Excerpt

## Table on contents

1. Introduction

2. Literature Review

a. Potential pitfalls

b. Results and conclusions

3. Empirical analysis

a. Data

b. Variable grouping

c. Variable selection

d. Regression

4. Conclusion

Appendix

List of literature

## Abstract

The purpose of this paper is to reinvestigate the linkage between microeconomic factors employed in the Carhart 4-factor model and macroeconomic variables for the German equity market from 2006 to 2015. This is achieved by relaxing the convention that the set of covariates has to be small and employing a comprehensive set of macroeconomic variables, which proxies for all potential macroeconomic risks inherent in the German equity market. To do so, I set up a three-tiered statistical procedure: First, I use prototype clustering to eliminate the risk of having highly intercorrelated variables in my model. Second, I select critical variables through the Least Absolute Shrinkage and Selection Operator (LASSO) and third, I employ selected variables to perform OLS estimations. I find that although the model established in this paper exhibits a reasonable higher R2 than conventional models, it doesn’t hold much explanatory power about underlying microeconomic factors. My bottom line is two minded: First, I conclude that microeconomic factors do not (or only partially) proxy for macroeconomic risks or that microeconomic factors act as hedge against macroeconomic events. Second, I conclude that there is a great need for more dynamic statistical models as conventional methods constitute a static approach and thus display low prediction accuracy when applied to different markets and different time intervals.

## List of figures

Figure 1: Cumulative factor development from 2006 to 2015 in Germany

Figure 2: Dendrogram of formed clusters including all investigated macroeconomic variables

Figure 3: Exemplary plot-output for one SMB cross validation

Figure 4: Histogram for lambda value frequency of thousand cross validations

Figure 5: Relationship between microeconomic factors and selected macroeconomic variables for path one

## List of tables

Table 1: Macroeconomic variables employed in prior research

Table 2: Macroeconomic variables chosen through LASSO for path one, two and three

Table 3: OLS estimation results

Table 4: Comparison between OLS estimations of this thesis and results based on variables used in prior research

Table 5: Macroeconomic variables selected manually from EIKON data base

Table 6: Macroeconomic variables selected through popularity among financial professionals

Table 7: R-Script

## 1. Introduction

Finding an investment opportunity set, which rewards the investor with an over-proportional return compared to the risk taken, is the main goal of every investment professional. Many concepts have been formed by practitioners and scholars over the last decades and today the Capital Asset Pricing Model (CAPM) (Lintner, 1965; Sharpe, 1964) is one of the most extensively used formulas in finance for pricing securities. The CAPM measures the systemic risk of a specific investment opportunity set by regressing past returns of the portfolio of interest on the market portfolio. According to the model, beta is the only risk factor that investors should take into account when making investment decisions. Portfolios exhibiting excess returns in market equilibrium embody a superior investment as they feature a better return to risk profile. In the long run, so the CAPM claims, no excess returns should be possible as they would attract investors as long as the former are brought down to the level of CAPM predictions. Over the last decades, however, practitioners and scholars have observed anomalies, which are not consistent with CAPM expectations. They found investment opportunity sets, which continuously performed better than the CAPM predicts (and some which performed worse). Merton (1973) extended the Capital Asset Pricing Model by adding a time component (ICAPM), claiming that there might be risks not captured by the original model, which the investor wishes to hedge. Different researchers tackled this issue from a micro and a macro perspective and since observing the first inconsistencies, researchers try to shed light on the underlying causes and deem this topic as relevant for further study.

At the micro level, the most prominent factors resulting in risk premia are the size, value and momentum factor. As this thesis focuses on those three anomalies, they deserve a more detailed explanation: The small firm effect or size effect holds that firms with a comparatively small market capitalization are expected to yield a higher return on average than their bigger rivals. Among other reasons, this can be explained by small firms facing more volatile business environments, which is tantamount to higher uncertainty for the investor. Value stocks are defined by a high ratio of book to market value. Displaying a high ratio often proxies for a high default risk and in fact, evidence was found that due to this risk, which is not captured by the original CAPM, value stocks outperform growth stocks around the world (Fama and French, 1998). So even if the historical beta of a number of value stocks average out at x, the beta in a recession could be much more than x due to higher levels of financial distress. This results in a higher required return for the portfolio as the risk of default affects the whole portfolio and not only a single security The risk is then said to be systemic and not only idiosyncratic. The momentum factor describes the tendency of rising asset prices to rise further and falling asset prices to keep depreciating. One reason for this phenomenon is that many traders today use various tools in their technical analysis to identify trends in an underlying security to jump on the bandwagon as long as the former is accelerating. This investment behavior is one of the reasons why trends in both directions, upwards and downwards, are amplified justifying the existence of the momentum effect. The size and value effect were put into the original CAPM formula and formed the popular 3-factor model (Fama and French, 1998, 1996, 1993, 1992). Later on, a 4-factor model was formed (Carhart, 2012) and today many other factor models gained a foothold in financial research. Figure 1 can be used to obtain a first overview of how the three microeconomic factors have cumulatively developed.

*Source: Own (underlying data provided by TU München)*

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*Fig1: Cumulative factor development from 2006 to 2015 in Germany (SMB ~ Small Minus Big; HML ~ High Minus Low; WML ~ Winner Minus Looser)*

At the macro level Chen, Roll and Ross (1986) were the first researchers, who developed a factor model, which consists of the following six macroeconomic factors: Excess return of the market, growth rate of industrial production, change of anticipated inflation, difference between current and anticipated inflation, unexpected change of the credit spread and unexpected change of the term spread. Analogous to the example made earlier in this thesis investment opportunity sets that are depending on one or more of those factors expose the investor to higher risk levels. Stocks, inter alia, strongly correlating with changes in GDP will most probably depreciate during market downturns. Tangjitprom (2012) reviewed various studies on macroeconomic factors and stock returns and classified the macroeconomic variables into four groups: Variables reflecting general economic conditions, variables related to interest rate and monetary policy, variables concerning price level, and variables concerning international activities. As those classes themselves consist of a great number of variables, the whole array of variables is extensive and will thus not be discussed in detail here.

Since the development of those models, a multitude of papers has been published which investigate the linkage between micro- and macroeconomic risk factors, which claim that microeconomic factors act as proxies for risk originating from macroeconomic activities. Liew and Vassalou (2000) found a linear dependency between the three microeconomic factors SMB, HML and WML and various macroeconomic variables and as a result made choices about which macroeconomic variables are to be used for explaining their microeconomic companions. The results of all papers, however, vary to a high degree as diverse markets, time intervals and statistical methodologies were used in the research. The endeavor to establish a linkage between micro- and macroeconomic factors, however, bears greater obstacles than one might think at the first glance. One problem (P1) arising from quantifying this linkage originates in the micro- and macroeconomic variables being timely offset. Thus, a linear regression analysis, which uses macroeconomic variables such as inflation, GDP growth etc., which are timely synchronic with the output variables might declare some microeconomic factors orthogonal, even if they are not. To circumvent this problem, Bergbrandt and Kelly (2015) used direct measures of investor expectations rather than the macroeconomic variables expressing a current state. Other researchers overcame this problem by linking macroeconomic variables to iterative timely staggered microeconomic factors or tried to establish a link by investigating the behavior of risk premia through short-term macroeconomic news announcements (Bestelmeyer et al., 2011). Another problem (P2) faced by researchers was the computational limitations in performing and testing regression analysis. Preceding literature has subsequently added new factors to the model in order to test whether the newly added variable contributes to the goodness of fit. As there exist a vast number of potential macroeconomic variables, a researcher is at odds with picking exactly the ones needed for his/her model. Moreover, when it comes to security returns, the ICAPM does not restrict the number of factors that could be used for the model. Especially, in light of every market having its own peculiarities, it would be illogical to restrict the number of variables ex ante. So, in fact, it was computational power which narrowed researchers down to a cumbersome statistical methodology and not empirical evidence.

Similar to Bergbrandt and Kelly (2015), I aim at circumventing P1 by also taking macroeconomic variables into account, which express investors’ future expectations rather than only a current state1). I aim at limiting the potential downsides of P2 by a three-tiered statistical procedure which requires higher levels of effort and is thus placed at the very center of this thesis. In the first stage, I tackle the problem of having a great number of highly intercorrelated variables in my model, which would, if not eliminated, lead to maintaining redundant variables and hence lead to biased estimations. I use prototype clustering (Bien and Tibshirani, 2011), which is an unsupervised classification algorithm building on agglomerative hierarchical clustering and stemming originally from the field of machine learning and data mining. It is unique in the sense that by using minimax linkage, it naturally associates a prototype (or clustroid) with every interior node of the dendrogram in a way that unlike previous hierarchical clustering methods, it exhibits a high degree of interpretability. In this thesis, I transform the correlation matrix of the normalized macroeconomic variables into a distance matrix in order to perform sound clustering. In the second stage of the process, I select the best possible variables being left in the basket after prior clustering through the LASSO (Tibshirani, 1996). LASSO is a method for estimating linear models by not only minimizing the residual sum of squares but also a penalty factor added to the equation, which consists of a constant multiplied by the sum of absolute values of the coefficients. In most cases it produces some coefficients that are exactly zero and thus provides the user with more interpretable and properly fitted models. To choose the best possible value for the constant of the penalty factor and to obtain the best fitted model I use cross validation. On the third stage selected variables are employed to perform OLS estimations. The idea of this three-tiered statistical procedure stems from Zhu, Basu, Jarrow and Wells (2018), who reinvestigated the estimation of multiple factor models by relaxing the prevalent consensus of the number of factors having to be small. In total, I investigate 383 distinct macroeconomic variables, which proxy for all possible variables in the German macroeconomic universe.

## 2. Literature Review

### a. Potential pitfalls

This chapter provides a conspectus of the main risks arising from setting up and interpreting statistical models which investigate the linkage between microeconomic factors and macroeconomic variables. These pitfalls are not further investigated in the course of this thesis but shall raise awareness of the risks inherent in a specific statistical procedure. The risks include the presence of mispricing, the effect of moderators, the danger of using statistical models for future predictions and the problem of reverse causality.

The presence of substantive mispricing can lead to biased findings as microeconomic factor loadings could exhibit orthogonality even if they are correlated in a market featuring perfect competition. In other words: A researcher is at odds with finding any correlation between micro- and macroeconomic variables if assets are continuously and significantly mispriced (given that the level of mispricing is steadily changing). With this in mind the question arises if mispricing is a factor the researcher should worry about. In fact, Griffin and Lemmon (2002) found evidence for inaccurate asset valuations as they observed high return reversals around earning announcements dates, especially for firms with low analyst coverage. These findings are supported by the study conducted by Lakonishok et al. (1994). They argue that value-based strategies yield higher returns not because they are fundamentally riskier but rather due to irrational investors behavior.

In statistics, moderators are variables, on which the impact of one variable on another is dependent. To provide an elucidatory example, let a and b be variables, which correlate with variable c. A fourth variable, called the moderator composed of the product of a and b, might help to explain c better than a or b could on an individual basis. The small firm effect, for instance, might neither be correlated to “unemployment” nor “wage increase” but instead to a combination of the two.

A set of statistical assumptions produced by any statistical model can nothing but determine dependencies observed in the past. However, the future might develop significantly dissimilar to the past and as a consequence the model will not hold. The CAPM, for instance, naturally requires higher rates of returns of risky assets than as of risk-free assets (such as government bonds) and thus implicitly predicts higher returns for risky assets in the long run. Elton (1999), however, discovered long term bonds being outperformed by the risk-free rate from 1927 to 1981 indicating CAPM predictions failed for more than 50 years. For this reason, dependencies among micro- and macroeconomic variables observed in the past do not guarantee the same dependencies for future developments. Especially in today’s fast pacing environment sound believed economic theories seem to fall through the cracks and dependencies observed in the past might differ from those in the future.

One of the first hurdles a researcher investigating dependencies has to overcome, is to identify cause and effect. In the context of this thesis some of the microeconomic factors could constitute the effect and some macroeconomic variables the cause and vice versa. As I investigate the impact of macroeconomic variables on microeconomic factors, I assume macroeconomic variables to be the cause. However, some researchers linked macroeconomic variables to microeconomic factors and argued this reverse causality to be true. Inter alia, there was evidence provided for SMB and HML containing information about future GDP growth (Harnhardt and Olcoz, 2008; Liew and Vassalou, 2000).

### b. Results and conclusions

This chapter provides an overview about the results and conclusions of prior research covering the linkage between the microeconomic factors SMB, HML as well as WML and various macroeconomic variables. The first two paragraphs address the question of which macroeconomic variables could successfully be linked (or not be linked) to the microeconomic factors and the third paragraph covers empirical findings why these connections could be drawn.

As discussed in the introduction, risk premia for microeconomic factors are mainly originating from beta values being potentially higher in prospective recessions and recoveries respectively (measured in absolute figures) than CAPM estimations predict. As coming economic downturns and upswings are strongly related to conjuncture movements, microeconomic factors are frequently linked to variables based on business cycle activities with GDP growth leading the way. Liew and Vassalou (2000) tested whether the profitability of SMB, HML and WML can be connected to future GDP growth for ten developed countries and found that even in the presence of changing business cycle states, SMB and HML are sound predictors for future GDP growth in some countries. At this, Liew and Vassalou (2000) state a positive relationship between SMB as well as HML and future economic growth. Thus, they support a risk-based explanation for those two factors. For WML, however, Liew and Vassalou could not establish any linkage to future GDP activities. Harnhardt and Olcoz (2008) built upon the research conducted by Liew and Vassalou and extend their geographical scope by investigating also 11 pan-Eurozone industries besides 16 European industries. Their findings: SMB and HML do exist in various industries as well as in many countries and serve as a leading indicator for future economic activity. Their results are therefore in line with those documented by Liew and Vassalou (2000). Davis (2001) reviewed the calculations undertaken by Liew and Vassalou (2000) and found evidence for their findings being poor for a number of markets. He especially underlines his results of the US financial market, where a long time series was available (1957 to 1998) and no linkage between future GDP growth and microeconomic factors was found. Unlike Liew and Vassalou (2000), who connected factor loadings to future GDP activities, Zhang et al. (2009) investigated the relationship between SMB as well as HML and current GDP growth. As in prior research they were able to document that both SMB and HML exhibit high loadings during periods of high GDP growth. Besides the endeavor to link microeconomic variables to economic recovery movements, other researchers also investigated microeconomic factor behavior during economic downturns. In fact, in addition to high loadings during upswings, evidence was provided for SMB as well as HML exhibiting high performance during recessions (Kalaycıoğlu, 2009; Kang et al., 2011).

Although GDP growth constitutes the principal driver for business cycle activities, many researcher also investigated a great number of other variables in order to explain microeconomic factor loadings. Aretz, et al. (2010) employed the inflation rate, the aggregate survival probability, the term structure of interest rates and the exchange rate beside GDP growth to explain factor loadings of SMB, HML and WML. They found that SMB and HML could be linked to the above mentioned macroeconomic variables and thus state that an asset pricing model’s performance based on those variables is comparable with the performance of the Fama-French 3-factor model. The momentum factor, however, just contains marginal information about asset prices according to their findings. Similar variables are employed by Zhang et al. (2009) who provide a risk-based explanation for size and value premia grounded on macroeconomic variables such as GDP growth, inflation, the interest rate and the term spread. They employed two distinct statistical methodologies, which support the same findings, and could establish a link to some of the variables. Apart from a positive linkage to current GDP growth, the researchers document a negative relationship between the size effect and inflation, a positive dependency between the value effect and inflation, a negative relationship between SMB as well as HML and the interest rate and a positive dependency between risk premia and the term spread. Similarly to Liew and Vassalou (2000), Fuerst (2006) endeavored to link the size and value effect to future economic activities. Fuerst (2006), however, did not only use GDP growth as a variable but also employed other economic measures such as the level of employment and durable orders. He finds that changes in SMB, and to a lesser extent changes in HML, initiate impacts on the real economy similar as those from monetary policy. Whereas findings regarding SMB and HML mostly coincide, documentations for WML are surprisingly different. Chordia and Shivakumar (2002) show that momentum strategies can be explained by an array of timely-staggered macroeconomic variables. Thus they support the explanation of momentum profits stemming from time-varying expected returns. On the contrary Griffin and Martin (2003) find evidence for momentum profits being neither explainable through timely synchronic nor timely lagged macroeconomic variables.

One of the most accepted reasons for the size and value effect exhibiting positive loadings is that small firms and firms displaying a high book to market ratio are subject to higher levels of financial distress. Fama and French (1998) found evidence for the existence of a value effect in twelve out of thirteen major markets and claim that the value effect proxies for a relative measure of financial distress. Black (2006) builds up on that study and finds that the unconditional variance of default premia contain critical information about the unconditional variance of value and small stock performance. Similar results were produced by Kalaycıoğlu (2009), who established a linkage between SMB and default risk as well as between HML and the term spread. He documents that the risk of financial distress among small-growth firms is particularly high in economic downturns and that SMB as well as HML loadings underly timely variation.

## 3. Empirical analysis

### a. Data

Currently there exist various research institutions offering free of charge time series data of the size, value and momentum factor for the German capital market. The individual data sets differ especially in respect to the timespan covered and some of them do not include the momentum factor. The explanatory power, however, of the factor loadings when compared over time might be low as accounting standards varied over the last decades. In Germany, most of financial statements published before 2005 were based on German Law (HGB) whereas after 2005 they were grounded on international financial reporting standards (IFRS) (Brückner et al., 2015). As HGB aims at protecting investors and IFRS tries to provide the investor with a true and fair view of financial performance, the accounts of a company might look very unequal when regarded through the lens of diverse accounting systems. Differences in asset valuations might, for instance, occur in the fields of hedge accounting, pension commitments and software development, which affect the profit and loss statement as well as the balance sheet. To overcome the potential risk of using, ceteris paribus, varying asset prices due to the use of diverse accounting standards I decided to use time series data from 2006 onwards. As Hanauer et al. (2013) provide time series data, which cover the longest period after 2005 including the momentum factor, I chose this data set to represent microeconomic factor loadings in my empirical analysis (the data set reaches from 2006 to 2015).

The underlying data used by Hanauer et al. (2013) was primarily extracted from Worldscope Financial Database and represent, with the exception of data of financial service providers (as banks and insurance companies), all monthly stock returns from the prime and general standard at Börse Frankfurt (CDAX). To overcome the potential problem of being subject to survivorship-bias, the data base is manually adjusted each year and is thus containing many currently delisted companies. To further improve the quality of the data set, various other corrections have been carried out on a two leveled filtering process based on suggestions of Ince and Porter (2006), who primarily address the issue of deficient data quality of single stock returns2).

For calculating corresponding factor loadings of SMB and HML, Hanauer et al. closely follow the methodology employed by Fama and French (1993). For obtaining SMB they arrange all stocks, previously selected by the above-mentioned filtering process, in ascending order of market capitalization and split the set of stocks into two groups using the median. Stocks belonging to the first group are defined as small (S), stocks in the second group as big (B). HML is computed through the 30 and 70 percent quantile and the three resulting groups are named Low (L), Medium (M) and High (H). Those two orderings are realized simultaneously and form the six portfolios S/H, S/M, S/L, B/H, B/M and B/L. Based on it’s value-weighted return from July of year (y-1) until June of year (y), a stock is reallocated to a corresponding portfolio in July of year (y). To compute loadings for the established factor-mimicking portfolios following formulas are used:

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where represents the time series vector for a corresponding portfolio. HML is calculated analogously to Carhart’s study (1997) and to Kenneth French’s Website (2012) on a monthly basis. Every month (t), stocks are listed in ascending order according to their cumulative returns starting from (t-12) to (t-2). Similar to calculating HML, stocks are grouped into three classes on the basis of a 30/70 percent quantile split and subsequently indexed as Winner (W), Neutral (N) or Loser (L). In connection with SMB-classification six portfolios are formed, now named S/W, S/N, S/L, B/W, B/N and B/L. WML is then computed as:

Time series data for macroeconomic variables were provided via the EIKON-database at the University of Passau. EIKON is a series of software products developed by Thomsen Reuters for researchers and financial professionals to extract and analyse financial information. In total I collected 383 distinct variables, which represent current state observations as well as investor expectations. All of them were carefully chosen by first, including most of the variables mentioned in previous papers, by second, using a comprehensive set of variables which I chose myself and third sampling out the most frequently used macroeconomic attributes employed by financial professionals.

A good starting-point for getting a conspectus of macroeconomic variables used in prior research is the review of macroeconomic factors and stock returns carried out by Tangjitprom (2012). Not all the variables empirically employed could be used in this thesis as some of them were not available in the EIKON data base. All available variables are listed in Table 1.

*Source: The Review of Macroeconomic Factors and Stock Returns (Tangjitprom, 2012)*

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*Table1: Macroeconomic variables employed in prior research*

**[...]**

- Quote paper
- Marwin Zimmermann (Author), 2018, The effect of macroeconomic variables on the size, value and momentum factor in Germany, Munich, GRIN Verlag, https://www.grin.com/document/449782

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