Is that factor just lucky? Australian evidence

Pacific-Basin Finance Journal 57 (2019) 101191

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Is that factor just lucky? Australian evidence Khoa Hoang , Damien Cannavan, Clive Gaunt, Ronghong Huang ⁎

T

UQ Business School, The University of Queensland, St Lucia, Brisbane, QLD 4072, Australia

ARTICLE INFO

ABSTRACT

Keywords: Multiple hypothesis testing Factor models Data-mining Bootstrap

Conventional test statistics do not account for data mining, leading to the identification of asset pricing factors that are the product of luck rather than being true risk factors. We construct a unique set of ninety-one risk factors for the period from January 1992 to December 2017 and test which factors contribute in an economically and statistically significant manner towards improving the cross-sectional performance of asset pricing models. Our results show that the market factor is the only factor in the Australian market that consistently survives the stringent approach to factor selection that takes into account the data-mining issue. Our results hold across individual stocks and portfolios and are robust to block bootstrapping and exclusion of small stocks.

JEL classifications: C15 G11 G12 G17

1. Introduction There is a growing Australian asset pricing literature which devotes much of its attention to assessing the applicability of asset pricing models, constructed originally on US data, to Australian stock returns.1 The number of risk factors investigated with Australian data is very limited relative to US evidence, and we fill this gap by constructing a set of ninety-one risk factors identified in prior research and testing them in the Australian market. In doing so we directly account for multiple hypothesis testing and address the issue of data mining as recognised recently in the empirical asset pricing literature (Harvey, 2017). We adopt the method of Harvey and Liu (2018) to better address these issues and assess the reliability of risk factors in Australian equity returns from the ninety-one candidate factors. The more recent Australian asset pricing literature has focused much of its attention on tests of the Fama and French (1993) threefactor model2 with conflicting results, and given limited attention to other potential factors. Halliwell et al. (1999) finds the model to be marginally superior to the Capital Asset Pricing model (CAPM) but that the value (hml) factor lacks explanatory power. Aided by a significantly larger dataset, Gaunt (2004) finds the three-factor model offers significantly increased explanatory power to the CAPM but much of that comes from size (smb), in addition to the market factor, and not hml. Faff (2001) constructs proxies for the three factors from ‘off the shelf’ style index data and reports strong support for the three-factor model though a significant negative size premium is at odds with expectations. Faff (2004) also finds support for the model and a negative size premium when using daily

Corresponding author. E-mail address: [email protected] (K. Hoang). 1 While non-US market research is under-represented in the top finance journals (Karolyi, 2016), out-of-sample tests of research first conducted using US data remain an important means of validating that original research. 2 Prior to this a number of Australian studies had undertaken tests of CAPM and its variants including Ball et al. (1976), Faff (1991), Faff and Lau (1997), Faff and Chan (1998), Faff et al. (2000), Dean and Faff (2001), and Durack et al. (2004). Australian tests of the Arbitrage Pricing Theory (APT) were undertaken by Faff (1988) and Faff (1992). ⁎

https://doi.org/10.1016/j.pacfin.2019.101191 Received 20 May 2019; Received in revised form 26 July 2019; Accepted 9 August 2019 Available online 10 August 2019 0927-538X/ © 2019 Published by Elsevier B.V.

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returns. Brailsford et al. (2012b) suggest the inconsistencies with US findings may be due to differences between the structure of the Australian and US equity markets. After adopting a new portfolio construction approach (following Brailsford et al., 2012a), they find all three factors to be positive and significant in cross-sectional tests, and the three-factor model to be superior to the CAPM. Following the US literature, more recent Australian studies investigate the validity of new asset pricing models. For example, Chiah et al. (2016) test the Fama and French (2015) five-factor model. They find the two new factors, profitability (rmw) and investment (cma), to be priced and the new five-factor model to be superior to the three-factor model, to the Carhart (1997) fourfactor model and to several models which incorporate US factors. Huynh (2018) find the five-factor model to be superior to the threefactor model in terms of its ability to explain 16 anomalies, however both models repeatedly fail the Gibbons, Ross and Shanken (GRS, 1989) test. Other studies have also incorporated additional variables into the baseline Fama and French asset pricing models. The role of liquidity has drawn much attention. Chan and Faff (2003, 2005) find that liquidity proxied by turnover has a negative relation with returns and is an important priced factor. Limkriangkrai et al. (2008) find that adding a liquidity factor improves on the Australian three factor model's ability to explain the cross-section of returns. They also extend prior research on the integration of the Australian and US capital markets3 reporting that a liquidity factor when added to the US three factor model is able to explain returns on small stocks. Using individual stock returns, Limkriangkrai et al. (2009) find results consistent with Durand et al. (2006) and Limkriangkrai et al. (2008). Gharghori et al. (2007) create several four-factor models incorporating the three Fama and French factors along with either a default factor, leverage factor, momentum factor or liquidity factor. They find that none of these significantly improve the explanatory power of the basic Fama and French three-factor model. Chai et al. (2013) add a liquidity factor to the Carhart four factor model but finds that it adds only minor explanatory power. Durand et al. (2016) also add momentum and liquidity factors to the three-factor model and observe a similar result to Chai et al. (2013). Chai et al. (2019) focus on the more investable large cap portion of the Australian equity market and test ten alternative models including the one (CAPM), three, four and five-factor models, the qfactor model, standalone US factor models and US factor models with Australian orthogonalised factors. They find the domestic fivefactor model to be the best model followed by the domestic q-factor model. In summary, at this point, the Fama and French five-factor model appears to best explain the cross-section of returns though there is some contention around the integration of the Australian market with international markets.4 However, only a relatively small number of potential factors have been tested using Australian data. In contrast, the asset pricing literature based largely on US stock returns, has documented hundreds of factors and strategies that purport to explain returns (Harvey et al., 2016; Green et al., 2013; Hou et al., 2017; Harvey and Liu, 2018). Among others, Elliot et al. (2018), Chan and Docherty (2016), Gray (2014), and Doan et al. (2010) highlight that the Australian market may be more representative of equity markets around the world, where firms tend to be smaller and concentrated in fewer industry sectors relative to the US. Also, as shown in Jacobs and Müller (2019), Australia has the highest average factor long-short returns among 39 countries studied, and does not show significant reduction post the publication of these factors. This suggests that the Australian market may be a useful laboratory for testing these factors as the persistence of relatively high returns suggest that they are more likely to be priced risk factors (Cochrane, 1999). However, to this point, the Australian research has been limited to the core five factors plus a handful of additional factors. To fill this gap we have constructed a set of ninety-one risk factors identified from the prior literature as potential candidate factors for the Australian market. Critically, there are several issues in prior studies of asset pricing models. First, the model comparison is somewhat ad-hoc. The common practice is to compare the cross-sectional adjusted R2 or GRS test statistics of the competing models. However, the GRS statistic is not designed to measure the relative performance of competing models (Harvey and Liu, 2018).5 Additionally, the asymptotic approximation behind these test statistics means that they cannot be applied to a sample with a large cross-section dimension and only a limited time-series dimension. Finally, none of the statistics accounts for the data-mining issue that is prevalent in academic research. As an example of this, suppose an analyst has tried and back-tested one hundred factors, yet he only reports strategies that are deemed to be statistically significant as judged by the traditional t-ratio of 1.96 (p-value of 0.05). Here the traditional cut-off point is inappropriate because there are variables, by luck, that will produce t-statistics greater than 1.96 among these one hundred factors (see for example, Foster et al., 1997 and Harvey and Liu, 2018). Past data-mining is now becoming evident as recent work has begun to question the empirical validity of previously identified strategies. For example, McLean and Pontiff (2016) reveal a decline in the profitability of trading strategies after the publication of research documenting them; Linnainmaa and Roberts (2018) demonstrate widespread deterioration in return anomalies out-ofsample; and Hou et al. (2017) find the majority of anomalies are insignificant in their large-scale replication study of 447 anomalies. Chordia et al. (2018) construct over two million strategies in their study and find that the majority of strategies are statistically insignificant and those that are significant are economically meaningless. Therefore, it is important to employ a testing framework that accounts for the data-mining issue and to adjust the cut-off point to determine statistical significance (Harvey, 2017).

3

See Durand et al. (2001), Faff and Mittoo (2003) and Durand et al. (2006). Other works incorporate macroeconomic factors. For example, Di Iorio and Faff (2002) implement and test a two-factor asset pricing model which augments the market factor with an exchange rate factor. Nguyen et al. (2009) add a GDP growth factor to the three-factor model and find it not to be priced. Faff et al. (2014) reported a similar result using non-nested testing techniques. 5 GRS test has been widely used in the Australian asset pricing studies (see for example, Chai et al., 2013 and Chai et al., 2019). For the comparison of non-nested models, Faff et al. (2014) employ the “J” test (Elyasiani and Nasseh, 2000) and the “C” test (Chen, 1983). 4

2

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Harvey and Liu (2018) develop a novel testing framework to simultaneously address multiple testing, variable selection, and test dependence in the context of risk factor models, and therefore tackle the data-mining issue directly. Instead of focusing on the extreme case that 100% of the expected return is explained by a factor model, Harvey and Liu (2018) declares a factor useful as long as the factor significantly contributes to the explanation of cross-sectional returns. The framework starts with a baseline model with no pre-selected variables, and then evaluates the incremental contribution of each factor, sequentially selecting factors that offer the largest reduction in pricing errors. The factor is transformed to ensure that it does not explain cross-sectional asset returns under the null hypothesis.6 The distribution of test statistics, obtained from bootstrapped samples generated under the null, is employed to statistically assess the contribution of the additional factor. As a result, the method does not rely on asymptotic approximation and therefore can be applied to a large cross-sectional dimension with a small time-series sample size. More importantly, the bootstrap based resampling techniques is robust to small-samples and the non-normality of asset returns, and allows us to conduct multiple hypothesis testing and therefore directly account for the data-mining issue. Empirically, we assess the ninety-one risk factors using both individual stocks and a large set of portfolios as test assets in the Australian market. Regardless of which type of test asset is used, we find the market factor to be the single dominating risk factor, when accounting for multiple hypothesis testing. The magnitude of reduction in pricing errors offered by the market factor is strikingly higher than that for any other factor considered. Our main results remain robust to the use of block bootstrapping to allow for time-series dependence in asset returns, the exclusion of microcap stocks, and the grouping of risk factors. Our research is important in the context of the prior Australian research which has provided inconsistent findings on the significance of the hml and smb factors and limited evidence on a range of other factors. We contribute to the existing Australian and international asset pricing literature in several ways. First, we assess the most comprehensive set of factors to date in the Australian context with ninety-one candidate risk factors tested. We find that the market factor is the only true risk factor. Second, by using the Harvey and Liu (2018) approach, we account for multiple hypothesis testing and address the endemic concern of data-mining. This is the first Australian and first non-US research that has addressed this issue. Further, we test the largest number of factors to date using this method for any market globally (the previous highest was 14 factors, for the US market in Harvey and Liu, 2018). Our empirical results suggest that risk factors identified to date, other than the market factor, are not true risk factors in the Australian market. The rest of the paper proceeds as follows. In Section 2, we elaborate on the method adopted in our study, which is an application of the approach of Harvey and Liu (2018). Section 3 provides details relating to our data and the construction of factors. The results of our study are presented in Section 4 and in Section 5 we conduct a number of robustness tests including block bootstrapping to allow for time-series dependence in asset returns, the exclusion of microcap stocks, and the grouping of highly correlated factors. Section 6 offers a conclusion. 2. Methodology 2.1. Approach of Harvey and Liu (2018) In this section, we offer a brief overview of the Harvey and Liu (2018) testing approach for cross-sectional returns. Consider the following time-series regression model: K

R i, t

Rf , t = ai +

bi, j f j, t +

i, t ,

i = 1, 2, 3…,N

(1)

j =1

where Ri, t, Rf, t and fj, t are the asset i, risk-free asset and factor j returns for time t. If the model is the true pricing model, the cross-section of the intercepts should not be statistically different from zero. This essentially is the testing approach of the GRS test. However, the stringent conditions of the GRS test mean that it almost always rejects the null. As a result, when comparing two alternative models, researchers use the GRS test statistics as a heuristic model performance measure and consider the one with the lower GRS statistic as the better model. Harvey and Liu (2018) point out that the GRS test is not designed to assess the relative performance of two models. The GRS test essentially focuses on the extreme case that 100% of the expected return is explained by the factor model. In contrast, the Harvey and Liu (2018) approach declares a factor useful as long as the factor significantly contributes to the explanation of cross-sectional returns. The framework does not aim to identify the factor model that completely explains cross-sectional returns. The approach starts with a baseline model with no pre-selected variables, and then evaluates the incremental contribution of each factor and sequentially selects factors that offer the largest reduction in pricing errors. To statistically assess the contribution of an additional factor, the basic idea is to transform the factor to ensure that it does not explain cross-sectional asset returns under the null hypothesis and obtain the distribution of test statistics through bootstrapping. The actual performance of the factor is then compared 6

For the one-factor model, the transformation involves demeaning the factor by subtracting the in-sample mean from its time-series. As the expected value of the transformed factor is zero, it has no impact on the cross-section of expected asset returns. This is because the cross-section of intercepts from time-series regressions will be the same as the cross-section of average excess returns that the factor is supposed to explain. However, the transformed factor maintains all the time-series properties. For models with more than one factor, the factor is first projected onto the preselected factors through a time-series regression. It is then transformed by subtracting the regression intercept. A more detailed discussion is provided in Section 2.1. 3

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to this distribution to statistically assess whether it has any significant additional contribution towards reducing the cross-sectional pricing errors. This method does not rely on asymptotic approximations and therefore can be applied to a large cross-sectional dimension with a small time-series sample size. More importantly, the bootstrap based resampling technique is also robust to smallsample and non-normality of asset returns, and able to flexibly account for multiple hypothesis testing. Specifically, we start with a baseline model with no pre-selected factors and generate the test statistics as follows. First, we orthogonalise the factor returns to ensure that the orthogonalised factors are not correlated with the cross-sectional returns under the null. For the one-factor model, the “pseudo” factor f1⁎ is defined by subtracting the in-sample mean of f1 from its time-series. If we take the unconditional expectation of the one-factor model, we will have

E (Ri, t

(2)

Rf , t ) = ai + bi1 E (f1t ), i = 1, 2, 3…,N

By definition, f1 is uncorrelated with the cross-sectional expected returns, but does preserve all the time-series properties and explanatory power of the original time-series. Since E(f1t∗) equals zero, the cross-section of intercepts (ai) from time-series regressions will be the same as the cross-section of average excess returns that the factor is supposed to explain. Second, after factors are orthogonalised, we use a bootstrap process to obtain the distribution of the cross-sectional pricing error statistic (e.g. regression intercepts). Finally, we are able to assess the statistical significance of the additional factor by comparing the realised statistic to this bootstrapped distribution. When a factor survives the test, it is declared as significant and added to the baseline model. The contribution of additional factors is then assessed against this new baseline model. For a model with multiple factors, the “de-mean” process mentioned above is accomplished by projecting the candidate factor onto the pre-selected factors through a time-series regression. In a one-factor model, the demeaning is essentially projecting the factor onto a constant. The new pseudo factor is then defined as the factor minus the regression intercept, and the testing process is carried out as described above. The sequential process only stops when no additional factor can significantly contribute to explaining the cross-section of returns, assessed by the multiple hypothesis test statistics. We discuss the test statistics next. ⁎

2.2. Test statistics Harvey and Liu (2018) emphasize the importance of test statistics that are both economically sensible and statistically sound, and focus on the reduction in pricing errors. Their approach consists of panel regressions, where {aib}iN= 1 and {aig }iN= 1 represent the crosssection of regression intercepts for the baseline model and the augmented model when an additional factor is considered, respectively. Further, {sib}iN= 1 represents the cross-section of standard errors for regression intercepts under the baseline model. The first test statistic is the scaled intercept (SI) measure, which is defined as the scaled difference in the absolute regression intercepts between the baseline and the augmented models. The standard error from the baseline model is used to scale both the augmented and baseline model intercepts to ensure the test statistics are exactly zero when the null hypothesis is forced to hold exactly in-sample (Harvey and Liu, 2018). This first test statistic is the equal-weighted mean absolute SI, constructed as

SIm, ew

1 N

N i=1

|aig | sib

1 N

N i=1

|aib | sib

/

1 N

N i=1

|aib |

=

sib

1 N

N i=1

(|aig |

1 N

|aib|)/sib |aib |

N i=1 s b i

(3)

The SI measure is negative if the augmented model improves the baseline model. The significance of any improvement is tested against the bootstrapped empirical distribution generated under the null hypothesis that the additional variable has no incremental contribution in explaining the cross-sectional returns. To address the potential outlier issue, we also calculate a SI measure using the scaled median absolute intercept. This second test statistic is given by

SImed, ew

median

|aig | sib

N

median i=1

|aib | sib

N

/median i=1

|aib | sib

N

i=1

(4)

As argued by Harvey and Liu (2018), using the scaled intercept approach has several advantages, including providing a better method to evaluate the economic significance of the factor, taking heterogeneity in return volatilities into account, and being consistent with recommended approaches for obtaining pivotal statistics for bootstrap hypothesis testing (Hall and Wilson, 1991). To account for the multiple testing issue, we need to jointly consider all the possible factors for statistical testing. In particular, for each bootstrapped sample, we need to obtain the statistics for all the orthogonalised factors and record the minimum test statistic SImin across them. SIm, ew and SImed, ew should be compared to this minimum statistic SIminwhich represents the largest intercept reduction among all the possible candidates. By comparing the realised minimum statistic with the bootstrapped distribution of minimum statistic, we effectively control for multiple hypothesis testing. Given that none of these orthogonalised factors should be correlated with the cross-section of expected returns, the minimum statistic represents the largest possible intercept reduction that can be observed just by chance if one examines all the available factors. We use the multiple hypothesis test statistic SImin to decide when to stop the factor selection process. 4

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3. Data and sampling The data for this study are collected from several sources. The sample of Australian stocks includes 3320 individual stocks. We collect firm-level data from Compustat Global. Our equity data includes all available common stocks (TPCI = 0) on the Xpressfeed Global daily security file for the Australian market between January 1992 and December 2017. In total, we have 359,654 non-missing monthly return observations. We construct 91 risk factors7 with monthly frequency for the period. Other than as indicated, we construct our own factors following the literature. We have also included Betting-Against-Beta (bab [id# 57]) (Frazzini and Pedersen, 2014), HML Devil (hmldevil [id# 80]) (Asness and Frazzini, 2013) and Quality-Minus-Junk (qmj [id# 85]) (Asness et al., 2019) factors.8 We follow Brailsford et al. (2012a) and construct the smb [id# 65] and hml [id# 79]factors using the methodology and spirit of Fama and French (1993). We collect the Fama and French (2015) five factors from Professor Kennth French's website. The 91 risk factors are summarised in the Appendix Table A.1, including the acronyms, descriptions, definitions, and the associated references. Following Hou et al. (2015) and Feng et al. (2017), Table A.1 also classifies factors into six groups, Momentum (M), Value-versus-Growth (V), Investment (I), Profitability (P), Intangibles (IN), and Trading Frictions (T). The monthly average returns and t-statistics for each of the risk factors are also reported in Table A.1. Equally and value weighted common idiosyncratic risk (ewivol [id# 88] and vwivol [id# 89]) are expressed in annualised standard deviations, risk-free rate (rf [id# 91]) is the monthly average of risk-free rate and liquidity [id# 90] is the equally weighted average of the liquidity measure of individual stocks within the month, rather than average factor monthly returns. Among the remaining 87 factors, 26 of them have an average absolute monthly return greater than 0.5%, and four risk factors (bab [id# 57], mom12m [id# 42], mom6m [id# 43] and qmj [id# 85]) greater than 1%, with qmj achieving 2.83% per month. However, qmj is not statistically significant at the conventional level. Only 34 factors' returns are statistically different from zero at the 5% level. 4. Empirical results 4.1. Individual stocks as test assets with equal-weighted scaled intercepts In this section, we present empirical results for when individual stocks are used as test assets and equal-weighted mean and median scaled intercepts are used as test statistics. Generally, individual stocks are seen to be too noisy for asset pricing model testing (see for example, Black et al., 1972 and Fama and Macbeth, 1973). Moreover, the use of individual stocks is often prohibitive for traditional testing methods given the problematic inversion of the large variance-covariance matrix of the return residuals. However, as argued by Harvey and Liu (2018), individual stocks provide an unbiased test of risk factors, whereas tests based on characteristicssorted portfolios are biased towards those risk factors used to construct those portfolios. Further, Lo and Mackinlay (1990) show that data-snooping bias can be stronger in portfolio-based asset pricing tests. Ang et al. (2018) also argue that the larger dispersion in beta across individual stocks can potentially increase the power of the test. Taken together, the issue with testing with individual stocks lies in the high level of noise and lack of suitable statistical tests, rather than individual stocks themselves. The framework offered in this paper alleviates the concern over the noise in individual stocks by allowing the use of test statistics that account for stock volatilities and market values. The framework also overcomes other challenges in the use of individual stocks as test assets, including unbalanced panel, cross-sectional dependence, extreme observations and a large cross-section relative to timeseries. The simulation results in Harvey and Liu (2018) suggest that the use of individual stocks does not necessarily result in loss of test power. Our results for the individual stocks are displayed in Table 1. Panel A presents the results for the first step of the testing procedure which has no pre-selected factors in the baseline model. This step tests whether any of the 91 factors is individually significant in explaining the expected cross-section of returns. We rank the factors based on the reduction in the equally weighted mean/median scaled absolute intercept, from largest to smallest. For brevity, we have only presented the top 20 and bottom 10 factors.9 The results show that the market factor is by far the best candidate in explaining the cross-sectional returns. It reduces the mean scaled absolute intercept by 17% and this magnitude is comparable to that reported in Harvey and Liu (2018) for the US market (19.2%). The next best candidate is betting-against-beta factor (bab [id# 57]), which reduces the SIm, ew by 8.7%. Notably, even without evaluating the statistical significance, only 13 out of 91 factors reduce SIm, ew compared to the baseline model of no factors, which suggest the majority of factors discovered to date do not offer a reduction in the cross-sectional individual stock pricing errors. The statistical significance of SIm, ew is evaluated using the procedure described in Section 2. Specifically, we orthogonalise each of the factors so that they do not have any impact on the cross-sectional returns. We then bootstrap to obtain the empirical distributions of SIm, ew. The 5th percentile of this distribution is reported in Table 1. The actual SIm, ew is then compared to the distribution to obtain 7 We construct the same set of candidate risk factors as in Feng et al. (2017), but we are unable to construct 13 out of 99 risk factors due to data limitations. The 13 factors are unexpected quarterly earnings, bid-ask spread, dividend initiation, dividend omission, new equity issue, industry adjusted size, illiquidity, abnormal earnings announcement volume, earnings announcement return, HXZ investment, intermediary capital risk, intermediary investment, and convertible debt indicator. Additionally, we add equally and value weighted common idiosyncratic risk, return on assets, risk-free rate and 6-month momentum to our list of factors. 8 These factors are obtained from the AQR website at https://www.aqr.com/Insights/Datasets. AQR is a global investment management firm which manages $194 billion as of 30 June 2019. A number of AQR studies have been published in academic journals, including Asness and Frazzini (2013), Frazzini and Pedersen (2014) and Asness et al. (2019). 9 Full results are available from the authors upon request.

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Table 1 Equal-weighted scaled intercepts tests – individual stocks as test assets. This table presents test results using individual stocks as test assets. The sample covers 3320 individual stocks in the Australian market. The sample period is from January 1992 to December 2017. The test statistics are the equal-weighted mean (SIm,ew) and median (SImed,ew) scaled absolute intercepts, as discussed in Section 2.2. The 5th-percentile statistics and the p-values (both single and multiple test) are obtained from the bootstrapped distribution, as discussed in Section 2.1. The definitions of 91 risk factors are provided in Table A.1. The factors are sorted based on the reduction in the scaled mean absolute intercept. For brevity, only the top 20 and bottom 10 factors are presented. Panel A: Baseline = no factor Factor

Single test SIm,ew

Multiple test

Single test

Multiple test

5th-percentile

p-value

p-value

SImed,ew

5th-percentile

p-value

p-value

Panel A.1 Top 20 factors mrp −0.170 bab −0.087 pricedelay −0.039 cash −0.038 acc −0.038 absacc −0.030 hire −0.018 chinv −0.017 pchcapx_ia −0.013 pchsale_pchxsga −0.008 herf −0.007 hmldevil −0.005 nincr −0.001 rsup 0.000 pchsale_pchrect 0.000 pchcurrat 0.001 pchquick 0.001 liquidity 0.002 pchgm_pchsale 0.005 mom36m 0.007

−0.104 −0.057 −0.040 −0.071 −0.032 −0.042 −0.036 −0.009 −0.024 −0.025 −0.014 −0.013 −0.015 −0.018 −0.009 −0.012 −0.013 −0.015 −0.031 −0.021

0.002 0.000 0.057 0.166 0.030 0.113 0.273 0.012 0.139 0.181 0.182 0.225 0.461 0.493 0.460 0.526 0.507 0.377 0.792 0.809

0.005

−0.183 −0.138 −0.056 −0.041 −0.058 −0.052 −0.024 −0.039 −0.013 0.005 −0.015 −0.011 0.005 0.002 0.006 0.005 0.009 −0.004 −0.000 0.005

−0.111 −0.071 −0.056 −0.091 −0.052 −0.063 −0.056 −0.017 −0.037 −0.033 −0.025 −0.021 −0.025 −0.027 −0.020 −0.020 −0.023 −0.026 −0.049 −0.033

0.005 0.000 0.051 0.205 0.030 0.079 0.342 0.003 0.208 0.572 0.161 0.160 0.707 0.544 0.670 0.640 0.723 0.294 0.676 0.713

0.025

Panel A.2 Bottom 10 factors salerec 0.185 age 0.192 invest 0.201 lgr 0.228 cma 0.258 egr 0.262 chcsho 0.372 ewivol 1.235 vwivol 1.995 rf 2.935

−0.017 −0.075 −0.026 −0.024 −0.049 −0.051 −0.064 0.422 0.060 0.241

1.000 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.253 0.250 0.258 0.304 0.334 0.312 0.489 1.146 2.056 2.842

−0.025 −0.092 −0.031 −0.033 −0.064 −0.067 −0.081 0.237 0.039 0.086

0.999 0.998 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Panel B: Baseline = mrp Factor

Single test SIm,ew

Panel B.1 Top 20 factors acc −0.009 nincr −0.006 pricedelay −0.005 hire −0.005 absacc −0.004 hmldevil −0.004 herf −0.001 chinv −0.001 rsup 0.000 pchcapx_ia 0.000 pchsale_pchrect 0.000 pchgm_pchsale 0.003 pchcurrat 0.004 pchquick 0.005 dolvol 0.006 mom36m 0.006 pchsale_pchxsga 0.006

Multiple test

Single test

Multiple test

5th-percentile

p-value

p-value

SImed,ew

5th-percentile

p-value

p-value

−0.012 −0.010 −0.011 −0.013 −0.015 −0.009 −0.009 −0.005 −0.013 −0.013 −0.008 −0.013 −0.005 −0.005 −0.014 −0.009 −0.009

0.101 0.118 0.147 0.237 0.279 0.212 0.333 0.217 0.481 0.413 0.407 0.508 0.599 0.653 0.704 0.737 0.471

0.925

−0.028 −0.009 −0.004 0.001 −0.000 0.013 0.015 −0.009 0.007 0.011 −0.002 −0.006 0.018 0.014 0.034 0.009 0.014

−0.028 −0.020 −0.024 −0.026 −0.030 −0.020 −0.019 −0.015 −0.025 −0.025 −0.018 −0.028 −0.015 −0.016 −0.022 −0.022 −0.020

0.051 0.206 0.361 0.659 0.550 0.801 0.890 0.124 0.712 0.637 0.384 0.319 0.901 0.813 0.976 0.728 0.741

0.696

(continued on next page) 6

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Table 1 (continued) Panel B: Baseline = mrp Factor

Single test

Multiple test

Single test

Multiple test

p-value

SImed,ew

5th-percentile

p-value

SIm,ew

5th-percentile

p-value

0.008 0.009 0.010

−0.007 −0.021 −0.005

0.871 0.984 0.876

0.028 0.009 0.018

−0.017 −0.034 −0.015

0.979 0.874 0.878

Panel B.2 Bottom 10 factors invest 0.117 age 0.134 grltnoa 0.142 lgr 0.143 cma 0.149 egr 0.162 chcsho 0.267 vwivol 1.487 ewivol 1.633 rf 2.972

−0.005 −0.031 −0.009 −0.010 −0.014 −0.014 −0.023 0.077 0.592 0.289

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.185 0.214 0.255 0.208 0.227 0.215 0.395 1.365 1.625 2.621

−0.013 −0.044 −0.018 −0.018 −0.025 −0.026 −0.035 0.034 0.353 0.131

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

std_dolvol bab pchsale_pchinvt

p-value

the p-value. As shown in Panel A of Table 1, only market risk premium (mrp), betting against beta (bab [id# 57]), working capital accruals (acc [id# 1]) and change in inventory (chinv [id# 3]) achieve a reduction in SIm, ew at the 5% level or better. Notably, the best factor to be selected may not be the one with the lowest single test p-value. Panel A of Table 1 provides a good example of this. The market factor is selected despite bab [id# 57] having a lower p-value. This occurs because the test statistic selects the factor that has the lowest SIm, ew, being the highest percentage reduction in the mean scaled intercept. This approach is preferred as it gives weight to both economic and statistical significance (Harvey and Liu, 2018). To account for the multiple hypothesis testing issue, we bootstrap the empirical distribution of the minimum statistic SImin as discussed in Section 2.2. That is, for each bootstrapped sample, we record the minimum test statistics for all 91 orthogonalised factors. This minimum test statistic shows the maximum reduction in SIm, ew that can be achieved if one has access to all 91 factors. As the 91 factors are orthogonalised and have no impact on the cross-sectional returns, the largest reduction in SIm, ew is pure luck. By comparing the actual minimum SIm, ew to the distribution of this simulated minimum statistic SImin, multiple hypothesis testing is controlled for. As shown in Panel A of Table 1, the market factor remains statistically significant (with a p-value of 0.005) after taking into account the multiple hypothesis testing issue. After finding the market factor significant, the procedure is repeated, with the market factor included in the updated baseline model. Panel B displays the results of this second step. The best reduction in SIm, ew is achieved by acc [id# 1], with a relatively small economic magnitude, a 0.9% reduction in the pricing errors. Furthermore, acc [id# 1] is not statistically significant at the conventional 10% level for a single test, and even less significant under the multiple hypothesis testing. As no factor is significant in the second step, the procedure is terminated and concluded with a one-factor model. That is, the market factor is the only factor priced in the cross-section of individual stock returns. These results and conclusions are the same regardless of whether we rely on the scaled mean or scaled median test statistics. Our one (market) factor model using equally weighted test statistics is consistent with Harvey and Liu (2018) where the market factor is selected in the first step offering the largest reduction in pricing errors. However, they find both Fama and French (1993) variables (smb [id# 65] and hml [id# 79]) to be significant after the second and third steps respectively, although the additional contribution to the reduction in SIm, ew is relatively small (4.1% and 1.7% respectively) when compared to the market factor (19.2%). 4.2. Individual stocks as test assets with value-weighted scaled intercepts We next turn our attention to the results obtained by value weighting the scaled absolute intercepts. The statistic makes economic sense as, for example with two stocks generating the same regression intercept, the mispricing of the factor model should be more significant economically for the stock with higher market value. The use of a value-weighted scheme also reduces the impact of the extreme returns sometimes associated with smaller-sized stocks. Specifically, we modify the equal-weighted test statistic by applying N market value weighting. {mei,t}Tt = 1 represents the time-series market value for stock i, and MEt = i = 1 mei, t is the aggregate market value of all stocks at time t. The test statistics is then adjusted to reflect the market value weighting, and is as follows:

SIm, vw

T t=1

N mei, t i = 1 MEt

×

|aig | sib

T t=1

/T N mei, t i = 1 MEt

T t=1

×

|aib | sib

N mei, t i = 1 MEt

×

|aib | sib

/T

/T

(5)

Table 2 presents the results using the value-weighted test statistic. As shown in Panel A, the results for the value-weighted case are qualitatively the same as for the equal-weighted case for the base line model without a pre-selected variable. The market factor results in the highest reduction in SIm, vw. Having the market factor reduces the SIm, vw by 34.5%, which more than doubles the reduction 7

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Table 2 Value-weighted scaled intercepts tests – individual stocks as test assets. This table presents test results using individual stocks as test assets. The sample covers 3320 individual stocks in the Australian market. The sample period is from January 1992 to December 2017. The test statistic is the value-weighted mean (SIm,vw) scaled absolute intercept, as discussed in Section 4.2. The 5th-percentile statistics and the p-values (both single and multiple test) are obtained from the bootstrapped distribution, as discussed in Section 2.1. The definitions of 91 risk factors are provided in Table A.1. The factors are sorted based on the reduction in the scaled mean absolute intercept. For brevity, only the top 20 and bottom 10 factors are presented. Panel A: Baseline = no factor Factor

Single test

Multiple test

SIm,vw

5th-percentile

p-value

p-value

Panel A.1 Top 20 factors mrp bab pricedelay cash cashpr acc hmldevil absacc dolvol pchsale_pchxsga cfp_ia pchcurrat roa pchquick chinv hire tb cashdebt ms rsup

−0.345 −0.060 −0.037 −0.033 −0.031 −0.030 −0.029 −0.024 −0.010 −0.009 −0.008 −0.007 −0.006 −0.005 −0.004 −0.002 −0.002 −0.001 −0.001 −0.001

−0.207 −0.105 −0.072 −0.084 −0.018 −0.043 −0.046 −0.056 −0.044 −0.030 −0.033 −0.035 −0.040 −0.035 −0.014 −0.026 −0.018 −0.032 −0.020 −0.025

0.003 0.163 0.186 0.203 0.016 0.111 0.111 0.231 0.251 0.194 0.279 0.301 0.324 0.350 0.209 0.452 0.284 0.385 0.336 0.417

0.003

Panel A.2 Bottom 10 factors turn sgr invest mom6m cma chcsho egr ewivol rf vwivol

0.151 0.156 0.183 0.187 0.204 0.219 0.258 0.340 0.670 1.181

−0.104 −0.080 −0.072 −0.055 −0.063 −0.068 −0.075 −0.004 −0.017 −0.044

0.992 0.992 0.998 1.000 0.998 1.000 1.000 1.000 1.000 1.000

Panel B: Baseline = mrp Factor

Panel B.1 Top 20 factors lev sp hml cfp cashpr dy quick maxret idiovol age retvol rmw salecash chcsho roeq ep std_turn

Single test

Multiple test

SIm,vw

5th-percentile

p-value

p-value

−0.097 −0.095 −0.093 −0.090 −0.081 −0.078 −0.077 −0.073 −0.073 −0.072 −0.070 −0.069 −0.065 −0.059 −0.059 −0.056 −0.052

−0.051 −0.054 −0.047 −0.058 −0.037 −0.059 −0.047 −0.052 −0.053 −0.075 −0.048 −0.055 −0.053 −0.027 −0.046 −0.058 −0.044

0.004 0.002 0.003 0.008 0.001 0.019 0.005 0.009 0.012 0.056 0.010 0.025 0.030 0.000 0.024 0.052 0.025

0.130

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Table 2 (continued) Panel B: Baseline = mrp Factor

Single test

Multiple test

SIm,vw

5th-percentile

p-value

beta currat cma

−0.051 −0.049 −0.047

−0.047 −0.036 −0.032

0.043 0.023 0.022

Panel B.2 Bottom 10 factors mom1m depr chinv invest indmom12m mom12m mom6m vwivol ewivol rf

0.030 0.030 0.032 0.060 0.090 0.138 0.153 0.263 0.450 0.484

−0.022 −0.026 −0.028 −0.032 −0.040 −0.042 −0.044 −0.030 0.016 −0.024

0.998 0.980 0.964 0.999 1.000 1.000 1.000 1.000 1.000 1.000

p-value

compared to the use of equal weighting (17.0%). The results suggest that the market factor plays a more important role in explaining the cross-sectional returns of large stocks. This reduction is also highly statistically significant, with a p-value of 0.003 under both the single and multiple tests. Panel B presents the results with the market factor included in the baseline model. Compared to the second step results for the equal-weighted statistics, more factors are shown to reduce the value-weighted scaled absolute intercepts with greater magnitude. The result suggests that many of the factors can better explain the expected returns of large stocks than small stocks. Among the remaining factors, lev (id# 81) provides the largest further reduction in SIm, vw by 9.7%, which is significant under the single test. However, once multiple hypothesis testing is taken into account, lev is no longer significant, with a p-value of 0.130. As a result, we conclude that none of the remaining factors is significant when the market factor is included in the model, and our results support a one-factor (market) model. This conclusion is consistent with that for the equal-weighted results. Comparing our results for the value-weighted case with those of Harvey and Liu (2018) for the US market, we note that their study points to a two-factor model. Harvey and Liu (2018) include the market factor after the first iteration, the quality-minus-junk (qmj [id# 84]) factor of Asness et al. (2019) after the second iteration, then no further inclusions. Notably, the further reduction in SIm, vw from qmj [id# 85] (14.9%) is much smaller than that offered by the market factor (44.4%), which suggests that the market factor plays a dominant role in explaining cross-sectional expected returns. 4.3. Portfolios as test assets While the use of individual stocks as test assets is preferred, as argued by Harvey and Liu (2018), asset pricing implications can differ significantly between the use of individual stocks and portfolios as test assets (Avramov and Chordia, 2006). Further, characteristic-sorted portfolios have more stable betas, higher signal-to-noise ratios, and are less prone to missing data issues, despite the existence of a bias-variance trade-off between the choice of portfolios and individual assets (Feng et al., 2017). Consequently, in this section, we also report test results using portfolios as test assets. As discussed previously, selecting too few portfolios based on sorts of a handful of characteristics is likely to tilt the results in favor of these factors (Harvey and Liu, 2018). There might also be a loss in efficiency in using too few portfolios (Litzenberger and Ramaswamy, 1979). The ideal portfolios should also be constructed in a way that is likely to reduce individual noises while preserving the factor structure at the same time. Following Feng et al. (2017) and Lewellen et al. (2010), we base our analysis on a large cross section of characteristic-sorted portfolios, which helps strike a balance between having many individual stocks or a handful of portfolios. We use a total of 488 portfolios as test assets from the 3320 individual stocks. The same stock can appear multiple times in different portfolios. For each continuous factor, bivariate-sorted 3 × 2 portfolios are constructed by intersecting its three groups with those formed on size (market equity). There are 80 factors that are continuous at the firm level, and as a result, we have 480 portfolios.10 In addition, sin stocks (sin [id# 64]) and R&D increase (rd [id# 12]) are dummy variables, and we construct 2 × 2 bivariate-sorted portfolios for each of them. In total, there are 488 portfolios available as test assets. Across these 488 portfolios, the mean (median) number of constituents is 115 (69). For firm characteristics we rebalance portfolios annually except for momentum and trading volume, where we rebalance portfolios monthly. 10 The excluded 9 factors are smb (id# 65), rf, mrp, vwivol (id# 89), ewivol (id# 88), liquidity (id# 90), qmj (id# 85), bab (id# 57) and hmldevil (id# 80). The last three factors are directly from the AQR database, and as a result, we do not construct our own version.

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Table 3 Equal-weighted scaled intercepts tests – portfolios as test assets. This table presents test results using portfolios as test assets. The sample covers 488 portfolios in the Australian market. The construction of these portfolios is discussed in Section 4.3. The sample period is from January 1992 to December 2017. The test statistics are the equal-weighted mean (SIm,ew) and median (SImed,ew) scaled absolute intercepts, as discussed in Section 2.2. The 5th-percentile statistics and the p-values (both single and multiple test) are obtained from the bootstrapped distribution, as discussed in Section 2.1. The definitions of 91 risk factors are provided in Table A.1. The factors are sorted based on the reduction in the scaled mean absolute intercept. For brevity, only the top 20 and bottom 10 factors are presented. Panel A: Baseline = no factor Factor

Single test SIm,ew

Multiple test

Single test

Multiple test

5th-percentile

p-value

p-value

SImed,ew

5th-percentile

p-value

p-value

Panel A.1 Top 20 factors mrp −0.547 cash −0.113 bab −0.113 acc −0.098 pricedelay −0.087 absacc −0.072 chinv −0.051 pchcapx_ia −0.048 pchsale_pchxsga −0.044 hmldevil −0.036 ms −0.015 cfp_ia −0.014 cashdebt −0.011 liquidity −0.009 roa −0.007 qmj −0.006 pchquick −0.005 herf −0.005 pchcurrat −0.005 hire −0.002

−0.328 −0.246 −0.166 −0.105 −0.153 −0.151 −0.041 −0.082 −0.087 −0.063 −0.046 −0.047 −0.109 −0.062 −0.132 0.001 −0.058 −0.028 −0.044 −0.059

0.000 0.228 0.121 0.062 0.141 0.223 0.033 0.124 0.17 0.127 0.196 0.229 0.406 0.341 0.414 0.008 0.383 0.297 0.364 0.572

0.001

−0.599 −0.105 −0.096 −0.112 −0.078 −0.076 −0.043 −0.045 −0.043 −0.029 −0.009 −0.034 −0.004 −0.009 −0.003 −0.002 −0.006 −0.009 −0.010 −0.002

−0.338 −0.259 −0.181 −0.114 −0.155 −0.167 −0.044 −0.082 −0.093 −0.068 −0.048 −0.055 −0.119 −0.068 −0.145 −0.005 −0.066 −0.030 −0.052 −0.062

0.000 0.244 0.163 0.057 0.168 0.223 0.054 0.146 0.182 0.165 0.257 0.119 0.467 0.322 0.439 0.078 0.375 0.23 0.29 0.553

0.001

Panel A.2 Bottom 10 factors zerotrade 0.390 grcapx 0.419 age 0.464 lgr 0.488 invest 0.515 cma 0.660 egr 0.673 chcsho 0.673 rf 1.124 vwivol 2.487

−0.221 −0.186 −0.287 −0.125 −0.164 −0.171 −0.173 −0.179 −0.067 −0.081

0.975 0.988 0.953 0.998 0.993 0.998 1.000 0.999 1.000 1.000

0.376 0.428 0.428 0.488 0.546 0.673 0.683 0.645 1.011 2.397

−0.239 −0.204 −0.309 −0.136 −0.174 −0.179 −0.188 −0.180 −0.066 −0.087

0.954 0.975 0.915 0.992 0.990 0.994 0.997 0.998 1.000 1.000

Panel B: Baseline = mrp Factor

Single test SIm,ew

Panel B.1 Top 20 factors hmldevil −0.064 acc −0.056 cashpr −0.037 pctacc −0.030 bab −0.025 chtx −0.019 pchcapx_ia −0.017 cfp_ia −0.012 pchsale_pchxsga −0.009 absacc −0.007 pchsale_pchinvt −0.006 nincr −0.006 pricedelay −0.005 liquidity −0.003 rsup 0.000 pchcurrat 0.000

Multiple test

Single test

Multiple test

5th-percentile

p-value

p-value

SImed,ew

5th-percentile

p-value

p-value

−0.078 −0.055 −0.037 −0.031 −0.025 −0.038 −0.050 −0.045 −0.054 −0.057 −0.018 −0.043 −0.050 −0.039 −0.054 −0.022

0.071 0.048 0.051 0.058 0.051 0.141 0.209 0.244 0.325 0.383 0.177 0.292 0.411 0.382 0.562 0.526

0.773

−0.089 −0.074 −0.077 −0.023 −0.045 −0.009 −0.017 −0.016 0.003 −0.015 −0.009 −0.016 −0.008 −0.005 0.000 −0.001

−0.096 −0.074 −0.052 −0.043 −0.039 −0.041 −0.070 −0.065 −0.076 −0.082 −0.032 −0.072 −0.068 −0.050 −0.073 −0.034

0.056 0.050 0.017 0.143 0.041 0.276 0.274 0.241 0.562 0.354 0.230 0.281 0.405 0.380 0.542 0.516

0.766

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Table 3 (continued) Panel B: Baseline = mrp Factor

Single test

Multiple test

Single test

Multiple test

p-value

SImed,ew

5th-percentile

p-value

SIm,ew

5th-percentile

p-value

0.001 0.001 0.001 0.001

−0.018 −0.027 −0.016 −0.022

0.553 0.647 0.447 0.620

0.014 −0.005 0.010 0.003

−0.031 −0.038 −0.028 −0.039

0.850 0.405 0.698 0.589

Panel B.2 Bottom 10 factors idiovol 0.476 lgr 0.483 invest 0.530 rf 0.550 beta 0.563 chcsho 0.567 egr 0.587 cma 0.624 ewivol 0.634 vwivol 0.673

−0.177 −0.087 −0.113 −0.013 −0.187 −0.078 −0.092 −0.095 0.002 −0.014

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.591 0.650 0.701 0.517 0.725 0.665 0.713 0.799 0.725 0.420

−0.218 −0.116 −0.153 −0.025 −0.239 −0.105 −0.126 −0.130 −0.004 −0.029

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

pchquick hire pchsaleinv herf

p-value

Table 3 presents the results using portfolios as test assets. As can be seen in Panel A, the factor that has the lowest SIm, ew (i.e. the highest percentage reduction in the equal-weighted mean scaled intercept) when testing against the baseline model containing no pre-selected variables is the market factor, consistent with our findings using individual stocks. The market factor reduces the mean scaled absolute intercept by 54.7%, which is much higher than any other factor. Statistically, the market factor is significant both under single and multiple hypothesis testing, with a p-value less than 0.001 in each case. Compared to the results using individual stocks as test assets, there are more factors that can reduce SIm, ew with twenty factors that have negative SIm, ew and ten of them lower than −3.5%. The results are not surprising given the reduced noise at the portfolio level. The test results are similar using SImed, ew as the test statistic. Panel B reports the results for the identification of a second risk factor when the market factor is declared as significant in the first step. The results show that hmldevil (id# 80) offers the largest reduction in the mean/median scaled absolute intercepts. However, the magnitude of further pricing error reduction is greatly reduced when compared to the market factor in the first step, with SIm, ew of −6.4% and SImed, ew of −8.9% for hmldevil. While the reduction is statistically significant with a p-value of 0.071 (0.056) under the single test for SIm, ew (SImed, ew), it is not statistically significant (with a p-value of 0.77 for both) when multiple hypothesis testing is accounted for. As a result, we terminate the process and conclude that the market factor is the only pricing factor. This result is consistent with the conclusion from using individual stocks as test assets. Our results vary from those of Harvey and Liu (2018) in the US market, where they find the investment factor (cma id# 6) of Fama and French (2015) to be significant after their second step, although no further variables are subsequently included. 5. Robustness tests 5.1. Block bootstrapping When bootstrapping to obtain the empirical distribution of the test statistics, we have treated each time period as independent and sample with replacement. To allow for time dependence for both stock returns and factor returns, we re-run our tests using block bootstrapping. We use block sizes of 3, 6 and 12 months and as the results are qualitatively similar, we only report the results using a block size of 6. Panel A and B of Table 4 present the block bootstrap results for the equal-weighted and value-weighted scaled intercepts when individual stocks are used as test assets. The conclusion remains the same as for the independent bootstrap results shown earlier. In each case, the market factor is selected first for inclusion into the updated baseline model, and no further selections are required, after accounting for multiple hypothesis testing. Finally, in Panel C of Table 4, we show our block bootstrap results when portfolios are used as test assets. The results conform to our conclusion when independent bootstrapping is used. That is, the market factor is the only significant factor in explaining the cross-section of returns. 5.2. Microcaps removed When using individual stocks as test assets, the use of SImed, ew and SIm, vw as test statistics alleviates the concern over the effects of illiquidity, potential bias and extreme observations that are likely to be caused by small stocks. In this section, to allow for the

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Table 4 Scaled intercepts tests with block bootstrapping. This table presents test results using block bootstrapping of 6 months. Panel A and B use individual stocks as testing assets. The sample covers 3320 individual stocks in the Australian market. Panel C uses 488 portfolios as test assets. The sample period is from January 1992 to December 2017. The test statistics are the equal-weighted mean (SIm,ew, Panel A and C), value-weighted mean (SIm,vw, Panel B), and median (SImed,ew, Panel A and C) scaled absolute intercepts, as discussed in Section 2.2 and 4.2. The 5th-percentile statistics and the p-values (both single and multiple test) are obtained from the 6-month block bootstrapped distribution. The definitions of 91 risk factors are provided in Table A.1. For brevity, only the factor that offers the largest reduction in the scaled mean absolute intercept in each step is reported. Factor

Single test SIm,ew/vw

Multiple test

Single test

Multiple test

5th-percentile

p-value

p-value

SImed,ew

5th-percentile

p-value

p-value

Step 1: Baseline = no factor mrp −0.170

−0.125

0.012

0.029

−0.183

−0.139

0.010

0.057

Step 2: Baseline = mrp acc −0.009

−0.012

0.086

0.914

−0.028

−0.027

0.044

0.712

Step 1: Baseline = no factor mrp −0.345

−0.225

0.001

0.004

Step 2: Baseline = mrp lev −0.097

−0.069

0.014

0.225

Step 1: Baseline = no factor mrp −0.547

−0.321

0.000

0.004

−0.599

−0.332

0.000

0.003

Step 2: Baseline = mrp hmldevil −0.064

−0.079

0.090

0.785

−0.089

−0.099

0.068

0.787

Panel A: Individual stocks EW

Panel B: Individual stocks VW

Panel C: Portfolio

Table 5 Scaled intercepts tests with microcaps removed. This table presents test results using individual stocks as testing assets. The sample covers 3320 individual stocks in the Australian market. The sample period is from January 1992 to December 2017. The test statistics in Panel A are the equal-weighted mean (SIm,ew) and median (SImed,ew) scaled absolute intercepts, while the test statistic in Panel B is the value-weighted mean (SIm,vw) scaled absolute intercept. Details of the test statistics are discussed in Section 2.2 and 4.2. The 5th-percentile statistics and the p-values (both single and multiple test) are obtained from the bootstrapped distribution, as discussed in Section 2.1. The definitions of 91 risk factors are provided in Table A.1. For brevity, only the factor that offers the largest reduction in the scaled mean absolute intercept in each step is reported. Factor

Single test SIm,ew

5th-percentile

Multiple test

Single test

Multiple test

p-value

p-value

SImed,ew

5th-percentile

p-value

p-value

Panel A: Individual stocks equal-weighted Step 1: Baseline = no factor mrp −0.209

−0.133

0.002

0.003

−0.215

−0.151

0.009

0.017

Step 2: Baseline = mrp bab −0.025

−0.031

0.108

0.720

−0.012

−0.049

0.446

0.825

Panel B: Individual stocks value-weighted Step 1: Baseline = no factor mrp −0.349

−0.217

0.002

0.002

Step 2: Baseline = mrp lev −0.099

−0.053

0.002

0.155

potential bias induced by very small stocks as discussed in prior studies such as Gray (2014) and Elliot et al. (2018), we follow Brailsford et al. (2012a) and remove the bottom quintile of stocks by market capitalisation from the test assets.11 Panel A of Table 5 presents results for the equal-weighted approach. The overall results are consistent with the preceding tests in which the market factor is found to be the only significant pricing factor. Notably, when these small stocks are removed and hence

11 In a similar vein, Chai et al. (2019) focus on explaining the returns on large Australian stocks noting evidence that asset pricing models perform differently between large and small stocks, and that small stocks are generally not investable for major institutional investors.

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Table 6 Scaled intercepts tests with factor grouping. This table presents test results for factor grouping. Panel A and B use individual stocks as testing assets. The sample covers 3320 individual stocks in the Australian market. Panel C uses 488 portfolios as test assets. The sample period is from January 1992 to December 2017. The test statistics are the equal-weighted mean (SIm,ew, Panel A and C), value-weighted mean (SIm,vw, Panel B), and median (SImed,ew, Panel A and C) scaled absolute intercepts, as discussed in Section 2.2 and 4.2. The 5th-percentile statistics and the p-values (both single and multiple test) are obtained from the bootstrapped distribution, as discussed in Section 2.1. Risk factors are classified into Momentum (M), Value-versus-Growth (V), Investment (I), Profitability (P), Intangibles (IN), and Trading Frictions (T). Monthly returns for factors within each of the groups are averaged. Liquidty, vwivol and ewivol are not measured in monthly returns, and hence are included separately in the tests. Market risk premium and the risk-free rate factors are also reported separately. The factors are sorted based on the reduction in the scaled mean absolute intercept. For brevity, only the top 3 factors are reported. Factor

Single test SIm,ew/vw

5th-percentile

Multiple test

Single test

Multiple test

p-value

p-value

SImed,ew

5th-percentile

p-value

p-value

Panel A: individual stocks equal-weighted Step 1: Baseline = no factor mrp −0.170 average_T −0.022 average_I −0.002

−0.109 −0.064 −0.003

0.009 0.214 0.057

0.009

−0.183 −0.025 −0.007

−0.114 −0.086 −0.010

0.010 0.223 0.061

0.011

Step 2: Baseline = mrp average_IN 0.004 average_I 0.006 liquidity 0.014

−0.006 −0.002 −0.003

0.456 0.235 0.737

0.969

−0.004 0.001 0.026

−0.016 −0.012 −0.013

0.228 0.192 0.864

0.816

Panel B: Individual stocks value-weighted Step 1: Baseline = no factor mrp −0.345 average_T −0.023 average_I −0.002

−0.224 −0.048 −0.015

0.003 0.174 0.246

0.003

Step 2: Baseline = mrp average_P −0.032 average_V −0.010 average_IN −0.000

−0.049 −0.109 −0.011

0.120 0.397 0.411

0.516

Step 1: Baseline = no factor mrp −0.547 average_T −0.099 average_I −0.063

−0.338 −0.196 −0.040

0.000 0.208 0.021

0.000

−0.599 −0.085 −0.057

−0.341 −0.196 −0.046

0.000 0.240 0.034

0.000

Step 2: Baseline = mrp average_I −0.033 liquidity −0.003 average_IN 0.000

−0.027 −0.038 −0.018

0.034 0.404 0.507

0.515

−0.043 −0.005 −0.012

−0.040 −0.059 −0.033

0.040 0.367 0.205

0.536

Panel C: Portfolio

noise is reduced, the reduction in the mean scaled absolute intercepts has increased from 17.0% to 20.9%. Panel B of Table 5 presents results for the value-weighted approach. In the first step, for the baseline model without a pre-selected factor, the market factor continues to dominate in reducing the value-weighted mean scaled absolute intercept by 34.9%. The magnitude is comparable to the results when all stocks are included (34.5%), which suggests that the value-weighted test statistic is more robust to extreme observations caused by small stocks. We then continue the test with the market factor included in the baseline model. Lev [id# 81] is shown to reduce SIm, vw by about 9.9% and is statistically significant under the single test. However, once multiple hypothesis testing is accounted for, lev [id# 81] is no longer statistically significant. The results are qualitatively similar if we use block bootstrapping of 3, 6 and 12 months. As a result, we conclude that the single market factor model dominates all others. 5.3. Factor grouping Empirically, some factors have relatively high correlations as they capture similar underlying economics. For example, qmj [id# 85], rmw [id# 52] and roeq [id# 55] are representatives of the profitability group. In this section, we group similar factors, in an attempt to reduce noise and identify additional pricing factors beyond the market factor. As detailed in Section 3 and Table A.1, we classify risk factors into Momentum (M), Value-versus-Growth (V), Investment (I), Profitability (P), Intangibles (IN), and Trading Frictions (T). Monthly returns for factors within each of the groups are averaged, with the aim of reducing noise. Liquidity [id# 90], vwivol [id# 89] and ewivol [id# 88] are not measured in monthly returns, and hence are included separately in the tests. In addition, the market risk premium and the risk-free rate factors are reported separately. For brevity, we only report the top three factors in each of the steps. 13

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Panels A and B of Table 6 report the results using individual stocks as test assets for both the equal-weighted and value-weighted test statistics. The results consistently show that the market factor is the single dominating factor in explaining cross-sectional returns. None of the averaged factors offer a statistically significant reduction in the mean/median absolute intercepts after the market factor is included in the baseline model. The results are qualitatively similar if we use block bootstrapping of 3, 6 or 12 months. Panel C of Table 6 presents the test results using portfolios as test assets. The market factor continues to dominate in terms of scaled absolute intercept reduction in the first step of the test. After the market factor is included in the baseline model, the average investment (average_I) factor offers the highest further reduction in pricing errors, which is significant under single test. However, the economic magnitude is small when compared to market factor, and it does not survive the multiple test. Overall, the results remain consistent with our earlier conclusion that the market factor is the single dominant pricing factor. 6. Conclusion Hundreds of potential factors have been identified in prior research, with US stock returns being the basis of much of that work. In comparison there has been relatively little examination of the large range of candidate factors using Australian stock returns, with the focus of prior Australian research on the Fama and French three- and five-factor models. Notable in this Australian work is a lack of consistent support for these models. Harvey and Liu (2018) highlight several issues which may affect the validity of prior asset pricing model tests and perhaps help explain the lack of support in prior Australian studies for the Fama and French models. One important issue is that conventional test statistics do not account for data-mining making the bar too low for identifying actual risk factors. Harvey and Liu (2018) offer an alternative testing approach which directly accounts for data-mining. This alternative approach allows us to significantly increase the number of candidate factors tested in the Australian market to ninety-one, while properly accounting for data-mining. The consistent results across all our tests confirms that the market factor is the dominant risk factor in explaining the cross-sectional variation in returns in Australia. When individual stocks are used as test assets, with both equal and value weighted scaled intercepts, the market factor is the only factor that survives the iterative process of testing all ninety-one factors. While individual stocks are preferred as test assets we also use a comprehensive set of portfolios as test assets and again find the market factor to be the only priced factor. These results are robust to the use of block bootstrapping, the removal of microcap stocks and the grouping of factors. Our results are consistent with Harvey and Liu (2018) in that they too find the market factor to be the dominant factor. However, they also find smb and hml to be significant when using individual stocks as test assets with equal-weighting, and qmj with valueweighting. When portfolios are used as test assets, Harvey and Liu (2018) find cma to be significant in addition to the market factor. Our results account for the multiple hypothesis testing phenomenon and have important implications for the Australian asset pricing literature. The results show that the dominance of the market factor survives testing outside of the US, and that no other factors are important for explaining Australian stock returns. Acknowledgements This research receives funding from University of Queensland BEL Faculty New Staff Research Start-Up Fund (NSRSF). We are grateful for the valuable comments of Barry Oliver, Robert Faff, Tom Smith, and Jimmy Xu. We thank Helen Truong for excellent research assistance. All errors are our own. Appendix A. Appendix Table A.1

Factor descriptions and summary statistics. This table provides the descriptions and summary statistics for the 91 factors studied. The id, description, acronym, definition and reference are provided. Further, factors are classified into six different groups, Momentum (M), Value-versus-Growth (V), Investment (I), Profitability (P), Intangibles (IN), and Trading Frictions (T). The mean values and t-statistics are also reported. Id

Category

Description

Acronym

Definition

1

I

Working Capital Accruals

acc

2

I

chcsho

3

I

4

I

5

I

Change in Shares Outstanding Change in Inventory Change in Tax Expense Corporate Investment

Annual income before extraordinary items (ib) minus Sloan (1996) operating cash flows (oancf) divided by average total assets (at); if oancf is missing then set to change in act – change in che – change in lct + change in dlc + change in txp-dp. Annual percent change in shares outstanding (csho). Pontiff and Woodgate (2008) Change in inventory (inv) scaled by average total assets (at). Thomas and Zhang (2002) Percent change in total taxes (txtq) from quarter t-4 to t. Thomas and Zhang (2002) Change over one quarter in net PP&E (ppentq) divided by Titman et al. sales (saleq) – average of this variable for prior 3 quarters; if (2004) saleq = 0, then scale by 0.01

chinv chtx cinvest

Reference

t-stat.

Mean

−1.59

−0.24%

−4.76

−0.76%

−2.42

−0.29%

1.10

0.24%

1.75

0.47%

(continued on next page)

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Table A.1 (continued) Id

Category

Description

Acronym

Definition

6

I

Conservative Minus Aggressive

cma

7

I

egr

8

I

Growth in Common Shareholder Equity Growth in Capital Expenditures

Average return on the two conservative investment portfo- Fama and French lios minus average return on the two aggressive investment (2015) portfolios. Annual percent change in book value of equity (ceq). Richardson et al. (2005)

9

I

10

I

11

I

12

I

13

I

14 15

I I

16

IN

Absolute Accruals

17

IN

18 19

IN IN

# Years Since First age Compustat Coverage Cash Holdings cash Cash Flow to Debt cashdebt

20

IN

Current Ratio

currat

21

IN

herf

22

IN

23

IN

Industry Sales Concentration Capital Expenditures and Inventory Organizational Capital

24

IN

25

IN

26

IN

27

IN

28

IN

29

IN

30

IN

31

IN

32 33

Growth in Long Term Net Operating Assets Employee Growth Rate Growth in LongTerm Debt R&D Increase

grcapx grltnoa

Reference

Percent change in capital expenditures from year t-2 to year Anderson and t. Garcia-Feijoo (2006) Growth in long term net operating assets. Fairfield et al. (2003)

t-stat.

Mean

4.90

0.85%

−5.30

−0.80%

−2.54

−0.47%

−3.71

−0.55%

hire

Percent change in number of employees (emp).

Belo et al. (2014) 0.24

0.04%

lgr

Annual percent change in total liabilities (lt).

−4.50

−0.67%

rd

An indicator variable equal to 1 if R&D expense as a percentage of total assets has an increase greater than 5%. R&D expense divided by end-of-fiscal-year market capitalization. R&D expense divided by sales (xrd/sale). Annual earnings before interest and taxes (ebit) minus nonoperating income (nopi) divided by non-cash enterprise value (ceq + lt-che). Absolute value of acc.

Richardson et al. (2005) Eberhart et al. (2004) Guo et al. (2006)

0.37

0.07%

0.37

0.06%

Guo et al. (2006) Brown and Rowe (2007)

0.12 2.61

0.02% 0.47%

Bandyopadhyay et al. (2010) Jiang et al. (2005)

0.69

0.10%

1.64

0.34%

0.12 −0.13

0.03% −0.02%

−2.37

−0.36%

−0.10

−0.01%

−3.79

−0.72%

−0.06

−0.01%

−0.87

−0.21%

0.25

0.03%

1.02

0.13%

0.10

0.01%

0.85

0.12%

0.66

0.08%

−0.86

−0.18%

0.81

0.11%

−2.71

−0.44%

−0.78

−0.10%

R&D to Market rd_mve Capitalization R&D to Sales rd_sale Return on Invested roic Capital absacc

invest orgcap

IN

Industry Adjusted Change in Capital Expenditures Change in Current Ratio Change in Gross Margin – Change in Sales Change in Quick Ratio Change in Sales – Change in Inventory Change in Sales – Change in A/R Change in Sales – Change in SG&A Change Sales-toInventory Percent Accruals

pchcapx_ia

pctacc

IN

Price Delay

pricedelay

Number of years since first Compustat coverage. Cash and cash equivalents divided by average total assets. Earnings before depreciation and extraordinary items (ib + dp) divided by avg. total liabilities (lt). Current assets/current liabilities.

Palazzo (2012) Ou and Penman (1989) Ou and Penman (1989) 2-digit SIC – fiscal-year sales concentration (sum of squared Hou and Kimmel percent of sales in industry for each company). (2006) Annual change in gross property, plant, and equipment Chen and Zhang (ppegt) + annual change in inventories (invt) all scaled by (2010) lagged total assets (at). Capitalized SG&A expenses. Eisfeldt and Papanikolaou (2013) 2-digit SIC – fiscal-year mean adjusted percent change in Abarbanell and capital expenditures (capx). Bushee (1998)

pchcurrat

Percent change in currat.

pchgm_pchsale

Percent change in gross margin (sale-cogs) minus percent change in sales (sale).

pchquick

Percent change in quick.

pchsale_pchinvt pchsale_pchrect pchsale_pchxsga pchsaleinv

Ou and Penman (1989) Abarbanell and Bushee (1998)

Ou and Penman (1989) Annual percent change in sales (sale) minus annual percent Abarbanell and change in inventory (invt). Bushee (1998) Annual percent change in sales (sale) minus annual percent change in receivables (rect). Annual percent change in sales (sale) minus annual percent change in SG&A (xsga). Percent change in saleinv.

Abarbanell and Bushee (1998) Abarbanell and Bushee (1998) Ou and Penman (1989) Hafzalla et al. (2011)

Same as acc except that the numerator is divided by the absolute value of ib.; if ib. = 0 then ib. set to 0.01 for denominator. The proportion of variation in weekly returns for 36 months Hou and ending in month t explained by 4 lags of weekly market Moskowitz returns incremental to contemporaneous market return. (2005)

15

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Table A.1 (continued) Id

Category

Description

Acronym

Definition

34

IN

Quick Ratio

quick

(current assets – inventory)/current liabilities.

35

IN

Earnings Volatility

roavol

36

IN

Sales to Cash

salecash

37

IN

Sales to Inventory

saleinv

38

IN

salerec

39

IN

40

M

41

M

42

M

43

M

44

M

Sales to Receivables Debt Capacity/ Firm Tangibility Change in 6Month Momentum Industry Momentum Momentum 12 months Momentum 6 months Number of Earnings Increases

45

M

Revenue Surprise

rsup

46

P

chatoia

47

P

48

P

49

P

Industry-Adjusted Change in Asset Turnover Industry-Adjusted Change in Employees Industry-Adjusted Change in Profit Margin Gross Profitability

50

P

51

P

52

P

53

P

Retrun on Assets

roa

54

P

roaq

55

P

Return on Assets – quarterly HXZ Profitability

56

P

Sales Growth

sgr

57

T

bab

58

T

Betting Against Beta Beta

59

T

60

T

61

T

62

T

63

T

Financial Statement Performance Financial Statements Score Robust Minus Weak

Dollar Trading Volume Idiosyncratic Return Volatility Maximum Daily Return Short-Term Reversal Return Volatility

tang chmom indmom12m mom12m mom6m nincr

Reference

Ou and Penman (1989) Standard deviaiton for 16 quarters of income before extra- Francis et al. ordinary items (ibq) divided by average total assets (atq). (2004) Annual sales divided by cash and cash equivalents. Ou and Penman (1989) Annual sales divided by total inventory. Ou and Penman (1989) Annual sales divided by accounts receivable. Ou and Penman (1989) Cash holdings +0.715 × receivables +0.547 × inventory Almeida and +0.535 × PPE/ total assets. Campello (2007) Cumulative returns from months t-6 to t-1 minus months t- Gettleman and 12 to t-7. Marks (2006) Equal weighted average industry 12-month returns. Moskowitz and Grinblatt (1999) 11-month cumulative returns ending 1 month before month Carhart (1997) end. 5-month cumulative returns ending 1 month before month Jegadeesh and end. Titman (1993) Number of consecutive quarters (up to eight quarters) with Barth et al. an increase in earnings (ibq) over same quarter in the prior (1999) year. Sales from quarter t minus sales from quarter t-4 (saleq) Kama (2009) divided by fiscal-quarter- end market capitalization (cshoq * prccq). 2-digit SIC – fiscal-year mean adjusted change in sales (sale) Soliman (2008) divided by average

t-stat.

Mean

−2.48

−0.42%

−0.59

−0.24%

2.40

0.51%

1.91

0.29%

2.40

0.37%

−1.63

−0.31%

2.40

0.42%

4.01

0.87%

5.34

1.15%

5.70

1.25%

0.27

0.06%

−0.01

0.00%

1.16

0.17%

chempia

Industry-adjusted change in number of employees.

Asness et al. (2000)

1.53

0.24%

chpmia

2-digit SIC – fiscal-year mean adjusted change in income before extraordinary items (ib) divided by sales (sale).

Soliman (2008)

−0.77

−0.15%

gma

Revenues (revt) minus cost of goods sold (cogs) divided by Novy-Marx lagged total assets (at). (2013) Sum of 8 indicator variables for fundamental performance. Mohanram (2005)

0.96

0.18%

−0.65

−0.15%

Piotroski (2000)

1.52

0.29%

Fama and French (2015)

1.89

0.33%

Balakrishnan et al. (2010) Balakrishnan et al. (2010) Hou et al. (2015)

−0.23

−0.05%

2.01

0.62%

2.32

0.69%

−3.08

−0.47%

2.28

1.15%

−1.92

−0.59%

−1.22

−0.82%

−1.56

−0.41%

−1.80

−0.37%

5.03

0.88%

−1.87

−0.44%

ms ps rmw

roeq

beta dolvol idiovol maxret mom1m retvol

Sum of 9 indicator variables to form fundamental health score. Average return on the two robust operating profitability portfolios minus average return on the two weak operating profitability portfolios. Income before extraordinary items (ib) divided by 1 year lagged total assets (at). Income before extraordinary items (ibq) divided by one quarter lagged total assets (atq). Earnings before extraordinary items divided by lagged common shareholders' equity. Annual percent change in sales (sale).

Lakonishok et al. (1994) Average return on high-beta portfolio minus average return Frazzini and on low-beta portfolio. Pedersen (2014) Estimated market beta from weekly returns and equal Fama and weighted market returns for 3 years ending month t-1 with Macbeth (1973) at least 52 weeks of returns. Natural log of trading volume times price per share from Chordia et al. month t-2. (2001) Standard deviation of residuals of weekly returns on weekly Ali et al. (2003) equal weighted market returns for 3 years prior to month end. Maximum daily return from returns during calendar month Bali et al. (2011) t-1. 1-month cumulative return. Jegadeesh and Titman (1993) Standard deviation of daily returns from month t-1. Ang et al. (2006)

(continued on next page)

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Table A.1 (continued) Id

Category

Description

Acronym

Definition

Reference

t-stat.

Mean

64

T

Sin stocks

sin

0.00%

T

Small Minus Big

smb

−0.76

−0.15%

66

T

1.93

0.55%

67

T

Monthly standard deviation of daily share turnover.

−0.26%

T

Chordia et al. (2001) Bandyopadhyay et al. (2010)

−1.66

68

Volatility for std_dolvol Dollar Trading Volume Volatility for Share std_turn Turnover Accrual Volatility stdacc

Hong and Kacperczyk (2009) Fama and French (1993) Chordia et al. (2001)

0.01

65

An indicator variable equals to 1 if a company's primary industry classification is in smoke or tobacco, beer or alcohol, or gaming Average return on three small stock portfolios minus average return on three big stock portfolios. Monthly standard deviation of daily dollar trading volume.

−1.25

−0.42%

69

T

Cash Flow Volatility

stdcf

Huang (2009)

−1.38

−0.45%

70

T

Share Turnover

turn

−2.64

−0.55%

71

T

Zero Trading Days

zerotrade

Datar et al. (1998) Liu (2006)

2.50

0.53%

72

V

bme_ia

3.25

0.43%

73

V

Industry-Adjusted Book to Market Cash Productivity

−3.55

−0.59%

74

V

2.70

0.61%

75

V

0.81

0.14%

76

V

2.03

0.38%

77

2.19

0.51%

1.48

0.29%

3.04

0.57%

1.21

0.26%

Bhandari (1988)

2.97

0.55%

Debondt and Thaler (1985) Sharpe (1964)

−1.34

−0.27%

2.11

0.43%

1.79

0.23%

−0.85

−2.83%

2.74

0.59%

1.18

0.18%

16.44

68.61%

18.99

21.94%

0.13

0.00%

53.82

0.39%

cashpr

Asness et al. (2000) Fiscal year end market capitalization plus long term debt Chandrashekar (dltt) minus total assets (at) divided by cash and equivalents and Rao (2009) (che). Operating cash flows divided by fiscal-year-end market Desai et al. capitalization. (2004) Industry adjusted cfp. Asness et al. (2000)

depr

Depreciation divided by PP&E.

V

Cash Flow to Price Ratio Industry-Adjusted Cash Flow to Price Ratio Depreciation/PP& E Dividend to Price

dy

Total dividends (dvt) divided by market capitalization at fiscal year-end.

78

V

Earnings to Price

ep

79

V

High Minus Low

hml

80

V

HML Devil

hmldevil

81

V

Leverage

lev

82

V

mom36m

83

V

84

V

85

V

86

V

Long-Term Reversal Excess Market Return Change in Depreciation Quality Minus Junk Sales to Price

Annual income before extraordinary items (ib) divided by end of fiscal year market cap. Average return on the two value portfolios minus average return on the two growth portfolios. Average return on the two value portfolios minus average return on the two growth portfolios, where current prices are used to construct book-to-market ratio. Total liabilities (lt) divided by fiscal year end market capitalization. Cumulative returns from months t-36 to t-13.

87

V

88

Tax Income to Book Income

cfp

Standard deviation for 16 quarters of accruals (acc measured with quarterly Compustat) scaled by sales; if saleq = 0, then scale by 0.01. Standard deviation for 16 quarters of cash flows divided by sales (saleq); if saleq = 0, then scale by 0.01. Cash flows defined as ibq minus quarterly accruals. Average monthly trading volume for most recent 3 months scaled by number of shares outstanding in current month. Turnover weighted number of zero trading days for most recent 1 month. Industry adjusted book-to-market ratio.

cfp_ia

mrp pchdepr qmj sp tb

90

Common ewivol Idiosyncratic risk – equally weighted Comon vwivol Idiosyncratic risk -value weighted Liquidity liquidity

91

Risk-free rate

89

rf

Value-weighted excess market returns (All Ordinaries less 13-week treasury note yield). Percent change in depr.

Holthausen and Larcker (1992) Litzenberger and Ramaswamy (1982) Basu (1977) Fama and French (1993) Asness and Frazzini (2013)

Holthausen and Larcker (1992) Average return on two high-quality portfolios minus Asness et al. average return on the two low-quality (junk) portfolios. (2014) Annual revenue (sale) divided by fiscal-year-end market Barbee Jr et al. capitalization. (1996) Tax income, calculated from current tax expense divided by Lev and Nissim maximum federal tax rate, divided by income before (2004) extraordinary items. Equally-weighted variance of residuals of monthly excess Herskovic et al. stock returns on the returns of Fama French's five-factor (2016) model. Value-weighted variance of residuals of monthly excess Herskovic et al. stock returns on the returns of Fama French's five-factor (2016) model. Equally weighted average of the liquidity measures of Pastor and individual stocks using daily data within the month. Stambaugh (2003) 13-week treasury note yield Campbell (1987)

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