Foreign exchange risk in asset pricing models and the similarities
between the financial- and nonfinancial sector, empirical evidence
from Germany
Daan Eltjo Ruiter1 Thesis MSc. Finance University of Groningen
June 2015
Supervisor: Dr. M. A. Lamers
Abstract
This study investigates whether exchange rates explain average stock returns between 1999 and 2014 when incorporated in the three and four factor model in the German stock market. Additionally, the financial- and nonfinancial sector are compared separately. The financial sector is often excluded in asset pricing models without theoretical evidence although its influential size in Germany. Results show that exchange rates explain average returns in the financial sector, nonfinancial sector and whole German stock market and thus is a priced factor. The financial- and nonfinancial sector co-move in the market-, size- and exchange rate factors. However, the value and momentum factor vary enormously. Furthermore, the financial sector is better hedged against foreign exchange risk than the nonfinancial sector through higher returns and less sensitivity.
Key words: Empirical asset pricing, foreign exchange rate risk.
JEL Classifications: F31, G12, G14
1 University of Groningen
Faculty of Economics & Business Student number: s1882775
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1. Introduction
Fama & French (1992) investigated price anomalies besides the CAPM which resulted in the now famous three factor model. The three factor model shows that the premiums associated with market-, size-, and value risk explain a large fraction of the cross-sectional variation in stock returns. The momentum factor of Carhart (1997) explains even more variation by adding a factor that accounts for historic returns (momentum and is the most used addition to the three factor model. Both models are often used to examine price premiums associated with the risk factors in the stock market as has been done for the German stock market.2
Germany is one of the largest countries in the world concerning import and export and even the largest of the Eurozone (OECD, 2015). This introduces the risk of changing currency values. However, the start of the European Monetary Union (EMU) has made the risk of changing exchange rates disappear between member states. Germany benefited substantially from EMU member states since the start of the EMU, however, still over 40% is exported out of the European Union (Eurostat). The trade surplus that can be attributed to the euro is near 10% for Germany since the start of the EMU (Serlenga & Shin, 2013). The trade surplus are mostly due to foreign trades, however, trades with member states increased rapidly in recent years (Economic and Financial Affairs). Despite the fact that Germany is part of the EMU, foreign exchange rate risk still exists.
Kolari, Moorman, & Sorescu (2008) find that foreign exchange rates explain part of the cross-section of average stock returns in the U.S. This finding is based on adding an exchange rate factor to the three as well as the four factor model and the notion there is less variation left to explain. Apergis, Artikis, & Sorros (2011) find the same results for Germany using an index of 21 countries that affect the exchange rate of the euro. The method of Kolari Moorman, & Sorescu (2008) is more complete than the use of an index (Kolari, Moorman, & Sorescu, 2008). Both studies find that firms who experience the most exposure to foreign exchange on average realize the lowest returns. This study tries to validate the results found in Germany by Apergis, Artikis, & Sorros (2011) using a more complete method for measuring foreign exchange risk and tests if this anomaly is a priced factor in the three and four factor model.
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Using the methodology proposed by Fama & French (1993) for creating the factors of the asset pricing model, an interesting finding is the exclusion of the financial sector. Nowadays Fama & French (1998;2012) include the financial sector, however, a clear reasoning for the inclusion is omitted (Baek & Bilson, 2014). Brückner, Lehmann, Schmidt, & Stehle (2014) analyze four publicly available databases about factor models in Germany. They find four different ways of treating the financial sector. Foerster & Sapp (2005) find that the inclusion of the financial sector in the U.S. influences the amount of significant risk factors and coefficients as well as the acceptance of the Fama Macbeth regressions. This implies that an influential sector must be taken into account.
The Deutsche Börse, which is located in Frankfurt, is the largest stock market of Europe (World Exchanges, 2014) and the financial sector is a very important sector in the Deutsche Börse, creating more than 20% of the total value added in the domestic market (Assa, 2012). The largest bank in Europe is the Deutsche Bank and regulation for the European financial markets is to be determined in Frankfurt (Wójcik & MacDonald-Korth, 2014).
This all indicates the importance of the financial sector in Germany and especially the stock exchanges in Frankfurt. When such a large fraction of the market is the financial sector, this sector could influence the results of average stock returns in asset pricing models. There has not been an analysis on the influence these different treatments of the financial sector have on the asset pricing results in Germany. Following the path set by Barber & Lyon (1997), the coefficients and t-statistics of the financial- and nonfinancial sector are analyzed with regard to the similarity of the different risk factors.
This paper contributes to the existing literature in two ways; the first contribution is the calculation of an improved exchange rate factor, validating the results of Apergis, Artikis & Sorros (2011) to show that foreign exchange is a priced anomaly in the German stock market. The second contribution is the analysis of the financial- and nonfinancial sector in Germany. The analysis indicates whether the financial- and nonfinancial sector show similarities in the factor coefficients of the asset pricing models. This paper builds upon the work of Barber & Lyon (1997) and Baek & Bilson (2014), who analyzed the three factor model for the U.S., and extends their research with a four factor model and an exchange rate factor for the German stock market.
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well as foreign exchange risk should be included in asset pricing models due to its ability to reduce the unexplained variation, bringing the alpha closer to zero. This convergence is observed in the whole stock market as well as the independently measured financial- and nonfinancial sector. Other findings are the co-movement of the market-, size- and exchange rate factor between the financial- and nonfinancial sector meaning that these factors impact the sectors in a similar manner. However, a linear relation between the book-to-market factor and foreign exchange portfolios merely exists in the nonfinancial sector. Additionally, the momentum factor varies greatly between sectors with no real pattern visible. The financial sector has coefficients closer to zero, indicating that the financial sector is better hedged against changing currencies.
Remainder of the paper is structured as follows: Section 2 presents the theoretical literature on the factor models, exchange rates and the comparison of the financial- and nonfinancial sector, while Section 3 presents the methodology used. Thereafter, Section 4 presents the data & descriptives. The results are presented in Section 5 and discussed in Section 6 before I conclude in Section 7.
2. Literature review
2.1. Fama & French’s size- and value factor
Fama & French (1992) find that in the U.S. small firms earn higher returns than big firms (size factor) and that firms with a high book-to-market ratio perform better than firms with low book-to-market ratio (value factor). These two factors, together with the market factor account for more than 90% of the variance in a diversified portfolio of U.S. equities. (Fama & French, 1992, 1993, 2008).
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size factors, more variation in average stock returns is explained. Basu (1983) used the market and size factors as well, but added the E/P-ratio. He finds that this addition increases the explanatory power of explaining average stock returns. Rosenberg, Reid, & Lanstein (1985) find a positive relation between the firms book-to-market ratio and average stock returns and that it helps to explain the cross-section of average stock returns.
Since Fama & French (1992) included factors that are all scaled versions of price, they expected that some of these factors would be redundant. They created one database instead of multiple. One database has the advantage to be able to compare the influence of different factors on average stock returns. Fama & French (1992) excluded financial institutions from their database because financial institutions are leveraged for a different reason than nonfinancial firms, who are often in distress when being highly levered.3
Redundancy is found for the leverage and E/P-factors. Their impact is absorbed by the size and value factors. Leverage is redundant in that the slope of book leverage (market leverage) is always negative (positive) and cancels each other out by means of the book-to-market ratio. The explanatory power of E/P decreases when size is added to the factor model and even disappears when size and value are both added. These findings result in the now famous Fama & French three factor model (from now on denoted as FF) with significant risk premiums associated to the market-, size- and value factors.
2.2. Carhart’s momentum factor
The Carhart four factor model (from now on denoted as C4) extends the FF by adding momentum as a factor (Carhart, 1997). Jegadeesh & Titman (1993) were the first who describe momentum which is the phenomenon that stocks that increased (decreased) in price in recent history, keep rising (falling) in the near future. The C4 explains more variation in average stock returns by the addition of momentum. Therefore it is one of the most generally used asset pricing models next to the FF.
2.3. Four factor models for Germany
The analysis of a factor model in Germany has the advantage that comparative material exists in four publicly available databases.4 These databases all research the C4 and find a negative alpha which is non-significant in most cases. The market factor shows significant positive coefficients. Germany has a reversed size effect and a value effect that decreased in
3 Based on prior research of Bhandari (1988). 4
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strength after 2007. The most pronounced factor seems to be momentum, in which all researches find a premium above 1% per month. The R²-bar is around 0.80 with upward outliers.
2.4. The exchange rate
Fama & French (2012) mention the importance of exchange rates but do not correct for them, instead purchasing power parity is assumed. However, according to the International Capital Asset Pricing Model (ICAPM) the covariance of assets with the foreign exchange currency return should be a risk factor when the purchasing power parity is violated (Adler & Dumas, 1983). This happens when investors from different countries face different prices for goods and services. Since this is the case countries are exposure to exchange rates. The exposure to foreign exchange rate risk is defined as the sensitivity or correlation of an asset or liability to a change in exchange rates (Adler & Dumas, 1984).
Hovanov, Kolari, & Sokolov (2004) state that time-series’ results can be significantly affected by the choice of base currency due to currency fluctuations over time. Therefore, they create an invariant currency value index (ICVI) as base currency, which has the same value independent of base currencies choice. Quoting Adam Smith “… a commodity which
itself is continuing varying in its own value, can never be an accurate measure of the value of other commodities.” (Smith, 1976, p. 48). The ICVI overcomes this problem and sets a clear
path for researching exchange rates and creating a minimum variance portfolio. The benefit of a minimum variance portfolio is that changes in the numerator currency are the actual changes in currencies compared to a whole currency basket and not just the base currency (Hovanov, Kolari, & Sokolov, 2004). Hovanov, Kolari & Sokolov (2004) chose the currencies of countries that were major industrialized countries to be included in the currency basket. These major industrialized countries are the base of the Special Drawing Right.
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the euro (EUR) , the Japanese yen (JPY), the British pound (GBP) and the U.S.-dollar (USD) (IMF, 2015). It is used by the IMF to describe the financial activities of more than 180 countries.
However, the currencies weights are chosen once every five years based on the last three months of trading. Therefore, Hovanov, Kolari, & Sokolov (2004) investigate the applicability of using a minimum variance portfolio instead and conclude that their portfolio is a better choice as base currency since it has a lower standard deviation.
Kolari, Moorman, & Sorescu (2008) use the minimum variance portfolio introduced by Hovanov, Kolari & Sokolov (2004) to calculate a stable aggregate currency (SAC). They use the SAC to measure exchange rate exposure in order to calculate a factor that can be used in a factor model. Kolari, Moorman & Sorescu (2008) use the methodology proposed by Adler & Dumas (1984) to measure the exposure to foreign exchange rate risk by regressing the firms stock returns on the percentage change of the dollar against the SDR. The estimated coefficients represents every exposure an individual firms can have to fluctuations in the exchange rate (Adler & Dumas, 1984). Kolari, Moorman & Sorescu (2008) use the calculated exposure to sort firms into 25 portfolios, where the first portfolio exhibits the most negative exchange rate exposure and the last portfolio the most positive exposure. They find that firms with the most exposure to exchange rates have the lowest average stock return. The returns are therefore distributed in an inverted u-shape, which makes it nonlinear and not very suitable as a factor in asset pricing models.5 They overcome this problem by creating an exchange rate neutral portfolio by subtracting the extreme exposure portfolios (1 and 25) from the less extreme portfolios (2 to 24). They find that by including an exchange rate factor, the explanatory power of average stock returns increases.
This research is replicated in Germany by Apergis, Artikis & Sorros (2011). However, they use the effective exchange rate calculated by the European Central Bank instead of the SAC. The effective exchange rate of the euro is adjusted for inflation and is calculated for the whole euro area are as well as for the individual member states (Schmitz, De Clercq, Fidora, Lauro, & Pinheiro, 2012). Apergis, Artikis & Sorros (2011) use the effective exchange rate of the whole euro area and do not take intratrade of member states into account. Their results are highly similar to the results of Kolari, Moorman & Sorescu (2008) and show that exchange rate exposure follows an inverted u-shape across portfolios and is a priced factor in Germany.
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There is a higher negative price premium for U.S. and German firms that exhibit more exposure to exchange rates. However, the financial sector is excluded in the U.S. database (Kolari, Moorman, & Sorescu, 2008), while included in the German database (Apergis, Artikis, & Sorros, 2011).
Comparing these outcomes therefore is difficult since the exclusion of the large U.S. financial sector might influence the results.
2.5. The in- and exclusion of the financial sector
Arguments for in- or excluding the financial sector in asset pricing models remain limited (Baek & Bilson, 2014). The analysis of publicly available databases of asset pricing models across countries shows various treatments of the financial sector. The U.S. database changed from not including the financial sector to including them, the Canadian and U.K databases do not include the financial sector in their research and the Swiss database only includes investment companies.6 The different treatments are not grounded on theory. The comparison of four factor model papers based on Germany show a similar treatment of the financial sector. The papers (i) do not exclude the financial sector, (ii) exclude the financial sector because of the ease of obtaining data, (iii) only include the financial sector in the calculations for the market-, size- and value factors and (iv) exclude the financial sector due to data limitations.7
Excluding a sector can create bias in the results of asset pricing models (Barber & Lyon, 1997; Baek & Bilson, 2014), especially when that sector is of influential size, as Assa (2012) states that the financial sector in Germany is. Foerster & Sap (2005) find that the inclusion of the financial sector in the FF influences the amount of significant risk factors and the corresponding coefficients. Moreover, the Fama-MacBeth regressions fail to reject models when the financial sector is included and rejects these models when they are incorporated in the model. The exclusion of the financial sector is not grounded on theory, although gives reason to believe that it influences the results.
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The U.S. database is made available by Fama & French (2012), the Canadian database by Francoeur & Niyubahwe (2009), The U.K. database by Gregory, Tharyan, and Christidis (2013), and the Swiss database by Ammann & Steiner (2008).
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Barber & Lyon (1997) start the discussion about the possible bias in the result due to the exclusion of the financial sector in the FF. Their paper reacts on the arguments about the size and value factors being the result of data mining by treating the financial sector as natural holdout sample. Fama & French (1992) excluded the financial sector due to the different meaning of leverage. However, the leverage factor lost influence when size and value factors were incorporated. Barber & Lyon (1997) research the holdout sample on similarity in reactions of the factor coefficients. The financial- and nonfinancial sector were analyzed by calculating the t-statistic with a null-hypothesis assuming unequal variances. They find no t-statistic low enough to reject the null hypothesis in every decile portfolio formed on size and value.
Baek & Bilson (2014) replicate the research of Barber & Lyon (1997) with an extended database and find similar results. They extend existing research by adding tests for similar size and value betas. They react on the influence of the financial sector on the corresponding coefficients found by Foerster & Sap (2005). Baek & Bilson find that both sectors have size and value premia, although the financial sectors’ premia are less explicable. The theory of Modigliani & Miller (1958) states that an influence on the corresponding betas of the company, and therefore its risk, does not infringe the CAPM restrictions. Therefore, there is no reason not to incorporate the financial sector even if this sector affects the results of asset pricing models.
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3. Methodology
3.1.1. Market-, size- and value factors
The calculation procedures of the factors closely follow Fama & French (1993) with the inclusion of momentum (Carhart, 1997).8 Portfolios are created via double sorting on firm specific characteristics; size & book-to-market and size & momentum. Each firms value-weighted return is assigned to a portfolio based on these firm specific characteristics. These returns are then equally weighted per portfolio. Then the calculated difference from the extreme portfolios is taken to end up with the corresponding factor (see Appendix A for the detailed factor calculations). This indicates a long-short investment strategy resulting in a factor that is largely free of other influences than the anomaly for which the factor is created. There are two deviations from this procedure. The first deviation is the addition of an exchange rate factor. The second is the difference in the dependent variable, altering from the excess return of double-sorted portfolios on size and book-to-market to the excess returns on foreign exchange sensitive portfolios. These deviations are further explained in the Section 3.1 and result in Eq. (1) when included.
(𝑅𝑒𝑖− 𝑟𝑓)
𝑡 = 𝛼𝑖+ 𝛽1[𝑅𝑚𝑡− 𝑅𝑓𝑡]𝑡+ 𝛽2 𝑆𝑀𝐵𝑡+ 𝛽3𝐻𝑀𝐿𝑡+ 𝛽4𝑈𝑀𝐷𝑡+ 𝛽5𝐼𝑀𝑋 + 𝑒𝑖 (1) where (𝑅𝑒𝑖− 𝑟𝑓) is the excess return on the ten exchange rate sensitive portfolios and 𝑟𝑓 is the risk-free rate. The 𝛼𝑖 is the intercept of portfolio i and the factors included are the market factor [𝑅𝑚𝑡− 𝑅𝑓𝑡], the size factor SMB, the value factor HML, the momentum factor UMD and the exchange rate factor IMX measured at time t. 𝛽𝑛 is the coefficient to the corresponding factor of portfolio i and 𝑒𝑖 is the error term per portfolio.
3.1.2. The exchange rate factor
The methodology of Kolari, Moorman & Sorescu (2008) is used in calculating the exchange rate factor. The principles of Adler & Dumas (1983) and Jorion (1990) are used to measure exchange rate sensitivity. These principles imply that the stock returns of each firm are regressed against the percentage change of the domestic currency against a basket of foreign currencies. To specify; the domestic currencies log return is divided by the log return per unit of SAC, which results in the foreign exchange return. The calculation of the SAC is defined in Section 3.2.
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This method uses rolling betas to calculate the exposure of firm i to foreign exchange rate risk by estimating the coefficient that corresponds to the return of the euro compared to a basket of currencies using Eq. (2). The estimation of Eq. (2) is conducted using monthly intervals and two year rolling betas:
(𝑅𝑖𝑡 − 𝑟𝑓) = 𝛼𝑖𝑡+ 𝛽1[𝑅𝑚𝑡− 𝑅𝑓𝑡]𝑡+ 𝛽2 𝑆𝑀𝐵𝑡+ 𝛽3𝐻𝑀𝐿𝑡+ 𝛽4𝑈𝑀𝐷 + 𝛽5𝐼𝑀𝑋𝑡+ 𝑒𝑖𝑡 (2) where (𝑅𝑖𝑡− 𝑟𝑓) is the log return of firm i at time t over the risk-free rate, 𝛼𝑖𝑡 and 𝑒𝑖𝑡 are the constant and error term of firm i at time t. [𝑅𝑚𝑡− 𝑅𝑓𝑡] is the market factor, SMB the size factor, HML the value factor, UMD the momentum factor and IMX the exchange rate factor. The coefficient that measures the risk exposure of the corresponding factor is denoted by 𝛽𝑛. The exchange rate exposure is measured by 𝛽5. After obtaining the annual measures of firm-specific exchange rate exposure (𝛽5), firms are ranked into 10 portfolios. Portfolio 1 contains the firms with the most extreme negative foreign exchange rate exposure and 10 are the firms that have the most positive foreign exchange rate exposure, meaning that an increase in the euro has the most negative consequences for firms in portfolio 1 and increases the profits made by firms in portfolio 10. After being assigned to a portfolio, the value-weighted log return is calculated per firm, which is then converted into monthly average returns per portfolio. The factor remains based on returns since the returns of firms are used after being assigned to a portfolio based on exchange rate exposure.
I will verify whether the relation between the portfolios is monotonic, validating the results of Apergis, Artikis & Sorros (2011), and Kolari, Moorman & Sorescu (2008). If this restriction is not met, a monotonic relation is created by subtracting the two most extreme portfolios (1 & 10) from the more foreign exchange risk neutral portfolios (2-…-9) resulting in the IMX factor.
3.1.3. The dependent variable
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3.2. Creating a minimum variance portfolio
Results are always based on the value of two currencies if exchange rates are analysed. This creates a bias because the ratio is dependent on changes in the numerator as well as the denominator. The choice of the denominator, the base currency, is thus of importance. It is best if the base currency’s variance is as low as possible so changes only come from the numerator. The choice of base currency affects the results and dynamics of the time series of currency values and the correlation of currencies quite differs among different base currencies (Hovanov, Kolari, & Sokolov, 2004). For example; the Japanese yen is positively correlated with the British pound when taking the U.S-dollar as a base currency, while there is a negative correlation between these same currencies when taking the euro as the base currency. So comparisons are based on the base currency, which makes the exchange rate a difficult tool in time-series.
Hovanov, Kolari & Sokolov (2004) overcome this problem by introducing a new value indicator of exchange. They create a minimum variance portfolio which makes it easier to analyze exchange rates. Therefore, this paper uses their methodology. The value indicator of exchange is based on the fact that value in exchange is measured by a scale of ratios with a precision of a positive factor β as defined by Dieudonne (1960). This beta takes the value of the inverse of the geometric mean of the currency values.
𝐺𝑀𝑒𝑎𝑛(𝑉𝑎𝑙1𝑗, … 𝑉𝑎𝑙𝑛𝑗) = √∏𝑛 𝑛𝑟=1𝑉𝑎𝑙𝑟𝑗 (3.1)
The geometrical mean is calculated based on the currencies included in the currency basket. Using this geometrical mean as a new indicator of value, the base currency problem is solved. Calculating the exchange rate using the geometrical mean as denominator creates a new normalized value. This normalized value is calculated using the positive factor beta:
𝑁𝑉𝑎𝑙𝑖𝑗 = 𝛽𝑉𝑎𝑙𝑖𝑗 = 𝑉𝑎𝑙𝑖𝑗
𝐺𝑀𝑒𝑎𝑛(𝑉𝑎𝑙1𝑗,…𝑉𝑎𝑙𝑛𝑗)=
𝑉𝑎𝑙𝑖𝑗 √∏𝑛𝑟=1𝑉𝑎𝑙𝑟𝑗
𝑛 (3.2)
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𝑅𝑁𝑉𝑎𝑙𝑖(𝑡/𝑡0) = 𝑁𝑉𝑎𝑙𝑖(𝑡)
𝑁𝑉𝑎𝑙𝑖𝑡0 (3.3)
where 𝑡0 is 1999, the year the value of the euro was set before it was introduced in 2002. 𝑅𝑁𝑉𝑎𝑙𝑖(𝑡/𝑡0) has the advantage over the original exchange rate that it does not depend on the base currency. All currencies are divided by 𝑡0, which means they all start at the same amount at the same time: 𝑡0. Before, statements about the euro were depending on the base currency, so a decrease in the exchange rate of the euro could be due to an increase in the value of the euro or a decrease in the value of the base currency. This is problem is overcome when using 𝑅𝑁𝑉𝑎𝑙𝑖.
Before 𝑅𝑁𝑉𝑎𝑙𝑖(𝑡/𝑡0), the optimal currency weights were dependent on the base currency, therefore no possibility to obtain these weights with a precision of a positive homogeneous transformation existed. By using the 𝑅𝑁𝑉𝑎𝑙𝑖(𝑡/𝑡0) it is possible to create such an index. Using the value in exchange in the form of a weighted arithmetical mean, or the sum of the weights times the reduced normalized values:
𝐼𝑛𝑑(𝑤; 𝑡) = ∑𝑛 𝑤𝑖 𝑅𝑁𝑉𝑎𝑙𝑖(𝑡/𝑡0)
𝑖=1 (3.4)
where i=1,…,n, are the currencies used in the basket, w is a positive weight vector to the corresponding 𝑅𝑁𝑉𝑎𝑙𝑖 and t=1,…T is the time. The volatility of such an index can be calculated using:
𝑆2(𝑤) = 𝑣𝑎𝑟(𝑤) = 1
𝑇∑𝑇𝑡=1[𝐼𝑛𝑑(𝑤; 𝑡) − 𝑀𝐼𝑛𝑑(𝑤)]2 (3.5) The arithmetical mean is calculated using the following formula:
𝑀𝐼𝑛𝑑(𝑤) =1
𝑇∑ 𝐼𝑛𝑑(𝑤; 𝑡) 𝑇
𝑡=1 (3.6)
Minimizing the variance of the currency basket results in optimal weights. These weights have two restrictions (i) it is a non-negative weight vector w=(𝑤1, … , 𝑤𝑛) and (ii) the sum of all weights is one: 𝑤1+ ⋯ + 𝑤𝑛 = 1. The variance could also be written down as a function of the covariance (3.7):
𝑆2(𝑤) = ∑ 𝑤
𝑖𝑤𝑘𝑐𝑜𝑣(𝑖, 𝑘) = ∑𝑛𝑖=1𝑤𝑖2𝑠𝑖2+ 2 ∑𝑛𝑖,𝑘=1𝑤𝑖𝑤𝑘𝑐𝑜𝑣(𝑖, 𝑘) 𝑛
𝑖,𝑘=1 (3.7)
Minimizing this function while abiding the restrictions of the weights creates a minimum variance portfolio of 𝑅𝑁𝑉𝑎𝑙𝑖.9 This method calculates the optimal weights for a basket of
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currencies so that it minimizes the variance and is denoted as a stable aggregate currency (SAC). The optimal weights of the 𝑅𝑁𝑉𝑎𝑙𝑖(𝑡/𝑡0) are calculated for the EUR (𝑤1), JPY (𝑤2), the GBP (𝑤3) and the USD (𝑤4). These weights can then be recalculated to their official currency values using:
𝑞𝑖 =𝑐𝑤𝑖
𝑖𝑗(𝑡0)𝜇 (3.8)
where 𝑞𝑖 is the weight of currency i in the SAC, 𝑤𝑖 is the weight of currency i as a 𝑅𝑁𝑉𝑎𝑙𝑖, 𝑐𝑖𝑗(𝑡0) is the exchange rate of currency i against base currency j at time 𝑡0 and 𝜇 is an arbitrary positive constant for every unit of SAC.
The correlation between the SAC and the effective exchange rate of the euro is measured and shows how the stable aggregate currency moves along with the effective exchange rate compiled by the ECB.
3.3. Times-series approach.
After composing factors, asset-pricing models are often tested using a time series approach (Jagannathan, Schaumburg, & Zhou, 2010). In classical asset pricing models, the expected return and beta have a linear relation (Cochrane, 2005). Such a relation requires normality and therefore the errors need to be tested for autocorrelation and heteroskedasticity. The existence of autocorrelation is tested using Durbin-Watson test (Durbin & Watson, 1951). To keep all regressions unambiguous, a notion of autocorrelation in one regression will result in correcting for autocorrelation and heteroskedasticity by using the Newey-West standard errors in all regressions. All factors of this paper are based on returns, even the exchange rate factor. This justifies the use of OLS regressions to analyze the asset pricing models.10 The times-series multifactor regression model in expected-beta form is:
𝑟𝑖𝑡 = 𝛼𝑖+ ∑𝐾 𝛽𝑖𝑗𝑓𝑗,𝑡
𝑗=1 + 𝜀𝑖𝑡 (4)
where 𝑟𝑖𝑡 is excess return of the ith asset at time t, 𝛼𝑖 is the regression intercept, 𝛽𝑖𝑗 measures the risk exposure on N assets, 𝑓𝑗,𝑡 denotes the excess return on the jth factor portfolio at time t, and 𝜀𝑖𝑡 are the error terms. When applied to our case Eq. (4) results in Eq. (1).
The coefficients are analyzed per portfolio and per incorporated factor for significance, sign and magnitude. Such an analysis gives insight in how much exposure and variance a factor
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has in explaining average returns. The first analysis is a regression of portfolios based on size and book-to-market against exchange rate sensitive portfolios. This regression, seen in Eq. (5), is compared to the results of Apergis, Artikis & Sorros (2011).
𝑅𝑒𝑖= 𝛼
𝑖+ 𝑓𝑖𝐹𝑋𝑡+ 𝑒𝑖 (5)
where 𝑅𝑒𝑖 are the returns of the six double-sorted portfolios on size and book-to-market. 𝛼 𝑖 is the constant, 𝐹𝑋𝑡 are ten foreign exchange sensitive portfolios, 𝑓𝑖 is the corresponding coefficient and 𝑒𝑖 the error term. Additionally, four factor models are analyzed for the same reason as Eq. (5). The FF with- and without the exchange rate factor:
(𝑅𝑒𝑖− 𝑟𝑓)
𝑡 = 𝛼𝑖+ 𝛽𝑖[𝑅𝑚𝑡− 𝑅𝑓𝑡] + 𝑠𝑖 𝑆𝑀𝐵𝑡+ ℎ𝑖𝐻𝑀𝐿𝑡+ 𝑒𝑖 (6)
(𝑅𝑒𝑖− 𝑟𝑓)
𝑡 = 𝛼𝑖+ 𝛽𝑖[𝑅𝑚𝑡− 𝑅𝑓𝑡] + 𝑠𝑖 𝑆𝑀𝐵𝑡+ ℎ𝑖𝐻𝑀𝐿𝑡+ 𝑓𝑖𝐼𝑀𝑋 + 𝑒𝑖 (7) and the C4 with- and without the exchange rate factor:
(𝑅𝑒𝑖− 𝑟𝑓)
𝑡 = 𝛼𝑖+ 𝛽𝑖[𝑅𝑚𝑡− 𝑅𝑓𝑡] + 𝑠𝑖 𝑆𝑀𝐵𝑡+ ℎ𝑖𝐻𝑀𝐿𝑡+ 𝑚𝑖𝑈𝑀𝐷𝑡+ 𝑒𝑖 (8)
(𝑅𝑒𝑖− 𝑟𝑓)
𝑡 = 𝛼𝑖+ 𝛽𝑖[𝑅𝑚𝑡− 𝑅𝑓𝑡] + 𝑠𝑖 𝑆𝑀𝐵𝑡+ ℎ𝑖𝐻𝑀𝐿𝑡+ 𝑚𝑖𝑈𝑀𝐷𝑡+ 𝑓𝑖𝐼𝑀𝑋 + 𝑒𝑖 (9) where (𝑅𝑒𝑖− 𝑟𝑓) is the excess return on the ten exchange rate sensitive portfolios, 𝑟𝑓 is the risk-free rate. The 𝛼𝑖 is the intercept of portfolio i and the factors included are the market factor [𝑅𝑚𝑡− 𝑅𝑓𝑡], the size factor SMB, the value factor HML, the momentum factor UMD and the exchange rate factor IMX measured at time t. 𝛽𝑛 is the coefficient to the corresponding factor of portfolio i and 𝑒𝑖 is the error term per portfolio.
The amount of variation a model explains compared to the observed outcomes is measured by the 𝑅2. However, adding independent variables can increase the 𝑅2 by chance alone (Thiel, 1961). Thiel (1961) introduces the 𝑅2-bar to adjust for this phenomenon. 𝑅2-bar only increases when the new independent variable improves the 𝑅2 more than by chance alone. The inclusion of extra independent variables in the asset pricing models makes it necessary to adjust for this phenomenon. Eq. (10) shows the adjustment:
𝑅2− bar = 1 − (1 − 𝑅2) 𝑛−1
𝑛−𝑝−1 (10)
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The amount of variation across portfolios over time is analyzed. The analysis is conducted using the coefficients, significance of the coefficients and the R²-bar. This gives valuable insights on how specific factors react on exchange rates and the amount of explained variability.
3.4. GRS-test
The estimated risk premiums, 𝜆̂𝑗 , associated to the corresponding exposure to risk can be calculated by taking the mean of the expected factor coefficient 𝑓𝑗:
𝜆̂𝑗 = 𝐸𝑇( 𝑓𝑗) (11.1)
The risk premiums are then used to construct the linear beta pricing restriction on expected returns:
𝐸[𝑅𝑖𝑡] = 𝜆0+ ∑𝐾𝑗=1𝛽𝑖𝑗𝜆̂𝑗,𝑡 (11.2) where 𝐸[𝑅𝑖𝑡] is N vectors of expected excess return of ith asset and 𝜆 is a vector of K risk premiums. In asset pricing models the expected excess returns are based on replicating portfolios. The β’s are linear for the expected returns as stated in (11.3):
𝑟𝑖𝑡 = 𝐸(𝑅𝑖𝑡) (11.3)
If all existing risk factors of the stock market that explain anomalies are included in Eq. (11.2), then Eq. (11.3) holds with all the regression intercepts 𝛼 equal to zero. The pricing errors equal the regression intercepts when Eq. (11.2) and (11.3) both hold as well. If not all risk factors are included in Eq. (11.2), then 𝛼𝑖will restore equality by taking the amount that ensures 𝛽𝑖𝑗𝐸(𝑓𝑗,𝑡) is equal to 𝐸(𝑅𝑖𝑡) in order for Eq. (11.3) to hold.
Using the expected excess return in the calculations in order to derive the replicating portfolio has the advantage that both sides of the Eq. now consist of returns. The dependent variable (11.2) should be considered as a long position and one should go short in Eq. (4) in order to arrive at the value of 𝛼𝑖:
𝛼𝑖= 𝜆0+ ∑𝐾 𝛽𝑖𝑗
𝑗=1 (𝜆̂ − 𝐸(𝑓𝑗,𝑡 𝑗,𝑡)) (11.4)
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market line and the ex-post efficient frontier if all the alphas are equal to zero, indicating the model perfectly explains the variation. The GRS-statistic can only be tested if all dependent and independent variables are returns, which is the case in this paper. The GRS statistic is calculated as seen in Eq. (14):
𝑇−𝑁−1 𝑁 [1 + ( 𝐸𝑇( 𝑓𝑗) 𝜎 ̂( 𝑓𝑗)) 2 ] −1 𝛼̂′Σ̂−1𝛼 ̂ ~ 𝐹 𝑁 ,𝑇−𝑁−1 (12) where T is the number of observations, N equals the number of independent variables/regressions, K is the number of factors, and (𝐸𝜎̂( 𝑓𝑇( 𝑓𝑗)
𝑗)) is the Sharpe ratio of the ex
post tangency portfolio. The error terms should be uncorrelated and homoscedastic to use the GRS-test. Therefore, this paper applies a correction for these restrictions using Newey-West standard errors when performing the regressions (Newey & West, 1987). The lower the GRS-statistic, the more satisfactory a factor model explains stock market returns.
3.4 Financial versus nonfinancial sector.
After the whole market is analyzed the database is divided into two groups; a financial- and nonfinancial group. The financial group is called ‘F’, the nonfinancial group is ‘NF’ and the whole market is ‘ALL’. The independent variables remain the same and the effect of the factors are tested against different dependent variables. The methodology used for calculating the dependent of the whole market is applied to the financial- and nonfinancial sector separately. The firms are divided into ten portfolios based on foreign exchange risk. Subsequently, the firm’s value-weighted return is calculated and averaged per portfolio. The dependent variables are regressed against the factors using Eq. (6) to (9) and the coefficients per portfolio across sectors are analyzed following the methodology of Barber & Lyon (1997).
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4. Data & descriptive
The sample used in the empirical tests consists of companies that are traded on the Frankfurt Stock Exchange (FSE) between 1999 and 2014 and were still active in 2014. Firm data on stock returns (measured by return index RI), market capitalization, number of shares, share price and book-values are obtained from Thompson Reuters DataStream. These items are used to construct a new database. This is necessary because of the comparison of the financial- and nonfinancial sector, which is not possible using an existing database. The number of firms included per item per year are shown in Appendix B. The strong decrease in book-to-market ratio in 2014 may be due to non-availability of the book-value at the time of collection.
The rate of return on the market portfolio is obtained by using the market risk premium calculated by Brückner et al. (2014). This choice is based on the fact that the CDAX composition is not documented which may result in a selection bias. Also, the DAFOX is only available until 2004 (Brückner et al., 2014). The risk-free rate is obtained from the Deutsche Bundesbank (Bundesbank, 2015). Exchange rate data are obtained from IMF and based on the SDR-value per dollar as base currency.
Following the exchange rate principle of transitivity that: c(i,k)*c(k,j)=c(i,j), where c is the exchange rate between i, j or k (Hitrov & Hovanov, 1992) multiplying the USD/SDR by the USD/EUR exchange rate converts the SDR exchange rate values into EUR/SDR..
Firms must abide the following restrictions to be included as a firm in the portfolio construction of factors:
i. A positive BE (Book value of equity) as well as ME (Market value of equity) during the end of year 𝑡−1 to calculate the BE/ME ratio.
ii. A firm should have a market capitalization of at least five million euros at the end of June in the preceding year. If this requirement could not be met, a share price of more than 1 euro has to be observed during that same period. This restriction is due to frequently occurred price manipulation of small ‘penny stocks’ as warned by the German BaFin.11
iii. Stock returns of the preceding 12 months for calculating momentum.
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iv. Return outliers are removed if returns are above 100% for one month. Returns above 100% in consecutive months are used as normal returns as suggested by Brückner et al. (2014).
v. 3 outliers are removed due to their extreme outcomes compared to the rest of the outcomes and the effect they had on the mean outcomes.
The data in this database are subject to survivorship-bias, which can influence the results, because only firms that are currently active on the Frankfurt stock exchange are in the database. The sample does not contain firms that went bankrupt, which can have a negative influence on the average returns, firm size and would increase the average book-to-market ratio. The results can be biased upwards and need to be interpreted with caution. The possible effects are discussed in more depth in the Section 6.
The use of Thompson Reuters DataStream as primary data source gives rise to some systematic errors as indicated by Brückner (2013):
i. Selection bias may occur when DataStream classifies non-voting stocks as preferred stocks.
ii. Random errors are present in the return indexes provided caused by incorrect adjustments for dividends and corporate actions, this could have implications for this paper due to the fact that the time series are only 15 years.
iii. There are considerable errors in the number of shares that affect the sorting on size, the value-weighted returns and the estimates of firm size in general.
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5. Results
5.1. Summary statistics of comparable researches
The mean factors of this paper are shown in Table 1 next to the mean factors of four publicly available databases of comparable studies on factor models in Germany. The market factor from Brückner et al. (2014) is used, with an average of 0.44%. This paper has a negative size factor of -0.98%, which is more negative than the comparison studies. Therefore, one can conclude that big firms earn more than small firms on average. This finding is in line with the four comparison studies. The methodology of Brückner et al. (2014) is most similar to this paper; hence the finding that their size factor shows most resemblance seems valid. By constrast, HML and UMD only show slight similarities to the research of Bückner et al. (2014). HML is high with 1.04% compared to an average of 0.75% in the comparison studies. The momentum closely resembles the other studies, except Bückner et al. (2014), showing a monthly average of 1.30%. The deviations can be due to not taking survivorship bias into account and will be elaborated upon in the discussion section.
Table 1. Descriptive statistics of the mean factors of 4 comparative studies concerning the 4 factor model in Germany.
The first column shows the research from which the mean factor outcomes are withdrawn. Column two to five show the factors that are compared. MKT is the average realized return of the market portfolio in excess of the risk-free rate, SMB is the average realized return on a size neutral mimicking portfolio, HML is the average realized return on a value neutral mimicking portfolio and UMD is the average realized return on a momentum neutral mimicking portfolio in the period 1999-2014.
MKT (%) SMB (%) HML (%) UMD (%)
Factors of this study 0.44 -0.97 1.04 1.30
Artmann et al. (2012b) 0.53 -0.55 0.85 1.44
Brückner et al. (2014) 0.44 -0.80 0.76 1.05
Hanauer et al. (2011) 0.56 -0.71 0.74 1.19
Schmidt et al. (2011) 0.39 -0.28 0.55 1.31
5.2. Minimum variance portfolio
Using the methodology proposed by Hovanov, Kolari & Sokolov (2004), the optimal weights for the minimum variance portfolio are:
𝑤1=0.2216 𝑤2= 0.2320 𝑤3=0.2381 𝑤4=0.3083
Following the methodology of Hovanov, Kolari & Sokolov (2004), setting 𝜇 at 30, the optimal weights can be converted into optimal amounts for the SAC using Eq. (5.8). These amounts correspond to:
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This stable aggregate currency portfolio has a far lower standard deviation than the euro (0.00123 compared to 0.0529), resulting in a far more stable return with less outliers as shown in Fig. 1. The correlation between the SAC and the effective exchange rate is 0.395. So for every increase in one unit of SAC the effective exchange rate of the euro rises with 0.395. The SAC and effective exchange rate have the same sign, as one would expect, however, the correlation is not very large. The effective exchange rate consists of 21 countries including: Japan (10.5%), the United Kingdom (21.0%) and the United States (24%). These four countries contribute more than 50% of the basket although the correlation is only 40%. This shows that the other 17 countries that participate in the effective exchange rate currency basket play a more important role in their influence on the euro. This finding corresponds to the finding of Schmitz et al. (2012). They find that European economies without the euro as currency account for the largest variation in trade weights and thus for most fluctuations
5.3. Time-series’ asset pricing tests
Eq. (5) to (9) are tested for autocorrelation using the Durbin-Watson test and the results are shown in Appendix C. No sign of autocorrelation is found in Eq. (5) although Eq. (6) to (9) have some inconclusive tests. Newey-West standard errors are used to assure autocorrelation has no effect on the regressions. Another benefit of using Newey-West standard errors is that it corrects for heteroskedasticity.
The return of the euro per unit of SAC is used to estimate foreign exchange sensitivity per portfolio. The exposure is shown in Table 2. Portfolio 1 to 5 experience a negative exposure to foreign exchange rates, portfolio 6 to 10 react positive on an increasing euro. Firms at the extreme exposure portfolios perform less when looking at present and historic
-2.5% -1.5% -0.5% 0.5% 1.5% 2.5%
Figure 1: Return of the Standard Aggregate Currency basket compared to the euro
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returns, validating the results of Kolari, Moorman & Hovanov (2008) and Apergis, Artikis & Sorros (2011). Firms that react positive on an increasing euro are very likely to be importers, firms that can buy more goods with one euro than before. Vis-à-vis are the portfolios that lose from an increasing euro therefore probably importers. Moreover, smaller firms have on average more exposure to exchange rates looking at the average market capitalization between the portfolios. The book-to-market ratio is almost identical for every portfolio and all below zero, ranging from 0.8233 to 0.9109. This indicates that book-to-market ratios have a similar role across portfolios in explaining exposure to foreign exchange risk. This result varies greatly with that of Apergis, Artikis & Sorros (2011), who find all book-to-market ratios bigger than one and the biggest book-to-market ratios at the most extreme portfolios.
Table 2. Summary statistics per foreign exchange sensitive portfolio.
The average FX sensitivity is the foreign exchange sensitivity per portfolio. Returns are value-weighted and multiplied by 12 to represent annual results. B/M is the book-to-market ratio and historical returns are the average returns from 𝑡−2 to 𝑡−12.
The first regression examines the size- and book-to-market portfolios and their relation to foreign exchange sorted portfolios using Eq. (5). The outcomes are shown in Appendix D. Every coefficient is highly significant which could imply that foreign exchange is a priced factor. All coefficients show a positive relationship between size-, book-to-market- and foreign exchange risk. The results correspond to Asperigs, Artikis & Sorros (2011) in that small firms have a larger coefficient than big firms, thus small firms exhibit a larger sensitivity to foreign exchange fluctuations as was noticed in Table 2. The exposure may be due to the fact that small firms have less resources to hedge against exchange rates. As the results show, low book-to-market firms are most sensitive to currency fluctuations. This contradicts with Fama & French (1998), who find that high book-to-market firms outperform low book-to-market firms in an international asset pricing model in twelve major markets.
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Even although they do not measure foreign exchange risk, Apergis Artikis & Sorros (2011) find the same result measuring foreign exchange rate risk. This contradictory result is tested against the asset pricing models in Eq. (6) to (9) but could indicate that by including firms from the middle segment of the Frankfurt Stock Exchange, the sensitivity to foreign exchange rate movements changes from high book-to-market firms to low book-to-market firms.
Table 3. Multivariate regression of the Fama & French three factor model;
(𝑅𝑒𝑖− 𝑟𝑓)
𝑡= 𝛼𝑖+ 𝛽𝑖[𝑅𝑚𝑡− 𝑅𝑓𝑡] + 𝑠𝑖 𝑆𝑀𝐵𝑡+ ℎ𝑖𝐻𝑀𝐿𝑡+ 𝑒𝑖
This Table shows the regression as stated above and uses Newey-West standard errors. Where (𝑅𝑒𝑖− 𝑟𝑓)𝑡 is the excess
return on the ten portfolios ranked on exchange rate sensitivity at time t, 𝑟𝑓 is the risk-free rate, 𝛼𝑖 is the intercept, [𝑅𝑚𝑡−
𝑅𝑓𝑡] is the excess return of the market portfolio, SMB is the realized return on a size neutral mimicking portfolio and HML is
the realized return on a value neutral mimicking portfolio. 𝛽, s and h are the coefficients of the corresponding risk factor and 𝑒𝑖𝑡 is the error term. All returns are value-weighted and the portfolios are regressed monthly during the time period of
01/1999-12/2014. The R²-bar shows the amount of variation of the dependent variable that is explained by the independent variables over time, corrected for the amount of factors. t-statistics are shown in parentheses. ***, **, * denote the significance level of the p-value of the corresponding t-statistic at 1%, 5% or 10%.
Alpha Beta s h R²-bar
1 -0.01335 1.46832 1.20855 -0.19462 0.65 (-3.52)*** (17.24)*** (11.02)*** (-1.79)* 2 -0.00861 1.37548 0.98790 -0.00024 0.77 (-3.35)*** (23.88)*** (13.31)*** 0.00 3 -0.00400 1.30095 0.91267 0.09120 0.78 (-1.70)* (24.59)*** (13.39)*** (1.35) 4 -0.00302 1.08968 0.71969 0.11967 0.81 (-1.66)* (26.78)*** (13.73)*** (2.30**) 5 -0.00215 1.03765 0.60652 0.09597 0.78 (-1.12) (24.11)*** (10.94)** (1.75*) 6 -0.00121 1.09735 0.62055 -0.30016 0.70 (-0.45) (18.22)*** (8.00)*** (-3.90)*** 7 -0.00191 0.99671 0.65191 0.20570 0.79 (-1.11) (25.76) *** (13.08)*** (4.16)*** 8 -0.00545 1.09440 0.68350 0.22877 0.83 (-3.19)*** (28.58)*** (13.86)*** (4.68)*** 9 -0.00998 1.04333 0.74392 0.31500 0.73 ( -4.68)*** (21.81)*** (12.07)*** (5.16)*** 10 -0.01329 1.26510 0.98653 0.12615 0.73 (-5.18)*** (21.98)*** (13.31)*** (1.72)*
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significant and follows a u-shape, meaning that an increase in the value of the euro has the most impact in the extreme exposure portfolios. All coefficients are one or higher, thus for every percent the euro increases in value, the market portfolio benefits with a return of at least a euro or more. The same u-shape is noticed at the size factor, although more extreme. Analyzing the value factor coefficients, after excluding portfolio 6, shows a positive linear relationship with the exchange rate as seen in Appendix G4. This finding contradicts with the outcomes of Eq. (5) and are in line with the findings of Apergis, Artikis & Sorros (2011) and Fama & French (1998). This means that the other anomalies reverse the effect that was noticed in Eq. (5). The outcomes show that an increase in the euro decreases the returns of the first two portfolios. The other portfolios show a positive relation between an increase in the euro and the amount of variation explained by the value factor. Even the portfolios that have a negative exposure (e.g., portfolio 1 to 5) benefit on average from an increase in the euro. The 𝑅2-bar is relatively large ranging from 0.65 to 0.83 showing that the independent variables explain the dependent variable relatively well over time.
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encouraged to benefit from an increasing euro and inclined to take more risk due to the large potential benefit. The exporters, the portfolios that lose from an increasing euro (e.g., portfolio 1-5) are dependent on stable currency values and therefore they are more encouraged to hedge themselves against these risks as is seen by the coefficient closer to zero of the value factor. The foreign exchange factor itself is very significant and has the largest coefficients, which are negative, at the extreme exposure portfolios. The significance shows that the factor explains variation over time per portfolio, the signs and magnitudes across portfolios do not show a clear pattern. The coefficients reflect the relationship between the a risk factor and return. The inclusion of the foreign exchange factor decreases the coefficients of other risk factors, meaning that by including a factor for foreign exchange, the other factors are less influenced by a fluctuating euro.
Table 4. Multivariate regression of the Fama & French three factor model including exchange rate factor
(𝑅𝑒𝑖− 𝑟𝑓)
𝑡= 𝛼𝑖+ 𝛽𝑖[𝑅𝑚𝑡− 𝑅𝑓𝑡] + 𝑠𝑖 𝑆𝑀𝐵𝑡+ ℎ𝑖𝐻𝑀𝐿𝑡+ 𝑓𝑖𝐼𝑀𝑋 + 𝑒𝑖
This Table shows the regression as stated above and uses Newey-West standard errors. Where (𝑅𝑒𝑖− 𝑟𝑓)
𝑡 is the excess return on the ten portfolios ranked on exchange rate sensitivity at time t, 𝑟𝑓 is the risk-free rate, 𝛼𝑖 is the intercept, [𝑅𝑚𝑡− 𝑅𝑓𝑡] is the
excess return of the market portfolio, SMB is the realized return on a size neutral mimicking portfolio, HML is the realized return on a value neutral mimicking portfolio and IMX is the realized return on a foreign exchange neutral mimicking portfolio. 𝛽, s, h and f are the coefficients of the corresponding risk factor and 𝑒𝑖𝑡 is the error term. All returns are value-weighted and the
portfolios are regressed monthly during the time period of 01/1999-12/2014. The R²-bar shows the amount of variation of the dependent variable that is explained by the independent variables over time, corrected for the amount of factors. t-statistics are shown in parentheses. ***, **, * denote the significance level of the p-value of the corresponding t-statistic at 1%, 5% or 10%.
Alpha Beta s h f R²-bar
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The Carhart four factor model is regressed using Eq. (8). The addition of the momentum factor does not cause large differences in most factor, however, the changes caused by adding momentum are shown in Table 5.1. The complete regression results can be found in Appendix E. The constants move closer to zero compared to the FF indicating that more variation is explained by adding momentum. The coefficients of the market and size factor show the same pattern although momentum absorbs some of the magnitude concerning the coefficients. Foreign exchange exposure influences momentum in a negative way as seen by the negative signs in the coefficients. Especially the portfolios that are negatively influenced by an increasing euro. This indicates that an increase in the euro increases the amount of firms that exhibit negative momentum. Adding momentum to the FF increases the explained variation explained, nonetheless, the increase is fairly small as confirmed by the 𝑅2-bar.
Table 5. Multivariate regressions using different asset pricing models.
This table shows the Carhart regression with- and without exchange rate factor (IMX) and uses Newey-West standard errors. Where (𝑅𝑒𝑖− 𝑟𝑓)
𝑡 is the excess return on the ten portfolios ranked on exchange rate sensitivity at time t, 𝑟𝑓 is the risk-free rate, 𝛼𝑖 is the
intercept, [𝑅𝑚𝑡− 𝑅𝑓𝑡] is the excess return of the market portfolio, SMB is the realized return on a size neutral mimicking portfolio,
HML is the realized return on a value neutral mimicking portfolio, UMD is the realized return on a momentum neutral mimicking portfolio and IMX is the realized return on a foreign exchange neutral mimicking portfolio. 𝛽, s, h, m and f are the coefficients of the corresponding risk factor and 𝑒𝑖𝑡 is the error term. All returns are value-weighted and the portfolios are regressed monthly during the
time period of 01/1999-12/2014. The R²-nar shows the amount of variation of the dependent variable that is explained by the independent variables over time, corrected for the amount of factors. t-statistics are shown in parentheses. ***, **, * denote the significance level of the p-value of the corresponding t-statistic at 1%, 5% or 10%.
5.1 Carhart four factor model 5.2 Carhart four factor model + IMX (𝑅𝑒𝑖− 𝑟𝑓)
𝑡= 𝛼𝑖+ 𝛽𝑖[𝑅𝑚𝑡− 𝑅𝑓𝑡] + 𝑠𝑖 𝑆𝑀𝐵𝑡+ ℎ𝑖𝐻𝑀𝐿𝑡+ 𝑚𝑖𝑈𝑀𝐷𝑡+ 𝑒𝑖 (𝑅𝑒𝑖− 𝑟𝑓)𝑡= 𝛼𝑖+ 𝛽𝑖[𝑅𝑚𝑡− 𝑅𝑓𝑡] + 𝑠𝑖 𝑆𝑀𝐵𝑡+ ℎ𝑖𝐻𝑀𝐿𝑡+ 𝑚𝑖𝑈𝑀𝐷𝑡+ 𝑓𝑖𝐼𝑀𝑋 + 𝑒𝑖
Alpha m R²-bar Alpha m R²-bar
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The results of Eq. (9) largely resemble the results of Eq. (7). The only major differences are the constant and the momentum factor, which are shown in Table 5.2. All regression outcomes can be found in Appendix F. The constant becomes less significant and follows a random path. However, they move closer to zero compared to Eq. (6) to (8). Momentum is positively affected by the inclusion of the foreign exchange factor. The result is that all coefficients become less negative, with the most extreme effect in the outer portfolios. The foreign exchange rate factor thus reduces the negative effect exchange rate has on momentum. The market-, size-, value- and the foreign exchange factor’s pattern of the coefficients remain unchanged. However, although the effect is fairly small, the absolute coefficients move closer to zero.
A parsimonious asset pricing model can only exist if the constants are zero or statistically insignificant (Merton, 1973). The constant moving closer to zero and turning less significant show that adding momentum and an exchange rate factor enhances the predictability of the model. The increased 𝑅2-bar shows the same effect in explaining more variation over time. The increase is especially evident in the outer portfolios, validating the results of Apergis, Artikis & Sorros (2011).
5.4. GRS-statistics
The GRS statistic tests if the factor model captures all risk by using a t-test. This test tests the hypothesis that all alphas are equal to zero. The test is executed for Eq. (6) to (9) and is shown in Table 6. The results show that all GRS statistics are significant at 1% level and thus reject the hypothesis for every model. However, both GRS-statistic decrease in value after including a foreign exchange rate factor, meaning that the predictive power of the model increases. The model increases in explanatory power when momentum is included as a factor.
Table 6. GRS-statistics per asset pricing model.
The GRS statistics tests if the alphas of all ten foreign exchange portfolios are jointly equal to zero in the period 1999-2014. The GRS F-statistic shows the amount of variation that is left unexplained by the regression used and the GRS p-value tests the hypothesis if all alphas are jointly equal to zero.
Model GRS F-statistic GRS p-value
three factor model 5.892 0.00
three factor model + IMX 4.194 0.00
four factor model 5.652 0.00
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5.5 Comparing asset pricing models between the financial- and nonfinancial sector.
The following section shows the results of the asset pricing models in Eq. (6) to (9) for the financial-, as well as the nonfinancial sector and the results are compared for similarity. First, the summary statistics are shown in Table 7.
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Table 7. Summary statistics per portfolio per sector.
The average FX sensitivity is the foreign exchange sensitivity per portfolio. Returns are value-weighted and multiplied by 12 for annual results. B/M stands for book-to-market and historical returns are the average returns from 𝑡−2 to 𝑡−12.. Where F stands for the financial sector, NF for the nonfinancial sector and ALL for all stocks included in the database in the period 1999-2014.
Average FX sensitivity (fi) in % Average annual raw return Average MCAP (in millions €) Average B/M Historic annual past return
F NF ALL F NF ALL F NF ALL F NF ALL F NF ALL
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Table 8. Multivariate regressions of the financial- and nonfinancial sector using the Fama & French three factor model;
(𝑅𝑒𝑖− 𝑟𝑓)
𝑡= 𝛼𝑖+ 𝛽𝑖[𝑅𝑚𝑡− 𝑅𝑓𝑡] + 𝑠𝑖 𝑆𝑀𝐵𝑡+ ℎ𝑖𝐻𝑀𝐿𝑡+ 𝑒𝑖
This Table shows the regression as stated above and uses Newey-West standard errors for the financial sector in The F columns and the nonfinancial sector in the NF columns. Where (𝑅𝑒𝑖−
𝑟𝑓)𝑡 is the excess return on the ten portfolios ranked on exchange rate sensitivity at time t, 𝑟𝑓 is the risk-free rate, 𝛼𝑖 is the intercept, [𝑅𝑚𝑡− 𝑅𝑓𝑡] is the excess return of the market portfolio,
SMB is the realized return on a size neutral mimicking portfolio and HML is the realized return on a value neutral mimicking portfolio. 𝛽, s and h are the coefficients of the corresponding risk factor and 𝑒𝑖𝑡 is the error term. All returns are value-weighted and the portfolios are regressed monthly during the time period of 01/1999-12/2014. The R²-bar shows the amount of variation of the dependent variable that is explained by the independent variables over time, corrected for the amount of factors. t-statistics are shown in parentheses. ***, **, * denote the significance level of the p-value of the corresponding t-statistic at 1%, 5% or 10%.
Alpha Beta s h R²-bar
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The addition of an exchange rate factor to the FF results in Eq. (7). The changes compared to the FF are discussed in this section and can be seen in Table 9.1. The entire regression including results are shown in Appendix H.
The addition of an exchange rate factor decreases the constant for every portfolio in both sectors. This indicates there is less variation unexplained. Another change in the constant is found concerning the magnitudes of the coefficients in the financial sector. The biggest coefficients were first found at the extreme exposure portfolios and are now moved to the more foreign exchange neutral portfolios. This shows that part of the effect that was first stored in the constant of the financial sector is now absorbed by the foreign exchange factor. The nonfinancial foreign exchange factor has larger coefficients in the outer portfolios, as was noticed when analyzing the whole market. All negative coefficients in this sector are significant as well, indicating that they explain average returns relatively well over time in these portfolios. The financial sector shows no clear pattern, although eight out of ten coefficients are closer to zero compared to the corresponding portfolio in the nonfinancial sector. This observation could indicate that the financial sector is less affected by the foreign exchange risk than the nonfinancial sector. However, the absence of significant coefficients makes this observation less reliable. The foreign exchange factor explains variation over time really well in the most negative exposure portfolios. These exposure coefficients are the most negative in both sectors and the only significant coefficient in the financial sector at 1%. Financial firms in the portfolio most exposed to exchange rates exhibit the worst returns of all portfolios in both sectors when the euro increases in value. For the remainder the financial sector has higher average returns in every portfolio compared to the nonfinancial sector except for portfolio 8. The non-significance makes these interpretations less reliable and implies that the returns are not explained in a constant manner across time. Hedging works in the way that it creates a constant return no matter the change in currency value. The nonsignificance indicates that the financial sectors returns are not explained very well by exchange rate movements, which could indicate hedging. The 𝑅2–bar remains unchanged.
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Table 9. Multivariate regressions of the financial- and nonfinancial sector using stated model;
(𝑅𝑒𝑖− 𝑟𝑓)
𝑡= 𝛼𝑖+ 𝛽𝑖[𝑅𝑚𝑡− 𝑅𝑓𝑡] + 𝑠𝑖 𝑆𝑀𝐵𝑡+ ℎ𝑖𝐻𝑀𝐿𝑡+ 𝑚𝑖𝑈𝑀𝐷𝑡+ 𝑓𝑖𝐼𝑀𝑋 + 𝑒𝑖
The Fama & French model exists of the MKT-, SMB- and HML factors and the Carhart model includes the UMD factor, both using Newey-West standard errors. The financial sector are in the F columns and the nonfinancial sector in the NF columns. Where (𝑅𝑒𝑖− 𝑟𝑓)
𝑡 is the excess return on the ten portfolios ranked on exchange rate sensitivity at time t, 𝑟𝑓 is the risk-free rate,
𝛼𝑖 is the intercept, [𝑅𝑚𝑡− 𝑅𝑓𝑡] is the excess return of the market portfolio, SMB is the realized return on a size neutral
mimicking portfolio, HML is the realized return on a value neutral mimicking portfolio, UMD is the realized return on a momentum neutral mimicking portfolio and IMX is the realized return on a foreign exchange neutral mimicking portfolio. 𝛽, s,
h, m and f are the coefficients of the corresponding risk factor and 𝑒𝑖𝑡 is the error term. All returns are value-weighted and the portfolios are regressed monthly during the time period of 01/1999-12/2014. The R² shows the amount of variation of the dependent variable that is explained by the independent variables over time, corrected for the amount of factors. t-statistics are shown in parentheses. ***, **, * denote the significance level of the p-value of the corresponding t-statistic at 1%, 5% or 10%.
9.1 Fama & French three factor model + exchange rate factor
(𝑅𝑒𝑖− 𝑟𝑓) 𝑡= 𝛼𝑖+ 𝛽𝑖[𝑅𝑚𝑡− 𝑅𝑓𝑡] + 𝑠𝑖 𝑆𝑀𝐵𝑡+ ℎ𝑖𝐻𝑀𝐿𝑡+ 𝑓𝑖𝐼𝑀𝑋 + 𝑒𝑖 Alpha f F NF F NF 1 -0.0088 -0.0103 -0.7166 -0.5821 (-1.62) (-2.6)*** (-2.96)*** (-4.39)*** 2 -0.0169 -0.0061 -0.1107 -0.197 (-3.63)*** (-2.31)** (-0.72) (-2.23)** 3 -0.007 -0.0026 -0.0729 -0.2067 (-1.53) (-0.93) (-0.49) (-2.19)*** 4 -0.0064 -0.002 0.0587 0.0283 (-1.82)* (-1.01) (0.52) (0.42) 5 -0.01 -0.0024 0.0267 0.0236 (-2.78)*** (-1.1) (0.23) (0.33) 6 -0.0024 -0.0024 0.144 0.0969 (-0.69) (-1.4) (1.30) (1.67)* 7 -0.0029 -0.0026 0.2063 0.190 (-0.86_ (-1.4) (1.90)* (3.00)*** 8 -0.0067 -0.0054 -0.0403 0.0482 (-1.68)* (-3.04)*** (-0.31) (0.82) 9 -0.0086 -0.0108 0.119 0.0491 (-2.17)*** (-5.45)*** (0.93) (0.73) 10 -0.004 -0.0166 0.0717 -0.2145 (-0.93) (-5.67)*** (0.52) (-2.19)**
9.2 Carhart four factor model
