The value line timeliness ranking and the equivalence of analyst forecasts and market expectations.
The Value Line Enigma is so named to describe researchers' inability to reconcile the seemingly superior investment performance of Value Line's Timeliness rank with market efficiency (Copeland and Mayers 1982; Black 1973). A stock's Timeliness is the output of a quantitative model incorporating ex post data on earnings and stock prices in order to project 6-12 months ahead relative stock price performance. The scale ranges from one to five, with stocks ranked 1 (5) having the best (worst) relative prospects. Independent of Timeliness, Value Line's team of fundamental research analysts produces ex ante forecasts of stock prices and 23 financial variables for each of the stocks in its coverage universe. Scholarly research in equity valuation and implied expected returns has incorporated Value Line fundamental forecasts in valuation models on the maintained assumption of equivalence between analyst forecasts and market expectations. I investigate this equivalence in the context of valuation errors across Timeliness rank. I argue that a stock's Timeliness reflects the probable magnitude and direction of divergence between market expectations and analyst forecasts. The evidence suggests that equivalence holds for stocks of average Timeliness and that it does not hold for stocks above or below average Timeliness.
The question is important for research in equity valuation and implied expected returns, because inferences can be confounded when equivalence breaks down. Heinrichs et al. (2013), Devos et al. (2009), Courteau et al. (2001) and others compute stocks' intrinsic values with Value Line analyst forecasts as model inputs, but these valuations are biased predictors of stock price if the forecasts do not correspond to investor expectations. Botosan et al. (2011) evaluate alternative measures of implied expected return in terms of their correlation with firm-specific risk proxies that depend on stock price (e.g., the ratio of book-to-market value), and Brav et al. (2005) test asset pricing theories on the basis of correlations between implied returns and stock price-based risk measures. Because stock price responds quickly to new information and because both implied returns and measures such as book-to-market value depend on stock price, an observed correlation between them can be attributed to divergent expectations during the time in which analysts are working to update their forecasts. This correlation will be observed even when equivalence prevails on average in the cross section, and it calls into question the validity of the inferences reached in these and similar studies.
Value Line (hereafter, VL) publishes quarterly updates of analysts' fundamental forecasts for each of approximately 1,700 companies in its coverage universe; hence, there can be a lag of up to three months from, for example, the date of a quarterly earnings release to the date of its incorporation in updated forecasts. Analyst forecasts are stale when they do not incorporate all relevant information, and it is generally the case that at any time some forecasts are stale. A stock's Timeliness rank factors in recent earnings and stock price news, and stocks that have experienced recent positive (negative) news are more likely to be rated above (below) average. Because Timeliness is updated weekly, a stock's rank often reflects more recent information than do analyst forecasts. Consequently, market expectations most likely differ from analyst forecasts for stocks rated above or below average Timeliness. Further, because the magnitude and direction of divergence are related to Timeliness, valuations based on stale forecasts manifest in a systematic pattern of valuation errors across Timeliness ranks. Among average Timeliness stocks, analyst forecasts reflect recent news, and valuation errors tend to be near zero. Among stocks above (below) average Timeliness, stale forecasts tend not to reflect recent good (bad) news, and the valuations based on these forecasts tend to be biased downward (upward) vis-a-vis actual stock prices. Moreover, the magnitude of the bias increases the farther that Timeliness is above or below average.
I compute intrinsic values and analyze the valuation errors for a large sample of stocks covered by VL from 1988 through 2012. Valuation errors among stocks of average Timeliness are consistent with equivalence between analyst forecasts and market expectations; for these stocks valuation errors have zero mean, and intrinsic value appears to be an unbiased predictor of stock price with very high explanatory power. In contrast, analyst forecasts for stocks above and below average Timeliness appear not to fully incorporate recent news, and valuations appear to be biased downward (upward) among stocks above (below) average Timeliness. These conclusions are supported by regression-based tests, by decomposition of valuation errors and by the pattern of VL quarterly earnings forecast revisions. The results are important because they show that researchers cannot simply assume the unconditional equivalence of analyst forecasts and market expectations, and they show that in studies that make this assumption confounding effects can operate with varying magnitude and direction.
Relevant Literature Review
VALUE LINE FORECASTS IN EQUITY VALUATION
Value Line, Inc. is an independent investment advisory with substantial influence among investors. The Value Line Investment Survey, which has been in circulation since 1965, delivers research and recommendations on approximately 1,700 companies comprising roughly 90% of the market capitalization of U.S. common stocks. For each company, VL's team of analysts produces forecasts of 23 fundamental variables (sales, earnings, dividends, cash flow and more) for the current year, the following year and the three-to-five-year-ahead period. VL analysts also produce three-to-five-year-ahead Target Stock Price forecasts. Forecasts for individual companies are usually updated on a quarterly schedule.
A literature in accounting utilizes VL forecasts for the purpose of computing common stock intrinsic values. Courteau et al. (2001) compare the performance of dividend, free cash flow and residual income valuation models. They find that from 1992-1996 residual income valuation with VL Target Price in the terminal value expression dominates dividend and free cash flow models in terms of generating valuations with the smallest bias and inaccuracy. Mean (median) prediction error scaled by stock price is 8.4% (4.7%), and linear regression of stock price on intrinsic value produces an [R.sup.2] of .93. However, the intercept is significantly positive and the slope is significantly less than one, which indicate the presence of a prediction bias that varies with stock price. To compute equity capital costs, they use the CAPM with VL betas and a constant market risk premium of 6%.
Heinrichs et al. (2013) use VL forecasts in their evaluation of dividend, free cash flow and residual income models with corrections for deviations from ideal conditions. From 1977-2006, the residual income model again produces the smallest valuation errors among the three models, but the median error of 12% of stock price is substantially larger than that reported by Courteau et al. (2001). This marked deterioration of forecast performance is likely attributable in part to their use of terminal value expressions that assume constant, perpetual growth beyond the forecast period. Unlike Courteau et al. (2001), these authors do not use VL Target Prices in terminal value expressions. Rather, across their entire sample, they assume a constant terminal growth rate beyond their five-year forecast horizon. Although the model's explanatory power is not reported, the form of their terminal value expression suggests it is most likely lower than that reported by Courteau et al. (2001). Interestingly, after correcting for non-ideal conditions, their residual income model performs no better than their uncorrected model, which suggests that residual income valuation works well even under conditions that are less than ideal. These authors compute equity capital costs with the CAPM. They assume a constant market risk premium of 5%, and they estimate betas from returns in the prior five years.
Francis et al. (2000) compare the predictive performance of dividend, free cash flow and residual income models using VL forecasts from 1989-1993. They find that residual income outperforms other models. Similar to Heinrichs et al. (2013), they compute terminal values by assuming perpetual growth, and they obtain an [R.sup.2] of .71 in their regression of stock price on their intrinsic value estimates. Bernard (1995) is an early example of VL forecasts used in residual income modeling for the purpose of explaining stock prices. In a different research vein, Devos et al. (2009) compute enterprise values with VL forecasts in order to isolate sources of synergies in mergers and acquisitions.
Importantly, all of the studies mentioned in this and the next section proceed on the basis of two maintained assumptions of equivalence. The first, which is the one being investigated here, is the assumed equivalence between analyst forecasts and market expectations. The second is the assumed equivalence between the particular implementation of a valuation model chosen by the researcher and the process by which investors actually price stocks. The results presented by Heinrichs et al. (2013), Courteau et al. (2001), Francis et al. (2000) and Bernard (1995) indicate that residual income valuation using VL forecasts and the VL Target Price in the terminal value expression seems to most closely represent the actual process by which stocks are valued.
VALUE LINE FORECASTS AND IMPLIED EXPECTED RETURNS
A substantial body of research in the accounting and finance literatures utilizes implied expected returns. Implied returns are obtained by inserting analysts forecasts into valuation models, equating stock price to intrinsic value and solving for expected return. Implied returns have been employed in studies addressing a range of research questions. For example, Huang and Wei (2012) investigate advertising intensity, investor recognition and implied cost of capital, and Bhattacharya et al. (2012) pursue the linkages among earnings quality, information asymmetry and the cost of equity. Brav et al. (2005) use implied returns to test asset pricing theories.
Botosan et al. (2011) assess the validity of twelve proxies for expected return, nine of which are implied cost of capital estimates proposed in the literature and which are implemented with VL forecasts. Their comparative evaluation involves regressing expected return proxies on firm-specific risk characteristics that include market capitalization, book-to-market value of equity, debt-to-market value of equity, beta and growth. They recommend two proxies in part on the basis of significantly positive coefficients on book-to-market value and debt-to-market value, arguing that these results are consistent with theory. These results, however, can also be attributed to stale VL forecasts and their divergence from market expectations, which biases the regression coefficients away from zero. Even if equivalence holds on average, stock prices respond more quickly to new value-relevant information than equity analysts can update their forecasts. For VL analysts in particular there can be as much as a three-month lag between the arrival of new information and the incorporation of that information in published forecasts. Consequently, some subset of stock prices will always embed market expectations that differ from analyst forecasts, and this will induce a correlation between implied return and any variable the measurement of which depends on stock price. It seems probable, therefore, that the conclusions drawn by Huang and Wei (2012), Bhattacharya et al. (2012), Botosan et al. (2011), Brav et al. (2005) and others are confounded by measurement error.
THE VALUE LINE TIMELINESS RANK AND THE VALUE LINE ENIGMA
VL is perhaps best known for its Timeliness ranking system for relative 6-12 month stock price performance. In its coverage universe, stocks ranked 1 for Timeliness are 100 stocks with the best relative prospects, and those ranked 5 are 100 stocks with the worst relative prospects. Rank 2 includes 300 stocks with above-average prospects. Rank 3 includes approximately 900 stocks with average prospects, and Rank 4 comprises 300 stocks with below-average prospects. From 1965 through 2013, Timeliness 1 stocks returned a cumulative 49,440% versus 1,718% for the Dow Jones Industrial Average and -99% for Timeliness 5 stocks. The following excerpt from VL's website describes the model inputs:
The components of The Value Line Ranking System for Timeliness include factors such as the 10-year trend of relative earnings and prices, recent earnings and price changes, and earnings surprises. All data are actual and known. A computer program combines these elements into a forecast of the price change of each stock, relative to all other ranked stocks for the six to 12 months ahead (Value Line 2014).
Because the impressive record of the Timeliness system has been difficult to reconcile with market efficiency, it is dubbed the "Value Line Enigma" by Copeland and Mayers (1982). Since then the record has been subjected to extensive scholarly research, and the results are mixed. Holloway (1981) finds that VL Timeliness recommendations yield abnormal returns even after transactions costs. Stickel (1985) finds that changes in Timeliness impact stock prices, especially when Rank 2 stocks are upgraded to Rank 1. Huberman and Kandel (1987) find that within size-sorted quintiles the mean payoffs of costless positions constructed according to Timeliness are positive. Huberman and Kandel (1990) find that because Timeliness reflects systematic risk, abnormal returns represent compensation for undiversifiable risks. Affleck-Graves and Mendenhall (1992) find that the VL enigma is actually a manifestation of post-earnings announcement drift. Zhang et al. (2010) find that the enigma is still alive and that post-earnings announcement drift does not explain it. Nayar et al. (2011) find that the stock price drift following Timeliness upgrades cannot be explained solely in terms of post-earnings announcement drift.
TIMELINESS AND EQUIVALENCE
The description of VL's Timeliness model indicates that stocks ranked 1 and 2 (4 and 5) are often characterized by recent good (bad) news on earnings and rising (falling) stock prices. Given VL's schedules for updating its Timeliness ratings (weekly) and its fundamental forecasts (quarterly), the prices of stocks above and below average Timeliness reflect recent information that has yet to be incorporated in analyst forecasts. Consequently, analyst forecasts will tend to lag more optimistic market expectations for Timeliness 1 and 2 stocks and more pessimistic expectations for Timeliness 4 and 5 stocks. Because intrinsic values based on stale forecasts are poorer predictors of stock price, valuation errors will display a systematic pattern of bias and inaccuracy across Timeliness ranks. Valuations are biased downward (upward) among stocks above (below) average Timeliness, and the magnitude of this bias will increase as Timeliness is farther above or below average, i.e., as Timeliness increases from 2 to 1 or decreases from 4 to 5. Furthermore, forecast accuracy will decrease as Timeliness moves farther above or below average.
Methodology and Data
RESIDUAL INCOME VALUATION
Residual income valuation can be derived in a straightforward manner from dividend valuation with the added assumption that clean surplus accounting holds (see, e.g., Courteau et al. 2001). By the clean surplus relation, changes in book equity, apart from capital flows, flow through the income statement:
[B.sub.t] = [B.sub.t-1] + [NI.sub.t] - [D.sub.t] + [SI.sub.t] (1)
[B.sub.t] denotes book equity at time t, [NI.sub.t] denotes net income, [D.sub.t] equals cash dividends and [SI.sub.t] denotes share issuance (or repurchase if negative).
Residual income in period t ([RI.sub.t]) equals net income less a charge for the opportunity cost of beginning-of-period equity capital (r[B.sub.t-1]):
[RI.sub.t] = [NI.sub.t] - [rB.sub.t-1] (2)
The equity cost of capital is denoted by r. In residual income valuation, intrinsic value equals current book equity plus the present value of residual income. With an infinite forecast horizon, intrinsic value is given by equation (3):
[V.sub.t] = [B.sub.t] + [[infinity].summation over (s=1)] [RI.sub.t+s]/[(1 + r).sup.s] (3)
Courteau et al. (2001) operationalize equation (3) using VL forecasts for a five-year forecast horizon plus a terminal value expression that incorporates the Target Stock Price for the present value of residual income beyond the explicit forecast period. I do so as well:
[V.sub.t] = [B.sub.t] + [5.summation over (s=1)] [[RI.sub.t+s]/[(1 + r).sup.t+s]] + [[P.sub.t+5] - [B.sub.t+5]/[(1 + r).sup.5]] (4)
In equation (4), [P.sub.t+5] denotes VL's three-five-year-ahead Target Stock Price.
NON-DIVIDEND CAPITAL FLOWS AND OTHER COMPREHENSIVE INCOME
In a departure from prior research, I measure non-dividend capital flows between the firm and stockholders in terms of the dollar value of share issuance. Share issuance ([SI.sub.t]) in each forecast year equals the projected change in shares outstanding multiplied by projected stock price. Projected stock price equals projected earnings per share multiplied by the Target P/E ratio. Target P/E equals Target Stock Price divided by forecast three-to-five-year earnings per share. These calculations are represented in equation (5):
[SI.sub.t+s] = ([Shares.sub.t+s] - [Shares.sub.t+s-1]) ([E.sub.t+s]) ([P.sub.t+s]/[E.sub.t+5]) (5)
[E.sub.t+s] denotes the s-year-ahead earnings per share forecast, and [P.sub.t+5]/ [E.sub.t+5] is the target P/E ratio.
To construct a measure of earnings that accounts for violations of clear surplus, the right-hand side of equation (1) is augmented to include other comprehensive income ([CI.sub.t]):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1')
Adjusted net income ([NI.sup.*]), which includes other comprehensive income, is obtained from equation (1'):
[NI.sup.*.sub.t] = [NI.sub.t] + [CI.sub.t] = [B.sub.t] = [B.sub.t] - [B.sub.t-1] + [D.sub.t] - [SI.sub.t] (6)
Intrinsic value is calculated according to equation (4) with adjusted net income in the residual income calculation:
[RI.sub.t] + [NI.sup.*.sub.t] - r[B.sub.t-1] (2')
EQUITY COST OF CAPITAL
The equity cost of capital (r) is computed with the CAPM using VL betas and 10-year Treasury constant maturity yields. VL estimates betas from weekly data for the prior two to five years with the NYSE Composite as the market index. To account for mean reversion, VL adjusts estimated betas toward one. Unlike prior studies, the market risk premium is allowed to vary over time. It is calculated as [RP.sub.Mt] = .04 + ([y.sub.t.sup.Baa] - [y.sub.t.sup.T10]). [y.sub.t.sup.Baa] and [y.sub.t.sup.T10], respectively, denote the yield to maturity of Moody's Baa Corporate Bonds and the 10-year Treasury Note. These data are from Federal Reserve Release H.15. The constant is chosen to produce an average market risk premium consistent with prior research. The mean (median) market risk premium of 6.4% (6.3%) is very much in line with values typically assumed by researchers (see Table 1).
ANALYSIS OF VALUATION ERRORS
The valuation error for firm i in year t equals current stock price ([P.sub.it]) minus intrinsic value ([V.sub.it]):
[e.sub.it] + [P.sub.it] - [V.sub.it] (7)
The bias and accuracy of intrinsic value as a proxy for stock price are examined in terms of both mean and median signed and unsigned valuation errors and in terms of both per-share amounts and as scaled by intrinsic value. If VL forecasts for Timeliness 3 stocks more closely represent market expectations, then valuation errors for these stocks will show smaller bias and better accuracy than for stocks above or below average Timeliness. Further, as Timeliness increases (decreases) from average, errors will become increasingly positive (negative), indicating an increasingly downward (upward) bias.
TESTS BASED ON STOCK PRICE REGRESSIONS
If intrinsic value were a perfect proxy for stock price, then the following regression of [P.sub.it] on [V.sub.it] would produce an estimated constant term equal to zero, a slope equal to one and an R-square equal to one.
[P.sub.it] = [[alpha].sub.0] + [[alpha].sub.1] [V.sub.it] + [[epsilon].sub.it] (8)
A non-zero constant term reflects a bias that is uncorrelated with stock price. A significantly positive constant indicates a downward bias (P > V), and a significantly negative constant indicates an upward bias (P < V). A slope coefficient significantly greater (less) than one indicates a downward (upward) bias with magnitude increasing in stock price. If Timeliness captures the magnitude and direction of divergence between analyst forecasts and market expectations, then among ranks 1, 2 (4, 5) the intercept should be positive (negative), and the slope should be not less (greater) than one. For Rank 3 stocks, the intercept should equal zero and the slope should equal one. Further, the regression R-square should decrease as Timeliness increases or decreases from average.
As an additional test, I augment the base regression model equation (8) to include the price-earnings ratio as an explanatory variable. If analyst forecasts are stale then a stock-price based variable such as the P/E ratio will add explanatory power beyond that provided by intrinsic value, and its coefficient will increase the farther that Timeliness is above or below average.
MEAN SQUARE ERROR DECOMPOSITION
The mean square error of a prediction can be decomposed into three component sources of error. Utilizing regression results for (8), mean square error can be expressed as:
MSE = [summation] [e.sup.2.sub.it]/N = [([bar.P] - [bar.V]).sup.2] + [(1 - [[??].sub.1]).sup.2] [S.sup.2.sub.V] + (1 - [R.sup.2]) [S.sup.2.sub.P] (9)
The error [e.sub.it] is given by equation (7); it is not the regression residual. [bar.P] and [bar.V] denote the sample means of stock price and intrinsic value. [S.sup.2.sub.P] and [S.sup.2.sub.V] denote the sample variances of stock price and intrinsic value, respectively, [[??].sub.1] denotes the estimated slope, and [R.sup.2] is the unadjusted R-square. The first term on the right-hand side of (9) measures bias uncorrelated with stock price, the second term measures the bias correlated with stock price and the third term captures the error due to random noise. I refer to these terms as bias, inefficiency and noise, respectively. For an ideal forecast, bias and inefficiency equal zero, and the MSE is small and comprised entirely of noise.
If equivalence holds for stocks of average Timeliness but not for other stocks, then the MSE and its bias and inefficiency components should be smallest for Rank 3 stocks and increasing as Timeliness is further from average.
EARNINGS FORECAST REVISIONS
As a robustness check, I present additional evidence from VL quarterly earnings forecast revisions. If VL forecasts are increasingly stale for stocks above and below average Timeliness, then this staleness will be reflected in a pattern of quarterly earnings forecast revisions that is increasing in Timeliness. That is, conditional on prior-quarter forecast error, quarterly earnings forecast revisions will be most negative for Rank 5 stocks and increasing to most positive for Rank 1 stocks.
VL issues multiple earnings forecasts for a given quarter t, with initial forecasts typically published more than a year prior to quarter end. As quarter end approaches, forecasts are revised to incorporate new information. An important source of new information arises from prior-quarter earnings and earnings forecast errors. The forecast revision for quarter t is calculated as the change in the forecast from quarter t-1 to quarter t: [DELTA][F.sub.t] = [F.sup.t] - [F.sup.t-1.sub.t]. [F.sup.t.sub.t] is the latest forecast of quarter t earnings, and is the forecast of quarter t earnings as of quarter t-1. Because is issued before the prior-quarter forecast error is known, the forecast revision reflects information in the prior-quarter earnings release. Flence, it will be correlated with Timeliness; good (bad) prior-quarter news will tend to reflect in above (below) average Timeliness.
THE VALUE LINE EARNINGS AND PROJECTIONS FILE
Timeliness ranks and forecasts from 1988 through 2012 are obtained from the VL Earnings and Projections file. For the purpose of computing intrinsic value and valuation error, data are sampled once per year in January in order to maximize the number of usable observations while minimizing issues with serial correlation in stock prices. Only firms with December fiscal years are included. Financial firms (6000 [less than or equal to] SIC [less than or equal to] 6999) are excluded. In VL's January report for a December fiscal year firm, the current forecast year corresponds to the year ending in December one month prior; hence, that forecast combines actual results for the first three quarters with the most recent fourth quarter forecasts. Current equity book value is taken from this forecast. The next forecast year corresponds to the current calendar year ending in December eleven months hence. This is the first forecast year (i.e., s = 1) for the purpose of computing intrinsic value. In about one-half the sample, forecasts are also available for the following year. When available, these become the forecasts for the second year (s = 2). Forecasts for the three-to-five-year-ahead period are assumed to represent the fifth-year forecast (s = 5). Intermediate forecasts are obtained by linear interpolation. For the purpose of computing quarterly earnings forecasts and revisions, data are sampled quarterly.
Results and Discussion
Table 1 presents summary statistics for sample characteristics including stock price, intrinsic value, market capitalization, P/E, P/B, beta and cost of equity. The sample comprises 17,261 firm-year observations for 1,960 unique firms from 1988 through 2012. To mitigate effects of potential errors in the data, observations in the extreme .5% tails of the valuation error distribution are deleted. The difference between mean (median) stock price and intrinsic value is $0.45 ($0.09), which suggests that intrinsic value possesses good predictive power for stock price. Sample firms are primarily mid- and large-cap firms with mean (median) market capitalizations of $8.0 ($1.7) billion. P/E and P/B ratios appear to be somewhat above average for equities generally, and beta is also slightly above its expected value of 1.0. These results are similar to those reported in Brav et al. (2005) that stocks covered by VL tend toward large-cap growth stocks. The 10-year Treasury yield ranges from a low of 2.0% to a high of 9.1%, and the market risk premium ranges from 5.3% to 9.6% with mean (median) of 6.4% (6.3%). Mean (median) cost of equity is 11.9% (12.2%), which is in line with the mean (median) implied costs of equity for the proxies recommended by Botosan et al. (2011).
VALUATION ERRORS: SUMMARY STATISTICS
Panel A of Table 2 shows the number of observations annually ranges from 535 in 1990 to 835 in 2012. From 1988-2012, the mean valuation error is a statistically significant $0.45 per share, while the median error equals $0.09 and is not significantly different from zero. The proportions of positive and negative errors are almost equal with 51% positive and 49% negative. There is substantial year-to-year variation. The mean error ranges from -$6.97 in 2009 to $3.83 in 2006, and annual t-tests indicate the mean valuation error is significantly greater (less) than zero in 11 (five) years. The median error ranges from -$5.82 in 2009 to $3.07 in 2007, and annual sign tests indicate the median error is significantly greater (less) than zero in 10 (seven) years. The proportion of positive errors ([P.sub.it] > [V.sub.it]) ranges from 16% in 2009 to 72% in 1997 and 2004. Finally, mean (median) unsigned error equals $5.64 ($3.64) from 1988-2012. On balance, these results indicate that intrinsic value appears to be a good predictor of stock price over the entire 25-year period and that valuation errors in any single year vary substantially in magnitude and direction.
Table 2 Panel B shows results for valuation errors scaled by intrinsic value ([e.sub.it]/[V.sub.it]). From 1988-2012, the mean (median) error is 3.2% (0.6%) of intrinsic value. While the mean is statistically significant, the median is not. Substantial year-to-year variation is again apparent. Mean error ranges from -23% in 2009 to 19% in 2004, and the mean percent error is significantly greater (less) than zero in 13 (five) years. The median error ranges from -27% in 2009 to 13% in 1997 and 2004, and in 10 (seven) years it is significantly positive (negative). From 1988-2012, mean (median) absolute percent errors are 23% (18%). Again, these results indicate that as a percent of predicted stock, price signed errors are, on balance, small during the entire period, but they display sizable year-to-year variation.
Table 3 reports regression results of stock price on intrinsic value annually and for the entire sample period. The regression model is given by equation (8). If intrinsic value were a perfect predictor of stock price, the intercept would equal zero, the slope would equal one and the R-square would equal one. For the 1988-2012 period, the estimated intercept [[??].sub.0] = 0.55 is significantly greater than zero at the .01 level, and the estimated slope [[??].sub.1] = 0.996 is not significantly different from one. These results indicate that intrinsic value is downwardly biased by a constant $0.55 per share, and this bias is uncorrelated with stock price. The adjusted R-square equals 0.90, which indicates that intrinsic value explains a very large proportion of the variation in actual stock price. Nonetheless, the RMSE of $8.32 indicates the presence of large errors.
Annual regression results display year-to-year variation. The estimated intercept ranges from -2.96 in 2003 to 4.93 in 2004, and it is significantly positive (negative) in seven (six) years. The estimated slope ranges from 0.81 in 2009 to 1.12 in 1997, and it is significantly greater (less) than one in 10 (six) years. The R-square ranges from 0.74 in 2000 to 0.97 in 1993, and the RMSE ranges from 5.32 in 1989 to 10.65 in 2011.
VALUATION ERRORS BY TIMELINESS
Table 4 presents statistics on valuation errors by Timeliness. The results are as expected if equivalence holds among average Timeliness stocks but not among stocks above and below average. Panel A shows that among Rank 3 stocks, the mean (median) valuation error of $0.21 (-$0.07) is not significantly different from zero, and the proportion of positive errors (49%) is almost equal to the proportion of negative errors (51%). Mean absolute error is nearly minimized at $5.32 among Rank 3 stocks; only Rank 4 stocks show a smaller mean absolute error of $5.15. Median absolute error is minimized among Rank 3 stocks at $3.39. Results in Panel B, which reports statistics for errors scaled by intrinsic value, are consistent with those in Panel A. The mean (median) valuation error equals 1.8% (-0.5%) of intrinsic value. The mean is significantly positive, and the median is not significantly different from zero. These percentage errors are small when viewed in comparison with prior research, like Courteau et al. (2001), and with stocks of Rank 1, 2, 4 and 5. In addition, mean and median absolute percentage errors are also smallest for Rank 3 stocks. These results are consistent with the view that analyst forecasts equal market expectations for Rank 3 stocks.
The pattern of errors in Table 4 is also what one would expect if divergent expectations among stocks above (below) average Timeliness reflect recent good (bad) news on earnings and stock price that is not yet reflected in analyst forecasts. Mean and median signed valuation errors increase monotonically with Timeliness, and they are significantly positive (negative) for above (below) average stocks. In addition, the proportion of positive errors increases with Timeliness, and mean and median absolute errors increase with rank distance from average.
REGRESSION RESULTS BY TIMELINESS
Table 5 reports regression results of stock price on intrinsic value by Timeliness. Again, the results suggest that equivalence holds for Rank 3 stocks but not for other ranks. Among Rank 3 stocks, the estimated intercept is not significantly different from zero, and the estimated slope is not significantly different from one. The R-square = 0.92 exceeds that for the entire sample of 0.90, and the RMSE of 7.90 is below that of 8.32 for the entire sample. Table 5 shows that for stocks ranked 1 and 2 the estimated intercepts are both positive and statistically significant, and the estimated slopes are not significantly less than one. These results indicate that intrinsic value is a downwardly biased predictor of stock price and that, among Timeliness 1 stocks, the magnitude of the bias increases in stock price. Among stocks ranked 4 and 5 the estimated intercepts are both negative, significantly so for Rank 5 stocks, and the estimated slopes are both significantly less than one. For stocks below average Timeliness, intrinsic value is an upwardly biased predictor of stock price, and the magnitude of the bias increases in stock price.
If there is no staleness in analyst forecasts so that intrinsic value fully and accurately reflects market expectations, then any explanatory variables added to equation (8) should possess no explanatory power beyond that of intrinsic value. Table 6 reports regression results for the following augmented model of stock price on intrinsic value, size, growth, systematic risk and P/E ratio:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
As before, [P.sub.it] = stock price, and [V.sub.it] = intrinsic value. In order to assess the robustness of the results, control variables for size, growth and beta are included in several model permutations: [MC.sub.it] = log of market capitalization, [G.sub.it] = projected three-five year growth in revenue and [[beta].sub.it] = beta. [P/E.sub.it] = the ratio of stock price to earnings per share. [D.sub.it] is a dummy variable indicating Timeliness rank. This is explained further below.
Model 1 in Table 6 corresponds to the baseline regression of stock price on intrinsic value as in equation (8). In Model 2, the coefficient on P/E [[alpha].sub.5] = .050 is significantly positive at better than 1%, which points to staleness in analyst forecasts. To assess the location of the staleness across Timeliness ranks, Model 3 contains a dummy variable [D.sub.it], which equals zero for Rank 3 stocks and one for all other stocks. In Model 3, the coefficient on P/E among Rank 3 stocks is [[alpha].sub.5] = .042, which is smaller than in Model 2 but still significant. More importantly, the coefficient [[alpha].sub.6] = .020 is significantly positive, which suggests that the staleness in analyst forecasts is more substantial among stocks above or below average Timeliness. In Model 4, the dummy variable [D.sub.it] equals zero for stocks ranked 2, 3 or 4 and one for stocks ranked 1 or 5. The coefficient [[alpha].sub.6] = .066 is significantly positive and more than three times its magnitude in Model 3. These results suggest an increasing degree of staleness in analyst forecasts as Timeliness is farther above or below average. Among stocks above (below) average Timeliness, analyst forecasts appear to lag more optimistic (pessimistic) market expectations. As check on the robustness of these results, Models 5-7 include variables to control for size, growth and beta. The coefficients [[alpha].sub.5] and [[alpha].sub.6] and their statistical significance are little changed from inclusion of the controls. Hence, the conclusion appears robust that Timeliness proxies for the magnitude and direction of divergence of analyst forecasts and market expectations.
MSE DECOMPOSITION BY TIMELINESS
Table 7 reports results of the MSE decomposition by Timeliness. If equivalence prevails among average Timeliness stocks only, the MSE should be smallest for Timeliness 3 stocks. Table 7 shows, however, that MSE is generally increasing in Timeliness. It is smallest among stocks ranked 4 and 5 (54.90 and 59.39, respectively), increases modestly to 62.38 for Rank 3 stocks and increases rather dramatically among stocks ranked 2 (73.20) and stocks ranked 1 (120.91). These results alone suggest that divergent expectations primarily characterize stocks above average Timeliness. Results from the decomposition, however, paint a different picture. Among Rank 3 stocks, virtually all valuation error (99.93%) is attributable to random noise. Only 0.08% can be attributed to bias and inefficiency. These results provide strong support for the view that among stocks ranked 3, analyst forecasts align closely with market expectations. Among stocks above or below average Timeliness, the proportion of valuation error due to bias increases somewhat dramatically and very nearly symmetrically. These proportions are 4.99% and 4.87%, respectively, among stocks ranked 2 and 4, and they increase to 20.68% and 17.78%, respectively, among stocks ranked 1 and 5. These results clearly suggest that the staleness of analyst forecasts, and hence their divergence from market expectations, becomes more substantial with Timeliness ranks farther from average.
QUARTERLY EARNINGS FORECAST REVISIONS
VL quarterly earnings forecast revisions provide corroborating evidence that Timeliness proxies for staleness in analyst forecasts. If analyst forecasts lag more optimistic prospects for Rank 1 and 2 stocks and more pessimistic prospects for Rank 4 and 5 stocks, then VL earnings forecast revisions should be increasing in Timeliness. That is, conditional on prior-quarter forecast error, quarterly earnings forecast revisions will be most negative for Rank 5 stocks increasing to most positive for Rank 1 stocks. Table 8 reports VL mean quarterly earnings forecast revisions by Timeliness and by prior-quarter forecast error. Each table entry shows the mean revision and the number of revisions in each intersection of Timeliness and prior-quarter forecast error. For example, there are 1,237 instances of prior-quarter forecast error in the range of $0.00 to $0.05 among Rank 1 stocks. The mean revision among these stocks of $0,010 is statistically greater than zero at the .01 level of significance. Among 442 Rank 5 stocks in the same stratum of prior-quarter forecast error, the mean revision of -$0,057 is significantly less than zero. The mean difference between Rank 1 and Rank 5 stocks in this stratum of prior-quarter forecast error is statistically significant at the .01 level. This same clear pattern emerges in each stratum of prior-quarter forecast error; viz., mean forecast revision increases monotonically in Timeliness, and mean differences between Rank 1 and Rank 3, Rank 1 and Rank 5 and Rank 3 and Rank 5 are all statistically significant at the .01 level.
Summary and Conclusions
Research utilizing implied expected returns and equity valuation maintains assumptions of equivalence between analyst forecasts and market expectations and between the valuation models chosen by researchers and the process actually employed in the pricing of common stock by investors. The research question herein is whether, and under what conditions, equivalence holds between VL analyst forecasts and market expectations. I compute intrinsic values using VL forecasts in a residual income valuation model, and I investigate the valuation errors. Valuation errors arise from violation of one or both of the maintained hypotheses. Either analyst forecasts do not equal market expectations, or the valuation model chosen does not represent the process by which stocks are valued, or both. Prior research establishes that residual income valuation dominates other valuation models in terms of predictive power for stock price, so I assume that this model most closely represents the process by which stocks are priced. As a result, valuation errors can be attributed to divergence of analyst forecasts from market expectations. I argue that this divergence arises from stale analyst forecasts and that this staleness is captured in the VL Timeliness ranks, i.e., that Timeliness contains information about the magnitude and direction of divergence between analyst forecasts and market expectations. Hence, I investigate valuation errors for evidence of systematic patterns of bias and accuracy across Timeliness ranks. The stocks for which analyst forecasts are most likely closest to market expectations are those of average Timeliness (rank = 3), so that for these stocks, intrinsic value is a relatively unbiased predictor of stock price. Stocks above average Timeliness (rank = 1,2) are those for which market expectations are more optimistic than analyst forecasts, so that intrinsic value is downwardly biased. Stocks below average Timeliness (rank = 4, 5) are those for which market expectations are more pessimistic than analyst forecasts, so that intrinsic value is upwardly biased.
Empirical results indicate that Timeliness reflects the magnitude and direction of divergence between analyst forecasts and market expectations. Among Rank 3 stocks, valuation errors are generally not statistically different from zero, and regression-based tests and the decomposition of mean square error both indicate that intrinsic value is an unbiased predictor of stock price. Among Ranks 1 and 2 (4 and 5), valuation errors show that intrinsic value provides downwardly (upwardly) biased stock price predictions. In addition, regression-based tests and the decomposition of mean square error indicate the presence of prediction bias, that the magnitude of this bias increases as Timeliness rank is farther from average and that analysts' stale forecasts lag more optimistic (pessimistic) market expectations among stocks above (below) average Timeliness.
The equivalence of analyst forecasts and investor expectations underpins various research streams involving implied returns and computation of intrinsic values. Yet, to my knowledge, the question of whether and under what conditions this equivalence holds has not been addressed previously. The issue is important because the interpretation of empirical results is confounded when analyst forecasts and market expectations diverge, and the problem is likely widespread. Because stock prices respond to new value-relevant information more quickly than analysts can update their forecasts, it is almost certainly the case that at any point some proportion of analyst forecasts are stale, i.e., they fail to incorporate all relevant information about firms' future prospects. The consequence is that the implied returns and valuations calculated from these forecasts contain measurement error that impacts empirical results. For example, when a sample contains some stocks for which analyst forecasts are more optimistic than market expectations and some for which analyst forecasts are more pessimistic, the coefficients in a regression of implied returns on stock-price based variables (e.g., book-to-market value) are biased away from zero. This is the case even if equivalence holds on average in the sample.
My results also confirm prior findings that residual income valuation with VL forecasts and VL Target Price in the terminal value expression dominates other valuation approaches heretofore used by researchers. Both Courteau et al. (2001) and Heinrichs et al. (2013) demonstrate clearly that residual income valuation outperforms dividend and free cash flow models in terms of their predictive power for stock prices. On the basis of these authors' findings, we may conclude that residual income valuation, among available alternatives, most closely represents the process by which stocks are actually priced by investors. My results support and refine this conclusion by demonstrating that under certain conditions residual income valuation with VL forecasts and VL Target Price in the terminal value expression produces unbiased stock price predictions that possess very high explanatory power. These conditions are (a) that the measurement of residual income includes the effect of non-dividend capital flows and other comprehensive income and (b) that the set of stocks is limited to those for which analyst forecasts are at least approximately equivalent to market expectations.
Palumbo/Donahue School of Business, Duquesne University
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TABLE 1 Sample Characteristics Summary statistics on N = 17,261 firm-year observations for 1,960 unique firms from 1988 through 2012. Extreme .5% tails of the distribution of valuation error ([e.sub.it]) are deleted. Intrinsic value ([V.sub.it]) is calculated by residual income valuation as per equation (4). All data are from the Value Line Earnings and Projections File except the 10-year T-Note yield and the Market Risk Premium, which are obtained from Federal Reserve release H.15. The Market Risk Premium is calculated by adding .04 to the yield spread between Moody's Baa Corporate Bond average and the 10-year Treasury Note. Min Median Mean Max Stock Price ([P.sub.it]) 0.59 22.23 27.53 907.98 Intrinsic Value ([V.sub.it]) -34.00 22.13 27.08 905.84 Market Capitalization ($ millions) 0.79 1,682 7,999 483,337 P/E 1.50 17.20 20.26 79.90 P/B 0.07 2.19 3.39 294.8 Beta 0.05 1.05 1.08 3.00 Cost of Equity 0.056 0.119 0.122 0.309 10-yr Treasury Note Yield 0.020 0.048 0.053 0.091 Market Risk Premium 0.053 0.063 0.064 0.096 TABLE 2 Valuation Error Statistics Valuation error equals stock price minus intrinsic value: [e.sub.it] = [P.sub.it] - [V.sub.it]. Intrinsic value is calculated by equation (4). Extreme .5% tails of the distribution of valuation error ([e.sub.it]) are deleted. In Panel A, results are shown in terms of dollars per share. In Panel B, results are shown for valuation errors scaled by intrinsic value ([V.sub.it]) (#) indicates statistical significance at .01 level in 2-tailed t-test of [H.sub.0]: mean = 0 (^) indicates statistical significance at .01 level in sign test of [H.sub.0]: median = 0 Panel A Mean Median $ per share N [e.sub.it] [e.sub.it] 1988 582 -0.19 -0.26 1989 560 0.92 (#) 0.37 (^) 1990 535 0.77 (#) 0.27 (^) 1991 563 -2.58 (#) -1.85 (^) 1992 578 0.02 -0.11 1993 595 -0.02 -0.01 1994 579 1.25 (#) 0.65 (^) 1995 598 0.50 0.18 1996 618 0.34 0.07 1997 629 3.22 (#) 2.22 (^) 1998 644 1.64 (#) 0.94 (^) 1999 679 1.27 (#) -0.17 2000 708 0.87 -1.01 (^) 2001 690 -1.27 (#) -1.26 (^) 2002 738 0.05 0.10 2003 778 -2.81 (#) -2.15 (^) 2004 788 3.08 (#) 2.76 (^) 2005 787 2.11 (#) 1.75 (^) 2006 770 3.83 (#) 2.93 (^) 2007 779 3.68 (#) 3.07 (^) 2008 778 -0.30 -1.35 (^) 2009 791 -6.97 (#) -5.82 (^) 2010 830 -0.13 -0.09 2011 829 3.51 (#) 2.05 (^) 2012 835 -1.52 (#) -1.73 (^) 1988-2012 17261 0.45 (#) 0.09 Mean Median [absolute [absolute value of value of $ per share % Pos [e.sub.it]] [e.sub.it]] 1988 45% 3.25 1.95 1989 59 3.19 1.73 1990 56 3.37 1.89 1991 27 4.34 3.02 1992 49 4.23 2.75 1993 50 4.11 2.43 1994 57 4.07 2.53 1995 54 3.59 2.22 1996 51 4.10 2.64 1997 72 5.32 3.56 1998 56 5.44 3.70 1999 49 6.66 4.38 2000 44 7.42 5.21 2001 41 6.33 4.29 2002 51 4.90 3.37 2003 30 4.92 3.24 2004 72 5.46 4.10 2005 63 5.35 3.90 2006 66 7.04 4.87 2007 70 6.39 4.97 2008 41 7.31 5.17 2009 16 8.47 6.51 2010 49 5.96 3.78 2011 62 7.85 5.16 2012 41 7.49 5.05 1988-2012 51% 5.64 3.64 Panel B Mean Median [absolute [absolute Mean Median value of value of Scaled by [e.sub.it]/ [e.sub.it]/ [e.sub.it]/ [e.sub.it]/ [V.sub.it] N [V.sub.it] [V.sub.it] V.sub.it]] V.sub.it]] 1988 579 -0.02 -0.03 0.18 0.13 1989 559 0.05 (#) 0.03 (^) 0.17 0.12 1990 537 0.04 (#) 0.03 (^) 0.17 0.13 1991 563 -0.14 (#) -0.15 (^) 0.23 0.20 1992 581 0.02 -0.01 0.21 0.17 1993 597 0.02 0.00 0.22 0.17 1994 584 0.06 (#) 0.05 (^) 0.21 0.16 1995 598 0.03 0.01 0.18 0.14 1996 620 0.01 0.00 0.18 0.15 1997 633 0.15 (#) 0.13 (^) 0.24 0.18 1998 646 0.06 (#) 0.05 (^) 0.21 0.16 1999 679 0.04 (#) -0.01 0.26 0.21 2000 707 0.01 -0.06 (^) 0.28 0.23 2001 688 -0.04 (#) -0.06 (^) 0.23 0.19 2002 740 0.03 (#) 0.01 0.22 0.17 2003 766 -0.10 (#) -0.12 (^) 0.23 0.19 2004 778 0.19 (#) 0.13 (^) 0.26 0.18 2005 785 0.11 (#) 0.07 (^) 0.23 0.17 2006 765 0.17 (#) 0.12 (^) 0.27 0.19 2007 779 0.15 (#) 0.12 (^) 0.24 0.17 2008 782 -0.01 -0.05 (^) 0.24 0.18 2009 781 -0.23 (#) -0.27 (^) 0.30 0.29 2010 831 0.05 (#) 0.00 0.23 0.17 2011 840 0.15 (#) 0.10 (^) 0.28 0.21 2012 843 -0.04 (#) -0.07 (^) 0.22 0.19 1988-2012 17261 0.032 (#) 0.006 0.23 0.18 TABLE 3 Regression Results of Stock Price on Intrinsic Value The regression model is given by equation (8): [P.sub.it] = [[alpha].sub.0] + [[alpha].sub.1][V.sub.it] + [[epsilon].sub.it]. The table reports number of observations, adjusted R-square, root mean square error, estimated intercept and slope by year and for the entire sample 1988-2012. (#) indicates statistical significance at .01 level for [H.sub.0]: [[alpha].sub.0] = 0 (^) indicates statistical significance at .01 level for [H.sub.0]: [[alpha].sub.1] = 1 Year N [R.sup.2] RMSE [[??].sub.0] [[??].sub.1] 1988 582 0.95 5.37 -0.81 (#) 1.03 (^) 1989 560 0.95 5.32 -0.25 1.06 (^) 1990 535 0.96 5.48 -0.89 (#) 1.08 (^) 1991 563 0.96 5.78 -2.32 (#) 0.99 1992 578 0.89 6.22 1.02 0.95 (^) 1993 595 0.97 6.18 -0.88 (#) 1.04 (^) 1994 579 0.93 5.94 -0.83 1.09 (^) 1995 598 0.91 5.62 -0.21 1.03 (^) 1996 618 0.90 6.45 -0.75 1.05 (^) 1997 629 0.90 6.79 0.42 1.12 (^) 1998 644 0.82 7.72 0.40 1.05 (^) 1999 679 0.76 9.54 -0.53 1,07 (^) 2000 708 0.74 10.54 -0.48 1.05 2001 690 0.93 8.76 -0.65 0.98 2002 738 0.80 6.99 1.79 (#) 0.93 (^) 2003 778 0.95 6.57 -2.96 (#) 1.01 2004 788 0.86 6.73 4.93 (#) 0.93 (^) 2005 787 0.96 7.11 2.35 (#) 0.99 2006 770 0.93 8.96 3.43 (#) 1.01 2007 779 0.91 7.71 3.86 (#) 0.99 2008 778 0.91 10.07 0.76 0.97 (^) 2009 791 0.83 7.77 -1.48 (#) 0.81 (^) 2010 830 0.84 8.57 2.72 (#) 0.90 (^) 2011 829 0.83 10.65 3.98 (#) 0.99 2012 835 0.91 10.38 -0.63 0.98 1988-2012 17261 0.90 8.32 0.55 (#) 0.996 TABLE 4 Valuation Errors by Timeliness Valuation error equals stock price minus intrinsic value: [e.sub.it] = [P.sub.it] - [V.sub.it]. Intrinsic value [V.sub.it] is calculated by equation (4). Panel A presents results for valuation errors in terms of dollars per share. Panel B presents results for errors scaled by intrinsic value. In Panel A, the extreme .5% tails of the distribution of valuation error ([e.sub.it]) are deleted. In Panel B, the extreme .5% tails of the distribution of scaled valuation error ([e.sub.it]/[V.sub.it]) are deleted, (#) indicates statistical significance at .01 level in 2-tailed t-test of [H.sub.0]: mean = 0 (^) indicates statistical significance at .01 level in sign test of [H.sub.0]: median = 0 Panel A Mean Median Timeliness N [e.sub.it] [e.sub.it] 1 1081 5.00 (#) 3.33 (^) 2 3435 1.91 (#) 1.23 (^) 3 8371 0.21 -0.07 4 2974 -1.64 (#) -1.29 (^) 5 1024 -3.25 (#) -2.50 (^) 16885 Mean Median [absolute [absolute value of value of Timeliness % Pos [e.sub.it]] [e.sub.it]] 1 75% 7.55 4.91 2 62 5.85 3.78 3 49 5.32 3.39 4 37 5.15 3.44 5 31 5.47 4.08 Panel B Mean Median [e.sub.it]/ [e.sub.it]/ Timeliness N [V.sub.it] [V.sub.it] 1 1088 0.227 (#) 0.190 (^) 2 3418 0.106 (#) 0.070 (^) 3 8340 0.018 (#) -0.005 4 2917 -0.061 (#) -0.074 (^) 5 977 -0.125 (#) -0.145 (^) 16740 Mean Median [absolute [absolute value of value of [e.sub.it]/ [e.sub.it]/ Timeliness % Pos [V.sub.it]] [V.sub.it]] 1 7.55 4.91 2 5.85 3.78 3 5.32 3.39 4 5.15 3.44 5 5.47 4.08 TABLE 5 Regression Results of Stock Price on Intrinsic Value by Timeliness The regression model is given by equation (8): [P.sub.it] = [[alpha].sub.0] + [[alpha].sub.1][V.sub.it] + [[epsilon].sub.it]. The table reports number of observations, adjusted R-square, root mean square error, estimated intercept and slope by Timeliness over the entire sample period 1988-2012. (#) indicates statistical significance at .01 level for [H.sub.0]: [[alpha].sub.0] = 0 (^) indicates statistical significance at .01 level for [H.sub.0]: [[alpha].sub.1] = 1 Timeliness N [R.sup.2] RMSE [[??].sub.0] [[??].sub.1] 1 1081 0.87 9.74 3.77 (#) 1.044 (^) 2 3414 0.91 8.34 2.03 (#) 0.996 3 8314 0.92 7.90 0.27 0.998 4 2935 0.85 7.18 -0.42 0.952 (^) 5 999 0.93 6.82 -1.90 (#) 0.944 (^) 16743 TABLE 6 Regression Results of Stock Price on Intrinsic Value and P/E Ratio The full model is given by equation (10): [P.sub.it] = [[alpha].sub.0] + [[alpha].sub.1] [V.sub.it] + [[alpha].sub.2][MC.sub.it] + [[alpha].sub.3][G.sub.it] + [[alpha].sub.4][[beta].sub.it] + [[alpha].sub.5] [P/E.sub.it] + [[alpha].sub.6] ([D.sub.it] x [P/E.sub.it]) + [e.sub.it] [P.sub.it] = stock price. [V.sub.it] = intrinsic value. [MC.sub.it] = log of market capitalization. [G.sub.it] = projected three-to-five year growth in revenue. [[beta].sub.it] = beta. [P/E.sub.it] = the ratio of stock price to earnings per share. In models 3 and 6, [D.sub.it] = 0 for Timeliness 3 and 1 for Timeliness 1,2,4 and 5. In models 4 and 7, [D.sub.it] = 0 for Timeliness 2-4 and 1 for Timeliness 1 and 5. The table reports the estimated parameters over the sample period 1988-2012, number of observations, adjusted R-square and root mean square error. Reported below are the statistics for models one through seven, (#) indicates statistical significance at .01 level for [H.sub.0]: [[alpha].sub.1] = 0, k = 0, 2, ... ,6 (^) indicates statistical significance at .01 level for [H.sub.0]: [[alpha].sub.0] = 1 Models 1 2 3 4 [[alpha].sub.0] 0.610 (#) -0.476 (#) -0.511 (#) -0.559 (#) [[alpha].sub.1] 0.9952 (^) 0.9943 0.9945 0.9946 [[alpha].sub.2] [[alpha].sub.3] [[alpha].sub.4] [[alpha].sub.5] 0.050 (#) 0.042 (#) 0.045 (#) [[alpha].sub.6] 0.020 (#) 0.066 (#) N 15901 15901 15901 15901 [R.sup.2] 0.897 0.899 0.899 0.900 RMSE 8.137 8.036 8.030 8.012 Models 5 6 7 [[alpha].sub.0] -9.987 (#) -9.944 (#) -9.914 (#) [[alpha].sub.1] 0.9809 (^) 0.9812 (^) 0.9819 (^) [[alpha].sub.2] 0.918 (#) 0.917 (#) 0.912 (#) [[alpha].sub.3] -8.242 (#) -8.543 (#) -9.346 (#) [[alpha].sub.4] 3.409 (#) 3.357 (#) 3.357 (#) [[alpha].sub.5] 0.046 (#) 0.038 (#) 0.042 (#) [[alpha].sub.6] 0.018 (#) 0.067 (#) N 15901 15901 15901 [R.sup.2] 0.904 0.904 0.905 RMSE 7.831 7.825 7.806 TABLE 7 Mean Square Error Decomposition by Timeliness The MSE decomposition utilizes regression results from equation (8). The decomposition into Bias, Inefficiency and Noise is given by equation (9): MSE = [summation] [e.sup.2.sub.it]/N = [([bar.P] - [bar.V]).sup.2] + [(1 - [[??].sub.1]).sup.2] [S.sup.2.sub.V] + (1 - [R.sup.2]) [S.sup.2.sub.P] Valuation error [e.sub.it] is given by equation (7); it is not the regression residual. [bar.P] and [bar.V], respectively, denote sample means of stock price and intrinsic value. [S.sup.2.sub.P] and [S.sup.2.sub.V] denote sample variances of stock price and intrinsic value, respectively. [[??].sub.1] denotes the estimated slope coefficient from regressing stock price on intrinsic value, and [R.sup.2] denotes the unadjusted R-square of that regression. Timeliness MSE Bias Inefficiency Noise 1 120.91 25.00 1.073 94.81 2 73.20 3.66 0.014 69.53 3 62.38 0.04 0.004 62.33 4 54.90 2.67 0.756 51.47 5 59.39 10.56 2.329 46.49 Full Sample 69.41 0.21 0.008 69.20 Timeliness % Bias % Inefficiency % Noise 1 20.68 0.89 78.41 2 4.99 0.02 94.99 3 0.07 0.01 99.93 4 4.87 1.38 93.75 5 17.78 3.92 78.28 Full Sample 0.30 0.01 99.69 TABLE 8 Mean Quarterly Earnings Forecast Revision by Timeliness and Prior-Quarter Forecast Error For quarter t, the forecast revision equals the change in forecast from quarter t-1 to quarter t: [DELTA][F.sub.t] = [F.sup.t.sub.t] - [F.sup.t-1.sub.t]. [F.sup.t.sub.t] is the latest forecast of quarter t earnings. [F.sup.t-1.sub.t] is the forecast of quarter t earnings as of quarter t-1. [F.sup.t-1.sub.t] is issued before the release of quarter t-1 earnings and thus before the prior-quarter forecast error is known. Prior-quarter forecast error is F[E.sub.t-1] = [E.sub.t-1] - [F.sup.t-1.sub.t-1], where [E.sub.t-1] is prior-quarter earnings. Forecast error and forecast revision are in dollars per share. Table entries show the mean forecast revision and number of revisions in each intersection of Timeliness and prior-quarter forecast error. Within each stratum of prior-quarter forecast error, mean differences between Rank 1 and Rank 3, Rank 1 and Rank 5 and Rank 3 and Rank 5 are all statistically significant at the .01 level of significance. * denotes statistical significance at 1% in two-tailed t-test of [H.sub.0]: mean revision = 0 Timeliness F[E.sub.t-1] 1 2 3 4 5 N <-.10 -0.013 * -0.018 * -0.062 * -0.116 * -0.156 * 16353 542 1954 7287 4524 2046 (-.10, -.05) 0.009 -0.006 * -0.034 * -0.056 * -0.075 * 7916 279 1005 3794 2092 746 (-.05, 0) 0.002 -0.005 * -0.021 * -0.037 * -0.060 * 13535 453 1905 7230 3126 821 Zero 0.005 -0.003 -0.012 * -0.034 * -0.059 * 5058 227 985 2762 864 220 (0, .05) 0.010 * 0.004 * -0.008 * -0.027 * -0.057 * 16862 1237 4023 8916 2244 442 (.05, .10) 0.029 * 0.017 * 0.000 -0.036 * -0.064 * 6707 610 1643 3518 775 161 > .10 0.070 * 0.048 * 0.004 -0.031 * -0.107 * 8165 816 2261 4124 822 142 N 4164 13776 37631 14447 4578 74596
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|Publication:||Quarterly Journal of Finance and Accounting|
|Date:||Jan 1, 2015|
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