Australian School of Business School of Accounting. Semester 2, 2013

Transcription

1 Australian School of Business School of Accounting School of Accounting Seminar Series Semester 2, 2013 Mitigating the effects of forecast errors on estimates of the implied expected rate Peter Easton University of Notre Dame Date: Friday 30 th August 2013 Time: 3.00pm 4.30pm Venue: ASB 216

2 Mitigating the Effects of Forecast Errors on Estimates of the Implied Expected Rate of Return on Equity Capital Zhi Da Department of Finance Mendoza College of Business University of Notre Dame (574) Peter Easton Center for Accounting Research and Education Mendoza College of Business University of Notre Dame (574) Keejae Hong Department of Accounting Belk College of Business UNC Charlotte (704) March 2013 We thank Brad Badertscher, Robert Battalio, Jeff Burks, Shane Corwin, Megan Cosgrove, Paul Healy, Xue Jia, Stephannie Larocque, Karthik Ramana, Georgios Serafeim, Suraj Srinivasan, Tom Stober, Gary Taylor, Stephen Taylor, Marcel Tuijn, and workshop participants at Harvard Business School, Tilburg University, the University of Alabama, the University of Notre Dame, and the University of Technology, Sydney for helpful comments and suggestions on an earlier draft.

3 Mitigating the Effects of Forecast Errors on Estimates of the Implied Expected Rate of Return on Equity Capital Abstract We suggest an alternative source of prices, which can be used in reverse engineering accounting based valuation models to obtain estimates of the implied expected rate of return on equity capital. We replace market prices with the present value of analysts forecasts of dividends and (target) prices; in other words, we estimate the expected rate of return that equates the present value of expected dividends and target prices with the present value of the pay-offs from earnings-based valuation models. We predict and show that the positive correlation between errors in forecasts of earnings and errors in forecasts of target prices may be used to obtain estimates of the implied expected rate of return that are less affected by measurement error than estimates based on market prices (as in the extant literature). Our paper compliments a growing literature, which has the goal of improving estimates of the cost of capital that are based on analysts earnings forecasts by removing predictable errors from these forecasts. We show that removing these errors does not lead to improvement for the estimates based on forecasts of changes in earnings (e.g., those based on the PEG model), but, use of our alternative source of prices does lead to improvement in estimates based on these models. Keywords: analysts forecasts, target prices, cost of capital

4 1. Introduction A vast recent literature reverse engineers accounting-based valuation models to obtain estimates of the implied expected rate of return on equity capital. The inputs to these models are, either: (1) market prices, book values, and analysts forecasts of earnings; 1 or, (2) market prices, forecasts of dividends, and target prices. 2 The implied expected rate of return is either: (1) the rate that equates market prices and the present value of the forecasted earnings-based payoffs; or, (2) the rate that equates market prices and the present value of forecasts of dividends and target prices. In addition to the considerable empirical evidence of errors and bias in analysts forecasts, 3 the observation that analysts use these forecasts to make buy/sell recommendations suggests that these earnings forecasts are not the earnings expectations that are implicit in market prices. It follows that (mis-) matching prices to analysts forecasts when reverse engineering accountingbased valuation models may be a source of error in estimates of the implied expected rate of return. We circumvent this issue by replacing market prices with an intrinsic value calculated as the present value of analysts forecasts of dividends and target prices. In other words, we use an apples-to-apples match; we estimate the expected rate of return that equates the present value of expected dividends and target prices and the present value of the pay-offs from earnings-based valuation models. We show that, for some sub-samples of observations, our estimates of the implied expected rate of return where market prices are replaced by the present value of expected dividends and target prices are less affected by measurement error than those based on observed market prices. 1 See Easton (2007) for a review of these models in the literature prior to Recent examples include Pastor et al. (2008), Chava and Purnanadam (2009), and Lee et al. (2009). 2 See, for example, Francis et al. (2004), Botosan and Plumlee (2005), Brav et al. (2005), and Botosan, et al. (2011). 3 See, for example, O Brien (1988); Mendenhall (1991); Brown (1993); Dugar and Nathan (1995); Das, Levine, and Sivaramakrishnan (1998), and Easton and Sommers (2007)). 1

5 We refer to estimates of the expected rate of return implied by market prices and earnings forecasts as market price based estimates (denoted MPER). Estimates of the implied expected rate of return when we replace market prices with the present value of expected dividends and target prices are referred to as target price based estimates (denoted TPER). We observe a significant positive correlation among: errors in forecasts of shorter run future earnings, errors in forecasts of longer run future earnings, and, errors in forecasts of target prices. 4 These correlations have three effects when these forecasts are used in reverse engineering: (1) the correlations between errors in forecasted earnings and errors in forecasted target prices mitigate the effect of these errors because they tend to cancel one another in the estimation of TPER; (2) the correlation between errors in shorter run forecasts of earnings and errors in longer run forecasts: (a) exacerbates the effect of errors in either one of these variables when the level (rather than changes) in these earnings forecasts are used in estimating the implied expected rate of return; 5 but, (b) mitigates the effect of these errors when forecasts of changes in earnings are used in estimating the implied expected rate of return because they tend to cancel one another. 6 We consider the effects of each of these correlations on estimates of the implied expected rate of return. The empirical trade-off between, on the one hand, mis-matching market prices and analysts forecasts of earnings in the market price based estimates (MPERs) and introducing another 4 Our forecast of earnings change is the difference between the forecast of two-year-ahead earnings and the forecast of one-year-ahead earnings. As a practical matter, forecasts of the earnings are generally available for the next two years and then the forecasted growth rate is used to grow from the year 2 base to obtain forecasts for the next three years. For convenience we use the correlation between the errors in year 2 forecasts and the errors in year 3 forecasts as the indicator of the correlation between errors in shorter run earnings forecasts and errors in longer run earnings forecasts. This correlation is likely primarily due to the fact that forecasts of longer run earnings are based on the forecast of shorter run earnings. 5 Methods that rely on the level of forecast of earnings, include, Gebhardt, et al. (2001), Claus and Thomas (2001) and Easton and Monahan (2005). 6 Methods that rely on forecasted changes in earnings include the PEG method and the Modified PEG method from Easton (2004) and the Gode and Mohanram (2003) method. These models are all restricted forms of the Ohlson and Juettner-Nautoth (2005) abnormal growth in earnings valuation model. 2

6 source of error when target prices are used instead of market prices in the target price based estimates (TPERs) on the other, depends on the relative magnitude of the forecast errors in earnings and target price and on the correlation among these errors. We identify, a priori, samples where we expect the canceling effect of the correlation among the errors to be most evident. These samples are formed by comparing MPERs with TPERs. Much higher MPERs than TPERs suggest that the earnings forecasts have a large optimistic bias (leading to high MPERs) and that the effect of the bias in the target price on the estimate of the implied expected rate of return is similar to or less than the effect of the bias in these earnings forecasts (leading to lower TPERs). For these observations, the mitigating effect of the positive correlation between the errors in forecasts of earnings and the errors in forecasts of target prices, likely leads to TPERs that are less affected by measurement errors when compared to MPERs. On the other hand, observations where the MPERs are much lower than the TPERs are likely to have much less optimistic bias in the forecasts of earnings (leading to lower MPERs). Consistent with the results in Easton and Monahan (2005), which show that the effect of measurement error is lower when the earnings forecast error is lower, we expect the MPERs to be less affected by measurement error for these observations. Also, for these observations, we expect the bias in the forecasts of target prices to be high relative to the bias in the forecast of earnings (the higher TPER relative to MPER suggests relatively higher bias in target price and relatively lower bias in earnings forecasts). It follows that, for these observations, substituting target prices for market prices will likely exacerbate rather than mitigate the effect of errors in earnings forecasts. We explore these predictions by partitioning our sample into terciles based on the difference between the MPERs and the TPERs. 3

7 Consistent with our expectations, we find that, for the tercile of observations where the MPERs are much higher than the TPERs, the TPERs are less affected by measurement error. On the other hand, when the MPERs are much lower than the TPERs, the MPERs are less affected by measurement error. The main point of these analyses is that researchers may use these straight-forward partitions of the data to obtain superior estimates of the implied expected rate of return. It is important to point out that these partitions identify forecast errors ex ante; in other words, researchers can form the partitions before the actual earnings and future prices are known. Our results suggest that, when MPERs are much higher than TPERs, researchers should consider using TPERs instead of MPERs. We show that the superior performance of the TPERs for the sample where the MPER is much higher than the TPER is attributable to the facts that, for this sample: (1) the errors and biases in forecasts of earnings and in forecasts of target prices are greater than in the other terciles; (2) the effects of the errors in forecasts of earnings are of a similar magnitude to the effects of the errors in forecast of target prices; 7 and, (3) the errors in forecasts of earnings are positively correlated with the errors in forecasts of target prices. These features of the forecasts suggest that: (1) noise introduced by matching market prices with forecasts of earnings is considerable; and, (2) the effect of errors in analysts forecasts of earnings opposes, and therefore mitigates, the effect of errors in analysts forecasts of target prices when we reverse engineer to estimate the implied expected rate of return. In short, this partition identifies a sub-sample of observations for which the errors in the forecast is large. This is the sub-sample for which Easton and Monahan [2005] show that all of the extant methods of estimating the implied expected rate of return perform poorly. Our results are important because we show that it is for this sub-sample that substituting the present value of forecasts of dividends and target prices for 7 We will elaborate on this important point in the empirical analyses. 4

8 market prices mitigates the measurement error effects, leading to superior performance of the estimates. We show that the superior performance of the MPERs for the sample where the MPER is much lower than the TPER is primarily attributable to the fact that, for this tercile, the errors and biases in forecasts of earnings and target prices are much smaller than in the other terciles. Thus, for this tercile, noise introduced by matching market prices with forecasts of earnings is low and, although the errors in forecasts of earnings are positively correlated with the errors in forecasts of target prices, introducing this second source of error leads to inferior estimates of the implied expected rate of return. Although the main thrust of our analyses is on demonstrating the possibility of exploiting the correlation between the errors in the forecasts of earnings and the errors in the forecasts of target prices in order to obtain improved estimates of the implied expected rate of return, we also examine the implications of the positive correlation among the shorter run and longer run forecasts of earnings for estimates of the implied expected rate of return. Our additional analyses show that the use of only shorter run forecasts of earnings (i.e. only two years as opposed to five years) in the estimation of the implied expected rate of return mitigates the effect of measurement error. Some of the extant methods of estimation of the implied expected rate of return are based on five years of forecasted earnings generally estimated using: (1) specific forecasts of earnings per share for the next two years; and, (2) growing the second-year earnings at the analysts forecast of the long term earnings growth rate. A positive correlation between errors in forecasts of earnings and errors in forecasts of earnings growth amplifies the effect of error in these earnings forecasts. 5

9 Claus and Thomas (2001), for example, reverse engineer a model based on five years of earnings forecasts. We show that reverse engineering a similar model based on just two years of earnings forecasts (thereby removing the effect of the correlation in the errors in short run forecasts of earnings and errors in longer run forecasts of earnings) yields estimates of the implied expected rate of return that are less affected by measurement error. On the other hand, Gode and Mohanram (2003) reverse engineer a model based on two years of earnings forecasts. We show that reverse engineering a similar model based on five years of earnings forecasts (thereby introducing the effect of the correlation in the errors in shorter run forecasts of earnings and errors in longer run forecasts) yields estimates of the implied expected rate of return that are more affected by measurement error. Interestingly, however, when we use five years of forecasts of earnings in the Gebhardt et al. (2001) model rather than just two years (as in their original analyses), the change in the effect of measurement error is not significant. Our paper is part of a growing literature, which has the goal of developing better estimates of the cost of capital. We compliment and extend recent studies, which examine the effects of removal of predictable errors in analysts forecasts on estimates of the implied expected rate of return (Larocque (2012) and Mohanram and Gode (2012)). An implication of the correlation between errors in forecasts of one-period ahead earnings and errors in forecasts of two-period ahead earnings is that the effect of errors in the forecasts may be much lower for estimates of the implied expected rate of return based on forecasts of change in earnings when compared to those based on the forecasted level of earnings. Furthermore, in the process of removing the so-called predictable errors, new errors may be introduced, potentially leading to inferior estimates of the implied expected rate of return. 6

10 We show that the removal of predictable forecast errors using the methods in Larocque (2012) and Mohanram and Gode (2012) does not lead to improved estimates of the implied expected rate of return for estimates that are based on forecasts of changes in earnings (e.g., those based on the PEG method). These estimates are inferior on at least three dimensions: (1) for more than 30 percent of the observations, the forecasts of change in earnings becomes negative after removing the predictable errors and, hence, the implied expected rate of return cannot be calculated at all for these observations; (2) the correlations between the implied expected rate of return and realized returns are lower for the estimates where predictable forecast errors have been removed; and, (3) the effect of measurement error in the estimates of the implied expected rate of return based on forecasted changes in earnings with the predictable error removed is greater than the effect of measurement error before the removal of the predictable error; this result is evident for all three terciles of observations formed on the difference between TPER and MPER. The practical implication of these results is that removing predictable forecast error is not a viable way of improving estimates of the implied expected rate of return that are based on forecasts of changes in earnings and the only viable alternative of which we are aware is the method suggested by the results in this paper: (1) form groups of observations based on the difference between TPER and MPER; and (2) use TPER for the subsample where the difference between MPER and TPER is high and use MPER where the difference is small. An alternative, often used, estimate of the implied expected rate of return is the rate that equates market prices and the present value of forecasts of dividends and target prices; we refer to these estimates as r div. 8 As an important aside, we compare these r div estimates with the 8 See, for example, Francis et al. 2004; Botosan and Plumlee 2005; Brav et al. 2005; and Botosan et al

11 MPERs and the TPERs and we observe that the measurement error effect on these r div estimates is much greater than the measurement error effect on any of the MPERs and TPERs. The remainder of the paper is organized as follows. Section 2 describes our sources of data and our sample selection procedure. Section 3 describes the data with a particular emphasis on the correlations among the errors in forecasts of earnings and target prices. Section 4 examines the correlations among the forecast errors. Section 5 describes our methods of estimating the implied expected rate of return. Section 6 describes our method for evaluation of the estimates of the implied expected rate of return. Section 7 reports our estimates of the implied expected rate of return and presents the results of the evaluation of the estimates. Section 8 identifies subsamples where we predict (and find) that use of target prices instead of market prices in the reverse engineering of the accounting-based valuation models leads to superior estimates of the implied expected rate of return. Section 9 analyses the effectiveness of attempts to improve estimates of the implied expected rate of return by removing predictable forecast errors. Section 10 demonstrates the effect of the positive correlation between the errors in the short run forecasts of earnings and the errors in longer run forecasts on estimates of the implied expected rate of return that are based a two year forecast horizon compared with those based on a five year forecast horizon. Section 11 presents a summary and our conclusions. 2. Sources of Data and Sample Selection Our source of forecast data is Value Line Inc. Value Line provides forecasts of earnings in 13 week cycles such that a forecast is provided for each covered firm in each quarter of the year; a set of firms is covered in each weekly report and once covered the firm is generally not covered again until 13 weeks later. For each firm-year observation, we select the first Value Line 8

12 forecast of earnings for the current year t that is made after the announcement of earnings of the prior year t-1. With this forecast, we also extract from Value Line the forecasts of earnings and dividends for year t, earnings and dividends for year t+1, forecasts of long-term earnings growth and long term dividend growth and target prices. The price at the close of trade two days after the forecasts are made is obtained from the CRSP daily price file. Book value per share is the book value as at the end of year t-1 divided by number of shares used to calculate earnings per share and is obtained from the Compustat annual file. We delete any firm-year observation with an estimate of the implied expected rate of return missing, smaller than 0.01 percent, or greater than percent. Methods used to estimate the implied expected rate of return are described in section 5 of the paper. The Value Line data used in this study have four features, which make them more suitable for our evaluation of estimates of the implied expected rate of return, than those of the more commonly used consensus earnings forecasts issued by sell-side equity analysts. First, target price forecasts are readily available for a long time period. Second, consensus sell-side earnings forecasts obtained from other often-used data such as I/B/E/S are averages of forecasts issued at different points in time by different analysts. The resulting non-synchronicity may lead to additional error in the estimates of the implied expected rate of return. Third, the fact that revisions in forecasts may be due to changes in analyst coverage does not arise with Value Line forecasts because there is only one analyst issuing the forecast on any date. Finally, forecasts issued by sell-side analysts may be affected by pressure from investment banking relations and career concerns of the analyst (Lin and McNichols, 1998); Value Line forecasts are likely less affected by such considerations. 9

13 Table 1 reports the descriptive statistics for our final sample. Overall, our final sample contains 9,238 firm-year observations from 1991 to 2007 and it covers approximately 543 stocks each year. 9 In terms of number of stocks covered, our sample is small compared to the entire CRSP universe. We note that the stocks covered by Value Line are typically larger, with a mean market capitalization of $8,784 million compared with $2,259 million for the entire CRSP universe. On average over the sample period, our sample covers 39 percent of the entire U.S. CRSP equity universe in terms of market capitalization. As a comparison, I/B/E/S covers more than twice the number of stocks on average and its average market capitalization coverage is 78 percent of the CRSP universe. The next section of the paper is devoted to an analysis of forecast errors. Errors in forecast of earnings per share are calculated as the difference between the corresponding actual earnings as reported by Value Line and the forecast; errors in the forecasts of dividends per share are calculated as the difference between the corresponding actual dividends obtained from the Compustat and the forecast; and, the error in the forecast of target price is calculated as the difference between the end of year price reported on CRSP corresponding to the target price (i.e., the actual price is taken from CRSP four years after the forecast) and the target price. 3. Bias and Errors in Forecasts Because the correlations among the forecast errors are at the core of the thesis of our study, we analyze these errors and their correlations in some detail. We begin, in this section with a description of the magnitude of these errors and as a part of this description; we observe 9 All results are reported for this sample of observations. Our analyses require four years of realizations of earnings and prices; hence, although our data extends through 2010, we can complete our analyses only for years 1991 to To ensure that we have the same sample for all thirteen different estimates of the implied expected rate of return on equity, which we compare in this study, we delete observations when any one of the estimates of implied expected rate of return on equity capital is missing or smaller than 0.01 percent or greater than percent. 10

14 correlations at the portfolio level. We report and discuss the firm-level correlations in the next section. We, initially, scale forecast errors by the stock price two days after the forecasts are made. These scaled forecast errors are then sorted each year into deciles, where the lowest (highest) decile portfolio has the most pessimistic (optimistic) error and we calculate the mean and median error for each decile. Table 2 reports the average price-scaled error (i.e., bias) in forecasts of earnings, target prices and dividends. Two observations are immediately apparent. First, the mean (median) error (i.e., bias) in earnings per share for the current year, eps t, shows optimism in the forecasts (-0.7 percent of stock price and -0.5 percent of stock price) as does the mean (median) error in forecasts of earnings per share for the next year eps t+1 (-0.9 percent and -0.3 percent of stock price). Forecasts of two-year hence earnings per share eps t+2 are also optimistic (mean and median errors of -1.5 percent and -0.9 percent of stock price). Forecasts of target prices (TP) are also optimistic (mean and median errors of percent and percent of stock price). At first glance it may appear that forecasts of target prices are much more optimistic that forecasts of earnings but this is not so; a mean error in forecasts of earnings of -1.5% and a mean error in forecasts of target prices of -14.5% implies a price-earnings ratio of about ten, which is not high. 10 The tendency for forecasts to be optimistically biased is also seen in the forecasts of dividends (e.g., mean and median errors in forecasts of dividends per share dps t+3 of percent and percent of stock price). Second, the errors in eps t, eps t+1, eps t+2 and target price are positively and monotonically related to each other; the mean price-scaled error in forecasts of eps t+1, eps t+2 and target price decrease monotonically from pessimistic (5.1 percent, 3.6 percent and 48.3 percent) for the 10 We will discuss this point in much more detail in section

15 decile of observations for which the forecast of eps t is most pessimistic to optimistic (-6.9 percent, -5.4 percent and percent) for the decile of stocks for which the forecast of eps t is most optimistic. That is, a stock with a more optimistic (pessimistic) eps t, is also likely to have more optimistic (pessimistic) eps t+1 and eps t+2 as well as more optimistic (pessimistic) forecasts of target price. These correlations suggest that basing estimates of the implied expected rate of return on these forecasts of both earnings and target prices (i.e., TPERs) may yield estimates that are superior to those based on earnings forecasts and market prices (i.e., MPERs). Although the pattern of dividend forecast errors is not monotonically related to the error in forecast of eps t, the positive correlation is still evident. In our TPERs, these dividends are discounted with the target price and hence also serve to potentially mitigate the effects of the errors in forecasts of earnings. Since the correlation among the forecasts is a central issue in our paper, we dwell on these correlations and conduct three sets of further analyses. First, in section 4 we examine the correlation among the forecast errors at the firm (rather than at the decile) level. Second, in section 7.2, we examine the relative magnitude of the errors and biases in the earnings forecasts and in the target prices (in other words, we ask the question: in light of the fact that prices have a much greater magnitude than earnings, is there relatively more/less forecast error and bias in the target prices than in the earnings forecasts?). Third, Cheong and Thomas (2011) make the observation that analysts forecast error does not vary with price per share. If this is, indeed, so for our data, the results in Table 2 may (all) be due to deflation of the forecast error by price. We show that this is not so for our sample by including mean and median price per share for each of the price-scaled eps t deciles in the last column of Table 2. The fact that the mean and median prices are similar across the deciles confirms that these results are not due to fact that we form 12

16 deciles based on scaled forecast errors. Nevertheless, in the next section, we report correlations among both scaled and un-scaled forecast errors, which are quite similar. 4. Correlations among the Forecast Errors Correlations among the forecast errors are summarized in Table 3. Correlations are calculated for each year of available data and then averaged across years. The positive correlations that were seen in the analyses of deciles of observations are also seen in the correlations among the individual forecast errors. Each of the errors in forecasts of earnings are correlated with each other; for example, the Pearson correlations between the unscaled error in forecast of eps t and the unscaled error in forecast of eps t+1 (eps t+2 ) are 0.45 (0.30) and the Pearson correlation between the unscaled error in the forecast of eps t and the unscaled error in the forecast of target price is The Spearman correlations are quite similar as are the correlations among the scaled forecasts errors. The positive correlation between the errors in the forecasts of eps t (and eps t+1 ) and the errors in the forecast of eps t+2 suggests that we are likely to find that using just two years of forecasted earnings in the reverse engineering exercise rather than adding extra years of forecasts by growing the eps t+1 forecasts by the forecast of the long-term earnings growth rate is likely to reduce the effect of measurement errors on the forecasts. On the other hand, the positive correlation between the errors in the forecasts of eps t (and eps t+1 ) and target price suggests that replacing market price with target prices in the reverse engineering exercise may reduce the effect of measurement error on the resultant estimates of the implied expected rate of return. 13

17 5. Methods of Estimating the Implied Expected Rate of Return 5.1. Extant Accounting-Based Estimates of the Implied Expected Rate of Return Using Market Prices We divide the extant methods of estimating the implied expected rate of return into two groups; those based on forecasts of earnings levels (Claus and Thomas (2001), Gebhardt et al. (2001) and Easton and Monahan (2005)), and those based on forecasts of earnings changes (Gode and Mohanram (2003) and the PEG and Modified PEG methods from Easton (2004)). We make this distinction because we expect that efforts to remove predictable forecasts errors (as done by, for example, Larocque (2012) and Mohanram and Gode (2012)) will be more effective for the models based on forecasts of earnings levels. Descriptions of these estimates are presented in Table 4. We begin by describing the details of our reverse engineering procedure based on the valuation model that is similar to that in Claus and Thomas (2001) because: (1) this method has most requirements for forecast data; and, (2) we underscore the use of forecasts of earnings and earnings growth, which is particularly relevant in this model. When these data are used in other estimates of the implied expected rate of return, they are obtained from the same sources and the timing of the collection of each data item is the same Estimates Based on Forecasts of Earnings Levels When reverse engineering the model based on Claus and Thomas (2001), our data are combined as follows (the time-line representing the chronology of each of the inputs to the model is shown in Figure 1): P in P it-1 = 1+r ct n/365 = bps it-1+ (ROE it+τ -r ct) bps it+τ-1 1+r ct τ 1 5 τ=0 + (ROE it+5 -r ct) bps it+4 (1+γ) r ct -γ (1+r ct ) 5 (1) 14

18 where i denotes firms, t and τ denote years and n denotes days, ROE it+τ =eps it+τ /bps it+τ-1, P in is the CRSP closing price two days after the earnings forecast, n is the number of days between the end of the previous fiscal year and two days after the earnings forecast, bps it-1 is the book value per share at the end of year t-1 (i.e., beginning of year t) obtained from Compustat, eps it is the Value Line forecast of earnings per share for year t, eps it+1 is the Value Line forecast of earnings per share for year t+1, eps it+τ =eps it+1 (1+ltgeps i ) τ-1 τ>1. ltgeps i is the Value Line forecast of the growth rate in earnings per share, bps it+τ =bps it+τ-1 +eps it+τ -dps it+τ, dps it is the Value Line forecast of dividends per share for year t, dps it+1 is the Value Line forecast of dividends per share for year t+1, dps it+τ =dps it+1 (1+ltgdiv i ) τ-1 τ>1. ltgdiv i is the Value Line forecast of the growth rate in dividends per share, γ is the yield on a ten-year government bond less 3 percent, r ct is numerically estimated. Gebhardt et al. (2001) also base their model on the residual income valuation model: where 1 P it-1 =bps it-1 + (ROE it+τ -r gls) bps it+τ-1 1+r gls τ + TVi τ=0 10 TVi = (ROE it+τ -r gls) bps it+τ-1 1+r gls τ τ=2 + (ROE it+11 -r gls) bps it+10 r gls 1+r gls 11, (2) ROE it+τ =eps it+τ /bps it+τ-1 for τ=0,1. ROE it+τ =ROE it+τ-1 -fade τ>1, fade =(ROE it+1 HIROE t) /10, HIROE t is the historical industry median ROE for all firm-years in the same industry spanning year t-5 through year t-1 with positive earnings and equity book values. The industry definitions are those in Fama and French (1997). r gls is numerically estimated. 15

19 Finally, we include an estimate, r pe, that is based on the assumption that expected cumdividend aggregate earnings for the next two years are sufficient for valuation purposes; i.e., the estimate of the implied expected rate of return is equal to the inverse of the two-year price-toforward-earnings ratio. This model is described in Table 4 and analyzed in detail in Easton and Monahan (2005) Estimates Based on Forecasts of Change in Earnings Three estimates of the implied expected rate of return (r peg, r mpeg, and r gm ) are each derived from the finite-horizon versions of the earnings, earnings growth model developed by Ohlson and Juettner-Naurouth (2005), Gode and Mohanram (2003) and Easton (2004). Details of these models are provided in Table 4. All of these methods are, essentially, based on forecasts of earnings changes Accounting-Based Estimates of the Implied Expected Rate of Return Using Target Prices; an Apples-to-Apples Match When estimates of the implied expected rate of return are computed using market prices and contemporaneous analyst earnings forecasts, there are three potential problems, which will not arise if the estimate is based on target prices rather than market prices. First, analysts earnings forecasts may differ from market expectations. As a result, earnings forecast errors, which are not in the market price, will, by construction, affect the estimate of the implied expected rate of return. Second, since the earnings forecasts and the market prices are often not observed at exactly the same point in time, the resulting non-synchronicity may lead to additional error in the estimates of the implied expected rate of return. Third, analysts may be forecasting earnings, which they believe are not aligned with market expectations and market prices, so that matching market prices with these analysts forecasts is not an apples-to-apples match. 16

20 Matching implicit values based on the present value of forecasts of dividends and target prices and on implicit values based on forecasts of accounting earnings is, however, an apples-toapples match; forecasts are matched with forecasts. Further, a positive correlation between the errors in the forecasts of earnings and errors in the forecasts of target prices may mitigate the effect of each of these errors inasmuch as they affect the estimate of the expected rate of return in opposite directions (a negative error in the forecast of earnings i.e., the market s expectation of earnings is greater than the analyst s forecast will bias the estimate of the implied expected rate of return upward while a negative forecast error in target prices will tend to bias the estimate of the implied expected rate of return downward). In the reverse engineering based on target prices (i.e., estimating the TPERs), the left-handside (market price based) variable, is replaced with an estimate of the intrinsic value: 4 dps it+τ-1 1+r div τ τ=1 + TP i (1+r div ) 4 (3) Where TP i is the average of the Value Line forecast of the highest target price to be met between years 3 and 5 and the lowest estimate of this target price; we assume that this price is met at the end of year 4. That is, when estimating the implied expected rate of return based on target prices and accounting-based valuation models, we substitute the present value of expected dividends and target prices in lieu of market prices Modifications to Examine the Effect of Correlations among Earnings Forecasts Claus and Thomas (2001) use forecasts of earnings for the next five years; the forecasts for the last three of these years are based on the forecast of earnings for the second year grown at the analyst s forecast of the long-term earnings growth rate. If, as we show, shorter run forecast errors of earnings and these (calculated) longer run forecast errors of earnings are positively 17

21 correlated, this positive correlation may exacerbate the effects of each of these errors. On the other hand, the methods in Gebhardt et al. (2001) and Gode and Mohanram (2003) rely on just two years of earnings forecasts and, thus, cannot be affected by the positive correlation between the errors in the forecasts of short-run earnings and the errors in the forecasts of longer run earnings. We investigate the effect of use of forecasts of growth to extend the forecast horizon by modifying the Claus and Thomas (2001), the Gebhardt et al. (2001), and the Gode and Mohanram (2003) estimates. We base the modified Claus and Thomas (2001) estimate on just two years of earnings forecasts rather than five, growing residual income at the expected rate of inflation thereafter (Claus and Thomas (2001) invoke this growth rate after five years. Removing the effect of the forecasts of earnings growth from the analyses may have two opposing effects: (1) the forecast of growth may, indeed, lead to earnings forecasts that improve estimates of the implied expected rate of return; and/or, (2) the fact that the errors in forecasts of earnings for the next two years and errors in forecasts of longer run earnings are positively correlated may exacerbate the effect of each of these errors on estimates of the implied expected rate of return. We base the modified Gebhardt et al. (2001) estimate on five years of earnings forecasts rather than two, fading to the industry median return of equity thereafter. The modified Gode and Mohanram (2003) estimate (see Table 4) is based on five years of earnings forecasts rather than two. These modified estimates are examined in Section 10 and Table

22 5.4. Estimates of the Implied Expected Rate of Return Obtained by Matching Market Prices and Target Prices A number of studies (see, for example, Francis et al. 2004; Botosan and Plumlee 2005; Brav et al. 2005; and Botosan et al. 2011), reverse engineer the following finite horizon version of the dividend capitalization model: 4 P it-1 = dps it+τ-1 1+r div τ τ=1 + TP i (1+r div ) 4 (4) Like the models that match current market prices with forecasts of accounting earnings, described in section 5.1, there is a possible mis-match because analysts forecasts of target prices will differ from the expectations implicit in the market prices. We compare the properties of the implied expected rate of return based in this model with those based on the models described in in sections 5.1 and Evaluating Estimates of the Implied Expected Rate of Return We evaluate the estimates of the implied expected rates of return in two ways. First, we compare the correlations with future realized returns based on the compelling intuition that these estimates of expectations should be correlated with realizations. The second method is based on the procedure developed in Easton and Monahan (2005), who argue, as do others, that focusing on the simple correlation of estimates of expected return with realized return fails to take account of possibly correlated omitted news variables. 11 We find that the ranking of the estimates is the same under either method. We briefly describe the Easton and Monahan (2005) method in this 11 Elton (1999) states: The use of average realized returns as a proxy for expected returns relies on a belief that information surprises tend to cancel out over the period of a study and realized returns are therefore an unbiased measure of expected returns. However, I believe there is ample evidence that this belief is misplaced. Fama and French (2002) provide evidence that suggests the abnormally large equity premium observed during the post-war era was attributable to information surprises that took the form of consistent downward revisions in expected future discount rates. 19

23 section. Interested readers are referred to Easton and Monahan (2005) for a detailed description and explanation. We start with the return loglinearization of Campbell and Shiller (1988) and Vuolteenaho (2002). The continuously compounded return in period t, r it, can be decomposed into three components: (1) expected return, er it (estimated at the beginning of the period); (2) changes in expectations about future cash flows (or cash flow news, cn it ); and, (3) changes in expectations about future discount rates (return news, rn it ): r it er it +cn it -rn it (5) er it corresponds to the log of one-plus an estimate of the implied expected rate of return. cn it and rn it can be estimated using Value Line forecasts as follows. The proxy for cash flow news is: ρ cn it = roe it - froe it-1,t + froe 1-ρ ω it,t+1 - froe it-1,t+1 t (6) where roe it =ln (1+ROE it ), ROE it =eps it /bps it-1, froe ijk =ln (1+ROE i,k ), ROE ijk denotes the forecast of return on equity for fiscal year k and is based on the Value Line forecast made at the end of year j. ω t is estimated on an annual basis via the pooled, cross-sectional regression of roe iτ+1 =ω 0t +ω t roe iτ. The proxy for return news is: rn it = ρ 1-ρ er it+1-er it (7) ρ is a capitalization factor to account for the change in the expected cash flow and the discount factor over an infinite future horizon. The details of the estimation of ρ are provided in Easton and Monahan (2005). Equation (7) allows us to evaluate the relation between realized returns (r it ) and expected returns (er it ) after controlling for changes in expectations about future cash flows and future 20

24 discount rates. One can run cross-sectional regressions of the following form as the empirical counterpart of (7): r it = α 0t +α 1t er it+α 2t cn it +α 3t rn it +ε it (8) If er it, cn it and rn it are measured without error, the estimate of α 1t will be one because equation (7) is a tautology. In practice, however, measurement errors in all the regressors will result in a bias in α 1t. Since the bias is unknown, we cannot draw meaningful inferences from the parameter estimates in regression (8). Thus, Easton and Monahan (2005) isolate the portion of the bias in α 1t that is solely attributable to the measurement error in er it ; a superior estimate of the implied expected rate of return has lower measurement error bias. The Easton and Monahan (2005) procedure is based on annual Fama-MacBeth (1973) cross-sectional regressions of the following form: υ M Cit = δ 0t +δ 1t ε A 1it +δ 2t ε A 2it +δ 3t ε A 3it +μ it (9) Variables specific to equation (9) are defined as follows: υ M Cit = υ Cit - σ er it,cn it +σ er it,rn it ε A 1it, where υ Cit = r 1t -er it- cn it + rn it and ε A 1it = ε 1it-β 10t, ε σ 2 (ε 1it ) 2it A = ε 2it-β 20t,ε σ 2 (ε 2it ) 3it A = ε 3it-β 30t. σ er it,cn σ 2 (ε 3it ) it (σ er it,rn it ) is the covariance between the estimate of the implied expected rate of return and the estimate of cash flow news (return news). The β, ε, and σ 2 (ε) terms are from the following first-stage regressions: er it= β 10t +β 11t cn it +β 12t rn it +ε 1it cn it = β 20t +β 21t er it+β 22t rn it +ε 2it rn it = β 30t +β 31t er it+β 32t cn it +ε 31it (10) A Thus, ε 1it captures the contribution of the error in the estimate of the implied expected rate of return (to the total error (captured in υ M Cit )) after removing (via regression (10)) the error in the 21

25 A cash flow news and return news estimates. Similarly, ε 2it (and ε A 3it ) capture the contribution of the error in the estimate of cash flow news (and return news) after removing the error in the estimate of the implied expected rate of return and return news (and the error in the estimate of the implied expected rate of return and cash flow news). As discussed in detail in Easton and Monahan (2005), it follows that the estimate of the implied expected rate of return with the smallest estimate of δ 1 contains the least measurement error and is the most reliable. From our analyses based on equation (9), we report only estimates of δ 1 since this is the focus of our interest. 7. Empirical Results We begin with descriptive statistics on the estimates of the implied expected rate of return. Next, these descriptive statistics provide an opportunity for us to gain a sense of the relative magnitudes of errors in forecasts of earnings and errors in forecasts of target prices, which may affect the effectiveness of the use of target prices in lieu of market prices. Finally we provide two forms of evaluation of the estimates of the implied expected rate of return. 7.1 Estimates of the Implied Expected Rate of Return Descriptive statistics for each of the estimates of the implied expected rate of return are reported in Table 5. Panel A describes the estimates based on market prices and forecasts of earnings similar to those in the extant literature (i.e., MPERs); Panel B describes the estimates based on target prices (i.e., TPERs). In each panel, the estimates of the implied expected rate of return are slightly smaller than those reported in Easton and Monahan (2005). The patterns across the estimates are, however, 22

26 very similar to those reported in Easton and Monahan (2005). 12 For example, in Panel A, the estimates based on forecasts of earnings changes show a predictable pattern; the median estimate (r peg ), which does not include forecasts of dividends and implicitly assumes that the short-run forecasted change in earnings will continue in perpetuity, is 9.5 percent and this increases to 10.7 percent when dividends (r mpeg ) are taken into consideration. The estimates based on the method suggested by Gode and Mohanram (2003) rely on assumptions about growth in abnormal growth in earnings; since the assumed growth rate is positive on average, the median value of r gm (11.2 percent) exceeds the median value of other estimates of the implied expected rate of return that rely only on forecasts of earnings and dividends. The median estimates based on earnings levels -- r ct, r gls and r pe are 9.2 percent, 9.1 percent and 6.7 percent, respectively. The estimate based on forecasted dividends and target-price r div has a median of 13.3 percent; the higher estimate most likely reflects optimism in analysts forecasts of target price. 13 The estimates of the implied expected rate of return based on earnings forecasts and target prices (i.e., TPERs) reported in Panel B show a very similar pattern to those reported in Panel A (i.e., MPERs) but the mean and median of all TPERs are smaller than the corresponding MPERs reflecting: (1) on the one hand matching of market prices and generally optimistically biased forecasts of earnings; and (2) matching target prices, which are also optimistically biased, with earnings forecasts, on the other hand. 12 The average of seven estimates of the implied expected rate or return in Easton and Monahan (2005) is 11.0 percent, and the average estimate in this study is 10.6 percent. We believe this difference is mainly due to different sample composition between Easton and Monahan (2005) and this study. For example, Easton and Monahan (2005) covers the data from 1981 to 1998 while this study covers from 1991 to Moreover, Easton and Monahan (2005) relies on the sample from I/B/E/S data while this study makes use of Value Line data, which covers fewer and larger firms. 13 We will discuss this larger estimate later in the paper. 23

27 7.2. Comparison of the Magnitude of the Bias and Errors in Forecasts of Earnings and Target Prices The effectiveness of replacing market prices with target prices when reverse engineering the accounting-based valuation models may be affected by the extent to which the magnitudes of the bias and errors in the forecasts of earnings are in keeping with the magnitudes of the bias and errors in the forecasts of target prices. If, for example, the magnitude of the bias in the forecasts of target prices is considerably greater than the magnitude of the bias that would be in line with the magnitude of the bias in the forecast of earnings, the effectiveness of using target prices rather than market prices may be questionable. At first glance it may seem, from the results in Table 2, that the bias and error in the forecasts of target prices are obviously greater than the bias and error in the forecasts of earnings. But some, perhaps most, of this difference in magnitude of the forecasts errors may simply be due to differences in the magnitude of the variable that is being forecasted. We noted earlier that a back-of-the-envelope analysis suggests that these errors in forecasts of target prices are, indeed, not at all out-of-line with the errors in forecasts of earnings; a mean error in forecasts of earnings of -1.5% and a mean error in forecasts of target prices of -14.5% (see Table 2) implies a price-earnings ratio of about ten, which is not high. A comparison of estimates of the implied expected rate of return determined by matching market prices to forecast of dividends and target prices (r div ) with those based on market prices and forecasts of earnings (i.e., MPERs) provides a more formal comparison of the effects of differences in the magnitudes of the errors. Intuitively, if prices are, say, 10 times earnings, a price-deflated error in a forecast of earnings of one percent would be equivalent in magnitude to a 10 percent price-deflated error in a forecast of target prices. Our analyses extend this intuition by comparing forecasts of target 24