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Journal of Financial Economics

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International tests of a five-factor asset pricing model Eugene F. Fama a , Kenneth R. French b , ∗a University of Chicago Booth School of Business, United States b Tuck School of Business, Dartmouth College, United Statesa r t i c l e i n f o Article history: Received 25 December 2015 Revised 11 March 2016 Accepted 9 June 2016 Available online 23 November 2016 JEL classification: G15 Keywords: International asset pricing Multifactor models Dividend discount modela b s t r a c t Average stock returns for North America, Europe, and Asia Pacific increase with the book- to-market ratio ( B/M ) and profitability and are negatively related to investment. For Japan, the relation between average returns and B/M is strong, but average returns show little relation to profitability or investment. A five-factor model that adds profitability and in- vestment factors to the three-factor model of Fama and French (1993) largely absorbs the patterns in average returns. As in Fama and French (2015, 2016) , the model’s prime prob- lem is failure to capture fully the low average returns of small stocks whose returns behave like those of low profitability firms that invest aggressively. ©2016 Elsevier B.V. All rights reserved.1.IntroductionMotivated by the dividend discount valuation model, Fama and French (FF) (2015) test a five-factor asset pric- ing model that adds profitability and investment factors to the market, Size , and value-growth factors of the Fama and French (1993) three-factor model. In FF (2015) , the left- hand-side (LHS) assets used to test the five-factor model are portfolios formed using sorts on Size (market capital- ization or market cap) and combinations of the book-to- market equity ratio ( B / M ), profitability ( OP ), and invest- ment ( Inv ). The LHS portfolios are thus finer sorts on the variables used to construct the factors. To test the robust- ness of the five-factor model, FF (2016) use LHS portfolios formed on anomaly variables not directly targeted by the model. Here we study international markets, specifically, Eugene F. Fama and Kenneth R. French are consultants to, board membersof, and shareholders in Dimensional Fund Advisors. Thanks to Stan-ley Black, Savina Rizova, and the research group at Dimensional Fund Ad-visors for constructing the data files. Thanks also to the Journal of Finan-cial Economics referee for two rounds of excellent comments.∗ Corresponding author.E-mail address: kfrench@dartmouth.edu (K.R. French).the four regions –North America (NA), Europe, Japan, and Asia Pacific (AP) –examined in Fama and French (2012) . The goal is out-of-sample tests of the US results in FF (2015) . Our tests use variants of the five-factor time-series re- gression, R it −R F t = a i + b i M k t t + s i SM B t + h i HM L t + r i RM W t + c i CM A t + e it . (1) We take the perspective of a US investor and measure all returns in dollars. R it is the dollar return on asset i for month t, R Ft is the risk-free rate (the one-month US Trea- sury bill rate) , Mkt t is the value-weight (VW) market port- folio return minus the risk-free rate, and e it is a zero-mean residual. The remaining right-hand-side (RHS) variables are differences between the returns on diversified portfolios of small and big stocks ( SMB t ), high and low B/M stocks ( HML t ), stocks with robust and weak profitability ( RMW t ), and stocks of low and high investment firms (conservative minus aggressive, CMA t ). If the true values of the factor ex- posures, b i , s i , h i , r i , and c i , capture all differences in ex- pected returns, the intercept a i in (1) is indistinguishable from zero for all LHS assets i . http://dx.doi.org/10.1016/j.jfineco.2016.11.0040304-405X/©2016ElsevierB.V.Allrightsreserved.442 E.F. Fama, K.R. French / Journal of Financial Economics 123 (2017) 441–463Chan, Hamao, and Lakonishok (1991), Fama and French (1998, 2012) , Griffin (2002) , Hou, Karolyi, and Kho (2011) , and others identify Size and B/M patterns in international stock returns. We are also not the first to study how international returns relate to profitability and investment. Titman, Wei, and Xie (2013) show that high investment is followed by low average returns in many markets. Sun, Wei, and Xie (2013) and Watanabe, Yu, Yao, and Yu (2013) confirm this result and show that higher profitabil- ity is associated with higher future returns. These papers do not study in detail how the profitability and investment patterns in average returns vary across Size groups, and they do not attempt to capture these patterns in average returns in an asset pricing model. We show that, as in the case of B/M , small stocks pose the most serious asset pricing challenges related to profitability and investment. Our asset pricing tests ask whether the five-factor model and variants of the model explain the Size , B/M , OP , and Inv patterns in international returns. Thus, as in FF (2015) , the LHS portfolios are finer sorts of the variables used to form the RHS factors. This choice of LHS portfo- lios stacks the deck against rejection. The tests neverthe- less provide strong challenges to the models we consider. We examine local versions of the models, in which the returns to be explained and the factors to explain them are from the same region. The relations between average re- turns and profitability or investment are largely missed by local versions of the three-factor model of FF (1993) , which (dropping the time subscript) include only Mkt , SMB , and HML in Eq. (1) . Though also typically rejected in formal tests, local versions of the five-factor model absorb most of the OP and Inv patterns in average returns. We also pro- vide evidence, brief and negative, on the performance of a global version of the five-factor model. Our LHS portfolios reveal novel results about returns in international markets. Among the most interesting are the low average returns in Europe and Asia Pacific for small stocks with factor loadings like those of unprofitable firms that invest a lot. When we sort on profitability and invest- ment, for example, the 1990–2015 average excess return (relative to the risk-free rate) for a value-weight portfo- lio of small European stocks in the lowest OP and high- est Inv groups is −0.65% per month and the average for AP stocks is −0.71%. The average excess return for the anal- ogous North American portfolio is low but less extreme, 0.12% per month. In tests on US returns, FF (2015) find that the average returns on portfolios of small stocks with factor loadings like those of firms that invest a lot despite low profitability are usually much lower than predicted by the five-factor model. This result is also prominent in the anomaly sorts in FF (2016) . Although the low average re- turns for these stocks are more extreme in Europe and AP, in some sorts the five-factor model captures them. We start ( Section 2 ) with descriptions of the left-hand- side portfolios and right-hand-side factors used in esti- mates of Eq. (1) . Section 3 tests whether asset pricing in the four international regions conforms to a global ver- sion of Eq. (1) . The answer is a strong no, and the rest of the paper focuses on tests in which we use regional fac- tors to capture LHS returns for the same region. Sections 4 and 5 present summary statistics for regional RHS andLHS returns. Sections 6–8 are the main event –tests of as- set pricing models. Section 6 tests whether any regional factors are redundant in the sense that their average re- turns are captured by their exposures to other factors. Section 7 presents summary statistics for regression inter- cepts that flesh out the implications of the spanning tests. Section 8 details the intercept improvements produced by adding RMW and CMA to the FF (1993) three-factor model. Finally, the dividend discount model that motivates Eq. (1) is useful for suggesting variables related to differences in expected asset returns, but it is silent on economic or behavioral explanations of the differences. One interpreta- tion of Eq. (1) is that it is the regression equation for a multifactor version of Merton’s ( 1973 ) intertemporal cap- ital asset pricing model (ICAPM). In this view, SMB , HML , RMW , and CMA are not themselves state variable mimick- ing portfolios, but their long and short ends are [in the terminology of Fama (1996) ] multifactor minimum vari- ance portfolios that together capture the effects of state variables on returns. A less ambitious interpretation of Eq. (1) is that it is the regression equation of an empirical asset pricing model designed to span the mean-variance- efficient tangency portfolio and so capture expected asset returns. We return to interpretation issues in the conclu- sions, Section 9 . 2.Data and variablesOur international stock returns and accounting data are primarily from Bloomberg, supplemented by Datastream and Worldscope. The sample period, July 1990 to Decem- ber 2015 (henceforth 1990–2015), is constrained by data availability and the desire to have broad coverage of small and big stocks in the markets we examine. To increase the power of the tests, we use diversified LHS portfolios in the regressions. Diversification improves regression fits, which increases the precision of the intercepts that are the focus of the asset pricing tests. As in FF (2012), to diversify the LHS portfolios, we combine 23 developed markets into four regions: (1) North America (United States and Canada); (2) Japan; (3) Asia Pacific (Australia, New Zealand, Hong Kong, and Singapore); and (4) Europe (Austria, Belgium, Den- mark, Finland, France, Germany, Greece, Ireland, Italy, the Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, and the United Kingdom). We also examine Global portfo- lios that combine the four regions. Parsimony in the choice of regions is important for the power of the tests, but we want regions in which market integration is a plausible assumption. It is reasonable to assume that the US and Canada are close to one market for goods and securities during our sample period ( Mittoo, 1992 ). For much of our sample, almost all the countries of Europe are members of the European Union (EU), and those that are not (e.g., Switzerland) participate in most of the EU’s open market provisions. Our tests suggest that market integration is most questionable in the Asia Pacific region, which on average accounts for only about 4% of the market cap of the four regions, versus about 48% for NA, 30% for Europe, and 18% for Japan. In each region, we sort stocks on Size and combinations of B/M (ratio of book equity to market equity), OP (ratioE.F. Fama, K.R. French / Journal of Financial Economics 123 (2017) 441–463 443of operating profits –sales minus the cost of goods sold, selling, general, and administrative expenses, and interest expense –to book equity), and Inv (annual growth rate of assets). In our work on US stocks (e.g., FF, 1993) we use NYSE breakpoints for Size and other variables to avoid sorts that are dominated by tiny but plentiful AMEX and Nasdaq stocks. For the same reason, the B/M , OP , and Inv break- points here are based on large stocks, and the Size break- points are percentiles of aggregate market cap chosen to avoid undue weight on tiny stocks.2.1. RHS factorsThe RHS explanatory returns in our tests are for port- folios constructed from 2 ×3 sorts on Size and B/M, OP , or Inv . At the end of June each year t , we sort the stocks in a region on market cap. Big stocks are those in the top 90% of market cap for the region, and small stocks are in the bottom 10%. For North America, 90% of market cap corre- sponds roughly to the NYSE median used to define small and big stocks in FF (1993). The B/M , OP , and Inv break- points in the 2 ×3 sorts for a region are the 30th and 70th percentiles of the variable for the big stocks of the region. As in FF (1993), the accounting variables are for the fiscal year ending in calendar year t −1 and market cap in B/M is for the end of December of calendar year t −1. Global portfolios use global Size breaks, but to mitigate any ef- fects of differences in accounting rules, we use each re- gion’s breakpoints for B/M , OP , and Inv to allocate its stocks to the Global portfolios. For each region, the intersection of the independent 2 ×3 sorts on Size and B/M produces six portfolios –SG , SN , SV , BG , BN , and BV , where S and B indicate small or big and G , N , and V indicate growth, neutral, and value (bot- tom 30%, middle 40%, and top 30% of B/M ), respectively. We compute monthly VW returns for each portfolio from July of year t to June of t + 1. The Size factor, SMB B/M , for a region is the equal-weight (EW) average of the returns on the three small stock portfolios from the 2 ×3 Size – B/M sorts for the region minus the average of the returns on the three big stock portfolios. For each region, we con- struct value minus growth returns for small and big stocks, HML S =SV –SG and HML B =BV –BG , and HML is the av- erage of HML S and HML B . The profitability and investment factors, RMW and CMA , are constructed in the same way as HML except the second sort is on either profitability (ro- bust minus weak) or investment (conservative minus ag- gressive). The 2 ×3 sorts used to construct RMW and CMA pro- duce two additional Size factors, SMB OP and SMB Inv . The overall Size factor SMB is the average of SMB B/M , SMB OP , and SMB Inv . Equivalently, SMB is the average of the returns on the nine small stock portfolios of the three 2 ×3 sorts mi- nus the average of the returns on the nine big stock port- folios. The variables used to construct HML , RMW , and CMA are correlated. High B/M value stocks, for example, tend to have low profitability and investment, and low B/M growth stocks, especially large low B/M stocks, tend to be prof- itable and invest aggressively (Fama and French, 1995) . The correlations imply that the value, profitability, and invest-ment factors, HML , RMW , and CMA , are different mixes of value, profitability, and investment effects in returns. The breakpoints used in the factors in Eq. (1) mimic those of our previous work (e.g., FF, 1993) but are never- theless arbitrary. The experiments in FF (2012) suggest that the performance of international models is not sensitive to choice of factor breakpoints. Since all models are im- perfect, however, future work will likely refine the defini- tions of the factors in models like Eq. (1) . For example, Ball, Gerakos, Linnainmaa, and Nikolaev (2015) find that a prof- itability factor based on cash profitability captures average returns associated with accruals better than the RMW fac- tor of Eq. (1) , which uses operating profitability. We also expect that future work will introduce additional factors. For example, a momentum factor is a common addition to the FF (1993) three-factor model. In the interests of par- simony, we are also hopeful that future work will identify redundant factors. The tests here suggest, for example, that the investment factor is on shaky ground.2.2. LHS portfoliosAt the end of June each year, we construct 25 Size – B/M , 25 Size – OP , and 25 Size – Inv portfolios for each region, to use as LHS assets in asset pricing regressions. The Size breaks for a region are the 3rd, 7th, 13th, and 25th per- centiles of the region’s aggregate market cap. These corre- spond roughly to the average market caps for NYSE quin- tile breakpoints for Size used in FF (1993, 2015). The B/M , OP , and Inv breakpoints in the 5 ×5 sorts follow the same rules as the 2 ×3 sorts, except we use quintile breakpoints (instead of 30–40–30 splits) for big stocks (top 90% of mar- ket cap) in each region to allocate the region’s big and small stocks. The 25 VW Size – B/M , Size – OP , and Size – Inv portfolios for a region are the intersections of the indepen- dent 5 ×5 Size and B/M , Size and OP , and Size and Inv sorts. Since B/M , OP , and Inv are correlated, Size – B/M , Size – OP , and Size – Inv portfolios do not isolate value, profitability, and investment effects in average returns. To disentangle the dimensions of average returns, we would like to sort jointly on Size , B/M , OP , and Inv , but even a 3 ×3 ×3 ×3 sort produces 81 portfolios, many poorly diversified. We compromise with 2 ×4 ×4 sorts on Size and pairs of B/M , OP , and Inv . We form two Size groups, big and small, again defined as the top 90% and bottom 10% of the market cap of a region, and we use quartiles to form four groups for each of the other two sort variables. This process yields 32 Size-B/M-OP , 32 Size-B/M-Inv , and 32 Size-OP-Inv port- folios to use as LHS portfolios in asset pricing tests. The sorts on B/M , OP , and Inv are independent, but to spread stocks more evenly in the 2 ×4 ×4 sorts, we use separate breakpoints for B/M , OP , and Inv for small and big stocks. 3.Tests of a global modelFama and French (2012) find that a Global version of the FF (1993) three-factor model does not explain regional expected returns. A simple spanning test produces the same conclusion for the five-factor model. We summarize the results but do not present a table.4 4 4 E.F. Fama, K.R. French / Journal of Financial Economics 123 (2017) 441–463We estimate 20 regressions in which the LHS returns are for regional factors (five local factors for each of four regions) and the RHS returns are the five Global factors. If the Global model describes expected returns, the 20 regression intercepts are indistinguishable from zero: the Global factors span the regional factors. In fact, the inter- cept for the NA Mkt regression, 0.43%, is more than four standard errors from zero ( t = 4.10), five intercepts are more than three standard errors from zero, and seven are more than two. The GRS test ( Gibbons, Ross, and Shanken, 1989 ) says the probability the true intercepts are zero is zero to at least five decimal places. Thus, adding the re- gional factors to the Global factors produces a reliably large increase in the maximum Sharpe ratio from the Global fac- tors alone. In short, the Global factors do not span the re- gional factors. Tests on other sets of LHS portfolios also confirm that the Global five-factor model performs poorly when LHS portfolios are regional. For example the average return for the NA market factor, 0.62% per month, is greater than the mean for Global Mkt , 0.43%, and when Global factors are used to explain a wide range of NA portfolio returns, most intercepts are strongly positive. The average Mkt return for Japan is close to zero, 0.01%, and strong negative inter- cepts are the rule when Global factors are used to explain Japanese portfolio returns. We can infer that the Global five-factor model is a poor choice in applications in which the LHS portfolios are regional. The Global five-factor model may fail because markets are not globally integrated or because we have the wrong Global model. Our goal, however, is to follow the trail of the five-factor model as far as it will go. Since the Global model fails badly when the LHS assets are regional portfo- lio returns, we close that branch of the trail. The asset pric- ing tests to come focus on five-factor models that use re- gional factors to explain LHS portfolio returns for the same region. To set the stage, we start with summary statistics for returns to the regional RHS factors ( Section 4 ) and LHS portfolios ( Section 5 ). 4.Summary statistics for RHS factor returnsPanel A of Table 1 shows that the equity premium (av- erage Mkt return) for Japan for 1990–2015 is near zero (0.01% per month, t = 0.04). Equity premiums for the other regions are large (0.62%, t = 2.53 for NA; 0.47%, t = 1.64 for Europe; and 0.71%, t = 2.05 for AP). The 1990–2015 Size premium (average SMB return) is close to zero in all regions. The largest Size premium is 0.17% per month ( t = 1.05) for NA. Large value premiums (average HML returns) are the rule for 1990–2015. Only the value premium for North America, 0.20% ( t = 1.08), is less than 2.19 standard errors from zero. The value premium for Japan, 0.36% per month ( t = 2.19), is the only Japanese premium not close to zero–an important result for the asset pricing tests to come. Profitability premiums (average RMW returns) for NA and Europe are substantial (0.34%, t = 2.46 for NA; 0.41%, t = 4.76 for Europe). The average RMW return for Asia Pacific is lower, 0.21% per month ( t = 1.25). Investment premiums (average CMA returns) for NA, Europe, and AP are 0.20%to 0.39% per month and 1.86–2.60 standard errors from zero. Panel B of Table 1 confirms the evidence in FF (2012) that, for NA, Europe, and AP, the value premium is larger for small stocks. For Japan, however, average HML B , 0.41% per month ( t = 1.97), is larger than average HML S , 0.30% ( t = 1.67), but the average difference is only 0.51 stan- dard errors from zero. The new evidence is that, except for Japan, where there are no reliable profitability and invest- ment premiums, average RMW and CMA returns are also larger for small stocks. For NA, Europe, and AP, the aver- age values of RMW S range from 1.80 (AP) to 7.79 (Europe) standard errors from zero. The average values of CMA S are all more than 2.9 standard errors from zero. Average RMW B and CMA B returns are all positive, but none break the two standard error barrier. The average values of the difference RMW S-B =RMW S –RMW B are 0.61 (AP) to 2.09 (Europe) standard errors from zero, so the evidence that the ex- pected profitability premium is larger for small stocks is weak. The evidence that the expected investment premium is larger for small stocks is stronger, at least for the two major regions, North America and Europe. The average val- ues of CMA S for NA (0.45%, t = 3.03) and Europe (0.31%, t = 2.95) are about three times those of CMA B (0.13%, t = 0.70 for NA, and 0.09%, t = 0.68 for Europe), and the average values of the spread, CMA S-B =CMA S –CMA B , are 2.59 (NA) and 2.14 (Europe) standard errors above zero. For AP, av- erage CMA S (0.51%) is almost twice average CMA B (0.28%), but the average difference is only 1.04 standard errors from zero. Panel C of Table 1 shows correlations across regions for each of the five factors. Market factors ( Mkt ) are most correlated. The correlation between the market factors of NA and Europe is 0.80, and the correlations of AP with NA and Europe are 0.72 and 0.75. The correlations of the Japanese market factor with the other three are lower, 0.42–0.51. Correlations are lower for other factors. Europe and NA on average account for almost 80% of the mar- ket cap of the sample, so their correlations are of spe- cial interest. Of the four non-market factors, the HML re- turns of Europe and NA are most correlated (0.61), next is CMA (0.57), then SMB (0.31) and RMW (0.21). The prof- itability factor RMW is least correlated across regions. The 0.21 correlation for Europe and NA is the largest in the matrix. Average premiums for 1990–2015 (306 months) have large sampling errors as estimates of expected premiums. For example, the standard error of the average Mkt (eq- uity premium) return for Japan is 0.34%, and the plus and minus two standard error range about the 0.01% mean is wide, from −0.67% to 0.69%. More important, average factor returns are interesting, but the contribution of a factor to a model depends on its marginal information about average returns (the in- tercept in the spanning regression of the factor’s return on the returns of the model’s other factors), not on its average return. We examine factor spanning regressions in Section 6 , after discussing the LHS average returns of portfolios formed on Size , B/M , profitability, and invest- ment that variants of the five-factor model attempt to explain.E.F. Fama, K.R. French / Journal of Financial Economics 123 (2017) 441–463 445Table 1Summary statistics for factor returns: July 1990–December 2015, 306 months.We construct regional factors for North America (NA), Europe, Japan, and Asia Pacific (AP) (excluding Japan). To construct HML , we form portfolios at theend of June of each year t by sorting stocks in a region into two market cap and three book-to-market equity ( B/M ) groups. Big stocks are those in thetop 90% of June market cap for the region, and small stocks are those in the bottom 10%. The B/M breakpoints for the four regions are the 30th and 70thpercentiles of lagged (fiscal year t − 1) B/M for the big stocks of a region. The independent 2 ×3 sorts on Size and B/M produce six value-weight portfolios,SG , SN , SV , BG , BN , and BV , where S and B indicate small or big and G, N, and V indicate growth, neutral, and value (bottom 30%, middle 40%, and top30% of B/M ). SMB B/M is the equal-weight average of the returns on the three small stock portfolios for the region minus the average of the returns on thethree big stock portfolios. We construct value – growth returns for small and big stocks, HML S =SV – SG and HML B =BV – BG , and HML is the equal-weightaverage of HML S and HML B . The profitability and investment factors, RMW and CMA, and their small and big components are constructed in the same wayas HML except the second sort is either on OP (the ratio of operating profits to book equity, sorted from robust to weak) or investment (the rate of growthof total assets, sorted from conservative to aggressive). The 2 ×3 sorts used to construct RMW and CMA produce two additional Size factors, SMB OP andSMB Inv . The Size factor SMB is the average of the returns on the nine small stock portfolios of the three 2 ×3 sorts minus the average of the returns on thenine big stock portfolios. All returns are in US dollars. Mkt is the return on a region’s value-weight market portfolio minus the US one-month T-bill rate.The average T-bill rate is 0.28%. Mean and Std Dev are the mean and standard deviation of the return, and t -Mean is the ratio of Mean to its standard error.The table shows Mean , Std Dev , and t – Mean for factor returns (Panel A), the small and big components of the factors and the difference between them(Panel B), e.g., HML S , HML B , and HML SB =HML S – HML B , and the correlations of each factor across regions (Panel C).
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