Excerpt

## Table of Contents

Abbreviations

List of Figures

List of Tables

1 Introduction

2 Theoretical Framework

2.1 The Arithmetic of Active Management

2.2 Information Efficiency of Financial Markets

2.3 Factor Models

3 Previous Empirical Findings

4 Method

4.1 Factor Creation

4.2 Data

4.3 Forming the Dataset and Limitations

4.4 Approach

4.5 Separating Skill from Luck

4.6 Summary Statistics

5 Empirical Results

5.1 Net Returns

5.2 Gross Returns

5. 3 Beating Individual Benchmarks

6 Critical Review

7 Conclusions

Appendix A: CAPM Bootstrap Simulations

Appendix B: R – Code

References

## Abbreviations

Abbildung in dieser Leseprobe nicht enthalten

## List of Figures

Figure 1: Difference between Active and Passive Management Styles

Figure 2: Kernel Density Plot of Estimated Risk Adjusted Performance (Net)

Figure 3: Kernel Density Plot of Estimated Risk Adjusted Performance (Gross)

Figure 4: Simulated and Actual CDF of Four-Factor t(α) Percentiles (Net)

Figure 5: Simulated and Actual CDF of Four-Factor t(α) Percentiles (Gross)

Figure 6: Simulated and Actual CDF of Individual Benchmark t(α) Percentiles (Net)

Figure 7: Simulated and Actual CDF of Individual Benchmark t(α) Percentiles (Gross)

Figure 8: Simulated and Actual CDF of CAPM t(α) Percentiles (Net)

Figure 9: Simulated and Actual CDF of CAPM t(α) Percentiles (Gross)

## List of Tables

Table 1: Summary Statistics

Table 2: Correlation Matrix and VIF

Table 3: Significance of Alphas

Table 4: Intercepts and Slopes in Variants of Regression (1) for EW Portfolios

Table 5: Actual and Simulated Percentiles of t(α) Estimates (Net)

Table 6: Actual and Simulated Percentiles of t(α) Estimates (Gross)

Table 7: Actual and Simulated Percentiles of Individual Benchmark t(α) Estimates (Net & Gross)

Table 8: Actual and Simulated Percentiles of CAPM t(α) Estimates (Net & Gross)

## Abstract

Financial markets are as complex as ever due to an accelerating development in the last decades. Especially evaluations of mutual fund performance have been a subject of interest since the introduction of financial services.In this thesis, a study on the performance of mutual funds investing in German equity from July 1995 to June 2015 is conducted. The aim is to find out if fund managers have sufficient skill to generate risk adjusted return in order to cover the cost imposed on the investors. Another purpose is to provide investors with relevant results. Inter alia, Jensen one-factor, Fama and French three-factor and the Carhartfour-factor model are used as different benchmark models for performance. Paired bootstrap simulations suggest that, net of cost, a small fraction of fund managers do have sufficient skill to cover cost. For the bottom ranked funds, there is statistical evidence that their poor performance is caused by bad management, rather than by bad luck. The results for gross returns show that there is an unneglectable fraction of fund managers with good performance not due to luck. Compared to net returns, there is stronger evidence of skill, negative as well as positive. Form an investor’s point of view it seems rather beneficial to invest in passively managed vehicles. High costs eat into the return, and they are the main reason why the majority of actively managed funds end up with sub-par performance.

**Keywords:** fund performance, active, passive, German equity funds, fund management, luck, skill, bootstrapping, Fama and French, Carhart, factor model, CAPM, OLS regression

## 1 Introduction

Ever since the formation of capital markets, investors wanted to outperform their competitors, the market. Investors want to obtain the highest returns but contemporaneously they want to commit themselves to as little risk as possible. Different styles of investment try to fulfil everybody’s perception of an ideal investment. Hence, the debate between active and passive fund management is still ongoing. In Europe, especially in Germany, passive investment strategies such as investing in exchange traded funds have been growing in popularity since the last decade. During this period, ETFs became a major conduit for private investment in Germany.Nowadays many people consider them as a save deposit for their retirement.The setback in popularity of mutual funds might originate in a presumed deadability to provide returns, net of costs, which are higher than the returns of passive investment strategies.Stated in the words of Gruber (1996): „The question remains as to why mutual funds and in particular actively managed mutual funds have grown so fast, when their performance on average has been inferior to that of index funds.”^{1} Hence, the question arises if the professional service and consulting which comes with active management really adds large enough value to compensate for the high costs and whether there are skilled enough fund managerswho are able to provide their outperformance with a given certainty and persistence. Another important aspect considering investments is the level of risk an investor is exposed to. Passively managed funds just track a specified index and therefore inhabit less risk than their actively managed counterparts inhabit.On the contrary, active management is forcedproduce returns large enough to offset its higher risks and fees.

Nonetheless, mutual funds also offer important advantages. Minimizing idiosyncratic risk, investors can obtain better diversification opportunities and theyacquire access to asset classes, which they would not be able to invest in otherwise, too. Basically, “a mutual fund is a company that pools money from many investors and invests the money in stocks, bonds, short-term money-market instruments, other securities or assets, or some combination of these investments”^{2}.Mutual funds have two primary functions: providing liquid access to a portfolio of diversified stocks and exploiting economies of scale in information gathering and processing.

Given the size and growth of the mutual fund industry as well as the questionable theoretical and practical raison d’être for active management, the aim of this thesis is to use ordinary least squares estimations in order to investigate if active mutual fund managers possess the ability to generate risk adjusted abnormal excess return. If they are capable to produce an excess return, is it due to skill or luck?Therefore, the monthly net and gross returns of several mutual funds that invest in German equity are examined and processed with the help of bootstrap simulations.An evaluation of typical econometric measures and problems arising whilst working with financial data will also be given.

One of the most known studies on the subject of fund performance is the “Luck vs. Skill” research paper from Fama and French^{3}. In their study on U.S. mutual funds, they have been using bootstrap simulations in order to find that only few fund managers have skill at all and even fewer fund managers are able to generate enough return to cover the expenses of the investors. Since the U.S. mutual fund market is still the largest fund market in the world, a lot of research has been done on the U.S. fund market. Although the German mutual fund market experienced a vast development in the last decade, there are, in comparison, hardly any similar studies on the German mutual fund market.

Besides the introduction, the remainder of this thesis is structured as follows: First, some basics about the theoretical framework and the concept of capital markets are discussed, in particular the Efficient Market Hypothesis, in chapter two. Then the factors models, which are used in this thesis, are characterized as well. A brief overview about previous empirical findings is given in chapter three and then the construction of the necessary factors is described in the beginning of chapter four. In the rest of chapter four, data limitations and the actual approach of the analyses including the bootstrap procedure and potential problems arising due to the conditions, which need to be fulfilled when working with OLS regressions, are discussed. Finally, the results for net and gross returns of the funds in the sample are given in chapter five and a short critical review of the whole process is conducted in chapter six.

## 2 Theoretical Framework

Within the approach of active portfolio management, a fund manager selects securities in order to exploit inefficiencies on the capital market. The objective is to outperform a specified benchmark. Most of the time the benchmark is a certain index but the active investment may also be compared with the general market or a segment of the market. According to active fund management, a return is achievable, which is higher than the return one would have received with an investment in an adequate passive investment. On the other side, the potential loss is also higher considering active management, which leads to a greater variability.Nonetheless, mutual funds might also create less volatility than the benchmark index due to a fund manager’s ability to actively pick certain stocks.Active management can also stand for the management of volatility and investing in less-risky and high quality assets.For his administration, a fund manager charges management fees.When investing in mutual funds, investors have to pay sales charges, annual fees and other management fees. Some of these charges also incur when thinking about passive management. Figure 1 therefore graphically contrasts the difference in the rate of return between passive and active portfolio management. The long-term return may vary within the full range plotted.

**Figure 1 : Difference between Active and Passive Management Styles**

Abbildung in dieser Leseprobe nicht enthalten

**Source: after (Ambachtsheer & Farrell, 1979)**

Because of the minor management charges and plain replication of the market portfolio, a passive investment generates an average long-term return just below the return on such a portfolio.

### 2.1 The Arithmetic of Active Management

In 1991, William Sharpe came up with the framework of equilibrium accounting or as he called it: “the arithmetic of active management”, a constraint on the returns on active investing: The average return of an active management will always be equal to the average return of a comparable passive management. At least gross returns on an actively managed investment will be equal to that of a passive managed investment. The net returns, after costs, on an average actively managed investment are less than the return on a passively managed investment. According to Sharpe, this framework must hold at any point in time whilst only being dependent on the laws of addition, subtraction, multiplication and division.^{4}

Since a passive investment is tracking a certain market, each passive investor will receive exactly the market return before cost. Such an investor is holding a so-called cap-weighted market portfolio. Active management, in contrast, provides the chance for individual investors to add value but only at the expense of other active investors. “The arithmetic of equilibrium accounting then implies that the deviations from cap weights in one active investor's portfolio must be absorbed by other active investors who take offsetting positions.”^{5} That implies a zero sum game. Individual active managers may still generate positive abnormal returns, but only at expense of other active investors. Unfortunately, there are costs to be considered when talking about active management. Hence, net of charges the active portfolio management can be described as a negative sum game.

### 2.2 Information Efficiency of Financial Markets

Humans behave completely rational and try to maximize their utility. That is the quintessence of the theory of the so-called “Homo economicus”. Since we all know that this behavior does not always take place, the question arises whether capital markets can be efficient or not. According to the Efficient Market Hypothesis, stocks always trade at their fair value since every market participant knows all relevant information one needs in order to charge such a fair price at all time. The prices on the market reflect all known information. Furthermore, the prices can react sufficiently fast if there is new information available.^{6} Because of that, it is said that it is impossible to outperform the market in the long run. Since all investors have the same information, outperformance is only due to luck.According to Bachelier (1900), stock prices follow a random walk. Accordingly, the best prediction for the future price of an equity is today’s price of that equity plus a random error term. For Fama, a market is efficient when prices always fully reflect available information.^{7} Hence, the proclaimed advantages of active management, like market timing and stock picking, seem to be pointless. Therefore, it is reasonable that the only way to gain higher return is to invest in riskier securities.Contrary to this, Grossman and Stiglitz (1980) argue that informed investors earn abnormal returns in an efficient market, but only enough to cover their expenses of becoming informed.^{8}

Of course, there are detractors of the EMH. One of the most important arguments states, that every individual interprets information differently and therefore would price a security adversely. One could also negotiate about the time it takes a market participant to discern new information and to process it. According to behavioral economics, the pricing of securities is also susceptible to emotional decisions since investors are prone to behavioral biases in their investment decisions. All of these anomalies can lead to an abnormal excess return. Otherwise, one would plainly invest in index funds since the market is considered as effective.

Realistically, the financial markets neither seem to be fully efficient nor completely inefficient, so a possible abnormal excess return is possible. If the markets would be at least efficient in a semi-strong way, then there would not be a practical raison d’être for active management, according to Fama and his article in 1970.

### 2.3 Factor Models

There are different factor models, which can be chosen as a benchmark to evaluate a fund’s performance. Probably the most common is the Capital Asset Pricing Model. It is a development of Markowitz’s portfolio theory improved by Sharpe, Treynor, Lintner and Mossin. The key message of the CAPM is the relationship between risk and return: The expected return increases with proportion to an increasing level of risk and vice versa. In order to make the CAPM work, there are different key assumptions that need to be fulfilled. According to Markowitz, all investors are risk averse, have homogenous expectations regarding risk, volatility, correlations and expected return.They also have access to the exact same investment opportunities and only hold efficient portfolios, i.e. portfolios of securities that yield the maximum expected return for a given level of volatility. Another assumption postulates that these securities are bought at a competitive market price and lend at a risk-free interest rate.^{9} The CAPM suggests that an optimal portfolio is a combination of a risk-free asset and the market portfolio. Hence, the performance of a fund, using the CAPM as a benchmark can be estimated with this market model:

[Abbildung in dieser Leseprobe nicht enthalten]is the expected return of the i-th fund ( ) at time t ( ), stands for the risk-free rate which is the rate where the threat of losing money is theoretically zero and is the expected market return, the return on the reference portfolio at time t.The , the intercept of the model, is interpreted as the vertical distance to the security market line.It is the difference between an asset’s expected return and its benchmark return. is a measure of systematic risk that cannot eliminated by diversification.It indicates the volatility of an asset in relation to the overall market. The market has a beta equal to one.If the chosen portfolio has a beta higher than one, the portfolio will be more volatile than the returns of the market index and as such, bear a bigger risk compared to the market. Beta is calculated as follows:

Abbildung in dieser Leseprobe nicht enthalten

Therefore, one can see that the only way of increasing the return is increasing the risk. Under the CAPM, investors are only compensated for taking necessary risks, but not for unnecessary ones.Black et al. (1972) provided a critical view of this relationship. Their study on U.S. equities showed that equities with low beta performed better than the CAPM suggested and equities with high beta performed worse.^{10} At last, is a measure for the random error, ideally with mean zero in order to make OLS estimations work. It is the white noise error term. It stands for the part of the security returns that cannot be explained by the market model.

Since the traditional CAPM is based on some unrealistic assumptions and does not explain sufficient cross-sectional return variation, economists extended it to a more useful multifactor model: The Fama and French three-factor model and the derived Carhart four-factor model. In contrast to the CAPM, the Fama and French factor model uses three factors to describe stock returns or as used in this thesis, fund returns. According to Fama and French, the two factors that are added reflect the portfolio’s exposure to stocks of companies with low total value of issued shares and stocks with high book-to-market value.

Abbildung in dieser Leseprobe nicht enthalten

(small minus big) is a size factor,which results from the returns from buying small-cap stocks and short-selling large-cap stocks each year.

(high minus low), the value factor is generated by simulating the return history of a long-short strategy that every year bought low price-to-book (value) stocks and short-sold high price-to-book (growth) stocks.

Carhart’sfour-factor model complements the three-factor model by the factor momentum, MOM:

[Abbildung in dieser Leseprobe nicht enthalten] is a version of Carhart’s momentum return, it simulates the returns of a monthly strategy that bought the best-performing stocks by trailing 12-month returns, excluding the most recent month, and short-selling the worst-performing stocks.

The important figure, these factor models are supposed to deliver is the so-called Jensen alpha ( ). It is a measure of the abnormal performance. The abnormal performance is defined as all return exceeding a certain benchmark. So alpha is obtained when comparing a fund’s, or any other security’s, actual return with the theoretical return according to one of the already mentioned factor models. It is a measure of how well a fund manager performs.^{11} In its simplest form it is the intercept of a regression of the fund's excess return on the excess return of an appropriate equity index.Alpha then is the intercept in a regression of the fund excess return on the market excess return and the other factors mentioned before. Alpha is calculated as follows:

SMB, HML and MOM are determined as risk factors. Hence, it is indeed appropriate to include them in the OLS regressions. However, the interpretation in risk terms is not clear.Fama and French (2010) state that, “[...] there is no need to take a stance on this issue. We can simply interpret SMB, HML, and MOM as diversified passive benchmark returns that capture patterns in average returns during our sample period, whatever the source of the average returns”^{12}.

## 3 Previous Empirical Findings

The empirical approach of this thesis is adopted from Kosowski et al. (2006)^{13} and “Luck vs. Skill in a Cross-Section Mutual Fund Performance” by Eugene Fama and Kenneth French.Fama and French did a study of mutual fund performance from the perspective of the, already mentioned, equilibrium accounting: A positive abnormal expected return is at expense of other investors, after cost a negative sum game is adopted. Therefore their aim was to find out if a positive abnormal excess return can be realized and if this α is due to skill or luck. The period examined was 1984 through 2006. Most of the funds that were used in the research primarily invested in U.S. equities. Fama and French used their own three-factor model as well as Carhart’sfour-factor model as benchmarks. At first, an equal-weighted and a value-weighted portfolio of active funds on the aggregate level were constructed. The value-weighted portfolio was close to the market portfolio and produced an alpha, which is not significantly different from zero.Than they used bootstrap simulations to distinguish skill from luck. Hence, alpha was set to zero. The alphas and t-statistics of the regressions on each fund in the portfolio were compared to respective simulated cross-sectional distributions obtained from the bootstrap simulations. These simulations were performed jointly on all funds and with both, net and gross returns.

On the one hand, the simulations with the net returns suggest that only a few funds produce a risk adjusted return that is large enough to compensate for the costs imposed on the investor. On the other hand, there are skilled managers but their performance was absorbed by the high costs according to the simulations with gross returns. It is said that“…, the fund managers do not have enough skill to produce risk-adjusted expected returns that cover trading costs."^{14} Another problem is to distinguish between funds with an alpha that is due to real skill and funds that are just lucky enough to generate a high abnormal return: "…, these good funds are indistinguishable from the lucky bad funds that land in the top percentiles...”^{15}

Kosowski et al.did a research on U.S. equity mutual funds as well. It was the first study to use cross-section bootstrap simulations to analyze fund performance during the period 1975 through 2002. The Carhart four factor model was used as a benchmark.A cross-section bootstrap procedure was used to separate skill from luck for individual ordered funds, when idiosyncratic risks were highly non-normal. They compared their original alpha estimates to the distribution of bootstrap alpha estimates after 10,000 simulations. The simulations suggested that there are skilled managers as well as managers with bad skill. Hence, the funds at the bottom part of the distribution did not just have unlucky managers and vice versa for the upper part of the simulated distribution. The authors found that many funds in the fifth percentile have significant positive bootstrapped alphas net of expenses. Hence, the conclusions drawn are more positive than the ones in the study of Fama and French. According to critics, this is because of the simulation approach used in the study. In contrast to Fama and French, they carry out individual simulations on each fund. Therefore, correlations of alpha estimates were not considered. Another criticism mentions the fact that Kosowski et al. did not have a sample free of survivorship-bias. This falsifies the estimates of the sample to a greater extend.

One of the sparse non-U.S. equity studies has been conducted by Cuthbertson et al.^{16}. They formed a portfolio of UK equity funds spanning 1976 – 2002. Once again, the same procedure as in Kosowski et al. has been used in order to distinguish between lucky and skilled funds. The authors applied a cross-section bootstrap on unit trusts, a special form of open-ended mutual funds existing in the UK. In contrast to the research mentioned above, there is no survivorship-bias in this sample. The simulations suggest that there are some funds with stock-picking ability and there are bad performing funds due to bad skill again. Furthermore, the abnormal performance of most of the funds could be attributed to plain luck.

Otten and Bams^{17} did a study on European equity funds. They constructed a portfolio consisting of funds from France, Italy, UK, Germany, and the Netherlands. With the aid of the Carhart four-factor model, they concluded that only the UK funds outperform the used benchmark significantly, at a 5% confidence level. When investigating the German funds an insignificantly negative alpha, net of cost, was observed. According to the authors, the only funds in their sample with skilled managers are UK funds.

Considering the rare studies on German funds, the paper of Cuthbertson and Nietzsche^{18} has to be mentioned. They analyzed equity funds with their domicile in Germany, no matter in which kind of equity these mutual funds invested during the period from 1990 – 2009. In order to estimate the funds alphas the Famaand French three-factor model has been used. Once again, a cross-sectional bootstrap procedure was applied to separate skill from luck. The conclusion they drew is that a big portion of the funds has an alpha, no matter if positive or negative, solely due to luck rather than exhibiting skill. There is a big part of unskilled funds with negative abnormal excess return. Nearly the same result applied after usage of gross instead of net figures.

To sum it up, the studies mentioned find relatively weak evidence of positive alpha performance in the extreme right tail of the cross-section performance distribution, especially after the deduction of management fees. Hence, investors do net benefit from an active investment. Nevertheless, there are funds with abnormal risk adjusted return which is not due to plain luck and which exists net of costs, too.

However, there is relatively strong evidence of negative alpha performance throughout the left tail. Hence, outperformance due to real skill is rare and a passive management seems to be more appropriate for an investor. French (2008) finds that U.S. investors could increase their annual returns by switching to a passive strategy.^{19} Unfortunately, most of the studies only contain a time-period before the financial crisis around the year 2008. Therefore, the result of this thesis might differ since active funds could or could not have performed well throughout the rough times of financial crisis. Maybe active fund managers had perfect market timing abilities throughout the crisis and therefore picked securities, which led to huge outperformance. Nonetheless, the other way around is also possible.

## 4 Method

The method applied in this study can be described as similar to the approaches from Kosowski et al. and Fama/French. At first, a dataset of funds is created. Then a benchmark consisting of the three Fama and French factors as well as the momentum factor for evaluating aggregate fund performance is set up. Later on, 10,000 bootstrap simulations are used to separate plain luck from real skill concerning the individual funds. Therefore a so-called “luck-distribution” is formed.

### 4.1 Factor Creation

The Humboldt Universitätzu Berlinprovides up-to-date and accurate time-series for the SMB, HML and MOM factors. Hence, these factor time-series are used in this thesis.In the following, the creation of these factors and their peculiarities are described. The creation process is aligned with the instruction of Martin H. Schmidt et al. from the HU Berlin^{20}.The risk free rate and the market portfolio follow either an own or a different creation which is also described in the following. These changes are made to adjust the model to a more real life and practical situation.

Almost all factor sets follow a typical building strategy. In order to create the SMB, HML and MOM factors and the excess return on the market portfolio, RMRF, data on equities of companies listed on the Frankfurt Stock Exchange is needed.Not all companies reported by the FSE at any specific point in time are listed: closed-end funds, real estate investment trusts and profit participation bonds are not included.As described in Cremers et al.^{21} this eliminates one major mistake made in the CRSP database, which is often used to construct the Carhartmomentum factor. Firms solely listed on other stock exchanges are not included as well as companies listed in the lowest segment, “Freiverkehr”, of the FSE.

In the period from 1995-2000, the so-called “Körperschaftssteuergutschrift”, corporate income tax voucher, is not considered in the construction of the factors: Additionally to the cash dividend, the companies received a voucher in the amount of tax they paid on these dividends. Hence, those companies actually received a tax credit, which might be taken into account since this voucher technically rises the dividend. Of course, this only applies to companies that pay dividends but otherwise the rate of return of their stocks is biased downwards during that period.

All of the companies, no matter if small, large, blend or growth style are weighted equally in the factor set. Financial firms are excluded from the construction of the SMB, HML and MOM factors. Nevertheless, they are part of the market portfolio time-series. This is due to Fama and French who said, “We exclude financial firms because the high leverage that is normal for these firms probably does not have the same meaning as for nonfinancial firms, where high leverage more likely indicates distress.”^{22} Others include financial firms in the construction of their risk factors as well as in the construction of the market portfolio. Anyways, according to Viale/Kolari/Fraser (2009)^{23} there is no evidence, that stocks of financial firms have any exposure to size and book-to-market. So it does not really seem to matter if they are included or not.The rates of return on individual stocks like cash dividends, stock dividends or additional stocks, if a stock split happens, are accounted for as well.

Importance is also attached to penny stocks like Infineon in early 2009. Since penny stocks have a high standard deviation, stocks with a share price less than one are not included in the calculation.

A good proxy for the German market portfolio is the publicly available CDAX index. The CDAX includes all companies listed in the Prime Standard or General Standard of the “Frankfurter Wertpapierbörse”. It is also one of the most popular proxies for the German market portfolio. However, the CDAX changed its composition. Until September 1998, it only contained stocks from the top market segment, “AmtlicherMarkt”. In the period from 1998 to 2003, the index also includes stocks listed in the segment called “NeuerMarkt”. Hence, only after the year 1998 the CDAX can be considered as a completely fair approximation for the German market portfolio. In the years before 1998, the index only consisted of stocks from the highest segment of the German stock market. Since the sample of this thesis starts in 1995, the market portfolio factor created by the Center for Financial Research Cologne is used as a proxy. Unfortunately, the CFR stopped reporting this time series at the end of 2013. Hence, in the period until June 2015 the plain CDAX performance index is utilized. It can be accessed via the website of the “Deutsche Bundesbank” with the code BBK01.WU018A. Using an index with only German stocks as constituents is a better approximation than using a potentially downward-biased value weighted market portfolio like Fama and French propose.Investigating the construction of U.S. factor models, Cremers et al. said: “Instead, it would be more appropriate to benchmark actively managed portfolios that only hold U.S. equities with a market portfolio consisting of U.S. equities.”^{24}

Since the construction of risk factors using local MSCI indices is either not possible due to the history of the sample compared to the history of the indices or does not seem to produce robust results, the construction of the SMB and HML factors follows Fama/French (1993)^{25} as precise as possible.Building the MOM factor time series follows Fama and French (2012)^{26}.Nevertheless, the MSCI Germany Index is interesting for usage as a proxy for the market portfolio. There is a high correlation of the market portfolio used in this thesis, the CDAX performance index, and the MSCI Germany index. In their study of German mutual funds, Cuthbertson and Nitzsche use the local MSCI indices for constructing their factor set.

Aside from German peculiarities like the “Körperschaftssteuer”, there is another deviation compared to the Fama/French approach of calculating the risk factors. Fama and French use the New York Stock Exchange breakpoints to differentiate between small and big stocks. According to them, the median market capitalization of all stocks listed at the NYSE is used to allocate the small stocks from other exchanges to each size portfolio. They do not utilize the median market capitalization of all stocks included. In German factor sets, usually a size breakpoint of 0.5 is applied.

For the SMB and HML factor, stocks are sorted into two groups, small size and big size, at the end of June of each year in the sample. The size refers to the market capitalization of the stock. Sorting the stocks every June is also the reason why the sample of funds in this thesis ends with June 2015. Now the book-to-market ratio is needed. Than the stocks are sorted in three groups, where particular quantiles are used as an intersection: low (bottom 30% of BE/ME), medium (middle 40% of BE/ME) and high (top 30% of BE/ME). A return history of at least twelve months is taken as a premise. The sorting of the stocks into size groups, mentioned above, is based on its ME and its median. All companies with ME larger than the ME median are big the others are small. Six value-weighted portfolios are created: S/L, S/M, S/H, B/L, B/M, B/H, where, e.g. portfolio B/H represents stocks that are in the big capitalization group as well as in the high BE/ME group. The portfolios are reweighted at the end of June every year.

Now the size-to-momentum portfolios still need to be constructed. In contrast to the book-to-market ratio, this ratio is built on a monthly basis. For each recent month, the stock’s past return for the previous eleven months is calculated. Than the stocks are rated according to their past performance deriving from those returns. Again, three groups, with the 0.3 and 0.7 quantiles as intersections, are established: losers (bottom 30%), normal (middle 40%) and winners (top 30%). Now portfolios are allocated based on the companies past return and ME from previous June: S/Loser, S/Normal, S/Winner and B/Loser, B/Normal, B/Winner. So for example, S/Winner stands for “small winners”. Than the return of a company in the recent month is weighed by its market value of all listed shares, expressed as a percentage of the market value of all listed shares, from the end of the previous month. Therefore, a value weighted return for the portfolios that were just created is estimated.

The final factors, SMB, HML and MOM are calculated as follows for each month t:

Abbildung in dieser Leseprobe nicht enthalten

The proxy for the risk free rate, which is necessary to calculate the excess returns of each fund and the excess return of the market portfolio or other possible benchmarks consists of the average one-month money market rates, the so-called “Monatsgeld”, reported by Frankfurt banks. The time series of monthly averages of “Monatsgeld” can be accessed, at “Deutsche Bundesbank” online database using the code: BBK01.SU0104. Since the “Deutsche Bundesbank” stopped reporting this interest rate in 06/2012, the one moth EURIBOR rate BBK01.SU0310 is used for the rest of the needed period. These time series are stated as percentage per annum. Hence, each monthly value still needs to be divided by twelve.

### 4.2 Data

Since the aim is to evaluate the performance of open-ended mutual funds investing in German equity during the period from 1995 to 2015, it is necessary to derive the amount of funds, which can be used. Another requirement is that there is hardly any survivorship-bias in the dataset. According to Morningstar, there are 300 suitable funds, which can both be categorized, as active and non-active. However, Morningstar also lists the funds that are not the oldest share class. Since we do only want to include the parent share class there are 100 funds left. All the other spin-offs are just an almost exact version of the same fund again. They usually differ in their fee structure. So not properly handling this factor might lead to a potential bias. Although a slightly more accurate way to handle the share classes might be to build an aggregate portfolio of all the share classes of a fund, only the oldest share class will be used in this thesis. As one can see, the number of share classes is almost three times the number of original funds. A well-known example is the CRSP database created by Kenneth French and used in the study of Fama and French mentioned above. Their database treats each share class of a fund as a separate, new observation. Since Morningstar does not list dead or merged funds, one has to search for these types using other databases.

The financial database “Datastream” is used to download the monthly return of each fund. Since there are funds that pay a dividend and funds that do not pay a dividend, a problem of comparability might arise. Therefore, the so-called total return index utilized.

Where is the return index on day t and the return index on the previous day. and are the price indices on the particular day t. The dividend yield in percentage on day t is describe by and stands for the number of working day in each considered period. Hence, possible dividends are assumed to be reinvested to purchase additional units of the security at the closing price applicable on the ex-dividend date.

Ethical funds as well as funds that simply track an index or explicitly state that they are an ETF are excluded from the research. Another restriction is applied in order to get results that are more robust: Only funds with a return history of at least 18 observations, which equals 18 months out of the overall 240 months, are included. The returns available in Datastream are calculated net of management fees. Therefore, the expense ratio of every fund at any given point of time in the data history is needed to retrieve the gross returns. Unfortunately, this data is not available in Datastream. Instead, the management fees were obtained from a combination of different sources: Morningstar, finanzen.net and the adequate factsheet of a fund. The data is not reported for every fund in the sample. For these funds, one can conduct an average management fee. Since this concerns only a minor number of funds the method can still be used.The gross fund returns are before the costs in expense ratios, including management fees, but they are net of other costs, primarily trading costs.The total amount of funds in the sample is 125, since there are funds that match the criteria and that are not reported by Morningstar, like dead funds.

### 4.3 Forming the Dataset and Limitations

As mentioned above, the return index is used to compute the monthly returns of the funds, since the returns in a pure form are not available in Datastream.The fund returns are measured gross of taxes of dividends and net of management fees. In order to create gross returns, of the management fee of the particular fund is added to its monthly return. If there is missing data on the management fees of a particular fund than an average management fee of 1.35% is used. This is the average management fee of all available fee data of funds in the sample.

Since negative market returns can be observed in some months, which might lead to negative risk premiums, the Sharpe-Lintner-Black version of the CAPM is utilized instead of the Sharpe-Lintner version.

Ideally, the assets under management (AUM) of each fund would have been needed to create a value-weighted portfolio. With a value-weighted portfolio, inferences could have been made about the fate of the aggregate wealth invested in the funds. Unfortunately, Datastream does not provide this kind of data.

Another restriction might result from the declaration of the type of fund. Equity funds, investing the majority of their wealth in German equity are utilized in this thesis. To make the selection of funds Morningstar and other fund databases were used.Hence, it could still be that possibly not all funds matching the criteria were found.

### 4.4 Approach

To prepare for the bootstrap method, OLS- estimated alphas, factor loadings and residuals are calculated, using the time-series of monthly excess returns of each fund in collaboration with the market model version of the CAPM, the Fama-French model and the Carhartmodel. The detailed results for the CAPM and the three-factor model are provided in the appendix, only equations for the four-factor model are given here. First, an equal-weighted portfolio of funds, an umbrella like fund of funds, is built. Therefore the funds are weighted equally every beginning of a new month in order to be balanced throughout the whole portfolio. Then the performance models are applied. The returns are both measured before and after cost i.e. net and gross. The alpha or intercept of the equal-weighted portfolio of net returns gives information about the skill of active managers on average: Are they skilled enough to produce risk adjusted abnormal excess return to cover the costs imposed on the investor. The intercept of the portfolio built with gross returns tells us if a fund manager has any skill at all, which causes expected returns to differ from the potential return of a comparable passive benchmark. However, an abnormal performance exists if the estimated alpha significantly deviates from zero.

The first thing to do is to adjust for factor exposures. Thereby, the return series of fund i will be used in the regression of the respective factor model. To derive the alpha of each fund, time-series regressions from July 1995 to June 2015 are performed in which the excessreturns of the fund are used as the dependent variable and factor returns as explanatory variables applying the estimated version of regression ( 4 ):

Now the estimated alphas, factor loadings, residuals and t-statistics of alpha are saved and the value of the cross-sectional t(α) distribution at each percentile can be calculated.

### 4.5 Separating Skill from Luck

Bootstrapping is a computation intensive method for estimating the distribution of an estimator or a test statistic by resampling the data at hand.The overall aim of the bootstrap is to use histories of individual fund returns and simulations of these return histories to infer the existence of superior and inferior funds.Kosowski et al. (2006) showed in their study that a large part of the mutual funds in their sample had 𝛼’s from non-normal distribution and concluded that normality is a poor approximation in practice even for large mutual fund performance. The bootstrap can improve this approximation. According to the Jarque - Bera Test on normal distribution, the major part of the original estimated alphas in the sample also show non-normal distributions. Because of these non-normalities, bootstrapping is necessary in order to state clear results.Apart from a longer time series of performance, the bootstrap also has the advantage that no assumptions are made about the distribution of returns and alphas of funds.

To follow the bootstrap approach proposed by Fama and French, the t-statistic of a funds alpha and its distribution are examined. The distribution of the actual t(α) and the simulated t(α) are compared to infer whether the actual distribution is generated by mere luck or whether some manager exhibits skill.Therefore, it is not a severe problem if a fund appears with fewer observations compared to another fund during one simulation run. The only difference is in degrees of freedom. More degrees of freedom lead to thinner extreme tails in the distribution compared to the original distribution of t(α) for the actual returns of the funds.

The advantage of bootstrapping is that within a simulation run and within the whole procedure, the oversampling should compensate the undersampling. So a representative sample of t(α) estimates is obtained.Hence, the basic idea is to build the chance distribution of estimates resulting from a cloned population of funds. These are funds, which inhabit the exact same features as the actual population of funds, like fat tails compared to a normal distribution and correlation. The only exception is that true alpha is set to zero.

The bootstrap procedure of this thesis uses resampling of the number of months under the null hypothesis of no outperformance and consists of the following four steps:

**[...]**

^{1} (Gruber, 1996, S. 783).

^{2} (SEC, 2008).

^{3} Cf. (Fama & French, 2010).

^{4} Cf. (Sharpe, 1991).

^{5} (Fama & French, Fama/French Forum, 2009).

^{6} Cf. (Morningstar.com, 2016).

^{7} Cf. (Fama, 1970, p. 383).

^{8} Cf. (Grossman & Stiglitz, 1980).

^{9} Cf. (Markowitz, 1952).

^{10} Cf. (Black, Jensen, & Scholes, 1972).

^{11} Cf. (Jensen, 1968).

^{12} (Fama & French, 2010).

^{13} Cf. (Kosowski, Timmermann, Wermers, & White, 2006).

^{14} (Fama & French, 2010, p. 1933).

^{15} (Fama & French, 2010, p. 1933).

^{16} Cf. (Cuthbertson, Nitzsche, & O'Sullivan, 2008).

^{17} Cf. (Otten & Bams, 2002).

^{18} Cf. (Cuthbertson, Nitzsche, & O'Sullivan, 2008).

^{19} Cf. (French, 2008).

^{20} Cf. (Schmidt, Brückner, Lehmann, & Stehle, 2015)

^{21} Cf. (Cremers, Petajisto, & Zitzewitz, 2012, p. 3).

^{22} (Fama & French, 1992, p. 429).

^{23} Cf. (Viale, Kolari, & Fraser, 2009).

^{24} (Cremers, Petajisto, & Zitzewitz, 2012, p. 18).

^{25} Cf. (Fama & French, 1993).

^{26} Cf. (Fama & French, 2012).

- Quote paper
- Carsten Fritz (Author), 2016, Empirical Analysis of Mutual Funds investing in German Equity (1995-2015), Munich, GRIN Verlag, https://www.grin.com/document/342479

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