# How are MF returns calculated

## Factor investing. Rule-based and quantitatively driven capital investment

### Table of Contents

I. List of Abbreviations

II. List of Figures

III. List of tables

IV. List of formulas

V. Symbol directory

1 Introduction

1.1 Problem

1.2 Approach and objective

2 Theoretical Foundations

2.1 The determination of the performance

2.1.1 Yield

2.1.2 Risk

2.1.3 Risk and return in the context of the financial markets

2.2 Critical questioning of information efficiency in financial markets

2.3 The Capital Asset Pricing Model

2.4 From CAPM to Factor Investing

3 factor investing

3.1 The factors

3.2 The choice of factors

3.2.1 Momentum

3.2.2 Low volatility

3.2.3 Value

3.3 Implementation of the factor strategies

3.3.1 Style factors in the context of the business cycle

3.3.2 The long-only and long-short debate

3.4 Past performance of the factors

4 The implementation of the factor strategies on the Euro Stoxx 50

4.1 The architecture of index products

4.1.1 Portfolio reassembly and rebalancing

4.1.2 The number of individual values

4.1.3 The weighting of the individual values

4.2 Data and methodology

4.3 Single factor strategies

4.3.1 Momentum strategy

4.3.2 Low volatility strategy

4.3.3 Value strategy

4.4 Multi-factor strategies

4.4.1 Top-Down Approach

4.4.2 Bottom-up approach

5 Analysis of the strategies implemented

5.1 Alpha factor and information ratio

5.2 The benchmark

5.3 Regression and Jensen Alpha

5.4 Sharpe ratio

5.5 Performance analysis

5.5.1 Benchmark

5.5.2 Single factor strategies

5.5.2.1 Momentum

5.5.2.2 Low volatility

5.5.2.3 Value

5.5.3 Multi-factor strategies

5.5.3.1 Top-Down Approach

5.5.3.2 Bottom-up approach

5.6 Overall result and critical appraisal

6 Conclusion

VI. Appendix list

VII. Literature and list of sources

### I. List of Abbreviations

Figure not included in this excerpt

### II. List of Figures

Figure 1: Risk / return relationship of various asset classes

Figure 2: Findings from the CAPM for factor investing

Figure 3: Business cycle and factor premiums

Figure 4: Risk and return of the factors (1975-2014)

Figure 5: The investment process of a long-term investor

Figure 6: Standard deviation and number of stocks in the portfolio

Figure 7: Characteristics of the weighting according to market capitalization

Figure 8: Weighting according to factor characteristics

Figure 9: Momentum strategy

Figure 10: Low volatility strategy

Figure 11: Value strategy

Figure 12: Correlation between the factors

Figure 13: Top-down approach in multifactor investing

Figure 14: Bottom-up approach in multifactor investing

Figure 15: Risk / return diagram: benchmark (2010-2020)

Figure 16: Risk / return diagram: Momentum (2010-2020)

Figure 17: Risk / return diagram: Low volatility (2010-2020)

Figure 18: Risk / return diagram: Value (2010-2020)

Figure 19: Risk / return diagram: MF top-down (2010-2020)

Figure 20: Risk / return diagram: MF bottom-up (2010-2020)

Figure 21: General overview (2010-2020)

Figure 22: Eurozone gross domestic product

Figure 23: General overview (2010-2015)

Figure 24: General overview (2015-2020)

### III. List of tables

Table 1: The implementation of the momentum strategy

Table 2: Momentum Portfolio (June 30, 2019)

Table 3: The implementation of the low volatility strategy

Table 4: Low Volatility Portfolio (06/30/2019)

Table 5: The implementation of the low volatility strategy

Table 6: Value Portfolio (01/01/2019)

Table 7: Multifactor bottom-up portfolio (06/30/2019)

Table 8: CAPM regression: benchmark (2010-2020)

Table 9: CAPM regression: Momentum (2010-2020)

Table 10: CAPM Regression: Low Volatility (2010-2020)

Table 11: CAPM regression: Value (2010-2020)

Table 12: CAPM regression: MF top-down (2010-2020)

Table 13: CAPM regression: MF bottom-up (2010-2020)

Table 14: Evaluation of the portfolios (2010-2020)

### IV. List of formulas

formula *(1):* performance

Shape *l (2):* Gross return

Shape *l (3):* Discrete return

shape *el (4)*: Steady return

shape *el (5)*: Reshaping steady return

formula *(6):* Standard deviation

formula *(7):* Anualized standard deviation

formula *(8)*: Expected return

formula *(9)*: Weighting of a share

formula *(10)*: Alpha factor

formula *(11):* Information ratio

formula *(12)*: CAPM regression

formula *(13)*: Sharpe ratio

### V. Symbol directory

Figure not included in this excerpt

### 1 Introduction

### 1.1 Problem

There has been a profound change in mutual fund investing since the financial crisis. The actively managed financial products, which were often routinely prepared using templates, brought investors results during the financial crisis with which they could not be satisfied. Classic, actively managed funds are also exposed to great competition from the passively managed and exchange-traded index funds.^{1} These so-called ETFs are very popular. However, the mass euphoria surrounding the much advertised ETF products is almost exaggerated. Such funds are dominated by overvalued and sector focused stocks.^{2}

In order to counteract the dissatisfaction of the investors, the industry responds with factor investing. This has been very popular in recent times. Factor investing is a strategy for active and passive investing.^{3} The investment approach is intended to deliver higher risk-adjusted returns. This is achieved through the systematic selection of securities that have certain properties.^{4}

As factor investing becomes more prevalent, concerns have been raised that the higher risk-adjusted returns on factors may fade over time. One reason for this is that factor investing relies on anomalies that arbitrage could resolve. If, as a result of the further spread of factor investing, a lot of money flows into certain stocks, their return potential can be reduced and the valuation ratio increased. An investor with the idea of factor investing is also not equally successful at all times. The individual factors do not always provide higher risk-adjusted returns.^{5}

It is therefore important to deal extensively with the topic of factor investing, as it requires a lot of specialist knowledge and thought. In order to counteract the problems mentioned, a few things should be considered when selecting the factors.

### 1.2 Approach and objective

The work on factor investing begins with an explanation of the theoretical basics required for the reader. The focus is on the performance measurement of an investment and its two components. After developing the formulas commonly used in practice for measuring performance, the capital asset pricing model and information efficiency in the financial markets are discussed. The execution via the capital asset pricing model forms the transition to the core of the work, factor investing.

An explanation of the terms is followed by a precise explanation of which factors are suitable for factor investing. The factors momentum, low volatility and value suitable for factor investing are then described in more detail. After that the question is answered as are the factors in a portfolio^{6} of an investor can be implemented. The dependence of the performance of the factors on the economy is also discussed here, as is the conflicting goals between the effectiveness of the implementation of the factors and the possibility of practical implementation.

The question of implementation is specified in the next sub-item of the work and explained for each factor. Afterwards, eligible stocks for the factor strategies are presented. These are the individual values of the Euro Stoxx 50, which serves as an investment universe. The structure of a portfolio based on an index is described below. In addition to the problems that a long-term investor must be aware of, the selection of individual stocks and their weighting is also described in detail. This is followed by the practical implementation of the individual portfolios of the factor strategies. After the portfolios of the individual factor strategies have been created, a transition to the multifactor strategies takes place. This is a link between the individual factor strategies. Portfolios are also created for this purpose.

At the end of the thesis there is a detailed description of the methods of performance measurement used in the thesis. The benchmark for the portfolios of the factor strategies is determined and key performance indicators are described. Thereafter, the theoretical description goes into the measurement of the performance of all portfolios of the factor strategies against the specified benchmark using the key figures. The performance measurement ends with a summary and an overview of all portfolios and includes a critical appraisal. A conclusion forms the end of the work.

The aim of the work is to check whether the portfolios of the factor strategies deliver a better performance than their benchmark. The focus is on comparing the theoretical literature with the practical implementation of factor investing.

### 2 Theoretical Foundations

### 2.1 The determination of the performance

Various investment characteristics are considered when constructing a portfolio. The risk-return relationship of the individual assets plays a role here^{7}, which is presented in capital market theory, plays a major role.^{8} Together, the two components are referred to as performance. The respective importance of the two components mentioned depends on the investor's attitude to risk.^{9}

The first component of performance, the return, represents the profitability of an asset. This is a percentage value that shows the return on the capital employed. If an investor buys a share, the amount of the return is fraught with uncertainty. Because whoever buys and holds a share cannot say whether he will be able to sell it again at a profit in the future. Since the return data of the future are not available, an investor who is faced with the question of whether it is worthwhile to invest in certain stocks has to make other means. In doing so, he can look at past returns to get an idea of future returns. The historical returns thus serve as an estimate for future returns.^{10} Risk, the second component of performance, takes account of the uncertainty described. It denotes the possibility of deviations from a planned size. Risk is understood as both a negative and a positive deviation from the planned size.^{11} The two-dimensional target performance just explained is mathematically represented as the return on investment divided by the risk associated with the investment. The following expression is used to determine the performance:

Figure not included in this excerpt

Source: Bruns, C./Meyer-Bullerdiek, F. (2020), p. 2.

By considering return and risk together with values of the same dimensions, a simultaneous optimization of the two components is possible in terms of portfolio construction.^{12}

#### 2.1.1 Yield

The return on capital investments is also made up of two components. On the one hand, there is periodic income, such as coupons on bonds or dividends on stocks, and on the other hand, capital gains and losses due to price changes. Returns can be calculated for multiple periods or just over a single period.^{13} Returns can be specified for any length of time. However, the returns on different assets can only be compared with one another if they refer to periods with the same duration. The common comparison period for returns usually extends over a period of one year. The term of return used in the following is referred to as the gross nominal return, since transaction costs, taxes and inflation are not taken into account.^{14} The gross return of a single period to can be shown as follows:

Figure not included in this excerpt

Source: Ang, A. (2014), p. 625.

The price of the capital investment is at the beginning of the considered period and specifies the price of the capital investment at the end of the considered period. The variable includes all dividends and interest payments that are paid up to the end of the period under review.^{15} This consideration is a period that can be delimited by two individual points in time. For this reason, the return calculated using the above formula is also known as the discrete return. The formula assumes that dividends and interest payments will be retained, or in other words, withdrawn, with no further loss or interest until the end of the period.^{16} The interest rate effect is therefore not taken into account. The discrete return described here can, if it is determined over several periods for the same investment, be linked to one another in a multiplicative manner. The result then shows the gross return over the entire period under consideration.^{17}

The discrete return has the further advantage that it is additive when considering different assets. To do this, it just needs to be converted into the form defined by the following formula:

Figure not included in this excerpt

Source: Ang, A. (2014), p. 626.

With the additive link, the individual discrete returns of the assets must be multiplied by their weighting in the portfolio and then added to a total return.^{18}

In addition to the discrete returns, there are also the so-called steady returns. Continuous returns can be described as a series of discrete returns at very short intervals, whereby the continuous return mathematically corresponds to the limit value from the discrete return when the duration of the investment period tends towards a minimum.^{19} The steady return can be shown as follows:

Figure not included in this excerpt

Source: Ang, A. (2014), p. 626.

Figure not included in this excerpt

Source: Ang, A. (2014), p. 626.

With the steady return, there is a continuous interest rate between the beginning and the end of the period, which takes the compound interest effect into account. The steady returns of an asset can be additively linked to calculate the total return on an asset.^{20} Another advantage of the steady returns is that they can be viewed approximately as normally distributed. They are therefore suitable for the calculation of models in which the normal distribution is assumed.^{21}

The use of steady returns or discrete returns should be based on suitability for practical applicability. Despite their differences, they are equivalent instruments for calculating returns. Ang (2014) compares the use of the two approaches with the units of measurement in the recipe of an apple pie. It doesn't matter whether the ingredients are measured in pounds or kilos. However, you should not switch freely between the methods in order to avoid errors.^{22} Discrete returns are increasingly used in portfolio theory. The described properties of the multiplicative linkage of the returns on an asset and the additive linkage of several assets are well suited for this.^{23}

Return considerations over several periods can be compared with one another by converting the individual returns for each period into an average return. There is a difference between the arithmetic and the geometric average return. When calculating the arithmetic average return, the individual returns are added and then divided by the number of periods considered.^{24} The individual steady returns are used here, as they are additive for an asset, as described above. In the case of discrete returns, the geometric average return is formed. These meet the requirement required here to be multiply linkable via an asset. In the case of the geometric average return, the individual discrete returns are multiplicatively linked and then the root is extracted according to the periods under consideration.^{25}

#### 2.1.2 Risk

The risk of an investment also consists of two components. The overall risk is made up of the systematic and the unsystematic risk. Systematic risks are risks that affect an entire asset class. Unsystematic risks, on the other hand, relate to a single investment.^{26} The risk in the form of uncertainty relates to future price movements. They can turn out differently in the future, as different environmental influences can occur. The environmental influences can be assigned probabilities of occurrence. In this way, a probability distribution for achievable returns can be created. In these circumstances, the returns are random numbers, and the distributions of which can be metrics calculated.^{27}

These key figures thus serve as a measure of risk, the most common key figure for financial market risks being volatility. The overall risk can be calculated using the volatility expressed as a percentage. Volatility is nothing more than the term used in the financial markets for the annualized standard deviation.Mathematically, the volatility is based on the variance, which represents the squared average return deviations of the mean. Since the variance is not a percentage, it has to be converted into the standard deviation, which is the square root of the variance. Thus, the measure of risk and the return now have the same unit.^{28}

The assumption of normal distribution is essential for volatility. With the two parameters, expected population return *𝜇* and standard deviation *𝜎*, only the normal distribution can be fully described. Since the steady returns come closer to the normal distribution than the discrete returns, the formula for the standard deviation *𝜎* be modified.^{29} The formula then looks like this:

Figure not included in this excerpt

Source: Own illustration based on Steiner, M. et al. (2017), p. 60.

Stands for the number of return observations used and for the steady return. The expected rate of return of the population *𝜇* reflects the average of historical returns. The advantage of the approach with constant returns is that the expected return can be calculated as the arithmetic mean of the constant return.^{30} When considering a sample of the population, the denominator of the formula is reduced by one. With the denominator *T-1 * an unbiased estimate can be guaranteed.^{31}

In order to make the standard deviations of different investments comparable, they are converted into the annualized volatility. To do this, the standard deviation must be multiplied by the square root of the number of calculation periods.^{32} The following formula results when calculating the standard deviation using monthly returns:

Figure not included in this excerpt

Source: Own illustration based on Mondello, E. (2018), p. 7.

When using monthly returns, the standard deviation is thus multiplied for the annualization with the square root of the number twelve. The calculation of volatility from historical data shown here is used in practice as a forecast for the future and thus contributes to the making of investment decisions.^{33}

#### 2.1.3 Risk and return in the context of the financial markets

It is important for an investor to define good phases and times of crisis. Times of crisis are defined, for example, by a recession, rising inflation or financial crises. During such times, high-risk investments tend to perform worse than low-risk investments.^{34} Investors must be compensated for the risk. Otherwise, no investor would be holding risky assets such as stocks. The compensation for the risk taken is the so-called risk premium. In normal times, investors are rewarded with this premium.^{35} Figure 1 on page 18 shows the return and risk of various asset classes in 19 countries^{36} over the period between 1900 and 2010.

Figure not included in this excerpt

Figure 1: Risk / return relationship of various asset classes

*Source: Ang, A. (2014), p. 241.*

Over this period, stocks have a higher return per year and a higher volatility than bonds. Over the long term, stocks have a higher risk premium than bonds. The factors presented later in chapter three are the driving forces behind the risk premiums.^{37}

### 2.2 Critical questioning of information efficiency in financial markets

A widely discussed question is whether financial markets are efficient or inefficient.^{38} Information efficiency in financial markets relates to the reaction of asset prices to new information. An information-efficient market is characterized by the rational pricing of new information. All current and past information is processed in the current system prices. With efficient markets, a manager will not be able to permanently beat the market.^{39} The market efficiency hypothesis describes the theory that states that the price of a security always corresponds to the true value of the security through efficient markets.^{40} Fama (1970) names three sufficient prerequisites for efficient financial markets:

- No transaction costs when trading securities

- All information is free and accessible to all investors

- The investors agree with the given implications of the information.^{41}

Fama describes the difficulty of fulfilling all three requirements in practice. He does not name the three prerequisites as necessary prerequisites for efficient markets, but instead names the lack of one of the prerequisites as a reason for possible inefficiency. Fama divides information efficiency into three different levels based on the level of information included in the prices.^{42}

Many economists do not believe in the existence of markets with perfect information efficiency. The reason for this is that information comes with a cost. If the markets were information efficient, it would never occur to anyone to spend money on information. But if nobody is willing to pay money for information, it cannot be priced in either.^{43}

Grossmann and Stiglitz (1980) describe a scenario in which the acquisition of information is associated with costs. Managers use the paid information to look for market inefficiencies. The elimination of the inefficiencies that results from this results in an almost perfect information-efficient market. Excess return can be achieved by investors willing to spend the money on the information as well.^{44}

The market efficiency hypothesis is still important these days. For managers, it forms the basis for conducting research to identify possible inefficiencies. These would ultimately lead to excess returns compared to the market. Inefficiencies in the financial markets can be due to two reasons. They can be explained on the one hand with rational and on the other hand with behavioral approaches.^{45}

### 2.3 The Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM)^{46} is considered the first theory in the field of factor investing. Before the model was released, it was widely believed that volatility determines the risk and return of a security. The CAPM showed new insights into the concept of risk and the reward for the risk taken.^{47} The CAPM takes up the idea of portfolio theory to influence the risk of securities through diversification.^{48} With CAPM, the risk is not considered on the basis of an isolated asset, but rather the movement of the asset in comparison to other assets and to the entire market.^{49} The volatility of a single, isolated share is therefore irrelevant. The risk measure used, the beta factor, measures the fluctuation of an asset compared to the market.^{50} The beta factor reflects the systematic risk of an asset which, in contrast to the unsystematic risk, cannot be diversified away. One assumption of the CAPM is the investment and borrowing of money at an identical, risk-free interest rate. In the CAPM, investors hold a combination of the risk-free interest rate and the risky part of the investment.^{51} The risky part of the investment is the market portfolio. It contains all available securities in relation to their market capitalization.^{52} The following additional assumptions are made in the CAPM:

- All investors have homogeneous expectations

- The capital market is perfect

- The capital market is in equilibrium

- The same prices and thus the same returns apply to all investors.^{53}

Under the given assumptions, every investor invests in the same, risky and optimal portfolio, the market portfolio.^{54} The return on a single risky investment can be calculated in the CAPM using the following formula:

Figure not included in this excerpt

Source: Bruns, C./Meyer-Bullerdiek, F. (2020), p. 95.

It denotes the expected return value of the security and the expected return value of the market portfolio. The variable stands for the beta of the security in question, the variable for the risk-free interest rate. There is a linear relationship between the return and the beta factor of the investment. The return results from the risk-free interest rate and the risk premium.^{55} In the CAPM, the market risk is the only return driver and therefore the only factor considered. The return grows linearly with the risk assumed by the investor. Losses that occur when prices fall are compensated for by a higher return. There are no other factors that systematically influence expected stock returns.^{56}

### 2.4 From CAPM to Factor Investing

The simplifying assumptions of the CAPM ignore the complexity of the financial markets. The model suggests a world with a perfectly efficient financial market. Market inefficiencies are excluded by the assumption that information can be provided and accessed free of charge from the model. Section 2.2 explains that economists do not believe in the existence of such perfect, efficient markets.^{57} The CAPM defines the risk of an asset using the beta factor. This market risk is the only risk that is rewarded with a risk premium. The CAPM is therefore a single factor model. Over time, however, it became clear that the CAPM cannot adequately explain the returns on assets. Financial market researchers then looked for other factors in addition to market risk in order to better explain the returns on an asset.^{58} Despite the simplified assumptions, the CAPM is useful for implementing the factors. Figure 2 on page 22 shows some of the findings that can be transferred from the CAPM to multifactor models.

Figure not included in this excerpt

Figure 2: Findings from the CAPM for factor investing

Source: Ang, A. (2014), p. 205 f.

Investors should hold the factor instead of individual, individual assets and integrate it into their portfolio. If you hold individual, non-diversified assets, incorporate unsystematic risk into your portfolio that is not rewarded with a risk premium. By diversifying, you can exclude unsystematic risk from your portfolio. In the CAPM, the market portfolio is the most diversified portfolio that an investor can hold. In CAPM, the considerations relate to the market factor, i.e. the market risk entered into. The consideration can, however, be transferred to any other tradable factor.^{59} Investors can decide for themselves how much they want to benefit from the factors. Depending on the respective risk appetite of an investor, the portfolio consists of various combinations of the market portfolio and the risk-free investment.^{60} The risk of an individual asset is determined by its beta. Assets with a high beta have a higher risk, but also a higher risk premium to compensate for the higher losses in bad times.^{61} The theory of factors and recognized factors alongside the market factor are discussed in the next section.

### 3 factor investing

### 3.1 The factors

Stocks have risk premiums because they are subject to various risks. An example of a risk is the market risk described in the CAPM. Market risk is a factor. Many such factors have been identified over the years that explain the differences in risk and return between stocks.^{62} Factors are comparable to the nutritional values contained in food. Just as different foods have different nutritional values, so do the factors of individual stocks. And just as nutritional values are the drivers behind food, so are the factors that drive an asset's performance.^{63} Ang, Goetzmann and Schaefer (2009) were commissioned to examine and assess the Norwegian state fund. You find out that the good performance of the fund can be explained by the factors. They do not attribute the good performance of the fund to active management, but to the factors implemented in the portfolio. They recommend that the management of the fund focus on better and more efficient implementation of the factors.^{64} The extensive investigation is considered a breakthrough in factor investing.^{65}

Investing in factors leads to good performance over a longer period of time. However, in bad times it can also lead to underperformance. As with CAPM, in which the risk is the falling market, the risk is compensated with a risk premium. For each factor, however, there is an individual definition of a crisis in which the strategy leads to underperformance.^{66}

A distinction is made between two types of factors. There are macroeconomic factors such as inflation or economic growth that affect all asset classes. The second type of factor is represented by the so-called style factors. They explain the return and the risk within an asset class.^{67}

### 3.2 The choice of factors

Before an investor wants to invest in factors, he must think about which factors he is considering. It is important to follow a few principles consistently. Ang (2014) recommends paying attention to the following four properties of the factors:

- Legitimation through scientific research

- The factors have generated significant premiums in the past that will continue to exist in the future

- There must be historical studies of the factors, also for times of crisis

- They must be convertible in liquid and tradable instruments.^{68}

Satisfying the four characteristics prevents factors that represent a trend from being excluded. The focus is therefore on the recognized and extensively examined and documented factors.^{69}

While style factors are tradable investment strategies, investing in macroeconomic factors is far more difficult. This is because individual asset classes do not move one-to-one with them. Movements of the assets in relation to the macroeconomic factors are often not intuitive.^{70} The focus is therefore on the style factors of stocks. The interaction of the style factors with the macroeconomic factors is discussed in Section 3.3.1. Recognized style factors that meet all characteristics are momentum, low volatility and value.^{71}

#### 3.2.1 Momentum

The momentum factor relates to the past returns on stocks. According to this, stocks with positive price trends will continue to have a positive price movement in the future. On the other hand, stocks with below-average returns will continue to be among the losers in the future.^{72} The considerations on the momentum factor have been around for a very long time, but empirical studies came late. Today the momentum factor is a recognized and widespread factor.^{73} The existence of the momentum factor has been proven in many studies.

Jegadeesh and Titman (1993) were one of the first to demonstrate the effect. The basis of their research over the period from 1965 to 1989 are those on the NYSE^{74} listed stocks. The stocks are selected based on the historical return over a certain period of time. After that, the stocks segmented in this way are held for a predetermined period of time. They find that the strategy allows them to outperform the benchmark.^{75} Glaser and Weber (2002) published their detailed studies of the momentum effect on the German stock market and demonstrated it there.^{76}

The cause of the momentum factor is in herd behavior^{77} seen by man. This behavior explains why people instinctively follow a trend.^{78} Momentum generated by herd behavior can be seen in the spread of positive company news as an example. The starting point for this is the processing of the information from the various individuals. Momentum can now be generated if there is a delay in overreacting to the positive news and the price rises continuously as a direct consequence. Momentum can also be generated by an underreaction. Because in the event of an underreaction, if this is noticed, the price can rise at a later point in time.^{79}

#### 3.2.2 Low volatility

The low volatility factor relates to the risk of the stocks. With the factor, a lower risk is rewarded with a positive risk-adjusted return. This contradicts traditional financial literature, which associates higher risks with higher returns.^{80} Numerous empirical studies have found that stocks with a lower volatility perform better than stocks with a high volatility.^{81}

Haugen and Heins (1972) conducted studies of the risk and return relationship of stocks listed on the NYSE for the period from 1926 to 1971. During this period, they find that stocks with a low beta have a high alpha^{82} have.^{83} They also find that an investor who takes a higher risk may not necessarily expect a higher risk premium.

Baker and Haugen (2012) review the low volatility factor premium on the stock market for 21 industrialized and twelve emerging countries in the period from 1990 to 2011.You can prove the premium on the examined markets over the period under consideration. During the investigation, they implement the stocks in their portfolio that showed the least volatility in the last 24 months prior to the buying processes. Based on the results, they question the previous teaching on the relationship between risk and return.^{84}

The factor premium can be explained based on investor behavior. They are more likely to buy stocks with high volatility than stocks with low volatility. The reason for this is that many investors think that they can make a lot of money with high-risk investments. Also, they often only look at stocks that have seen large gains in the past and ignore the fact that this may change in the future. Due to the behavior of investors, there is no arbitrage process for the anomaly. The factor premium remains.^{85} The structure of the market provides another explanation. Investors who have high return targets, such as fund managers, are more likely to invest in volatile stocks. As a result, the volatile stocks have higher prices and lower returns relative to lower-risk stocks.^{86} Also the specification of an investor, the tracking error^{87} Keeping it low ensures that the low volatility factor exists.^{88}

**[...]**

^{1} See www.faz.net (2018).

^{2} See Seethamraju, C. (2017), p. 2.

^{3} See www.faz.net (2018).

^{4} See www.bnpparibas-am.de (no year).

^{5} See www.robeco.com (2019).

^{6} A portfolio is a collection of individual systems that is put together with the aim of optimizing individual aspects. See Bruns, C./Meyer-Bullerdiek, F. (2020) p. 2.

^{7} Synonymous with the term capital investment.

^{8} See Mondello, E. (2015), p. 1.

^{9} See Bruns, C./Meyer-Bullerdiek, F. (2020), p. 1.

^{10} See Beike, R./Schlütz, J. (2015), p. 72.

^{11} See Steiner, M. et al. (2017), p. 55.

^{12} See Bruns, C./Meyer-Bullerdiek, F. (2020), p. 1 f.

^{13} See Mondello, E. (2015), p. 2.

^{14} See Franzen, D./Schäfer, K. (2018), pp. 72-74.

^{15} See Ang, A. (2014), p. 625.

^{16} See Spremann, K. (2008), pp. 72-74.

^{17} See Mondello, E. (2015), p. 3 f.

^{18} See Ang, A. (2014), p. 625 f.

^{19} See Franzen, D./Schäfer, K. (2018), p. 75 f.

^{20} See Ang, A (2014), p. 626 f.

^{21} See Steiner, M. et al. (2017), p. 54.

^{22} See Ang, A (2014), p. 627 f.

^{23} See Steiner, M. et al. (2017), p. 55 f.

^{24} See Mondello, E. (2018), p. 7.

^{25} See Franzen, D./Schäfer, K. (2018), p. 78 f.

^{26} See Steiner, M. et al. (2017), p. 56 f.

^{27} See Franzen, D./Schäfer, K. (2018), p. 84 f.

^{28} See Mondello, E. (2018), pp. 25-27.

^{29} See Steiner, M. et al. (2017), p. 59.

^{30} See Mondello, E. (2018), p. 27 f.

^{31} See Steiner, M. et al. (2017), p. 60.

^{32} See Steiner, M. et al. (2017), p. 60.

^{33} See Steiner, M. et al. (2017), p. 61 f.

^{34} See Ghayur, K. et al. (2019), p. 38.

^{35} See Ang, A. (2014), p. 195 f.

^{36} The countries considered are Australia, Belgium, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Canada, the Netherlands, New Zealand, Norway, South Africa, Spain, Sweden, Switzerland, Great Britain and the USA.

^{37} See Ang, A. (2014), pp. 194, 240 f.

^{38} See Islam, S. M. N./Watanapalachaikul, S. (2005), p. 53.

^{39} See Mondello, E. (2018), p. 68.

^{40} See Islam, S. M. N./Watanapalachaikul, S. (2005), p. 54 f.

^{41} See Fama, E. F. (1970), 387 f.

^{42} See Fama, E. F. (1970), 413-416.

^{43} See Ang, A. (2014), pp. 209 f.

^{44} See Grossmann, S. J / Stiglitz, J. E. (1980), p. 393.

^{45} See Ang, A. (2014), p. 210.

^{46} The CAPM builds on the assumptions and findings of portfolio theory. It was developed independently in the 1960s by William F. Sharpe, John Lintner, and Jan Mossin. See Sharp, W. F. (1964); Lintner, J. (1965); Mossin, J. (1966).

^{47} See Ghayur, K. et al. (2019), p. 37.

^{48} See Steiner, M. et al. (2017), p. 22.

^{49} See Ghayur, K. et al. (2019), p. 37.

^{50} See Ang, A. (2014), p. 196 f.

^{51} See Mondello, E. (2015), 245 f.

^{52} See Steiner, M. et al. (2017), p. 22.

^{53} See Franzen, D./Schäfer, K. (2018), p. 209.

^{54} See Mondello, E. (2015), p. 250.

^{55} See Steiner, M. et al. (2017), pp. 22, 26-27.

^{56} See Ang, A. (2014), p. 197 f.

^{57} See Mondello, E. (2015), pp. 236-239; See Ang, A. (2014), pp. 207-209.

^{58} See Lumholdt, H. (2018), p. 118.

^{59} See Ang, A. (2014), p. 197 f.

^{60} See Ang, A. (2014), p. 199 f.

^{61} See Ghayur, K. et al. (2019), p. 37f.

^{62} See Ghayur, K. et al. (2019), p. 35 f.

^{63} See Ang, A. (2014), p. 193 f.

^{64} See Ang, A. et al. (2009), pp. 24-26.

^{65} See www.robeco.com (no year).

^{66} See Ang, A. (2014), pp. 193, 444.

^{67} See www.blackrock.com (no year).

^{68} See Ang, A. (2014), p. 455 f.

^{69} See Ang, A. (2014), p. 456.

^{70} See Ang, A. (2014), p. 474.

^{71} See Ghayur, K. et al. (2019), p. 40.

^{72} See Lumholdt, H. (2018), p. 127.

^{73} See Zaher, F. (2019), pp. 118-120.

^{74} The New York Stock Exchange (NYSE) is the largest stock exchange in the world. She is based on Wall Street, New York. See www.handelsblatt.com (no year).

^{75} See Jegadeesh, N./Titman. S. (1993), pp. 67-68, 89.

^{76} See Glaser, M./Weber, M. (2002), p. 19 f.

^{77} Further information on herd behavior in Johnson, F. (2018).

^{78} See Zaher, F. (2019), p. 127 f.

^{79} See Ang, A. (2014), p. 238.

^{80} See Zaher, F. (2019), p. 99.

^{81} See Lumholdt, H. (2018), p. 128.

^{82} "The alpha factor of a share corresponds to {...} the difference between the total return of a share and the systematic return." Steiner, M. et al. (2017), p. 311.

^{83} See Haugen, R. A./Heins, A. J. (1972), pp. 21-26.

^{84} See Baker, N. L./Haugen A. (2012), pp. 1-2, 4, 16.

^{85} See Zaher, F. (2019), p. 107 f.

^{86} See Bruns, C./Meyer-Bullerdiek, F. (2020), p. 312.

^{87} Tracking error plays a role in terms of passive portfolio management. Here it is a measure of the quality of the replication of the benchmark. See Steiner, M. et al. (2017), p. 73.

^{88} See Zaher, F. (2019), p. 109 f.

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