Journal of Accounting Research The Higher Moments of Future Return on Equity For Review Only

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1 Journal of Accounting Research The Higher Moments of Future Return on Euity Journal: Journal of Accounting Research Manuscript ID: JOAR Wiley - Manuscript type: Original Article Dimension 1 (Method): Empirical-archival Dimension 2 (Topic area): Financial accounting Capital markets/asset pricing, Financial accounting Corporate finance and contracting Abstract: We use uantile regressions to evaluate the higher moments of future return on euity, ROE. First, we evaluate the in-sample relations between current firm-level attributes and the moments of lead ROE. We show that: (1) as current profitability increases lead ROE tends to increase, become more disperse, and more leptokurtic; (2) loss firms tend to have lower, more disperse, and more left-skewed lead ROE; (3) as accruals increase lead ROE tends to decrease and become more disperse; and, (4) firms with higher leverage and/or lower payout ratios tend to have greater dispersion in lead ROE. Second, we evaluate the reliability of out-of-sample estimates that are based on the in-sample relations. We begin by evaluating firmyear estimates; and, we show that our estimates of the probability of a future loss and the standard deviation of lead ROE are reliable. Next, we form industry-year portfolios; and, we show that the in-sample relations yield reliable out-of-sample predictions of the freuency of losses within an industry and the within-industry standard deviation, skewness, and kurtosis of lead ROE.

2 Page 1 of 63 Journal of Accounting Research The Higher Moments of Future Return on Euity Abstract We use uantile regressions to evaluate the higher moments of future return on euity, ROE. First, we evaluate the in-sample relations between current firm-level attributes and the moments of lead ROE. We show that: (1) as current profitability increases lead ROE tends to increase, become more disperse, and more leptokurtic; (2) loss firms tend to have lower, more disperse, and more left-skewed lead ROE; (3) as accruals increase lead ROE tends to decrease and become more disperse; and, (4) firms with higher leverage and/or lower payout ratios tend to have greater dispersion in lead ROE. Second, we evaluate the reliability of out-of-sample estimates that are based on the in-sample relations. We begin by evaluating firm-year estimates; and, we show that our estimates of the probability of a future loss and the standard deviation of lead ROE are reliable. Next, we form industry-year portfolios; and, we show that the in-sample relations yield reliable out-of-sample predictions of the freuency of losses within an industry and the withinindustry standard deviation, skewness, and kurtosis of lead ROE. 1

3 Journal of Accounting Research Page 2 of Introduction Return on euity, ROE, is a key economic variable. Ceteris paribus, higher expected future ROE implies higher euity market value; many contracts contain provisions that are based on the level of future ROE or other earnings-based measures; etc. Hence, numerous academic studies focus on ROE. For example, there is a vast literature in accounting and finance that evaluates the relation between ROE and firm-level attributes. There is also an extensive literature that evaluates different approaches for forecasting ROE. A limitation of extant studies is that virtually all of them focus (implicitly or explicitly) on expected i.e., the mean of future ROE. Although the mean is an important moment of the distribution, higher moments (e.g., the variance) are important too. For example, as the variance of future ROE increases, the likelihood of violating an earnings-based contract provision (e.g., a loan covenant) increases; undiversified investors (e.g., the owners of a private company) face greater risk; the likelihood that the firm will violate an earnings-based regulatory constraint (e.g., limits on bank regulatory capital) increases; etc. With the above in mind, we investigate two related uestions. First, what are the insample relations between firm-level attributes and the higher moments of lead ROE? Second, can these in-sample relations be used to develop reliable out-of-sample estimates of the higher moments of lead ROE? To provide answers to the uestions posed above we use a novel research design that is based on uantile regressions. Quantile regressions are particularly appropriate in our setting for two reasons. First, as discussed in Buchinsky [1998], when a regression is estimated for the th uantile, the coefficient on a particular regressor is a consistent estimate of the marginal change in the th conditional uantile of the dependent variable given a marginal change in the regressor 1

4 Page 3 of 63 Journal of Accounting Research of interest. Hence, we use the coefficients obtained from regressions estimated for a seuence of uantiles to infer the relation between firm-level attributes and the location and shape of the distribution of lead ROE. Second, the fitted value obtained from a regression estimated for the th uantile is a consistent estimator of the th conditional uantile of the dependent variable. Hence, for each firm-year in our sample we calculate out-of-sample estimates of the conditional uantiles of lead ROE for a seuence of uantiles. Next, we combine these out-of-sample estimates to form an estimate of the conditional cumulative distribution function, cdf, of lead ROE. Finally, we use the cdf to infer the conditional probability that lead ROE is negative i.e., the probability of a future loss as well as the conditional mean, standard deviation, skewness, and kurtosis of lead ROE. We analyze the in-sample relations between the moments of lead ROE and firm-level attributes that fall into two categories: (1) attributes of current ROE and (2) attributes of current financial policy. Regarding the relations between lead ROE and the attributes of current ROE, three results are noteworthy. First, we show that for profitable firms current ROE is positively associated with the mean and median of lead ROE. However, as current profitability increases, the distribution of lead ROE spreads out and becomes more leptokurtic i.e., has fatter tails. Hence, although higher current profitability implies higher expected lead ROE, it also implies greater risk. Second, we show that, relative to profitable firms, loss firms have lower mean and median lead ROE. Loss firms also have distributions of lead ROE that exhibit relatively high dispersion and left skewness. In addition, as the magnitude of current losses increase, the mean 2

5 Journal of Accounting Research Page 4 of 63 and median of lead ROE decline and the variance of lead ROE increases. Hence, current losses are associated with lower, riskier lead ROE. Finally, we show that current accruals are negatively associated with the mean and median of lead ROE. Moreover, there is a positive association between current accruals and the variance of lead ROE. This result is particularly interesting as it is consistent with the notion that extreme positive accruals are an indicator of low earnings uality i.e., ceteris paribus, large income-increasing accruals imply riskier future earnings. Two of the relations between the moments of lead ROE and firms financial policies are also noteworthy. First, we show that the distribution of lead ROE becomes more disperse and more leptokurtic as current leverage increases. This is consistent with the well-known result described in Modigliani and Miller [1958]: euity becomes riskier as leverage increases. Second, we show that, relative to non-dividend-paying firms, dividend-paying firms have distributions of lead ROE that are less disperse and less left-skewed. Hence, firms that pay dividends are less risky, which is consistent with the notion that managers want to avoid cutting dividends and, thus, choose payout ratios that are sustainable given the degree of uncertainty about future earnings (e.g., Lintner [1956] and Miller and Rock [1985]). After documenting the in-sample relations described above we evaluate whether they can be used to develop reliable out-of-sample estimates. First, we evaluate estimates formed in year t of the probability of firm i experiencing a loss in year t+h, PROB_LOSS i,t,t+h, and the standard deviation of firm i s ROE in year t+h, STD_ROE i,t,t+h. We show that these two variables are positively associated with the occurrence of a loss in year t+h and the absolute value of the realized forecast error for firm i in year t+h. 1 For example, when h euals one, the average of the 1 A firm s realized forecast error in year t+h euals the difference between the firm s realized ROE in year t+h and our year t estimate of the mean of the firm s ROE in year t+h. 3

6 Page 5 of 63 Journal of Accounting Research annual Pearson (Spearman) correlations between PROB_LOSS i,t,t+h and an indicator that euals one (zero) if firm i incurred (did not incur) a loss in year t+h is (0.519); and, the average of the annual Pearson (Spearman) correlations between ROE_STD i,t,t+h and the absolute value of the forecast error for firm i in year t+h is (0.462). Second, we evaluate the reliability of out-of-sample, industry-year estimates. We begin by showing that for every forecast horizon we consider there is a positive association between the industry-average of PROB_LOSS i,t,t+h and the fraction of industry members that report losses in year t+h. For example, when h euals one, the average of the annual Pearson (Spearman) correlations between these two variables is (0.708). Next, we use the law of total moments to develop year t predictions of the withinindustry-year standard deviation, skewness, and kurtosis of year t+h ROE. As discussed in Brillinger [1969], the law of total moments is a generalization of the law of total variance. Hence, it allows us to predict the cross-sectional properties of lead ROE for a particular industryyear. For example, we can use it to predict whether ROE will vary widely across the members of an industry. After forming the industry-level predictions described above, we compare them to the realized within-industry-year standard deviation, skewness, and kurtosis of ROE. We find that for most forecast horizons each of the predicted statistics is strongly correlated with the realized value in year t+h. For example, when h euals one, the average of the annual Pearson (Spearman) correlations between predicted and realized standard deviation, skewness, and kurtosis are: (0.624), (0.372), and (0.442). We make three contributions to the extant literature. First, we illustrate how to use uantile regressions to investigate and predict the cdf of future ROE. Hence, we develop a 4

7 Journal of Accounting Research Page 6 of 63 general approach that can be used to study the economic relevance of the higher moments of future ROE as well as other variables such as return on invested capital, earnings growth, accruals, etc. Second, we provide interesting evidence about the relation between the higher moments of lead ROE and key firm-level attributes such as the properties of current earnings and current financial policy. Finally, we show that our methodology yields reliable: (1) firm-level estimates of the probability of a future loss and the standard deviation of lead ROE; and, (2) predictions of the freuency of losses within an industry as well as the within-industry-year standard deviation, skewness, and kurtosis of lead ROE. Hence, we provide a reliable approach for estimating risk on an ex ante basis. In the next section, we provide a brief literature review. In section three, we describe our research design. In section four, we explain how we construct our sample and we provide descriptive statistics. In section five, we discuss the results of our analyses of the in-sample relations between firm-level attributes and the moments of lead ROE; and, in section six, we discuss the results of our analyses of the reliability of our out-of-sample estimates. We provide concluding comments in section seven, which is followed by an appendix. 2. Literature Review ROE plays a central role in many economic decisions. Nonetheless, there is very little evidence about the relation between firm-level attributes and the higher moments of future ROE. Moreover, whether it is possible to develop reliable out-of-sample estimates of the higher moments of future ROE is an open uestion. In fact, we know of only one other study that sheds light on these issues. Specifically, in a recent study Konstantinidi and Pope [2012] (KP hereafter) evaluate the properties of the extreme uantiles of lead return on assets, ROA. There 5

8 Page 7 of 63 Journal of Accounting Research are three key differences between our study and the study by KP. First, KP only evaluate the extreme uantiles i.e., the fifth uantile, the 95 th uantile, and the difference between the two of lead ROA. On the other hand, we develop a general methodology for estimating the entire cdf. Being able to estimate the entire cdf is important because it allows us, and other researchers, to evaluate and predict the higher moments of lead ROE as well as the probability of a future loss. Second, we use uantile regression coefficients to infer the relations between firm-level attributes and the moments of lead ROE. KP, on the other hand, do not focus on these relations. Third, when estimating the extreme uantiles, KP base their estimates solely on the attributes of firms earnings. On the other hand, we also consider financial-policy attributes, which are important given well-known results about the relation between leverage, payout policy, and risk. Finally, there are two other less significant yet relevant differences between our study and the study by KP. First, our sample period spans 1973 through 2011 whereas KP only consider the period between 1998 and 2009; and, our sample is more than three times larger than KP s sample. Second, KP only consider a one-year forecast horizon whereas we consider horizons between one and five years. 3. Research Design We begin by providing a general overview of uantile regressions and how to use them to evaluate and estimate the conditional moments of a random variable. Next, we provide details about how we model the relation between firm-level attributes and the higher moments of lead ROE. Please note that in this section we focus on providing an intuitive description of uantile 6

9 Journal of Accounting Research Page 8 of 63 regressions and how we use them. We relegate the discussion of technical details to the appendix. 3.1 General Overview of Quantile Regressions As discussed in Buchinsky [1998], the estimation of a uantile regression involves choosing the coefficient vector Β = β, K 0, βk that solves the minimization problem shown in euation (1). 1 argmin N 0 k i : y Β = β,..., β k β i, t j= 0 y i, t k k β j xi, t h, j + ( 1 ) yi, t β j xi, t h, j (1) j= 0 k j= 0 i: y i t j x, β i, t h, j j= 0 j xi,, t h, j < In euation (1), y i,t is the year t i.e., the lead value of the dependent variable for observation i [1,N], x i,t-h,j is the year t-h i.e., the current value of the j th independent variable (j [0,k]) for observation i, and (0,1) denotes the th uantile. Euation (1) has a similar structure as the minimization problem underlying an ordinary least suares, OLS, regression. For example, the objective function involves choosing regression coefficients that minimize a function of the residuals. However, unlike the OLS minimization problem in which eual weight is put on each of the suared residuals, the estimation of a uantile regression involves assigning weights that depend on the sign of the residual. In particular, the weight put on a positive residual is (1-) orders of magnitude of the weight put on a negative residual. This implies that the coefficient vector Β is chosen so that for each positive residual there are (1-) negative residuals; and, conseuently, (1-) 100 ( 100) percent of the residuals will lie above (below) the fitted value, which euals β k j = 0 x j i, t h, j. Hence, similar to OLS regressions, which yield fitted values that are consistent estimates of the 7

10 Page 9 of 63 Journal of Accounting Research conditional mean of y i,t, uantile regressions yield fitted values that are consistent estimates of the th conditional uantile of y i,t, which we refer to as ( ) QUANT.2 The coefficients obtained from a uantile regression can also be interpreted in a similar manner as the coefficients obtained from an OLS regression. In particular, as discussed in Buchinsky [1998], β j is a consistent estimate of QUANT x i, t h, j y i,t ( y ) i, t. Hence, the coefficients obtained from a uantile regression reflect marginal effects. However, unlike the coefficients obtained from an OLS regression, which eual the marginal change in the conditional mean of y i,t given a marginal change in x i,t-h,j, coefficients obtained from a uantile regression eual the marginal change in the th conditional uantile of y i,t given a marginal change in x i,t-h,j. The facts described above have two important implications. First, the fact that the coefficients obtained from a uantile regression reflect marginal effects implies that we can use them to infer the association between x i,t-h,j and higher moments of y i,t. To do this we begin by solving the minimization problem shown in euation (1) for a seuence of Q uantiles. This yields Q estimates of β. Next, we evaluate how j β varies with and we infer the effect of j x i,t-h,j on the moments of y i,t. For example, suppose that β = c > 0 for all. This implies that as x i,t-h,j increases all of the uantiles shift to the right by an eual amount. Hence, x i,t-h,j is associated with the location, but not the shape, of the distribution of y i,t. On the other hand, suppose that for β j is positive for all and an increasing function of. Hence, although increases in x i,t-h,j lead to increases in all of the uantiles of y i,t, the upper uantiles increase by a larger amount. This implies that x i,t-h,j is positively associated with both the conditional mean j 2 A formal discussion of this result is provided in Buchinsky [1998]. 8

11 Journal of Accounting Research Page 10 of 63 and conditional variance of y i,t i.e., as x i,t-h,j increases the conditional distribution of y i,t shifts to the right and becomes more disperse. Second, the fact that uantile regressions yield fitted values that are consistent estimates of ( ) QUANT implies that we can use the fitted values to develop out-of-sample estimates y i,t of the moments of y i,t+h. To do this we begin by solving the minimization problem shown in euation (1) for a seuence of Q uantiles. This yields Q estimates of the coefficient vector B. These Q estimates reflect the in-sample relations between the conditional uantiles of the dependent variable measured at year t and the year t-h values of the independent variables. However, we want out-of-sample estimates of the year t+h uantiles. Hence, we combine the year t values of the independent variables with the estimated coefficient vector to predict the year t+h th k ^ conditional uantile of the dependent variable i.e., x QUANT ( y ) = + j i, t, j i, t h j= 1 β. Next, we combine the Q predicted uantiles to arrive at a discrete estimate of the conditional cdf of y i,t+h. Finally, we use standard statistical formulas to impute the conditional moments of y i,t+h from the cdf. 3.1 Modeling the Higher Moments of ROE For each estimation year EY we assume the following linear relation between the th conditional uantile of ROE for year t and firm-level attributes measured at year t-h. We refer to year t as the lead year and we refer to year t-h as the current year. QUANT ( ROE ) = β + β ROE + β LOSS + β ( ROE LOSS ) i, t + β 0, EY 4, EY 1, EY ACC i, t h + β i, t h 5, EY LEV 2, EY i, t h + β i, t h 6, EY 3, EY PAYER i, t h + β i, t h 7, EY i, t h PAYOUT i, t h (2) The variables in euation (2) are described in the table shown below. 9

12 Page 11 of 63 Journal of Accounting Research Variable Name Description ROE i,t Earnings of firm i during year t divided by firm i s year t-h euity book value ROE i,t-h Earnings of firm i during year t-h divided by firm i s year t-h euity book value LOSS i,t-h An indicator variable that euals one (zero) if ROE i,t-h < 0 (ROE i,t-h 0) ACC i,t-h Accruals reported by firm i during year t-h divided by firm i s year t-h euity book value LEV i,t-h Total assets of firm i for year t-h divided by firm i s year t-h euity book value PAYER i,t-h An indicator variable that euals one (zero) if PAYOUT i,t-h > 0 (PAYOUT i,t-h = 0) PAYOUT i,t-h Dividends paid by firm i during year t-h divided by firm i s year t-h euity book value Our model is similar to the model used by Hou, Van Dijk, and Zhang [2012] (HVZ hereafter), who focus on forecasting the mean of ROE. However, there are two differences. First, HVZ do not deflate by euity book value. Second, HVZ do not include the interaction term ROE i,t-h LOSS i,t-h. The motivation for the independent variables in euation (2) is straightforward. First, it is well-known (e.g., Freeman et al. [1982]) that ROE is persistent; hence, we include ROE i,t-h in our model. Second, there is ample evidence (e.g., Basu [1997]) that losses follow a different time-series process than profits; thus, we allow the coefficient on ROE i,t-h to vary with the sign of ROE i,t-h. Third, evidence provided by Sloan [1996] implies that accruals are less persistent that cash flows. Conseuently, we control for the portion of year t-h ROE that is attributable to year t-h accruals, ACC i,t-h. Finally, well-known results in finance (e.g., Lintner [1956], Modigliani and Miller [1958], and Miller and Rock [1985]) show that firms capital structure and payout policies are associated with the level and dispersion of ROE. Hence, we include LEV i,t-h, PAYER i,t-h, and PAYOUT i,t-h in our model. In addition to being intuitively appealing and comparable to extant models such as that used by HVZ, our model has two advantages. First, it is parsimonious and tractable. Second, it is superior to a number of more elaborate models. In particular, we evaluate models in which we 10

13 Journal of Accounting Research Page 12 of 63 add the following variables to euation (2): the log of sales, SIZE i,t-h ; an indicator for extreme ROE that euals one (zero) if ROE i,t-h is (is not) in the top or bottom tenth percentile of the annual distribution, XTRM_ROE i,t-h ; the interaction between XTRM_ROE i,t-h and ROE i,t-h ; the lagged change in earnings deflated by euity book value, ROE i,t-h ; an indicator for extreme changes in ROE that euals one (zero) if ROE i,t-h is (is not) in the top or bottom tenth percentile of the annual distribution, XTRM_ ROE i,t-h ; the interaction between the XTRM_ ROE i,t-h and ROE i,t-h ; the interaction between XTRM_ ROE i,t-h and ROE i,t-h ; the measure of unconditional conservatism described in Penman and Zhang [2002], CONS i,t-h ; the interaction between CONS i,t-h and ROE i,t-h ; and, various combinations of these aforementioned variables. In a set of untabulated results we show that none of these models generate better out-of-sample estimates than the estimates derived from euation (2). Moreover, adding these additional variables to euation (2) does not change the tenor of our results regarding the in-sample relations between the independent variables shown in euation (2) and the higher moments of lead ROE. Our research design involves the following three steps. First, for each estimation year EY we obtain estimates of the coefficient vector Β for 150 different values of EY = β0, EY, K, β7, EY (0,1). 3 To obtain the coefficient vector for a particular value of we solve the minimization problem shown in euation (1). When doing this we use a mix of time-series and cross-sectional data (i.e., panel data). We reuire that each panel contains at least five years of data; however, we never use more than ten years of data to construct a panel. For example, suppose the estimation year is 1990 (i.e., EY = 1990) and the forecast horizon is 3 (i.e., h = 3), we use values 3 The 150 values of are in seuential order and all the pairs of consecutive values of are euidistant. The number 150 is a function of our sample size and number of covariates. It represents the maximum number of uantile regressions that we can estimate while guaranteeing that the numerical estimates converge. As shown in the appendix, as the sample size increases and the number of uantile regressions increases, the estimates of the predicted moments converge in probability to the moments of lead ROE. 11

14 Page 13 of 63 Journal of Accounting Research of the dependent variable that fall between 1981 and 1990 and we use values of the independent variables that fall between 1978 and We include a firm in the panel if it has at least one valid observation during the relevant time span. Second, we evaluate the in-sample relations between the higher moments of lead ROE and firm-level attributes. For each estimation year EY, value of, and firm-level attribute j we obtain the relevant coefficient estimate (i.e., β j,ey ) and we compute the average of β j,ey, which we refer to as β j,avg. Next, assuming a lag length of ten, we calculate the Newey-West adjusted standard error of β j,avg ; and, we use the standard error to form a 95 percent confidence interval around β j,avg. We then graph β and its confidence interval on. j,avg Finally, we develop our out-of-sample estimates. For a particular year t we obtain the contemporaneous (i.e., EY = t) estimated coefficient vector for each of the 150 values of. We then predict the th conditional uantile of ROE i,t+h by calculating the inner product of the coefficient vector and a vector containing the contemporaneous (i.e., year t) values of the independent variables for firm i. Next, we combine the 150 predicted conditional uantiles to form our estimate of the conditional distribution of ROE i,t+h, which we use to infer the conditional probability of a loss and the other conditional moments Sample Construction and Descriptive Statistics 4.1 Sample Construction We obtain our data from the Compustat North America Annual file. We use the Compustat variable IB, Income Before Extraordinary Items, as our measure of earnings. The 4 We define the conditional probability of a loss as the largest value of for which ( ) < 0 12 ^ QUANT ROE i, t +h.

15 Journal of Accounting Research Page 14 of 63 Compustat variable CEQ, Common/Ordinary Euity - Total, is our measure of euity book value. We use the balance sheet approach described in Sloan [1996] to estimate accruals. Total assets euals Compustat variable AT, Assets - Total; and, dividends euals Compustat variable DVPSX_F, Dividends per Share - Ex-Date - Fiscal. For each forecast horizon h, we form two samples: (1) the estimation sample and (2) the prediction sample. The estimation sample contains observations that are used to estimate the coefficients shown in euation (2). The prediction sample contains observations for which we develop out-of-sample, firm-level predictions of the cdf of lead ROE. To form the estimation sample we identify all observations that have non-missing values of the variables shown in euation (2) and positive values of euity book value in year t-h. Next, we delete extreme observations, which we define as observations for which: ROE i,t > 2, ROE i,t-h > 2, ACC i,t-h > 2, LEV i,t-h [1,20], and PAYOUT i,t-h [0,1]. When h euals 1 (i.e., a one-year forecast horizon), the estimation sample contains 174,215 firm-years with independent (dependent) variables drawn from the time-period spanning 1963 to 2010 (1964 to 2011). The sample size decreases as h increases; for example, when h euals 5, the estimation sample contains 122,935 firm-years with independent (dependent) variables drawn from the time-period spanning 1963 to 2006 (1968 to 2011). To form our prediction sample we identify all firm-years with positive euity book value in year t and non-missing values of ROE i,t = IB i,t CEQ i,t, LOSS i,t, ROE i,t LOSS i,t, ACC i,t, LEV i,t, PAYER i,t, and PAYOUT i,t. We do not remove extreme observations nor do we remove observations with missing values of lead ROE. We limit our prediction sample to firm-years drawn from 1973 to The prediction sample contains 170,522 firm-years. However, 13

16 Page 15 of 63 Journal of Accounting Research because some of our tests involve comparing ex ante predictions to ex post realizations, the number of observations varies. 4.2 Descriptive Statistics and Correlations In Panel A of Table One we provide descriptive statistics for the estimation sample pertaining to a one-year forecast horizon (i.e., h = 1). Descriptive statistics for estimation samples pertaining to the other forecast horizons are available upon reuest. The mean (median) of ROE i,t is (0.101). Twenty-four percent of the sample observations have negative ROE i,t-1. The mean (median) of ACC i,t-h is (-0.054). LEV i,t-h has a mean (median) of (1.970), 43.8 percent of the observations pay dividends, and the average payout ratio is Panel B of Table One contains the correlation structure of the variables shown in euation (2). Pearson (Spearman) correlations are shown above (below) the diagonal. The correlations shown in the table eual the average of the annual correlations. The t-statistics eual the average correlation divided by its temporal standard error. We tabulate results for the estimation sample that pertains to a one-year forecast horizon. Results for estimation samples pertaining to the other forecast horizons are available upon reuest. Several correlations warrant discussion. First, the Pearson (Spearman) correlation between lead ROE (i.e., ROE i,t ) and current ROE (i.e., ROE i,t-1 ) is 0.60 (0.70); hence, shocks to ROE have high persistence. Second, firms that are currently experiencing losses have lower lead ROE; in particular, the Pearson (Spearman) correlation between ROE i,t and LOSS i,t-1 is (-0.43). Third, the Pearson (Spearman) correlation between lead ROE and current accruals (i.e., ACC i,t-1 ) is 0.11 (0.12), which implies that accruals are less persistent than cash flows. Current leverage (i.e., LEV i,t-1 ) is uncorrelated with lead ROE i,t. However, the Pearson (Spearman) 14

17 Journal of Accounting Research Page 16 of 63 correlation between lead ROE and PAYER i,t-1 is 0.20 (0.23); and, the Pearson (Spearman) correlation between the current payout ratio (i.e., PAYOUT i,t-1 ) and lead ROE is 0.22 (0.31). 5. Analyses of In-sample Relations In this section we describe the relation between the independent variables shown in euation (2) and the moments of lead ROE. We use graphical evidence. In particular, for each estimation year EY, value of, and firm-level attribute j we obtain the relevant coefficient estimate i.e., β j,ey. We then compute the average of refer to as β j,avg, and the temporal standard error of β across estimation years, which we β j,ey j, AVG. When calculating the temporal standard error we make the Newey-West adjustment assuming a ten-year lag length. Next, we use the standard error to calculate a 95 percent confidence interval around the average; and, we graph β and its confidence interval on. For comparative purposes, we also graph the j, AVG average coefficient, which we refer to as β OLS j,avg, and the 95 percent confidence interval obtained from an OLS regression. We provide graphs of coefficients that relate to a one-year forecast horizon (i.e., h = 1). Graphs of coefficients that relate to other forecast horizons are available upon reuest. The graphs presented in this section are based on regressions that are estimated on demedianed independent variables. In particular, we set each of the independent variables eual to the difference between its raw value and its median value for the relevant panel. This demedianing makes it easier to interpret the coefficient on the constant term. In particular, when we use de-medianed independent variables the estimated constant for uantile i.e., β 0, EY 15

18 Page 17 of 63 Journal of Accounting Research euals the conditional th uantile for the typical observation. That is, the observation for which each of the independent variables is eual to the median of that variable for the panel. It is important to note that de-medianing only affects the estimate of the constant term and has no effect on the estimates of the slope coefficients. That is, the estimates of β, K β 1, 7 obtained from estimating euation (2) on the de-medianed data are identical to the estimates of β, K β obtained from estimating euation (2) on the original data. It is also important to 1, 7 note that we only use the de-medianed data to generate the graphs presented in this section: Our out-of-sample estimates are based on regressions estimated on the raw data. We show the graph of the constant term, β 0, AVG, in Figure One. As discussed above, β ( 0, AVG OLS β 0, AVG ) euals the conditional th uantile (conditional mean) of lead ROE for the typical observation. 5 As shown in Figure One, the typical observation has median (mean) lead ROE of (0.069). Untabulated results show that the interuartile range of lead ROE for the typical observation is Moreover, lead ROE for the typical observation is negative for all values of that are less than 0.23 i.e., there is a 23 percent probability that the typical observation will experience a loss in year t+1. Figure Two contains the graph of β 1, AVG, which is the coefficient on current ROE (i.e., ROE i,t-1 ). We provide an in-depth discussion of this graph so that we can: (1) discuss the specific relation between current ROE and the moments of lead ROE and (2) make some general points about how to interpret the graphs of the remaining coefficients. A natural starting point is to determine the relation between current ROE and the location of the distribution of lead ROE. To do this we evaluate the coefficient β ( β , AVG OLS 1, AVG ), which 5 The graph of β 0, AVG obtained from regressions estimated on the raw data is available upon reuest. 16

19 Journal of Accounting Research Page 18 of 63 euals the marginal change in the conditional median (mean) of lead ROE given a marginal change in current ROE. β ( β , AVG OLS 1, AVG ) euals 1.01 (0.853); hence, there is a positive association between current ROE and the median (mean) of lead ROE. Second, we consider the relation between current ROE and the variance of lead ROE. As shown in Figure Two, β is an increasing function of (i.e., β 0 ). This implies 1, AVG 1, AVG > that as current ROE increases the higher uantiles of lead ROE increase by larger amounts than the lower uantiles i.e., the distribution of lead ROE spreads out. Hence, there is a positive association between current ROE and the variance of lead ROE. Finally, we note that: (1) for values of < 0.80 the relation between β 1, AVG and is concave (i.e., β 0 ) but (2) for values of > 0.80 the relation between 1, AVG < β 1, AVG and is convex (i.e., β 0 ). Hence, firms with higher current ROE are more likely to have 1, AVG > extreme values of lead ROE. That is, these firms have more leptokurtic (i.e., fat-tailed) distributions of lead ROE. In light of the above, we conclude that firm s with high current ROE tend to have higher lead ROE that is more volatile and more extreme. Hence, although higher current profitability is associated with higher future profitability it also implies greater risk. Figure Three contains the graph of β 2, AVG, which is the coefficient on the loss indicator (i.e., LOSS i,t-1 ). The graph illustrates that, ceteris paribus, loss firms tend to have lower, more volatile lead ROE. In particular, β ( β , AVG OLS 2, AVG ) euals (-0.071) and β 2, AVG is increasing in. Loss firms are also more likely to experience extreme poor performance. Specifically, the relation between β 2, AVG and is a concave for most values of. Hence, loss firms have lead ROE that is more left-skewed. 17

20 Page 19 of 63 Journal of Accounting Research Figure Four contains the graph of 1 ( β + β ). 6 We are interested in the total 1, AVG 3, AVG relation between current losses and lead ROE; hence, we evaluate the sum of the coefficient on ROE i,t-1 (i.e., β 1, AVG ) and the coefficient on the interaction of ROE i,t-1 and LOSS i,t-1 (i.e., β 3, AVG ). We multiply the coefficients by negative one so that the graph shows the relation between larger losses (i.e., more negative ROE) and the uantiles of lead ROE. The graph illustrates that firms with higher current losses tend to have lower lead ROE. In particular, ( β + β ) OLS OLS ( 1 ( β + β )) euals (-0.440). In addition, 1 ( β + β ) 1, AVG 3, AVG 1, AVG 3, AVG 1 1, AVG 3, AVG is increasing in, which implies the magnitude of the current loss is positively associated with the variance of lead ROE. In Figure Five we graph the relation between the coefficient on current accruals, β 4, AVG, and. The results shown on the graph are consistent with the conventional wisdom that higher current accruals are an indicator of low earnings uality. First, higher current accruals are associated with lower median (mean) lead ROE. In particular, β ( β ) euals , AVG OLS 4, AVG (-0.057). Moreover, given β is increasing in, higher accruals also imply more volatile 4, AVG lead ROE. Figure Six contains the graph of β 5, AVG, which is the coefficient on current leverage (i.e., LEV i,t-1 ). As the graph shows, current leverage is not associated with the median (mean) of lead ROE. β 5, AVG is an increasing function of, however; hence, current leverage is positively associated with the variance of lead ROE. Moreover, for the lower uantiles of, the relation The confidence intervals shown in Figure Four relate to the standard error of the average of 1 ( β 1, EY + β3, EY ) OLS OLS and the average of 1 ( β + β ). That is, we use the standard error of the average of the sum not the sum 1, EY 3, EY of the standard errors of the averages. 18

21 Journal of Accounting Research Page 20 of 63 between β 5, AVG and is concave; however, for values of > 0.80 the relation is convex. Thus, firms with high current leverage have more leptokurtic (i.e., fat-tailed) distributions of lead ROE. These results are consistent with fundamental theorems in classical finance (i.e., Modigliani and Miller [1958]) that show that euity becomes riskier as leverage increases. In Figure Seven we show the graph of β 6, AVG, which is the coefficient on the dividend indicator (i.e., PAYER i,t-1 ). First, OLS β 6, AVG euals and β euals Hence, , AVG dividend-paying firms tend to have higher lead ROE; however, the effect primarily relates to the mean. Second, β 6, AVG is a decreasing function of, which implies that dividend-paying firms have less volatile lead ROE. Finally, for most values of, the relation between β 6, AVG and is convex. This implies that dividend-paying firms are less likely to exhibit extreme poor performance i.e., the distribution of lead ROE is less left-skewed. Figure Eight contains the graph of β 7, AVG, which is the coefficient on PAYOUT i,t-1. The graph illustrates that there is a complex relation between current payout ratios and the moments of lead ROE. First, regarding the location of lead ROE, higher current payout implies higher mean but lower median lead ROE. In particular, β ( , AVG OLS β 7, AVG ) euals (0.088). Second, for values of between 0.10 and 0.90, β is a decreasing function of 7, AVG ; however, for values of {(0,0.10) (0.90,1.00)}, β 7, AVG is an increasing function of. Hence, as the current payout ratio increases the middle 80 percent of the distribution of lead ROE clusters together but the extreme uantiles become more spread out. This implies that firms with high payout ratios tend to exhibit either relatively small or relatively large deviations 19

22 Page 21 of 63 Journal of Accounting Research from the mean of lead ROE. However, these firms rarely exhibit moderate deviations from the mean of lead ROE. Finally, in Figure Nine we show the pseudo r-suared from each of the uantile regressions and the r-suared from the OLS regression. The pseudo r-suared of a uantile regression measures the impact of the covariates on the ability of the uantile regression to explain the weighted sum of the absolute deviations. 7 (The weighted sum of the absolute deviations is the value of the objective function minimized in euation (1).) The pseudo r- suared is eual to zero if the model s explanatory variables do not explain more of the weighted absolute deviations than a model that contains only a constant term. On the other hand, the pseudo r-suared is eual to one if the model s predictions do not deviate from the realizations. The OLS r-suared is calculated in the usual way; and, it euals the fraction of the variance of ROE explained by the independent variables. The pseudo r-suared and the OLS r-suared are not directly comparable. The results shown on the graph imply that the covariates significantly improve the model s fit. The lowest pseudo r-suared is approximately 27 percent. The model s fit is better for the smallest uantiles (i.e. for values of below 0.50). 5. Analyses of Out-of-sample Estimates In this section we evaluate our out-of-sample estimates of the moments of lead ROE. We begin by discussing descriptive statistics and correlations. Next, we evaluate the reliability of our firm-year estimates of the probability of a future loss and the standard deviation of lead ROE. Finally, we evaluate our predictions of the freuency of losses within an industry and the 7 The pseudo r-suared shown in Figure Nine is the standard statistic reported by software packages such as STATA and SAS. 20

23 Journal of Accounting Research Page 22 of 63 within-industry-year standard deviation, skewness, and kurtosis of lead ROE. All of the analyses described in this section are based on observations drawn from the prediction sample. It is important to note that all of the estimates described in this section are out-of-sample. In particular, we develop a year t estimate of a variable in year t+h by combining regression coefficients with firm-specific values of the independent variables. The regression coefficients are obtained from regressions estimated on data that were available on or before the end of year t; and, the firm-specific values of the independent variables are measured at the end of year t. 5.1 Descriptive Statistics and Correlations We provide descriptive statistics and correlations for the variables shown below. Descriptive statistics and correlations relate to the out-of-sample forecasts made in year t of the moments of firm-level ROE in year t+1. Descriptive statistics for forecast horizons (i.e., values of h) between two and five are available upon reuest. Variable Name MEAN_ROE i,t,t+h Description Year t estimate of the mean of ROE i,t+h PROB_LOSS i,t,t+h Year t estimate of the probability that ROE i,t+h < 0 STD_ROE i,t,t+h SKEW_ROE i,t,t+h KURT_ROE i,t,t+h Year t estimate of the standard deviation of ROE i,t+h Year t estimate of the skewness of ROE i,t+h Year t estimate of the excess kurtosis of ROE i,t+h As discussed in section three, the variables shown above are inferred from our out-of-sample, firm-level estimates of QUANT ^ ( ) ROE i ^ predicted values of ( ), t + h. Specifically, for firm i in year t we obtain the QUANT ROE i, t + h for all 150 values of. Next, we calculate the sample mean, standard deviation, skewness, and kurtosis for this sample of 150 values; and, we set PROB_LOSS i,t,t+h eual to the largest value of for which ( ) < 0 ^ QUANT ROE. i, t +h 21

24 Page 23 of 63 Journal of Accounting Research We define skewness as the ratio of the third central moment to the third power of STD_ROE i,t,t+h ; and, we calculate kurtosis by subtracting three from the ratio of the fourth central moment to the fourth power of STD_ROE i,t,t+h. Hence, we evaluate standardized skewness and excess, standardized kurtosis. We do this for two reasons. First, by standardizing we eliminate the possibility that our measures of skewness and kurtosis are simply redundant measures of the variance. For example, if the standard deviation is high, non-standardized kurtosis will also be high even if the distribution is not leptokurtic. Second, the excess kurtosis of a normally distributed random variable is zero; hence, by providing descriptive statistics about excess kurtosis we can evaluate the extent to which our predicted cdf s differ from the normal distribution. Panel A of Table Two contains descriptive statistics. Several comments are warranted. First, the average (typical) firm has positive MEAN_ROE i,t,t+1 ; in particular, the mean (median) of MEAN_ROE i,t,t+h is (0.097). However, MEAN_ROE i,t,t+1 varies considerably across observations; for example, the standard deviation (interuartile range) of MEAN_ROE i,t,t+1 is (0.210). Moreover, untabulated results show that 27.8 percent of the observations have negative MEAN_ROE i,t,t+h. Second, there is also considerable within-sample variation in PROB_LOSS i,t,t+h. Although the sample average of PROB_LOSS i,t,t+h is 25.1 percent, the predicted probability of a loss in year t+1 for the typical firm is only 9.3 percent. Moreover, untabulated results show that 29.9 percent of the sample observations are predicted to be profitable with probability one (i.e., PROB_LOSS i,t,t+1 = 0). On the other hand, untabulated results show that 26.3 percent of the observations are more likely to suffer a loss than realize a profit (i.e., PROB_LOSS i,t,t+h > 0.50). 22

25 Journal of Accounting Research Page 24 of 63 Third, STD_ROE i,t,t+1 is large, which implies there is considerable uncertainty about future ROE. To understand the magnitude of STD_ROE i,t,t+1 better we evaluate the coefficient of variation, CV_ROE i,t,t+1, which euals the ratio of STD_ROE i,t,t+h to MEAN_ROE i,t,t+h. The mean (median) of CV_ROE i,t,t+1 is (0.716). Hence, for the average (typical) observation, the standard deviation of lead ROE is more than twice as large as (70 percent of) the mean of lead ROE. Moreover, untabulated results show that the coefficient of variation for 62.4 percent of the observations exceeds There is also considerable variation in the degree of uncertainty. For example, the interuartile range of CV_ROE i,t,t+1 is Finally, the median of SKEW_ROE i,t,t+1 (KURT_ROE i,t,t+1 ) is (1.748). Moreover, untabulated results show that 65.2 percent of the observations have distributions of lead ROE that are negatively skewed; and, 82.7 percent of the observations have distributions of lead ROE that are leptokurtic (i.e., fat-tailed). Hence, the typical observation in our sample has lead ROE that is drawn from a fat-tailed distribution with a long left tail. This implies that extreme deviations from the mean occur relatively often and that these deviations are more likely to be negative. In Panel B of Table Two we show the correlation structure of the variables. The correlations shown in the table eual the average of the annual cross-sectional correlations. The t-statistics eual the ratio of the average correlation to its temporal standard error. We tabulate results for the prediction sample that pertains to a one-year forecast horizon. Results for prediction samples pertaining to the other forecast horizons are available upon reuest. We discuss the Pearson correlations but the Spearman correlations lead to similar inferences. Several comments are warranted. First, the Pearson correlations between MEAN_ROE i,t,t+1 and PROB_LOSS i,t,t+1, STD_ROE i,t,t+1, SKEW_ROE i,t,t+1, and 23

26 Page 25 of 63 Journal of Accounting Research KURT_ROE i,t,t+1 are , , 0.199, and 0.265, respectively. Hence, firms with high mean lead ROE are less likely to experience a loss, have less volatile earnings, are more likely to experience an extreme deviation from the mean (i.e., lead ROE is more leptokurtic), and positive extreme deviations are more likely than negative extreme deviations (i.e., lead ROE is more positively skewed). Second, when the probability of experiencing a loss is high, lead ROE is more volatile, the likelihood of experiencing an extreme deviation from the mean is greater, and negative extreme deviations are more likely than positive extreme deviations. In particular, the Pearson correlations between PROB_LOSS i,t,t+1 and STD_ROE i,t,t+1, SKEW_ROE i,t,t+1, and KURT_ROE i,t,t+1 are 0.784, , and , respectively. Third, the Pearson correlation between STD_ROE i,t,t+1 and SKEW_ROE i,t,t+1 is ; and, the correlation between STD_ROE i,t,t+1 and KURT i,t,t+1 is Taken together, these two facts imply that as the variance of lead ROE increases the distribution becomes more (less) negatively (positively) skewed. Finally, the Pearson correlation between SKEW_ROE i,t,t+1 and KURT_ROE i,t,t+1 is Hence, when extreme deviations from the mean are likely, extreme positive deviations are more likely than extreme negative deviations. 5.2 Reliability of Firm-year Estimates In this section we evaluate the construct validity of our out-of-sample, firm-year estimates of the probability of a loss and the standard deviation of lead ROE Reliability of Estimates of the Probability of a Future Loss For each holding period h we test whether observations with relatively high values of PROB_LOSS i,t,t+h are more likely to realize a loss in year t+h. We provide two types of evidence. First, we determine the cross-sectional correlation between PROB_LOSS i,t,t+h and 24

27 Journal of Accounting Research Page 26 of 63 LOSS i,t+h. We show the results of this test in Panel A of Table Three. We report the average of the annual cross-sectional correlations and t-statistics that eual the ratio of the average to its standard error. As documented in the table, PROB_LOSS i,t,t+h and LOSS i,t+h are positively correlated regardless of the value of h considered. For example, when h euals one (i.e., a oneyear holding period) the Pearson (Spearman) correlation is (0.519). Although the correlations become smaller as h becomes larger, the correlation pertaining to five-year holding period is still statistically significant and nontrivial: the Pearson (Spearman) correlation euals (0.340). Second, for each sample year and each value of h we form deciles on the basis of PROB_LOSS i,t,t+h. Next, for each decile we calculate the average of: (1) PROB_LOSS i,t,t+h ; (2) LOSS i,t+h ; and, (3) OUT_COMP i,t+h, which euals one (zero) if firm i is no longer (still) a member of the Compustat population in year t+h. We report these averages in Panel B of Table Three. The table contains five main columns each of which pertains to the different values of h. Each main column is divided into three sub-columns named PROB, LOSS, and COMP, which contain the decile averages of PROB_LOSS i,t,t+h, LOSS i,t+h, and OUT_COMP i,t+h, respectively. The results in Panel B illustrate a strong, positive relation between PROB_LOSS i,t,t+h and realized future losses as well as attrition. Although the relations are not monotonic, they are nearly so. Moreover, the differences between the top decile and the bottom decile are economically significant Reliability of Estimates of the Standard Deviation of Lead ROE In this section we evaluate the relation between STD_ROE i,t,t+h and FERR i,t,t+h. FERR i,t,t+h euals the difference between ROE i,t+h and MEAN_ROE i,t,t+h. In Panel A of Table Four we show the average of the annual Pearson (Spearman) cross-sectional correlations 25