How to Construct Fundamental Risk Factors?

Transcription

1 EDHEC-Risk Institute promenade des Anglais Nice Cedex 3 Tel.: +33 (0) research@edhec-risk.com Web: How to Construct Fundamental Risk Factors? January 2011 Georges Hübner Affiliate Professor of Finance, EDHEC Business School Marie Lambert Researcher, School of Business and Economics, Maastricht University

2 Abstract This paper proposes an alternative way to construct the Fama and French (1993) empirical risk factors. Without losing in significance power, in beta consistency or in factor efficiency compared to the Fama and French factors, our technique insulates the effects of other sources of risk as much as possible when evaluating one risk factor. Consequently, the approach is neater and leads to risk premiums that may not necessarily be used jointly in a regression-based model, unlike the original Fama and French factors whose risk exposures are highly sensitive to the inclusion of the other factors in the regression. This property is very useful for stepwise factor selection procedures. Besides, the methodology developed in this paper is easily extendable to price risk fundamentals other than the empirical size, book-to-market and momentum effects and to other markets (even to small exchange markets as the technique requires less in terms of data histories). Concluding, this paper creates a theoretical framework for pricing the returns attached to a unit exposure to any particular source of risk. Keywords: Fama and French Factors, Momentum, Hedge/Mimicking Portfolios, Market Risk Jel codes: G11, G12 This paper has benefited from comments by Antonio Cosma, Dan Galai, Martin Gruber, Thorsten Lehnert, Pierre Armand Michel, Patrick Navatte, Christian Wolff, as well as Luxembourg School of Finance seminar participants, the 2010 French Finance Association Conference (St Malo, France), the 2010 EFMA Conference (Aarhus, Denmark), the 2010 annual joint Maastricht-Liège seminar and the 1st World Finance Conference 2010 (Viana do Castelo, Portugal) attendees. This paper is a substantially revised version of chapter 3 of the first author s Ph.D dissertation at the Universities of Liège and Luxembourg. The present project is supported by the National Research Fund, Luxembourg and co-funded under the Marie Curie Actions of the European Commission. Georges Hübner acknowledges financial support of Deloitte Luxembourg. All remaining errors are ours. EDHEC is one of the top five business schools in France. Its reputation is built on the high quality of its faculty and the privileged relationship with professionals that the school has cultivated since its establishment in EDHEC Business School has decided to draw on its extensive knowledge of the professional environment and has therefore focused its research on themes that satisfy the needs of professionals. 2 EDHEC pursues an active research policy in the field of finance. EDHEC-Risk Institute carries out numerous research programmes in the areas of asset allocation and risk management in both the traditional and alternative investment universes. Copyright 2011 EDHEC

3 Introduction Fundamental factor models use market fundamentals or firm characteristics to construct factor betas. Among this class of models, the empirical three-factor model of Fama and French (1993) and the four-factor Carhart (1997) model capture the size, book-to-market, and momentum effects. They have largely been used in the literature for explaining stock or mutual fund returns. With the growth of the alternative investments universe since the mid-nineties, scholars have also tried to estimate the returns attached to non-linear co-movements of asset returns with market index returns. Their idea is to use the levels of skewness (3rd moment) and excess kurtosis (4th moment) of stock, mutual fund, or hedge fund return distributions as factor betas. 1 The challenge in such models is to constitute mimicking or hedge portfolios that are able to capture the marginal returns associated with a unit exposure to each attribute. The factor construction method developed by Fama and French (1993) has become a standard in constructing fundamental risk factors. Using a set of data from CRSP (The Center for Research in Security Prices), Fama and French consider two ways of scaling US stocks, an annual two-way sort on market equity and an annual three-way sort on book-to-market according to NYSE breakpoints (quantiles). They then construct six value-weighted (two-dimensional) portfolios at the intersections of the annual rankings (performed each June of year y according to the fundamentals displayed in December of year y-1). The size factor or SMB factor ( Small minus Big ) measures the return differential between the average small cap and the average big cap portfolios, while the book-to-market factor or HML factor ( High minus Low ) measures the return differential between the average value and the average growth portfolios. 2 Carhart (1997) completes the Fama and French threefactor model by computing, along a similar method, a momentum (i.e. a 1-year prior-return) or UMD ( Up minus Down ) factor that reflects the return differential between the highest and the lowest prior-return portfolios. On his online data library, French replaces the book-to-market risk dimension by the momentum risk dimension and provides an estimation of the momentum factor. The set of 2x3 size/momentum-sorted portfolios is rebalanced on a monthly basis. The Fama and French (henceforth F&F) methodology specifically sticks to the US stock market, and cannot be extended as such to other contexts for two reasons. First, their factor construction method relies only on a limited equity market segment (the NYSE stocks) to define the sorting breakpoints in order not to be tilted towards the numerous small stocks of the NASDAQ and AMEX exchanges. The F&F methodology is also rather heuristic regarding the way risk fundamentals are priced. The authors suggest performing a double sorting for size and a triple sorting for booktomarket on the judgmental basis that size is less informative than book-to-market. The recent works by Cremers et al. (2008) and Huij and Verbeek (2009) have both shown that the F&F method tends to misevaluate some of the premiums. According to Cremers et al. (2008), the value premium is overestimated in the F&F framework as the latter methodology puts the same weight on the small and big size portfolios while the value effect is in fact more important in small caps than in big caps. As a corrective action, Cremers et al. (2008) consider small and big caps separately when pricing the value factor. Following Huij and Verbeek (2009), an overestimation of the value premium and an underestimation of the momentum factor should be related to the ignorance of transaction costs when using stock returns for constructing the mimicking portfolios. In this paper, we address and confirm the issues raised in the paper of Cremers et al. (2008) and Huij and Verbeek (2009) concerning the overvaluation of the HML premium. We argue that the F&F premiums are contaminated by cross-effects that are not well taken into account when performing an independent sorting procedure (causing correlation between the rankings). In our view, the independent ranking procedure does not optimally diversify the other sources of risk than the one to be priced or does it sufficiently take into account the correlations across risk dimensions See a.o. the analysis of Moreno and Rodriguez (2009) for mutual funds, and the models of Kat and Miffre (2006) and Agarwal et al. (2009) for hedge funds. 2 - French has made these series available for download on his website

4 If our view is true, the F&F methodology is not appropriate for pricing market fundamentals that would be highly correlated as it would lead to a seriously unbalanced set of portfolios due to the high correlation levels between the rankings. Therefore, we propose to replace the independent sorting procedure by a sequential sorting procedure (the sort on the risk dimension to be priced is made conditional on the sorts over the two control risk dimensions) in order to produce pure estimates of the returns associated with each risk exposure. Moreover, as the purpose is to define general guidelines that could be valid for any market fundamentals and for any markets, we review some of the methodological choices made by Fama and French that, according to us, are specific to the US stock market and to the analysis of the empirical sources of risk. We particularly rely on a cubic representation of the risk exposures: we consider three sources of risk and measure them using a three-way sort, forming a cube. Two breakpoints are used for all fundamentals and are based on the whole equity market. Hence, 27 portfolios, instead of 6 in the original F&F methodology, are formed per cube. Relying on a systematic and sequential sorting technique has two main advantages. First, our factor construction method enables us to maximise the dispersion in the related source of risk while keeping minimal dispersion in correlated sources of risk. Thus, it better captures the return spread exclusively related with the source of risk to be priced. We expect our method to produce more consistent estimates of the returns attached to any risk exposures than the ones produced by F&F. Second, the cubic technique leads to risk premiums with a much lower level of correlations. By not conditioning the use of a risk premium to the inclusions of all other factors in the model, we circumvent a strong limitation of the original Fama and French factors. Their risk exposures are indeed highly sensitive to the inclusion of the other risk factors in the regression because of the levels of cross-correlations. By contrast, the cubic risk factors ought not necessarily be used jointly in a regression-based model. Besides, since our premiums do not rely on market-specific guidelines, we anticipate them to be more easily applicable to other markets and other risk fundamentals. Based on the same empirical risk factors example as F&F, we test our guidelines against those of F&F. We conduct our analysis on a sample of monthly data downloaded from Thomson Financial Datastream Inc 3, and on a recent time period: the actual sample for the risk premiums range from May 1980 to April 2007, i.e. a total of 324 monthly observations. An analysis of the statistical properties of both samples of risk premiums shows that the cubic and the original F&F factors are highly correlated (at about 70%). They aim at capturing the same type of risk, but at the same time display very different descriptive statistics. In the F&F original analysis, the size factor is considered to be less important than the momentum and the bookto-market effects. On the contrary, only the size and the momentum factors (and thus not the book-to-market effect) generate a significant risk premium in the cubic framework. Through a very simple example, we show that the F&F method does not deliver pure estimates of the return attached to each type of risk. The exposures to the F&F empirical factors become erratic when all three premiums are not considered together in one single regression-based analysis. We find evidence that the cubic construction method better isolates the cross-effects between premiums and, as a consequence, delivers purer estimates for the factors. Beyond the study of each method s own pricing abilities, we devote the second part of the study to test the dominance of one method over the other. It is first shown that the cubic premiums more efficiently price the returns of the F&F set of 2x3 portfolios sorted on size and book-tomarket than the F&F factors do. Nevertheless, the impact of correlations between the F&F risk factors does not contaminate the ordering of their coefficient from one portfolio to another when performing the time series analysis as shown from the analysis of a set of 5x5 portfolios The use of alternative databases for the same market does not influence our results.

5 sorted on size and book-to-market. We also point out that the cubic factors proportionally better explain and provide less specification error than the F&F premiums do in explaining a set of 11,377 stock returns. In conclusion, we argue that if one has to choose one specification or the other, all evidence indicates that the cubic construction should be preferred. The rest of the paper is organised as follows. Through one simple example, the first section addresses the problems related with the F&F methodology. Section 2 presents the alternative cubic methodology proposed in the paper. Section 3 carries out the analysis of the properties of the cubic and the F&F samples of empirical risk factors. Section 4 performs comparative tests about the specification power of each pair of premiums. Section 5 concludes. 1. Preliminary Evidence: An Acid Test on Factor Exposures We discuss one intuitive illustration showing concrete pitfalls with the use of the F&F premiums as they stand. Table 1 displays the results of an acid test on a set of F&F portfolios made available on French s website. These portfolios are based on a 2x3 sort of stocks into size and book-to-market. For instance, the Low/Mid portfolio stands for a portfolio made of stocks with low market capitalisation and medium levels of book-to-market. For each of these portfolios, the table considers the original 4-factor F&F and Carhart model (model M.1) and evaluates the changes in the regression coefficients when successively eliminating one risk factor (models M.2 to M.4), and then a second one (models M.5 to M.7). SMB ff, HML ff, and UMD ff, stand respectively for the F&F estimates of the size, book-to-market and momentum premiums. All changes superior to 80% with regard to the 4-factor model are reported in bold. Table 1: Independence test of the F&F empirical risk factors Table 1 indicates that the exposures to the SMB ff factor displayed by the High/High (541.18%) and High/Mid (87.39%) portfolios are highly sensitive to the inclusion of the HML ff factor in the regression-based analysis. Out of the analysis of the Low/Low and Low/Mid portfolios, the loadings on the HML ff factor also appear to be unstable when other risk factors are not included, 5

6 particularly the SMB ff factor (Δ HML ff =124.62% for the L/L portfolio, 96.47% for the L/M portfolio when SMB ff is not included in the regression). Finally, the UMD ff factor is sensitive to the inclusion of the two other empirical factors in the regression for 5 out of 6 portfolios. Although the sort is not performed on momentum, the UMD ff factor of F&F is significant in the 4-factor Carhart model for all 2x3 portfolios, but the significance of the coefficient vanishes for small cap portfolios in the absence of the size premium. From this table, it appears that the exposures to the F&F empirical factors become erratic when all three premiums are not considered together in one single regression-based analysis. Despite the fact that the portfolios chosen in our example are supposed to reflect the size and value dimensions, the F&F method does not deliver pure estimates of the returns attached to each type of risk. 2. The Cubic Method Although it does not suggest any method to improve the F&F shortcomings, the previous example clearly outlines the need for a method that delivers stable and consistent fundamental risk premiums. In this section, we propose an alternative approach that leads to the purification of the risk factors by ensuring the homogeneity of each constructed portfolio on all three fundamental risk dimensions. We call this the cubic method, by analogy to the creation of a cube built with 27 identical cubic components The principles The cubic approach differs from the F&F methodology on various points. First, we consider a comprehensive approach that jointly analyses the three empirical dimensions of risks: size, booktomarket and momentum. Each form of risk is equally considered. Besides, we propose a consistent and systematic sorting of all listed stocks, while F&F perform a heuristic split according to NYSE stocks only. Second, a monthly rebalancing of the portfolios is more realistic in capturing the returns associated with some time-varying dimensions of risk like higher-moment exposures. Third and lastly, our sequential sort avoids spurious significance in risk factors due to any correlation between the rankings underlying the construction of the benchmarks. The following subsections go into the details of the construction of this cube. i. Three-way sort We consider the cross-section of US stock returns and model this risk space as a cube. We split the sample according to three levels of size, BTM, and momentum. 4 Two breakpoints (1/3th and 2/3th percentiles) are used for all fundamentals. Thus, not 6 but 27 portfolios are formed. The breakpoints are based on all US markets, not only on NYSE stocks. ii. Monthly rebalancing To comply with a monthly rebalancing strategy, we assume that market participants refer to the last quarterly reporting to form their expectations about each stock. Therefore, we use a linear interpolation to transpose annual debt and asset values into quarterly data, as this is the usual publishing frequency on the US markets: (1) (2) We borrow Jegadeesh and Titman s (2001) and Carhart s (1997) definition of momentum. The one-year momentum anomaly for month t is defined as the trailing eleven-month returns lagged one month (t-11 to t-1). Stocks that do not have a price at the end of month t-12 are not considered for that period. Our momentum strategy benefits from the outperformance of winners and from the underperformance of losers by combining long positions in winners and short positions in losers.

7 for k = 3,6,9,12, i.e. k th month of year y. Second, we ignore unrealistic values 5 of BTM for the US markets, i.e. higher than 12.5, in line with the empirical study of Mahajan and Tartaroglu (2008). iii. Sequential sorting procedure Our objective is to detect whether, when it is made conditional on two of the three risk dimensions, there is still enough variation related to the third risk criterion. Therefore, we substitute the F&F independent sort with a sequential or conditional sort, i.e. a multi-stage sorting procedure. To be precise, we perform three sorts successively. The first two sorts are operated on control risk dimensions, while we end with the risk dimension to be priced. The sequential sorting produces 27 portfolios capturing the return related to a low, medium, or a high level on the risk factor, conditional on the levels registered on the two control risk dimensions. Taking the simple average of the differences between the portfolios scoring high and low on the risk dimension to be priced, but scoring at the same levels for the two control risk dimensions, we obtain the return variation related to the risk under consideration. Figure 1 illustrates this procedure. Figure 1. Sequential three-stage sorting procedure. This figure illustrates the sequential 3-stage sorting procedure. The stocks are first sorted into 3 portfolios according to one control risk dimension. Within each portfolio. the stocks are sorted into 3 portfolios according to another control risk dimension. Finally. the stocks within the 9 portfolios are sorted into 3 portfolios according to the risk dimension to be priced. Out of the 27 portfolios. we take the 9 return spreads on the risk dimension to be priced and compute the simple average of these 9 portfolios. 5 - We allow a variation of up to one standard deviation around the US average BTM. 7

8 In the sequential sort, we end up with the risk dimension to be priced. Therefore, there are only two possible ways to create the risk premiums, depending on the ordering of the first two sorts. We choose the one that maximizes the number of stocks into the smallest final portfolio. 6 First, the size-sorted portfolios are formed by successively performing a 3-stage sequential sorting procedure on book-to-market, momentum and market capitalisation. Second, the book-tomarket-sorted portfolios are formed by successively performing a 3-stage sequential sorting procedure on market capitalisation, momentum and book-to-market. Finally, the momentumsorted portfolios are formed by successively performing a 3-stage sequential sorting procedure on book-to-market, market capitalisation and momentum. 2.2 The setup The sample used in this paper is formed of all NYSE, AMEX and NASDAQ stocks collected from Thomson Financial Datastream for which the following information is available: 7 company annual total debt, 8 the company annual total asset, 9 the official monthly closing price adjusted for subsequent capital actions and the monthly market value. Monthly returns and market values 10 are then recorded for observations whose stock return does not exceed 100% and whose market values are strictly positive. This is to avoid outliers that could result from errors in the data collection process. We then define the book value of equity as the net accounting value of the company assets, i.e. the value of the assets net of all debt obligations. From a total of 25,463 dead and 7,094 live stocks available as of August 2008, we retain 6,579 dead and 4,798 live stocks with all criteria respected for the period ranging from February 1973 to June The usable sample for the risk premiums ranges from May 1980 to April 2007 due to some missing accounting data. The analysis covers 324 monthly observations. The market risk premium inferred from this space corresponds to the value-weighted return on all US stocks minus the one-month T-Bill rate. We illustrate our methodology with the HML factor construction. We start by breaking up the NYSE, AMEX and NASDAQ stocks into three groups according to the market capitalisation criterion. We then successively scale each of the three size-portfolios into three classes according to their 2-12 prior return. Each of these 9 portfolios is in turn split in three new portfolios according to their book-to-market fundamentals. We end up with 27 value-weighted portfolios. The rebalancing is made on a monthly basis. For each month t, every stock is ranked on the selected risk dimensions. It integrates one side, then one row, then one cell of the cube and thus enters one and only one portfolio. The stock specific value-weighted return in the month following the ranking is then related to the reward of the risks incurred in this portfolio. To create a risk factor, we only consider, among the 27 portfolios inferred from the cubic risk space, the 18 that score at a high or a low level on the risk dimension. 9 portfolios are then constituted from the difference between high and low scored portfolios, which display the same ranking on the size and momentum dimensions (used as control variables). Finally, the HML cubic risk factor is computed as the arithmetic average of these 9 portfolios The acid test revisited This subsection revisits the preliminary evidence presented in the previous section and contrasts it with a similar acid test on the exposures obtained with the cubic premiums. Table 2 reproduces the analysis presented at Table 1 using a cubic 4-factor Carhart model. SMB c, HML c, and UMD c, stand respectively for the cubic estimates of the size, book-to-market and momentum premiums We assume that the larger the portfolio, the better the accuracy of the risk premiums. Our conjecture is confirmed by the empirical results. The control check tests are available upon request. 7 - As for the risk space of F&F, temporary data non-availability just excludes the stock from the analysis at that time. 8 - The company total debt at year y (D) concerns all interest bearing and capitalised lease obligations (long and short term debt) at the end of the year. These variables have been collected on Compustat. 9 - The company total asset at year y (A) is the sum of current and long term assets owned by the company for that year. These variables have been collected on Compustat We designate by market value at month t, the quoted share price multiplied by the number of ordinary shares of common stock outstanding at that moment. As in Fama-French (1993), negative or zero book values that result from particular cases of persistently negative earnings are excluded from the analysis.

9 Table 2: Independence test of the cubic empirical risk factors We perform a 4-factor Carhart analysis (M.1) on the F&F 2x3 portfolios sorted on size and book-to-market. In Models M.2 to M.4, we estimate the exposures to SMB c, HML c or UMD c when one of these factors is removed from the regression. In Models M.5 to M.7, we estimate the exposure to SMB c, HML c or UMD c when two factors are removed from the regression. We report in parentheses the % of change of the estimated coefficient. The average relative change reports, for each factor, the average percentage of change (in absolute value) when one other factor is removed. The total average relative change reports, for each factor, the average percentage of change (in absolute value) when either one or two other factors are removed from the regression. Percentages over 80% are reported in bold. T-tests are performed over the estimated coefficients: *,**, and *** stand for significant at 10%,5%, and 1%, respectively. Contrary to the SMB ff factor, the cubic estimate of the size premium resists much better to the inclusion of the other empirical risk factors. In Table 1, the total average relative difference (last column) in the SMB ff coefficient resulting from removing one or two factors ranges from 6.99% to 384.6%, with a mean change of 88.2%. The corresponding range in Table 2 is 1.08% to 34.45%, with a mean of 14.14%. The exposure to HML c factor appears as well to remain fairly insensitive to the inclusion of the SMBc factor in the regression for all portfolios but the Low/Mid one. The UMD c factor is also largely independent from the other two empirical risk factors: the UMD c portfolio is sensitive to the inclusion of the SMBc factor only for one out of 6 portfolios. 9

10 Such results suggest that the SMB c, HML c, and UMD c factors could be used separately in regressionbased analyses. Their exposures do not substantially vary when the other factors are introduced or eliminated from the regression, unlike the SMB ff, HML ff, and UMD ff factors. The maximum relative change in Table 1 is % from model M.1 to M.2-M.7, while it is only % in Table 2. The average instability is 52.97% when removing the first factor in the F&F analysis while it is only 23.83% in the cubic framework. When removing the second factor, the difference is even more important: the average instability evaluated from the absolute percentage of change when 2 factors are jointly removed rises to 87.30% in the F&F analysis (Table 1), whereas it is only % in Table 2 with the cubic premiums. 3. Properties of the Cubic and the Original F&F Factors The previous sections only present preliminary evidence over the stability of the F&F factors compared to the cubic risk premiums. We now turn to a deeper and more systematic analysis of the properties of the competing sets of factors. Table 3 below displays a full table of descriptive statistics for both sets of premiums. The table covers the period May 1980-April Table 3: Descriptive statistics over the empirical risk premiums (May 1980-April 2007) Table 3 displays descriptive statistics for size (SMB), book-to-market (HML), and momentum (UMD) premiums over the period ranging from May 1980 to April T-tests of the significance of the different time-series are conducted. The values of the t-stats have been corrected for the presence of autocorrelation in the time-series. Panel A presents the statistics for the empirical risk premiums of F&F, while Panel B presents the statistics for the updated F&F premiums built along our cubic methodology. *,**,and *** stand for significant at 10%, 5%, and 1%. respectively. All the F&F premiums display positive average returns over the period, but only the HML ff and UMD ff premiums are significantly positive over the period (at the usual significance levels). The momentum strategy has the strongest returns, with an average value that is more than five times higher than the one displayed by the size premium and almost the double that of the one displayed by the HML ff strategy. The momentum premium is also more volatile. Regarding the cubic version of the premiums, not all premiums present a positive average return. The HML c premium displays a very small, insignificant negative average return over the total period. The importance of the SMB c premium becomes similar to the one of the momentum strategy. They report approximately the same (significant) positive average return over the period. The UMD c premium presents characteristics very similar to the corresponding F&F premium. 10 In order to analyse the impact of our modifications on the original F&F method and the differences in descriptive statistics between the F&F and the cubic risk premiums reported in Table 3, we examine the 9 return spreads that result from each of our 3-stage sequential sorting procedures

11 and the return spreads that result from the F&F construction. This analysis helps us to understand the differences resulting from applying the alternative methodologies. Table 4 reports descriptive statistics for the 3 sets of 9 return spreads related respectively to the SMB c, the HML c, and the UMD c factors. For each panel, the ordering sequence ends up with the dimension to be priced, as explained in the methodological section. We closely examine the correlations of these 3 sets of 9 portfolios with the SMB c, the HML c, and the UMD c factors. 11 Table 4: Descriptive statistics over the return spread portfolios forming each cubic risk factor Table 4 displays descriptive statistics for the 9 return spreads forming the SMB c, HML c, and UMD c factors. The correlations (in %) of each spread portfolio with the SMB c, HML c, and UMD c factors are reported. The last column reports the average and the standard deviation of the statistics for the different portfolios. The size (resp. book-to-market, resp. momentum) spread portfolios are formed by performing a 3-stage sequential sorting procedure on, successively, book-to-market, momentum and market capitalization (resp. market capitalisation, momentum, and finally book-to-market; resp. book-to-market, market capitalisation, and momentum). Each spread portfolio is defined from a difference between two portfolios defined by 3 letters describing the 3-stage sequential sorting procedure. L stands for a low scoring portfolio, M for a medium scoring portfolio, and H for a high scoring portfolio. S.D. = Standard Deviation. The row corresponding to the dimension sought after by the spread portfolios is grayed. Panel A of Table 4 shows that each of the 9 return spreads related to the SMB c factor values the premium related to the size effect equivalently. All portfolios offer comparable levels of mean returns and volatility. The coefficient of variation of the series of average returns across portfolios is quite low (i.e. CV=0.26/0.88 or 0.30). Besides, the portfolios seem to display strong correlation with the SMB c factor but weak correlations (inferior to 30%) with the HML c and UMD c factors. Panel B shows that the 9 differences are correlated on average at 54.77% with the 11 - Note that all correlations are significantly different from 1 at the usual significance levels. 11

12 HML c factor, but display only weak correlation (less than 30%) with other types of risk. The bookto-market risk premium is the highest in portfolios formed of stocks of low (resp. medium) market capitalisations and presenting low (resp. medium) levels of prior returns. The table shows very large variations within the series of mean returns across the different book-to-market spreads. As there appears to be no stable BTM effect in the cubic framework, this might indicate that the sort on BTM captures noisy returns that could not be related to a source of risk priced on the market. The BTM effect might thus be overvalued in the F&F framework. Panel C shows that the effect of momentum is decreasing with market capitalisation. The momentum spreads tend to be the highest in stocks presenting small or medium levels of market capitalisation. Table 5 repeats the same analysis on the 2 spread portfolios respectively forming the F&F HML ff and UMD ff factors and the 3 spread portfolios forming the F&F SMB ff factor. Table 5: Descriptive statistics over the return spread portfolios forming each F&F risk factor Table 5 displays descriptive statistics for the return spreads forming the SMB ff (Panel A), HML ff (Panel B), and UMD ff (Panel C). The correlation (in %) of each spread portfolio with the SMB ff, HML ff, and UMD ff factors are reported. The last line in each panel reports the average and the standard deviation of the statistics for the different portfolios.the size spread portfolios are formed from the return spreads between small and big caps for 3 levels of book-to-market. The book-tomarket (resp. momentum) spread portfolios are formed from the return spreads between high and low levels of book-to-market (resp. momentum) for two levels of market capitalisation. Each spread portfolio is defined from a difference between two portfolios formed at the intersection of a two-way sort of stocks on size and a three-way sort on book-tomarket or on momentum. S.D. = Standard Deviation. J-B = Jarque-Bera. The column corresponding to the dimension sought after by the spread portfolios is grayed. The size spreads forming the SMB ff factor displayed in Table 5 are all strongly correlated with the SMB ff factor but, contrary to the return spreads forming the SMB c factor, they also display substantial correlations with the HML ff factor (superior to 30% for all 3 portfolios). Besides, while our specification delivers portfolios which are quite homogeneous regarding the return spreads related to size, here the low book-to-market-sorted portfolios display an average size spread very different compared to the ones of the two other portfolios. The coefficient of variation even increases from 0.30 to 2.14 (i.e. CV=0.30/0.14). Similarly, the two book-to-market spreads forming the HML ff factor display strong correlation with the HML ff factor, but still present moderate levels of correlation with the SMB ff factor. The characteristics of the book-to-market return spread portfolios confirm evidence that the book-to-market effect is the highest in low size portfolio. Only the two return spread portfolios underlying the UMD ff factor appear to be almost only correlated with the UMD ff factor, even though their mean values are again very different. The coefficient of variation of the series of cross-sectional mean returns is evaluated at 0.80 while it stands as a moderate 0.55 when considering the cubic sorts. 12 Finally, the cubic construction method induces a large correlation of the post-formation spread portfolios with the related factor but at the same time isolates the effects of the other two sources of risk. The F&F factors do not seem to purely price the returns attached to the size and book-to-market effects respectively, but appear to be contaminated with correlated sources of

13 risks. Our analysis even suggests that the book-to-market effect does not capture any kind of systematic risk priced on the stock market. The comparison of Table 4 against Table 5 explains much of the differences in the risk properties between the cubic and the original F&F SMB, HML and UMD factors displayed in Table 3. Evidence shown in Table 5 reflects that the F&F empirical size factor is contaminated by a bookto-market effect, as indicated by the values taken by the cross-correlations between the size return spreads and the HML ff factor. It even results in a negative size return spread in the low book-to-market portfolio (where the reward associated with the book-to-market effect is in fact negative). The book-to-market effect does not systematically affect average returns across the 9 cubic spread portfolios constructed on size (Table 4, Panel A). Consequently, our specification gives a premium whose average return is superior to the F&F equivalent. Besides, as already mentioned, our size factor is formed from the return differential between portfolios of extremely small caps and portfolios of big stocks. By considering all the NYSE, NASDAQ and AMEX stocks, our breakpoints are tilted towards small caps compared to the F&F premium. This could also explain the larger average spread observed for this premium. Second, as to the HML factor, the decomposition of the cubic premium into 9 portfolios shows that the negative sign of the premium comes mostly from the return spread in stocks displaying big capitalisation and high levels of prior returns. When pricing the HML factor, the cubic specification associates highly negative return to this risk in a high size/high momentum-portfolio. This particular portfolio concentrates high performing companies that demonstrate persistence in performance over several periods. Since the HML factor somehow rewards a contrarian strategy, it is thus not surprising to find a negative reward for the book-to-market effect in this portfolio. Besides, by sorting stocks according to the NYSE only, the F&F method tends to over-represent small stocks by comparison with big stocks as both the size and BTM quantiles are higher in the F&F framework than it would be based on a median sort. The HML ff factor overstates the returns attached to the book-to-market effect in the small size portfolio due to its contamination through the size effect. In turn, by taking a simple average of these two portfolios, the F&F premium tends to overestimate the return spread attached to the book-to-market effect. Finally, the momentum constructed according to our cubic specification displays half the level of volatility compared to the F&F UMD ff factor. Substantial variation exists in returns related to the F&F momentum risk between small and big capitalisations. The returns are more stable across the 9 cubic difference portfolios, showing that the size effect has been eliminated. Correlation analysis Table 6 displays the correlation matrix of these two sets of premiums. Table 6: Correlations matrix of the empirical risk premiums (May 1980-April 2007) Table 6 reports the paired correlations (in %) among the cubic and among the F&F empirical risk premiums, as well as across these two sets of factors. Tests over the significance of the pair-wise correlations are performed: *,**,and *** stand for significant at 10%, 5%, and 1%. respectively. 13

14 The bottom-left corner displays the cross-correlations between the two sets of premiums. The SMB c and HML c factors are correlated at 67.16% and 68.25% respectively with their F&F counterparts. These levels indicate that, although the original and the modified size and value premiums are intended to price the same risk, approximately one third of their variation provides different information. The analyses conducted in Tables 4 and 5 have highlighted the potential reasons for this difference. The momentum premium displays a higher correlation with the UMD ff factor. Contrary to the SMB ff and HML ff factors, the French s momentum premium does not exactly follow the Fama and French (1993) methodology. The premium is rebalanced monthly rather than annually. It differs from our momentum premium only with regard to the breakpoints used for the rankings and the sequential sorting. The bottom-right corner presents the intra-correlations among the F&F premiums. The SMB ff and HML ff factors are highly negatively correlated over the period (-40.83%). The UMD ff premium also displays a negative correlation with the HML ff factor, but a positive correlation with the SMB ff factor. Such evidence contrasts with the top-left corner that presents the intra-correlations among the cubic premiums. The signs are consistent with the ones displayed by the F&F premiums but the levels of the correlations are considerably lower, which is consistent with our objective of designing uncorrelated premiums. The intra-correlations among the F&F premiums are all statistically significant, whereas the correlations among our alternative factors are only significant (but at an inferior level) between the SMB ff and HML ff factors Specification Tests The three types of specification tests proposed in this section obey a progressive logic. To begin with, we perform a basic efficiency test on the asset pricing model to ensure that both the F&F and the cubic specifications are worth being examined further. The second test is a relevance check on the individual betas of each specification. We study whether the risk factor exposures of portfolios that are supposed to borrow certain risk characteristics faithfully reflect these characteristics. The third test allows us to carry out a direct and rigorous comparison of the competing models. The procedure features a test of non-nested models on individual stocks. The outcome of this test delivers the proportions of stock return series for which there is a statistical dominance of one specification over the other Factor efficiency test We evaluate the specification error displayed by a cubic or an original 4-factor Carhart analysis on the set of 2x3 F&F portfolios. These portfolios are constructed on the basis of a two-way sort into size and a three-way sort into book-to-market. The time-series are downloadable on French s website. We consider the following multivariate linear regression and test the values taken by the alphas: Instead of testing N univariate t-statistics based on each equation, we use the Gibbons et al. (1989) test on the joint significance of the estimated values for α p across all N equations: H 0 : α p = 0 for p=1,,n (4) (3) 14 Following Gibbons et al. (1989), under the null hypothesis (H 0 ) that α p is equal to 0 for all N portfolios, the statistics follows a central F distribution with degrees of freedom N and (T-N-L), where is a vector of sample means for the L factors, is the sample variance-covariance matrix for, is a vector of the least squares estimates of the α p across the N equations Note that all correlations are statistically different from 1 at the usual significance levels.

15 We apply the Gibbons et al. methodology on the set of size and book-to-market-sorted portfolios using returns from May 1980 to April We consider the case where L = 4 (the market index, the SMB, the HML, and the UMD factor) and N = 6 for the 6 independent portfolios. The F statistic to test hypothesis (4) when using the set of F&F premiums is , so we cannot reject efficiency of the F&F model at the usual levels of significance. When using the cubic premiums, the F statistic is even reduced to Thus, both sets of premiums seem to efficiently explain stock returns, with a slight advantage to the cubic approach. In other words, the different changes performed on the original F&F methodology does not seem to affect the efficiency of the factors. We investigate the properties of the alternative methods further below. 4.2 Test on the magnitudes of betas We perform a cubic and an original 4-factor Carhart analysis and test the value of the loadings for different portfolios sorted on size and book-to-market. We expect the loadings on the size premium to be highly significant and high in magnitude for portfolios made of stocks of small capitalisation. Besides, we expect the exposures to the HML factor to be higher in portfolios with a high book-to-market level. In order to have more variability in the exposures, we consider the F&F set of 5x5 portfolios sorted into size and book-to-market. Table 7 displays the values of the loadings on the market, size, book-to-market and momentum premiums for the 2 models. Table 7: Cubic and F&F empirical 4-factor model: Magnitude of the betas for a set of 5x5 portfolios sorted on size and BTM 15

16 We estimate a cubic and a F&F 4-factor Carhart model on a set of 5x5 F&F portfolios sorted on size and book-to-market. Table 7 reports the estimates of the beta loadings for the different portfolios. T-tests over the significance of the beta coefficients are reported: *,**,and *** stand for significant at 10%, 5%, and 1%, respectively. Average betas and their standard deviations are also reported for the 5 size- and the 5 BTM- portfolios. For instance, the beta of the portfolio of Size 1 (resp. BTM 1) corresponds to the average beta of the 5 book-to-market- (resp. size-) portfolios with a size 1 (resp. BTM 1). The left part of the table displays the results for the cubic premiums. The portfolios display consistent loading on the size premium: presenting positive and significant exposure in small portfolios, medium level of exposure in mid portfolios and finally negative loadings for portfolios made of big stocks. While the premium is statistically significant for almost all portfolios, the factor seems to matter only for the first two sets of size-sorted portfolios, the exposures being considerably reduced as expected for medium size-sorted portfolios. In the F&F framework (right half of the table), the size premium, while consistently significant across all portfolios, is still highly priced in medium size-sorted portfolios of levels 3 or 4. The F&F size premium captures more than the risks embedded in small size stocks. For both sets of premiums, the HML premiums behave consistently with the book-to-market level of the related portfolio. The exposures to the HML premium increase within each size portfolio with the level of book-to-market. As expected, the book-to-market effect is less important in the cubic framework than in the F&F analysis. Consistently with the sort underlying the construction of the 5x5 portfolios, the exposure to momentum is close to 0 for most of the portfolios. The table also analyses the stability of the exposures to the SMB, HML and UMD factors within each of the size-portfolios for different levels of book-to-market and within each book-to-market portfolios for different levels of market capitalisation. The estimates of the beta loadings on the HML factor are less volatile in the cubic framework than in the F&F one. The volatility of the SMB loadings are however comparable in the cubic and the original F&F framework Non-nested models on individual stocks. This sub-section attempts to identify the potential superiority of one set of empirical premiums (either the F&F ones or our updated version of the premiums) over the other one. We follow the literature on model specification tests against non-nested alternatives (MacKinnon, 1983; Davidson and MacKinnon, 1981, 1984). Such tests have already been used in financial macroeconomic literature Bernanke et al. (1986) and Elyasiani and Nasseh (1994), among others, use non-nested models to compare some model specifications about investment and U.S. money demand, respectively. Elyasiani and Nasseh (2000) differentiate between the performance of the CAPM and of the consumption CAPM through non-nested econometric procedures. Al-Muraikhi and Moosa (2008) test the impact of the actions of traders who act on the basis of fundamental or of technical analysis on financial prices based on non-nested models.

17 We consider the following two models: 1. M1 or the F&F model: 2. M2 or the cubic model : (5) (6) where R i stands for the excess return on asset i, μ for the market premium, X for the F&F premiums, and Z for the cubic risk premiums. Two tests are jointly conducted. First, the model to be tested is M1, and the alternative model M2. To test the model specification, we set up a composite model within which both models are nested. The composite model (M3) writes: Under the null hypothesis θ i1 = 0, M3 reduces to M1; if θ i1 0, M1 is rejected. Tests are conducted on the value of θ i1. Davidson and MacKinnon (1981, 1984) prove that under H 0, can be replaced by its OLS estimate from M2 so that θ i1 and δ i (and α 3,i, β i ) are estimated jointly. This procedure is called the J-test. We define δ i * = (1 - θ i1 ) δ i so that M3 can be rewritten as follows: To test M2, we reverse the roles of the two models. We construct model M4: We replace δ i by its estimate along M1 ( ) and estimate γ* i (and α 4,i, β i ) jointly with θ i,2. If θ i,2, M4 reduces to M2; if θ i,2 0, M2 is rejected. Tests are conducted on the value of θ i,2. We evaluate the goodness-of-fit of the two alternative asset pricing models on 11,377 individual stocks. The following hypotheses are jointly tested on all the individual test assets: Hypothesis I: H 0 : θ i,1 = 0 against H 1 : θ i,1 # 0; Hypothesis II: H' 0 : θ i,2 = 0 against H' 1 : θ i,2 # 0 Each θ j follows a normal distribution with mean θ j and volatility σ j. Therefore, under the null hypothesis, the statistics follows a Student distribution with 315 degrees of freedom the number of observation in each time-series (i.e. 324) minus the number of factors in each regression (i.e. 9: the constant, the market portfolio, the 2 sets of 3 empirical premiums and the θ estimate). Among the four possible scenarios, we consider the following two cases: (H 0, H' 1 ), M1 is not rejected but M2 is; (H' 0, H 1 ), M2 is not rejected but M1 is. 14 Table 8 presents the results of the tests over the significance of θ 1 and θ 2 across the assets, for different confidence levels. We perform the following tests about the value of θ 1 and θ 2 : (7) (8) (9) (10) 14 - Note that the rejection of H 0 does not reveal anything about the validity of H 0. 17

18 Table 8: Tests over the value of θ 1 and θ 2 in the Nested Models M3 and M4 for individual assets Table 8 estimates the models M3 (testing the F&F model) and M4 (testing the cubic model) for the 11,377 individual assets (only 11,087 were available for the analysis). It jointly tests the significance of θ 1 and θ 2 using equation (10). The table reports, for different levels of significance, the number of assets (and the frequency) for which both models are accepted, rejected, or accepted while the other one rejected. The table displays, for different levels of significance, the frequency of non-rejections of the F&F model, i.e. H 0 (resp. of the cubic model, H 0 ), while rejecting the cubic premiums, i.e. H 1 (resp. of the F&F premiums, H 1 ). It also reports the frequency of assets for which M1 and M2 are both rejected or not. The first quarter of the table reflects the performance of the cubic model (Not reject M2 & Reject M1), while the second quarter identifies the frequency of dominance of the original F&F model. We report evidence that the cubic version is less frequently rejected and the F&F premiums are more often rejected for individual stocks than the opposite. The gap is largest at the 10% significance level, where the test leads to the non-rejection of the cubic premiums 6.54% more often than with F&F premiums. Overall, the non-nested econometric analysis shows that in most cases the F&F and the cubic models are both not rejected when compared to the augmented model. This result, which does not imply that any of the models provides a good fit (as it is not the scope of the test), could be expected from a database of individual stocks. For a limited subset of stocks (up to ca. one third), we can discriminate between these models. Our cubic premiums seem to outperform the F&F specification. The extent of this superiority is economically quite important, as the adoption of F&F factors instead of the cubic ones would (statistically) be a wrong choice for almost 4,000 individual stocks. 5. Concluding Remarks This paper proposes an alternative way to construct the empirical risk factors of Fama and French (1993). The original F&F method performs a 2x3 sort of US stocks on market capitalisation and on book-to-market and forms six two-dimensional portfolios at the intersections of the two independent rankings. The premiums are defined as the spread between the average low- and high-scoring portfolios. Our paper aims to address the drawbacks of this heuristic approach to constructing risk factors, and to tackle an important gap in literature: how to best construct fundamental risk factors. It has become standard practice to use the Fama and French method (F&F) to construct multiple size and BTM portfolios and to use them in the cross-sectional asset-pricing literature to evaluate models (Daniel and Titman, 1997; Ahn et al., 2009; Lewellen et al., 2009). But there are, to our knowledge, only very few articles that use such multiple portfolio sorts for pricing fundamental risk premiums. The usefulness of such an analysis is obvious for at least two reasons. First, a method that could be systematically applied enables us to apply the method to other exchange markets or to price other risk fundamentals. Second, by insulating the effects of other sources of risk as much as possible when evaluating one risk factor, each of them can be used independently of the others. This property is very useful for stepwise factor selection procedures, for instance in style analysis or hedge fund models. 18

19 With regard to these objectives, we raise three main issues in applying the F&F methodology. First, the annual rebalancing is consumptive in long time-series which sometimes simply do not exist for small exchange markets. Besides, it would also not easily capture the time-varying dimension of risk such as higher-moment exposures. Second, the independent sorting procedure underlying the formation of the 6 F&F two-dimensional portfolios causes moderate, but economically significant levels of correlation between premiums. Finally, the breakpoints as defined along the NYSE stocks produce an over-representation of small caps in portfolios. The main innovations of our premiums reside in a monthly rebalancing of the portfolios underlying the construction of the risk premiums, and in a conditional sorting of stocks into portfolios. We consider three risk dimensions. The conditional sorting procedure answers the question whether there is still return variation related to the third risk criterion after having controlled for two other risk dimensions. It consists in performing a sequential sort in three stages. The first two sorts are performed on control risks, while we end by the risk dimension to be priced. Compared to the F&F method, our factor construction method better captures the return spread associated with the source of risk to be priced as it is able to maximise the dispersion in the related source of risk while keeping minimal dispersion in correlated sources of risk. Put another way, without losing in significance power, in beta consistency or in factor efficiency, the cubic technique is neater and leads to risk premiums that may not necessarily be used jointly in a regression-based model, unlike the original F&F factors whose risk exposures are highly sensitive to the inclusion of the other F&F risk factors in the regression. In conclusion, as in Cremers et al. (2008) and in Huij and Verbeek (2009), we argue that the book-to- market premium of F&F is overvalued. Moreover, we argue that a sequential sorting procedure could be more appropriate to take the contamination effects between the premiums into consideration. We show that the premiums constructed in this way deliver more consistent risk properties while having at least the same specification as the F&F premiums. As shown is the very last analysis, there is still some risk to be captured in both sets of premiums. As there is evidence that empirical risk premiums could be proxies for higher-order risks (Barone- Adesi et al., 2004; Chung et al., 2006, Hung, 2007; Nguyen and Puri, 2009), future research must consider the incremental value of higher-order moment-related factors for benchmark models. References Agarwal, V., G. Bakshi and J. Huij Do Higher-Moment Equity Risks Explain Hedge Fund Returns? Unpublished working paper, Georgia State University, RSM Erasmus University, Smith School of Business. Ahn, D.H., J. Conrad and R. F. Dittmar Basis Assets. Review of Financial Studies 22 (12): Al-Muraikhi, H. and I. A. Moosa The Role of Technicians and Fundamentalists in Emerging Financial Markets: A case Study of Kuwait. International Research Journal of Finance and Economics 13: Barone-Adesi, G., P. Gagliardini and G. Urga Testing Asset Pricing Models with Coskewness. Journal of Business and Economic Statistics 22 (4): Bernanke, B., H. Bohn and P. C. Reiss Alternative Nonnested Specification Tests of Time- Series Investment Models. Unpublished working paper, NBER Research Paper: 890. Carhart, M. M On Persistence in Mutual Fund Performance. Journal of Finance 52 (1):

20 Chung, Y. P., H. Johnson and M. J. Schill Asset Pricing when Returns are Nonnormal: Fama- French Factors versus Higher-Order Systematic Comoments. Journal of Business 79 (2): Cremers, M., A. Petajusto and E. Zitzewitz Should Benchmark Indices Have Alpha? Revisiting Performance Evaluation. Unpublished working paper, Yale School of Management, AFA 2010 Atlanta Meetings Paper. Daniel, K. and S. Titman Evidence on the Characteristics of Cross Sectional Variation in Stock Returns. Journal of Finance 52 (1): Davidson, R. and J. G. MacKinnon Several Tests for Model Specification in the Presence of Alternative Hypotheses. Econometrica 49: Davidson, R. and J. G. MacKinnon Model Specification Tests based On Artificial Linear Regressions. International Economic Review 25: Elyasiani, E. and A. Nasseh The Appropriate Scale Variable in the U.S Money Demand: An Application of Nonnested Tests of Consumption versus Income Measures. Journal of Business and Economic Statistics 12 (1): Elyasiani, E. and A. Nasseh Nonnested Procedures In Econometric Tests Of Asset Pricing Theories. Journal of Financial Research 23 (1): Fama, E. F. and K. R. French Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics 33 (1): Gibbons, M. R., S. A. Ross and J. Shanken A Test of the Efficiency of a Given Portfolio. Econometrica 57 (5): Huij, J. J. and M. J. Verbeek On the Use of Multifactor Models to Evaluate Mutual Fund Performance. Financial Management 38 (1): Hung, D. C.H Momentum, Size and Value Factors versus Systematic Comoments in Stock Returns. Unpublished working paper, Durham University. Jegadeesh, N. and S. Titman Profitability of Momentum Strategies: An Evaluation of Alternative Explanations. Journal of Finance 56 (2): Kat, H. and J. Miffre Hedge Fund Performance: The Role of Non-Normality Risks and Conditional Asset Allocation. Unpublished working paper, Cass Business School, City University, London Faculty of Finance. Lewellen, J., S. Nagel and J. Shanken, A Skeptical Appraisal of Asset-Pricing Anomalies. Journal of Financial Economics 96 (2): MacKinnon, J. G Model Specification Tests against Non-nested Alternatives. Econometric Review 2: Mahajan, A. and S. Tartaroglu Equity Market-Timing and Capital Structure: International Evidence. Journal of Banking and Finance 32 (5): Moreno, D. and R. Rodriguez The Coskewness Factor: Implications for Performance Evaluation. Journal of Banking and Finance 33 (9): Nguyen, D. and T. N. Puri Higher Order Systematic Comoments and Asset Pricing: New Evidence. The Financial Review 44 (3):