6A.2 Earnings surprises and post-earnings-announcement drift

The expectation model is $\mathbb E_{q-1}[E_q] = E_{q-4} + \hat\delta_q$: last year's same-quarter EPS plus the usual year-on-year change. Two alternatives are common. The analyst surprise replaces $E_{q-4} + \hat\delta_q$ with the consensus forecast and scales by price; in India, consensus coverage is thin outside the top 200 or so stocks and the published consensus mixes stale forecasts. The earnings announcement return (EAR) skips the accounting model altogether and uses the abnormal return over $[-1, +1]$ as the measure of surprise. EAR captures revenue, margins and guidance, which EPS alone misses.

The evidence

• Ball and Brown (1968) first showed that returns after the announcement continue in the direction of the earnings news.
• Bernard and Thomas (1989) sorted US firms into SUE deciles each quarter over 1974–1985. The long-extreme-good, short-extreme-bad portfolio earned a positive abnormal return over the following 60 trading days in most quarters; secondary summaries of their Table 1 put the 60-day spread at about 4.2% and positive in 41 of 48 quarters¹. They argued that the drift is a delayed response, not a risk premium, because it does not line up with any identifiable risk.
• Bernard and Thomas (1990) found the mechanism: seasonal differences of quarterly earnings are positively autocorrelated at lags 1–3 and negatively at lag 4, and a disproportionate share of the drift arrives in the days around the next announcements, with signs matching those autocorrelations. Prices behave as if investors forecast with a seasonal random walk, ignoring the autocorrelation.
• Later work explains persistence by limits to arbitrage and inattention. Mendenhall (2004) finds the drift larger where idiosyncratic risk makes arbitrage expensive. DellaVigna and Pollet (2009) find weaker immediate responses and stronger drift for Friday announcements. Hirshleifer, Lim and Teoh (2009) find the same on days with many competing announcements.
• The US drift has weakened sharply. Martineau (2022, "Rest in peace post-earnings announcement drift") argues it has largely disappeared in recent US data, especially for larger stocks. Treat PEAD as a well-documented historical regularity whose current size must be measured, not assumed.

India The evidence is mixed and comes from small samples. An IIM Bangalore working paper (Agarwal and Sridharan, 2013) reports drift in NSE stocks over 120 days, together with pre-announcement movement. Gupta and Dhusia (2021, Trends in Economics and Management, 100 NSE firms, 2014–2018) conclude that prices are efficient with respect to earnings announcements in their sample. A dissertation on BSE 500 firms (FY2007–2012) finds drift after bad news but not after good news [all S: abstracts only]. There is no large, survivorship-free Indian PEAD study with SUE deciles comparable to Bernard–Thomas in the sources checked here. Replicating one is a good research exercise (Problem 6A.3 and m06a_pead.py).

Proof

At announcement $q+1$ the market's surprise is the realised $\tilde d_{q+1}$ minus the naive expectation 0, so the price moves by $c\,\tilde d_{q+1}$. Conditional on information at $q$, $\mathbb E[\tilde d_{q+1} \mid \tilde d_q] = \phi\,\tilde d_q$ because $\eta$ has mean zero. For lag $k$ with a general stationary process, the linear projection of $\tilde d_{q+k}$ on $\tilde d_q$ is $\rho_k\,\tilde d_q$ (unit variance).∎

A rational investor who knew $\phi$ would already have moved the price by $c\phi\,\text{SUE}_q$ on day 0, leaving nothing predictable. The drift is the gap between the naive and the correct forecast. With $c$ = 1.5% per unit SUE and $\phi$ = 0.35, the next announcement carries 0.53% per unit SUE, about ±0.92% for the extreme deciles (Problem 6A.2).

Proof

For $Z \sim N(0,1)$ and threshold $c$, $\mathbb E[Z \mid Z > c] = \int_c^\infty z\varphi(z)\,dz / P(Z > c) = \varphi(c)/P(Z > c)$, because $\varphi'(z) = -z\varphi(z)$. With $c = z_{1-1/n}$, $P(Z > c) = 1/n$. By symmetry the bottom group has mean $-\varphi(c)\,n$. Multiply the difference of conditional means by $\lambda$ using the tower property.∎

So a decile spread of 4.2% corresponds to $\lambda \approx 1.2\%$ of drift per unit of SUE. The synthetic panel in m06a_pead.py injects exactly this and checks that your estimator recovers it.

Measuring the reaction: an event-study primer#

Every statement about how prices react to results ("the stock fell 6% on the numbers", "PEAD is 4% over 60 days") is a comparison with what the stock would have done anyway. This primer gives the machinery used in the rest of the module. M12 · 12.1–12.3 later extends it (multi-event standardised tests, event-induced variance, geopolitical events) and refers back here.

Windows Index trading days by event time $\tau$, with $\tau = 0$ the first session in which the market could trade on the news (6A.1). The estimation window $\tau \in [T_0 + 1, T_1]$ has length $L_1$, for example $[-250, -11]$ with $L_1 = 240$. The event window $[T_1 + 1, T_2]$ has length $L_2$, for example $[0, +1]$ for a release plus its concall. The gap keeps pre-announcement leakage out of the benchmark.

Market model and abnormal returns For stock $i$ and a market proxy $m$ (the Nifty 50, or the sector index when the question is "beyond the sector"),

Let $X$ be the $L_1 \times 2$ estimation-window matrix with rows $(1, R_{m\tau})$ and $R_i$ the vector of stock returns. OLS gives $\hat\theta_i = (\hat\alpha_i, \hat\beta_i)' = (X'X)^{-1}X'R_i$ and $\hat\sigma_i^2 = \lVert R_i - X\hat\theta_i\rVert^2/(L_1 - 2)$. In the event window, with regressor matrix $X^*$ and return vector $R_i^*$,

where $\gamma$ is the $L_2$-vector of ones (or of ones on a sub-window). The market-adjusted return $R_{i\tau} - R_{m\tau}$ is the special case $\alpha = 0$, $\beta = 1$ with nothing estimated.

Proof

Under the null, $R_i^* = X^*\theta_i + \varepsilon_i^*$ and $\hat\theta_i - \theta_i = (X'X)^{-1}X'\varepsilon_i$, so $AR_i^* = \varepsilon_i^* - X^*(X'X)^{-1}X'\varepsilon_i$: today's noise minus the estimation error of the benchmark. The two windows do not overlap, so $\varepsilon_i^*$ and $\varepsilon_i$ are independent normal vectors, and $AR_i^*$ is normal with mean zero and covariance $\sigma_i^2 I + X^*(X'X)^{-1}X'(\sigma_i^2 I)X(X'X)^{-1}X^{*\prime} = \sigma_i^2[I + X^*(X'X)^{-1}X^{*\prime}]$. Pre- and post-multiplying by $\gamma$ gives the variance of $CAR_i$. For the studentisation, the residual vector is $M\varepsilon_i$ with $M = I - X(X'X)^{-1}X'$ idempotent of rank $L_1 - 2$, so $(L_1 - 2)\hat\sigma_i^2/\sigma_i^2 \sim \chi^2_{L_1-2}$. Since $MX = 0$, $M\varepsilon_i$ is uncorrelated with, hence (jointly normal) independent of, $(X'X)^{-1}X'\varepsilon_i$, and it is independent of $\varepsilon_i^*$. So $CAR_i/(\sigma_i\sqrt{v_i}) \sim N(0,1)$ is independent of $\hat\sigma_i^2$, and the ratio is $t_{L_1-2}$ by definition. The one-day factor is the $(\tau,\tau)$ element of $I + X^*(X'X)^{-1}X^{*\prime}$, using the explicit inverse of the $2\times2$ moment matrix.∎

The estimation-error term is small ($1/L_1$) unless the market itself moved a lot on the event day, so the quick version $t \approx AR_0/\hat\sigma_i$ is usually adequate for one stock (Solution 6A.18(c) uses it). With $L_1 = 240$ the 5% two-sided critical value of $t_{238}$ is 1.970.

Many announcements For $N$ results events, average the CARs: $\overline{CAR} = N^{-1}\sum_i CAR_i$. If the events were independent, $\operatorname{Var}(\overline{CAR}) = N^{-2}\sum_i \sigma_i^2 v_i$. The cross-sectional $t$, $\overline{CAR}/(s_{CAR}/\sqrt N)$ with $s_{CAR}$ the sample standard deviation of the $CAR_i$, is robust to results days being more volatile than estimation days. Results seasons, however, cluster in calendar time: a sector reports within the same fortnight, so abnormal returns are correlated. With common variance $\sigma^2$ and average pairwise correlation $\bar\rho$,

so a test that assumes independence overstates $t$ by the factor $\sqrt{1 + (N-1)\bar\rho}$: with $N = 100$ and $\bar\rho = 0.05$ that is 2.4. The standard remedy is a calendar-time portfolio: each day, average the abnormal returns of all stocks inside their event windows, then test the mean of that daily series with its own time-series standard error.

Measuring drift properly Use market-model or characteristic-matched abnormal returns as above; $\tau = 0$ from exchange timestamps; SUE computed only from information available before the announcement; groups formed using breakpoints from the cross-section of that quarter; calendar-time portfolios for the standard error; and buy-and-hold rather than summed returns when moves are large (6A.13.4).