Further Evidence on the Relation between Analysts’ Forecast Dispersion and Stock Returns

We find that 1 changes in dispersion capture primarily changes in consensus or changes in information asymmetry whereas the level of dispersion captures primarily the level of unsystemat

Further Evidence on the Relation between Analysts’ Forecast Dispersion and Stock Returns

Orie E BarronAssociate ProfessorPennsylvania State UniversitySmeal College of Business AdministrationUniversity Park, PA 16802-1912

814-863-3230

Mary StanfordAssociate Professor, Nobel Faculty Fellow

Texas Christian UniversityM.J Neeley School of BusinessFort Worth, TX 76129

Yong YuPennsylvania State UniversitySmeal College of Business AdministrationUniversity Park, PA 16802-1912

September 29, 2005

(This is a preliminary revised draft of a manuscript requiring a major revision for publication Please

do not quote without permission)

We gratefully acknowledge the contribution of I/B/E/S International Inc for providing earnings per share forecast data, available through the Institutional Brokers' Estimate System These data have been provided as part of a broad academic program to encourage earnings expectation research

For their helpful comments we thank Richard Schnieble and workshop participants at the CUNY Baruchand Penn State University

Previous research finds a negative association between the level of dispersion in analysts’

earnings forecasts and subsequent stock returns This finding can be consistent with either unsystematic uncertainty that is priced positively because it increases a firm’s option value (Johnson, 2004) or

overpricing due to a lack` of consensus among investors that limits the short sales of the most

pessimistic traders (Diether et al, 2002) We use the Barron et al (1998) framework to measure the two theoretical variables that can both cause dispersion and cause it to be priced These are (1) uncertainty

or (2) a lack of consensus (i.e., a relatively high level of private information among analysts) A high level of uncertainty may be priced positively or negatively depending on whether it is systematic or unsystematic In contrast, the private information that causes a lack of consensus is likely to be priced negatively because it reflects information asymmetry and a higher cost of capital (Botosan, Plumlee, andXie 2004)

The evidence we report helps distinguish between the different explanations for forecast

dispersion and what drives its relation to stock returns We find that (1) changes in dispersion capture primarily changes in consensus (or changes in information asymmetry) whereas the level of dispersion captures primarily the level of unsystematic uncertainty (and not the level of information asymmetry), and (2) the uncertainty in forecast dispersion is negatively associated with future stock returns, but the lack of consensus (or information asymmetry) is positively associated with future stock returns These findings support Johnson’s option value explanation but do not support Diether et al.’s overpricing explanation because lower levels of consensus do not lead to lower future returns

In addition, our evidence that levels and changes in dispersion reflect fundamentally different constructs reconciles the evidence on changes in dispersion presented by L’Her and Suret (1996) with the conclusions of Diether et al.(2002) and Johnson (2004), both of which are inconsistent with L’Her and Suret’s argument that forecast dispersion represents uncertainty that is priced negatively We find that increases in forecast dispersion coincide with more negative stock returns because there is less consensus (and thus more information asymmetry that adversely affects firms’ cost of capital)

1 Introduction

Prior research posits opposing explanations for the empirically documented relation between dispersion in analysts’ forecasts and stock returns Diether et al (2002) argue that the negative relation between dispersion levels and future returns is due to overpricing resulting from investor disagreement and limits on short sales that lead to optimistic current stock prices and lower future stock returns Johnson (2004) presents a model suggesting that this negative relation may also be due to the uncertainty (or risk) reflected in dispersion which, although unsystematic in nature, increases the option value

of the firm and leads to lower future returns Further clouding the issue, Deither et al.’s arguments suggest a positive relation between changes in dispersion and

contemporaneous stock returns This is contrary to evidence presented by L’Her and Suret (1996) that increases in forecast dispersion are negatively associated with stock returns

Using the Barron, Kim, Lim, and Stevens (1998) (hereafter BKLS) decomposition

of dispersion into uncertainty and lack of consensus we are able to distinguish between these opposing explanations Our evidence provides increased empirical support for Johnson’s conclusion that the level of dispersion analysts’ forecasts reflects unsystematic risk (uncertainty) In addition, consistent with L’Her and Suret’s (1996) findings, we find

a negative relation between changes in dispersions and stock returns, which we show is likely caused by increased information asymmetry between informed and uninformed investors reflected in a decrease in consensus

Diether et al (2002) argue that when investors disagree, limitations on trading,

pessimists This suggests a negative relation between dispersion levels and future stock returns when the overpricing is corrected and a positive relation between changes in dispersions and stock returns In addition, based on tests explaining dispersion with several measures of risk Diether et al conclude “…our results strongly reject the

interpretation of dispersion in analysts’ forecasts as a measure of risk.” (p 2115)

By contrast, Johnson (2004) provides a pricing model in which the negative relation between dispersion levels and stock returns may be due to a form of information risk (uncertainty) where dispersion reflects nonsystematic risk (idiosyncratic uncertainty) that increases the option value of the firm and lowers expected future returns However,

as the author notes, his model is not inconsistent with Diether et al.’s overpricing

argument, i.e., both may explain the relation between dispersion and returns

We begin by empirically separating dispersion into its theoretical components Theoretically, in order for dispersion in analysts’ forecasts to exist there must be both (1) some uncertainty regarding future performance and (2) some lack of consensus due to thediversity of private information (Barry and Jennings 1992; Abarbanell, Lanen, and Verrecchia 1995; Barron, et al 1998) Thus, it is unclear the degree to which forecast dispersion reflects uncertainty or a lack of consensus Finding that dispersion levels reflect uncertainty rather than consensus would provide some support for Johnson’s (2004) hypothesis that dispersion levels reflect information risk Finding that changes in dispersion reflect changes in consensus would serve to increase understanding of the findings of L’Her and Suret (1996) and to reconcile these findings with those of Johnson (2004) and Diether et al (2002)

We provide evidence on whether dispersion in analysts’ forecasts reflects

uncertainty or a lack of consensus using the BKLS empirical proxies for these theoretical constructs We examine both the level of dispersion prior to an earnings announcement and the change in dispersion around earnings announcements.1 We find that the level of pre-announcement forecast dispersion reflects primarily uncertainty rather than a lack of consensus By contrast, changes in forecast dispersion reflect primarily changes in consensus rather than changes in uncertainty

Our finding that levels of dispersion reflect uncertainty is consistent with

Johnson’s (2004) conclusion that the negative relation between dispersion levels and future stock returns is driven by uncertainty In further analysis, we examine market data

to determine whether the level of forecast dispersion reflects primarily systematic or unsystematic uncertainty about future performance We show that higher levels

dispersion are associated with higher idiosyncratic risk and lower future returns This combined evidence provides support for Johnson’s argument that the negative relation between future returns and dispersion is due to investors’ unsystematic uncertainty and not overpricing

Although this evidence lends support to Johnson’s theory it does not completely rule out Deither et al.’s (2002) conclusion that the negative relation results from

overpricing due to investor disagreement However, Deither et al.’s argument that

dispersion is negatively associated with future returns implies a positive (negative) relation between consensus (lack of consensus) and future stock returns because

dispersion increases as consensus (lack of consensus) decreases (increases) In contrast

to the positive relation implied by the overpricing argument, we find a negative (positive)

relation between consensus (lack of consensus) and future returns This finding also increases understanding of L’Her and Suret’s (1996) evidence that increases in dispersion coincide with decreases in stock returns Specifically, we show that increases in forecast dispersion coincide with decreases in consensus that reflect information asymmetry between informed and uninformed investors and that increases in dispersion coincides with decreases in stock returns Because low consensus stocks have high information asymmetry, this is also consistent with the positive relation between information

asymmetry and the cost of equity capital hypothesized in Amihud and Mendelson (1986 and 1989), King et al (1990), Diamond and Verrecchia (1991), among others, and

documented in previous studies (see Barron et al 2005 for further discussion)

Uninformed investors demand a return premium to compensate for their risk of trading with privately informed investors This risk is not diversifiable since uninformed

investors are always at a disadvantage relative to informed investors (O’Hara 2003) and demand to be compensated with higher expected future returns

This evidence is of interest to accounting and finance researchers wishing to understand the relation between forecasts dispersion and stock returns For example, understanding that levels and changes in dispersion reflect different theoretical constructscan help researchers choose the appropriate proxy In addition, to the extent that

dispersion is easily measured by investors while the BKLS measures are more complex and cannot be measured ex-ante our evidence allows investors as well as researchers to more precisely interpret the meaning of levels versus changes in forecast dispersion.2

2 Determining what forecast dispersion reflects the most is important to for methodological reasons For example, over fifty empirical studies published in selected top tier accounting and finance journals use

analyst forecast dispersion as an empirical proxy for various firm characteristics Appendix 1 lists papers

published in The Accounting Review (15 papers), Journal of Accounting Research (11 papers), Journal of

The discussion proceeds as follows Section 2 describes our empirical proxies and research design as it relates to the strength of the associations between both levels of and changes in forecast dispersion, analysts’ uncertainty, and analysts’ lack of consensus (or diversity of information) Section 3 investigates the relation between the two

components of dispersion levels and future stock returns then provides evidence on the relation between dispersion levels and both systematic and unsystematic risk Section 4 reconciles our evidence on changes in dispersion with prior research Section 5 discusses robustness checks on the BKLS measures and alternate specifications Finally, section 6 contains our conclusions

2 Forecast Dispersion: Earnings Uncertainty or Lack of Consensus

BKLS show how one can measure the theoretical constructs uncertainty and

consensus by exploiting the fact that forecast dispersion and error in analysts’ forecasts

reflect these theoretical constructs differently The intuition underlying their results stemsfrom the fact that forecast dispersion and error differentially reflect error in analysts' common and idiosyncratic information The BKLS empirical proxies for consensus and uncertainty are:

DISPERSION = V(1- ) (1)

Where:

D = dispersion in analysts’ forecasts, i.e., the sample variance of the

individual forecasts (FCi ) around the mean forecast ( C F ),

FC

1

2 ( 1))

Accounting and Economics (7 papers), Journal of Finance (4 papers) , Journal of Financial Economics (5

V = Uncertainty, i.e., the mean of the squared differences between

individual analysts’ forecasts (FCi ) and reported earnings per

share (EPS) measured as





n i

FC

1

2)

From equation (1), dispersion is the product of uncertainty (V) and lack of consensus ρ) Thus, forecast dispersion is simultaneously determined by both uncertainty and lack

(1-of consensus

To understand the intuition for these measures it is helpful to consider the extreme

examples where CONSENSUS is zero or one and a large number of forecasts (n) exist as

described in Barron, Harris, and Stanford (2005) With a large number of forecast, the

difference between the mean forecast ( C F ) and realized earnings per share (EPS) only

reflects error due to common information because idiosyncratic error is averaged out of the mean When the mean forecast equals realized earnings CONSENSUS equals zero and D/V = 1 When this is true, the BKLS model suggests that forecasts are based entirely on private information because all forecast error is idiosyncratic The difference

between individual forecasts (FC i ) and the mean forecast ( C F ) reflects error due to

private information When all individual forecasts are exactly equal to the mean forecast CONSENSUS equals one and D/V = 0 When this is true, the BKLS model suggests that forecasts are based entirely on common information because all forecast error is

common Consistent with V reflecting overall uncertainty, if all forecasts exactly equal

3 The relations we report between dispersion, consensus, and uncertainty are not merely mechaincal

Theoretically, which component, V or (1-ρ), has more explainatory power for dispersion is, ex ante, not

clear (Barron et al 1998) Also see Section 3 for empirical evidence this relation is not merely mechanical

realized earnings then V is equal to zero, consistent with perfectly accurate information, i.e., no uncertainty.

We investigate both the level of dispersion prior to earnings announcements and the change in dispersion estimated around earnings announcements and non-

announcement dates Specifically, we estimate the following models and use a Vuong test

to compare the explanatory power of equation (3) and (4) to determine whether the level

of dispersion in analysts’ forecasts is more highly associated with lack of consensus or uncertainty prior to the earnings announcement Similarly, comparing the explanatory power of equations (5) and (6) tests whether the change in dispersion around earnings announcements is better explained by changes in consensus or changes in uncertainty

Log(D/P) = natural log of dispersion D scaled by the stock price P D is

pre-announcement dispersion in analysts’ forecasts measured as the variance of analysts’ earnings forecasts issued within 30 days prior to the earnings announcement;

Log(V/P) = natural log of overall uncertainty V scaled by the stock price P V is pre-announcement uncertainty estimated with equation (2) using forecasts issued within 30 days prior to the earnings announcement

Log(1-CONSENSUS) = natural log of one minus pre-announcement consensus (i.e.,lack of consensus) estimated with equation (1) using forecasts issued within 30 days prior to the earnings announcement

Δlog(V/P) = change in natural log of overall uncertainty V scaled by the stock price

P, estimated with equation (1) using forecasts issued within the 30-day

pre-announcement window and a 30-day post-pre-announcement window

Δlog(D/P) = change in the log-transformed dispersion D scaled by the stock price P, measured as the variance of the annual earnings forecast issued within the 30-day pre-announcement window and a 30-day post-announcement window;

Δlog(1-CONSENSUS) = change in the log-transformed lack of consensus (1 minus consensus), where consensus is estimated with equation (1) using forecasts issued within the 30-day pre-announcement window and a 30-day post-announcement window;

Reported results scale dispersion (D) and uncertainty (V) by the stock price (P) measured

at the end of the prior fiscal quarter We take the natural log of the variables for tworeasons: first, BKLS demonstrates that dispersion is equal to the product of uncertaintyand lack of consensus (i.e D=V(1- CONSENSUS)) Thus, it is natural to make thisrelation linear by taking the natural log; the second purpose is to mitigate skewnessproblems with dispersion and uncertainty. 4

2.1 Sample Selection and Empirical Results

The sample consists of quarterly and annual earnings per share forecasts from 1986 to

2003 Analysts’ earnings forecasts and actual earnings per share data are obtained from Institutional Brokers Estimate (I/B/E/S).5 Earnings announcement dates and other

financial data are obtained from the quarterly COMPUSTAT Primary, Supplementary, or Tertiary file We investigate one-quarter-ahead forecast and two-year-ahead forecasts The one-quarter-ahead sample consists of quarterly forecasts measured within 30 days before the current quarterly earnings announcement The two-year-ahead sample consists

of annual earnings forecasts measured within 30 days before the prior annual earnings announcement.6 To be included in the pre-announcement (levels) sample, two or more

4 Our results and inferences are the same when we do not scale by price and when we do not log transform the variables.

5 IBES forecasts and actual data are adjusted historically for stock splits and rounded to two decimals in the summary file and four decimals in the detail file This rounding will introduce measurement errors into our main variables, e.g., artificially reducing forecast dispersion (see Payne and Thomas 2003 for a detailed discussion) To avoid this problem, we conduct our analyses on the raw forecast data, unadjusted for stock splits.

6 All results and inferences are the same for a one-year-ahead sample We report the two-year sample to emphasize short-run versus long-run uncertainty and consensus.

individual analysts must have issued forecasts within a 30-day pre-announcement

window To be included in the change sample two or more individual analysts must have issued forecasts within a 30-day pre-announcement window and these same analysts musthave revised their forecast within a 30-day post-announcement window

Table 1 reports descriptive statistics and tests of the determinants of the level

analysts’ forecast dispersion measured for one-quarter- and two-year-ahead forecasts From panel A, the quarterly forecast sample consist of relatively large firms with a mean (median) market value of equity $6,073 ($1,450) million The annual forecast sample, although still large, exhibits a slightly wider range of firm size with a mean (median) market value of equity of $5,667 ($1,288) million (Panel B) With respect to the variables

of interest, both dispersion and uncertainty are less at the median for the quarterly

forecast sample (0.001 and 0.002) than for the annual forecast sample (0.011 and 0.157) This is consistent with dispersion and uncertainty increasing with the forecast time horizon Note that the standard deviation of the log-transformed uncertainty (D/V) is much smaller than the raw variable, e.g., 2.516 versus 85.535 for the quarterly forecasts Thus, in addition to being the correct empirical specification given the multiplicative relation between dispersion, consensus and uncertainty suggested by BKLS, the log transformation corrects for the fact that uncertainty is not naturally scaled Finally, consensus (), which ranges from zero to one, is lower for the quarterly sample, 0.530 versus 0.912, at the median This is consistent with analysts’ relying on common, i.e., public information for longer range forecasts

In Table 1, Vuong’s (1989) Z-statistic tests which independent variable exhibits a greater association with the dependent variable We use the procedures outlined in

Dechow (1994) to compute the Z-statistic; a positive (negative) Z-statistic indicates that lack of consensus (uncertainty) has a greater association with the dependent variable Panel A reports results of estimating equations (3) and (4) for the one-quarter ahead sample As expected, the coefficients on pre-announcement consensus (measured as lack

of consensus, 1-) and preannouncement uncertainty are both significantly positive The adjusted R2 for the model with preannouncement uncertainty is 55.30% while the

adjusted R2 for the model with prior consensus is 3.99% The Vuong’s Z-statistic is -68.92 ( = 0.001) indicating that the level of preannouncement uncertainty explains more of the variation in preannouncement dispersion levels than variation in consensus for quarterly forecasts Panel B leads to the same conclusion for the annual forecast sample Specifically, the adjusted R2 for the model with preannouncement uncertainty is 38.04% while the adjusted R2 for the model with prior lack of consensus is 7.20% The Vuong’s Z-statistic is -29.13 ( = 0.001) indicating that preannouncement uncertainty levels explain more of the variation in preannouncement dispersion levels than variation

in consensus for both long- and short-range forecasts

Table 2 reports descriptive statistics and tests of the determinants of changes in

analysts’ forecast dispersion around both quarterly and annual earnings announcements The requirement that the same analysts that provided a forecast prior to an earnings announcement revise that forecast within 30 days after the announcement results in a much smaller sample of relatively large firms Panel A describes the sample of quarterly earnings forecast updates around quarterly earnings announcements (n=10,150)

Dispersion, uncertainty and consensus all decline after quarterly earnings announcements

at both the mean and median Panels B describe the sample of annual earnings forecast

updates around annual earnings announcements (n=4,493) Consistent with the quarterly results, dispersion, uncertainty and consensus all decline after annual earnings

announcements at both the mean and median The decrease in consensus is consistent with that reported by Barron, Byard, and Kim (2002) They argue that consensus

declines because analysts have incentives to use their own private knowledge to create private information (or private interpretations) from earnings announcements (see also Kim and Verrecchia 1994; 1997 and Fischer and Verrecchia 1998)

Panel A of Table 2 also reports the results of estimating equations (5) and (6) for quarterly earnings announcement As expected, the coefficients on both changes in uncertainty and change in lack of consensus are significantly positive The adjusted R2 forthe model with change in uncertainty is 14.21% while the adjusted R2 for the model with change in lack of consensus is 51.79% The Vuong Z-statistic is 23.07 ( = 0.001), indicating that changes in analysts’ lack of consensus explain more of the variation in changes in dispersion in analysts’ forecasts than changes in uncertainty Panel B reports the results of estimating equations (5) and (6) for the annual earnings announcement The results are consistent with those in Panel A with changes in lack of consensus explaining approximately 74% of changes in dispersion

Overall, the evidence indicates that pre-announcement levels of dispersion in analysts’ forecasts proxy much more for pre-announcement uncertainty levels than for pre-announcement levels of lack of consensus, which supports the arguments in previous studies (e.g., Barron and Stuerke 1998; Johnson 2004) In contrast, changes in forecast dispersion around earnings announcements dates proxy much more for changes in

analysts’ lack consensus than for changes in uncertainty, which supports the conjectures

in Ziebart (1990)

Whether cross-sectional variation in forecast dispersion levels is driven primarily

by cross-sectional variation in uncertainty levels or variation in consensus levels was an empirical issue that, from an ex-ante perspective, could have gone either way Ex-post, the descriptive evidence that cross-sectional variation in uncertainty is relatively high compared to cross-sectional variation in consensus largely explains why cross-sectional variation in forecast dispersion levels is mostly attributable to variation in uncertainty levels For example, in panel A of table 1 cross sectional variation measured by the standard deviations of log transformed dispersion (D), uncertainty (V), and lack of consensus (1-) is 2.214, 2.516, and 1.717, respectively Cross-sectional variation in annual forecast sample is similar for dispersion and uncertainty but somewhat higher for lack of consensus

From an ex-ante perspective it was reasonable to expect that the evidence might differ for dispersion changes versus levels This is due to the nature of the information analysts’ acquire over time As a forecasted event approaches, analysts acquire

information that is either common (public) or private in nature (perhaps because it is complementary to publicly conveyed information) Although both types of information reduce uncertainty, private information decreases consensus while common information increases consensus This suggests that uncertainty levels trend steadily downward over time as the forecast event approaches due to the arrival of both types of information By contrast, consensus levels can increase for some firms and decrease for others depending

on the nature of the information analysts acquire during a particular time period As a

result, cross-sectional variation in changes in uncertainty may be quite small compared tocross-sectional variation in changes in consensus Descriptive statistics presented in table

2 are consistent with this In panel A of table 2, the cross-sectional variation in changes inuncertainty (standard deviation) is 1.391which is lower than the cross sectional variation

in changes in lack of consensus, 1.855 To underscore these statistics we conducted

supplementary analyses of the correlation between the relative position of individual

firms’ uncertainty within the distribution of all firms’ uncertainty before and after

earnings announcements This correlation is 0.93, which suggests that around earnings announcements changes in the relative positions of firms’ uncertainty is minimal In contrast this same correlation for lack of consensus is 0.37, suggesting there is a lot of jumbling in firms’ consensus around earnings announcements Thus, the cross sectional variation in both changes in analysts’ lack of consensus and changes in dispersion are greater around earnings announcements than the variation in changes in uncertainty Thissuggests why changes in consensus explain most of the cross-sectional variation in changes in dispersion

3 Evidence on Forecast Dispersion Levels: Revisiting Johnson (2004) and Diether et al (2002)

Evidence in the previous section shows that the level of dispersion in analysts’ forecasts primarily reflects uncertainty In this section we present three sets of analyses First, in order to differentiate the two competing explanations in Diether et al (2002) and Johnson (2004) we examine whether the negative relation between dispersion levels and stock returns is due to uncertainty or lack of consensus We do this by separately

examining the relation between returns and uncertainty and returns and consensus Next

we examine the relation between returns and each component of dispersion after

controlling for the other Finally, we investigate whether the uncertainty reflected in dispersion levels is systematic (non-diversifiable) or unsystematic (diversifiable)

3.1 Are stock returns associated with uncertainty or consensus?

In this section we replicate the analysis in Diether et al (2002) after replacing dispersion with it components, uncertainty and lack of consensus Diether et al argue thatinvestor disagreement and costly arbitrage lead to overpricing because current stock prices are optimistic and that when this is corrected prices fall resulting in a negative association between dispersions and future returns In contrast, Johnson (2004) poses an alternate explanation for the negative relation between forecasts dispersion and returns

He suggests that dispersion reflects information uncertainty (also referred to as parameterrisk) that is idiosyncratic in nature and that this risk is priced because it increases the option value of the firm Our evidence that the level of dispersion primarily reflects

uncertainty is consistent with Johnson’s arguments However, we cannot, ex ante, rule

out the possibility that the much smaller explanatory power of the consensus component drives the negative relation between dispersion and stock returns

In order to replicate Diether et al (2002)’s we follow their sample selection and portfolio analyses.7 We obtain stock returns from CRSP monthly stock file, and analysts’ annual forecasts from the unadjusted IBES database Each month we assign stocks to 5 quintiles based on dispersion in the previous month Dispersion is defined as the standarddeviation of earnings forecast scaled by the absolute value of the mean earnings forecast

If the mean earnings forecast is zero, then the stock is assigned to the highest dispersion

7 Following Diether et al (2002) the sample period is from January 1983 until December 2000.

category We calculate the monthly portfolio return for each quintile as the

equal-weighted average of the returns of all the stocks in the portfolio In untabulated analyses, the monthly return on the low minus high dispersion strategy is 0.74 percent (t=2.94), which is very similar to the 0.79 percent (t=2.88) reported in Diether et al (2002) We also find results very similar to Diether et al (2002) when we assign stocks based on size and then dispersion

To investigate whether uncertainty or consensus drives these results we repeat Diether et al.’s (2002) portfolio analyses replacing dispersion first with uncertainty (V) and then with CONSENSUS We scale uncertainty by the square of the mean forecast.8Table 3 reports the results of replicating Deither et al.’s analysis substituting uncertainty for forecast dispersion Consistent with Diether et al., we find a strong negative relation between uncertainty and future stock return The monthly return on the low minus high uncertainty strategy is 1.87 percent per month Also consistent with finding in Deither et al., this strategy is more profitable among smaller stocks with a return of 3.65 percent

Table 4 reports the results of replicating Deither et al.’s analysis substituting CONSENSUS for forecast dispersion Note that the analyses reported in Tables 1 and 2 explained dispersion with uncertainty and lack of consensus (1-) because dispersion increases as lack of consensus increases In Table 4 we use consensus ( because the negative relation between dispersion levels and future returns implies a positive

(negative) relation between consensus (lack of consensus) and future stock returns

8 We choose this scalar to be consistent with BKLS model: D=V(1-ρ) Note that BKLS dispersion D is the variance of analysts’ forecasts The dispersion defined in the portfolio tests is the standard deviation of analysts’ forecasts scaled by the absolute value of the mean forecast to be consistenet with Diether et al (2002) This defintion is equavalent to the BKLS dispersion D scaled by the square of the mean forecast

because dispersion increases as consensus (lack of consensus) decreases (increases) In contrast to the positive relation implied by the overpricing argument in Diether et al., we

find a negative (positive) relation between consensus (lack of consensus) and future

returns On average the stocks in the lowest consensus quintile outperform those in the highest consensus quintile by 1.33 percent per month.9

3.2 Further evidence on Dispersion Levels and Uncertainty

It is possible that negative (positive) relation between consensus (lack of

consensus) and future returns documented in Table 4 might be driven by the positive correlation between consensus and uncertainty To investigate this, in Panel A of Table 5, each month we assign stocks to one of five quintiles based on uncertainty in the previous month Then we rank stocks in each uncertainty quintile into five further quintiles based

on consensus in the previous month After controlling uncertainty, there is a negative relation between consensus and future returns In Panel B, each month we assign stocks

to one of five quintiles based on consensus in the previous month Then we rank stocks ineach consensus quintile into five further quintiles based on uncertainty in the previous month The negative relation between uncertainty and future stock returns still holds aftercontrolling for consensus

To provide further evidence in addition to the portfolio tests in Table 5, we

estimate Fama-MacBeth regressions to test the relation between one dispersion

component and future stock return after controlling for the other:

t t

In each month, all the stocks are assigned a quintile rank (1-5) based on uncertainty (V), lack of consensus (1-ρ) and size independently The cross section of monthly stock

returns (Ret ) is regressed on the quintile ranks of uncertainty ( t1 Rank _ V t) and

consensus (Rank_(1 t)) and size (Rank _ Size t) which are measured as of the

previous month Fama and Macbeth (1973) cross-sectional regressions are run every month for totally 216 continuous months from January 1983 till December 2000 T-statistics in parentheses are calculated using the coefficients from monthly regressions and also adjusted for autocorrelation using Newey-West standard errors Table 6 reports the results with and without controlling for size The coefficient on lack of consensus (disagreement) is significantly positive after controlling for uncertainty, which is

consistent with the portfolio results in Table 4 where returns decrease as consensus increases

In summary, interpretation of the evidence of a negative relation between

dispersions levels and stock returns is clouded by the fact that dispersion reflects both uncertainty and a lack of consensus We show that this negative relation is driven by uncertainty rather than lack of consensus This is an important finding because it allows

us to begin to differentiate between the two explanations for this relation suggested in the finance literature Evidence in Table 3 supports Johnson’s argument that the negative relation between dispersion and future returns is due to investors’ uncertainty Evidence

in Table 4 of a positive relation between lack of consensus and stock returns is

inconsistent with Diether etl al.’s overpricing explanation for the negative relation

between dispersion and returns Evidence in Tables 5 and 6 strengthens our conclusion

that the level of dispersion mainly reflects the level of uncertainty and that this

uncertainty drives the relation between dispersion and returns

3.3 Does forecast dispersion reflect systematic or unsystematic uncertainty/risk?

To provide further evidence on the relation between analysts’ forecast dispersion and firm risk, we explore how firms’ systematic risk and idiosyncratic risk explains the variation in dispersion We use market beta as a proxy for systematic risk and mean squared errors (MSE) - from the estimation of the market model as a proxy for

idiosyncratic risk Dispersion is measured as before, defined as the ratio of the standard deviation of analysts’ forecasts to the absolute value of the mean forecast for each firm month We estimate the market model to obtain beta and mean squared errors using a minimum of 36 month and a maximum of 60 months of return data prior to each firm month when we measure forecast dispersion To mitigate the high skewness problem in dispersion and mean squared errors, we log transform these two variables in our

regression Our sample covers all the firm months with valid earnings forecasts from IBES and monthly return data from CRSP and consists of 382,789 firm-month

observations

In Table 7, Panel A reports descriptive statistics for the variables used in the regression and Panel B reports correlations between these variables The variables, Dispersion and MSE, are highly skewed and the log transformation avoids this problem Dispersion is positively correlated with both beta and MSE (p-values < 0.001),

suggesting that both systematic and idiosyncratic firm risks may be determinants of dispersion In Panel B, we regress log-transformed dispersion on beta and log-

transformed MSE separately (Model 1 and 2) and then on both variables together (Model

3) We find that beta, MSE and both variables explain 2.5%, 5.8% and 5.9% of the variation in dispersion respectively The small change in adjusted R2, from 5.8% to 5.9%,when MSE is added to the model with beta indicates that systematic risk has very little incremental explanatory power for the variation in dispersion.10

Finally, we perform a robustness check using a rank regression framework Each month, we rank all the firms into deciles based on dispersion, beta and MSE

independently We regress decile ranks of dispersion on decile ranks of beta and MSE The results are reported in Panel C Again, we find that beta provides almost no

incremental explanatory power for dispersion over MSE

In summary, the evidence in Tables 3 through 6 suggests that the negative relationbetween dispersion levels and stock returns documented in prior studies reflects a relationbetween uncertainty levels and returns rather than a lack of consensus The evidence in Table 7 suggests that this uncertainty is idiosyncratic firm risk rather than systematic risk.Both results support the conclusions in Johnson (2004)

4 Reconciling evidence on Changes versus Levels of Dispersion

This section reconciles our evidence on changes in dispersion with prior research

In addition, because we find that changes in dispersion reflect analysts’ lack of consensus rather than uncertainty we discuss the link between consensus and stock prices

L’Her and Suret (1996) document a negative relation between changes in

dispersion and contemporaneous stock returns (as dispersion increases stock returns

10 To further gauge the economic significance of the relation between dispersions and MSE, we determined that one standard deviation increase in MSE from the median MSE is associated with the increase in