Market reactions to announcements to expense options

Abstract

The joint hypotheses of informationally efficient markets, transparent financial statements, and adequate accounting disclosure suggest that announcements of changes in the accounting treatment of employee stock options from footnote disclosure to expense recognition should not trigger stock price reactions because free-cash-flows will not change. Event study results from a sample of 241 firms that announce such changes reveal statistically significant negative price changes followed by positive price changes about equal in magnitude. We propose the learning, sophisticated investor, neglected firm, and firm size hypotheses to explain the observed announcement-period stock price reaction.

Appendix

We use event-study methodology to determine announcement effects. If security returns follow a single-factor market model, returns can be expressed as

$$R_{jt} = \alpha _j + \beta _j R_{mt} + \varepsilon _{jt,} $$

(1)

where R _jt is the realized return of stock j on day t, R _mt is the market return on day t; ɛ _jt is a random variable that is homoskedastic, uncorrelated with R _jt and R _mt , and has an expected value of zero. β _j measures the sensitivity of R _jt to the market for stock j. The estimated coefficients are used to compute abnormal daily returns for the announcement period where event day +t (−t) represents the tth trading day after (before) the announcement date (t = 0). AR _jt is the abnormal return of stock j on day t and measures the difference between the observed and expected daily returns. Formally, abnormal returns of stock j on day t are

$$A_{{jt}} = R_{{jt}} - (\ifmmode\expandafter\hat\else\expandafter\^\fi{\alpha }_{j} + \widehat{\beta }_{j} R_{{mt}} )$$

(2)

where R _jt is the realized return of stock j on day t, R _mt is the market return on day t, and $\hat \alpha _j$ and $\hat \beta _j$ are the market model parameter estimates for stock j.

The average abnormal return (AAR _t ) across announcing firms on day t is

$${\text{AAR}}_t = \frac{{\sum\limits_{j = 1}^N {A_{jt} } }}{N}$$

(3)

where N is the number of firms (j) making announcements.

The cumulative average abnormal return (CAAR) between event day T ₁ and day T ₂ is

$${\text{CAAR}}_{T_1 ,T_2 } = \frac{1}{N}\sum\limits_{j = 1}^N {\sum\limits_{t = T_1 }^{T_2 } {A_{jt} } } $$

(4)

where T ₁ and T ₂ are the beginning and ending days of the event period, respectively.

Because abnormal returns are linearly related to beta, we use two methods of estimating betas. First, we utilize OLS regression to estimate coefficients and the estimate of the beta for security j is

$$\beta _j = \frac{{{\text{Cov}}_{j,m} }}{{\sigma _m^2 }}$$

(5)

where Cov _{j ,m} is the covariance of the returns between security j and the market m.

Scholes and Williams (1977) show that errors in variables problems can occur with high-frequency (daily) data when OLS is used to estimate coefficients. However, consistent estimators can be computed as

$$\hat \alpha _j = \frac{1}{{T - 2}}\sum\limits_{t - 2}^{T - 1} {R_{jt} - } \hat \beta _j \frac{1}{{T - 2}}\sum\limits_{t - 2}^{T - 1} {R_{mt} } $$

(6)

and

$$\hat \beta _j^ * = \frac{{\hat \beta _j^ - + \hat \beta _j^0 + \hat \beta _j^ + }}{{1 + 2\hat \rho _m }}$$

(7)

where $\hat \alpha _j$ is the Scholes-Williams OLS intercept estimator, $\hat \beta _j^ -$ is the slope estimate from the OLS regression of R _jt on R _{mt −1}; $\hat \beta _j^0$ is the slope estimate from the OLS regression of R _jt on R _mt , $\hat \beta _j^ +$ is the slope estimate from the OLS regression of R _jt on R _{mt +1}, and $\hat \rho _m$ is the estimated first order autocorrelation of R _m .

As another test of robustness, we use generalized autoregressive conditional heteroskedasticity (GARCH) and exponential generalized autoregressive conditional heteroskedasticity (EGARCH) models. With GARCH and EGARCH models, returns continue to follow the form $R_{jt} = \alpha _j + \beta _j R_{mt} + \varepsilon _{jt,}$. However, they differ in the assumption of the behavior of the error term. In GARCH and EGARCH, $\left. {\varepsilon _{jt} } \right|\Psi _{t - 1} \sim \left( {0,h_{jt} } \right)$ and Ψ denotes all information available at time t − 1. Using maximum likelihood to estimate the parameters, the conditional variance for the GARCH model is $h_{jt} = \omega _j \;\delta _j \;h_{jt - 1} + \gamma _j \;\varepsilon _{jt - 1}^2$ with ω _j  > 0, γ _j  > 0, δ _j  ≥ 0, and γ _j  + δ _j  < 1 and the conditional variance for the EGARCH model is where

$$z_{jt} = {\raise0.7ex\hbox{${\varepsilon _{jt} }$} \!\mathord{\left/ {\vphantom {{\varepsilon _{jt} } {\sqrt {h_{jt} } }}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${\sqrt {h_{jt} } }$}}.$$

Boehmer et al. (1991) report that if an event causes event-induced variance, commonly used tests reject the null hypothesis too frequently. Their corrected test requires the residuals to be standardized by the estimation period standard deviation before using the ordinary cross-sectional technique.

Formally, the standardized cross-sectional test statistic, Z _t , is

$$z_t = \frac{{TSAR_t }}{{N^{\frac{1}{2}} \left( {S_{SAR \cdot t} } \right)}}$$

(8)

where TSAR _t is the average event-period standardized residual for event day t, N is the number of firms in the sample, and √(N × S_{SAR ·t} ) is the contemporaneous cross-sectional standard error. TSAR _t is

$${\text{TSAR}}_t = \sum\limits_{j = 1}^N {{\text{SAR}}_{jt} } $$

(9)

where SAR _jt is the standardized error for firm j on day t.

$S_{{\text{SAR}}_{ \cdot {\text{t}}} }^{\text{2}}$ is

$$S_{{\text{SAR}} \cdot t}^2 = \frac{1}{{N - 1}}\sum\limits_{i = 1}^N {\left( {{\text{SAR}}_{it} - \frac{1}{N}\sum\limits_{j = 1}^N {{\text{SAR}}_{jt} } } \right)^2 .} $$

(10)

The standardized cumulative abnormal, SCAR, return for stock j is

$${\text{SCAR}}_{T_{1j} ,T_{2j} } = \left( {{\raise0.7ex\hbox{${{\text{CAR}}_{T_{1j} ,T_{2j} } }$} \!\mathord{\left/ {\vphantom {{{\text{CAR}}_{T_{1j} ,T_{2j} } } {s_{{\text{CAR}}_{T_{1j} ,T_{2j} } } }}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${s_{{\text{CAR}}_{T_{1j} ,T_{2j} } } }$}}} \right)$$

(11)

and $s_{{\text{CAR}}_{T1j,t2j} }$ is

$$S_{{\text{CAR}}_{t_{1j} ,t_{2j} } }^2 = S_{Aj}^2 \left\{ {L_j \left[ {1 + \frac{{L_j }}{{D_j }} + \frac{{\left( {\sum\limits_{t = T_{1j} }^{T_{2j} } {R_{mt} - L_j \overline R _m } } \right)^2 }}{{\sum\limits_{k = 1}^{D_j } {\left( {R_{mt} - \overline R _m } \right)^2 } }}} \right]} \right\}.$$

(12)

where $S_{A_j }^2$ is

$$S_{A_j }^2 = \frac{{\sum\limits_{k = T_{Db} }^{T_{De} } {A_{jk}^2 } }}{{D_j - 2}}$$

(13)

D _j is the number of trading day returns in the D-day interval, $T_{D_b }$ through $T_{D_e }$, that are utilized to estimate firm j’s parameters. L _j is length of the trading-day event period and $\bar R_m$ is the average market return over the estimation period.

The standardized cross-sectional test statistic is

$$z_t = \frac{{\sum\limits_{i = 1}^N {{\text{SCAR}}_{T_{1j,} T_{2j} } } }}{{N^{\frac{1}{2}} \left( {s_{{\text{SAR}} \cdot t} } \right)}}$$

(14)

where

$$S_{{\text{SAR}} \cdot t}^2 = \frac{1}{{N - 1}}\sum\limits_{i = 1}^N {\left( {{\text{SCAR}}_{T_{1i} ,T_{2i} } - \frac{1}{N}\sum\limits_{j = 1}^N {{\text{SCAR}}_{T_{1j} ,T_{2j} } } } \right)^2 } .$$

(15)

To enhance the robustness of our results, we use several nonparametric tests to confirm parametric results. The generalized sign Z-test examines the number of securities with positive and negative average abnormal returns during estimation and event periods under the null hypothesis that the fraction of positive returns during the event period is the same as the fraction of positive returns during the estimation period. The fraction of positive returns under the null hypothesis is

$$\hat \rho = \frac{1}{N}\sum\limits_{j = 1}^N {\frac{1}{{T_j }}} \sum {\varphi _{jt} } $$

(16)

and the test statistic is

$$Z = {{\left( {w - n\hat \rho } \right)} \mathord{\left/ {\vphantom {{\left( {w - n\hat \rho } \right)} {\sqrt {\left[ {n\hat \rho \left( {1 - \hat \rho } \right)} \right]} }}} \right. \kern-\nulldelimiterspace} {\sqrt {\left[ {n\hat \rho \left( {1 - \hat \rho } \right)} \right]} }}.$$

(17)

Corrado (1989) developed the rank Z-test that treats the combined estimation period and event period as a single set of returns and assigns a rank to each daily return for each firm. K _jt represents the rank of abnormal return of stock j during the combined estimation and event period, D _j  + E _j , respectively. The average rank across the combined estimation and event period is

$$\tilde K = \frac{{D + E + 1}}{2}.$$

(18)

The rank Z-test statistic for days T ₁ through T ₂ is

$$Z_r = \left( {T_2 - T_1 + 1} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \left\{ {\frac{{\overline {K_{T_1 T_2 } } - \tilde K}}{{\left[ {{{\sum\limits_{t = 1}^{D + E} {\left( {\overline {K_1 } - \tilde K} \right)} ^2 } \mathord{\left/ {\vphantom {{\sum\limits_{t = 1}^{D + E} {\left( {\overline {K_1 } - \tilde K} \right)} ^2 } {\left( {D + E} \right)^{\frac{1}{2}} }}} \right. \kern-\nulldelimiterspace} {\left( {D + E} \right)^{\frac{1}{2}} }}} \right]}}} \right\},$$

(19)

where

$$\overline {K_{T_1 T_2 } } = \frac{1}{{T_2 - T_1 + 1}}\sum\limits_{t = T_1 }^{T_2 } {\frac{1}{n}\sum\limits_{j = 1}^n {K_{jt} .} } $$

(20)

The jackknife Z-test introduced by Giaccotto and Sfridis (1996) incorporates the standardized abnormal return for each stock j, using the event period sample standard deviation. The standardized abnormal return of day t is

$$\hat \theta = \frac{{A_{jt} }}{{\widetilde{\sigma _{A_{jt} } }}}$$

(21)

where

$$\widetilde{\sigma _{A_{jt} } } = \left\{ {\sum\limits_{t = T_b }^{T_e } {\frac{{\left( {A_{jt} - \bar A_j^2 } \right)}}{{E_j }}} } \right\}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$

(22)

and $\overline {A_j }$ is the event period, $E = T_e - T_b + {\text{1}}$ days, mean abnormal return of stock j. If an event-induced, transient variance change on event day t occurs, $\tilde \sigma _{A_{jt} }$ and $\hat \theta$ become biased. The procedure for reducing the bias is to jackknife the $\hat \theta$ values. To perform the jackknife, we first sequentially delete one abnormal return A _jTs from Eq. 23 and re-compute $\tilde \sigma _{A_{jt} }$, using the new value to re-compute $\hat \theta$ using Eq. 22. We define the latter value $\hat \theta _{\left( { - s} \right)}$ and form pseudo-values

$$\hat \theta _{\left( { - s} \right)} = \left( {E_j } \right)\hat \theta - \left( {E_j - 1} \right)\hat \theta _{\left( { - s} \right)} .$$

(23)

The jackknife estimator for stock j on day t is the mean of the pseudo-values

$$\theta _{jt} = \frac{1}{{E_j }}\;\sum\limits_{t = T_b }^{T_e } {\theta _{\left( { - s} \right)} .} $$

(24)

Then, we average the estimates across the sample of stocks to gain efficiency

$$\overline {\Theta _t } = \frac{1}{N}\;\sum\limits_{j = 1}^N {\theta _{jt} ,} $$

(25)

and the jackknife Z-test statistic for the sample of stocks on day t is

$$t_{{{\text{Jackknife}}}} = \frac{{\overline{{\Theta _{t} }} }}{{{S_{{{\text{Jackknife,}}t}} } \mathord{\left/ {\vphantom {{S_{{{\text{Jackknife,}}t}} } {{\sqrt N }}}} \right. \kern-\nulldelimiterspace} {{\sqrt N }}}}$$

(26)

where

$$S_{{\text{Jackknife}}} = \left[ {\frac{1}{{N - 1}}\sum\limits_{i = 1}^N {\left( {\theta _{jt} - \overline {\Theta _t } } \right)^2 } } \right]^{\frac{1}{2}} $$

(27)

The distribution of t _Jackknife under the null hypothesis is approximately normal with mean zero and unit variance.

To test the significance of the cumulative average abnormal return over date T ₁ through date T ₂, define

$$\hat \theta _{T_1 ,T_2 } = \frac{{\sum\nolimits_{t = T_1 }^{T_2 } {A_{jt} } }}{{\left( {T_2 - T_1 + 1} \right)^{\frac{1}{2}} \widetilde{\sigma _{A_{jt} } }}}.$$

(28)

Sequentially we delete one abnormal return A _jTs from Eq. 22 and re-compute $\tilde \sigma _{A_{jt} }$, using the new value in turn to re-compute $\hat \theta$ using Eq. 28. Now, define the latter value $\hat \theta _{\left( { - s} \right),T_1 ,T_2 }$ and form pseudo-values

$$\hat \theta _{\left( { - s} \right),T_1 ,T_2 } = \left( {{\text{E}}_{\text{j}} } \right)\hat \theta _{T_1 ,T_2 } - \left( {{\text{E}}_{\text{j}} - 1} \right)\hat \theta _{\left( { - s} \right),T_1 ,T_2 } .$$

(29)

The jackknife estimator for stock j during the period (T ₁, T ₂) is the average of the pseudo-values

$$\hat \theta _{j,T_1 ,T_2 } = \frac{1}{{E_j }}\sum\limits_{t = E_b }^{E_ \in } {\theta _{\left( { - s} \right)} .} $$

(30)

We average the estimates across the sample of stocks,

$$\overline {\Theta _{T,T_{2_1 } } } = \frac{1}{N}\;\sum\limits_{j = 1}^N {\theta _{j,T_1 ,T_2 } } $$

(31)

and the jackknife test statistic is

$$t_{{\text{Jackknife}}} = \frac{{\overline {\Theta _{T_1 ,T_2 } } }}{{{{S_{{\text{Jackknife}},T_1 ,T_2 } } \mathord{\left/ {\vphantom {{S_{{\text{Jackknife}},T_1 ,T_2 } } {\sqrt N }}} \right. \kern-\nulldelimiterspace} {\sqrt N }}}},$$

(32)

where

$$S_{{\text{Jackknife}},T_1 ,T_2 } = \left[ {\frac{1}{{N - 1}}\sum\limits_{i = 1}^N {\left( {\theta _{j,T_1 ,T_2 } - \overline {\Theta T_1 ,T_2 } } \right)^2 } } \right]^{\frac{1}{2}} .$$

(33)