# Characteristic-Based Benchmark Returns and Corporate Events

Characteristic-Based Benchmark Returns and Corporate Events Abstract We propose that fitted values from market-wide regressions of firm returns on lagged firm characteristics provide useful benchmarks for assessing whether average returns to certain stocks are abnormal. To illustrate, we study eight documented events with abnormal returns, including credit rating and analyst recommendation downgrades, initial and seasoned public equity offerings, mergers and acquisitions, dividend initiations, share repurchases, and stock splits. We show that the apparently abnormal returns in the months after these events are substantially reduced or eliminated when compared to characteristic-based benchmarks. Characteristic-based benchmarks perform better in explaining post-event returns than do recent four- and five-factor models. Received September 19, 2016; editorial decision February 16, 2018 by Editor Andrew Karolyi. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web Site next to the link to the final published paper online. The assessment of whether average returns to certain securities during time intervals of interest are abnormal is one of the most frequently encountered empirical exercises in the field of Finance. Of course, such assessments require that researchers specify normal or benchmark returns. As noted by Kothari and Warner (2007), this requirement is of relatively minor importance for studies that focus on abnormal returns over short time intervals, such as 2 or 3 days, but can be of first-order importance for studies that measure abnormal returns over longer horizons. The methods used to assess whether returns to firms of interest (“event firms”) are abnormal often rely on strong assumptions regarding normal or benchmark returns. For example, comparing returns for event firms to returns for control firms selected based on characteristics such as size or market-to-book ratio implicitly makes the strong assumption that benchmark returns depend only on the characteristics used to select control firms. Similarly, estimates of abnormal returns obtained as intercepts (“alphas”) estimated by regressions of event firm returns on certain market-wide factors implicitly assume that benchmark returns depend on only those factor outcomes and firm return sensitivities. In practice, finance researchers have documented that equity returns are systematically related to a large number of observable variables. Haugen and Baker (1996) demonstrate that a set of 46 firm-level variables has significant forecast power for the next month’s stock returns. Lewellen (2015) shows for a more recent sample that predicted returns based on 15 firm characteristics forecast the next month’s realized returns. Green, Hand, and Zhang (2017) report that 24 “return predictive variables” forecast stock returns in multivariate cross-sectional regressions, each with $$t$$-statistics in excess of 3, that is, the threshold recommended by Harvey, Liu, and Zhu (2016). Collectively, the literature shows that firm characteristics have significant explanatory and predictive power for the cross-section of stock returns. At the same time, firms undergoing important events may differ from nonevent firms in terms of some of the same characteristics that explain the cross-section of returns.1 In this paper, we propose an explicit characteristic-based benchmark to evaluate event firm stock returns. In particular, we follow Lewellen (2015) and Haugen and Baker (1996) in estimating relations between returns and prior-month characteristics, using all common stocks. We then assess whether returns to event firms are abnormal by comparing realized returns for event firms to the same firms’ characteristic-based predicted returns. In essence, we assess whether average returns for event firms lie on the estimated hyperplane that relates returns to characteristics for the full stock market. This approach accommodates both relations between characteristics and predicted returns that exist for the broad stock market and the fact that event firms often differ from nonevent firms in terms of relevant characteristics. The method we propose is similar in intent to the widely used method where control stocks are matched to event stocks in terms of observable firm characteristics such as size or market-to-book ratio, since these characteristics are selected precisely because they are known to be related to returns. However, the proposed method can be used to control for as many observable characteristics as desired, while identifying firms that are well matched to event firms based on a large number of characteristics may not be practical. Bessembinder and Zhang (2013) also consider the role of firm characteristics, by focusing on intercepts estimated in regressions of differences in returns across event and control firms on differences in characteristics. The method proposed here is more direct. In particular, we eliminate the need to identify control firms entirely, as we compare realized to predicted returns. Further, the method here relies on cross-sectional relations between returns and characteristics estimated for the market as a whole, while the Bessembinder and Zhang regressions are estimated using only event and control firms. As we show in Section 1.1, the use of information from the broad stock market results in strong statistical power for the tests in this paper. To illustrate the method, we study several important corporate events, including credit rating downgrades, analyst recommendation downgrades, initial and secondary public equity offerings, mergers and acquisitions, dividend initiations, share repurchases and stock splits. While we believe that the methods proposed here will be useful in a broad array of applications, these events provide a useful illustration of the method because of the considerable prior research attention they have received. We focus in particular on log returns (each defined as the natural logarithm of the gross simple return) because of the direct linkage between each time series mean log return and the corresponding cumulative return to a buy-and-hold investor, and because average log returns appear to be anomalous even in our updated sample. We document that 14 firm characteristics drawn from Lewellen (2015) have substantial explanatory power for the apparently abnormal returns in the months after the eight corporate events studied here. With the exception of credit rating downgrade events, the data does not reject the hypothesis that the 14 characteristics fully explain average post-event returns. Hou, Xue, and Zhang (2015) and Fama and French (2016) show that their recently proposed risk factor models have significant success in explaining returns to portfolios sorted based on firm characteristics, including net share issuance. In light of this evidence, we assess the performance of these factor models in explaining event-firm returns in the months after the eight corporate events considered in this study. We find that characteristic-based benchmarks are significantly more effective in explaining post-event returns than either the Fama-French (2015) or the Hou-Xue-Zhang (2015) factor models. Further, when we decompose each characteristic into a portion explained by Fama-French and Hou-Xue-Zhang factor loadings (betas) and an unexplained portion, we find that the portion orthogonal to the factor loadings has significantly more explanatory power. The economic interpretations of the findings in this paper are inseparable from the unresolved question of why firm characteristics can significantly explain the cross-section of equity returns. Our results show that the 14 characteristics outperform loadings on observable factors drawn from recent models in terms of explaining event firm returns, whereas Chordia, Goyal, and Shanken (2015) show that characteristics outperform loadings from the same models in explaining the cross-section of individual stock returns. Although some might be inclined to interpret this evidence as supportive of market inefficiencies, Kozak, Nagel, and Santosh (2017) emphasize the difficulty of distinguishing between rational and behavioral explanations for asset pricing on the basis of “horse races” between factors and characteristics. Kelly, Pruitt, and Su (2018) present evidence that much of the explanatory power attributed to observable firm characteristics arises because the characteristics act as instrumental variables for time-varying loadings on a small set of unobservable common factors. Our reliance on 14 characteristics is based mainly on the evidence provided by Lewellen (2015) that these characteristics have explanatory and predictive power for returns in the recent data. However, we do not take a stand as to the optimal set of characteristics, as this issue also remains the focus of ongoing research. Freyberger, Neuhierl, and Weber (2017) adopt a nonparametric approach, and report that fifteen characteristics have independent explanatory power in their full sample. Kelly, Pruitt, and Su (2018) report that a set of just eight characteristics is responsible for the success of their latent factor model. An advantage of our proposed method is that it focuses on a relatively simple question, the answer to which is of interest independent of the outcome of ongoing research regarding the economic interpretation of characteristics’ ability to explain returns. In particular, our method answers the question of whether average returns to event firms differ significantly from the level that would be anticipated based on event firm characteristics and the estimated market-wide relation between returns and firm characteristics. As such, the proposed method clarifies whether event firm returns require any explanation beyond the characteristics of the firms engaging in the events. 1. The Proposed Method We propose a two-stage method to assess whether average returns to certain firms during time periods of interest are abnormal. In the first stage, we follow Lewellen (2015) and Haugen and Baker (1996) in estimating predicted returns for all (not just event) stocks by means of cross-sectional regressions of firm returns on lagged firm characteristics. Specifically, we estimate the following cross-sectional regression for each month $$t$$: $$\label{eq1} R_{it}=\alpha_{t}+\beta_{t}X_{i,t-1}+\epsilon _{it},$$ (1) where $$R_{it}$$ is stock $$i$$’s realized log (or simple) return in month t, and $$X_{i,t-1}$$ is a vector of firm $$i$$’s characteristics measured at the end of month $$t$$-1. Also following the two studies, we estimate predicted returns using rolling averages of past slope coefficients. In particular the predicted return for month $$t$$ is the average intercept over the prior 12 months plus the sum of products of average slope coefficients over the prior 12 months and month $$t$$-1 characteristics:2 $$E\left[ R_{it}\vert I_{t-1} \right]=\frac{1}{12}\sum\nolimits_{s=t-12}^{t-1} \hat{\alpha }_{s} +\left(\frac{1}{12}\sum\nolimits_{s=t-12}^{t-1} \hat{\beta }_{s} \right)X_{i,t-1},$$ (2) where $$\hat{\alpha }_{s}$$ and $$\hat{\beta }_{s}$$ are the coefficients estimated from Equation (1) in month $$s$$. In the second stage we regress, again using all stocks, differences between realized and predicted returns on a constant and on indicator variables that are set to one for firm/months of interest, and zero for other firm/months: $$\label{eq2} R_{it}-E\left[ R_{it}\vert I_{t-1} \right]=a+\sum\nolimits_{k=1}^K {b_{k}\times D_{itk}} +u_{it},$$ (3) where $$D_{itk}$$ is an indicator that equals one if firm i experienced corporate event $$k$$ during a specified horizon prior to month $$t$$, and zero otherwise. For example, the dividend initiation indicator in month $$t$$ equals one for those firms that initiated dividends between months $$t$$-1 and $$t$$-36, and zero otherwise. Coefficient estimates on the indicator variables ($$b_{k})$$ reveal the extent to which the average difference between realized returns and predicted returns differs during the specified post-event horizon for event firms as compared to nonevent firms. This approach is flexible and offers several advantages. First, the performance of the characteristic-based benchmark can readily be compared to alternative benchmarks by simply deducting the alternative benchmark return from the realized return instead. Second, the method can be adapted to either place equal weight on each event, by estimating the second stage as a pooled regression, or equal weight on each time period, by estimating the second stage by the well-known Fama-MacBeth (1973) approach.3 Third, authors can study simple or continuously compounded (log) returns as preferred. Fourth, the predicted return is based on information available prior to the return date, which potentially facilitates trading on or hedging against the unexpected returns identified by the method. Fifth, as we show, the method has strong statistical power, attributable at least in part to the fact that it estimates relations between returns and characteristics using the full cross-section of stocks, not just event firms. Sixth, the approach can be applied to event returns measured over any horizon, ranging from a single day to long horizons, such as 3 or 5 years. Finally, the method can allow for as many characteristics as a researcher deems appropriate. Regardless of the number of characteristics considered, their effect is captured through a single metric, the characteristic-based predicted return for the firm and month.4 We mainly emphasize results obtained with a set of 14 characteristics drawn from the fifteen that Lewellen (2015) shows to successfully predict future stock returns. The exception is that we exclude stock issuance as a variable to estimate predicted returns, because we study herein stock returns after equity offerings. Table A1 defines the set of 14 characteristics, which we denote C14. Table A1 Definition of the C5 and C14 firm characteristics Characteristics in the C5 model log size Natural log of market capitalization, which is stock price (prc in CRSP monthly stock file) times number of shares outstanding (shrout), at the end of the prior month log book-to-market ratio Natural log of the book-to-market ratio at the end of the prior month. Book value is the firm’s common equity (Compustat item ceq) in the latest annual report. Market value is the firm’s market capitalization (prc times shrout) at the end of the prior month reported in CRSP Momentum Buy-and-hold stock returns over months ($$-$$12, $$-$$2) before the month of interest ROA Income before extraordinary items (ib) divided by average total assets (at) in the year Asset growth Natural log of the ratio of total assets (at) at the end of the year to total assets at the beginning of the year, following Cooper, Gulen, and Schill (2008) Additional nine characteristics in the C14 model Beta Market beta estimated using monthly excess stock returns and market risk premiums over the preceding 60 months. We require a minimum of six data points for the accuracy of the estimation Accrual Change in working capital from the last year minus depreciation and amortization (dp), divided by average total assets (at) in the year, following Sloan (1996). Working capital equals current assets (act) minus cash and short-term investment (che) minus current liabilities (lct) plus debt in current liabilities (dlc) plus income taxes payable (txp). Missing act, che, lct, dlc, txp, and dp are replaced with zero Dividend Dividends per share over the prior 12 months divided by the price at the end of the prior month log LR return Natural log of buy-and-hold stock returns over months ($$-$$13, $$-$$36) before the month of interest Idiosyncratic risk In each month, we compute the standard deviation of the residual daily stock returns in the Fama and French (1993) three factor regression, following Ang et al. (2006). Idiosyncratic risk is the average standard deviation over the prior 12 months Illiquidity The average daily ratio of absolute stock return to dollar trading volume during the prior 12 months, as defined by Amihud (2002) Turnover Average monthly turnover (shares traded divided by shares outstanding) during the prior 12 months Leverage Debt in current liabilities (dlc) plus long-term debt (dltt), divided by market capitalization (prc times shrout in CRSP) at the end of the last month. Missing dlc and dltt are replaced with zero Sales/price Sales (sale) divided by market capitalization (prc times shrout in CRSP) at the end of the last month Characteristics in the C5 model log size Natural log of market capitalization, which is stock price (prc in CRSP monthly stock file) times number of shares outstanding (shrout), at the end of the prior month log book-to-market ratio Natural log of the book-to-market ratio at the end of the prior month. Book value is the firm’s common equity (Compustat item ceq) in the latest annual report. Market value is the firm’s market capitalization (prc times shrout) at the end of the prior month reported in CRSP Momentum Buy-and-hold stock returns over months ($$-$$12, $$-$$2) before the month of interest ROA Income before extraordinary items (ib) divided by average total assets (at) in the year Asset growth Natural log of the ratio of total assets (at) at the end of the year to total assets at the beginning of the year, following Cooper, Gulen, and Schill (2008) Additional nine characteristics in the C14 model Beta Market beta estimated using monthly excess stock returns and market risk premiums over the preceding 60 months. We require a minimum of six data points for the accuracy of the estimation Accrual Change in working capital from the last year minus depreciation and amortization (dp), divided by average total assets (at) in the year, following Sloan (1996). Working capital equals current assets (act) minus cash and short-term investment (che) minus current liabilities (lct) plus debt in current liabilities (dlc) plus income taxes payable (txp). Missing act, che, lct, dlc, txp, and dp are replaced with zero Dividend Dividends per share over the prior 12 months divided by the price at the end of the prior month log LR return Natural log of buy-and-hold stock returns over months ($$-$$13, $$-$$36) before the month of interest Idiosyncratic risk In each month, we compute the standard deviation of the residual daily stock returns in the Fama and French (1993) three factor regression, following Ang et al. (2006). Idiosyncratic risk is the average standard deviation over the prior 12 months Illiquidity The average daily ratio of absolute stock return to dollar trading volume during the prior 12 months, as defined by Amihud (2002) Turnover Average monthly turnover (shares traded divided by shares outstanding) during the prior 12 months Leverage Debt in current liabilities (dlc) plus long-term debt (dltt), divided by market capitalization (prc times shrout in CRSP) at the end of the last month. Missing dlc and dltt are replaced with zero Sales/price Sales (sale) divided by market capitalization (prc times shrout in CRSP) at the end of the last month We measure these characteristics following Lewellen (2015). All variables are created using data from the CRSP stock price files and the Compustat annual data. Accounting data are assumed to be available 4 months after the fiscal year end. Table A1 Definition of the C5 and C14 firm characteristics Characteristics in the C5 model log size Natural log of market capitalization, which is stock price (prc in CRSP monthly stock file) times number of shares outstanding (shrout), at the end of the prior month log book-to-market ratio Natural log of the book-to-market ratio at the end of the prior month. Book value is the firm’s common equity (Compustat item ceq) in the latest annual report. Market value is the firm’s market capitalization (prc times shrout) at the end of the prior month reported in CRSP Momentum Buy-and-hold stock returns over months ($$-$$12, $$-$$2) before the month of interest ROA Income before extraordinary items (ib) divided by average total assets (at) in the year Asset growth Natural log of the ratio of total assets (at) at the end of the year to total assets at the beginning of the year, following Cooper, Gulen, and Schill (2008) Additional nine characteristics in the C14 model Beta Market beta estimated using monthly excess stock returns and market risk premiums over the preceding 60 months. We require a minimum of six data points for the accuracy of the estimation Accrual Change in working capital from the last year minus depreciation and amortization (dp), divided by average total assets (at) in the year, following Sloan (1996). Working capital equals current assets (act) minus cash and short-term investment (che) minus current liabilities (lct) plus debt in current liabilities (dlc) plus income taxes payable (txp). Missing act, che, lct, dlc, txp, and dp are replaced with zero Dividend Dividends per share over the prior 12 months divided by the price at the end of the prior month log LR return Natural log of buy-and-hold stock returns over months ($$-$$13, $$-$$36) before the month of interest Idiosyncratic risk In each month, we compute the standard deviation of the residual daily stock returns in the Fama and French (1993) three factor regression, following Ang et al. (2006). Idiosyncratic risk is the average standard deviation over the prior 12 months Illiquidity The average daily ratio of absolute stock return to dollar trading volume during the prior 12 months, as defined by Amihud (2002) Turnover Average monthly turnover (shares traded divided by shares outstanding) during the prior 12 months Leverage Debt in current liabilities (dlc) plus long-term debt (dltt), divided by market capitalization (prc times shrout in CRSP) at the end of the last month. Missing dlc and dltt are replaced with zero Sales/price Sales (sale) divided by market capitalization (prc times shrout in CRSP) at the end of the last month Characteristics in the C5 model log size Natural log of market capitalization, which is stock price (prc in CRSP monthly stock file) times number of shares outstanding (shrout), at the end of the prior month log book-to-market ratio Natural log of the book-to-market ratio at the end of the prior month. Book value is the firm’s common equity (Compustat item ceq) in the latest annual report. Market value is the firm’s market capitalization (prc times shrout) at the end of the prior month reported in CRSP Momentum Buy-and-hold stock returns over months ($$-$$12, $$-$$2) before the month of interest ROA Income before extraordinary items (ib) divided by average total assets (at) in the year Asset growth Natural log of the ratio of total assets (at) at the end of the year to total assets at the beginning of the year, following Cooper, Gulen, and Schill (2008) Additional nine characteristics in the C14 model Beta Market beta estimated using monthly excess stock returns and market risk premiums over the preceding 60 months. We require a minimum of six data points for the accuracy of the estimation Accrual Change in working capital from the last year minus depreciation and amortization (dp), divided by average total assets (at) in the year, following Sloan (1996). Working capital equals current assets (act) minus cash and short-term investment (che) minus current liabilities (lct) plus debt in current liabilities (dlc) plus income taxes payable (txp). Missing act, che, lct, dlc, txp, and dp are replaced with zero Dividend Dividends per share over the prior 12 months divided by the price at the end of the prior month log LR return Natural log of buy-and-hold stock returns over months ($$-$$13, $$-$$36) before the month of interest Idiosyncratic risk In each month, we compute the standard deviation of the residual daily stock returns in the Fama and French (1993) three factor regression, following Ang et al. (2006). Idiosyncratic risk is the average standard deviation over the prior 12 months Illiquidity The average daily ratio of absolute stock return to dollar trading volume during the prior 12 months, as defined by Amihud (2002) Turnover Average monthly turnover (shares traded divided by shares outstanding) during the prior 12 months Leverage Debt in current liabilities (dlc) plus long-term debt (dltt), divided by market capitalization (prc times shrout in CRSP) at the end of the last month. Missing dlc and dltt are replaced with zero Sales/price Sales (sale) divided by market capitalization (prc times shrout in CRSP) at the end of the last month We measure these characteristics following Lewellen (2015). All variables are created using data from the CRSP stock price files and the Compustat annual data. Accounting data are assumed to be available 4 months after the fiscal year end. In addition, we study a subset of only five firm characteristics: firm size, book-to-market ratio, stock returns over months $$t$$-2 to $$t$$-12, profitability as measured by return on assets (ROA), and the firm’s rate of investment as measured by year-on-year growth in total assets, which we denote C5. These five characteristics correspond to the risk factors in the recently proposed asset pricing models of Fama and French (2015) and Hou, Xue, and Zhang (2015), except that we include momentum based on the evidence in Carhart (1997) and subsequent studies, and exclude firms’ market beta.5 We focus on the period from January 1970 to December 2014, because the corporate event samples we study start in 1980, and, in some specifications, we rely on up to 10 years of prior data to estimate predicted stock returns. Table A-2 of the Internet Appendix presents summary statistics regarding the firm characteristics, each measured on a monthly basis. To make coefficients on firm characteristics comparable across characteristics and time, we normalize each firm characteristic in each month by subtracting the cross-sectional mean for the month and dividing by the cross-sectional standard deviation for the month. That is, all firm characteristics have mean of zero and standard deviation of one each month. In Table 1 we report average coefficients on the firm characteristics obtained when estimating expression (1) over the period January 1970 to December 2014, for the C5 and C14 characteristics. (Corresponding results for the 46 Haugen and Baker (1996) characteristics, denoted C46, are contained in Table A-3 of the Internet Appendix.) Outcomes are broadly similar to those previously reported by Lewellen (2015) and Haugen and Baker (1996), although we note that the sign and significance on some of the coefficients (e.g., size and ROA) are sensitive to use of simple versus log returns. Table 1 Average coefficients on each firm characteristic across the sample period, January 1970 to December 2014 (1) (2) (3) (4) C5 C14 C5 C14 Dep. var. Simple return log return log size –0.2191*** –0.2373*** 0.2543*** –0.1313** (–2.82) (–4.35) (3.59) (–2.55) log book-to-market ratio 0.5177*** 0.4071*** 0.5765*** 0.4348*** (7.48) (7.74) (8.97) (9.40) Momentum 0.3894*** 0.3866*** 0.5875*** 0.5688*** (5.98) (7.43) (9.35) (11.57) ROA 0.0905 0.0774* 0.5674*** 0.3339*** (1.29) (1.74) (8.93) (8.88) Asset growth –0.3078*** –0.2217*** –0.4014*** –0.2788*** (–8.44) (–7.95) (–8.83) (–9.24) Beta 0.0859* 0.0328 (1.84) (0.67) Accrual –0.1137*** –0.1171*** (–5.82) (–5.98) Dividend 0.0083 0.0339 (0.21) (0.87) log LR return –0.0368 0.0120 (–0.92) (0.33) Idio. risk –0.2298*** –0.9077*** (–2.63) (–10.21) Illiquidity 0.2970*** 0.2992*** (6.36) (7.36) Turnover –0.0016 –0.1713*** (–0.04) (–4.38) Leverage –0.1037*** –0.2107*** (–2.87) (–5.43) Sales/price 0.1696*** 0.0855** (3.68) (2.20) Constant 1.2668*** 1.2668*** –0.0032 –0.0032 (4.36) (4.36) (–0.01) (–0.01) R2 0.036 0.062 0.042 0.071 (1) (2) (3) (4) C5 C14 C5 C14 Dep. var. Simple return log return log size –0.2191*** –0.2373*** 0.2543*** –0.1313** (–2.82) (–4.35) (3.59) (–2.55) log book-to-market ratio 0.5177*** 0.4071*** 0.5765*** 0.4348*** (7.48) (7.74) (8.97) (9.40) Momentum 0.3894*** 0.3866*** 0.5875*** 0.5688*** (5.98) (7.43) (9.35) (11.57) ROA 0.0905 0.0774* 0.5674*** 0.3339*** (1.29) (1.74) (8.93) (8.88) Asset growth –0.3078*** –0.2217*** –0.4014*** –0.2788*** (–8.44) (–7.95) (–8.83) (–9.24) Beta 0.0859* 0.0328 (1.84) (0.67) Accrual –0.1137*** –0.1171*** (–5.82) (–5.98) Dividend 0.0083 0.0339 (0.21) (0.87) log LR return –0.0368 0.0120 (–0.92) (0.33) Idio. risk –0.2298*** –0.9077*** (–2.63) (–10.21) Illiquidity 0.2970*** 0.2992*** (6.36) (7.36) Turnover –0.0016 –0.1713*** (–0.04) (–4.38) Leverage –0.1037*** –0.2107*** (–2.87) (–5.43) Sales/price 0.1696*** 0.0855** (3.68) (2.20) Constant 1.2668*** 1.2668*** –0.0032 –0.0032 (4.36) (4.36) (–0.01) (–0.01) R2 0.036 0.062 0.042 0.071 Each month, we estimate cross-sectional regressions of firm monthly simple and log stock returns on characteristics measured at the end of the preceding month as specified in Equation (1). This table presents average coefficients over time. Firm characteristics are winsorized within each month at the upper and the lower 1% and are normalized by subtracting the mean and dividing by the standard deviation. See Table A1 for variable definition. The Fama-MacBeth standard errors are based on the time-series variability of the estimates, incorporating a Newey-West correction with four lags. The associated $$t$$-statistics are reported in the parentheses below each coefficient. ***, **, and * correspond to statistical significance at the 1%, 5%, and 10% levels, respectively. Table 1 Average coefficients on each firm characteristic across the sample period, January 1970 to December 2014 (1) (2) (3) (4) C5 C14 C5 C14 Dep. var. Simple return log return log size –0.2191*** –0.2373*** 0.2543*** –0.1313** (–2.82) (–4.35) (3.59) (–2.55) log book-to-market ratio 0.5177*** 0.4071*** 0.5765*** 0.4348*** (7.48) (7.74) (8.97) (9.40) Momentum 0.3894*** 0.3866*** 0.5875*** 0.5688*** (5.98) (7.43) (9.35) (11.57) ROA 0.0905 0.0774* 0.5674*** 0.3339*** (1.29) (1.74) (8.93) (8.88) Asset growth –0.3078*** –0.2217*** –0.4014*** –0.2788*** (–8.44) (–7.95) (–8.83) (–9.24) Beta 0.0859* 0.0328 (1.84) (0.67) Accrual –0.1137*** –0.1171*** (–5.82) (–5.98) Dividend 0.0083 0.0339 (0.21) (0.87) log LR return –0.0368 0.0120 (–0.92) (0.33) Idio. risk –0.2298*** –0.9077*** (–2.63) (–10.21) Illiquidity 0.2970*** 0.2992*** (6.36) (7.36) Turnover –0.0016 –0.1713*** (–0.04) (–4.38) Leverage –0.1037*** –0.2107*** (–2.87) (–5.43) Sales/price 0.1696*** 0.0855** (3.68) (2.20) Constant 1.2668*** 1.2668*** –0.0032 –0.0032 (4.36) (4.36) (–0.01) (–0.01) R2 0.036 0.062 0.042 0.071 (1) (2) (3) (4) C5 C14 C5 C14 Dep. var. Simple return log return log size –0.2191*** –0.2373*** 0.2543*** –0.1313** (–2.82) (–4.35) (3.59) (–2.55) log book-to-market ratio 0.5177*** 0.4071*** 0.5765*** 0.4348*** (7.48) (7.74) (8.97) (9.40) Momentum 0.3894*** 0.3866*** 0.5875*** 0.5688*** (5.98) (7.43) (9.35) (11.57) ROA 0.0905 0.0774* 0.5674*** 0.3339*** (1.29) (1.74) (8.93) (8.88) Asset growth –0.3078*** –0.2217*** –0.4014*** –0.2788*** (–8.44) (–7.95) (–8.83) (–9.24) Beta 0.0859* 0.0328 (1.84) (0.67) Accrual –0.1137*** –0.1171*** (–5.82) (–5.98) Dividend 0.0083 0.0339 (0.21) (0.87) log LR return –0.0368 0.0120 (–0.92) (0.33) Idio. risk –0.2298*** –0.9077*** (–2.63) (–10.21) Illiquidity 0.2970*** 0.2992*** (6.36) (7.36) Turnover –0.0016 –0.1713*** (–0.04) (–4.38) Leverage –0.1037*** –0.2107*** (–2.87) (–5.43) Sales/price 0.1696*** 0.0855** (3.68) (2.20) Constant 1.2668*** 1.2668*** –0.0032 –0.0032 (4.36) (4.36) (–0.01) (–0.01) R2 0.036 0.062 0.042 0.071 Each month, we estimate cross-sectional regressions of firm monthly simple and log stock returns on characteristics measured at the end of the preceding month as specified in Equation (1). This table presents average coefficients over time. Firm characteristics are winsorized within each month at the upper and the lower 1% and are normalized by subtracting the mean and dividing by the standard deviation. See Table A1 for variable definition. The Fama-MacBeth standard errors are based on the time-series variability of the estimates, incorporating a Newey-West correction with four lags. The associated $$t$$-statistics are reported in the parentheses below each coefficient. ***, **, and * correspond to statistical significance at the 1%, 5%, and 10% levels, respectively. We also verify that, consistent with the results reported by these studies, predicted returns based on the C46 and C14 characteristics do indeed have statistically significant explanatory power for the next month’s realized returns in the broad stock market. We report in Table 2 the estimates for the C5 and C14 characteristics, while we report in Table A-4 of the Internet Appendix the corresponding estimates for the C46 characteristics. Table 2 Predicted stock return and realized stock return A. Fama-MacBeth regression of actual return on predicted return, January 1980 to December 2014 C5 C14 C5 C14 Dependent var. Simple return log return Expected return 0.8000*** 0.5418*** 0.8044*** 0.7540*** (3.72) (8.24) (10.65) (12.32) [-0.93] [-6.97] [-2.59] [-4.02] Constant 1.0231 0.4718* –0.0947 –0.0799 (1.11) (1.71) (–0.35) (–0.31) N 1,886,673 1,886,673 1,886,673 1,886,673 # months 420 420 420 420 R2 0.012 0.016 0.018 0.025 A. Fama-MacBeth regression of actual return on predicted return, January 1980 to December 2014 C5 C14 C5 C14 Dependent var. Simple return log return Expected return 0.8000*** 0.5418*** 0.8044*** 0.7540*** (3.72) (8.24) (10.65) (12.32) [-0.93] [-6.97] [-2.59] [-4.02] Constant 1.0231 0.4718* –0.0947 –0.0799 (1.11) (1.71) (–0.35) (–0.31) N 1,886,673 1,886,673 1,886,673 1,886,673 # months 420 420 420 420 R2 0.012 0.016 0.018 0.025 B. Returns to portfolios sorted on predicted simple return C5 C14 C5 C14 Decile Ret SD Ret SD Ret SD Ret SD Equal-weighted Value-weighted Low –0.18 7.29 –0.35 7.85 0.36 6.64 0.12 7.61 2 0.47 6.23 0.46 6.55 0.73** 5.95 0.73** 6.16 3 0.79*** 5.71 0.73** 5.81 1.00*** 5.45 0.91*** 5.60 4 0.92*** 5.41 0.94*** 5.36 0.97*** 5.39 1.04*** 5.07 5 1.08*** 5.18 1.11*** 5.23 1.05*** 5.60 1.19*** 5.09 6 1.22*** 5.08 1.31*** 5.10 1.26*** 5.52 1.18*** 5.12 7 1.37*** 5.08 1.38*** 5.22 1.37*** 5.48 1.23*** 5.45 8 1.60*** 5.41 1.67*** 5.56 1.59*** 6.24 1.43*** 5.97 9 1.93*** 6.36 1.84*** 6.17 1.53*** 6.92 1.52*** 6.47 High 2.33*** 7.93 2.37*** 7.80 1.89*** 8.35 1.75*** 8.03 H - L 2.51*** 7.21 2.72*** 8.09 1.53*** 8.14 1.63*** 8.70 C. Returns to portfolios sorted on predicted log return Low –0.08 8.83 –0.33 9.41 –0.22 8.57 –0.34 9.48 2 0.48 7.03 0.53 7.40 0.58 7.36 0.37 8.07 3 0.86*** 6.12 0.82*** 6.43 0.65** 6.72 0.67** 6.62 4 1.04*** 5.52 1.05*** 5.74 0.71** 5.91 0.89*** 6.06 5 1.16*** 5.12 1.17*** 5.31 0.97*** 5.80 0.87*** 5.35 6 1.26*** 4.88 1.32*** 4.94 0.91*** 5.60 1.05*** 5.02 7 1.44*** 4.94 1.42*** 4.89 0.98*** 5.41 1.18*** 4.80 8 1.62*** 4.92 1.64*** 5.02 1.17*** 5.16 1.29*** 4.95 9 1.78*** 5.58 1.81*** 5.25 1.28*** 5.28 1.34*** 5.19 High 1.94*** 6.81 2.07*** 6.15 1.39*** 6.03 1.61*** 5.94 H - L 2.02*** 7.30 2.40*** 8.45 1.62*** 7.60 1.95*** 9.09 B. Returns to portfolios sorted on predicted simple return C5 C14 C5 C14 Decile Ret SD Ret SD Ret SD Ret SD Equal-weighted Value-weighted Low –0.18 7.29 –0.35 7.85 0.36 6.64 0.12 7.61 2 0.47 6.23 0.46 6.55 0.73** 5.95 0.73** 6.16 3 0.79*** 5.71 0.73** 5.81 1.00*** 5.45 0.91*** 5.60 4 0.92*** 5.41 0.94*** 5.36 0.97*** 5.39 1.04*** 5.07 5 1.08*** 5.18 1.11*** 5.23 1.05*** 5.60 1.19*** 5.09 6 1.22*** 5.08 1.31*** 5.10 1.26*** 5.52 1.18*** 5.12 7 1.37*** 5.08 1.38*** 5.22 1.37*** 5.48 1.23*** 5.45 8 1.60*** 5.41 1.67*** 5.56 1.59*** 6.24 1.43*** 5.97 9 1.93*** 6.36 1.84*** 6.17 1.53*** 6.92 1.52*** 6.47 High 2.33*** 7.93 2.37*** 7.80 1.89*** 8.35 1.75*** 8.03 H - L 2.51*** 7.21 2.72*** 8.09 1.53*** 8.14 1.63*** 8.70 C. Returns to portfolios sorted on predicted log return Low –0.08 8.83 –0.33 9.41 –0.22 8.57 –0.34 9.48 2 0.48 7.03 0.53 7.40 0.58 7.36 0.37 8.07 3 0.86*** 6.12 0.82*** 6.43 0.65** 6.72 0.67** 6.62 4 1.04*** 5.52 1.05*** 5.74 0.71** 5.91 0.89*** 6.06 5 1.16*** 5.12 1.17*** 5.31 0.97*** 5.80 0.87*** 5.35 6 1.26*** 4.88 1.32*** 4.94 0.91*** 5.60 1.05*** 5.02 7 1.44*** 4.94 1.42*** 4.89 0.98*** 5.41 1.18*** 4.80 8 1.62*** 4.92 1.64*** 5.02 1.17*** 5.16 1.29*** 4.95 9 1.78*** 5.58 1.81*** 5.25 1.28*** 5.28 1.34*** 5.19 High 1.94*** 6.81 2.07*** 6.15 1.39*** 6.03 1.61*** 5.94 H - L 2.02*** 7.30 2.40*** 8.45 1.62*** 7.60 1.95*** 9.09 Panel A presents the results of Fama-MacBeth regressions where the dependent variable is the realized monthly simple or log return and the explanatory variable is the predicted simple or log return. The predicted simple (log) return is obtained from the regression of month $$t$$ simple (log) returns on month $$t$$-1 characteristics specified in Table 1: the 5-characteristic model (C5) or the 14-characteristic model (C14). The predicted simple or log return for month $$t$$ is the average regression intercept over the prior 12 months plus the sum of products of average slope coefficients over the prior 12 months and month $$t$$-1 characteristics. See Table A1 for definitions of the characteristics. The Fama-MacBeth standard errors are based on the time-series variability of the estimates, incorporating a Newey-West correction with four lags. $$t$$-statistics for tests of whether the estimated coefficient equals zero (one) are reported in parentheses (brackets). In each month from January 1980 to December 2014, stocks are sorted into deciles based on their predicted simple or log return. Panel B presents equal- and value-weighted returns to portfolios sorted on predicted simple return, and panel C presents the same information for the portfolios sorted on predicted log return. ***, **, and * correspond to statistical significance at the 1%, 5%, and 10% levels, respectively. Table 2 Predicted stock return and realized stock return A. Fama-MacBeth regression of actual return on predicted return, January 1980 to December 2014 C5 C14 C5 C14 Dependent var. Simple return log return Expected return 0.8000*** 0.5418*** 0.8044*** 0.7540*** (3.72) (8.24) (10.65) (12.32) [-0.93] [-6.97] [-2.59] [-4.02] Constant 1.0231 0.4718* –0.0947 –0.0799 (1.11) (1.71) (–0.35) (–0.31) N 1,886,673 1,886,673 1,886,673 1,886,673 # months 420 420 420 420 R2 0.012 0.016 0.018 0.025 A. Fama-MacBeth regression of actual return on predicted return, January 1980 to December 2014 C5 C14 C5 C14 Dependent var. Simple return log return Expected return 0.8000*** 0.5418*** 0.8044*** 0.7540*** (3.72) (8.24) (10.65) (12.32) [-0.93] [-6.97] [-2.59] [-4.02] Constant 1.0231 0.4718* –0.0947 –0.0799 (1.11) (1.71) (–0.35) (–0.31) N 1,886,673 1,886,673 1,886,673 1,886,673 # months 420 420 420 420 R2 0.012 0.016 0.018 0.025 B. Returns to portfolios sorted on predicted simple return C5 C14 C5 C14 Decile Ret SD Ret SD Ret SD Ret SD Equal-weighted Value-weighted Low –0.18 7.29 –0.35 7.85 0.36 6.64 0.12 7.61 2 0.47 6.23 0.46 6.55 0.73** 5.95 0.73** 6.16 3 0.79*** 5.71 0.73** 5.81 1.00*** 5.45 0.91*** 5.60 4 0.92*** 5.41 0.94*** 5.36 0.97*** 5.39 1.04*** 5.07 5 1.08*** 5.18 1.11*** 5.23 1.05*** 5.60 1.19*** 5.09 6 1.22*** 5.08 1.31*** 5.10 1.26*** 5.52 1.18*** 5.12 7 1.37*** 5.08 1.38*** 5.22 1.37*** 5.48 1.23*** 5.45 8 1.60*** 5.41 1.67*** 5.56 1.59*** 6.24 1.43*** 5.97 9 1.93*** 6.36 1.84*** 6.17 1.53*** 6.92 1.52*** 6.47 High 2.33*** 7.93 2.37*** 7.80 1.89*** 8.35 1.75*** 8.03 H - L 2.51*** 7.21 2.72*** 8.09 1.53*** 8.14 1.63*** 8.70 C. Returns to portfolios sorted on predicted log return Low –0.08 8.83 –0.33 9.41 –0.22 8.57 –0.34 9.48 2 0.48 7.03 0.53 7.40 0.58 7.36 0.37 8.07 3 0.86*** 6.12 0.82*** 6.43 0.65** 6.72 0.67** 6.62 4 1.04*** 5.52 1.05*** 5.74 0.71** 5.91 0.89*** 6.06 5 1.16*** 5.12 1.17*** 5.31 0.97*** 5.80 0.87*** 5.35 6 1.26*** 4.88 1.32*** 4.94 0.91*** 5.60 1.05*** 5.02 7 1.44*** 4.94 1.42*** 4.89 0.98*** 5.41 1.18*** 4.80 8 1.62*** 4.92 1.64*** 5.02 1.17*** 5.16 1.29*** 4.95 9 1.78*** 5.58 1.81*** 5.25 1.28*** 5.28 1.34*** 5.19 High 1.94*** 6.81 2.07*** 6.15 1.39*** 6.03 1.61*** 5.94 H - L 2.02*** 7.30 2.40*** 8.45 1.62*** 7.60 1.95*** 9.09 B. Returns to portfolios sorted on predicted simple return C5 C14 C5 C14 Decile Ret SD Ret SD Ret SD Ret SD Equal-weighted Value-weighted Low –0.18 7.29 –0.35 7.85 0.36 6.64 0.12 7.61 2 0.47 6.23 0.46 6.55 0.73** 5.95 0.73** 6.16 3 0.79*** 5.71 0.73** 5.81 1.00*** 5.45 0.91*** 5.60 4 0.92*** 5.41 0.94*** 5.36 0.97*** 5.39 1.04*** 5.07 5 1.08*** 5.18 1.11*** 5.23 1.05*** 5.60 1.19*** 5.09 6 1.22*** 5.08 1.31*** 5.10 1.26*** 5.52 1.18*** 5.12 7 1.37*** 5.08 1.38*** 5.22 1.37*** 5.48 1.23*** 5.45 8 1.60*** 5.41 1.67*** 5.56 1.59*** 6.24 1.43*** 5.97 9 1.93*** 6.36 1.84*** 6.17 1.53*** 6.92 1.52*** 6.47 High 2.33*** 7.93 2.37*** 7.80 1.89*** 8.35 1.75*** 8.03 H - L 2.51*** 7.21 2.72*** 8.09 1.53*** 8.14 1.63*** 8.70 C. Returns to portfolios sorted on predicted log return Low –0.08 8.83 –0.33 9.41 –0.22 8.57 –0.34 9.48 2 0.48 7.03 0.53 7.40 0.58 7.36 0.37 8.07 3 0.86*** 6.12 0.82*** 6.43 0.65** 6.72 0.67** 6.62 4 1.04*** 5.52 1.05*** 5.74 0.71** 5.91 0.89*** 6.06 5 1.16*** 5.12 1.17*** 5.31 0.97*** 5.80 0.87*** 5.35 6 1.26*** 4.88 1.32*** 4.94 0.91*** 5.60 1.05*** 5.02 7 1.44*** 4.94 1.42*** 4.89 0.98*** 5.41 1.18*** 4.80 8 1.62*** 4.92 1.64*** 5.02 1.17*** 5.16 1.29*** 4.95 9 1.78*** 5.58 1.81*** 5.25 1.28*** 5.28 1.34*** 5.19 High 1.94*** 6.81 2.07*** 6.15 1.39*** 6.03 1.61*** 5.94 H - L 2.02*** 7.30 2.40*** 8.45 1.62*** 7.60 1.95*** 9.09 Panel A presents the results of Fama-MacBeth regressions where the dependent variable is the realized monthly simple or log return and the explanatory variable is the predicted simple or log return. The predicted simple (log) return is obtained from the regression of month $$t$$ simple (log) returns on month $$t$$-1 characteristics specified in Table 1: the 5-characteristic model (C5) or the 14-characteristic model (C14). The predicted simple or log return for month $$t$$ is the average regression intercept over the prior 12 months plus the sum of products of average slope coefficients over the prior 12 months and month $$t$$-1 characteristics. See Table A1 for definitions of the characteristics. The Fama-MacBeth standard errors are based on the time-series variability of the estimates, incorporating a Newey-West correction with four lags. $$t$$-statistics for tests of whether the estimated coefficient equals zero (one) are reported in parentheses (brackets). In each month from January 1980 to December 2014, stocks are sorted into deciles based on their predicted simple or log return. Panel B presents equal- and value-weighted returns to portfolios sorted on predicted simple return, and panel C presents the same information for the portfolios sorted on predicted log return. ***, **, and * correspond to statistical significance at the 1%, 5%, and 10% levels, respectively. 1.1 Simulation evidence regarding the proposed method Before implementing the proposed method in actual data, we assess its statistical properties by means of bootstrap simulations. For each month from January 1980 to December 2014, we randomly choose ten pseudo-event firms. We obtain first-stage estimates of predicted log returns for all CRSP common stocks by cross-sectional regressions of log firm returns on lagged C5 or C14 characteristics. In the second stage, we deduct the predicted log return for every stock from the realized log return, and estimate Fama-MacBeth regressions of the resultant abnormal returns on a constant and an indicator variable that equals one if the firm was selected as a pseudo-event firm during any of the prior 36 months, and zero otherwise. Since the pseudo-event firms were selected at random, the coefficient estimate on the event indicator variable should not systematically differ from zero. We repeat the simulation 1,000 times and report in Table 3, under the column heading “zero,” the percentage of the 1,000 simulations in which the null hypothesis is rejected at the 5% significance level. The results indicate that the null hypothesis is rejected with approximately the correct frequencies, in particular in 5.5% of the simulations when relying on the C5 characteristics, and in 3.3% of the simulations when relying on the C14 characteristics. Table 3 Size and power of the tests based on characteristics-based predicted return Induced abnormal returns per month (%) $$-$$1 $$-$$0.5 $$-$$0.25 0 0.25 0.5 1 C5 100 100 99.8 5.5 99.9 100 100 C14 100 100 100 3.3 99.9 100 100 Induced abnormal returns per month (%) $$-$$1 $$-$$0.5 $$-$$0.25 0 0.25 0.5 1 C5 100 100 99.8 5.5 99.9 100 100 C14 100 100 100 3.3 99.9 100 100 In each month from January 1980 to December 2014, we randomly choose ten pseudo-event firms. We then estimate a Fama-MacBeth regression for all common stocks in CRSP, where the dependent variable is the difference between a firm’s realized log return and the predicted log return obtained from the regression of month $$t$$ log returns on month $$t$$-1 characteristics specified in Table 1: the 5-characteristic model (C5) or the 14-characteristic model (C14). The predicted log return for month $$t$$ is the average regression intercept over the prior 12 months plus the sum of products of average slope coefficients over the prior 12 months and month $$t$$-1 characteristics. See Table A1 for definitions of the characteristics. The independent variable is the post-event dummy that equals one if the firm has been chosen as a pseudo-event firm over any of the preceding 36 months. We then test the null hypothesis that the coefficient on the post-event dummy is zero at the 5% significance level. We carry out the simulations 1,000 times and report in this table the percentage of the 1,000 simulations in which the null hypothesis is rejected at different levels of induced abnormal returns (ranging from $$-$$1% to $$+$$ 1%) to the pseudo-event firms. Table 3 Size and power of the tests based on characteristics-based predicted return Induced abnormal returns per month (%) $$-$$1 $$-$$0.5 $$-$$0.25 0 0.25 0.5 1 C5 100 100 99.8 5.5 99.9 100 100 C14 100 100 100 3.3 99.9 100 100 Induced abnormal returns per month (%) $$-$$1 $$-$$0.5 $$-$$0.25 0 0.25 0.5 1 C5 100 100 99.8 5.5 99.9 100 100 C14 100 100 100 3.3 99.9 100 100 In each month from January 1980 to December 2014, we randomly choose ten pseudo-event firms. We then estimate a Fama-MacBeth regression for all common stocks in CRSP, where the dependent variable is the difference between a firm’s realized log return and the predicted log return obtained from the regression of month $$t$$ log returns on month $$t$$-1 characteristics specified in Table 1: the 5-characteristic model (C5) or the 14-characteristic model (C14). The predicted log return for month $$t$$ is the average regression intercept over the prior 12 months plus the sum of products of average slope coefficients over the prior 12 months and month $$t$$-1 characteristics. See Table A1 for definitions of the characteristics. The independent variable is the post-event dummy that equals on