An Evaluation of Accounting Based Measures of Expected …



An Evaluation of Accounting Based Measures of Expected Returns(

Peter D. Easton

University of Notre Dame

305A Mendoza College of Business

Notre Dame, IN 46556

peaston@nd.edu

and

Steven J. Monahan((

INSEAD – Accounting and Control Area

Boulevard de Constance

CEDEP No. 11

F-7705 Fontainebleau Cedex, 77305

France

steven.monahan@insead.edu

October 2003

An Evaluation of Accounting Based Measures of Expected Returns

Abstract

We develop and implement a method for comparing the measurement error in estimates of the expected rate of return on equity. We combine the Vuolteenaho [2002] return decomposition with the econometric method described in Garber and Klepper [1980] and Barth [1991] to infer cross-sectional measurement error variances. We evaluate a variety of estimates of expected returns that are discussed in the extant accounting literature (e.g., Gebhardt, Lee, and Swaminathan [2001], Easton [2004], and Gode and Mohanram [2003]). Our results show that the estimate that is based on the simplest model (i.e., price-to-forward earnings) is as reliable as more sophisticated proxies. This result is also observed when the analyses are repeated for portfolios of observations. Predicted values based on instrumental variables that are, a priori, expected to be correlated with the true expected return but uncorrelated with the measurement error have considerably lower measurement error variances. Nonetheless, the crudest proxy still performs at least as well as more sophisticated proxies.

I. Introduction

We develop and implement a method for comparing the measurement error in estimates of the expected rate of return on equity capital. We use our method to evaluate several accounting based proxies that are imputed from market prices and contemporaneous analyst earnings forecasts.[1] The method combines Vuolteenaho’s [2002] linear return decomposition with the econometric method described in Garber and Klepper [1980] and Barth [1991] to infer cross-sectional measurement error variances. We show that a crude measure of expected returns (the inverse of price to forward earnings) has a measurement error variance that is never greater (and often smaller) than that of more sophisticated proxies.

Vuolteenaho [2002] rigorously demonstrates that realized return is equal to the sum of expected return and changes in expectations about future cash flows (i.e., cash flow news) less changes in expectations about future discount rates (i.e., return news). Hence, he provides a theoretical foundation for a regression of realized returns on proxies for expected returns, cash flow news, and return news. Any difference between the estimated coefficients and one is attributable to measurement error in the proxies. We adapt the method described in Garber and Klepper [1980] and Barth [1991] to infer the measurement error variances from these differences. These error variances serve as the basis for evaluating the relative reliability of accounting based measures of expected returns.

Our approach to ranking estimates of expected returns has several advantages. First, since Vuolteenaho’s [2002] return decomposition is based on an identity we avoid the circular reasoning implicit in tests that rely on asset-pricing models. Judging the validity of an expected return proxy on the basis of its relation with a set of risk factors requires the researcher to make the implicit assumption that these factors are correct and exhaustive. As a practical matter this assumption is not descriptive; hence, the results of our tests nicely complement results presented in Botosan and Plumlee [2002b], Gebhardt, Lee and Swaminathan [2001], and Gode and Mohanram [2003].

Second, Voulteenaho’s [2002] model of the relation between realized return and changes in expectations about future cash flows and future discount rates provides a means of controlling for bias and noise in realized returns that is attributable to “information surprises.”[2] This is particularly pertinent to our study because the results of our empirical tests suggest that cash flow news and return news are also associated with expected returns. It follows that simple regressions of realized returns on expected return proxies yield spurious inferences because of omitted correlated variables bias; hence, results presented in Guay, Kothari and Shu [2003] must be interpreted with caution.

Third, by combining the Vuolteenaho [2002] return decomposition with the econometric methods described in Garber and Klepper [1980] and Barth [1991] we are able to draw clear-cut inferences about the relative reliability of the expected return proxies even though the cash flow news and return news proxies are also measured with error. Generally, the nature of the bias attributable to measurement error is unknown when more than one variable in a multivariate regression contains measurement error (e.g., Rao [1973]). However, the return decomposition demonstrates that if the components of realized returns are measured without error, the estimates of the slope coefficients in a regression of realized returns on expected returns, cash flow news, and return news are unambiguously equal to one. Thus, the bias in the coefficients corresponding to our empirical proxies is well-defined, which, in turn, implies we can develop unbiased estimates of the relative measurement error in each of our expected return proxies.

Finally, cross-sectional measurement error variance (i.e., the amount of cross-sectional variation in estimates of expected return that is attributable to measurement error) is a natural criterion for ranking expected return proxies and choosing the proxy that is the most reliable for conducting research and making practical decisions. For example, in order to maximize the power of their tests and avoid spurious inferences, researchers interested in evaluating the cross-sectional determinants of expected returns require an expected return proxy that has the least amount of variation attributable to error. Less “noisy” measures of expected return will also lead to better decisions when used in valuation, capital budgeting, and residual income based performance evaluation.

In addition to developing a method for evaluating measures of expected returns, we make two other contributions to the extant literature. First, we document that a crude measure of expected returns that is equal to the inverse of the price to forward earnings ratio is never worse (and often better) than more sophisticated proxies (e.g., the proxy developed by Gebhardt, Lee and Swaminathan [2001]). This result is quite robust. For example, even after using instrumental variables to reduce the measurement error in each of the expected return proxies, we continue to find that the simplest proxy is as good as, if not better, than the remaining proxies. In light of the fact that the price to forward earnings ratio is a naïve valuation multiple that relies on restrictive assumptions about future earnings growth, this result suggests that the absolute magnitude of the measurement error in each of the expected return proxies we evaluate is large.

Second, we evaluate the effectiveness of two commonly used methods for mitigating measurement error: instrumental variables, and grouping. We show that a simple instrumental variables technique reduces the measurement error: fitted values taken from regressions of expected return proxies on beta, standard deviation of return, equity market value, book to market, and industry type have lower measurement error variances than the raw proxies. This suggests that when developing firm level estimates of expected return for decision-making purposes (e.g., capital budgeting) it is better to use an instrumental variables approach rather than the actual value imputed from a particular valuation model. In addition, researchers who need expected return as a control variable in their empirical analysis may obtain more descriptive inferences if they use predicted values from an instrumental variables regression as their empirical measure of expected return.

While instrumental variables estimation combines a set of variables that are, a priori, expected to be correlated with expected returns but uncorrelated with the measurement error, grouping focuses on each instrument separately. We group observations into portfolios on the basis of variables designed to minimize the within portfolio variation in the construct of interest (e.g., beta) while maximizing the cross-portfolio variation. We then repeat all of our analyses using portfolio averages (i.e., average realized return, average expected return proxy, etc.). As expected, the measurement error variances observed at the portfolio level are lower than those observed at the firm specific level, which is consistent with the notion that portfolio level data provide a more appropriate basis for testing hypotheses about the cross-sectional determinants of expected returns. Nonetheless, we continue to find that the expected return proxy based on the price to forward earnings ratio is at least as good as proxies that are based on more complicated valuation models. We suggest that a reason for this result is the difficulty in forecasting earnings growth beyond one year – the forecasts of growth appear to introduce noise into the estimation of the expected rate of return.

The remainder of the paper unfolds in the following manner. In the next two sections we describe our empirical method, discuss our proxies of interest, and describe our sample. Our main empirical results are presented in section IV. The results of our instrumental variables and grouping analyses are discussed in section V. We provide concluding comments in section VI.

II. Empirical Method

II.1. The Components of Realized Returns

Voulteenaho [2002] demonstrates that realized return can be decomposed into three components: (1) expected return, (2) changes in expectations about future cash flows (cash flow news), and (3) changes in expectations about future discount rates (return news). In particular,

[pic] (1)

where rt denotes continuously compounded returns for time t (i.e., the natural log of one plus annual raw return), Et[.] is conditional expectation operator, cfnt denotes cash flow news, and r_newst denotes expected return news.[3] A detailed description of Vuolteenaho’s [2002] return decomposition, which is similar to the well-known return decomposition developed by Campbell [1991], is provided in Appendix A. Empirical proxies for expected return [pic], cash flow news [pic], and return news [pic] are described in section III.

Three observations about equation (1) warrant mentioning. First, equation (1) is derived from a tautology; hence, our analyses do not rely on implicit or explicit assumptions about investor rationality, the nature of market equilibrium, transactions costs, etc. Second, the linear return decompositions developed by Campbell [1991] and Vuolteenaho [2002] are well accepted. For example, a number of studies in finance use variations of equation (1) as a means of evaluating the determinants of realized returns (comprehensive literature reviews may be found in chapter seven of Campbell, Lo and MacKinlay [1997] and chapter 20 of Cochrane [2001]). Finally, since equation (1) reflects the effect of changes in expectations about future cash flows and future discount rates on realized returns, it provides a direct means of dealing with Elton’s [1999] argument that information surprises cause realized returns to be a biased and noisy measure of expected returns.

The third point is especially pertinent to our study. Bias in realized returns implies that the estimate of the slope coefficient taken from a simple regression of realized returns on expected returns may also be biased. If changes in expectations about future cash flows (discount rates) are associated with contemporaneous expected returns, the coefficient on expected returns will be affected by correlated omitted variables bias. This is quite plausible. For example, an explanation for the equity premium puzzle is that during the post-war period the US (and other western nations) experienced an unprecedented run of “good luck.”[4] Hence, the expected future rate of return required by investors as compensation for holding the market portfolio steadily declined (i.e., economy-wide r_newst was negative), which, as per equation (1), caused the realized equity premium to be consistently larger than expected.[5] This, in turn, led to higher than expected realized returns on individual stocks. Moreover, the magnitude of the bias at the individual stock level was arguably increasing in the covariance between a stock’s return and the return on the market portfolio. It follows that changes in expectations about future discount rates (i.e., r_newst) were correlated with both realized and expected returns (i.e., rt and [pic]), and the coefficient on [pic] is biased if r_newst is omitted from the regression.[6]

II.2. The Regression Based on Vuolteenaho’s Return Decomposition

We begin our analyses by estimating the following regression for each expected return proxy:

[pic] (2)

In equation (2) [pic], [pic], and [pic] represent the expected return proxy, the cash flow news proxy, and the proxy for the additive inverse of return news (i.e., the return news proxy), respectively.[7] If these empirical proxies are measured without error, α1, α2 and α3 are equal to one and α0 is equal to zero. Hence, one means of evaluating a particular measure of expected returns is to conduct a test of the difference between α1 and one. Unfortunately, these tests do not lead to clear-cut inferences. Because we are unable to observe expectations or revisions in expectations each of the regressors in equation (2) contains error, which implies the bias in a particular regression coefficient is a complex function of the measurement errors in all of the regressors (e.g., Rao [1973]). To circumvent this problem we use a refinement of the approach discussed in Garber and Klepper [1980] and Barth [1991] to isolate the portion of the bias in α1 that is solely attributable to the measurement error in [pic].

II.3. Measurement Error Analysis

In this subsection we describe the intuition underlying the method we use to isolate the portion of the bias in α1 that is solely attributable to measurement error in [pic]. Appendix B contains a rigorous description of our econometric approach, which is centered on the following regression:

[pic] (3)

In equation (3) [pic] equals the difference between observed realized return and the sum of the empirical measures of its components (i.e., [pic]); thus, [pic] equals the combined (or total) error in our empirical proxies. Each regressor, [pic], essentially equals the error from a regression of proxy i on the remaining two proxies (e.g., [pic] is obtained by regressing [pic] on [pic] and [pic]). Hence, each regression coefficient in equation (3) measures the relation between the error in a particular proxy and the total error in all the proxies. For example, the expression for the regression coefficient corresponding to [pic], which we refer to as the “noise variable:”[8]

[pic] (4)

In equation (4) [pic] is the variance of the measurement error in the expected return proxy, [pic] ([pic]) denotes the covariance between the measurement error in [pic] and the measurement error in [pic] ([pic]), and [pic] ([pic]) is the covariance between true expected return and the measurement error in [pic] ([pic]). If the covariance terms are constant across expected return proxies, variation across proxies in the noise variable is solely attributable to variation in the measurement error in the expected return proxies (i.e., [pic]). This implies that the expected return proxy with the smallest noise variable contains the least measurement error and is the most reliable. Hence, we begin our measurement error analyses by estimating equation (4) for each of the expected return proxies of interest.

A problem with using the noise variables to rank our expected return proxies is that it is unlikely that the covariance terms shown in equation (4) are constant across expected return proxies. Because each of our measures of the expected rate of return is based on a unique set of assumptions about dividends, future earnings growth, and terminal profitability it is likely that the relation between the error in each proxy and the errors in the remaining independent variables is also unique. This implies that inferences based on the relative magnitudes of the different estimates of the noise variable may not provide a meaningful basis for inferring the relative magnitudes of [pic]. To circumvent this problem we refine the econometric approach developed by Garber and Klepper [1980] and Barth [1991] and estimate “modified noise variables:”[9]

[pic] (5)

In equation (5) [pic] equals the covariance between true expected return and sum of true cash flow news and true return news. As discussed in section II.1, there is reason to believe this covariance does not equal zero; however, since it only involves the true values of the constructs it is constant across proxies. Hence, if differences in [pic] are second order, which is a reasonable assumption, differences in the modified noise variables will be primarily attributable to differences in the measurement error variances of the proxies.[10] In light of these observations, we base the bulk of our inferences on the modified noise variables (i.e., [pic]).

II.4 Summary

To summarize, bias and noise attributable to information surprises imply that simple regressions of realized returns on expected return proxies may yield spurious inferences that are attributable to omitted correlated variables. Hence, we include measures of cash flow news and return news in our regressions (i.e., equation (3)). However, because all of the regressors in equation (3) contain measurement error, the regression coefficient corresponding to the expected return proxy is not a clear-cut indicator of the reliability of this proxy. To overcome this problem we use a refinement of the method described in Garber and Klepper [1980] and Barth [1991] to estimate the measurement error variances.

III. Empirical Proxies and Sample Construction

III.1. Accounting Based Measures of Expected Return

We examine six expected return proxies each of which is imputed from prices and contemporaneous earnings forecasts. Our first proxy is based on the assumption that expected cum-dividend aggregate earnings for the next two years are valuation sufficient. Hence, it essentially equals the inverse of the price to forward earnings ratio. For this reason we refer to it as rpe. The purpose of including rpe in our analyses is to provide a naïve benchmark that is based on restrictive assumptions about future earnings growth.[11]

Our next four expected return proxies are each derived from the finite-horizon version of the earnings, earnings growth model developed by Ohlson and Juettner-Nauroth [2001] and described in Easton [2004]:

[pic] (6)

In equation (6) Pt is price at the end of year t, epst+τ is the year t forecast of year t+τ earnings per share, dpst+1 is the year t forecast of dividends paid in year t+1, r is the discount rate, and Δagr is a growth rate. Following Easton [2004], we refer to the difference between expected year-two cum-dividend accounting earnings (i.e., epst+2 + r×dpst+1) and normal accounting earnings that would be expected given earnings of period one (i.e., (1+r)×epst+1) as “abnormal growth in earnings” or agrt+1. Hence, Δagr equals the perpetual rate of change in abnormal growth in earnings beyond the forecast horizon.

The first proxy derived from equation (6) embeds the assumption that no dividends are paid in year t+1 and that Δagr equals zero. As shown in Easton [2004] this proxy is equal to the square root of the inverse of the PEG ratio; hence, we refer to it as rpeg.[12] Relaxing the assumption that dpst+1 is equal to zero yields a modified version of the PEG ratio and a proxy we refer to as rmpeg.[13] A criticism of rpeg and rmpeg is that the assumption of constant abnormal growth in earnings is too restrictive. Gode and Mohanram [2003] avoid this criticism by assuming Δagr is a cross-sectional constant equal to the difference between the risk free rate of interest and three percent. We refer to their proxy as rgm. Easton [2004], on the other hand, simultaneously estimates r and Δagr for portfolios of stocks allowing for cross-sectional variation in Δagr. We refer to his proxy as rΔagr.

Our final measure of expected return is the proxy developed by Gebhardt, Lee, and Swaminathan [2001], which we refer to as rgls. Gebhardt, Lee, and Swaminathan [2001] combine the well-known residual income valuation model with the assumption that accounting return on equity linearly fades to the historical industry median and remains constant thereafter.

III.2. Cash Flow News and Return News Proxies

In the Vuolteenaho [2002] return decomposition (equation (1)), cash flow news cfnt is the component of realized returns corresponding to the change in investors’ expectations about future cash flows. As shown in Appendix A, cash flow news is defined as follows:

[pic] (7)

In equation (7) ΔEt[.] equals Et[.]-Et-1[.], ρ is a number slightly less than one, and roe is the natural log of one plus the accounting rate of return on equity. We use the following formula to estimate our cash flow news proxies:

[pic] (8)

In equation (8) froei,j denotes forecasted roe for fiscal year j and is based on the consensus I/B/E/S forecast of epsj made in December of year i, ωt-1 is the expected persistence of roe as of time t-1.[14]

We estimate ωt-1 via the following pooled cross-sectional and time-series regression:

[pic] (9)

In equation (9) ( is a number between t-1 and t-10 and the sample includes all firm-years with requisite data.[15]

In the Vuolteenaho [2002] return decomposition (equation (1)), return news r_newst is the component of realized returns corresponding to the change in investors’ expectations about future discount rates. As shown in Appendix A, return news is defined as follows:

[pic] (10)

We use the following formula to estimate our expected return news proxy:

[pic] (11)

In equation (11) [pic] is the difference between the imputed r as of time t for a particular valuation model and the imputed r derived from the same valuation model at time t-1; thus, [pic] varies across valuation models. The expected return proxy embeds the assumption that changes in the discount rate are permanent (i.e., the discount rate follows a random walk). This assumption is consistent with the fact that each of our expected return proxies is the geometric average of all expected future discount rates (i.e., our estimates do not explicitly accommodate temporal variation in expected returns).

III.3. Sample Construction

Price at fiscal year-end, dividends, and number of shares outstanding are obtained from the 1999 COMPUSTAT annual primary, secondary, tertiary, and full coverage research files. Earnings forecasts are derived from the summary 2000 I/B/E/S tape. We determine median forecasts from the available analysts’ forecasts on the I/B/E/S file that is released on the third Thursday of December. We delete firms with non-December fiscal year-ends so that the market implied discount rate and growth rate are estimated at the same point in time for each firm-year observation. For observations in 1995, for example, the December forecasts became available on the 21st day of the month. These data included forecasts for a fiscal year ending 10 days later (i.e., December 31, 1995) and either an earnings forecast for each of the fiscal years ending December 31, 1996 and 1997 (i.e., epst and epst+1) or the forecast for the fiscal year ending December 31, 1996 (i.e., epst) and a forecast of growth in earnings per share for the subsequent years. When available, we use the actual forecasts for the subsequent year (in this example, 1997) as our proxy for epst+1. When actual forecasts are unavailable we use I/B/E/S forecasts of growth in earnings as the basis for developing our proxy for epst+1.[16] Realized returns for the year following the earnings forecast date (in the example, January 1, 1996 to December 31, 1996) are obtained from the Center for Research in Security Prices Monthly Return file. These selection criteria lead to a sample of 13,415 firm-years spanning the years 1981 to 1998.

IV. Main Empirical Results

In this section we discuss the univariate statistics and the results of our main empirical tests. As discussed above and in Appendix B, the Vuolteenaho [2002] return decomposition pertains to continuously compounded returns; hence, we transform each of the measures of expected return by taking the log of one plus the proxy of interest. Analyses of raw values lead to similar inferences. Our univariate statistics are based on the definition of r_news shown in equation (11). However, the value of r_news underlying our multivariate tests equals the additive inverse of the solution to equation (11) (i.e., [pic]).

IV.1. Univariate Analysis

Descriptive statistics are shown on Table 1. The median estimate of rpe is 8.4 percent, which is considerably less than the median realized return of 12.6 percent. This difference is potentially attributable to the combination of conservative accounting and positive growth. The adjustment provided by taking short-term earnings growth into account (rpeg) causes the median estimate of expected returns to increase by slightly more than two percentage points whereas inclusion of dividends in the expected pay-off (rmpeg) causes the median to rise to 11.6 percent.

The estimation procedure used by Gode and Mohanram [2003] relies on the assumption that abnormal growth in earnings increases after year t+1 and the method used by Easton [2004] leads to estimates of increases in abnormal growth in earnings that are, on average, positive. Hence, the median value of rgm (12.4 percent) and rΔagr (12.2 percent) exceeds the median value of rmpeg. The median expected rate of return implied by the residual income valuation model (rgls) is 10.6 percent.

The median value of the proxy for cash flow news ([pic]) is –3.1 percent.[17] To the extent that this measure of cash flow news is attributable to analyst optimism (e.g., Richardson, Teoh and Wysoki [2001]) that is ignored by the market, the negative median value of cash flow news represents measurement error in our cash flow news proxies. Moreover, given that our expected return proxies and return news proxies are also derived from analysts’ forecasts, the negative median value of cash flow news implies that these constructs are also measured with error.

Descriptive statistics for our return news proxies are shown in panel B of Table 1. Since these estimates equal the change in the estimate of expected return over the realized return interval, they differ across the various estimates of expected returns. The median estimates of expected return news are consistently negative. For example, the median return news implied by the change in rpe is -4.4 percent and the median return news for rgls is -3.7 percent. In light of the fact that prices rose during our sample period, the decline in our expected return proxies suggests a coincident decline in the equity premium.

Table 2, panel A summarizes the correlations among realized returns, the expected return proxies, and the estimates of cash flow news. Pearson product moment (Spearman rank order) correlations are shown above (below) the diagonal. Correlations between our return news proxies and the remaining variables of interest are shown in panel B. The correlations are the temporal averages of the annual cross-sectional correlations. The t-statistics equal the ratio of the averages to their temporal standard errors. We focus our discussion on the Spearman correlations. Similar inferences may be drawn from the Pearson correlations.

The correlation between rgls and realized return is positive (0.064) and significant at the 0.01 level (t-statistic of 2.08). On the other hand, none of the correlations between the remaining expected return proxies and realized returns is statistically different from zero at the 0.05 level. Moreover, two (three) of the Spearman (Pearson) correlations are negative.[18] These negative correlations imply that the cash flow news and return news components of realized returns reflect more than random measurement error, which only causes attenuation bias and does not affect the sign of the correlation coefficient.[19] Rather, these correlations imply that the cash flow news and return news components of expected returns are correlated with our expected return proxies. Hence, in order to develop unbiased inferences about our proxies it is crucial that we control for variation in cash flow news and returns news.

As expected the correlation between realized returns and the estimate of cash flow news is positive and significant at the 0.01 level (Spearman correlation of 0.304 with a t-statistic of 14.86). The correlation between the estimate of cash flow news and each of the estimates of expected returns derived from the earnings, earnings growth model (i.e., equation (6)) are significantly negative, however. For example, the Spearman correlation between rpeg and [pic] is –0.211 (t-statistic of –19.23). These negative correlations suggest that firms with relatively high discount rates experienced larger than average negative information surprises about future cash flows during the time period under study. These correlations are also consistent with the notion that analysts’ forecasts are too extreme in the sense that high forecasts are too high and low forecasts are too low. For example, if year zero analysts’ forecasts are too high (too low), the implied discount rate will be biased upwards (downwards) and analysts’ forecast errors and revisions will be negative (positive).

The correlation between rgls and the cash flow news proxy is positive and significant, however (Spearman correlation of 0.075 with a t-statistic of 3.53). A rationale for this phenomenon is as follows. There are two essential differences between rgls and the remaining expected return proxies: (1) the residual income valuation model is anchored on year t equity book value, and (2) the terminal value correction used by Gebhardt, Lee, and Swaminathan [2001] is a function of historical industry median ROE. Thus, unlike the remaining estimates of expected returns, cross-sectional variation in rgls is primarily attributable to variation in the firm-specific book-to-market ratio and variation in industry ROE. These facts may explain the positive correlation between rgls and realized returns (via the well-known book-to-market and price-to-earnings phenomena). They may also explain the positive correlation between rgls and the estimate of cash flow news: variation in our cash flow news proxy is also a function of variation in equity book value and it is possible that analysts tend to revise their optimistic forecasts of ROE towards the industry ROE.

Table 2, panel B summarizes the Pearson and Spearman correlations among our expected return news proxies and realized return, the cash flow news proxy, and the corresponding estimates of expected return. As expected, the correlation between the each of the estimates of return news and realized returns are significantly less than zero (all the t-statistics are less than -5). In addition, the correlation between each of the estimates of return news and the estimate of cash flow news is positive; however, the correlation between the cash flow news proxy and rgls is not significantly different from zero.

The correlations between the estimates of expected return news and the corresponding estimates of expected returns are all negative and statistically significant. There are two non-mutually exclusive interpretations of these results. First, these results may be attributable to a decline in the equity premium during our sample period. As discussed in section II.1, if the equity premium fell during the sample period, stocks with high betas (and, thus, high expected returns) experienced larger than average downward revisions in expected future discount rates (i.e., r_newst). Alternatively, these negative correlations may be attributable to measurement error in our proxies. For example, a decline in analyst optimism may have occurred during our sample period, or extremely high (low) forecasts may be followed by larger than average downward (upward) revisions.

IV.2. Multivariate Analysis

Results pertaining to our multivariate tests are shown in tables 3 and 4. The regression coefficients shown in these tables are the temporal averages of the regression coefficients obtained from annual cross-sectional regressions. The t-statistics are computed via the approach described in Fama and MacBeth [1973].

Summary statistics from the estimation of the regression of realized returns on the empirical measures of its components (expected return, cash flow news, and expected return news) – equation (2) – are presented in table 3 for each of the expected return proxies. The estimates of the coefficient α1 on the expected return proxies are negative for each of the regressions except the regression based on rpe (e.g., the estimate of α1 on rgls is –0.608 with a t-statistic of –2.27). Un-tabulated results demonstrate that the estimate of the coefficient on rpe is statistically different from the coefficients on the remaining proxies, and all of the estimates of α1 are significantly less than one at the 0.01 level. These results provide an initial indication that there may be a considerable amount of measurement error in each of the expected return proxies. On the other hand, as discussed in section II and Appendix B, the observed bias in α1 may be attributable to other factors; hence, the need for additional analysis. As expected, the estimates of the coefficients on [pic] and [pic] are significantly positive. For example, for the regression based on rgls, the estimate of the coefficient on [pic] is 0.370 (t-statistic of 6.64) and the estimate of the coefficient on return news is 0.191 (t-statistic of 5.58). Nonetheless, un-tabulated results demonstrate that these estimates are also significantly less than one, which implies that these constructs may also be measured with error.

The results of estimating equation (3) for each of our return proxies are shown in Table 4. Before proceeding with a discussion of the regression coefficients it is interesting to note that equation (3) explains a considerable portion of the cross-sectional variation in the total measurement error in our estimates of expected returns (i.e., [pic]). For example, the R-square based on rgls is 0.80. This implies the return decomposition developed by Vuolteenaho [2002] provides a useful characterization of the components of realized returns.

Turning to the regression coefficients, several implications are immediate. First, the simplest proxy (i.e., rpe) contains the least measurement error ((1 equals 0.00144, t-statistic of 0.63). In addition, all of the remaining proxies contain statistically significant measurement error. Moreover, un-tabulated results demonstrate that the measurement error in rpe is significantly less than the measurement error in each of the remaining proxies. The facts discussed above imply that the measurement error introduced by making additional assumptions about the evolution of future profitability and growth outweighs the benefit of using a more elaborate valuation model. Second, rgls is the second most reliable proxy. Taken together with the observation rpe is the most reliable proxy and that both of these proxies place the least amount of reliance on epst+1, this observation implies that the error present in analysts’ forecasts increases with the forecast horizon.

All of the estimates of the (2 coefficients corresponding to the cash flow news proxy and all of the estimates of the (3 coefficients corresponding to the return news proxy are significantly positive, which implies that these proxies contain significant measurement error. The estimates of the (3 coefficients pertaining to the rpe and rgls models are lower than the estimates pertaining to the other valuation models, however. This further buttresses the notion that there is considerable error in analysts’ forecasts of epst+1. Finally, among the remaining estimates (i.e., rpeg, rmpeg, rgm, and rΔagr), the estimate that is based on rΔagr contains the least amount of measurement error. This suggests that the growth correction underlying rΔagr is potentially worthwhile (a similar, though much smaller difference, is seen in the estimates of (1).

Inferences based on variation in (1 are predicated on the assumption that the correlations between the measurement error in our expected return proxies and the measurement error in the remaining proxies is the same for each of the expected return proxies. As discussed in section II this may not be descriptive. Thus, we evaluate the modified noise variables. The results of these analyses are summarized on the right-hand-side of Panel A of table 4.

The expected return proxy with the lowest estimate of δ1M is rgls (0.00032, t-statistic of 1.52). However, the coefficient on rgls is only slightly less than the estimate of δ1M pertaining to rpe (0.00038, t-statistic of 1.25), and as per un-tabulated results the difference between these two coefficients is statistically insignificant. Moreover, un-tabulated results demonstrate that the coefficient on rpe is smaller than each of the coefficients on the remaining proxies (i.e., rpeg, rmpeg, rgm, and rΔagr), and two of these differences are statistically significant at the .10 level (rpeg and rgm). Hence, consistent with the results from regression (3), the expected return proxy that is imputed from the crudest valuation model (i.e., rpe) is as reliable as estimates of the expected rate of return that are based on more sophisticated models.

V. Analyses Based on Instrumental Variables and Grouping

Two common approaches for dealing with measurement error are: (1) instrumental variables, and (2) grouping. In this section we examine the effect these methods have on the measurement error variances of the expected return proxies. Our motivation for this analysis is two-fold. First, we provide prescriptive advice for business practitioners and researchers who require an empirical estimate of expected return. For example, we demonstrate that instrumental variables and grouping ameliorates the measurement error in the expected return proxies. Hence, business practitioners and researchers may want to use these methods as the basis for developing estimates of expected return. Second, evaluating the effect these methods have on the relative reliability provides us with evidence about the robustness of our previous results. In particular, we show that rpe continues to perform as well as more complicated proxies even after we attempt to purge all of the proxies of their measurement error. Hence, we conclude that each of the proxies contains considerable measurement error as none of them are more reliable than a crude measure that relies on fairly unpalatable assumptions.

V.1. Instrumental Variables

We implement the instrumental variables procedure in the following manner. First, each of the expected return proxies is regressed on instruments that are assumed to be correlated with true expected return but uncorrelated with the measurement error. Next, the predicted values from the instrumental variables regression are evaluated in the same manner as the underlying expected return proxies.

We select the following instruments: CAPM beta, market capitalization, the ratio of equity book value to equity market value, the standard deviation of past returns, and industry type. Before discussing the empirical results pertaining to the instrumental variables analysis, we briefly motivate our choice of instruments.

Well-known theoretical results developed by Sharpe [1964], Lintner [1965], and Mossin [1966] suggest that beta is an appropriate measure of systematic risk. Our firm-year estimates of beta are derived from regressions of monthly returns on contemporaneous excess returns (i.e., market return less the yield on a one month treasury bill) where the data are drawn from the 60-month period prior to the earnings forecast date. Malkiel [1997] focuses on total risk as measured by the variance of returns. We use the standard deviation of daily returns prior to the forecast date as a measure of total risk. A variety of studies (e.g., Banz [1981] and Fama and French [1992]) document a negative association between size and realized returns suggesting that market capitalization as of the earnings forecast date is a potential proxy for risk. Consistent with Fama and French [1992] and Berk, Green, and Naik [1999], we use the ratio of the book value of common equity to the market value of common equity at the earnings forecast date as another measure of risk. Finally, a number of papers (e.g., Fama and French [1997], and Gebhardt, Lee, and Swaminathan [2001]) show that the equity premium differs across industry. Our industry variable is the average of the firm-specific estimates of expected return – industry classifications are the same as those in Fama and French [1997].

The results of estimating regressions of each of the expected return proxies on the four risk proxies (beta, standard deviation of returns, size, and the book to market ratio) are shown on the left-hand-side of Table 5. As expected, for all of the expected return proxies other than rpe, the estimates of the coefficients on beta, the standard deviation of returns, and the book to market ratio, are positive and significant. For example, in the rpeg regression the estimate of the coefficient on beta is 0.013 (t-statistic of 6.45), the estimate of the coefficient on the standard deviation of returns is 0.76 (t-statistic of 12.66), and the estimate of the coefficient on the book to market ratio is 0.007 (t-statistic of 4.34). Contrary to expectations, the estimates of the coefficient on size are not significantly less than zero (at the 0.05 level) for all proxies other than rpe (0.000 with a t-statistic of –2.75). A possible explanation for these results is that the I/B/E/S sample on which our analyses are based tends to include larger firms. Also, contrary to our expectations, the estimates of the coefficients on beta and the standard deviation of returns are significantly negative for the regression based on rpe. The estimate of the coefficient on the book to market ratio is positive and significant, however.

It is evident from the results in the right hand side of Table 5 that industry type captures much of the variation in the expected returns proxies. For example, the estimate of the coefficient on industry return for the regression where the dependent variable is rpeg is 0.83 with a t-statistic of 44.44. Nonetheless, with the exception of the regression involving rpe, the estimates of the coefficients on beta, the standard deviation of returns, and the book to market ratio remain positive as expected.

These results suggest that if our instrumental variables are correlated with true expected return but uncorrelated with the measurement error, the predicted value from the instrumental variables regressions is a better estimate of expected returns than the expected return proxy. The results shown on Table 6 support this conjecture.[20] With the exception of rgls, the estimates of the measurement error variances δ1M are lower for all expected return proxies. For example, the estimate of δ1M for rpe decreases from 0.00038 for the raw proxies (table 5) to 0.00018 and 0.00030 for the predicted value (table 6). Interestingly, none of the estimates of the measurement error variance differ significantly from the estimate with the lowest variance (i.e., rpe). Hence, inferences based on instrumental variables analysis continue to suggest that the crudest proxy is no worse than more sophisticated expected return proxies.

Several studies (e.g., Botosan and Plumlee [2002b], Gebhardt, Lee, and Swaminathan [2001], and Gode and Mohanram [2003]) use coefficients from regressions of proxies for expected returns on various risk factors (or correlations between proxies for expected returns and risk factors) as means of ranking expected returns proxies. It is evident from the coefficient estimates on the right hand side of table 5 that the expected return proxy that is best as per the criteria adopted by these authors is rpeg. Specifically, in the rpeg regression, the t-statistic for the estimated coefficients on beta and the standard deviation of returns (6.06 and 13.91, respectively) are higher than the corresponding t-statistics in the other regressions, the estimate of the coefficient on size is negative (t-statistic of –1.89), and the coefficients on the book to market ratio and industry type are highly significant. Nonetheless, the evidence in table 6 demonstrates that rpeg is the worst proxy as it has the highest δ1M (0.00059). In other words, ranking expected returns on the basis of correlations with potential risk factors may lead to erroneous conclusions.

V.2. Grouping

The instrumental variables procedure combines variables that are, a priori, expected to be correlated with expected returns. Grouping, on the other hand, focuses on each variable separately. We group observations into annual portfolios of ten based on variables designed to minimize the within portfolio variation in the construct of interest (e.g., beta) and maximize the variation across portfolios.[21] All analyses that are conducted at the firm-specific level are repeated using portfolio averages (i.e., average realized return, average expected return, average cash flow news and average return news).

The results of our portfolio level analyses are summarized in Table 7. With the exception of the estimate of δ1M for rpe the estimate of the measurement error variance is smaller than the estimate of the variance at the individual firm level, which is the expected effect of aggregating the data. The rankings of the portfolio averages of the expected return proxies are, by and large, similar to the rankings of the firm-specific estimates. With the exception of the estimates for portfolios based on the book to market ratio, the simplest expected return proxy (rpe) has the lowest measurement error variance followed by rgls and then by the estimates that place a greater emphasis on analysts forecasts of long-tem earnings growth (rpeg, rmpeg, rgm, and rΔagr).

VI. Summary and Conclusions

We take a measurement error perspective to arrive at a method for evaluating the relative reliability of various accounting based measures of expected returns. Our results imply that the simplest model, which is predicated on the least reasonable assumptions ex ante, yields the most reliable estimate of expected returns. Moreover, this result is robust as it remains descriptive even after we attempt to purge the proxies of their measurement error via the use of instrumental variables and grouping. Hence, we conclude that all of the proxies we analyze contain considerable measurement error. While procedures aimed at purging the proxies of their measurement error do not affect their relative reliability, these procedures do affect the absolute magnitude of the measurement error in each proxy. This implies that measures of expected return derived from instrumental variables analysis or portfolio averages provide a better basis for making practical investment decisions and conducting academic research.

The approach we describe may be used in other contexts. For instance, the identity of the model of market equilibrium that best characterizes the manner in which economic agents trade off risk and return is of significant interest to many. Our approach provides a potentially useful means of evaluating this issue. In particular, the methodology we describe may be a viable means of evaluating the relative reliability of the different measures of expected returns suggested by extant asset pricing models.

Appendix A

Linear Decomposition of Realized Returns

In this appendix we describe Vuolteenaho’s [2002] linear return decomposition, which serves as the primary motivation for our empirical tests.

We begin by noting the following identities.

[pic] (A.1)

[pic] (A.2)

In equations (A.1) and (A.2) ROEt denotes accounting return on equity for period t, Bt is equity book value at the end of period t, Dt is dividends paid during period t, Rt is stock return for period t and Mt is equity market value at the end of period t. We assume clean surplus accounting; hence, equation (A.1) is an identity.

Dividing each side of (A.1) and (A.2) by Dt-1, rearranging and taking logs we obtain the following expressions for the ratio of equity book value to dividends and the price to dividend ratio.

[pic] (A.3)

[pic] (A.4)

In equations (A.3) and (A.4) lower case letters denote natural logs, (dt is the natural log of dividend growth at time t (i.e., dt-dt-1), roet denotes the natural log of one plus ROEt and rt denotes continuously compounded returns. Equation (A.4) illustrates that the price to dividend ratio is increasing in future dividend growth but decreasing in the future discount rate. An analogous interpretation holds for equation (A.3).

While equations (A.3) and (A.4) are identities, they are also non-linear (i.e., they are both functions of logged expressions). Hence, they cannot be directly converted into a linear expression for the book to market ratio, which is necessary for the development of linear decomposition of realized returns. To derive a linear expression for the book to market ratio, we approximate equations (A.3) and (A.4) by taking Taylor expansions of [pic] and [pic] about the point [pic] ([pic] is a number between zero and one and m-d and b-d denote unconditional means). Next, we subtract the approximation of equation (A.4) from the approximation of equation (A.3). This yields the following linear approximation of the book to market ratio.

[pic] (A.5)

In equation (A.5) [pic] is a number slightly less than one. To understand [pic] better it is useful to assume [pic] equals zero in which case [pic], which implies that [pic] is bounded between zero and one, and is increasing in the price to dividend ratio (note [pic], thus [pic] if [pic]). Based on evidence presented in Vuolteenaho [2002] we assume [pic] equals .967.

If we assume [pic]and iterate (A.5) forward, we arrive at the following approximate expression for the book to market ratio.

[pic] (A.6)

Equation (A.6) illustrates that the book to market ratio is increasing in future discount rates (i.e., prices are low when expected future discount rates are high) and decreasing in future accounting returns (i.e., prices are high when expected profitability is high).

Finally, we obtain a linear approximation of realized returns by evaluating the change in expectations of (A.6) from t-1 to t and rearranging.

[pic] (A.7)

In equation (A.7) Et-1[.] is the conditional expectation operator and ΔEt[.] equals Et[.]-Et-1[.]. Equation (A.7) illustrates the components of realized returns. In particular, changes in expected future roe (i.e., cash flow news) and changes in expected discount rates (i.e., expected return news) cause realized returns to differ from expected returns. Hence, in order to draw meaningful inferences about the veracity of a particular measure of expected returns, we must control for cash flow news and expected return news.

Appendix B

Statistical Approach for Estimating Measurement Error Variances

As discussed in section II.2 of the main text, the bias in estimates of α1 taken from equation (2) does not provide clear-cut evidence about the relative measurement error in a particular expected return proxy. To understand better the bias in α1 it is helpful to consider the following re-expression of equation (1) from the main text.

[pic] (B.1)

In equation (B.1) [pic], [pic] and [pic] denote the measurement error in [pic], [pic] and [pic], respectively. We assume the measurement error in a particular proxy is uncorrelated with the true underlying construct but may be correlated with the true values of the other constructs and the measurement errors in the remaining proxies. While we are primarily interested in [pic], it is not observable. However, we can use the combined measurement error [pic] to evaluate the extent to which [pic] contributes to the bias in α1. Results presented in Rao [1973], Garber and Klepper [1980] and Barth [1991] demonstrate that [pic] and the bias in α1 are related in the following manner:

[pic] (B.2)

In equation (B.2) βi,j is the estimated coefficient corresponding to variable j taken from a first stage regression of variable i (one of the proxies) on the remaining independent variables (the other two proxies) in equation (2) from the main text, [pic] is the variance of the residuals from this first stage regression and [pic] is a function of the variance of the measurement error in proxy k; hence, we refer to the [pic] terms as “noise variables.” The relation between the noise variables and the covariance structure of the measurement errors is shown below:

[pic] (B.3)

In equation (B.3) [pic] is the variance of the measurement error in variable j, [pic] denotes the covariance between the measurement error in variable i with the measurement error in variable j, and [pic] is the covariance between the true value of construct i with the measurement error in construct j.

Equations (B.2) and (B.3) demonstrate that the difference between one and α1 is a complex function of the covariance structure of the independent variables and the covariance structure of their measurement errors. Hence, this difference is not solely attributable to measurement error in the expected return proxy (i.e., [pic]). However, we can use equations (B.2) and (B.3) to infer the portion of the bias in α1 that is attributable to [pic]. We follow the two-stage process described in Barth [1991].

In a set of first stage regressions we estimate the coefficients and residual variances shown in equation (B.2) (i.e., βi,j and [pic]). Next, these estimates are combined with the independent variables shown in equation (2) of the main text to develop a set of constructs that are used as the regressors in the following second stage regression, which is obtained by rearranging equation (B.2):

[pic] (B.4)

Ranking on the basis of [pic] embeds the assumption that [pic] and [pic], which may not be descriptive. To circumvent this problem we refine the regression shown in equation (B.4) by replacing the regressand (i.e., [pic]) with [pic], which is defined below.

[pic] (B.5)

In equation (B.5) [pic] ([pic]) denotes the correlation between the expected return proxy of interest and the cash flow news (return news) proxy, and [pic] is the regressor corresponding to [pic] in equation (B.4). The coefficient on [pic] obtained via estimation of B.5 is equal to the following expression:[22]

[pic] (B.6)

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Table 1

Descriptive statistics for firm-years spanning the time period 1981-1998.

Panel A – Descriptive statistics for realized returns, the expected return proxies, and the cash flow news proxy

| |MEAN |STD |MEDIAN |

|rt |0.106 |0.341 |0.126 |

|rpe |0.090 |0.034 |0.084 |

|rpeg |0.109 |0.031 |0.105 |

|rmpeg |0.121 |0.031 |0.116 |

|rgm |0.128 |0.033 |0.124 |

|rΔagr |0.127 |0.027 |0.123 |

|rgls |0.109 |0.031 |0.106 |

|[pic] |-0.008 |0.296 |-0.031 |

Panel B – Descriptive statistics for the return news proxies

| |MEAN |STD |MEDIAN |

|[pic](pe) |-0.004 |0.849 |-0.044 |

|[pic] (peg) |-0.015 |0.997 |-0.028 |

|[pic] (mpeg) |-0.013 |1.011 |-0.034 |

|[pic] (gm) |-0.016 |1.064 |-0.034 |

|[pic] (Δagr) |-0.011 |0.907 |-0.029 |

|[pic] (gls) |-0.016 |0.720 |-0.037 |

Outliers (observed values in the top or bottom 0.5 percentile) are removed. rt is the realized return over the year subsequent to the date on which the estimates of expected return are formed. rpe is the estimate of the expected rate of return imputed from the price to forward earnings model. rpeg is the estimate of the expected rate of return implied by the PEG ratio. rmpeg is the estimate of the expected rate of return derived from a the modified PEG ratio. rgm and rΔagr are the estimates of the expected rate of return imputed from Gode and Mohanram’s [2003] and Easton’s [2004] implementation of the Ohlson and Jeuttner-Nauroth [2001] model, respectively. rgls is the estimate of the expected rate of return based on the Gebhardt, Lee, and Swaminathan [2001] implementation of the residual income valuation model. [pic] is the estimate of cash flow news. [pic](xxx) is the estimate of expected return news for model xxx. MEAN, STD, MEDIAN are, respectively, the mean, standard deviation, and median of the distributions of the variables. N is the number of firm-years. All data are in logged form.

Table 2

Correlations among key variables

Panel A – Correlations among realized returns, the expected return proxies, and the cash flow news proxy

| |rt |rp|rpeg |

| | |e | |

| |rt |[pic] |cfn | |rt |

|rpe |0.09 |0.23 |0.39 |0.14 |0.22 |

| |(2.67) |(0.73) |(6.31) |(6.34) | |

|rpeg |0.18 |-0.58 |0.34 |0.06 |0.13 |

| |(6.20) |(-2.33) |(6.93) |(4.91) | |

|rmpeg |0.14 |-0.22 |0.34 |0.07 |0.14 |

| |(3.12) |(-0.68) |(7.18) |(5.84) | |

|rgm |0.16 |-0.40 |0.33 |0.06 |0.12 |

| |(4.92) |(-1.76) |(7.08) |(5.32) | |

|rΔagr |0.19 |-0.63 |0.33 |0.06 |0.12 |

| |(5.42) |(-2.16) |(6.94) |(4.71) | |

|rgls |0.18 |-0.61 |0.37 |0.19 |0.24 |

| |(7.58) |(-2.37) |(6.64) |(5.58) | |

Separate regressions are estimated for each of the 18 annual cross-sections of data. Parameter estimates equal the average of the annual regression coefficients. t-statistics (in parentheses) equal the ratio of the parameter estimates to their temporal standard errors. R2 is the mean r-square from the annual regressions. Outliers (observed values in the top or bottom 0.5 percentile) are removed. [pic], [pic], [pic], and rt represent the expected return proxy, the cash flow news proxy, the return news proxy, and the realized return over the year subsequent to the date on which the proxies are formed. rpe is the estimate of the expected rate of return imputed from the price to forward earnings model. rpeg is the estimate of the expected rate of return implied by the PEG ratio. rmpeg is the estimate of the expected rate of return derived from a the modified PEG ratio. rgm and rΔagr are the estimates of the expected rate of return imputed from Gode and Mohanram’s [2003] and Easton’s [2004] implementation of the Ohlson and Jeuttner-Nauroth [2001] model, respectively. rgls is the estimate of the expected rate of return based on the Gebhardt, Lee, and Swaminathan [2001] implementation of the residual income valuation model. All data are in logged form.

Table 4

Estimation of Measurement Error in Expected Return Proxies

Regressions:

[pic]

[pic]

| |Unadjusted Noise Variables | |Adjusted Noise Variables |

| |([pic] is the regressand) | |([pic] is the regressand) |

| | | | |

|[pic] |intercept |beta |

|Model |δ1M |δ1M |

|pe |0.00018 |0.00030 |

|  |(1.50) |(1.31) |

|peg |0.00043 |0.00059 |

|  |(2.95) |(3.41) |

|mpeg |0.00019 |0.00026 |

|  |(2.10) |(2.08) |

|gm |0.00022 |0.00030 |

|  |(2.45) |(2.58) |

|Δagr |0.00026 |0.00034 |

|  |(2.49) |(2.74) |

|gls |0.00033 |0.00045 |

|  |(2.96) |(2.72) |

Separate regressions are estimated for each of the 18 annual cross-sections of data. Parameter estimates equal the average of the annual regression coefficients. t-statistics (in parentheses) equal the ratio of the parameter estimates to their temporal standard errors. R2 is the mean r-square from the annual regressions. Outliers (observed values in the top or bottom 0.5 percentile) are removed. All data are in logged form.

[pic]

[pic], [pic], [pic], and rt represent the expected return proxy, the cash flow news proxy, the return news proxy, and the realized return over the year subsequent to the date on which the proxies are formed. βi,j is the estimated coefficient corresponding to variable j taken from a first stage regression of variable i (one of the proxies) on the remaining independent variables (the other two proxies) in equation (2), [pic] is the variance of the residuals from this first stage regression.

[pic]

Table 7

Estimation of Measurement Error in Expected Return Proxies for Portfolios of Observations

Regression:

[pic]

|  |Beta Portfolios |SD Portfolios |Size Portfolios |BP Portfolios |

|Model |δ1M |δ1M |δ1M |δ1M |

|pe |-0.00005 |-0.00017 |0.00005 |0.00064 |

|  |(-0.46) |(-1.74) |(0.64) |(6.79) |

|peg |0.00033 |0.00042 |0.00021 |0.00028 |

|  |(3.04) |(2.87) |(3.17) |(5.96) |

|mpeg |0.00009 |0.00009 |0.00012 |0.00039 |

|  |(1.29) |(0.86) |(2.41) |(6.01) |

|gm |0.00018 |0.00018 |0.00015 |0.00032 |

|  |(2.58) |(1.71) |(3.33) |(5.63) |

|Δagr |0.00017 |0.00024 |0.00012 |0.00022 |

|  |(2.09) |(2.13) |(2.59) |(4.37) |

|gls |0.00003 |0.00008 |0.0001 |0.00026 |

|  |(0.50) |(0.74) |(1.36) |(3.43) |

Separate regressions are estimated for each of the 18 annual cross-sections of data. Parameter estimates equal the average of the annual regression coefficients. t-statistics (in parentheses) equal the ratio of the parameter estimates to their temporal standard errors. R2 is the mean r-square from the annual regressions. Outliers (observed values in the top or bottom 0.5 percentile) are removed. All data are in logged form.

[pic]

[pic], [pic], [pic] , and rt represents the portfolio average of the expected return proxy, the cash flow news proxy, the return news proxy, and realized return over the year subsequent to the date on which the proxies are formed. Portfolios of 10 are formed by ranking the variables on the basis of the variable of interest: capm beta (beta), standard deviation of returns (SD), market capitalization (Size), and book to market (BP). Outliers (observed values in the top or bottom 0.5 percentile) are removed. All data are in logged form. βi,j is the estimated coefficient corresponding to variable j taken from a first stage regression of variable i (one of the proxies) on the remaining independent variables (the other two proxies) in equation (2), [pic] is the variance of the residuals from this first stage regression.

[pic]

-----------------------

( We thank the University of Notre Dame, Mendoza College of Business and INSEAD for research support. Workshop participants at the American Accounting Association annual meetings, Columbia University, INSEAD, the University of Alabama, the University of Arizona, the University of Chicago, the University of Houston, and the University of Iowa provided valuable comments on an earlier draft. We thank Ray Ball, Phil Berger, Bruce Johnson, Lubos Pastor, Marlene Plumlee, and Shiva Sivaramakrisknan for helpful comments and suggestions. Earnings forecast data are from I/B/E/S. This paper was formerly distributed under the title “An Evaluation of the Reliability of Accounting Based Measures of Expected Returns: A Measurement Error Perspective.”

(( Corresponding author

[1] A number of studies use accounting based expected return proxies to evaluate the cross-sectional determinants of expected returns. For example, Botosan [1997] and Botosan and Plumlee [2002a] use an estimate of the expected return that is implied by a variant of the dividend discount model to study the effect of disclosure quality on the cost of capital. Gebhardt, Lee and Swaminathan [2001] use a variant of the discounted residual income valuation model to develop their expected return proxies, which they use to evaluate a number of potential risk factors. Francis, LaFond, Olsson, and Schipper [2003] use an expected return proxy implied by a variant of the dividend discount model to evaluate the relation between measures of earnings quality and firms’ costs of capital. Gode and Mohanram [2003] use variants of the valuation model developed by Ohlson and Juettner-Nauroth [2001] to estimate expected returns, which they use to evaluate the relation between estimates of expected returns and various risk proxies. Finally, Hail and Leuz [2003] use estimates of expected return derived from various models to evaluate the economic consequences of international differences in legal institutions and security regulations. Accounting based proxies are also used to estimate mean expected returns. In particular, Claus and Thomas [2001], and Easton, Taylor, Shroff and Sougiannis [2002] use estimates of expected return derived from variants of the discounted residual income valuation model to estimate the equity premium. Our approach does not provide evidence about the bias in expected return proxies, however. Hence, our results do not have direct implications for the latter two studies.

[2] Elton [1999] states (p. 1199): “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.” In section II.1 we elaborate on Elton’s argument and its implications for our empirical design.

[3] Firm subscripts are omitted for notational convenience.

[4] See chapter 21 of Cochrane [2001] for a discussion of the equity premium puzzle. The discussion under the heading Luck and a Lower Target on pages 460 through 462 is particularly relevant.

[5] Claus and Thomas [2001] make a similar argument.

[6] A similar argument can be made for the inclusion of cfnt in the regression. In particular, if changes in expectations about future cash flows are correlated with investment opportunities, which, in turn, are correlated with expected returns, cfnt will be correlated with both realized and expected return. Berk, Green and Naik [1999] develop a model in which firms’ optimal investment choices are associated with expected returns.

[7] We describe the expected return proxies in section III.1, and the cash flow news and return news proxies in section III.2.

[8] As shown in Appendix B, the expressions for [pic] and [pic] are similar. Given our primary interest relates to the measurement error in the expected return proxies we choose to focus the discussion in the body of the paper on [pic].

[9] The refinements we make to the approach developed by Garber and Klepper [1980] and Barth [1991] are described in Appendix B.

[10] The assumption that [pic] is zero or of second order magnitude is similar to assumptions made by Garber and Klepper [1980] and Barth [1991].

[11] To be precise, the valuation model underlying rpe relies on the assumption that after year t+2 cum-dividend aggregate earnings grow at a rate equal to the cost of capital. We use price to forward aggregate earnings rather than a simple price to earnings multiple so that rpe embeds the same analyst forecast information as the other proxies.

[12] The PEG ratio, which is equal to the PE ratio divided by the short-term earnings growth rate is a common means of comparing stocks in analysts’ reports.

[13] Since I/B/E/S does not provide forecasts of dividends, we assume dpst+1 equals dpst (i.e., dividends per share paid in year t).

[14] Based on the evidence shown in Vuolteenaho [2002], we assume Á equals Vuolteenaho [2002], we assume ρ equals 0.967. Our cash flow news proxy embeds the assumption that roe follows a first order autoregressive process after year t+1. This assumption is consistent with evidence presented in Beaver [1970], Freeman, Ohlson and Penman [1982], and Sloan [1996].

[15] We obtain similar results when we perform our analyses using estimates of ωt-1 that are based on industry groups formed on the basis of the industry classification in Fama and French [1997].

[16] In this example, firm-year observations with a negative forecast of earnings for 1996 are deleted because growth from a negative base is not meaningful in this context. The number of observations deleted for this reason in each of the years 1981 to 1999 is 11, 22, 14, 30, 35, 58, 55, 47, 50, 73, 72, 88, 109, 121, 134, 287, 322, 363, and 487.

[17] The median value of ωt-1 used in the calculation of [pic] is 0.57.

[18] Similar evidence may be found in Guay, Kothari and Shu [2003]. In particular, they estimate simple regressions of realized return on empirical measures of expected returns and document negative slope coefficients for two of the four accounting based proxies they evaluate.

[19] The correlation between realized return and an expected return proxy equals [pic]. If cash flow news and return news are simply random measurement error, the second term in the numerator equals zero and the sign is equal to the sign of the first term. The sign of the first term is positive unless the covariance between true expected return and the measurement error in the expected return proxy is negative and has an absolute value greater than the variance of true expected returns. This is unlikely.

[20] Our primary interest is the relative measurement error of the expected return proxies; hence, we only tabulate the estimates of δ1M obtained from the instrumental variables and grouping analyses.

[21] Analyses based on portfolios of size twenty lead to similar inferences.

[22] To derive the expression for [pic] we begin by expanding the covariance terms in [pic]:

[pic]

Next, using the definition of [pic] shown in equation (B.3), we arrive at the following expression.

[pic]

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