The Cross Section of Long-Term Expected Returns *

Transcription

1 The Cross Section of Long-Term Expected Returns * Zhongjin Lu June 5, 2016 Abstract Eective capital budgeting decisions require reliable estimates of long-term expected returns. I extract a predictor for long-term returns from the book-to-market ratio (B/M) by using a cross-sectional regression to control for B/M's variation associated with long-term expected protability. The resulting predictor reliably describes the cross section of returns out-of-sample for at least ve years, even when focusing exclusively on large-cap stocks and controlling for expected-return estimates generated by implied cost of capital models, factor models, and characteristic-based models. My results have important implications for improving our cost of equity estimates. Keywords: out-of-sample, cross section, long-term returns, expected protability, book-to-market ratio, cost of equity * First Version: Mar 21, I thank Kent Daniel, Robert Hodrick, Sara Holland, Harold Mulherin, David Ng, Bradley Paye, Robert Resutek, Tano Santos and seminar participants at University of Georgia for detailed comments on the rst draft. I thank Zhi Da, Peter Easton, Andrey Ermolov, Pengjie Gao, Paul Irvine, Dana Kiku, Lars Lochstoer, Harold Mulherin, Annette Poulsen, Gil Sadka, Sophie Shive, Paul Tetlock, one referee from Texas Finance Festival conference program committee and participants at numerous seminars and conferences for their helpful comments. I thank Ken French, Yan Li, and Chen Xue for providing data. Assistant Professor of Finance, Terry College of Business, University of Georgia, Athens, GA zlu15@uga.edu. 1

2 1 Introduction Corporate managers make investment decisions by discounting future cash ows at the opportunity cost of capital. Since the typical useful life for corporate investments is between 5 and 10 years for xed assets and 15 years for intangible assets, 1 discounting the associated cash ows requires an estimate of long-term expected returns. However, asset pricing models are commonly tested on the grounds of explaining short-term expected returns. Due to the term structure of risk premia (e.g., Van Binsbergen, Brandt, and Koijen, 2012), models that describe short-term expected returns may not accurately describe long-term expected returns. In fact, although the ability to predict long-term returns is a necessary property of the discount rate desired for eective corporate capital budgeting decisions, there are only a handful of studies that focus on long-term return predictability at the rm level. In this paper, I develop prediction of long-term expected returns based on a present value model. The resulting predictor can reliably describe the cross section of returns for at least ve yearsthe lower bound of the typical useful life for corporate investments. My approach exploits the fact that the book-to-market ratio (hereafter B/M) reects both longterm expected returns and long-term expected protability. I rst estimate the variation in B/M associated with long-term expected protability by running a cross-sectional regression of B/M on variables commonly used in earnings forecast models. Subtracting the estimated variation from B/M, the resulting residual is a cleaner measure of the B/M's information about long-term expected returns. I name the residual the implied discount rate (IDR) and use the IDR to forecast future returns in the second stage regression. My approach to forecast long-term returns has several distinct features. First and foremost, my regression model is used to control for expected protability and the discount rate is then implied from the present value relation. The choice of predictors in my regression model is less susceptible to data mining because the economic mechanisms of earnings predictors are better understood than the economic mechanisms of popular return predictors. 1 See IRS Publication 946 (2014) and IRS Regulation Ÿ1.167(a)-3, respectively. 2

3 As a result, my approach may result in more robust predictions. 2 Second, my approach diers from the implied cost of capital (ICC) approaches, which also infer the discount rate from the stock price using the present value relation, in how I control for long-term expected protability. 3 Unlike the commonly used ICC approaches, my approach does not rely on analysts' earnings forecasts, which are known to have a general upward bias (e.g., Easton and Sommers, 2007) and are potentially cross-sectionally dierentially biased (Bradshaw, Richardson, and Sloan, 2001). Moreover, not requiring analyst coverage also allows me to study the full cross section of rms. The benet of this is highlighted in Hou, Dijk, and Zhang (2012), who propose a novel ICC approach that uses predicted earnings from a regression model instead of analysts' forecasts and show the resulting ICC has improved accuracy in forecasting subsequent returns. 4 However, since analysts rarely forecast earnings beyond 5 years and since the earnings regressions in Hou, Dijk, and Zhang (2012) quickly run into survivorship bias when one tries to forecast earnings multiple periods ahead, all ICC methods have to extrapolate the predicted short-term earnings to an innite horizon by imposing ad hoc assumptions on the terminal growth rate. In contrast, my approach does not need any extrapolation since the variation in B/M captured by my cross-sectional model is inherently linked to the long-term protability. Third, my computation of the IDR purges the variation in B/M predicted by earnings forecasting variables to arrive at a cleaner predictor of long-term returns. The IDR thus does not utilize the expected-return information contained in those earnings forecasting variables. In Section 2, I formally discuss the trade-o between noise reduction and the use of limited information associated with my approach. In Section 6.3, I explore a vector autoregression (VAR) approach in which I jointly estimate the B/M information about expected return and the B/M information about expected protability. The disadvantage of the VAR approach 2 Mclean and Ponti (2006) nd many return predictors lose a substantial amount of their predictive power out-of-sample. 3 Both the ICC approaches and my approach build on the present value relation and thus are silent about whether dierences in expected returns are driven by risk or irrational cash ow expectations. 4 Hou, Dijk, and Zhang's (2012) approach builds on Fama and French (2000, 2006), who rst advocate using cross-sectional models to predict future protability. 3

4 compared to my approach is that the VAR imposes a potentially restrictive structure on the long run dynamics, which can hurt the out-of-sample predictive power. To explore whether my IDR approach helps improve the existing description of the cross section of long-term returns, I rst contrast the IDR to the closely related ICC measures, which are also inferred from stock prices by controlling for long-term protability. To put results into a broader perspective, I also examine expected-return estimates from two other prominent types of models: factor-based models and a characteristic-based model. There are three challenges in testing long-term predictive power. First, explanatory variables have estimation errors because they are estimates of the expected returns. Second, a substantial number of rms are delisted over the 5-year investment period. Third, because the 40-year time span of my sample is short relative to the 5-year holding period, autocorrelations of overlapping holding period returns may not be accurately estimated. I use two approaches, portfolio sorts and Fama-MacBeth (1973) regressions, to confront these challenges. 5 Both the portfolio sorts and Fama-MacBeth regressions demonstrate that the IDR is a far more reliable predictor for long-term returns than the existing ICC measures considered. I also nd poor long-term predictive power associated with expected-return estimates generated by the factor models considered. The seven-characteristic model in Lewellen (2014) is the only model considered that has similarly strong predictive power when compared to the IDR. Furthermore, when the IDR and an expected-return estimate from each of the other models are used to predict returns in bivariate regressions, I nd that the IDR remains highly signicant in all regressions. Finally, since the bulk of the literature focuses on one-month holding period returns, I run spanning tests of the monthly returns of portfolios sorted on the IDR. Results reveal that the IDR has signicant predictive power over a one-month horizon and this predictive 5 To ensure that corporate managers can ultimately use these estimates as real-time guidance for their investments, I use only prevailing information to calculate all expected-return estimates in order to avoid look-ahead bias. 4

5 power cannot be fully explained by exposures to the two latest factor models (Fama and French (2015) and Hou, Xue, Zhang (2014)). 1.1 Related literature A relatively limited amount of literature studies the long-term predictability in the cross section of stock returns (see Fama and French (1997), Hou, Dijk, and Zhang (2012), Levi and Welch (2014) and Lyle and Wang (2015)). I contribute to this literature by proposing a new approach that reliably forecasts the cross section of long-term returns. Additionally, I present a comprehensive study of existing models in terms of their ability to forecast longterm returns, and I nd most of them exhibit weak long-term predictive power. My work builds on the B/M decomposition in Vuolteenaho (2002). Cohen, Polk, and Vuolteenaho (2003), which is closely related, apply the same B/M decomposition to both a panel of U.S. rms and a panel of international rms and nd that the cross-sectional variation in B/M is dominated by the variation in expected protability. In a recent paper, Lochstoer and Tetlock (2016) extend Vuolteenaho's (2002) VAR approach and also nd B/M information is mostly about long-term protability. This is consistent with my nding that controlling for the variation in B/M associated with expected protability can drastically improve our ability to forecast long-term returns. The present-value relation underlying Vuolteenaho's B/M decomposition is also widely used to study the association between returns, cash ows, and the valuation ratio (e.g., see Fama and French (2006, 2015), Hou, Xue, Zhang (2014), and Novy-Marx (2013)). Apart from the important distinction of the long-term focus, my approach is unusual in that I use B/M as the left-hand variable in the rst stage regression and my predictor for returns is orthogonalized to variables related to expected protability. In contrast, B/M and protability measures are often used together in the right-hand side of a regression to predict returns. In such regressions, a positive relation between protability measures and subsequent returns leads to two possible interpretations. The more often cited is that high protability is asso- 5

6 ciated with high risk and thus high expected returns. 6 The other is that these protability measures can have nothing to do with expected returns but nevertheless show up as a significant regressor, because they serve as a control variable for the variation in B/M associated with expected protability. My empirical results highlight the relevance of the second case. 7 The rest of the paper is organized as follows. Section 2 lays out the theoretical motivation of my new approach. Section 3 describes the two empirical steps used to compute the new predictor for long-term returns. Section 4 tests the long-term predictive power of various expected-return estimates. Section 5 discusses the implication for cost of equity estimates. Section 6 presents a battery of robustness tests that address the questions regarding the control for expected protability, diminishing predictive power over long horizons, and the comparison to a VAR model. Section 7 presents the spanning test of monthly portfolio returns sorted on the IDR. Section 8 provides a conclusion. 2 Theoretical motivation 2.1 Theoretical framework My approach builds on the decomposition of B/M of Vuolteenaho (2002) ( ) ( ) bm t E t ρ τ 1 r t+τ E t ρ τ 1 e t+τ. (1) τ=1 In equation (1), bm t is the log B/M at time t, r t+τ is the log return, e t+τ is the log returnon-equity (hereafter ROE), ρ is a discount coecient related to the average payout ratio, which is often set to be around 0.96 in the literature. Equation (1) shows that the variation in B/M is not only driven by variation in expected returns but also variation in expected protability. It highlights the signal-extraction 6 Fama and French (2015) and Hou, Xue, Zhang (2014) thus include protability as one risk factor in their factor models. Nevertheless, Novy-Marx (2013) comments that the positive relation is inconsistent with explanations based on the distress risk, operating leverage risk, and duration risk. 7 This possibility is not a new theoretical prediction. It is consistent with both the dividend discount model and the q-theory. 6 τ=1

7 problem faced by researchers. To extract variations regarding expected returns from B/M, researchers will need to control for variations associated with expected protability. Without loss of generality, I assume X r t and X e t are time t vectors of state variables that are associated with long-term expected returns (E t ( τ=1 ρ τ 1 r t+τ )) and long-term expected protability (E t ( τ=1 ρ τ 1 e t+τ )), respectively. 8 E t ( τ=1 ρ τ 1 r t+τ ) E t ( τ=1 ρ τ 1 e t+τ ) bm t = 1 α r,e α e,r 1 θ r θ e X r t X e t, (2) where θ r = 1 α e,r and θ e = 1 α r,e due to the present value relation in Eq (1). The regression coecient of long-term expected returns on X r t and the coecent of long-term expected protability on X e t are normalized to be 1. I assume that state variables in X r t are independent of state variables in X e t. This assumption is not restrictive because expected returns and expected protability can have an arbitrary correlation through their loadings on X r t and X e t. For illustration purpose, in the following discussion X r t and X e t are treated as scalars. The derivation for the case in which X r t and X e t are vectors is similar and available upon request. 2.2 B/M as a noisy signal for long-term expected returns One common approach to capture long-term expected returns is to regress them on B/M. ( ) E t ρ τ 1 r t+τ τ=1 = a bm + b bm bm t + η bm t (3) b bm = θ rv X r α r,e θ e V X e, (4) θrv 2 X r + θev 2 X e in which V X r and V X e are the variances of X r t and X e t, respectively. We can evaluate the 8 Since this study focuses on the cross-sectional predictability, variance and covariance of variables refer to those of the cross-sectional demeaned variables. 7

8 forecast accuracy of this model via the mean squared forecast errors (MSFE) MSF E bm = E ( ) ηt bm 2 = [1 b bm θ r ] 2 V X r + [α r,e + b bm θ e ] 2 V X e (5) It is useful to illustrate the intuition of Eqs (4) and (5) through a polar case in which α e,r = 0 and α r,e = 0. That is, all useful information about expected returns is in X r t and all useful information about expected protability is in X e t. In this case, Eq (4) becomes b bm = V X r, (6) V X r + V X e The regression coecient b bm is smaller than 1 because the regression model in Eq (3) extracts signals about expected returns from bm t, which contains both the signal X r t and the noise X e t. 9 When the variance of the noise increases relative to the variance of the signal, that is signal-to-noise ratio V X r V X e decreases, the regression coecient b bm will decrease towards 0 and bm t becomes a bad predictor for expected returns. This eect can been seen more directly from the MSFE. Under this simplifying assumption, MSF E bm = V X r V X r + 1 (7) V X e If much of variation in B/M comes from variation in expected protability, that is, the signal to noise ratio V X r V X e is low, the forecast accuracy of the model in Eq (3) is poor. Cohen, Polk, and Vuolteenaho (2003) nd that expected protability causes about 75% to 80% of the B/M variance and a recent paper by Lochstoer and Tetlock (2016) nd similar results. I now discuss my new approach that aims to derive a purer signal about expected returns by controlling the B/M variation associated with expected protability. 9 The categorization of signal versus noise is dictated by the objective of forecasting returns. 8

9 2.3 Noise reduction I control for the B/M variation due to expected protability by regressing bm t on X e t in the rst stage regression: bm t = c + βx e t + ɛ bm t (8) According to Eq (2), the residual ɛ bm t with expected returns. I name ɛ bm t is a proxy for X r t, the state variable associated the implied discount rate (IDR) because it is not a direct estimate of the discount rate but rather a predictor implied by the B/M. I posit that the IDR can be a better predictor for long-term returns than the original B/M. To investigate the forecast accuracy of this model, I regress long-term expected returns on the IDR and compute the associated MSFE: ( ) E t ρ τ 1 r t+τ τ=1 = a IDR + b IDR IDR t + η IDR t (9) From Eq (2), IDR = θ r X r t and b IDR = 1 θ r (10) MSF E IDR = E ( ) ηt IDR 2 = α 2 r,e V X e (11) The intuition from Eq (11) is straightforward. The rst stage regression in Eq (8) eliminates the variation in bm t associated with Xt e, a variable associated with expected profitability, and the resulting IDR is a purer signal about expected returns. However, if Xt e is also correlated with the expected returns (i.e., α r,e is not zero), then the IDR misses the useful information about expected returns contained in X e t and the forecasting model in Eq (9) suers a loss of α 2 r,ev X e in forecast accuracy. Therefore, my new approach involves a trade-o between noise reduction and the use of limited information. My approach operates under the premise that it is challenging to directly model expected returns. Therefore, we 9

10 only know state variables X e t that are associated with expected protability and we are agnostic about X r t. The use of limited information can be overcome if I instead assume that we know X r t. Then a VAR model can be used to jointly model dynamics of expected returns and expected protability. 10 However, since protability can more reliably forecasted than returns, operating under the constraint that X r t is not observable can yield predictors with potentially more robust out-of-sample predictive power. To guide the empirical analysis, I take the dierence between the MSFEs under the two forecasting models (Eqs (5) and (11)) to gauge the relative forecast accuracy, MSF E bm MSF E IDR = [1 b bm θ r ] 2 V X r + ( [α r,e + b bm θ e ] 2 α 2 r,e) VX e (12) Therefore, as long as α r,e + b bm θ e > α r,e, my new approach will have better forecast accuracy. Given the consensus in the literature that b bm > 0 (see, e.g., Cohen, Polk, Vuolteenaho (2003) and Cochrane (2011)), my new approach will have higher forecast accuracy if α r,e θ e > 0. That is, the correlation between Xt e and expected returns and the correlation between Xt e and B/M have opposite signs (See Eq (2)). 11 This condition is further simplied if the potential correlation between X e t and expected returns is positive. In this case, we have θ e = 1 α r,e > 0 and thus α r,e θ e > 0 as long as X e t predominately captures the B/M information about expected protability. 3 Empirical implementation 3.1 The cross-sectional model I use the following model to control for the variation in B/M associated with expected protability. I keep my benchmark model parsimonious on purpose and leave extensions of 10 I explore this in Section A similar condition can be derived under the more complicated case in which X r and X g are vectors. 10

11 the model to future research. bm i,t = b 1,t roe i,t + b 2,t D i,t + b 3,t (D i,t roe i,t ) + b 4,t op i,t + b 5,t rnd i,t + ɛ i,t. (13) Variables are natural logarithmic to be consistent with Eq (1). bm is the log of B/M; roe is the log of (1+ROE); D is the decile ranking of the deviation of a rm's ROE from its median non-zero ROE over the past three years; 12 op is the log of (1+operating protability); and rnd is the log of (1+R&D intensity). A detailed description of variables is provided in the Data Appendix. These variables are associated with future protability on both theoretical and empirical grounds. Current ROE is a natural choice for predicting future protability because ROE is persistent. Fama and French (2000) nd strong evidence of a faster rate of mean reversion for extreme protability. To allow for this possibility, I add an interaction term of current ROE and its deviation from the average ROE. The next variable, operating protability (hereafter OP) measures current revenue relative to current expenses (e.g., Novy-Marx 2013; Ball, Gerakos, Linnainmaa, and Nikolaev 2015). It has incremental forecasting power for future protability because, unlike ROE, operating protability is not contaminated by accruals. Finally, R&D expenditures, like investments, have the potential to generate future earnings, even though they are treated as current expenses under the GAAP accounting rule. Holding ROEs the same, evidence suggests that rms with high R&D expenditures have high future protability (e.g., Lev and Sougiannis, 1996; Resutek, 2015). Hence, it is important to account for this eect in the cross-sectional regression. I eliminate B/M information about expected protability by removing the tted value in Eq (13) from bm, and call the resulting estimate the implied discount rate (IDR): IDR t = bm t bm t X e t. (14) 12 The median non-zero ROE serves as a proxy for the equilibrium ROE. Using the industry median ROE instead of the rm historical median ROE as a proxy for the equilibrium ROE also yields similar results. 11

12 where bm t X e t is the tted value from the cross-sectional model in Eq (13). 3.2 Data Accounting data are collected from the COMPUSTAT Industrial Annual le and price related data are collected from the CRSP monthly stock le, including all common stocks traded in NYSE, Amex, and NASDAQ. The COMPUSTAT and CRSP data are merged using the CRSP-COMPUSTAT linking table. To avoid look-ahead bias, I follow Fama and French (1993) and assume that accounting data for scal years ending in year t-1 are known at the end of June of year t. I use CRSP delisting returns if available. Otherwise, I impute the delisting return following Beaver, McNichols, and Price (2007). 13 Detailed descriptions of the variables are in the Data Appendix. I estimate the cross-sectional model in Eq (13) starting from 1971 because fewer than 500 rms have non-missing COMPUSTAT value for R&D expenditures prior to To avoid the inuential observation problems, prior research imposes data lters that exclude rms with extreme observations (e.g., Fama and French (2006), Vuolteenaho (2002)). I follow this approach and exclude rms with ROE and OP smaller (larger) than NYSE 1st (99th) percentile or with R&D intensity larger than its NYSE 99th percentile in the same year. There are on average 3,510 rms per year in my sample. Panel A of Table 1 presents the summary statistics for the main variables used in my cross-sectional model (observations with missing R&D value are excluded when calculating R&D summary statistics). The average rm has a market equity larger than its book equity, with both ROE and OP being positive. ROE has substantial negative skewness, while R&D intensity has substantial positive skewness because R&D expenditures are non-negative. 13 Alternatively, I impute the delisting return following Shumway (1997) when CRSP delisting returns are unavailable. Results are almost identical since the frequency of missing CRSP delisting returns upon delisting is about 0.004% in my sample. 14 After that, on average, approximately 48% of the rms have non-missing COMPUSTAT value for R&D expenditures. I follow the standard treatment of missing data by setting missing R&D expenditures to zero and add a dummy variable to indicate a missing value. This approach is equivalent to ignoring these missing values in multiple regressions. See William H. Greene, Econometric Analysis, 5th Edition for more detail. 12

13 Panel B of Table 1 shows the time-series average of the cross-sectional correlations between B/M and the three explanatory variables. Consistent with the economic intuition that rms with high protability and high R&D intensity have high market valuation, I nd high ROE, OP, and R&D intensity to be associated with low B/M. 3.3 Fama-MacBeth full sample estimation results for the crosssectional model Table 2 reports the full sample mean (across years) of cross-sectional slope estimates of the rst stage regression model in Eq (13). The associated t-statistics are computed using Newey-West standard errors with 5 lags to take account of serial correlations. I run separate regressions with and without microcap stocks. Fama and French (2008) nd that the microcap stocks (dened as stocks smaller than 20th percentile of NYSE size distribution) on average account for 60% of the total number of stocks but only 3% of the market cap. Running separate regressions helps to detect whether including microcap stocks skews the estimation results. The column labeled All presents the regressions including all stocks. The average slope for ROE is negative and highly signicant (-7.26 with a t-stat of -12.4). The slope for the interaction term of ROE and the ROE deviation ranking (D) is positive and also highly signicant (0.61 with a t-stat of 17.3). These two slopes imply that the overall slope for ROE is about -3.9 for rms with the mean ROE deviation ranking (i.e., D=5.5). A negative overall slope for ROE indicates that a high ROE is associated with high market valuation (low B/M). This negative overall slope is consistent with the extant evidence that high current protability is associated with high future protability (e.g. Fama and French, 2000) and thus leads to higher valuation and lower B/M. Furthermore, a back-of-the-envelope calculation shows that if the ρ in Eq (1) is 0.967, as proposed by Vuolteenaho (2002), then the overall slope for ROE of -3.9 implies a rst-order autoregressive (AR(1)) coecient of

14 for ROE. 15 This number is not too far from the one estimated by predictive regressions of one-year ahead protability in Fama and French (2000, Table 1). 16 While my new approach allows for more complicated dynamics of protability than AR(1) process, nding a similar level of rst-order autocorrelation of protability lends credibility to the new approach. The positive slope for the interaction term of ROE and the ROE deviation ranking indicates that B/M responds less strongly to extreme ROE. This is also consistent with the nding in Fama and French (2000, Table 1) in which they nd extreme protability reverts faster and thus would have weaker eect on B/M. Slopes for OP and R&D intensity are both negative and signicant. Like the negative slopes for ROE, these negative slopes are consistent with the working hypothesis that OP and R&D intensity are positively correlated with expected protability and thus are associated with lower B/M. The column labeled Non-microcap in Table 2 presents the regressions for the sample excluding microcap stocks. The slopes for ROE and the interaction term in this non-microcap sample change only slightly. The slopes for OP and R&D intensity have more noticeable changes. The slope for OP becomes more negative, from to -2.06, indicating larger impact on valuation from OP in the non-microcap sample. This can explained by a more persistent operating protability for larger rms, which tend to have more stable business revenue. The slope for R&D intensity becomes less negative, from to -3.30, indicating smaller impact on valuation from R&D intensity in the non-microcap sample. Lastly, the time-series average of R 2 is 31.3% for the full sample and 40.0% for the non-microcap sample, suggesting that my baseline cross-sectional model in Eq (13) describes the expected protability of larger rms more accurately. 17 The literature suggests several reasons why a possible positive correlation can exist be- 15 The 0.82 number is computed under the simplifying assumption that ROE follows AR(1) process. 16 Fama and French (2000) deate the earnings by total asset. Their specications in Panel B of Table 1 imply an AR(1) coecient of protability around When I augment the baseline model with an interaction term of OP (R&D intensity) and size, the resulting IDR shows noticeable improved predictive power for subsequent returns. 14

15 tween the regressors in my cross-sectional model and expected returns. 18 Barring any cash ow eect, this positive correlation predicts positive coecients of B/M on ROE, OP, and R&D intensity, opposite to what we nd in the data. Together with the negative correlations between B/M and these regressors in Panel B of Table 1, empirical patterns are consistent with my working hypothesis that these regressors predominately capture B/M variation due to expected protability. Nevertheless, it is ultimately an empirical question whether the tted value from the cross-sectional model indeed controls for expected protability and the resulting IDR has superior predictive power for long-term returns. I now turn to investigate this question. 4 Testing long-term predictive power I rst use the portfolio sorts approach and use Fama-MacBeth regressions to examine the long-term predictive power of various expected-return estimates. Portfolio sorts and Fama- MacBeth regressions are complementary approaches. The portfolio approach requires few distributional assumptions. In contrast, the regression approach imposes parametric assumptions, and thus the estimation results are more sensitive to observations in the tails of the distributions. But if the assumptions are satised, regressions give us sharper results than those provided by the portfolio sorts. 4.1 Expected-return estimates I compute the IDR in two steps. In the rst step, I run the cross-sectional regression (Eq (13)) at the end of June every year t. I compute the mean (across years) of slope estimates up to year t (to avoid look-ahead bias) and multiply the average slopes by the year t value of explanatory variables to get the tted value in year t. In the second step, I subtract the 18 E.g., see Haugen and Baker (1996), Chan, Lakonishok, and Sougiannis (2001), Fama and French (2006), and Novy-Marx (2013) for a positive correlation between returns and ROE, R&D-to-sales, and operating protability, respectively.haugen and Baker (1996); Chan, Lakonishok, and Sougiannis (2001) 15

16 tted value from bm t to in order to arrive at the IDR, as dened according in Eq (14). I compare my IDR to expected-return estimates from three types of existing models: ICC models, factor models, and a characteristic-based model. I choose two recent ICC models (Hou, Dijk, and Zhang 2012, and Li, Ng, and Swaminathan 2013) from the large body of ICC literature because they represent the state-of-art ICC estimation techniques. Within the factor models, in addition to the commonly studied Fama-French three-factor model, I examine the expected-return estimates from the newly-proposed Fama-French ve-factor model (FF5) and the Hou, Xue, and Zhang four-factor model (HXZ4). Within the characteristic-based models, I examine the expected-return estimate from a seven-characteristic model (Lewellen, 2014) that includes size, B/M, OP, share issuance, momentum, and asset growth. I focus on the seven-characteristic model as Lewellen (2014) has also tested a three-characteristic model and a fteen-characteristic model and concludes that the seven-characteristic model outperforms the three-characteristic model while the fteen-characteristic model shows insignicant improvement over the seven-characteristic model. 19 I compute the expected-return estimates for all the models considered with the only exception of the ICC measure in Li, Ng, and Swaminathan (2013), which is provided by the authors. Appendix A provides a detailed description of the construction of these expectedreturn estimates. I use these expected-return estimates to predict returns for the period from July 1977 to June 2014 because the ICC measure from Li, Ng, and Swaminathan (2013) is not available before Hou, Dijk, and Zhang (2012) explore several versions of ICC models. I choose their Gordon growth version because it is one of the best performing one. Levi and Welch (2014) oer an extensive examination of numerous implementations of two factor models, CAPM and Fama-French three-factor model. They cannot reject the null that these models have no predictive power. Lyle and Wang (2015) proposes a characteristic-based model for returns with two characteristics, B/M and ROE. Their model exhibits weaker predictive power than the seven-characteristic model in most cases, which is not surprising given that the seven-characteristic model uses more characteristics. 20 Periods before 1977 will be used as a training period to compute expected return estimates. 16

17 4.2 Portfolio sorts based on expected-return estimates Forming portfolios minimizes the error-in-variables problem associated with the sorting variable (e.g., Black, Jensen, and Scholes, 1972; Fama and MacBeth, 1973). To minimize survivorship bias, I reinvest the proceeds of delisted stocks back into their original portfolios because delisted stocks and the remaining stocks in the same portfolio have similar expected-return estimates and thus should have similar expected returns if these estimates are accurate. 21 Lastly, I use the technique in Jegadeesh and Titman (1993) to examine the long-term predictive power. Every year, I form univariate sorted portfolios based on one expected-return estimate and hold the portfolios for 5 years. I compute the average return per annum over the 5-year holding period in the following way. Take the highest ranked portfolio as an example. I rst compute the year t + 1 return of a strategy that invests equal amount in the ve highest ranked portfolios that are formed on year t, year t 1,..., year t 4, respectively. The time-series average of the annual returns of this strategy is the estimate of the average return per annum over a 5-year holding period. Since annual returns have very low serial correlations, I can compute standard errors of this estimate without estimating autocorrelations. Unlike Jegadeesh and Titman (1993), I do not rebalance these portfolios (back to equal weights) in the holding period, making them comparable to illiquid corporate investments. 22 To balance the concern that the numerous tiny-cap stocks may have undue inuence in the equal-weighted portfolio results and the concern that the value-weighted portfolio results are dominated by mega-cap stocks, I sort stocks into three size groups. To make the number of stocks relatively the same across size groups, I dene three size groups, large-cap, small-cap, and tiny-cap using the NYSE 50th and 10th percentiles as breakpoints. 23 Within 21 Under the alternative assumption that the expected-return estimates are inaccurate, a proper reinvestment strategy is to invest delisting proceeds to the market portfolios. As a robustness check, I use this alternative reinvestment strategy in the regression approach. 22 This approach is in spirit similar to Hodrick (1992, Eq (11)), where he recommends avoiding the summation of autocovariances associated with regressions of long-term returns by regressing one-period return on a series of lagged regressors. 23 Large-cap, small-cap, and tiny-cap stocks on average account for approximately 92%, 7%, and less than 17

18 each size group, I further sort stocks into equal-weighted decile portfolios (P1 to P10) in the ascending order of an expected-return estimate. Table 3 presents the time-series means and the standard errors of P10-minus-P1 return spreads. Compared with the two ICC measures, the IDR is the most reliable predictor for cross-sectional dierences in long-term returns. For large-cap, small-cap, and tiny-cap stocks, respectively, the IDR generates an average P10-minus-P1 return spread of 5.1% (t=3.6), 9.2% (t=4.9), and 16.2% (t=5.2) per annum in the 5-year holding period. The return spread generated within the tiny-cap stocks is about three times as large as that in the large-cap, though the t-value is only about 65% higher. This suggests that the timeseries variability of return spreads in the tiny-cap sample is much larger. To put these return spreads into perspective, the corresponding P10-minus-P1 spreads generated by the ICC measure from Hou, Dijk, and Zhang (2012) are 1.6% (t=0.5), 4.6% (t=1.4), and 6.9% (t=2.8) for large-cap, small-cap, and tiny-cap stocks, respectively. Alternatively, when I sort on the ICC measure from Li, Ng, and Swaminathan (2013), the P10-minus-P1 spreads are 1.2% (t=0.8), -0.2% (t=-0.1), and 2.0% (t=0.5) for large-cap, small-cap, and tiny-cap stocks, respectively. Consistent with results in Hou, Dijk, and Zhang (2012), I nd that their ICC estimate predicts returns more strongly than the ICC estimate based on analyst forecasts, such as the one in Li, Ng, and Swaminathan (2013). 24 However, the improvement concentrates in the small-cap and tiny-cap samples, and only in the tiny-cap sample does the ICC estimate based on Hou, Dijk, and Zhang's (2012) approach generate a statistically signicant P10-minus-P1 return spread. In comparison, the IDR shows consistent predictive power across all size groups, generating P10-minus-P1 annual return spreads and associated t-statistics that are far larger than those of the other two ICC estimates across three size groups. The predictive power of the IDR relative to the two existing ICC measures particularly 1% of the market cap in my sample, respectively. 24 See Easton and Monahan (2005), Lewellen (2010) and Richardson, Tuna, and Wysocki (2010) for reviews of the forecasting performance of ICC measures. 18

19 stands out in large-cap stocks (P10-minus-P1 returns 3 times as large and t-statistics 4 times as large). This is interesting given the continual debate regarding why we observe the predictable cross-sectional dierences in returns. Arguments attributing predictability to behavioral biases typically predict that, due to the eect of arbitrage, the anomalies would be weaker in large-cap stocks as well as in the long run. In contrast, risk-based explanations do not predict a weaker risk-return relation for large-cap stocks or for the long run. Consequently, the evidence that the IDR can predict long-term returns among large-cap stocks makes it more likely to be related to risk characteristics as opposed to mispricing. This hypothesis is interesting but requires analysis beyond the scope of this study. Table 3 also compares the expected-return estimates from the IDR approach to those generated by FF3, FF5, and HXZ4 factor models. Results show that the factor models do not have signicant predictive power for long-term returns. P10-minus-P1 return spreads generated by sorting on their expected-return estimates yield t-values that are smaller than 1.3 across all three models and across all three size groups. My nding of weak predictive power from factor models is consistent with the ndings in Levi and Welch (2014) and Lyle and Wang (2015). Finally, Table 3 shows that the seven-characteristic model is the most successful one among existing models tested. This model performs on par with the IDR, generating a P10- minus-p1 return spread of 5.1% per annum (t=2.9), 12.8% (t=7.1), and 17.4% (t=7.1) in the large-cap, small-cap, and tiny-cap samples, respectively. This section shows that the IDR can reliably predict 5-year holding period returns consistently across three size groups. In terms of the statistical signicance of P10-minus- P1 returns spreads, the predictive power of the IDR can be matched only by the sevencharacteristic model among the three types of existing models tested. A characteristic-based model is dened as a regression model of returns based on characteristics. Researchers are still in the process of understanding why characteristics directly predict returns and the lack of a coherent explanation hinders the search of more characteristics that can enhance the 19

20 predictive power of the characteristic-based model. In contrast, the IDR approach is more intuitive. It comes from the present value formula and builds on the fact that at the rm level protability is far easier to forecast than returns. More earnings forecasting variables can be incorporated to extend my benchmark model, and the better we can control for expected protability, the better expected-return estimate we can get. 4.3 Univariate Fama-MacBeth regressions to predict 5-year returns This section examines an alternative approach to the portfolio sorts, I run Fama-MacBeth regressions of overlapping 5-year returns on various expected-return estimates across the three size groups. Time-series variations in the slopes from Fama-MacBeth regressions reect the overall sampling error consisting of both estimation errors of expected-return estimates and the usual error term. To calculate the 5-year cumulative return, I reinvest the delisted proceeds into the CRSP value-weighted market portfolio. This reinvestment assumption is dierent from that in the portfolio sorts approach and serves as a robustness check to conrm that inference of long-term predictive power is not driven by a particular reinvesting assumption. I use Newey-West standard errors with four lags to adjust for the autocorrelation generated by overlapping returns. To address the concern that Newey-West standard errors may not account for autocorrelations accurately in a nite sample, in Section 8 I propose an alternative approach to estimate the slope coecients in which autocorrelations are less of an issue. Fama (1976, Chapter 9) shows that slope coecients from Fama-MacBeth regressions are portfolio returns. Specically, the slope estimate of regressing 5-year return r t,5 on a cross-sectionally demeaned regressor X t is the following: β t,5 = ( X tx t ) 1 X tr t,5 (15) = w tr t,5. (16) 20

21 Therefore, the slope estimates can be interpreted as the 5-year holding period return of a long-short portfolio with the weights implied by the Fama-MacBeth regression. 25 For ease of comparing the regression coecients across tested models, I scale the slopes of Fama- MacBeth regressions in a way that the portfolio weights implied by the scaled slopes are one-dollar long and one-dollar short and the scaled slopes can be interpreted as returns to a one-dollar long-short position. 26 Table 4 presents Fama-MacBeth univariate regressions using each of the expected-return estimates. Explanatory variables are winsorized to the 1st and 99th percentiles. Consistent with the portfolio sorts results, the IDR strongly predicts 5-year returns across all three size groups (t=4.0, 4.8, and 5.0 for large-cap, small-cap, tiny-cap, respectively). The magnitude of the scaled predictive slopes is similar to the P10-minus-P1 return spread in the portfolio sorts approach. Also consistent with the portfolio sorts results, the ICC models and the factor models shows much weaker predictive power. Among fteen regressions across these ve models and three size groups, the 5% level signicance appears only three times, twice of the ICC measure Hou, Dijk, and Zhang (2012) in the small-cap and tiny-cap samples (t=2.0 and 2.9 respectively and once for FF3 in the large-cap sample (t=3.0). Finally, similar to the portfolio sorts results, Fama-MacBeth regressions show that the seven-characteristic model has long-term predictive power similar to that of the IDR. The predictive slopes are highly signicantly across all size groups, with t-statistics of 3.8 in the large-cap sample, 7.8 in the small-cap sample, and 7.1 in the tiny-cap sample. Overall, there are only two material dierences in statistical inference between the Fama- MacBeth regression approach and the portfolio sorts approach: Fama-MacBeth regressions nd the ICC measure from Hou, Dijk, and Zhang (2012) to be statistically signicant in the small-cap sample and the expected-return estimate from FF3 to be signicant in the 25 Back, Kapadia, and Ostdiek (2015) show that Fama-MacBeth regression slopes are returns of the maximally diversied portfolios that have a unit pure exposure to the explanatory variables. 26 The portfolio weights implied by the unscaled Fama-MacBeth regression slopes depend on the crosssectional variation in the expected-return estimates and thus the magnitude of these portfolio returns cannot be readily compared across tested models. 21

22 large-cap sample, while the portfolio sorts do not nd such signicance. One reason is that the Fama-MacBeth regressions might have better power because they use information from all stocks whereas portfolio sorts only use the two extreme deciles. Alternatively, the Newey- West standard errors may not be accurate because they require estimates of autocorrelations from a relatively short sample. In Appendix B, I take account of the second possibility by using an alternative approach to estimate Fama-MacBeth regression slopes. The alternative approach reconciles the difference in statistical inferences between portfolio sorts and Fama-MacBeth regressions. The IDR exhibits slightly stronger statistical signicance under the alternative approach. To be conservative, I continue to use Newey-West standard errors to adjust for autocorrelations generated by overlapping returns. But in the robustness section, I will report separate Fama- MacBeth regressions for annual returns 1- to 5-years ahead and in these regressions returns no longer overlap. 4.4 Bivariate Fama-MacBeth regressions to predict 5-year returns Fama-MacBeth regressions also allow a straightforward way to examine whether the IDR has incremental explanatory power over the expected-return estimates from the existing models. In particular, since the characteristic-based model performs quite well and the characteristics contain protability measures, I explore whether the estimate from the characteristic-based model and the IDR subsume one another. Table 5 presents the results for bivariate regressions using the IDR along with expectedreturn estimates from each of the existing models considered. I nd that the IDR retains its signicance in all twenty-one cases (three size groups and seven estimates from existing models). The column labeled C7-JL shows that the expected-return estimate generated by the seven-characteristic model in Lewellen (2014) and the IDR have independent explanatory power in describing the cross section of 5-year returns in the tiny-cap and small-cap samples, all with t-values above 3.5. Interestingly, in the large-cap sample, the IDR remains signi- 22

23 cant (t=2.7) whereas the estimate from the seven-characteristic model becomes insignicant (t=1.3). 5 Implications for cost of equity 5.1 Decile portfolios sorted on the IDR Results in the previous section have shown that the IDR is a robust and powerful predictor for subsequent long-term returns. I now investigate whether the predictive power only comes from rms with extreme IDR value. If the predictive power comes solely from the tail distribution, although we can still use the IDR to construct long-short positions that earn signicant excess returns, the IDR will have little impact on our estimate of long-term expected returns, or cost of capital, for majority of the rms. Table 6 nds that the average returns of the decile portfolios sorted on the IDR are generally increasing with the IDR ranking. 27 In the large-cap sample, the average return per annum in a 5-year holding period increases by 2.6% when the IDR ranking increases from 1 to 6, and by another 2.5% when the ranking increases to 10. This suggests that, among large-cap rms, there is a dierence about 2.5% per annum in the discount rate over a 5-year investment horizon between rms in the lowest IDR decile and rms in the median IDR decile (or between rms in the median IDR decile and rms in the highest IDR decile). Although a 2.5% dierence in cost of capital is quite substantial in economic terms, the sampling error of average returns is large enough to render the 2.5% return spread statistically insignicant (t=1.5). The return spread only becomes statistically signicant when the change in rankings is large. The return dierences among small-cap and tiny-cap stocks are are much larger. Among the small-cap stocks, the average annual return over a 5-year holding period increases by 5.6% when the IDR ranking increases from 1 to 6 (t=2.9) and by another 3.5% when the ranking 27 These portfolios are rst introduced in Section

24 increases to 10. These return spreads are about 50% to 100% larger than the corresponding numbers in the large-cap sample. The corresponding return spreads among the tiny-cap stocks are even larger, typically more than 50% larger compared to the small-cap sample and 2 times larger compared to the large-cap sample. There are two interpretations for these results. First, the dierences in long-term expected returns are small among large-cap rms. Second, the IDR has stronger predictive power among small-cap and tiny-cap stocks. To sum up, average 5-year returns increase gradually with the IDR across decile portfolios. The pattern suggests that the IDR can be useful for improving our cost of capital estimate for most rms. Among large-cap stocks, we nd that although the IDR is a statistically signicant predictor for 5-year returns when considering the whole cross section, the predicted return dierence only becomes signicant if the rms' IDRs are suciently dierent from each other. On the other hand, return spreads generated by sorting on the IDR are markedly larger among tiny-cap stocks. Tiny-cap stocks in the lowest (highest) IDR decile earn around 8% per annum less (more) than tiny-cap stocks in the median IDR decile. If one holds the view that all return predictability captured by the IDR is due to mispricing, the tiny-cap results here indicate the relative mispricing between the lowest and the highest IDR deciles is 80% 16% per year for ve years. 5.2 Fama-French industry portfolios In the following analysis, I investigate how the new IDR measure can help improve our estimates of long-term expected returns. Running pooled time-series and cross-sectional regressions of future realized returns on predicted returns is a good way to gauge the predictive power of a model. The regression slope tells how informative the predictor is and the R 2 indicates how much variations in returns a model can explain More specically, the regression slope b is equivalent to the optimal weight a shrinkage estimator F s i,t = sf i,t should put on F i,t to minimize the mean-squared-forecast-error (MSFE). That is, b = arg min {s} MSF E (F s 1 i,t). MSFE of predicted returns F i,t from a particular model is N T i,t (R i,t F i,t ) 2. Both R i,t and F i,t are cross-sectionally demeaned. So F i,t is a better predictor than 0 (the cross-sectional [ ] 1 mean of F i,t ) if N T i,t (R i,t F i,t ) 2 Ri,t 2 1 < 0. That is, N T i,t (F i,t) 2 (1 2b) < 0. b is the re- 24