor model ( ( ) ) and the single-factor model ( ) for the time period January 1993 to December 2010. The correlations are calculated for every time period of three years. Jan. 1993 - Dec. 1995 Jan. 1996 - Dec. 1998 Jan. 1999 - Dec. 2001 Jan. 2002 - Dec. 2004 Jan. 2005 - Dec. 2007 Jan. 2008 - Dec. 2010 Correlation 0.738 0.643 0.350 0.437 0.410 0.811

What I want to prove with Table VI, is which impact the endogenous factor has. The results of the correlations shows if the endogenous factor influence the size of the alpha’s or is it the standard error which is affected by the endogenous factor. The rank correlations vary between 0.350 and 0,811 and are not close to one. This indicates that the implementation of the endogenous factor affects both the size of the alpha’s and also the standard error.

IV. Conclusion

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classification into style and magnitude are divided into groups. The groups of equity funds are based on the style and magnitude of the funds. The four-factor model is run for equal weighted group returns and also for the all the funds only. With the aid of the four-factor regression on group returns, I obtained a new variable. This is the endogenous factor and is used in the model by Hunter et al. (2011). With this newly obtained variable, the omitted factor model is established. A small adjustment to this model results in the correlated errors model as in the paper by Hunter et al. (2011). By the use of these two models, the endogenous factor is obtained. This research shows that the endogenous factor is better and more important in explaining the returns than the factors of the four-factor model. The omitted factors model and the correlated errors model are doing very well in explaining the fund returns. The endogenous factor is also much more significant at the 5% level than the factors of the four-factor model. Applying only the endogenous factor in a model leads to the single-factor model. This model gives more significant alpha’s than the four-factor model and it has also higher adjusted R2, on average 75% against an average of 45% by the four-factor model. Thus the adjusted R2 of the single-factor model is very high for a model with just one factor and it means that the model explains the fund returns very well. Consistent with the paper by Hunter et al. (2011), the endogenous factor affects not only the alpha in its size but also the standard error. The correlated errors model increases more than the omitted factors model the accuracy of the alpha’s. This is also in line with the paper by Hunter et al. (2011). Overall, I can conclude that the endogenous factor increases the ability to explain the Dutch mutual equity fund returns in the period January 1993 to December 2010.

A. Limitations and future research

Through the many missing funds due to the lack of some data and especially by the absence of much data about the style and magnitude of the equity funds, some groups were smaller than I hoped for. This is especially true for data of funds in the early years of my time sample. So results of groups with very few funds in a specific period might be influenced. Although, I think it have not much influence on my research, because the results of all the groups are all very similar, but I think it would not hurt if some groups had more funds. For future research it might be attractive to find out what the best group classification is. It can be tested by making several different group classifications on the same market of funds and compare the results.

Literature list

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Carhart, M. M. 1997. On persistence in mutual fund performance. Journal of Finance, Vol. 52, pp. 57- 82.

Fama, E. F. and French, K.R. 1992. The cross-section of expected stock returns. Journal of Finance, Vol. 47, pp. 427-465.

Fama, E. F. and French, K. R. 1993. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, Vol. 33, pp. 3-56.

Fama, E. F. and French, K. R. 2010. Luck versus skill in the cross-section of mutual fund returns. Journal of Finance, Vol. 65, pp. 1915-1947.

Garyn-Tal, S. and Lauterbach, B. 2011. Evaluating mutual fund's alpha via alternative frameworks: some new evidence and insights. Working paper.

Grinblatt, M. and Titman, S. 1992. The persistence of mutual fund performance. Journal of Finance, Vol. 47, pp. 1977-1984.

Grinblatt, M. and Titman, S. 1994. A study of monthly mutual fund returns and performance evaluation techniques. Journal of Financial & Quantitative Analysis, Vol. 29, pp. 419-444.

Gruber, M. J. 1996. Another puzzle: The growth in actively managed mutual funds. Journal of Finance, Vol. 51, pp. 783-810.

Hendricks, D., Patel, J. and Zeckhauser, R. 1993. Hot hands in mutual funds: Short-run persistence of relative performance, 1974-1988. Journal of Finance, Vol. 48, pp. 93-130.

Hunter, D. et al. 2011. Endogenous benchmarks. Working paper, University of Maryland.

Jegadeesh, N. and Titman, S. 1993. Returns to buying winners and selling losers: implications for stock market efficiency. Journal of Finance, Vol. 48, pp. 65-91.

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Lehmann, B. N. and Modest, D. M. 1987. Mutual fund performance evaluation: A comparison of benchmarks and benchmark comparisons. Journal of Finance, Vol. 42, pp. 233-265.

Lintner, J. 1965. The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics & Statistics, Vol. 47, pp. 13-25.

Malkiel, B. G. 1995. Returns from investing in equity mutual funds 1971 to 1991. Journal of Finance, Vol. 50, pp. 549-572.

Otten, R. and Bams, D. 2002. European mutual fund performance. European Financial Management, Vol. 8, pp. 75-101.

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Appendix A

The equity funds are grouped according to their style and magnitude. The first part of the group name refers to the style of the fund and the second part of the group name refers to the magnitude of the fund. A table is presented below where you can see how many funds are in which group. For example, there are 28 funds in the group 'Value Large'.

Table AI: Fund classification

The funds are classified according to their style on the left side from the table and to their magnitude on the upper side of the table.

Style

Magnitude

Large Middle Small

Growth 17 funds 5 funds 2 funds

Mixed 66 funds 5 funds 2 funds

Value 28 funds 4 funds 8 funds

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Appendix B

Fund name Style Magnitude Fund name Style Magnitude

Allianz Wereld 10 Growth Large AXA Paraplu Fonds AXA Model Fund IV Capital Mixed Large

BNP Paribas Strategie Fondsen Multi Manager Profiel Fonds 2 C Growth Large Banque Nationale de Paris Paribas AEX Index Fund Mixed Large BNP Paribas Paribas Strategie Fondsen Multi Manager Profiel Fonds 3 C Growth Large Banque Nationale de Paris Paribas Global Property Securities Fund Mixed Large BNP Paribas Strategie Fondsen Multi Manager Profiel Fonds 4 C Growth Large Banque Nationale de Paris Paribas Netherlands Fund Mixed Large

BNP Paribas Strategie Fondsen Multi Manager Profiel Fonds 5 C Growth Large Banque Nationale de Paris Paribas Obam N V Mixed Large

BNP Paribas Strategie Fondsen Multi Manager Profiel Fonds 6 C Growth Large Banque Nationale de Paris Paribas Property Securities Fund Far East Mixed Large

Delta Lloyd Investment Fund Growth Large Delta Lloyd Donau Fonds Mixed Large

Delta Lloyd Mix Fund Growth Large Delta Lloyd Europa Fonds Mixed Large

Himalayan Fund Growth Large Delta Lloyd NL Fonds Mixed Large

Holland Pacific Fund Growth Large Holl Europe Fund Mixed Large

ING Daily Consumer Goods Fund Growth Large Holland Fund Mixed Large

ING Emerging Europe Fund Growth Large IDB Equity Income Mixed Large

ING Global Opportunities Fund Growth Large ING Basic Materials Fund Distribution Mixed Large

ING Information Technology Fund Growth Large ING Dutch Fund Mixed Large

ING Luxury Consumer Goods Fund Growth Large ING Duurzaam Aandelen Fonds Mixed Large

Rolinco Growth Large ING Dynamic Mix I Capital Mixed Large

Sustainable Values Fund of Growth Large ING Dynamic Mix II Capital Mixed Large

ASN Milieu and Waterfonds 2 Growth Middle ING Dynamic Mix III Capital Mixed Large

IDB Euro Mid Cap Growth Middle ING Dynamic Mix IV Capital Mixed Large

ING Global Real Estate Fund Growth Middle ING Dynamic Mix V Capital Mixed Large

Kempen Sense Fund NV Growth Middle ING Energy Fund Mixed Large

RO Afrika Fonds Growth Middle ING Far East Fund Mixed Large

European Asset Trust Growth Small ING Global Emerging Markets Fund Mixed Large

Kempen European Smallcap Fund N Growth Small ING Global Fund Mixed Large

Achm NL Aand FD1 Mixed Large ING Health Care Fund Mixed Large

Aegon Europees Mix Fund C Mixed Large ING Japan Fund Mixed Large

Aegon Optimum Europees Aandelen Fonds Cap Mixed Large ING Lion Fund Mixed Large

Aegon Optimum Europees Mix Cap Mixed Large ING North America Fund Mixed Large

Aegon Optimum Wereld Mix Fonds Cap Mixed Large Insinger De Beaufort Multi Manager International Equity Mixed Large

Aegon Optimum Wereld Aandelenfonds Cap Mixed Large IVM Paraplufonds IVM Mix Fonds Cap Mixed Large

Ageon Wereldw Vastgoed Mixed Large Meesman Index Umbrella Fund Meesman Global Stock Index Fund Mixed Large

ASN DZ Aandelen Mixed Large Meesman Index Umbrella Fund Meesman Emerging Markets Stock Mixed Large

AXA Paraplu Fonds AXA Model Fund III Capital Mixed Large Optimix America Fund Mixed Large

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Fund name Style Magnitude Fund name Style Magnitude

Optimix Investment Funds NV Optimix Emerging Markets Fund Mixed Large AXA Paraplu Fonds AXA Model Fund V Capital Value Large

RO Life Cycle B 15 Mixed Large Banque Nationale de Paris Paribas ALL IN Fund Value Large

RO Life Cycle C 20 Mixed Large Banque Nationale de Paris Paribas ALL Income Fund Value Large

RO Life Cycle D 25 Mixed Large Banque Nationale de Paris Paribas Asia Pacific High Income Equity Fund Value Large

RO Life Cycle E 30 Mixed Large Banque Nationale de Paris Paribas Global High Income Equity Fund Value Large

RO Life Cycle F 35 Mixed Large Banque Nationale de Paris Paribas Premium Global Dividend Fund Value Large

RO Life Cycle G 40 Mixed Large Friesland Aandelenfonds Value Large

Robeco Balanced Mix Mixed Large ING Europe Fund Value Large

Robeco Dynamic Mix Mixed Large ING Financials Fund Value Large

Robeco Growth Mix Mixed Large ING Hoog Dividend Aandelen Fonds Value Large

Robeco Safe Mix Mixed Large ING Industrials Fund Value Large

Robeco Solid Mix Mixed Large ING Premium Dividend Fund Value Large

SNS America Aandelenfonds Mixed Large ING Telecom Services Fund Value Large

SNS Azie Aandelenfonds Mixed Large ING Utilities Fund Value Large

SNS Duurzaam Aanelenfonds Mixed Large Kempen European High Dividend Fund NV Value Large

SNS Euro Aandelend Fund Mixed Large Kempen Global High Dividend Fund NV Value Large

SNS Euro Mix Funds Mixed Large Optimix Europe Fund Value Large

SNS Opkomende Landen Aandelenfonds Capital Mixed Large Optimix Mix Fund Value Large

SNS Wereld Aandelenfonds Mixed Large Pyramidefonds Cap Value Large

T&P Allegretto Fund Mixed Large Robeco Duurzaam Aandelenfonds Value Large

VAN Lanschot Global Index Fund Groeigericht Mixed Large Robeco Hollands Bezit Value Large

VAN Lanschot Global Index Offensief Capital Mixed Large SNS Hoogdiv Aandelenfonds Value Large

VAN Lanschot Global Index Fund Defensief Mixed Large SNS Nederlands Aandelenfonds Value Large

VAN Lanschot Global Index Fund Neutraal Mixed Large Banque Nationale de Paris Paribas High Income Property Fund Value Middle

ASN DZ Smal / Midcap Mixed Middle Banque Nationale de Paris Paribas Property Securities Fund Europe Value Middle

Banque Nationale de Paris Paribas Property Securities Fund America Mixed Middle HOF Hoorneman European Value Fund Value Middle

Delta Lloyd Select Dividend Fonds NV Distribution Mixed Middle SNS Euro Vastgoedfonds Value Middle

ING Europe Small Caps Fund Mixed Middle Banque Nationale de Paris Paribas Small Companies Netherlands Fund Value Small

Kempen Best Selection European Property Fund NV Mixed Middle Delta Lloyd Deelnemingen Fonds Value Small

ADD Value Fund NV Capitalisation Mixed Small Delta Lloyd EUR DF Value Small

Kempen Oranje Participaties NV Mixed Small HOF Hoorneman Phoenix Fund Value Small

ACH EUR Aand FD2 Value Large HOF Hoorneman Value Fund Value Small

ACH Wereld AANDFD3 Value Large Intereffektem EM Africa of Value Small

Aegon Eurofonds Cap Value Large Kempen Orange Fund NV Value Small

AXA Paraplu Fonds AXA Model Fund I Capital Value Large MEI ROM EN BUL Fund Value Small

AXA Paraplu Fonds AXA Model Fund II Capital Value Large

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Appendix C

For the alpha’s of the four-factor model and the single-factor model the Spearman’s rank correlation is conducted. The Spearman’s rank correlation is given in equation (A1):

∑

( ) (A1)

Where is the correlation coefficient, the amount of observations and is shown in equation (A2):

(A2)

The four-factor model supplemented by an endogenous factor Dutch mutual funds

The four-factor model supplemented by an endogenous factor

Dutch mutual funds

Author: Sander van Laarhoven

Master thesis

Master of Science in Business Administration, specialization Finance

August 2012

Supervisor: dr. L. (Lammertjan) Dam

2

Supervisor: dr. A. (Auke) Plantinga

The four-factor model supplemented by an endogenous factor

Dutch mutual funds

Author: Sander van Laarhoven

Table of contents

Introduction

I. Literature review

II. Data and methodology

A. Data

B. Methodology

III. Results

A. Group results

B. Regressions of the various models on equity funds

C. Accuracy of the alpha’s

D. Impact of the endogenous factor

IV. Conclusion

A. Limitations and future research

Literature list

Appendix A

Appendix B

Appendix C