An Extrapolation of Company-Issued Guidance and IBES Estimates as a Model of Investor Earnings Expectations

Transcription

1 An Extrapolation of Company-Issued Guidance and IBES Estimates as a Model of Investor Earnings Expectations Svilen Kanev Grant Wonders John Casale Ben Enowitz Alex Zhu {skanev, wonders, jcasale, benowitz, azhu}@fas.harvard.edu Harvard College May 2, 2012 Abstract We evaluate the influence of company-issued-guidance and analyst estimates in the formulation of expectations for quarterly earnings reports. Recognizing incentives that produce a systematic bias on the part of managers and research analysts, we construct extrapolative regression models for S&P 500 companies, from the period of , that offer a superior estimation of earnings-per-share. In addition, we augment the dynamic regressions by proposing a novel Kalman filter mechanism that explicitly accounts for random variation in management and analyst input parameters. Evaluating our predictions, we find that our multivariate predictive regression model explains a significantly greater variation of post-earnings-announcement 1-day stock price returns. As such, we demonstrate that extrapolative regression techniques come far closer than analyst estimates in approximating market expectations and reaction sentiments to quarterly earnings results. 1

2 1 Introduction Analyst estimates are central to quarterly earnings reports and are used by both investors and the financial press as a benchmark for earnings results. Given the prominence of these analyst estimates, it is worthwhile examining whether or not they are truly a reliable metric for expectations. Company management also frequently contributes to earnings expectations by issuing guidance for the subsequent quarter. Both of these estimates are also prone to bias. For example, literature has consistently shown [9] that research firms with investment banking franchises have an incentive to manage expectations by systematically underestimating the performance of their clients. Similarly, managers, faced with performance-driven option-vesting schedules will have incentives to be aggressive or conservative in their communication of asymmetric guidance information [15]. There has been extensive literature on the influence of management guidance on analyst estimates, as well as the predictive power of analyst estimates. To our knowledge, no academic study has systematically attempted to combine both of these first-order factors in a single predictive model. In this paper we seek to combine these inputs with dynamic and time-dependent regressive models in order to derive a better estimate of earnings per share (EPS) results. We benchmark against Institutional Brokers Estimate System (IBES) reporting of consensus estimates as well as company issued guidance (CIG). First we build simple regressive models that fit a rolling period of trailing input factors on previously reported results (Section 4). These regression models dynamically estimate coefficient values to construct a prediction for the subsequent quarterly earnings reports of each company. We measure the predictive error for each stock by averaging the deviation of the predicted value versus those that are actually announced. This allows us to quantify the uncertainty of the predictive regression models. However, we also recognize that the uncertainty inherent in the input values themselves. As a result, in Section 5 we propose augmenting the regression models with a filter model that explicitly accounts for random variation in input parameters to derive estimates with superior precision and specified confidence intervals rather than simply trusting the inputs. We use a simple linear, time-dependent, Kalman filter to construct a predictive value for EPS with a range estimation. In Section 6 we show that our predictive models explain a greater fraction of post-announcement stock price reactions than analysts estimates. We back this claim up by explicitly demonstrating the aggregate reaction function to both our model, as well as consensus estimates. 2 Background Investors have numerous sources of information available to them, including an abundance of company historical data, financial analysts forecasts, and companyissued guidance. Financial analysts forecasts, in particular, are considered one of the most influential sources for investment decision-making [5]. Analysts forecasts were also shown to give better market expectations than time-series models [4]. Company-issued guidance is thought to grossly underestimate expected earnings in order to avoid negative earnings surprises when the actual earnings are announced [15]. Theoretically, though, a company has better idea of its own workings and might be able to better predict its own performance. Company-issued guidance is supposed to reveal value-relevant information to investors in order to reduce information asymmetry [10]. Moreover, management forecasting has been shown to influence investor behavior and stock prices when the company has established a long record of accurate forecasting in the past [6]. Investors tend to favor companies with accurate management forecasting. This provides incentives for companies to achieve or beat their expected earnings for the quarter. Some have suggested that this encourages companies to put additional resources into achieving their short-term earnings projections, sometimes sacrificing long-term research and development and growth [1]. Altogether, it seems that company-issued guidance, while overly conservative, is also taken seriously by investors and affects future stock prices. Managerial and financial analysts forecasts are actually inter-related and influence each other. It has been observed that financial analysts will quickly revise their forecasts in direct response to explicit management guidance [2]. Conversely, if a company believes that financial analysts are too optimistic and are not beatable or achievable, they will issue guidance in the hopes of lowering the financial analysts forecasts to achievable or beatable levels. This gaming interaction between managerial and financial analysts forecasts, which has been welldocumented and studied, suggests that the two ought to be considered together. We believe a model that integrates both forecasts may produce more accurate predictions than either component alone. 3 Datasets 3.1 Data Sources We examine quarterly earnings reports from S&P 500 companies from by using data from Baker Library at Harvard Business School. The Institutional 2

3 Brokers Estimate System (IBES) service collects data from research analysts to produce consensus quarterly earnings estimates. The dataset is managed by Thomson Reuters. The IBES data we use includes the most recent consensus EPS estimate prior to any quarterly earnings release, reporting a mean, median, high estimate, low estimate, and standard deviation. IBES also provides the actual result reported in the earnings release along with a date and timestamp (to the minute) of the announcement. For historical information on company-issued guidance (CIG) we use the Thomson Reuters First Call database. Management often provides guidance for the earnings results it expects for an upcoming quarter. Rather than stating a specific EPS value, many companies provide a range of expected earnings. In addition to EPS, forecasts of other information, such as expected revenue, gross margins, tax rate, and stock-based compensation (all of which can be informative to research analysts), are frequently disclosed. The announcement of guidance for the upcoming quarter is typically paired with the release of results from the previous earnings period, most typically in a press release of on the post-announcement earnings call. As such, unlike the IBES estimates, the guidance data typically precedes the subsequent earnings announcement by a period of three months. Despite the fact that guidance data is issued far in advance of analyst consensus estimates, (which are generally established over the three months leading up to the subsequent quarterly announcement) it may still offer useful data for earnings expectations. Although analysts can make forecasts about a company, the management have direct insight into the strength of a company s sales and supply chain. Moreover, analysts may systematically fail to fully incorporate the guidance data into the projections of their research model because management does not always disclose all of the important assumptions in their forecast. The data we use from the First Call dataset includes the guidance date and timestamp, the EPS estimate provided by management, and often the range assumptions that management provides as part of the EPS forecast. In cases where only a range of EPS is provided and an estimate is not explicitly stated by management, the CIG EPS estimate is taken as the mean/midpoint of the high and low. 3.2 Data Construction We made an extensive effort to combine the First Call and IBES databases by matching up quarterly guidance data and IBES estimates for the relevant quarterly fiscal period. We include only the most recent guidance data and consensus estimates prior to any earnings announcement date. Prior to merging the two sets, we comprehensively scrubbed the CIG data by including adjustments for any stock splits during the interval. Naturally, not every S &P company issues guidance, leading us to divide our collective data into two broad categories- 1) companies that issued guidance at anypoint during the interval of and 2) companies that never issued guidance for any of the quarterly period. For the companies that do issue guidance during the interval, many did not provide it for every quarter. Since we are specifically interested in quarters where both consensus estimates and analyst guidance is available, we merge the IBES data with the guidance and drop the quarterly observations where no CIG data is available. The combined dataset encompasses 389 companies with 5,970 unique quarterly observation points (with both CIG and IBES data). Given our goal to understanding the contribution of company-issued guidance to earnings expectations, we also construct another dataset as a benchmark which is exclusively limited to companies that never issued guidance during the period. For companies that never issued CIG we examine IBES estimates only, yielding a dataset that includes 84 companies with 2,782 unique quarterly observation points. 3.3 Data Overview In order to attain a high-level understanding about the properties of our datasets, we begin by running regressions of actual earnings on our various data inputs (CIG and IBES). Specifying a minimum of 12 quarterly observations, we proceed by running stock-specific regressions on two types of companies: 1) companies that are covered by analysts and have company-issued guidance (184 companies) and 2) companies that are covered by analysts, but have never issued guidance (79 companies). We use these datasets to produce three simple regressions: 1. Regression of actual earnings on mean CIG estimates for quarterly periods where company issued guidance is available 2. Regression of actual earnings on mean IBES estimates for quarterly periods where company issued guidance is available 3. Regression of actual earnings on mean IBES estimates for companies that never issued guidance at anypoint in the interval The beta coefficients and standard errors calculated by these three types of regressions are summarized in Table 1 and presented as histograms in Figure 1. The magnitude of the betas shows the degree to which analysts or management underestimates (Beta > 1) or overestimates (Beta < 1) actual earnings. The distribution of 3

4 Standard Errors of CIG Beta Coefficients of CIG Frequency Standard Error Standard Errors of IBES with CIG Frequency Standard Error Standard Errors of IBES without CIG Frequency Frequency Beta Beta Coefficients of IBES with CIG Frequency Beta Beta Coefficients for IBES without CIG Frequency Standard Error Beta Figure 1: Simple regressions of announced earnings results versus (i) guidance numbers, (ii) IBES estimates where management guidance is available, (iii) IBES estimates when no guidance is available. the betas within each regression specification offers evidence about the degree to which aggressive or conservative forecasts relative to actual earnings exhibit companyspecific variation. The beta coefficients for the IBES regression for companies with CIG ((Figure 1) are clustered around 1 (mean of 1.008); however, the majority of the betas are greater than 1, indicating the general trend that analysts tend to underestimate earnings. Financial literature suggests that analysts have an incentive to underestimate given that most large research groups have a corresponding investment banking division, leading to moral hazard. Clients, such as the CEOs of companies, prefer to have low analyst expectations so that they are more likely to beat the street estimates and reap a positive stock price return post earnings. Thus, to maintain a strong relationship with the CEO, it is suggested that investment banking management influences the research group to provide low expectations. The standard error for this regression is the lowest out of the three options, with an average standard error of 32. This means that analysts who have access to CIG data are more consistent in their estimates (either conservative or aggressive than analysts who do not have access to CIG data and even more consistent than management themselves. This implies two things: 1) the extra three months the analysts have to 4

5 CIG IBES with CIG IBES no CIG Beta Coefficient SD of Beta Standard Error of Beta Coefficient SD of Standard Error Tickers Table 1: Summary statistics from the OLS regression results summarized in Figure 1. incorporate new data and refine their prediction is a distinct advantage over management and 2) companies provide valuable information when issuing guidance. However, management in theory should be more accurate in that they have access to all inside information and have a better understanding of the companies financial status. The beta coefficients for the IBES regression for companies without CIG is approximately normally distributed around a mean of Similar to the other histograms, most of the data is skewed left, indicating that analysts tend to underestimate actual earnings. In fact, this regression had the highest average beta, meaning that analysts were the more conservative when lacking CIG data and more conservative than management. Given that analysts do not have the valuable information that management provides, it is likely that the analysts tend to more conservative. One big difference is that average standard errors for the beta coefficients is much higher than the IBES regression for companies without CIG than those with CIG, with a mean of 95 as opposed to 32, respectively. Thus, even though analysts without CIG are more conservative, they are less consistent in their estimates. This reflects the uncertainty that the analyst faces without access to CIG data. The beta coefficients for the CIG regression (Figure 1) are the most surprising. The average beta is and the standard deviation is The average beta is just around 1, but that is because there is a very long, fat left tail. While it is true that management most likely underestimates earnings (most likely due to asymmetric stock reaction when earnings are missed), there are many cases where management significantly overestimates earnings. This could be a result of guiding earnings well in advance of the actual announcement date. However, it also speaks to the incentives of management. There is sometimes an incentive to overestimate earnings, to signal confidence in the company. This is particularly common in quarters where management s stock options become exercisable. In short, the analysts who do not have access to CIG data underestimate the most (average beta of 1.027), but also have the small standard deviation of betas (0.102). However, these estimates are the least accurate and least consistent with the highest standard error and highest standard deviation of standard errors. 4 Regression Models to Approximate Market EPS Expectations As observed in Figure 1, for many stocks, analysts and management have tendencies (as well as incentives) to be conservative in their forecast. Nonetheless, given the range of standard errors around these beta coefficients observed at the stock-specific level, we can infer that there might be some consistency to the degree to which they underestimate earnings for a given company. In order to uncover any serial patterns that emerge in the guidance and estimate forecasts for each stock, we produce a variety of predictive and dynamic historical regressions. 4.1 Extrapolative Regressions for Companies that Issue Guidance We first examine the dataset of companies with quarterly earnings reports for which we have both IBES estimates as well as CIG data. To create a linear prediction model for earnings at time t, we regress actual earnings on mean IBES estimates from time t minus a rolling period to time t 1 for the same time period. The rolling period is the number of previous quarters that we use to train the model. In particular, we look at rolling periods ranging from 8 quarters (2 years) to 36 quarters (9 years), incrementing by multiples of 4 quarters (1 year). We use OLS regressions at the stock-specific level to produce coefficient estimates that are dynamically updated for each new time period based on the data that spans the rolling period. The first model ( Model 1 ) is a univariate regression of actual earnings on mean IBES estimates. The second model ( Model 2 ) is a multivariate regression of actual earnings on mean IBES estimates and CIG estimates. The third model ( Model 3 ) incorporates the 5

6 Frequency of observations Frequency of observations Prediction error (a) CIG Errors Prediction error (b) IBES Errors Figure 2: Prediction errors for CIG guidance and IBES estimates versus actual EPS. Figure (a) shows the absolute value of the percentage error of company-issued guidance and (b) shows the percentage error of mean analyst estimates. We use this particular distribution of errors from guidance and consensus estimates as a benchmark that can compared with the errors of the 32 rolling period models shown in Figure 3. two independent regressors from the second model, as well as a third variable which is the standard deviation of the IBES estimates within each quarterly period. Figure 3 plots a histogram of the prediction errors for each quarter for three different OLS regression models. The error is defined as the absolute percent difference between the actual earnings and the EPS predicted by the regression models. This particular figure depicts the error results for a rolling period of 32 quarters. A comparison of these histograms with Figure 2 shows that Model 1, Model 2, and Model 3 do a better job in predicting actual earnings relative to consensus estimates alone. The frequency for the lowest bin (0 to 2) has 120 observations for Model 1, 143 observations for Model 2, and 142 observations for Model 3. In contrast, using solely the mean IBES estimate ( IBES Model ) produces only 109 observations in this range. Overall, the CIG guidance and IBES estimates have greater means and standard deviation for error terms, suggesting that our regression models have predictive power. In addition to the rolling period of 32 quarters illustrated in the histograms, we also considered other potential trailing quarterly windows in which to run our regressive models. The result of those regressions and a summary of their prediction errors that is benchmarked against IBES estimates alone, is shown in Table 2. As the length of the rolling period is increased effectively increasing the size of the training period used to produce estimate coefficients the accuracy and reliability of the model, relative to IBES estimates alone, is progressively enhanced. 4.2 Extrapolative Regression for Companies with No Guidance Given our hypothesis on the importance of companyissued guidance, we also wanted to test the value of an extrapolative regression in cases where management provides no forecast of EPS. In Figure 4, we consider a model that corresponds with the database that contains companies that have IBES data, but do not have CIG data. There is only one regression model, identical to Model 1, that regresses mean IBES estimates on actual earnings. For the error histogram in Figure 4, we consider the same 32 quarter rolling period. Although the regression prediction model results in more observations that fall within the smallest error range (% to 5.0%), 178 as compared with 151 for IBES estimates, it does not yield either a smaller mean error or standard deviation of the error term. The mean error for the model and benchmark are comparable, each producing a mean error of 30%. However, the standard deviation of errors for the regression model is actually larger, implying a greater distribution of error terms that makes it less reliable. As a result, we conclude that our model cannot produce any improvement over consensus estimates for companies that do not issue guidance. A verification of this conclusion is also demonstrated across different rolling periods applied to the OLS regression, as seen in Table 3. Not only is the size of the mean analyst error far larger in an absolute sense than companies that do not issue guidance, but our extrapolative regression model also fails to offer any incremental improvement over the analyst forecast alone. Moreover, the standard deviation of the error 6

7 Frequency of observations Frequency of observations Frequency of observations Prediction error (a) Model 1 Errors Prediction error (b) Model 2 Errors Prediction error (c) Model 3 Errors Figure 3: Prediction errors of regression models versus actual EPS using a rolling period of 32 quarters. Model 1 uses a univariate OLS regression of actual earnings on IBES estimates. Model 2 employs a multivariate OLS regression of actual earnings on IBES and CIG estimates. Model 3 employs a multivariate OLS regression of actual earnings on IBES estimates, IBES standard deviation, and CIG estimates. The distribution of errors across all observations predicted by Model 2 (multivariate regression) exhibits the lowest mean error and standard deviation. term for the model is higher versus the IBES estimate across all rolling periods we sample. 4.3 Assessment of Various Rolling Periods In Figure 5, we examine the cumulative distribution function of prediction errors for each model over various rolling periods. The cumulative function is an integral of the distribution histograms, as depicted in Figure 3, that shows what fraction of the observations (y-axis) are accounted for by a specified error range (x-axis). These graphs correspond to the dataset where there is both IBES and CIG data. The predictive models across various rolling periods all exhibit a similar smooth and concave curve. On the tails of the distribution function we do observe an out-performance by the 36 quarter rolling period. 4.4 Regression Equations Against Benchmark As depicted in Table 2, an increased rolling period is generally associated with reduced mean errors and standard deviations of the error term. However, due to the heterogeneity of the residuals, we cannot be certain if this is a result of time-dependent error correlations or truly an improved estimation of regression model coefficients. In order to compare the significance of the forecast value in relative terms, we use IBES estimates as a benchmark that captures time-dependent variation in mean error across stocks. In Figure 6, we show the relative improvement of the multivariate regression model that combines historical CIG and IBES estimates over the estimates alone for various rolling periods. 4.5 Company-Issued Guidance Incentives Although we have reason to believe that management can offer valuable insight into earnings expectations, taking the CIG estimate alone produces by far the largest mean error and the highest standard deviation, as seen in Table 2. For example, the standard deviation of CIG errors ranges from 0.36 for a rolling period of 32 quarters to for a rolling period of 8 quarters. In contrast, the range of standard deviation for Model 1 (a univariate regression using historical IBES estimates) varies far less, from to 0.370, during the same respective rolling periods. The fact that the mean and standard deviation of the errors from company guidance are very high, while the median is relatively similar to IBES estimates, is indicative of the large number of outliers present in guidance forecast. Although managers generally attempt to precisely forecast their estimate through the use of internal data (as indicated by their relatively low median prediction error) they may also, on occasion, have incentives to guide aggressively for a given earnings cycle. Consequently, for certain quarters, the actual EPS result spectacularly misses the previously issued guidance. We posit three explanations for the existence of fat left tails (see Figure 2a) in company-issued guidance: 1. Management may attempt to influence their stock price ahead of performance-driven option vesting or planned stock sales. In order to avoid conflicts of insider trading that arise from the untimely sale of stock, management regularly schedules planned sales of holdings under SEC rule 10b5-1, especially in periods where performance compensation in the form options securities becomes exercisable as stock. Given the constraints around scheduled 7

8 Rolling 8 Quarters Model 1 Model 2 Model 3 IBES CIG 2 vs. IBES Median Mean Standard Deviation Number of Tickers Number of Observations Rolling 16 Quarters Model 1 Model 2 Model 3 IBES CIG 2 vs. IBES Median Mean Standard Deviation Number of Tickers Number of Observations Rolling 24 Quarters Model 1 Model 2 Model 3 IBES CIG 2 vs. IBES Median Mean Standard Deviation Number of Tickers Number of Observations Rolling 32 Quarters Model 1 Model 2 Model 3 IBES CIG 2 vs. IBES Median Mean Standard Deviation Number of Tickers Number of Observations Table 2: Presents the absolute value of errors from the predictive regression models, CIG forecasts, and IBES estimates versus actually reported EPS. The sample, aggregated across all observations, includes companies that issue guidance data in quarters where both CIG and IBES estimates are available. Model 1 is calculated as a univariate OLS regression, at the stock-specific level, with various rolling periods of actual EPS on IBES estimates. Model 2 is a multivariate regression of actual earnings on IBES estimates and CIG forecast. Model 3 additionally includes the standard deviation of IBES estimates as a predictor variable in the multivariate regression. As can be seen in the data, for companies that issue guidance, an extrapolative regression model to predict EPS leads to a significant improvement over analyst estimates. Moreover, as the number of periods used in calculating the prediction coefficients increases, the regression models generally become both more accurate and reliable. sale date, management may attempt to issue positive press releases (such as strong earnings guidance or a dividend announcement) ahead of a 10b5-1 sale. This may incentivize unrealistically aggressive forecasts of the following quarterly earnings report. 2. Economic shocks will also produce quarters where management will significantly miss (or beat) previously issued guidance. Although IBES estimates can be updated in the three-months leading up to earnings so as to incorporate information about expected losses (or gains), management will rarely offer additional intra-quarter guidance updates. As a result, the high (or low) guidance number may bias the data in cases where investors have already discounted (or inflated) its value as a result of new developments. 3. As a corollary, during periods when negative shocks hit, management may be incentivize to front-load losses onto that particular quarter. As the conventional Wall Street wisdom argues, If you are going to miss, miss big. For example, management might take advantage of a particularly weak quarter in order to further write down assets and recognize losses, thus creating an easier benchmark for sequential and year-over-year comparisons. 8

9 Frequency of observations Frequency of observations Prediction error (a) IBES Estimate Errors Prediction error (b) Model Prediction Errors Figure 4: Error distribution from predictive regression model and IBES estimates for companies that have never issued guidance data. The model employs a rolling period of 32 quarters for a univariate OLS regression of actual earnings on IBES estimates. The distribution of errors suggests that an extrapolative regression model (based on historical estimates versus actual) is not effective for reducing mean prediction error if a company does not issue guidance. Rolling 8 Quarters Rolling 16 Quarters Model IBES Model IBES Median Mean Standard Deviation Number of Tickers Number of Observations Rolling 24 Quarters Rolling 32 Quarters Model IBES Model IBES Median Mean Standard Deviation Number of Tickers Number of Observations Table 3: Errors of predictive regression model and IBES estimate across all quarterly observations of companies that have never issued guidance data. The model is calculated as a univariate OLS regression, at the stock-specific level, with various rolling periods of actual EPS on IBES estimates. As can be seen in the data (as well as in the histograms for the 32-quarter rolling period depicted in Figure 4), if companies do not issue guidance, an extrapolative regression model to predict EPS does not lead to significant improvement over analyst estimates. When no CIG data is present, analysts do a worse job of predicting EPS (i.e. garbage in ) and an extrapolation using historical analyst conservatism fails to offer any incremental gains to accuracy (measured by mean error) or reliability (standard deviation), yielding garbage out. 4.6 Constrained Regression Specification The effect of these incentives in management guidance is to produce certain quarters in which earnings fall far short of expectations (making old guidance numbers appear heavily aggressive). Recognizing that the presence of these guidance outliers can sometimes produce large negative coefficients in our multivariate regression equations (as in Model 2), we decided to employ a regression that constrained the calculation of beta coefficients as positive. By using a constrained regression, we explicitly construct an algorithm that ignores quarters in which management issues unrealistically aggressive es- 9

10 Distribution of observations Starting Quarter Average model prediction error (a) Model 1 Cumulative Distribution Starting Quarter Average model prediction error (b) Model 2 Cumulative Distribution Distribution of observations Distribution of observations Starting Quarter Average model prediction error (c) Model 3 Cumulative Distribution Figure 5: Cumulative distribution of prediction errors for models versus actual EPS using trailing rolling periods 8, 12, 16, 20, 24, 28, 32, 36 quarters for the predictive OLS regressions. Model 1 is a univariate regression of actual earnings on IBES and CIG estimates. Model 2 is a multivariate regression of actual earnings on IBES and CIG estimates. Model 3 is a multivariate regression of actual earnings on IBES estimates, IBES standard deviation, and CIG estimates. timates. In constraining the coefficients as positive, periods marked by outsized management aggression will instead shift the distribution of CIG weighting in the multivariate regression towards IBES estimates. The advantages of this constrained regression specifications in accounting for outliers in management guidance is particularly salient during narrow rolling windows in which an aggressive guidance number exerts substantial influence over the least-squares regression. In Figure 6, we show the incremental error improvements, across different rolling periods, of constrained and unconstrained multivariate regressions on IBES estimates alone. The unconstrained regression from Model 2 significantly underperforms IBES during short rolling periods when the small number of datapoints allows guidance outliers to produce negative regression coefficients. Alternatively, our constrained regression generally produces superior mean prediction errors, even in periods where the rolling window is short. The advantages of the constrained regression in minimizing the standard deviation of the error term are particularly pronounced in short rolling periods. 4.7 accurate. Nonetheless, we cannot be certain that the decision to guide is exogenous and the presence of another confounding factor should not be ruled out. For instance, managers of riskier companies or in more unpredictable industries may elect to forgo guidance because they do not want to risk missing their own estimates. Nonetheless, we strongly suspect that all disclosure can be helpful for improving accuracy of analyst estimates. Even if the analyst is not able to explicitly incorporate the company-guidance value for EPS, they will still be able to deploy other relevant disclosure information such as predicted operating margins, revenue estimates, and supply chain metrics. Although analysts remain conservative in their estimates across both datasets, our findings suggest that having access to the CIG data allows analysts to be more predictably consistent in the degree to which they underestimate. Consequently, our univariate regression can achieve lower mean error with higher reliability (lower standard deviation). Moreover, the explicit addition of the CIG value in the form of a multivariate regression made the predictive model even stronger, providing an incremental gain to both mean error and standard deviation across all rolling periods tested. This suggests that while analysts generally incorporate the majority of guidance data, for some companies, there is still valuable predictive power in the guidance number itself that is not reflected in the IBES consensus. Implications for Management Guidance The fact that a univariate regression of actual earnings on IBES estimates (Model 1) is significantly more accurate when companies issue guidance, is indicative of the causal impact that management exerts on the estimates of research analyst. Although neither regression directly employs CIG data, the large difference in mean prediction errors between estimates with and without guidance (on the order of 10% versus 30%), suggests that the mere presence of guidance makes analyst forecasts much more 5 Predictor-Corrector Model of EPS In this section, we describe the estimator scheme that we use to augment our linear regression models. Since all 10

11 Mean Error Imrovement: Model vs. IBES! Constrained Model! Unconstrained Model! 8.0%! 7.0%! 6.0%! 4.0%! 2.7%! 2.4%! 2.0%! 2.0%! 1.6%! 0.9%! 0.7%! %! %! 8! 12! 16! 20! 24! 28! 32! 36! -2.0%! SD of Error Improvement: Model vs. IBES! 15.0%! 1%! 5.0%! %! -5.0%! -1%! -15.0%! Constrained Model! Unconstrained Model! 11.1%! 8.6%! 5.9%! 2.7%! 1.1%! 8! 12! 16! 20! 24! 28! 32! 36! -3.2%! -8.2%! -9.1%! -4.0%! Rolling Period! -2%! -91.1%! Rolling Period! (a) Mean Error Improvement (b) SD of Error Improvement Figure 6: A comparison of the mean and standard deviation of the error term for multivariate regression Model 2 (which includes historical IBES and CIG as predictors of actual EPS) relative to IBES estimates alone. For (a), the y-axis tracks the incremental improvement in percentage terms of the prediction model over the IBES estimate in minimizing the the average prediction error. For (b), the y-axis tracks the incremental improvement in percentage terms of the prediction model over the IBES estimate in minimizing the the standard deviation of the error. In addition to demonstrating the value of longer rolling periods for formulating predictive coefficient estimates, the results also demonstrate the superior stability regression models that are constrained to non-negative coefficients. See Section 4.5 and Section 4.6 for a discussion of the guidance incentives that make the constrained specification preferable. input factors in our model have a respective uncertainty, we use a model that explicitly factors in such noise and provides a confidence interval for the filter s final prediction. We first describe the theory behind our estimator, before moving on to actual implementation details. We finally show that such a model performs on par with IBES predictions in terms of the accuracy of earnings estimation. It also provides a confidence interval, which captures the majority of EPS movement. 5.1 Kalman Filtering We use a Kalman filter [7] as a predictor-corrector mechanism that allows us to factor in the inherent noisiness of our input factors. The Kalman filter is a recursive linear estimator that can be used to generate a statistically optimal estimate from a set of noisy observations of a certain value. It is very widely used in control and sensing theory to fuse multiple fuzzy sensor values [3, 8, 14, 13], and has also found place in econometrics as a time-series model of stock market prices [11, 12, 16]. System model Kalman filters make assumptions about the underlying system that is being modeled. The model is of a linear dynamic discrete-time system modeled on a Markov chain and perturbed by normally-distributed noise. In formal terms, the assumption is that the system state x 1 k at time k is evolved from the system at time 1 Note the vector form of the system state and the matrix form of all coefficient matrices. The matrix-vector notation allows us to abstract (k 1) by the following: where: x k = F k 1 x k 1 + B k u k + w k (1) F k 1 is a transition matrix that captures the dependence of the system on its previous true state x k 1. It models the evolution of the system if it is left to its own devices. u k is a vector of control inputs that quantify outside external perturbances of the system. B k is the matrix coefficient that captures the dependence of the real state of the system on these control inputs. w k is the inherent system transition noise. It is assumed to be normally distributed w k N(0, Q k ). The real state of the system x k is hidden in the Markov sense. Its measurement process is also subject to noise, resulting in a measured state z k, such that: z k = H k x k + v k (2) Here H k is the measurement, or observation matrix model that maps the hidden true state of the system to the observed one, and v k is the normally distributed measurement noise term z k N(0, R k ). away the dimensions and input parameters of the filter model. We show the concrete conversion matrices we use in our model in Section

12 Filter model The actual filter is a recursive model, which uses only the previous time step state estimate and the measurement to form a new prediction. It is formally split in predict and update stages. The predict stage updates the filter s estimate at time (k 1) of the hidden state at time k ˆx k k 1 and its associated covariance matrix P k k 1. This prediction occurs before a measurement is conducted, based on the system evolution and external controls. Formally, it is shown in Equations (3) (4): x k k 1 = F k x k 1 k 1 + B k u k (3) P k k 1 = F k P k 1 k 1 F T k + Q k (4) Once the observation is available at time k, the update stage updates the state estimate and its covariance a posteriori by incorporating the observed measurement z k. This is formalized in Equations (5) (9). Equation (5) defines the state innovation as the difference between the noisy measurement and the previous predicted state. Equation (6) introduces the innovation in the state covariance by adding in the measurement noise R k. Finally, Equation (7) defines the optimal Kalman gain K k that guarantees filter stability and optimal state estimation. The derivation of these terms can be found in Kalman s original paper [7]. Finally, Equation (8) uses the optimal gain to weigh in the measurement innovation, and Equation (9) scales the state estimate covariance inversely with the filter gain. ỹ k = z k H kˆx k k 1 (5) S k = H k P k k 1 H T k + R k (6) K k = P k k 1 H T k S 1 k (7) x k k = x k k 1 + K k ỹ k (8) P k k = (I K k H k )P k k 1 H T k (9) Filter properties There are several important properties and assumptions of the theoretical Kalman filter model that make it attractive as an earnings estimator. The mere structure of the filter model as a predictorcorrector scheme is useful in the context of predicting actual earnings numbers based on different estimates. The fact that the model explicitly accounts for errors in both measurements and system evolution makes it appropriate for handling financial data, which is inherently noisy. Equations (8) (9) suggest that the error ranges of the different inputs will affect not only the predicted state error, but also the predicted state itself, which is desirable for subsequently predicting stock price reaction based on the predicted value and confidence of earnings estimates. On the other hand, the Kalman filter model makes some strong assumptions, which, if not met, can significantly deteriorate its predictive properties. It is a linear time-dependant model assuming that, left to itself, the system evolves linearly with time. This is not necessarily obvious for quarterly earnings numbers, which are considered independent in some models. We explicitly test how strong this time dependence is in Section 5.3. The other strong assumption that the model makes is of the normality of measurement and system errors. Since there is generally a small number of analysts that produce estimates, and at most a single range of management guidance numbers, it is hard to verify this normality assumption for our data set. 5.2 Application as an EPS Estimator We construct a practical predictive model from the theoretical model described in the previous section. We define the hidden state vector as having only a singledimension, which signifies the earnings per share value that we are trying to estimate: x k = ( ) EP S. This choice immediately constrains the choice of an observation matrix that converts observations to state values since ( ) the observed value is the same EPS number, H k = 1. Thus, the observation process of converting the real numbers to the real state is not significantly errorprone. We chose a constant small standard deviation ( value of ) 1% for the observed EPS, resulting in R k = Note that choosing hidden state that has equivalent semantics to the observation value significantly simplifies the filter design. We consider it a simplifying assumption future more elaborate models can include various other hidden parameters. We use the IBES analyst estimates and the CIG numbers at each time step as control inputs to the system. The rationale behind this choice is that there is a causeand-effect relationship between both IBES estimates and CIG numbers, and the real announced earnings numbers due to the pressure to beat both estimates described in Section 1. In this case, the full control input to the system is the vector u k = ( ) T 1 EP S IBES,k EP S CIG,k, including both estimate numbers for the upcoming quarter and a constant term to control for static earnings. This defines the shape of the control-to-state matrix B k = ( β 1 k β2 k β3 k ). In Section 4, we showed that there is a strong correlation between a rolling window of EP S IBES and EP S CIG numbers and the to-be-announced EP S k, and that a multivariate regression can be efficient in capturing that correlation. Therefore, at each prediction step 12

13 Earnings per share (EPS) announced EPS model estimate IBES estimate CIG estimate Earnings per share (EPS) announced EPS model estimate IBES estimate CIG estimate Quarter (a) AAPL Quarter (b) ADSK Figure 7: Samples of the proposed filter model performing reasonably ((a) AAPL) and with a significant prediction error ((b) ADSK). Analyst estimates and company-issued guidance are presented for comparison. Dashed lines represent the confidence interval of the filter model. we estimate the B k vector by regressing previous announced earnings against IBES and CIG predictions for a rolling window of size w. For quarters before w, or for those where a regression does not converge, we use a constant B k = ( ) 0 1 0, that is, we rely only on IBES estimates alone because of their demonstrated stronger correlation to the actual announcement number. In Section 5.3, we will show that the type of regression performed and the size of the rolling window both have a strong influence on the model s predictive qualities. As another simplifying assumption, we model independent noise distributions for EP S IBES and EP S CIG. In that case, the system covariance matrix is also compressed into a scalar Q k = ( ) σ 2 IBIS,k +σ2 CIG,k that represents the sum of IBES and CIG variances. For quarters where company management only issues a single EPS target and no range that allows us to estimate a σ CIG, we assume an artificially high standard deviation of 10%. The rationale behind this is that these are exceptional quarters for companies that regularly issue guidance ranges. By explicitly communicating a range of zero, managers are trying to swing markets in a certain direction, which, given management s documented propensity to misguide, is a sign of treating such quarters more cautiously. Note that the analysis in Section 4 suggests a causeand-effect relationship between CIG numbers and IBES estimates. Under such a relation, modeling independent noise for these two estimates might not be appropriate. Given Kalman filters high sensitivity to the choice of noise covariance, this could be a cause of significant prediction inaccuracy. We plan to address this issue in future work. Finally, we need to address the issue of system time evolution. Given our state space choice, the time dependence matrix is also reduced to a scalar: F k = ( β k ). It is not obvious how strongly EPS prices are correlated with EPS prices from the previous quarter. Our initial attempts at developing a filter with a relatively strong time dependence lead to significant instabilities in the filter predictions. Therefore, we assume a weak constant correlation and evaluate two filters in the following section a static one for which β k = 0 and a dynamic one with β k = Estimating Prediction Error As seen in the previous sections, creating a predictive model is based on a variety of assumptions, which need to be verified as well as possible. We start the verification process with anecdotal evidence Figure 7 shows model predictions for two companies against the announced earnings per share, as well as the two benchmark predictions IBES estimates and CIG. The black line represents the model prediction for the appropriate quarter, with calculated confidence intervals represented as dashed lines. Figure 7a presents a case where we consider the prediction qualitatively successful we can notice how the prediction line is better at tracking AAPL earnings announced later for each respective quarter than both benchmarks. This is especially evident in the last three quarters, where the difference is more than 20%. On the other hand, Figure 7b shows significantly worse performance for predicting ADSK. For example, notice the 13

14 Dynamic Static 0.2 IBES CIG Average model prediction error (a) 8-quarter window Distribution of individual models1.0 Dynamic Static 0.2 IBES CIG Average model prediction error (b) 12-quarter window Distribution of individual models1.0 Dynamic Static 0.2 IBES CIG Average model prediction error (c) 16-quarter window Distribution of individual models1.0 Dynamic Static 0.2 IBES CIG Average model prediction error (d) 20-quarter window Distribution of individual models1.0 Figure 8: Cumulative distribution of average model error over different stocks. IBES and CIG data are provided as benchmarks. Model parameters evaluated are time-dependence and regression window size. prediction at quarter 15, where over-reliance on CIG estimates causes a significantly wrong prediction. The significant difference between predicting earnings for these two cases suggests that generalizing model properties across a wide range of companies could hide a lot of the model stock-specific differences and can lead to incorrect conclusions. Therefore, in order to analyze model correctness in more detail, we look into the distribution of its accuracy over a range of companies. In order to do that, we create predictions for each individual ticker symbol in the dataset described in Section 3. We limit the analysis to datapoints with more than 24 quarters of reported earnings estimates in order to keep results statistically significant. We then calculate the absolute relative model error for each prediction for the respective company as the absolute percent difference between predicted and announced earnings. The average of this error is indicative of the model s accuracy in the specific case and the standard deviation of this error is indicative of the model s reliability in producing estimates with the average accuracy. Figure 8 shows the results of such accuracy analysis. The stock-specific accuracy obtained by the static and dynamic models described in Section 5.2 is compared against accuracy metrics of IBES estimates and CIG numbers. Figures 8a 8d represent different regression window sizes to estimate the model s control coefficients. The plots are cumulative distributions over the range of stocks in the dataset. Therefore, a faster rise of a single model implies that it predicts a larger number of EPS numbers for the same average prediction error. First, comparing IBES and CIG predictions in Figure 8 is another confirmation of the result in Section 4 stating that analyst estimates are a better predictor of earnings than guidance numbers. The cumulative his- 14