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Tests of investor learning models using earnings innovations and implied volatilities

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Abstract

This paper investigates alternative models of learning to explain changes in uncertainty surrounding earnings innovations. As a proxy for investor uncertainty, we use model-free implied volatilities; as a proxy for earnings innovations, representing signals of firm performance likely to drive investor perceptions of uncertainty, we use quarterly unexpected earnings benchmarked to the consensus forecast. We document that uncertainty declines on average after the release of quarterly earnings announcements and this decline is attenuated by the magnitude of the earnings innovation. This latter result is consistent with models that incorporate signal magnitude as a factor driving changes in uncertainty. Most important, we document that signals deviating sufficiently from expectations lead to net increases in uncertainty. Critically, this result suggests that models allowing for posterior variance to be greater than prior variance even after signal revelation [e.g., regime shifts in Pastor and Veronesi (Annu Rev Financ Econ 1:361–381, 2009)] better describe how investors incorporate new information.

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Notes

  1. Throughout the manuscript, we use the terms “magnitude” and “size” interchangeably, to capture a measurement property of the signal. In such models, signals are benchmarked to some prior expectation preceding the signal’s release.

  2. Following the results of Black and Scholes (1973), we use the terms “increase (decrease) in implied volatility,” “increase (decrease) in uncertainty about future firm value,” and “increase (decrease) in uncertainty about future realized returns” interchangeably.

  3. Normal–normal updating is a common assumption in theoretical models (e.g., Holthausen and Verrecchia 1990; Barron et al. 1998).

  4. For example, Billings et al. (2014, footnote 3) note that the multivariate tests of Rogers et al. (2009) do not provide evidence about the absolute effects of bad news warnings or positive/confirming forecasts on uncertainty levels.

  5. If the system evolves linearly like in a dynamic linear model, the prior uncertainty will be higher before the signal is received compared to the posterior variance after receiving the signal, and the posterior variance is not affected by the absolute size of the innovation (West and Harrison 1997, Chapter 3).

  6. Subramanyam (1996) presents a single-period model that uses a similar mathematical structure. However, without modifications, this type of model will eventually resolve all uncertainty; this appears inconsistent with market data.

  7. ∆VIX (measured using the values of days −4 and +4) may appear inconsistent with our calculation of IVOL_365 (which is averaged across days −5 to −3, and days +3 to +5). As discussed below, we use averages for IVOL_365 to minimize the effects of noise upon this firm-level construct. For the change in the macro-level measure of volatility (∆VIX), the effects of noise are likely much less severe. Indeed, the correlation for our definition of ∆VIX and an alternative measure using days −5 to −3 and days +3 to +5 exceeds 94 %. Not surprisingly, our results are unchanged to using this alternative definition of ∆VIX.

  8. A constant implied volatility measures the next n days of volatility and thus is not tied to any traded option maturity. Restated, a constant 30-day maturity volatility measured at time t measures the forecasted volatility from t to t + 30, and at time t + 1, the 30-day maturity volatility measures the forecasted volatility from t + 1 to t + 31. Thus the length of maturity is held constant.

  9. In the Heston (1993) option pricing model, the total variance of a stock is determined by (1) the current variance, (2) the long-term variance, and (3) the mean reversion rate. The shape of the implied volatility smile or smirk observed in the market, however, is determined by the “vol-of-vol” and correlation between stock and variance movements. Thus two stocks may actually have different ATM volatilities but have the same expected variance if they have the same values for the first set of parameters but different values for the second set. If, for example, two stocks had the same value for the first three parameters then the stock with the higher vol-of-vol parameter would have a lower ATM implied volatility but higher OTM implied volatilities, in general.

  10. One example is Sridharan (2015), who uses a limited sample of model-free volatilities as a robustness check.

  11. Note that we do replicate our primary analyses using shorter-term (30- and 60-day) option maturities. Consistent with our above discussion, we provide evidence consistent with attenuation (H1) but fail to find evidence of net volatility increases (H2). Again, this is expected, as such short-maturity options do not incorporate expectations of future earnings (i.e., signal) realizations, which is critical to our assessment of these Bayesian learning models. That is, these models center on investor learning about expected volatility with respect to future signal realizations.

  12. The correlation between the two experimental variables is 0.498, suggesting these capture similar but not overlapping, notions of earnings innovations.

  13. Specifically, when scaling by the mean absolute earnings forecast, we (1) first identify the mean analyst forecasts, (2) take the absolute value of these means, (3) delete firm-quarters for which the absolute forecast is <$0.01/share (to avoid small denominator effects), and (4) then scale absolute unexpected earnings.

  14. Note that the interaction of LEV × SUE includes the signed earnings announcement, while our experimental variable Abs_SUE is unsigned. This is intentional and consistent with theory underlying the inclusion of the respective variables. Specifically, signed earnings has a direct effect on leverage and thus is appropriate to use when assessing the effect of leverage on uncertainty. In contrast, the effect on uncertainty under Bayesian updating is not conditioned on the sign of the signal. Note that the Merton (1974) model of leverage suggests that a firm with no leverage should not see a change in its volatility due to a change in its asset value. Furthermore, we do not use the stock return as a proxy for the size of the surprise, as uncertainty/volatility and stock prices are jointly determined.

  15. To further control for pre-announcement uncertainty, we also estimate a regression including as an additional control variable, %REVISIONS. This is measured as the number of analysts revising earnings forecasts in the five (or alternatively, 10) days preceding the earnings announcement, divided by the total number of analysts issuing earnings forecasts. The resulting coefficients on %REVISIONS are generally negative, indicating that firms with more analyst activity before an earnings announcement have more uncertainty resolved by the announcement. However, results on all other variables are unchanged by the inclusion of %REVISIONS.

  16. Results are robust to alternative definitions of size, including the natural log of equity market capitalization.

  17. In addition, untabulated results replicating both Tables 3 and 5 are unchanged to using the alternative option maturities of 273- and 182-days.

  18. We do not tabulate these analyses with alternative leverage proxies, as they resemble those presented. However, the results are available upon request.

  19. Note that we do not conduct sensitivity analyses incorporating temporal fixed effects into Tables 4 and 5. This is because the research design for both tables relies on mean (intercept) shifts; thus (for example) inclusion of time fixed effects would preclude using the intercept to assess the mean change in uncertainty for firms having average values of each of the control variables. However, results are robust to alternatively clustering by reporting quarter and firm.

  20. The coefficient on Abs_SUE in Table 3 is significant and positive when using price as the scalar in both the expansionary and recessionary periods.

  21. In addition, the interaction term Abs_SUE x VIX is negative and significant when added to the column (1) and column (2) models of Table 3.

  22. However, we do note that unscaled absolute forecast error and dispersion are positively correlated with equity price in our sample.

  23. This procedure is quite close to that used by the CBOE to calculate the VIX. In addition, Zhang et al. (2012) show that, for the parameter estimates of Duffie et al. (2000), the model-free volatility estimates are much closer to the true values than at-the-money volatilities.

  24. The delta of an option is the partial derivative of the option’s price to the underlying’s price.

  25. There is also a precedent for using only at-the-money and out-of-the-money options in academic studies. Carr and Wu (2010) state: “Since out-of-the money options are more actively traded than in-the-money options, the quotes on out-of-the-money options are usually more reliable.”

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Acknowledgments

We appreciate useful comments and discussions on prior versions of this paper from Russell Lundholm (editor), an anonymous reviewer, Ana Albuquerque, Rui Albuquerque, Bill Baber, Zhaoyang Gu, Yen-Jung Lee, Lynn Li, Suming Lin, Chi-Chun Lui, Krish Menon, Michael Smith, Shu Yeh, Linhui Yu, Julie Zhu, and workshop participants at Boston University, Chinese University of Hong Kong, George Mason University, George Washington University, and National Taiwan University. The U.S. Securities and Exchange Commission, as a matter of policy, disclaims responsibility for any private publication or statement by any of its employees. The views expressed herein are those of the author and do not necessarily reflect the views of the commission or of the author’s colleagues on the staff of the commission.

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  1. Boston University, 595 Commonwealth Avenue, Boston, MA, 02215, USA

    Thaddeus Neururer & Edward J. Riedl

  2. U.S. Securities and Exchange Commission, Washington, DC, USA

    George Papadakis

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  1. Thaddeus Neururer
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Correspondence to Edward J. Riedl.

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Neururer, T., Papadakis, G. & Riedl, E.J. Tests of investor learning models using earnings innovations and implied volatilities. Rev Account Stud 21, 400–437 (2016). https://doi.org/10.1007/s11142-015-9348-5

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