Online Appendix to: Revealing Downturns

This appendix (i) shows how dispersion, correlation, and volatility are related, (ii) shows that the theoretical results are robust to state-dependent uncertainty, (iii) presents supplementary summary statistics, (iv) shows robustness of the main paper’s OLS results to subsamples, (v) provides alternative theoretical and empirical measures of earnings response coefficients (ERCs), (vi) distinguishes the “revealing downturns” effect from a “bad news” effect empirically, (vii) shows that the model can quantitatively account for the magnitude of ERC differences across upturns and downturns, and (viii) evaluates empirically how much of return skewness can be attributed to the “Revealing Downturns” mechanism. The main text of the paper can be downloaded here. ∗University of Michigan, Stephen M. Ross School of Business, schmalz@umich.edu. †University of Vienna, Department of Finance, sergey.zhuk@univie.ac. at. 1 Relation of Correlation, Dispersion, and Volatility The following example shows that increased dispersion in downturns does not conflict with the established fact that correlations increase in downturns. Consider a vector of stock returns r that is normally distributed, with mean vector a and variance-covariance matrix Σ. The individual returns ri are distributed normally with mean ai and variance σ2. The covariance between two stock returns ri and rj is Cov(ri, rj) = ρσ2, where ρ is the correlation coefficient. The cross-sectional variance, or dispersion, is thenDisp = (1−ρ)σ2. In downturns, correlations ρ are high, as are idiosyncratic volatilities σ2. Nothing is said about the effect on cross-sectional dispersion. The present paper shows that the joint effect is such that cross-sectional dispersion is higher in downturns because individual stocks react more strongly to news. 2 Robustness of Theoretical Predictions to State-Dependent Ex-Ante Uncertainty In the above results, we assumed the variance of idiosyncratic noise, σ2 ε , is constant and does not depend on the business cycle. If, however, it changes with the business cycle in the data, it could affect our empirical results. To alleviate that concern, we now derive an additional proposition that predicts how ERCs depend on the state of the economy if we control for the effect ex-ante uncertainty about earnings news, var [ Y i t ] , has on earnings response coefficients. ERCs as a function of the variance of earnings surprises can be written as λ ( ξt, var [ Y i t ]) = σ2 a − φσab − ( φσ2 b − σab ) ξt √ var [ Y i t ] . (2.1) Proposition 1. If we consider λ as function of var [ Y i t ] and ξt, then a positive cutoff σ̄λ ab = φσ2 b > 0 exists such that 1. λ is decreasing in ξt if σab < σ̄λ ab, and 2. λ is increasing in ξt if σab > σ̄λ ab. The prediction says that the inclusion of an additional control that proxies for the uncertainty of an announcement does not qualitatively change the results. Thus, we still expect to see an asymmetry in ERCs across downturns and upturns, similar to the earlier predictions. 3 Supplements to Summary Statistics We refer in the main text to histograms of cumulative announcement returns (CARs) and earnings surprises (ESs), and present them in Figure 1. The top row shows the histograms for