RELATIONSHIP BETWEEN FIRM S PE RATIO AND EARNINGS GROWTH RATE

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1 RELATIONSHIP BETWEEN FIRM S PE RATIO AND EARNINGS GROWTH RATE Yuanlong He, Department of Accounting, Economics, Finance, and Management Information Systems, The School of Business Administration and Economics, The College at Brockport, State University of New York, Brockport, NY 14420, USA. yhe@brockport.edu, Tel: , Fax: Pradyot K. Sen, School of Business, University of Washington Bothell, Bothell, WA 98011, USA. pksen@uwb.edu, Tel: Yin Yu, Department of Accounting and Finance, School of Business Administration, Oakland University, Rochester, MI 48309, USA. yu234@oakland.edu, ABSTRACT This study reexamines the relationship between firm s forward PE ratio and the expected earnings growth rate under the theoretical frame work of the abnormal earnings growth (AEG) model (Ohlson & Juettner-Nauroth (2005), OJ). We use the AEG model as a method to separate and control the two opposite effects of the future earnings growth on the PE ratio: the increasing effect due to a higher growth rate that adds value to firm, and the decreasing effect caused by greater uncertainty from the risky growth that requires a higher return. In addition, we test several interactive relationships among following variables: the PE ratio, cost of capital, shortterm earnings growth rate and long-term abnormal earnings growth rate. Keywords: Earnings growth PE ratio Equity valuation Cost of capital INTRODUCTION This study reinvestigates the relationship between the price to forward earnings ratio (PE ratio) and the expected future earnings growth rate under the abnormal earnings growth model (AEG), the fundamental firm equity evaluation framework developed by Ohlson & Juettner-Nauroth (2005) (This model is referred in this study as OJ model or AEG (Abnormal Earnings Growth) model thereafter.). By applying the AEG model, this study documents and reconciles the two opposite effects of the future earnings growth on the PE ratio argued by Penman & Reggiani (2012, working paper): the increasing effect of a higher growth rate that adds value to firm, and the decreasing effect caused by greater uncertainty from the risky growth that requires a higher return. In addition, this study tests the relationship between the PE ratio and the firm s expected rate of change in the abnormal long-term growth rate after controlling for the firm s cost of capital and short-term growth rate. Essentially, this study empirically tests the implications embedded in the Ohlson & Juettner-Nauroth s AEG model and offers evidence in support of the model. The firm s price to earnings ratio, or the PE ratio, is probably one of the most widely used and important market ratios in evaluating the firm s common stock by both professional investors and academia researchers. There is a large body of literature that investigates the correlation

2 between the PE ratio and other variables such as next period earnings, future earnings growth rate, firm specific risk, and sales revenues, etc. We may classify these studies into two main categories: one line tries to find the determinants of the PE ratio, the other line of research tests the predictive power of the PE ratio for the other variables. Assuming a constant dividend payout rate, Gordon (1962) dividend growth model indicates theoretically that firm s PE ratios are positively correlated with firm s growth rate and negatively correlated with discount rate, the measurement for the risk. Following Gordon (1962), earlier literature explains the PE ratio in different ways. One line of existing literature shows strong evidence that the PE ratio tends to converge in the long run (Beaver & Morse, 1978) and suggests we may use the PE ratio as the proxy for the firm s cost of capital or the earnings capitalization rate (Graham et al, 1962; Alford, 1992). Alternatively, Ball (1978) treats it as a risk measurement. Another line of literature interprets the PE ratio as earnings growth indicator (Litzenberger & Rao, 1971; Cragg & Malkiel, 1982). In accounting and finance literature, studies investigating the predictive power of the PE ratio for the company s future earnings growth rate show mixed results. For example, early literature examining the price to current earnings ratio concludes PE ratio s predictability in determining future earnings growth is low (Murphy & Stevenson, 1967; Beaver & Morse, 1978; Chan et al, 2003). However, some studies show empirical evidence that the PE ratio positively associates with the patterns of the future earnings growth rate (Fairfield, 1994; Penman, 1996). More recent studies use forward earnings (PE is calculated by the year-end closing stock price scaled by following period firm s earnings number instead of using the earnings number at the same period) and find a stronger positive relationship between the forward PE ratio and the future earnings growth rate. However, the findings are still not conclusive even after using the forward PE ratio (Thomas & Zhang, 2006; Wu, 2009 working paper). Penman & Reggiani (2012 working paper) point out the potential problem is due to the two contradictory effects of the earnings growth on the PE ratio. Higher earnings growth rate in future from relatively low earnings in current period creates additional value to the firm value, implying there is an increasing effect on the PE ratio. On the other side, the bigger the portion of the firm s value to be realized in the future, the higher the uncertainty for the realization of the value, which requires a higher discount rate, implying there is a decreasing effect on the PE ratio due to the risky growth. The theory of fundamental equity valuations, the Residual Income Valuation Model (RIV) in Ohlson (1995), and the AEG model in Ohlson & Juettner-Nauroth (2005), show a much clearer idea that theoretically, the PE ratio is a function of the firm s cost of capital and the earnings growth rate (both long-term and short-term), and provide the foundation to address the above confounding issue. According to equity valuation model such as RIV and AEG, prior empirical testing of the correlation between the PE ratio and the future earnings growth rate suffers from the following potential drawbacks. First, the use of the contemporaneous price and earnings ratio does not match the equity valuation theory in that price is a reflection of the market expectation of the firm s future earnings instead of the current reported earnings. Second, majority of the studies build portfolio of the firms observations according to their PE ratio and apply either univariate or multivariate method to explore their relationships. But according to equity valuation theory, the PE ratio is not only determined by firm s expected near future earnings growth rate, but also affected by firm s cost of capital and long-term abnormal earnings growth

3 rate, which means a linear relationship between PE and other variables may not be true. By building portfolio according to the quintiles of PE ratios without controlling for the other variables, such research design may suffer from pooling of different categories of the cost of capital and long term abnormal earnings growth rate into one quintile of PE, therefore resulting in potentially confounding inference. Overall, these studies fail to address the dual effects of the high growth rate on the PE ratio. This study differs from the previous studies in that it uses the theoretical inference from the fundamental valuation formula of AEG model. AEG model states that price of the equity is equal to the capitalized forward earnings, plus an adjustment for subsequent superior or abnormal growth in expected earnings. If we substitute the future abnormal growth of expected earnings with two components: near-term earnings growth rate ( as referred in Ohlson & Juettner- Nauroth, 2005; Ohlson & Gao, 2006) and asymptotic or perpetual growth in the rate of change in future earnings growth ( as referred in Ohlson & Juettner-Nauroth, 2005; Ohlson & Gao, 2006), we may have a very simple valuation model rewritten as following (developed in details later for the sake of completeness): or, where (Equation 1) is the stock closing price or firm value at period 0, is the firm s earnings at period 1, r is the firm s cost of capital, is the firm s short-term expected earnings growth rate, and γ is the firm s perpetual growth in the rate of change in future earnings growth (This γ and the rate of change in the firm s abnormal future earnings growth rate are different in that the former one is the latter plus 1, and for simplicity, we use these two interchangeably. Also we refer this rate simply as firm s long-term earnings growth rate or long-term growth rate. As for the short-term, we use short-term earnings growth rate and short-term growth rate interchangeably). As shown by the left side of the equation, firm s PE ratio should be calculated as current stock price divided by the next period expected earnings (Existing literature calls this ratio forward PE ratio). This equation also explains that a firm s PE ratio is determined by several factors: firm s cost of capital ( ), firm s short-term expected earnings growth rate ( ), and long-term earnings growth rate ( ). In order to test the relationship between the PE ratio and, we need to control the other two variables ( and γ). Controlling for the cost of capital addresses the risk factor in the future earnings growth. Otherwise, cross-sectional tests may lead to incorrect results and inferences. It is apparent that the relationship between PE,, and γ are not linear according to the above equation. In this study, we estimate the firm s cost of capital following Easton (2004) s methodology, which is based on the AEG model. We use out of sample estimation to avoid circularity issue. To check for robustness, we also adopt one year forward application of the estimated and γ for the same firm based on the assumption that these values for the same firm remain stable in short-run. Additionally, we use the calculation of r estimated based on CAPM model as further robustness check. For the main empirical tests, we at first test, for a special group of firm observations with a short-term earnings growth rate equal or close to the firm s cost of capital, whether the

4 decreasing effect due to the risk dominates the increasing effect. Then we reinvestigate the positive correlation between the PE ratio and growth rate by building portfolios according to the cost of capital, rather than based on firm s PE ratio. This method essentially allows us to test the relationship between PE ratio and the future earnings growth rate by separating firms into different categories of cost of capital. Within each group, firm observations have the same or approximately the same risk profile. For the sub group of firm observations with a equal or close to, test results show a negative (or positive) and significant correlation between the PE ratio (or the EP ratio, the reciprocal of the PE ratio) and the short-term earnings growth rate, where the decreasing effect of growth due to risk on the PE ratio dominates. This is opposite to what is commonly expected, but exactly what is implied from the theoretical model. In addition, test results show a positive (or negative) and significant correlation between firm s PE (or EP) and the short-term earnings growth rate, after controlling for the firm s cost of capital, which cannot be shown by simply building portfolios according to firm s PE ratio. After controlling for the r and growth rate, the test shows the greater the γ, the higher the PE ratio. Multivariate results show a negative and significant correlation between the PE ratio and cost of capital, a negative and significant correlation between the PE ratio and difference of the cost of capital and longterm growth rate, and a positive and significant correlation between the PE ratio and the difference of the short-term and long-term growth rate, all for transformed variables. This study makes several contributions to the existing literature. First, a new methodology is developed to test and show empirically the correlation between the PE ratio and the short-term earnings growth rate according to the implication from the equity valuation theory. Second, we empirically show a special case where the risk effect from the high growth dominates the value addition effect. Third, we reestablish the positive correlation between the PE ratio and short-term earnings growth rate after controlling for the cost of capital and long-term growth rate. Our empirical results confirm the theoretical implication from the valuation model. Furthermore, we investigate and emphasize the effect of long-term abnormal earnings growth rate on the PE ratio, which previous literature does not address. Lastly, the overall test results confirm empirically the inference from the AEG model. The rest of the study is organized as follows. We start with a review of Ohlson & Juettner- Nauroth (2005) valuation theory (AEG model), and its inference. The hypothesis development follows. Then we outline the research design and measurement issues. Next we discuss the sample selection procedure. Test results are provided after the sample section, followed by robustness checking. The last section provides some concluding remarks. A REVIEW OF OHLSON AND JUETTNER-NAUROTH (2005) VALUATION THEORY AND OUR EMPIRICAL SPECIFICATION According to the firm s equity valuation model, we may derive an equation from Ohlson & Juettner-Nauroth (2005) (AEG model), which can serve as a theoretical background for exploring and testing the correlation between the PE ratio and the future earnings growth rate. Ohlson & Gao (2006) already provide a detailed derivation for the formula. Below is the theoretical derivation process: starting from PVED model, the value, or price of the firm is equal to the present value of future expected dividends

5 (PVED) (Equation 2) In PVED equation, P 0 is the firm value or price of the firm at t=0. is the dividend that is paid out for period t. equals to 1 plus and is the internal rate of return, or the firm s cost of capital. Then we may use the following algebraic zero-sum equality. where: (Equation 3) By adding both sides of this equality to PVED formula respectively, we obtain: (Equation 4) In above equations, yt is the book value of equity ending at the time t. earnings defined as. zt is the abnormal If we replace y 0 with (capitalized forward earnings) into above equation, then we derive the AEG model as equation 5, which states the price of the firm stock equals to the capitalized forward earnings,, plus an adjustment for subsequent superior or abnormal growth in expected earnings ( ), as measured by capitalized future earnings change adjusted by earnings due to earnings retained. Where: (Equation 5) According to general investment community s practice and AEG model, if we define short-term earnings growth rate as: and make an assumption about the time-series distribution of such that:, where (, where ) is the growth parameter. We can rewrite the AEG model to the following forms: or or (Equation 6) From the assumption of and the definition of with further assumption:, for all, then we will have: and (Equation 7) is the measure of the near-term growth in expected earnings, and can be interpret as a measurement for long-term growth in earnings. Ohlson and Gao (2006) shows that is both the measure of asymptotic growth for earnings and asymptotic growth for future dividend payout

6 growth ratio, which provides theoretical basis for the empirical use of some economy wide growth measurements, such as the GDP growth rate as the proxy for γ. Now from above, theoretically, we observe that firm s PE ratio should be calculated as current stock price divided by the next period expected earnings, as shown in the left side of the equation. When we are examining the relationship between firm s PE ratio and earnings growth rate, we need also consider several factors: firm s cost of capital ( ), firm s perpetual growth in the rate of change in future abnormal earnings growth ( ), together with firm s short-term expected earnings growth rate ( ). The model emphasizes the difference between the short-term earnings growth rate and long-term abnormal earnings growth rate, which earlier literature does not differentiate. In sum, in order to test the relationship between the PE ratio and, the model shows the necessity to control the other variables. Otherwise, cross-sectional tests may lead to incorrect results and inferences. HYPOTHESIS DEVELOPMENT Hypotheses follow directly from the implication of the AEG model as described in equation 6. From equation 6, we may illustrate the relationship between forward PE ratio and earnings growth rate under various types of restricted conditions as shown in the graph (taken and embellished from Ohlson & Gao, 2006). Graph: PE ration and growth rate P 0 /X 1 γ 2 (γ 2 > γ 1 ) γ 1 1/r 1/r 2 1 Slope= 1 r( R ) -1 r r g 2 From the equation 6: = +, the line with γ 1 is for firms with the same r and γ. The upward sloping from left to right is defined as, and the higher the g 2, the higher the PE ratio. The line with γ 2 is for firms with the same r but a different γ, which has a steeper slope if we assume γ 2 is greater than. Two lines must intersect at point 1 where and. If we change to a smaller, we may have dash lines with upward sloping from left to right representing firms with same and different γ. Dash lines also must intersect at point 2 where and. If we connect all intersecting points, the downward sloping dash line from left to right represents firm observations where. The location of point 2( relative to the left of point 1) and the downward sloping of the dash line connecting point 1 and 2 is the result of the assumption that

7 From equation 6, a special case happens when firm s short-term growth rate equals the firm s cost of capital, then the equation leads equal to. It states that firm s forward PE ratio equal to the reciprocal of firm s cost of capital when firm s short-term earnings growth rate equal to its cost of capital. Since we assume that firm s short-term growth rate equals to the cost of capital, we can rewrite the above equation into = or =. This equation states that firm s PE (or EP) ratio is negatively (or positively) related to the short-term growth rate, when firm s short-term growth rate equals to the firm s cost of capital, which is the opposite to the conventional conception. Referring back to above graph, this prediction can be illustrated by comparing point 1 with point 2. Point 2 has higher PE ratio than point 1. The negative relationship is shown along the dash line with backward slope formed by two points. This reasoning leads to the following hypothesis (stated in the alternative form). H1: When firms have short-term growth rate equal to their cost of capital, there is a negative (or positive) correlation between firm s PE (or EP) ratio and firm s short-term growth rate, which indicates that decreasing effect from risk aspect of growth dominates the increasing effect from value-adding aspect of growth. According to equation 6, firm s short-term growth rate is positively correlated with the forward PE ratio. In order to empirically test this relationship, we need to control firm s cost of capital and. As for, one way would be to place firms into a group with the exact ; another way would be to relax the assumption that firms specific are equal and treat all firm with a perpetual growth (of change in future abnormal earnings growth) equal to the GDP growth rate (Ohlson, 1995). H2: For those firms with the same r and, there is a positive relationship between firm s PE ratio and short-term growth rate. If we separate firm s observations into two groups, one group includes firms whose PE ratios are greater than the reciprocal of the cost of capital and the other group includes firms whose PE ratios are less than reciprocal of the cost of capital, then for the first group, due to the difference of manifested in the slope difference, controlling for the same cost of the capital and shortterm growth rates, firms that have higher long-term growth rate will have a higher PE ratio than those with a lower long-term growth rate. On the contrary, for the group of firms with PE ratio less than the reciprocal of the cost of capital, firms that have higher long-term growth rate will have a lower PE ratio than those with a lower long-term growth rate, under the assumption that those firms have the same cost of the capital and short-term growth rate. But empirically, since we do not have many of firm observations in this category, we omit the test for this sub-group. H3: For those firms with the same cost of capital r and short-term growth rate, if firms have, then firm with higher will have higher PE than firm with lower. From above hypothesis, we identify that the design and testing issue related to the correlation between PE and firm s earnings growth rate is to control the effect from cost of capital. Mathematical transformation enables us test directly the cross-sectional relationship between

8 firm s PE and earnings growth rate. If we take the natural logarithm on both sides of equation 6, we have:. After rewrite the right side, then we have:. (Equation 8) This is an empirical specification where we expect the coefficients of the independent variables to be negative, negative and positive respectively. It leads to the last hypothesis as follows: H4: In log form, we expect PE is negatively associated with the cost of capital, negatively associated with the difference between the cost of capital and long-term abnormal growth rate, and positively associated with the difference between the short-term and long-term growth rate. Estimation of the cost of capital and gamma METHODOLOGY Following Easton (2004), the estimation of firm specific cost of capital and Gamma is based on the portfolio approach built on the same PEG ratio. This method essentially assumes 20 firms in the same portfolio have the same cost of capital and Gamma. The estimation equation from Easton (2004) is shown as:, where:,, is the forecast of twoperiod-ahead cum-dividend earnings. This equation is directly derived from the AEG model, which was discussed in Easton (2004). To avoid the circularity issue that may arise due to the estimated r included in the test, we adopt an out of sample approach. Each portfolio includes 40 firm observations according to PEG ratio; 20 firm observations are included in the estimation, and the rest of 20 firm observations are included in the tests by applying the estimated r and γ. Out-of-sample test is commonly conducted by splitting a given data set into two parts: in-sample period and out-of-sample period. In-sample period is used for initial parameter estimation and model selection. Out-of-sample period is used to evaluate forecast performance. Out-of-sample test performance is generally considered more trustworthy than the test performance based on insample performance because it is more sensitive to outliers and data mining (White, 2000). So far there is no broadly accepted guideline for how to select the sample split (Hansen & Timmermann, 2012). Welch & Goyal (2008) state it is not clear how to choose the periods over which a regression model is estimated and subsequently evaluated. Stock & Watson (2007) recommend Pick a date near the end of the sample, estimate the forecasting model using data up

9 to that date, and then use that estimated model to make a forecast. In literature, researchers have adopted a variety of practical approaches for out-of-sample testing. Out-of-sample predications have been widely used in Accounting and Finance literature. For example, Dechow et al. (1998) develop a model by applying out-of-sample forecast and imply earnings can better predict future operating cash flow than current operating cash flows and the difference varies with the operating cash cycle. Nam et al. (2012) revisit the cash and accrual component of earnings in predicting future cash flows by using out-of-sample predictions and market value of equity as a good proxy for all future cash flows. Control of the cost of capital and gamma The main issue of tests is to control the cost of capital (or Gamma) for testing the relationship between the PE ratio and the short-term/long-term earnings growth rate. Previous literature commonly build portfolios according to the quintile of firm s PE ratio, then test the difference among the mean (or median) of the short-term growth rates of the different portfolios (Beaver & Morse, 1978; Fairfield, 1994; Penman, 1996). Previous literature s approach did not control the cost of capital effect that may have led to mix different observations belong to different risk category into a single group. Here is an example to illustrate the issue: Chart: an illustration of the negative correlation between PE and growth Firm Group PE ratio Short-term growth rate Cost of capital A B C To make the illustration simple, from equation 6, if we assume for each case that, then we have:. Grouping by the PE ratio cannot lead to the conclusion there is a positive relationship between the PE ratio and the short-term growth rate. On the contrary, compare A to B, higher PE ratio (firm A) has lower short-term growth rate due to the difference in cost of capital. In order to control other variables in the equation two, we take different approaches for several hypotheses as following. For H1, in order to get the firm observations with cost of capital equal to the short-term growth rate, we take an approximation approach. We define two values are the same if the absolute value of their difference is within the certain small range (for example, 0.01). Since the predicted relationship in H1 only applies to the case when and are close, we do not make any prediction nor provide tests when the distance becomes too large, which means the simple linear relationship no longer holds. For H2, we make the assumption that all firms long-term growth parameter Gammas are equal to GDP growth rate. The portfolio is built according to the quintile of the firm s cost of capital for testing H2. We assume firms in the same quintile have the same cost of capital. Within each portfolio, the correlation between firm s PE ratio and short-term growth rate is tested

10 For H3, We use the Easton (2004) estimated r and γ for individual firm and match the firm in pairs based on r and short-term growth rate, then test if among all pairs higher PE is positively correlated with higher Gamma for those observations with PE greater than. Timeline of variables used in the study The stock price used to calculate the EP or PE ratio is the stock closing price at year 0, while the analyst earnings forecasts are the most recent forecast for subsequent year one, which was made immediately before the ending of the calendar year 0. The expected short-term earnings growth rate is the expected growth rate between year 1 and year 2, also forecasted right before the ending of the calendar year 0. The timeline is shown as following: Figure: timeline of variables Year 0 Year 1 Year 2 Dec. 31, 2000 Dec. 31, 2001 Dec. 31, 2002 Price 0 Mean analysts forecast announced and for year 1 (X 1 ), year 2 (g 2 ) Estimated long-term γ SAMPLE Our sample consists of firms with fiscal year end December 31 st from 1986 to This enables us to make the cost of capital estimation in the same calendar year period for each portfolio. Following Easton (2004), we simultaneously estimate the firm s cost of capital and long-term change in the firm s abnormal earnings growth rate ( ). The estimation of cost of capital requires the year-ending closing price, dividend from the Compustat database. Analysts earnings forecast number is either directly from or calculated from IBES database. In addition, firms with SICs between 6000 and 6999 are excluded. Out-of-sample estimation is applied to estimate the firm s cost of capital and. Initial sample includes 17,220 observations that have cost of capital and long-term abnormal earnings growth rate based on Easton (2004) estimation method. As for firm s forward price earnings ratio, the price is the year-end closing price obtained from the Compustat, and the expected earnings is the most recent analysts earnings forecast for the following year ( ), made immediately before current year fiscal year end and obtained from IBES. In order to be consistent with previous literature for comparison purpose, we use the EP ratio to avoid the zero denominators and require stock price greater or equal to 2. The value of the EP ratio is restricted between zero and one. This procedure reduces the sample size to 17,036 firm observations. To test the relationship between firm s PE ratio and earnings growth rate, we use the expected one-year-ahead earnings growth rate as calculated from the IBES analysts earnings forecast (ex-ante). We calculate the expected earnings growth rate from two adjacent year earnings forecast taken from IBES. If earnings forecast number is not available then we use

11 forecast earnings growth rate. When calculate expected earnings growth rate, we adjust foregone earnings by dividend payout. Extreme observations for all variables at the top and bottom 1% for the entire sample are deleted. The final data consists of 15,973 firm year observations and 5,190 firms. Table 1 reports the descriptive statistics for all variables. The sample has a wide range of difference regarding the distribution of the total assets, sale revenue and year end closing price per share. For example, the mean value of the total assets is 9, million dollars while the medium is only million. Regarding main variables of interest, the calculated expected firm s earnings growth rate ( 2) has a mean value of 23.58% and medium value of 16.00%. The mean for the EP ratios is with a median of , which is about 20 for the PE value. The mean value of the cost of capital is 12.26% with the median value of 11.32%. The mean value for Gamma is 3.44% with the median value of 2.53%. The estimated cost of capital and Gamma are close to the Easton (2004) estimated value. Since the normality test for these main variables shows their distributions are not normally distributed, we also perform the nonparametric tests for the matched observations for hypotheses testing, and the results are similar to those from the parametric tests. Table 1 Descriptive statistics Variables Mean Std. Dev Min 25% Median 75% Max Total assets(mm$) Sales(MM$) Expected Growth rate ( ) EP ratio Gamma ( ) Cost of capital ( ) This table reports summary statistics for the main variables in the study. There are total 15,973 firm year observations and 5,190 firms included. In the study, we use annual data and only choose those firms with December 31 st fiscal year end. Total assets, sales and price per share are from Compustat annual database. All variables are defined as in Appendix. Table 2 reports the yearly distribution of the mean and medium of the main variables. The total observations are smaller for the earlier years than for the latter years. The mean and medium value for the EP ratio is stable across years, which ranges from 0.04 to 0.07, with majority of the value around Similar pattern exists for the expected growth rate, both long-term and shortterm, and for the estimated cost of capital. Overall, the sample shows a stable pattern for all four variables. Table 3 reports the Spearman correlations of the main variables. There is a negative (positive) correlation between firm s forward EP ratio (PE ratio) and short-term growth rate. The table also reports a positive correlation between EP and the cost of capital while a negative correlation between EP and Gamma. There is a positive correlation between cost of capital and Gamma

12 estimated, while the correlation between and growth rate is also positive. Finally, there is a negative correlation between the earnings growth rate short-term and long-term. Table 2 Yearly distribution of mean (median) of main variables Year Obs. EP Gamma r Mean Median Mean Median Mean Median Mean Median This table presents the mean and median value of main interested variables:,, Gamma ( ) and. There are total 15,973 firm year observations and 5,190 firms included. All variables are defined as in Appendix. Table 3 Spearman correlation for main variables EP Gamma EP < Gamma < < < < This table presents spearman correlation among main variables:,, Gamma ( ) and. All variables are defined as in Appendix

13 RESULTS Table 4 shows test results for H1. There is a positive correlation between EP and when firms have a short-term growth rate equal or approximately equal to the firm s cost of capital. Column 1 represents the distance between and. For example, second row of table consists of 868 firm observations with the difference between and less than (absolute value). The coefficient for the independent variable g 2 from the regression equals to with a p-value less than and adjusted R-square equals to As moving down from the top to the bottom, the differences of values between and the cost of capital increase. The absolute value of the difference between g 2 and r increase to the range between and Regression results show that coefficients on across all partition are significantly positive with EP at 1% level (except the last group), which confirms H1 s prediction. Coefficients and adjusted R- squares decrease as the difference between and becomes larger. If we keep building on the table by expanding the difference between and, the coefficient finally becomes negative and significant, which we do not include in the tabulated results. We may also design the test by including the observations in previous group into the following group of. We have the same result but the significance declines more slowly due to the effect of including previous more significant observations. Table 4 Test results of the positive relationship between EP and g 2 for firms that have a short-term growth rate equal or near equal to the cost of capital r Obs. No. (-0.005,0.005) 868 (-0.010, ] or [0.005, 0.010) 892 (-0.015, ] or [0.010, 0.015) 879 (-0.020, ] or [0.015, 0.020) 799 (-0.025, ] or [0.020, 0.025) 711 (-0.030, ] or [0.025, 0.030) 720 (-0.035, ] or [0.030, 0.035) 699 (-0.040, ] or [0.035, 0.040) 657 (-0.045, ] or [0.040, 0.045) 564 Regression of dependent (0.8474) on Adj r-sq This table presents the results of testing H1 in terms of positive relationship between and after controlling for firms shortterm growth rate equal to the firms cost of capital. P-value is in parenthesis. Sample size includes 6,789 firm year observations

14 is the distance between r and earnings growth rate. The difference between and is defined at an increment of (both negative and positive). The portfolio starts at the difference centered around zero. Then the absolute value of the difference between g 2 and r gradually becomes larger. All variables are defined as in Appendix. Table 5 reports the results of the correlation between EP and growth rate for portfolios built on the EP ratio ranked from lowest to highest. Each portfolio includes 798 or 799 firm year observation. Second column reports the mean (median) of. For example, portfolio 1 (the lowest EP portfolio) has a mean at and median at Portfolio 20 is the highest EP portfolio with the mean value of at and median value at There is a nearly monotonically decreasing trend for as EP becomes bigger. But this trend only exists if portfolios are apart from each other far enough. For those portfolios that are close, for example, portfolio 7 and 8, 8 and 9, 10 and 11, etc., mean and median values of are almost the same. The two samples t-test of difference of is performed and results are reported in column 3, 4, and 5. P-values indicate the significance of the difference of mean of between two comparing portfolios. For example, column 3 shows for the difference of for those EP portfolios that are adjacent. The value reported are taking the difference of between portfolio 1 and portfolio 2, portfolio 2 and portfolio 3, portfolio 3 and portfolio 4, so on and so forth. The results from 17 comparisons out of 19 are not different from each other at 5% level. The difference of between portfolio 1 and portfolio 2 equals to with a P-value at The second pair which has significant difference of is between portfolio 3 and portfolio 4. Column 4 compares portfolios that are one group apart, for example, portfolio 1 comparing to 3, and 2 comparing to 4. Result shows 9 out of 18 of these comparisons for are not significant from each other at 5% level. Results from Column 5 show there are increasing and significant difference in terms of between EP portfolios that are apart from each other far enough. For example, portfolio 9 comparing to portfolio 12 has a difference in equal to , indicating their are significantly different from each other. However, there are still some of the comparisons are insignificant. In sum, table 5 indicates several points: it confirms previous literature s finding that there are no definite correlations between firm s EP ratio and growth rate; it also partially explains the mixed results of extant literature regarding EP and earnings growth. Table 6 conducts the test for H2 by constructing portfolios based on the firm s cost of capital rather than based on EP. This approach makes the assumption that observations in the same portfolio have the same or similar cost of capital. Thus for each group, we essentially control the cost of capital and then run the regression of firm s EP ratio on short-term growth rate. For most of the portfolios, the difference of inside the portfolio is about 0.005, except for the extreme high and low cost of capital portfolios. For example, portfolio 1 has a range of r between and and portfolio 2 has a range of between and , while portfolio 20 has the highest value of, ranging from to Overall, column 2 shows that the difference of cost of capital across each cost of capital portfolio is relatively small, which validates our assumption for running the regression. Across all portfolios, test results show coefficients on are all negative and significant at 1% level, with adjusted R-squares ranging from 19% to 45%. For example, portfolio 16, coefficients on is , indicating is negatively associated with EP when firms have close enough cost of capital fallen into the specified range

15 Table 5 Test results of significance of the difference of according to the portfolios formed based on EP. (Replication of Penman (1996), Wu (2009, working paper)) portfolio 1 (low) (high) Mean (median) (0.2500) (0.2318) (0.2122) (0.1975) (0.2000) (0.1826) (0.1757) (0.1770) (0.1650) (0.1611) (0.1500) (0.1473) (0.1404) (0.1300) (0.1333) (0.1326) (0.1389) (0.1350) (0.1400) (0.1467) Two samples t-test of difference of (Pr> t ) 1-2,2-3,etc 1-3,2-4,etc 1-4,2-5,etc (0.0044) (0.0995) (0.0127) (0.0006) (0.9714) (0.2961) (0.0025) (0.3052) (0.0023) (0.0008) (0.0594) (0.0298) (0.0005) (0.7610) (0.0924) (0.0205) (0.1690) (0.0444) (0.0009) (0.5250) (0.0449) (0.0098) (0.1680) (0.0522) (0.0003) (0.6147) (0.0274) (0.0004) (0.0719) (0.0012) (0.1570) (0.1146) (0.7364) (0.6686) (0.0647) (0.2528) (0.6615) (0.4566) (0.1032) (0.0099) (0.4025) (0.0696) (0.8512) (0.2699) (0.4885) (0.4772) (0.0808) (0.0886) (0.9549) This table presents the test results of replication of previous literature. The portfolio is formed based on EP. All variables are defined as in Appendix. Sample size includes 15,973 firm year observations. Two samples t-test is done to compare the mean of the g 2 of portfolio 1 with portfolio2, portfolio 2 with portfolio 3, portfolio 3 with portfolio 4, etc. Then we also perform the test across two portfolios apart, for example compare portfolio 1 with portfolio 4, portfolio 2 with portfolio 5, portfolio 3 with portfolio 6, etc. P-value is in parenthesis. Wilcoxon-Mann-Whitney test is also performed to address the non-normal data with similar results

16 Table 6 Test results of regression on short-term growth rate g 2 after controlling for cost of capital r estimated by Easton (2004) method portfolios Range of Regression of on p-value Adj r-sq 1 (low) < < < < < < < < < < < < < < < < < < < (high) < This table presents the test results of testing H2. The portfolio is formed based on the cost of capital. Each regression has 798 or 799 observations. sample size includes 15,973 firm year observations. All variables are defined as in Appendix. Table 7 represents test results for the correlation between EP and Gamma after controlling for the cost of capital and short-term growth rate ( ) for firms with PE greater than. Firm observations are first matched in pairs based on and. We calculate for each pair the difference between matched firms Gamma and the difference between matched firms EP ratio. Then the spearman correlation test is run between two calculated differences. In H3, we predict the higher the Gamma, the higher the PE (or lower the EP) for firms with the same and. The results from both the parametric and non-parametric tests (spearman correlation with p-value in parenthesis) are significant and negative at 1% level. For example, the correlation between the difference of Gamma and the difference of EP ratio equals to with a P-value less than

17 Table 7 Test results of the relationship between EP and Gamma for matched firm observations with the same cost of capital and short-term expected growth rate for those firms with (or ) Difference between matched firm s Gamma Non-parametric difference between matched firm s Gamma Difference between matched firm s EP ratio Non-parametric difference between matched firm s EP ratio Difference between matched firm s Gamma 1 Non-parametric difference between matched firm s Gamma Difference between matched firm s EP ratio Non-parametric difference between matched firm s EP ratio This table presents the results of testing H3. Values reported in the table are the spearman correlation with P-value in parenthesis. Firm observations are separated according to their PE ratio with. Firms are matched based on the cost of capital estimated and their short-term earnings growth rate. Difference between matched firm s Gamma equals to Gamma estimated for the firm minus the Gamma of the matching firm. Non-parametric difference between the matched Gammas takes value of 1 if difference between the Gammas is greater than zero, zero if less than zero. Difference between matched firm s EP ratio is calculated by subtracting the matched firm s EP ratio from the matching firm s EP ratio. Non-parametric difference between matched firm s EP ratio takes value of 1 if difference is greater than zero, zero if less than zero. There are 6,087 pairs of observations for this test. Table 8 presents the test results for H4. The multivariate regressions is run with dependent variable of PE on independent variables of r, difference between and, difference between and, all natural logarithm transformed. The regression is run for the entire sample and also run by yearly. The second row of the table represents the result for entire sample, which consists of 15,973 observations. For the entire sample, the coefficient on the transformed r equals to , the coefficient on the transformed difference between r and γ equals to , and the coefficient on the transformed difference between and γ equals to , all significant at 0.01 level. The adjusted R-square for the first regression run on entire sample equals to Regarding yearly run regressions, except for the year 1995 and 2001 that coefficients on log(r) are not significant at 1% level, all of coefficients estimated have predicted signs and are significant at 1% level. The adjusted R-squares for regressions range from 54% to 82%

18 Table 8 Test results of multivariate analysis on the relationship between PE and Gamma, r, and Year Obs. Adj r-sq Entire sample (0.0003) (0.8829) (0.1614) (0.0010) (0.0010) (0.0158) (0.0012) (0.0097) This table presents the results of testing H4. All variables included in the regression are natural logarithm transformed. The regression is run for the entire sample and also run by yearly. P-value is reported in the parenthesis. Sample size includes 15,973 firm year observations. All variables are defined as in Appendix

19 Different measurements for the growth rate ROBUSTNESS CHECK We use additional measurements for earnings growth rate as a robustness check for the above tests. For realized earnings growth rate, we use reported yearly earnings numbers adjusted by dividend from Compustat, and reported yearly earnings numbers without dividend adjustment from IBES. Earnings growth rate calculated from Compustat is ex-post and is used to test the predictability of firm s PE ratio for the future growth rate. IBES earnings growth rate is ex-ante and is used to test whether expected growth rate could determine firm s PE ratio. As we combine three growth rates together, the final test sample is smaller than the original one. The results are similar, however the R-square from the regression is much smaller for the realized earnings growth rate comparing to the expected growth rate. Correlation between contemporaneous calculated PE ratio and following period growth rate Since existing literature explores the relationship between contemporaneous PE and following period earnings growth rate, we also conduct the test after controlling for the firm s cost of capital. We expect to see a more significant correlation after controlling for comparing to the results in previous existing literature, but a less significant correlation comparing to the correlation when the forward PE ratio and the following period earnings growth rate are used, after controlling for. The test shows contemporaneous PE is significantly correlated with forward PE for the most of portfolios but such correlation is much weaker as indicated by a very small coefficient and very low adjusted R-square. Cost of capital estimated by CAPM model Besides the cost of capital calculated from the Easton (2004) method, we also estimate the cost of capital by using the CAPM model with a 24-month rolling window. We use the r estimated by CAMP for testing H1 and H2. Empirical test results are robust, although the R-square from the regression is much smaller. Applying estimated cost of capital from the Easton (2004) method one year forward We apply estimated cost of capital from the Easton (2004) method one year forward to avoid the circularity issue under the assumption that cost of capital for a firm stays the same in short term. Test results are consistent but the coefficient and adjusted R-square are relatively lower. One possible explanation is firm s cost of capital does change within two years. SUMMARY AND CONCLUSIONS This study reexamines the relationship between firm s forward PE ratio and the expected earnings growth rate under the theoretical valuation framework of the AEG model. The equation derived from AEG model states that firm s PE ratio is determined by three components: firm s cost of capital, short-term expected earnings growth rate, and long-term abnormal earnings