Fundamental Factor Models and Macroeconomic Risks - An Orthogonal Decomposition

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1 Fundamental Factor Models and Macroeconomic Risks - An Orthogonal Decomposition Wolfgang Bessler a, Thomas Conlon b, a Center for Finance and Banking, Justus-Liebig University Giessen, Germany. b Smurfit Graduate School of Business, University College Dublin, Ireland. Abstract Orthogonalized macroeconomic variables are shown to play a significant role in explaining fundamental factor returns including the Fama-French fivefactors and Carhart momentum factor. Variance decomposition suggests that a small number of dominant explanatory variables capture much of the variation in fundamental factor returns, but pronounced dynamics in exposure attribution are evident. Using quantile regression, we find evidence of heterogeneous exposures of fundamental factors to macroeconomic variables at extremes of the return distribution. A series of robustness analyses support our initial findings. US macroeconomic factors are also found to account for a substantial proportion of variation in international fundamental factor returns. Keywords: Fama-French five-factor model, momentum, macroeconomic fundamentals, variance decomposition, quantile regression JEL Codes: G10, G12 Corresponding Author. conlon.thomas@ucd.ie, Tel: The authors would like to thank Chris Adcock, Bart Frijns, Ana-Maria Fuertes, David McMillan, Jijo Lukose P.J., Adam Zaremba and participants at the 2018 Midwest Finance Association annual meeting, 2018 Applied Financial Modelling Conference and 2017 IN- FINITI Conference on International Finance for helpful comments and suggestions. The usual caveat applies. Conlon would like to acknowledge the financial support of Science Foundation Ireland under Grant Number 16/SPP/3347.

2 1. Introduction Fama and French (2015) propose a five-factor model purporting to explain the cross-section of average stock returns. This model builds upon the Fama and French (1993) three-factor model, adding profitability [RMW] and investment [CMA] factors. In addition to improved explanatory power for the cross-section of US stock returns, these risk factors explain some well-known asset pricing anomalies (Fama and French, 2016) and hold for international markets (Fama and French, 2017). Hou et al. (2015) also document factors closely linked to RMW and CMA to help untangle the cross-section of stock returns. The ubiquitous application of the original Fama and French (1993) threefactor model throughout the asset pricing literature has led to a vibrant debate surrounding their economic significance and meaning (Petkova, 2006). Previous research linked the original three Fama-French factors to a set of macroeconomic variables (Bergbrant and Kelly, 2016; Aretz et al., 2010; Petkova, 2006; Hahn and Lee, 2006; Vassalou, 2003). Furthermore, a multitude of papers have analyzed links between specific macroeconomic variables and stock characteristics such as size, book-to-market and momentum (see, for example, Liu and Zhang (2008); Hahn and Lee (2006); Chen et al. (1986)). Evidence for the relative explanatory power of macroeconomic and fundamental factor models is, however, mixed (Aretz et al., 2010; Connor, 1995). In this paper, we contribute to the aforementioned debate by examining the individual contribution of othogonalized macroeconomic variables in explaining the Fama and French (2015) and Carhart (1997) [FFC] factors. While many variables in a model may be significant, researchers often employ a decomposition of the explained variation to glean some insight as to the relative importance of variables. A variety of decomposition techniques are available but one challenge relates to how the covariance between factors is apportioned. One possible approach is that of hierarchical regression, where variables are added sequentially to the model to isolate any change 2

3 in r-square. In Appendix A, we highlight the challenges surrounding the traditional hierarchical approach to decomposition by examining the relationship between innovations to macroeconomic state variables and market returns. Considerable differences between the proportion of r-squared attributed to each macroeconomic variable are observed, dependent upon the order in which the variables enter the model. To overcome this problem, we apply an orthogonal transformation with a number of attractive properties. Specifically, we employ the democratic decomposition recently outlined by Klein and Chow (2013) and having origins in the physical sciences (Schweinler and Wigner, 1970; Löwdin, 1950). This technique involves a transformation of the original variables by the gentlest shift to produce orthogonal variables. This provides two benefits. First, in a similar vein to the orthogonalization approach employed by Fama and French (1993) to enable reasonable interpretation, this orthogonal transformation helps us in extracting standalone orthogonal components of macroeconomic variables while maintaining an optimal relationship with the underlying variables. Second, the approach allows for direct decomposition of the coefficient of determination, independent of the orthogonalization sequence chosen (hence, democratic decomposition). This aids us in understanding the relative contribution of each macroeconomic variable in explaining the FFC factors and overcomes many of the problems associated with hierarchical orthogonalization and other approaches. The use of the democratic decomposition to examine whether FFC factors act as proxies for macroeconomic state variables is the first contribution of this paper. The next important contribution is to provide an analysis of the macroeconomic exposures of the Fama and French (2015) RMW and CMA factors, in addition to the Hou et al. (2015) q-factors. RMW is found to proxy for a number of macroeconomic variables. Interestingly, this is the only one of the FFC factors examined to reveal a significant relationship with unexpected inflation over the period For the RMW factor, the aggregate 3

4 firm survival probability variable (DSV) is found to account for more than 35% of explained variation, perhaps suggesting that operating profitability acts as a proxy for firm distress. After controlling for the impact of the market on CMA, we document no significant relationship with any of the macroeconomic variables under consideration. Over 91% of explained variation for this factor is attributable to the market factor. In addition to the recent Fama and French (2015) factors, we also assess the relative importance of macroeconomic variables in explaining the original three Fama and French (1993) factors and the Carhart (1997) momentum factor. Consistent with our findings for RMW many variables are statistically significant but only a small number of macroeconomic variables account for a substantial proportion of explained variation. In a further contribution, we decompose the coefficient of determination over time, using a moving-window approach. This highlights the considerable time variability of the factor exposures and the relative importance of macroeconomic variables in explaining the FFC factors over time. Across the factors examined, we find that no state variable is consistently associated with any of the FFC factors, perhaps helping to explain the relatively low explanatory power observed in the unconditional models. Furthermore, we employ quantile regression to understand the relationships between the FFC factors and macroeconomic variables at different return quantiles. While our findings are in general agreement with those presented earlier, we find some distinction at extremes of the return distribution. In particular, aggregate dividend yield and STS are found to be associated with SMB and CMA, respectively, at the lower quantiles. This analysis suggests that FFC factors may proxy for macroeconomic variables during different market states. We further analyze the macroeconomic exposures of the long and short portfolios comprising the FFC factors. In all cases, we find exposure to MYP and DSV, highlighting the importance of these two variables in explaining the FFC factors. In a robustness analysis, we further demonstrate that our 4

5 findings are not a consequence of the so-called January effect and show that variable-selection methods also specify a small number of macroeconomic state variables linked with the FFC factors. Our works relates to but is clearly distinguishable from previous papers examining whether the FFC factors act as proxies for macroeconomic state variables. Our focus on identifying the state variables primarily responsible for the explained variation differs from the approach of Aretz et al. (2010) and Petkova (2006), where variable significance is the primary criterion. This paper also relates to Maio and Philip (2015), where the asymptotic principal component analysis of Connor and Korajczyk (1986) was used to isolate the factors driving variation of the realized market return from amongst 124 macroeconomic variables. We also present new findings regarding the relationship between macroeconomic state variables and the recently proposed Fama and French (2012) RMW and CMA factors. We build further on Aretz et al. (2010) and Petkova (2006) by examining changes in exposures to macroeconomic state variables over time and assessing their relevance at extremes of the distribution using quantile regression. Our findings offer the new insight that macroeconomic exposures are dynamic and occasionally limited to the upper or lower quantiles. Finally, this paper builds upon the work of Klein and Chow (2013), Bessler and Kurmann (2014) and Bessler et al. (2015), in utilizing the LSW transformation to give an economic intuition behind the relative importance of state variables. The remainder of the paper is organized as follows. In Section 2 we provide a description of the methodology employed and the data studied. Section 3 details our empirical results, while Section 4 concludes. 2. Methodology and Data 2.1. Vector Autoregression Campbell (1996) highlights that priced factors should not be based upon important macroeconomic state variables, but rather that only the unex- 5

6 pected component of the state variables should command a risk premium. Accordingly, we specify a vector autoregressive (VAR) process for the demeaned state variables, represented by the k-element vector z t. Following the approach of Campbell (1996) and Petkova (2006), we define a first-order VAR containing returns for the market, macroeconomic state variables and the FFC factors, z t = Az t 1 + u t, (1) where A is a matrix of exposures. The residuals, u t, are a k-element vector of innovation terms associated with each state variable that proxy for changes in the investment opportunity set. The choice of a first-order VAR model is in line with Boons (2016), Petkova (2006) and Campbell (1996). 1 We present details on the estimation of the VAR model in Section The ICAPM Framework Following Aretz et al. (2010) and Petkova (2006), we adopt the intertemporal CAPM (ICAPM) framework of Merton (1973), where exposures to state variables, which forecast the future stock return distribution, are priced in addition to market beta. 2 As described by Fama (1996), the market factor rewards investors for risk not explained by the other state variables. In this setting, we estimate the relationship between the FFC factors and the unexpected components of macroeconomic state variables using the time-series regression: K F j,t = β 0 + β M R M + βj k u k j,t + ɛ j,t (2) where F j,t is the return on FFC factor j, R M is the return to the market and u k j,t correspond to surprises in the k th macroeconomic state variable. βj k is k=1 1 Models examining VAR models with further lags were also tested, but with little quantitative impact on results. Findings available upon request from the authors. 2 See Maio and Santa-Clara (2012) for a detailed treatise of the role of the ICAPM in asset pricing. 6

7 the coefficient on the state variable k for the FFC factor j. While our focus is on the macroeconomic exposures of the FFC factors, it is important to note that the formulation of Equation 2 is also applicable for stock portfolios (e.g. Fama-French 25 book and size sorted portfolios) and individual stocks (Boons, 2016). The FFC factors and macroeconomic state variables employed are described in Table 1. We estimate Equation 2 using OLS with Newey-West autocorrelation and heteroscedasticity adjusted standard errors. A generalized method of moments (GMM) approach is often employed in asset pricing studies to simultaneously estimate the covariances and price of risk. As our focus here is on the sensitivity of FFC factors to macroeconomic variables, rather than pricing of risk, we follow Petkova (2006) in using OLS. Indeed, combining OLS with democratic decomposition, described in the following section, further allows us to decompose the coefficient of determination into constituent contributors, providing a level of intuition not previously detailed in the literature Democratic Decomposition As the macroeconomic innovations (residuals) from the VAR described in Equation 1 are often highly correlated with the market, previous research has followed an orthogonalization process to purge the market effect (Boons, 2016; Petkova, 2006). This process, however, ignores possible correlations between macroeconomic innovations, as any supplemental orthogonalization could add noise through the arbitrary ordering of the variables (Boons, 2016). As highlighted in Appendix A, one such approach, hierarchical orthogonalization, results in an alternative attribution of variation depending upon the ordering of the orthogonalization sequence. In this paper, we implement an approach allowing us to determine the relative importance of each macroeconomic state variable, while retaining the original factor interpretation. To accomplish this, we orthogonalize the macroeconomic state variables and the market simultaneously using an ap- 7

8 proach originally proposed in the physical sciences by Löwdin (1950) and Schweinler and Wigner (1970) (referred to as the LSW transformation henceforth.). Klein and Chow (2013) develop this methodology in the context of examining the contribution of the Fama and French (1993) three factors to the return of risky assets and refer to the associated variance decomposition as a democratic decomposition. This title highlights that the use of the LSW transformation treats all input variables equally. In previous applications, Bessler et al. (2015) and Bessler and Kurmann (2014) employ the democratic decomposition to characterize the economic determinants of bank stock returns. We next outline the benefits of the LSW decomposition and describe how we implement this approach to provide a decomposition of the coefficient of determination. In Appendix B, we present the mathematical details underpinning the LSW transformation. The LSW orthogonalization presents a variety of benefits relative to alternative approaches. First, as shown by Schweinler and Wigner (1970), the LSW transformation produces orthogonalized factors which in a least-squares sense are closest to the original factors amongst all possible orthogonalizations. This is of particular importance for our application, as we focus on preserving an economic interpretation of our results. The LSW transformed factors also retain the same symmetry as the original factors, leading to the description as a symmetric othogonalization. Second, the LSW approach is democratic, in the sense that there is no requirement to select an ordering of the factors. Third, this decomposition allows us to decompose the systematic variation of the FFC factors with respect to each macroeconomic factor. In contrast to the sequential approach to decomposing systematic risk [r-squared] employed by Fama and French (1993), the orthogonalization resulting from the LSW transformation is independent of any imposed ordering of the variables. Finally, variances of orthogonalized variables resulting from the LSW decomposition are identical to those of the original variables. Gathering the set of innovations corresponding to the market and macroe- 8

9 conomic state variables into a vector u t, the systematic return variation associated with asset i can then be measured as σ 2 s i = K K β j k βj l Cov(uk, u l ). (3) l=1 k=1 The coefficient of determination, r-squared or R 2, then results as the proportion of total variation explained by the systematic risk variables relative to the total variation, σ 2 s j /σ 2 j. Equation 3 demonstrates that the decomposition of systematic variation into components is dependent not only on the relative importance of the beta coefficients, but also the variance-covariance matrix of the factors. Klein and Chow (2013) develop a democratic decomposition of the coefficient of determination, to determine the contribution of each factor to overall r-squared, building on earlier work (Schweinler and Wigner, 1970; Löwdin, 1970, 1950). An important implication of this transformation is the ability to partition the coefficient of determination into components, without having to consider the order in which variables enter the model. r-squared is given by R 2 j = K k=1 ( ˆβ k,t The decomposition of the ) 2 ˆσ u k (4) ˆσ j where ˆβ k,t is the estimated coefficient using the orthogonal factors, and ˆσ u k and ˆσ j are the estimated variance of variable k and asset j respectively. ( ) ˆσ 2 Considering each of the terms ˆβ u k k,t ˆσ j independently, we can determine the relative contribution of each factor to the explained variation Data In this paper, we focus on understanding the relative importance of macroeconomic variables in explaining FFC factors corresponding to US mar- 9

10 kets. Return data relating to the FFC factors are for the period The constructed FFC factors are the difference between portfolios with high and low exposures to the underlying factor. Thus, any macroeconomic exposures could be a result of differential underlying exposures to the macroeconomic variables for the FFC long and short portfolios. Consequently, we also reconstruct the underlying long and short portfolios, and individually model their sensitivity to macroeconomic state variables. We initially follow Aretz et al. (2010) to identify suitable macroeconomic variables relating to the FFC factors. The first macroeconomic variable, MY P t,t+12, is the change in expected industrial production growth over the following 12 months, estimated using a mimicking portfolio of traded base assets and control variables following Aretz et al. (2010) and Vassalou (2003). UI t 1,t is unexpected inflation, approximated by the difference between realized inflation and an ARMA[1,1] model estimated value (Ang et al., 2007). DSV t 1,t is the change in aggregate firm survival probability based upon the contingent claims model of Merton (1974). 4 AT S t 1,t and ST S t 1,t are changes in the average level and slope of the term structure, respectively. F X t 1,t is the change in a composite US dollar exchange rate index. This list of variables is not exhaustive and others have been proposed. 5 Petkova (2006), for example, suggested alternative variables as a means to explain the FFC factors. We focus on two of her proposed variables, DIV t 1,t, 3 We are grateful to Kenneth French for making this data available: dartmouth.edu/pages/faculty/ken.french/data_library.html. 4 The time period examined in this paper is limited by the availability of data relating to DSV over the period We are grateful to Söhnke Bartram for making this data available: faculty1/sohnke_bartram/data/. 5 For example, other proposed variables include the aggregate price earnings ratio of Campbell and Shiller (1988), the value spread, or difference between monthly book-tomarket ratios on small-value and small-growth stocks (Campbell and Vuolteenaho, 2004) and the Cochrane and Piazzesi (2005) bond risk premia factor extracted from forward rates. We examine the variables used in Aretz et al. (2010) and Petkova (2006) as these papers are closest to ours in terms of application. 10

11 the dividend yield of the CRSP value-weighted portfolio and DEF t 1,t, the difference between the yield of a BAA-rated long-term corporate bond and a 10-year government bond. 6 We take this set of eight variables as representative of the group of macroeconomic explanatory variables. The FFC factors and the macroeconomic state variables are described in Table 1. [Table 1 about here.] As a complement to the analysis of US markets, we also consider the FFC exposures for other markets including the World (excluding US), Europe, Japan and Asia-Pacific (excluding Japan). In each case, the data is in US dollar terms, as we examine exposure to US macroeconomic state variables. The data we analyze covers the period Empirical Results 3.1. Summary Statistics We first present summary statistics and correlations relating to the FFC factors and macroeconomic variables. Panel (i) of Table 2, reports summary statistics for the FFC factors. Highest (lowest) monthly average returns are evident for MOM (SMB), while MKT (CMA) has the highest (lowest) monthly standard deviation of returns. The MKT, RMW and MOM each have negative skewness and all factors present excess kurtosis. MOM is the factor with the most substantial one-month loss of 34.58% in April The worst month observed for the market is October 1987, surrounding the Black Monday crash, where the loss was 23.24%. The null hypothesis of normality is rejected at a 5% level for all factors by the Jacque-Bera statistic. [Table 2 about here.] 6 We exclude the term structure and short-term interest rate variables from Petkova (2006), as these are highly correlated with the AT S and ST S variables outlined. 11

12 Panel (ii) of Table 2 describes summary statistics for each of the macroeconomic state variables. MYP has a positive expected change year-on-year of 2.7%, DSV has increased year-on-year over the sample and ATS has decreased on average. The average dividend yield, DIV, over the sample was 3.30%. The Jacque-Bera test rejects the null hypothesis of normality for all macroeconomic state variables. Correlations between FFC factors and macroeconomic variables are detailed in panel (i) of Table 3. Only DSV has a significant correlation with all FFC factors. MYP is related to all with the exception of HML, whereas STS has positive (negative) relationships with HML and CMA (RMW and MOM). Significant correlations for UI and FX are limited to RMW and MKT, respectively. Large and significant correlations are observed between the MKT factor and both DSV and MYP, highlighting the importance of orthogonalizing the market factor with respect to these variables, to isolate the component of MKT that is unrelated to the state variables (Aretz et al., 2010). [Table 3 about here.] In Panel (ii) of Table 3, we report the correlations between our macroeconomic state variables. MYP has a significant relationship with all other macroeconomic variables, while significant links are evident in many cases for other variables. MYP and DSV have a correlation of suggesting that they capture common information. ATS and DEF have a negative correlation of , both variables partially reflecting information on the 10-year government bond. The presence of these, and other significant relationships, partially motivates our use of the LSW orthogonalization. This approach allows us to isolate the macroeconomic drivers of the FFC factors, while retaining maximal resemblance to the original factors. The results presented in Panel (iii) of Table 3 also provide support for our use of the LSW decomposition, by highlighting the correlation between 12

13 the orthogonal factors and their original counterparts. As described earlier, the LSW transformation provides orthogonalized factors which are minimally perturbed from the original factors. The lowest correlation observed between orthogonalized and original factors is in the case of DSV. The orthogonalized DEF has a correlation of with the original factor, while all other factors have a correlation of greater than By employing the LSW transformation we gain the ability to democratically decompose r-squared, in addition to other benefits previously outlined, but in such a way that the orthogonal factors strongly resemble the original factors. Finally, to distinguish our approach from that of principal component analysis (PCA), we provide a comparison in Panel (iv) of Table 3. In each case, the maximum correlation between the original factor and one of the PCA factors is detailed. 7 Average correlation between original and PCA factors is 0.859, compared to for the orthogonal LSW factors. The average value, however, masks some significant differences. For example, the maximal correlation between DSV and the PCA factors is 0.672, compared to for the corresponding orthogonal LSW factor. These findings provide further support for the application of the LSW transformed factors in the following analysis, relative to other approaches such as PCA Macroeconomic Risk Exposures Following Campbell (1996), we specify a first-order VAR and extract the unexpected component (residual) for each month as the state variables used to determine the exposures of the FFC factors. Findings for the VAR(1) model are detailed in Table 4. The results indicate that lagged market returns are significant predictors for all the FFC formed portfolios. Moreover, many of the macroeconomic state variables are significantly linked with the lagged market factor. Beyond this, we observe only limited explanatory power for 7 Note: In some cases, original factors have large and significant correlations with more than one PCA factor, making interpretation difficult. 13

14 the FFC factors, with the notable exception of momentum, for which we find an r-squared of As highlighted in previous research, DIV is persistent, achieving an r-squared close to [Table 4 about here.] We now consider the macroeconomic risk exposures of the FFC factors. In order to better derive a superior understanding of these exposures, we examine various models. First, we determine the unconditional link between FFC factors and Aretz et al. (2010) macroeconomic variables. We then consider and include additional state variables proposed by Petkova (2006). Second, using a moving-window approach, we outline how the sensitivity to each macroeconomic variable evolves in time. Third, using a quantile regression approach, we analyze whether macroeconomic exposures of the FFC factors are robust at various points of the return distribution. Fourth, to isolate the source of macroeconomic exposures, we examine the long and short portfolios underlying the FFC factors. Fifth, we ensure that our results are robust to the January effect by excluding January returns from our analysis and we examine whether model selection approaches provide similar inference. Finally, we determine the extent to which US macroeconomic variables explain variation in FFC factors across international markets Unconditional Exposures The unconditional exposures of the FFC factors to our macroeconomic state variables and the market over the period are outlined in Table 5. Note that the market and the macroeconomic state variables are simultaneously orthogonalized, in contrast to the limited market orthogonalization adopted in previous papers (Boons, 2016). SMB, a proxy for the small firm effect, has a significant relationship with MYP, DSV and ATS, with these variables accounting for 76.6% of explained variation and the market being responsible for much of the remaining explained variation. We 14

15 observe little change in overall model r-squared by expanding the set of state variables by including the variables DIV and DEF. In the presence of DEF, however, the ATS variable becomes insignificant with the former capturing 1.38% of total variation. [Table 5 about here.] Across the FFC factors examined, HML has the lowest total r-squared, with the Aretz et al. (2010) macroeconomic variables only explaining around 10% of total variation. In fact, for the two specifications detailed, only ATS is significant with the market responsible for the majority of explained variation. Of particular interest, the DSV and MYP variables capture little variation, in contrast to the results found for SMB, RMW and MOM. We next analyze the macroeconomic exposures of the recently proposed profitability [RMW] and investment [CMA] factors. RMW is related to all of the Aretz et al. (2010) macroeconomic factors with the exception of FX. Furthermore, it is the only FFC factor with a significant relationship with unexpected inflation [UI]. MYP, DSV and STS together account for 51% of the explained variation. No significant relationship between CMA and any of the macroeconomic variables is found and these only account for 8.4% of total explained variation, with the remainder explained by the market. Finally, we consider whether the momentum factor, MOM, relates to our set of macroeconomic state variables. Two such variables, DSV and STS are significant and contribute 9.06% out of a total r-squared of 10.24%. MOM is the only FFC factor for which the market is not a significant explanatory variable. Contrasting the relationships identified in Table 5, a number of macroeconomic variables stand out. First, MYP and DSV are each significant in explaining SMB, RMW and MOM. For all of these FFC factors, however, the explained r-squared due to MYP is less than 1.12%. DSV, in contrast, explains between 6.23% and 8.14% of total variation across these factors. Finally, STS is responsible for a substantial proportion of explained variation 15

16 for the RMW and MOM factors, but is not found to be important for the remaining FFC factors. None of the FFC factors are related to DIV, while DEF is positively linked with SMB and negatively with RMW. In light of previous findings highlighting the importance of DIV and DEF, the relatively small attribution of r-squared to these variables is surprising. A number of issues may influence these findings; First, common variation with other macroeconomic variables may be removed by the orthogonalization process invoked allowing a clear interpretation of the contribution of each variable. Second, these variables (and others previously described) may be time-varying, a concern we address next. Third, certain variables may be of greater relative importance during times of extreme returns. We address the latter point through a quantile regression Conditional Exposures Overall, the explanatory power of macroeconomic variables for the FFC factors is limited, having a maximum r-squared of 17.74% in the case of RMW (including all macroeconomic variables and the market). These findings are in line with those previously outlined by Aretz et al. (2010) for the SMB, HML and MOM portfolios. One possible explanation for the relatively low observed r-squared might be the dynamic exposures of the FFC factors to macroeconomic variables, perhaps brought about by changing prices of risk. Hence, we use a moving-window framework to examine how the sensitivities of the FFC factors to our set of macroeconomic state variables change over time. For these reasons, we re-estimate Equation 2 using moving windows of 60 months and report the proportion of variation explained by each of the orthogonal macroeconomic factors plus the market. The conditional market exposures for each of these factors are presented in Figure 1. 8 Two points are 8 As, in many cases, the r-squared associated with the market dominates the other factors, we detail these results separately. 16

17 worth noting. First, the systematic variation of the FFC factors attributable to the market shows substantial dynamics. For example, the r-squared associated with the market in the case of HML peaks at a level of in August Thereafter it drops consistently to a level of in April Second, the coefficient of determination for the FFC factors associated with the market can be large, up to 0.45 in the case of CMA (in 2001). This indicates that the FFC factors are not perfectly hedged but instead capture facets of market risk. Given the previously presented macroeconomic exposures of the market, we can isolate the pure FFC exposures to the macroeconomic state variable only by controlling for an orthogonal market variable. [Figure 1 about here.] Next, we examine the conditional macroeconomic sensitivities of the FFC factors in turn. In Figure 2, we plot the decomposition of total r-squared for the small minus big factor, SMB (i) and the high minus low factor, HML (ii). We find strong evidence for conditional macroeconomic exposures of the FFC factors. Considering SMB first, two macroeconomic variables, DSV and DEF, dominate, accounting for an average of 7.8% and 4.3% of variation respectively. For both variables, however, we observe sharp discontinuities. For example, during the period 1987 through 1990 we find limited evidence of exposure to DSV, with similar findings over the period encompassing the global financial crisis. For the remaining macroeconomic variables, we observe periods of exposure with high variability and little consistency. In the case of the HML factor, exposures are dynamic with regular and sharp discontinuities. This may help to explain the low overall unconditional r-squared documented for the HML factor in Table 5. [Figure 2 about here.] Turning our attention to Figure 3, we note that RMW has exposure to unexpected inflation at various points, especially during the period of high 17

18 inflation in the early 1980s. While many discontinuities are evident, DSV captures up to 21.1% of total variation. The total variation attributable to macroeconomic state variables ranges from a low of (February 2006) to a high of (August 1980). [Figure 3 about here.] Earlier unconditional findings highlighted that the CMA had no significantly exposure to any of the macroeconomic variables examined. The evidence presented in Figure 3, suggests that this may be a consequence of substantial exposure dynamics over the period Total variation explained by macroeconomic state variables ranges from (June 1993) to (August 1980). We find little evidence of a dominant role for any of the macroeconomic variables for more than a short period. Finally, we examine the momentum factor, MOM, of Carhart (1997). The results are similar to the other factors, with substantial dynamics in macroeconomic exposures are evident and two factors dominating. DSV and ATS account for an average of 5.78% and 4.07% of total variation. Considering other variables, FX accounts for 3.8% of variation over the period 1988 through 1993 but less than 1% elsewhere. MYP appears to have three distinct phases of relative importance, from 1983 through 1987, from 1990 through 1996 and from 2000 through By applying the LSW transformation to macroeconomic state variables, we are able to highlight the conditional exposures of the FFC factors to macroeconomic variables. The results provide some initial guidance regarding the relative importance of macroeconomic state variables in explaining the variation in the time series of returns of the FFC factors. While the conditional exposures of the FFC factors to macroeconomic variables helps in explaining the relative lack of importance of many variables when examined from an unconditional perspective, other considerations may also be at play. 18

19 3.5. Quantile Regression The unconditional results outlined above deemphasize the role of macroeconomic variables in explaining the FFC factors during times of extreme and unusual price movements. As highlighted in Table 2, we reject the assumption of normality for all the FFC factors. It may be that certain macroeconomic variables are of greater relative importance during periods of extremely high or extremely low price movements. To this end, we are among the first to consider quantile regressions between the FFC factors and orthogonalized macroeconomic variables. In Tables 6 and 7, we consider the sensitivity of FFC factors to orthogonal macroeconomic state variables at the 5 th, 25 th, 50 th, 75 th and 95 th quantiles over the period While findings for the dominant variables MYP and DSV are generally consistent across quantiles, some distinctions are evident for other macroeconomic variables at different quantiles. Examining SMB, the ATS variable was close to significant for the baseline analysis, Table 5. Using quantile regression, we find a significant relationship at the 75 th and 95 th quantiles. Similarly, STS is found to be significant at the 25 th and 50 th quantiles, but not at higher. DIV has a positive and significant relationship with SMB at 25 th and 50 th quantiles, but a negative and insignificant relationship at high quantiles. [Table 6 about here.] [Table 7 about here.] In the earlier baseline unconditional analysis, we report limited evidence of any significant relationships between HML and any of the macroeconomic variables under consideration. Using quantile regression, we find that HML is related to many of the macroeconomic state variables under consideration, but that these relationships only hold at specific quantiles. MYP has a positive and significant relationship at the 25 th and 50 th but not at higher 19

20 quantiles. In fact, the majority of macroeconomic variables are significant when returns to HML are above the median at the 75 th quantile. RMW is related to many of the macroeconomic state variables at the median, Table 7, but findings are less consistent at upper and lower quantiles. While baseline results for CMA revealed little, a significant exposure to STS is evident at the 25 th and 50 th quantiles, and to other macroeconomic variables at the 75 th quantile. Finally, in addition to a consistent negative exposure to DSV at quantiles greater than the 5 th, MOM has a negative exposure to UI at the 25 th quantile and a positive relationship at the 95 th, highlighting a phase change contributing to the lack of inferred relationship for baseline findings. MOM is also found to be related to DIV at the highest quantile. These results highlight that important relationships between the FFC factors and macroeconomic state variables may be masked when extreme or unusual price movements are not considered explicitly Long and Short FFC Portfolios While previous literature has examined the macroeconomic drivers of the net (long minus short risk) FFC factors, little attention has been paid to the portfolios from which these net factors are determined. To this end, we reconstruct the underlying long and short portfolios for each of the FFC factors and examine each of these in turn. We continue using orthogonalized macroeconomic risk factors and report the decomposition of total r-squared into components associated with each macroeconomic variable. Results are detailed in Table 8. First, we note the differential exposures to the market between long and short FFC portfolios. For example, the short RMW portfolio has a beta of 1.14, while the long portfolio has beta These findings again highlight the importance of controlling for the market factor when considering the FFC macroeconomic exposures. Without the market factor, a macroeconomic analysis of FFC factors may pick up the strong macroeconomic exposures of the market already detailed. [Table 8 about here.] 20

21 Second, MYP and DSV have a positive and significant relationship with the FFC factors in all cases. Together, these two state variables account for between 2.49 (SMB short portfolio) and (MOM short portfolio) of total variation. Beyond these factors, only FX and DEF stand out as having a regular significant relationship with the FFC factors. The significant relationship notwithstanding, the maximum r-squared associated with FX and DEF are 0.30 (SMB long portfolio) and 0.19 (MOM long portfolio), respectively. These findings suggest that many of the significant relationships observed for the FFC net factors are a consequence of differential exposures between the long and short portfolios to macroeconomic state variables Q-Factor Model Hou et al. (2015) propose the q-factor model as an alternative to the Fama and French (1993) characteristics, suggesting that it better summarizes the cross-section of average stock returns. Similar to our earlier analysis of the Fama and French (2015) characteristic returns, we now determine whether the Hou et al. (2015) factors are linked with macroeconomic fundamentals. Results, outlined in Table 9, illustrate that the ME factor is positively related to the market, while I/A and ROE are negatively related to the market, in keeping with the findings of Hou et al. (2015). In particular, the market accounts for 83.5% of explained variation for the I/A factor. Aside from the market, only DSV, MYP and DEF are found to account for more than 1% of total variation. Linking with our earlier findings for SMB and RMW, we observe congruent results in terms of the significant macroeconomic variables for ME and ROE. In contrast, the macroeconomic explanatory variables significantly associated with I/A diverge from our findings for the Fama and French (2015) five factor model. [Table 9 about here.] 21

22 3.8. Other Tests Much evidence exists for seasonality in stock returns, with specific emphasis on the so-called January effect (Keloharju et al., 2016; Kramer, 1994). Momentum returns, in particular, have been shown to differ in their macroeconomic factor loadings in January (Ji et al., 2017). In Table 10, we reconsider the relationship between the FFC factors and orthogonalized macroeconomic state variables after removing all returns associated with January. The majority of findings detailed in Table 5 still hold, suggesting that our results are not a consequence of a January effect. [Table 10 about here.] Finally, we examine whether our finding that FFC factors are related to a small number of important macroeconomic state variables using alternative methodologies. Stepwise regression uses a systematic approach to add and remove variables from a multivariate regression based upon their statistical significance. The least absolute shrinkage and selection operator (LASSO) penalizes the absolute size of the regression coefficients. 9 For a larger penalty, coefficients estimates may be shrunk towards zero, allowing identification of a reduced set of significant variables. Here we employ 10 folds cross-validation to select the model of interest. [Table 11 about here.] Table 11 details results using the variable selection techniques. While not decomposing the total R2 amongst the various macroeconomic variables, findings again illustrate the importance of a small number of such variables in explaining the FFC factors. The specific variables of importance, while similar, present some differences with those presented in Table 5 using LSW 9 Further details on the LASSO method can be found in Nazemi and Fabozzi (2018). 22

23 decomposition. For example, MKT and MYP were both significant, and account for 3.05% of variation, using the LSW approach, but do not appear in the models selected using stepwise or LASSO regression. Moreover, agreement on the choice of appropriate variables between the stepwise and LASSO models is only found for the RMW factor FFC International Exposures to Macroeconomic Variables The previous literature has largely concentrated on the macroeconomic exposures of FFC factors relating to the United States. We expand upon this literature, by investigating the exposure of international markets to orthogonal US macroeconomic state variables. Any influence of US macroeconomic data on international markets would provide an indication of spill-overs from the US economy to world markets. In all cases, international markets are studied in US dollar terms. Results are detailed in Table 12, for FFC factors representative of four distinct areas, Europe, Japan, Asia-Pacific excluding Japan and International Markets excluding US. The FFC factors are linked with macroeconomic variables in many cases, but findings are less consistent than those detailed for the US. The attribution of r-squared associated with MYP has a maximum value of 4.99 in the case of the Japanese SMB factor (the sign of the coefficient, however, is opposite to that previously obtained for the US). For all other regions and factors, the r-squared associated with MYP is less than 1.0. DSV is significant in a variety of cases, most notably for the CMA and MOM factors, with a maximum r-squared attribution of 5.56 (CMA for International ex. US). [Table 12 about here.] For the remaining macroeconomic variables, a number of notable findings emerge. First, we find a significant relationship between UI and a number of the FFC factors. Second, MOM has a negative relationship with STS across 23

24 all markets with the exception of Japan, in keeping with previous results for the US. Third, we identify a negative relationship between SMB and STS in the majority of cases. Finally, findings for Japan stand out, as the direction of the relationship between FFC factors and macroeconomic variables is often different in sign. These findings relate to those documented by Fama and French (2017, 2012), where Japan was highlighted as anomalous across many dimensions for FFC factors. In summary, little consistent evidence of spillovers is observed across the FFC factors. 4. Conclusions In this paper, we examine the sensitivity of the Fama and French (2015) five factors and the Carhart (1997) momentum factor to orthogonalized macroeconomic state variables. Employing the optimal LSW transformation, we extract standalone orthogonal components of macroeconomic state variables, allowing us to isolate the distinct contribution associated with each variable. Using these orthogonal state-variables, a democratic decomposition of the coefficient of determination is possible, independent of any orthogonalization sequence. While many independent variables in a model may be significant, this decomposition allows us to isolate those variables having the largest explanatory power; specifically, those accounting for the largest proportion of model r-squared. Linking the FFC factors to macroeconomic state variables, we demonstrate that a small number of variables, including MYP, DSV and STS, account for a large proportion of explained variation. Considering the conditional sensitivities to orthogonal macroeconomic variables, the proportion of r-squared attributed to each variable is time varying, often exhibiting sharp discontinuities. We also investigate the role of macroeconomic state variables in explaining the FFC factors during periods of relatively high or low price movements using a quantile regression approach. While findings for MYP and DSV are consistent across quantiles, other variables are shown to be 24

25 significant only at particular quantiles. Next, we analyze the sensitivity of international markets to orthogonal US macroeconomic variables. While findings for international markets highlight spillovers from the US for the market factor, we observe less consistency across the remaining FFC factors. This paper is among the first to consider a decomposition of the r-squared (explanatory power) associated with macroeconomic variables in explaining the returns associated with fundamental factor models. While previous literature has provided evidence of significant relationships between FFC factors and macroeconomic state variables, we demonstrate that a small number of variables are responsible for much explained variation. 25

26 Appendix A. Macroeconomic Risks and the Market In this section, we provide a comparison between the traditional hierarchical approach to decomposition of the coefficient of variation (r-squared) and the democratic decomposition applied in this paper. To this end, we consider a rolling decomposition of the market factor (MKT) using 60-month rolling windows and the set of macroeconomic state variables described in Table 1. The model employed to describe the dynamics follows the approach proposed by Chen et al. (1986) and followed by others since (for example, see Flannery and Protopapadakis (2002)) and can be written as: K R m,t = β 0 + βj k u k j,t + ɛ j,t k=1 (A.1) where R m,t is the return on the market, R M is the return to the market and u k j,t correspond to surprises in the k th macroeconomic state variable. βj k is the coefficient on the state variable k for the FFC factor j. Results are detailed in Figure 4 for three different approaches to decompose the explained r-squared. The first approach, Panel (i), uses hierarchical regression adding variables to the model in the order MYP, UI, DSV, ATS, STS, FX, DIV, DEF to measure the contribution of each. The second specification, Panel (ii), also uses hierarchical regression but the variables are added to the model in reverse order. Finally, in Panel (iii), we isolate the variation associated with each macroeconomic variable using democratic decomposition. [Figure 4 about here.] Contrasting first the two hierarchical approaches to decompose explained variation, differences in attribution to the various macroeconomic variables are emphatic. In the first approach, MYP is the dominant variable, baring a period centered on the late 1980s where DSV is dominant. In contrast, 26

27 for the second hierarchical decomposition, DSV is the dominant variable, with MYP contributing on average 3.5% of total variation. The democratic decomposition, in contrast, does not depend upon the ordering of variables. While the dynamics resemble those of the first approach, there are marked differences. For example, while DSV accounted for an average 10.8% of variation for the first hierarchical regression, under the democratic approach this increases to 17.3%. In this paper, we apply the democratic decomposition to avoid the variable ordering distortion associated with traditional approaches and also because of the many other attractive properties highlighted in the paper. Appendix B. Democratic Orthogonalization The requirement of an orthonormal set of linearly independent vectors is common across many disciplines. One method, commonly employed in the literature, is the Gram-Schmidt procedure (See Ramboud et. al. (2009) for an application to finance). This approach is sequential, meaning that the ordering of the vectors to be orthogonalized impacts upon the resulting orthonormal basis. For the purposes of this study, such an approach is analogous to a hierarchical regression, shown earlier to result in significant variation in the attribution of R 2 amongst factors. Instead, we adopt an approach attributed to Lowdin (1950) in the quantum chemistry literature and to Schweinler and Wigner (1970) in the wavelet literature. This procedure treats the members of the set of linearly independent vectors equally (or democratically). In this sense, there is no requirement to select an ordering of the variables prior to orthogonalization. A significant benefit of this methodology, as detailed by Schwinler and Wigner (1970), is that the resulting orthogonalized vectors are closest in a least squares sense to the original vectors. To understand how the orthogonal basis vectors are determined, we follow the description of Klein and Chow (2013). Starting with a set of K vectors, 27

28 each a time-series of length T, we first create a demeaned matrix F T K = [ f k t f k] k=1,2,...,k t=1,2,...,t. (B.1) Next, define a linear transformation, F T K = F T K S K K, (B.2) where S is an invertible matrix. Given a positive definite Hermitian matrix, the variance-covariance matrix, formed from the inner product of the factors returns, M = ( FT K ) FT K, we can then write ( ) F T K F T K = S K KM K K S K K, (B.3) F T K is an orthonormal basis provided that S K K M K KS K K = I, or equivalently, M 1 K K = S K KS K K. The general solution is given by S K K = M 1 2 K K C K K, (B.4) where C K K is an arbitrary unitary matrix. If C K K = I k K, the orthogonalization is symmetric, in other words the orthogonal basis vectors have the same symmetry as the original ones (known as the Slater-Koster theorem). Under the LSW orthogonalization approach, S is chosen as S K K = U K K P 1 2 U K K. (B.5) U K K is the eigenvector of M K K, P is a diagonal matrix containing the eigenvalues of M K K. To retain the variance of the original factors, a rescal- 28

29 ing is required σ σ S K K S K K T σ K, (B.6) where σ k is the standard deviation of the original factor k. Substituting S K K into equation B.2 results in the symmetric orthogonal transformation of the demeaned matrix, FT K. An addition transformation is required, to recover F T K, as follows: F T K+1 T 1 F1 K S K K = ( FT K + 1 T 1 F1 K ) S K K = F T K S K K = F T K (B.7) 29

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34 Figure 1: Rolling Variance Decomposition Market Sensitivity [ ] Variance of the Fama-French and Carhart factors explained by market exposures is shown using a rolling variance decomposition over the period The proportion of the coefficient of determination associated with the market factor is presented on the Y-axis. 34

35 Figure 2: Rolling Variance Decomposition Macroeconomic Variables [ ] Rolling window [60 month] democratic variance decomposition for market, size [small minus big] and value [high minus low] factors. Macroeconomic variables are as given in Table 1. The market factor is also controlled for in the case of size and value factors, but associated r-squared is not plotted. The decomposition of the coefficient of determination is presented on the Y-axis. 35

36 Figure 3: Rolling Variance Decomposition 36 Macroeconomic Variables [ ] Rolling window [60 month] democratic variance decomposition for profitability [robust minus weak], investment [conservative minus aggressive] and momentum factors. Macroeconomic variables are as given in Table 1. The market factor is also controlled for in each case, but associated r-squared is not plotted. The decomposition of the coefficient of determination is presented on the Y-axis.

37 Figure 4: Market Factor Rolling Variance Decomposition Macroeconomic Variables [ ] Variance decomposition of the market factor (MKT) using (i) a hierarchical regression where the variance are added in the order MYP, UI, DSV, ATS, STS, FX, DIV and DEF, (ii) a hierarchical regression variance decomposition where the variables 37 are added in the reverse order, (iii) democratic decomposition. Macroeconomic variables are as given in Table 1. The decomposition of the coefficient of determination is presented on the Y-axis.