Excerpt

## Table of Contents

1 Introduction to Risk and Return

2 Modeling Risk

2.1 Capital Asset Pricing Model

2.2 Fama-French Three-Factor-Model

2.3 Modified Fama-French Models

3 Methodology, Portfolios and Data

3.1 Time-Series Data

3.2 Procedure of the Regression

3.3 Conducting the Regression

4 Evaluation of the Regression Results

5 Conclusion

List of References

Figures

Tables

List of Abbreviations

Appendix

Second Examination Period (1990 – 2014)

R-Code

## 1 Introduction to Risk and Return

For a long time there has been the search for the true model that explains the cross section of asset returns^{[1]}. It is common knowledge in Finance that the exposure to risk and the return of an investment are connected. The relationship between risk and return appears to be straight-forward: to gain expected return one has to take up risk and vice versa^{[2]}. Since investors like higher expected return but they dislike risk, investors face a trade-off between the risk they take and the expected return they earn^{[3]}. Considering that there are various definitions of risk, that kind of risk has to be defined that is entitled to a risk premium. Thus, this kind of risk is essential to construct models to explain returns.

Risk is often just regarded as the possibility of a negative development that leads to a loss^{[4]}. In the context of investment decisions, it was more useful to see it as a deviation from expectation^{[5]}. Within Markowitz's Portfolio selection the risk of deviating from expectation was defined as the standard deviation^{[6]}. Initially, this led to the set-up of efficient portfolios and the choice between a risk-free asset and risky assets in which the investor could invest^{[7]}. The aim of efficient portfolio is the diversification effect, which is based on a correlation between the assets of less than one^{[8]}. In simple terms, if stocks don`t move perfectly with each other, the variance of a portfolio will be lower than the weighted variances of the single assets in the portfolio. Constructing portfolios based on the optimal diversification leads to the efficient frontier, which is predicated on the mean-variance dominance criterion. Thus, creating an efficient portfolio means that all other possible portfolios with the same risk (variance) have a lower expected return or that with the same expected return, all other portfolios have a higher risk^{[9]}.

Even though the Markowitz model for portfolio selection and diversification appears to be useful, it is only rarely applied. It is predicated on the fact that investors utility is solely based on the expected risk and return of assets^{[10]}. Diversification aims at reducing and ideally vanishing idiosyncratic risk, which is the risk that is peculiar to a stock^{[11]}. But in modern asset management an asset has to be seen and evaluated in portfolio context^{[12]}. In a diversified portfolio, the idiosyncratic risk can in general be neglected^{[13]}. On account of this, the investor is not entitled to a risk premium for bearing idiosyncratic risk^{[14]}.

But the total risk consists of two forms of risk, the already discussed idiosyncratic risk (specific risk) and the systematic risk (market risk)^{[15]}.

Abbildung in dieser Leseprobe nicht enthalten

Figure 1: Decomposition of Total Risk^{[16]}

Market risk is associated with market-wide variations, hence it reflects macro events^{[17]}. For instance, changes in the interest rate, government spending, oil prices, foreign exchange rates and other macroeconomic events will affect almost all companies in the market^{[18]}. Consequently, this risk can't be diversified away and is therefore the only kind of risk that is in portfolio context entitled to a risk premium^{[19]}. Since the vulnerability of a company towards those factors is not diversifiable, it implies that an asset earns a systematic risk premium for each risk-factor (beta-factor) it is exposed to^{[20]}. The difficulty is to determine which risk-factors contain systematic risk so that they`re entitled to a risk premium. Solely these factors should be essential to explain and forecast portfolio returns.

The literature contains research of a variety of possible risk factors, e.g. labor income^{[21]}, growth in GDP, future consumption and real investment^{[22]}, consumption-to-wealth ratio^{[23]} or the labor-income-to-consumption ratio^{[24]}.

In the following, the Capital Asset Pricing Model (CAPM), a single risk factor model, and the Fama-French Three Factor Model and modifications of it are introduced. The Fama-French Model and the modifications of it are representatives of the Arbitrage Pricing Theory, which includes not only one but several systematic risk factors to explain excess returns^{[25]}.

## 2 Modeling Risk

### 2.1 Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM) is an approach that was developed by Sharpe, Lintner and Mossin^{[26]}. It states that the only relevant risk for assets is the systematic risk, since investors can diversify idiosyncratic risk^{[27]}. On account of this, the expected return of an asset in the CAPM is based on its systematic risk and the risk-free rate^{[28]}. In particular, systematic risk in the CAPM is measured by one factor, the sensitivity of an asset to the market^{[29]}. This approach is premised on the basic thought that some stocks are more affected by fluctuations of the market than other stocks and thus have a higher systematic risk^{[30]}. Hence, risk depends on the exposure of assets to macroeconomic events^{[31]}. This sensitivity to macroeconomic events of an asset or stock is measured in comparison to the market and defined as beta^{[32]}. In technical terms, this means that beta is the covariance of the stock and the market, divided by the variance of the market^{[33]}.

A beta of zero means that an asset or portfolio has no sensitivity to the market and no market risk^{[34]}. On the other hand, a beta of one states that the asset is moving exactly with the market^{[35]}. Accordingly, an asset with a beta greater one is expected to react overproportionally to the market (aggressive stock) and with a beta of less than one underproportionally^{[36]}. Since a higher beta represents higher systematic risk, it leads to a higher return in comparison to an asset or portfolio with lower beta^{[37]}.

Again, the relationship between the risk of an investment and its return is a positive relationship. In the CAPM this relationship exists between the systematic risk (beta) and the return^{[38]}. Consequently, the CAPM can be illustrated by the security market line (SML), which shows the average excess return of a stock / portfolio given by its beta^{[39]}. The corresponding equation of the SML is^{[40]}:

Abbildung in dieser Leseprobe nicht enthalten

The corresponding econometric model for excess returns can be illustrated the following way:

Abbildung in dieser Leseprobe nicht enthalten

The formula describes that the expected excess return of an asset or portfolio i at time t (Abbildung in dieser Leseprobe nicht enthalten) is determined by an alpha-factor (Abbildung in dieser Leseprobe nicht enthalten) and beta times the excess return of the market at t compared to the risk free rate. The term Abbildung in dieser Leseprobe nicht enthalten thereby refers to the market risk premium at time t, which multiplied by beta yields the (systematic) risk premium^{[41]}. It should be noted that the expectation of the alpha-factor is zero, if the CAPM holds.

On account of this, the CAPM can explain and predict the expected excess return on the individual stock related to the expected excess return of the market portfolio^{[42]}. This can be used for e.g. determining a required rate of return for an investment^{[43]}, the marginal contribution of a stock to the market portfolio (with beta one)^{[44]} or a risk-adjusted performance measurement^{[45]}.

Before continuing with the depiction of the other two models, the author wants to stress, that the CAPM as any model is based on simplification with makes the model to simple to capture reality^{[46]}. Returns have repeatedly been observed that were not explainable by CAPM and can therefore be regarded as anomalies^{[47]}. Although earlier research was in favor of the CAPM, anomalies that have been observed in the classical CAPM were taken as evidence against the model^{[48]}. For example, regressing the excess return (risk premium) of an asset / stock against the market excess return should just lead to the beta coefficient. Nevertheless, it will often also lead to a constant (alpha). Jensen defined this alpha as the average excess return that is earned above the excess return of an asset with a comparable risk^{[49]}. Since only beta should have explanatory power concerning the excess return, this fact is not explained by the classical CAPM^{[50]}. Other anomalies were found concerning the spread of average returns, e.g. "small firm effect"^{[51]}, and the effect of book-to-market value^{[52]}. A further point of critique was made by Roll (1977), who claimed that the relationship between beta and the return of an asset or portfolio depends on the chosen market portfolio^{[53]}.

Since the critique is suggesting that the single factor model CAPM is not sufficient to model and forecast returns, a multi-factor could also be exploited^{[54]}.

### 2.2 Fama-French Three-Factor-Model

The Fama-French Three-Factor-Model (TFM) is based on the Arbitrage Pricing Theory (APT) and is one of the most famous models. The Arbitrage Pricing Theory states that systematic risk is of multidimensional character and is therefore dependent on different economic risk factors^{[55]}. The CAPM in comparison to the TFM was only dependent on beta as the sensitivity to the market. Fama and French added to the beta factor two additional factors that shall have a relationship to the return of an asset or portfolio^{[56]}.

Fama / French observed that companies with different market capitalization but the same beta showed highly differing returns^{[57]}. Moreover, the return also appeared to be dependent on the book-to-market value^{[58]}.

Hence, Fama French developed the TFM that explained the US-Stock returns based on three factors: the market return, the return on a "size" portfolio and the return on a "high-minus-low book-to-market" portfolio^{[59]}. The difference between the small cap portfolios to the big cap ones is abbreviated SMB and the difference between the high book-to-market to the low book-to-market is called HML^{[60]}.

The equation for the TFM on excess returns can be illustrated in the following way^{[61]}:

Abbildung in dieser Leseprobe nicht enthalten

In econometrical writing this model is:

Abbildung in dieser Leseprobe nicht enthalten

It states that the expected excess return at a point t (Abbildung in dieser Leseprobe nicht enthalten) can be determined by a constant (Abbildung in dieser Leseprobe nicht enthalten) and the return on the market, SMB and HML multiplied by the respective sensitivity of the asset or portfolio to those factors. This implies that an asset earns a systematic risk premium for each beta-factor it is exposed to^{[62]}. Each beta-coefficient depends on the sensitivity of an asset to the specified systematic risk factor. Similar to the CAPM, the expectation of the alpha factor is zero.

### 2.3 Modified Fama-French Models

The modified Fama-French Model (MFFM) is constructed by the author to test whether returns can be more precisely explained and predicted by enhancing the Fama-French Three-Factor-Model with autocorrelation of the returns. The author decided to test this hypothesis because some studies suggest that returns follow a short-term trend before returning to their long-term trend (mean reversion)^{[63]}. The theoretical background for this deviation may be explained by time-varying risk premia or irrational behavior of market participants (behavioral finance)^{[64]}. Based on these explanations, returns might be autocorrelated in a certain order.

Autocorrelation is illustrated by an autoregressive process, which emphasizes that the current value of a variable depends on values that the variable took in previous periods^{[65]}. The author assumes that the effect of autocorrelation of the returns solely exists in the first or up to the second order. This is premised on the expectation that the influence of more recent past values on the future return appears to be more plausible than of less current past values of the return. Therefore, a model with an autoregressive process of order one [AR(1)] and one with an autoregressive process of order two [AR(2)] have been created.

The equations can be written in the following way:

Abbildung in dieser Leseprobe nicht enthalten

## 3 Methodology, Portfolios and Data

### 3.1 Time-Series Data

The author uses for the regressions of the CAPM, TFM and the MFFM the data that are provided by Fama / French^{[66]}. Those data include first of all the market return and the risk-free rate from July 1926 to January 1990 with monthly frequency. Additionally, the author uses data for the six portfolios built up by Fama / French to determine their Three-Factor model. Since the test includes an AR(2) process, t =1 is assigned to September 1926.

The six portfolios are characterized in the first observation period by the following means and standard deviations:

Abbildung in dieser Leseprobe nicht enthalten

Table 1: Portfolio Returns September 1929 until January 1990

The mean between the portfolios ranges between 0.887 and 1.558, whereas the standard deviation for the six portfolios has a range between 5.662 and 9.106. It is not surprising that the first to moments of their distribution deviate from each other to such an extent since those portfolios were constructed by Fama-French with different characteristics concerning the size of the firms in each portfolio as well as different book-to-market values. The distributions for the regarded period can be illustrated the following way:

Abbildung in dieser Leseprobe nicht enthalten

Figure 2: Histogram of Market Returns between 1926 and 1990

The six portfolio distributions look as follows:

Abbildung in dieser Leseprobe nicht enthalten

Figure 3: Histograms of Portfolio 1 - 6 Returns between 1926 and 1990

As observed by the first moment, the mean of each distribution lies above zero. Furthermore, at first glance all distributions appear to be normal distributed but with slightly different skewness and kurtosis. Since this is not in the centre of focus for this paper, the distributions won`t be discussed in detail.

### 3.2 Procedure of the Regression

The methodology for this paper is to conduct an in-sample regression for the coefficients of the explanatory variables for the given observation period. Moreover, the model-fit of the regression models to the observed data has to be evaluated. For the evaluation of the model, the author applies the adjusted R² and the Akaike information criterion (AIC). This is done due to the fact that in general R² increases with increasing amount of explanatory variables. It does therefore not necessarily show if an explanatory variable should be in a regression or not^{[67]}. On account of this O'Doherty, Savin & Tiwari (2012) argue that the support of models by a high R² is weak^{[68]}. Hence, an information criterion shall be applied. The adjusted R² can be seen as a weak information criterion that takes into account the loss of degrees of freedom (df) when an explanatory variable is added^{[69]}. Thus, the increase in R² has at least to offset the decrease in df, which means that R² does not always increase if a variable is added to the model^{[70]}.

In addition to that, the Akaike information criterion will be applied. Information criteria differ in the strictness of their penalty terms^{[71]}. The author aims to prefer models that have more explanatory power even though they might be more elaborate. The AIC contains a soft penalty term that doesn`t punish bigger models much^{[72]}. Therefore, the AIC appears to be a suitable information criterion to be applied.

### 3.3 Conducting the Regression

First of all, the regression is conducted based on the Capital Asset Pricing Model. The results of the linear regression based on the CAPM are:

**Abbildung in dieser Leseprobe nicht enthalten**

Table 2: CAPM (September 1926 - January 1990)

All six portfolios have p-values for the beta of zero, which means that the hypothesis of beta being zero can always be rejected. Therefore, beta is in this regression always highly significant for the excess returns of all six portfolios. Since the CAPM predicts that excess returns are solely based on beta as the sensitivity to the market return, the expected value of the constant should be zero^{[73]}. Nevertheless, portfolio three shows a p-value of 0.023, meaning a significance of the constant with a significance level of 2.3%. In other words, the constant in portfolio three is significant in predicting the excess returns in portfolio three. A constant that is included in the regression of the CAPM can be seen as a Jensen alpha. The Jensen alpha can be interpreted as the excess return (“outperformance” that portfolio lies above the prediction of the CAPM^{[74]}. The existence of Jensen`s alpha can be judged as a falsification of the CAPM. Due to the fact that the author’s focus is on the precise explanation of excess returns and is not on the search for a proof of CAPM, Jensen`s alpha is accepted and evaluated as significant for the explanation of the excess returns. Notwithstanding, the existence of a significant Jensen alpha is problematical if a model for excess returns without a constant is required.

**[...]**

^{[1]} c.p. O'Doherty, Savin & Tiwari (2012) p.1331

^{[2]} c.p. Markowitz (1952) p.79ff

^{[3]} c.p. Cuthbertson & Nitzsche (2007) p.115; c.p. Markowitz (1952) p.77

^{[4]} c.p. Lüth (2010) p.32; c.p. Hannemann, Schneider & Hanenberg (2008) p.90; c.p. Schulte & Horsch (2002) p.15; c.p. Diederichs (2010) p.9; c.p. Wall (2003) p.665f.

^{[5]} c.p. Diederichs (2010) p.10

^{[6]} c.p. Markowitz (1952) p.77-91

^{[7]} c.p. Cuthbertson & Nitzsche (2007) p.115, p.126f

^{[8]} c.p. Zimmermann (2009) p.58

^{[9]} c.p. Hockmann & Thießen (2012) p.639; c.p. Lüscher-Marty (2012) p.57ff.; c.p. Brealey, Myers & Allen (2011) p.189

^{[10]} c.p. Michaud (1998) p.3; c.p. Hockmann & Thießen (2012) p.648

^{[11]} c.p. Brealey, Myers & Allen (2011) p.185; c.p. Brealey, Myers & Marcus (2012) p.334ff

^{[12]} c.p. Hockmann & Thießen (2012) p.646

^{[13]} c.p. Zimmermann (2009) p.186

^{[14]} c.p. Cuthbertson & Nitzsche (2007) p.182

^{[15]} c.p. Brealey, Myers & Allen (2011) p.185ff; c.p. Back (2010) p.112

^{[16]} c.p. Brealey, Myers & Allen (2011) p.170

^{[17]} c.p. Brealey, Myers & Allen (2011) p.185f; c.p. Brealey, Myers & Marcus (2012) p.346

^{[18]} c.p. Brealey, Myers & Marcus (2012) p.346

^{[19]} c.p. Brealey, Myers & Allen (2011) p.185; c.p. Cuthbertson & Nitzsche (2007) p.118, p.136; c.p. Back (2010) p.112

^{[20]} c.p. Hockmann & Thießen (2012) p.697; c.p. Brealey, Myers & Allen (2011) p.199ff

^{[21]} c.p. Heaton & Lucas (2000) p.1163 - 1198

^{[22]} c.p. Cochrane (1996) p. 572 - 621

^{[23]} c.p. Lettau & Ludvigson (2001) p.1238 - 1287

^{[24]} c.p. Santos & Veronesi (2006) p.1 - 44

^{[25]} c.p. Cuthbertson & Nitzsche (2007) p.182ff

^{[26]} c.p. Hockmann & Thießen (2012) p.660; c.p. Lintner (1965) p.13-37; c.p. Sharpe (1964) p.425-442

^{[27]} c.p. Hockmann & Thießen (2012) p.660

^{[28]} c.p. Hackl (2005) p.329; c.p. Brealey, Myers & Allen (2011) p.192

^{[29]} c.p. Brealey, Myers & Allen (2011) p.185

^{[30]} c.p. Brealey, Myers & Marcus (2012) p.346

^{[31]} c.p. Brealey, Myers & Marcus (2012) p.346f

^{[32]} c.p. Brealey, Myers & Allen (2011) p.185

^{[33]} c.p. Cuthbertson & Nitzsche (2007) p.117, p.134

^{[34]} c.p. Zimmermann (2009) p.188)

^{[35]} c.p. Cuthbertson & Nitzsche (2007) p.119

^{[36]} c.p. Zimmermann (2009) p.184f; c.p. Cuthbertson & Nitzsche (2007) p.119; c.p. Brealey, Myers & Allen (2011) p.185

^{[37]} c.p. Hockmann & Thießen (2012) p.661

^{[38]} c.p. Zimmermann (2009) p.187

^{[39]} c.p. Cuthbertson & Nitzsche (2007) p.134, p.190; c.p. Zimmermann (2009) p.187ff

^{[40]} c.p. Brealey, Myers & Marcus (2012) p.354; c.p. Lüscher-Marty (2012) p.364f; c.p. Back (2010) p.105; Note: The econometric formula for the excess return can be found in: Von Auer (2013) p.375

^{[41]} c.p. Zimmermann (2009) p.181; c.p. Von Auer (2013) p.375

^{[42]} c.p. Cuthbertson & Nitzsche (2007) p.117, p.190

^{[43]} c.p. Zimmermann (2009) p.179)

^{[44]} c.p. Brealey, Myers & Allen (2011) p.194

^{[45]} c.p. Zimmermann (2009) p.191

^{[46]} c.p. Brealey, Myers & Marcus (2012) p.358

^{[47]} c.p. Hockmann & Thießen (2012) p.695

^{[48]} c.p. Zimmermann (2009) p.199

^{[49]} c.p. Jensen (1968) p.389-419; c.p. Jensen (1969) p.167-247; c.p. Zimmermann (2009) p.197

^{[50]} c.p. Brealey, Myers & Allen (2011) p.198, p.204

^{[51]} c.p. Banz (1981) p.3-18; c.p. Cuthbertson & Nitzsche (2007) p.202; c.p. Hockmann & Thießen (2012) p.662

^{[52]} c.p. Cuthbertson & Nitzsche (2007) p.193

^{[53]} c.p. Roll (1977) p.129-176

^{[54]} c.p. Hockmann & Thießen (2012) p.696; c.p. Zimmermann (2009) p.276

^{[55]} c.p. Ross (1976) p.341-360; c.p. Zimmermann (2009) p.179, 206, 275f; c.p. Back (2010) p.112; c.p. Copeland, Weston & Shastri (2005) p.176f

^{[56]} c.p. Fama & French (1993)p.3-56; c.p. Zimmermann (2009) p.209ff, p.268f; c.p. Hockmann & Thießen (2012) p.695f

^{[57]} c.p. Fama & French (1992)p.427-465; c.p. Zimmermann (2009) p.203

^{[58]} c.p. Brealey, Myers & Allen (2011) p.201ff

^{[59]} c.p. Cuthbertson & Nitzsche (2007) p.194

^{[60]} c.p. Zimmermann (2009) p.209

^{[61]} c.p. Fama & French (1993)p.21f; c.p. Cuthbertson & Nitzsche (2007) p.195; c.p. Brealey, Myers & Allen (2011) p.202

^{[62]} c.p. Hockmann & Thießen (2012) p.697

^{[63]} c.p. Hockmann & Thießen (2012) p.662

^{[64]} c.p. Hockmann & Thießen (2012) p.662f

^{[65]} c.p. Brooks (2008) p.215

^{[66]} c.p. Fama & French (2014) Current Research Returns

^{[67]} c.p. Brooks (2008) p.110

^{[68]} c.p. O’Doherty, Savin & Tiwari (2012) p.1331

^{[69]} c.p. Brooks (2008) p.110ff

^{[70]} c.p. Brooks (2008) p.110ff

^{[71]} c.p. Brooks (2008) p.232f

^{[72]} c.p. Brooks (2008) p.233

^{[73]} c.p. Cuthbertson & Nitzsche (2007) p.175

^{[74]} c.p. Cuthbertson & Nitzsche (2007) p.170, 175f

- Quote paper
- Christoph Lohrmann (Author), 2014, Comparison of the CAPM, the Fama-French Three Factor Model and Modifications, Munich, GRIN Verlag, https://www.grin.com/document/304738

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