# Practical way to estimate price sensitivity to unexpected earnings (i.e., post-earnings drift)?

Post-earnings announcement drift is a well documented anomaly in financial research. In 2017 May NBER paper, Replicating Anomalies, the authors found that anomalies related to standardized unexpected earnings to be among the most robust quantitative factors. However, I am somewhat unsatisfied with canonical approaches to model calibration.

Canonically, post-earnings announcement drift is usually measured as a standardized score, akin to a Z-score, as such:

$$\mathbb{SUE} = \frac{(E_{act,t}-E_{exp,t})-\mathbb{E}^T_t[E_{act}-E_{exp}]}{\sqrt{\mathbb{E}^T_t[(E_{act}-E_{exp})^2]}}$$

where: $\mathbb{SUE}$ is standardized unexpected earnings; and $E$ is earnings.

Yet, I believe that an astute observer should also take into account the post-earnings drift. But do so, I would think we need to know price elasticity (i.e., "sensitivity") of an expected change in price (i.e, "drift") to an expected change in earnings, such that:

$$\mathbb{E}[\mu] = \frac{E_{act}-E_{exp}}{P} *\frac{dP}{dE} - \frac{dP}{P}$$

where:

$\mathbb{E}[\mu]$ is the expected residual logarithmic drift in an asset price following the announcement actual EPS ($E_{act}$) in light of previous analyst expectations ($E_{exp}$);

$\frac{dP}{dE}$ is the constant of price elasticity (i.e., price "Beta") to a change in earnings;

$\frac{dP}{P}$ is the post-earnings logarithmic price drift following an announcement.

Thus the question: what are some practical ways to estimate the constant of price elasticity, $\frac{dP}{dE}$ (i.e., the expected change in price for a change in earnings)? An expected price-to-earnings ratio might be a decent first approximation, but there are problems with this assumption (e.g., it is a very indirect measure; meaningless in cases of negative earnings and asymptotically low/high PEs; etc).

• If x% is the amount of unexpected earnings z% is how much it moved at that announcement, and n% is the current unexpected earnings, and your goal is to predict price movement m%, then wouldn't your formula be (z/x) * n? – ChaimG May 17 '17 at 3:43
• Thanks. I think you're saying that $\frac{dP}{dE}$ can be discretized as follows: $$\frac{dP}{dE} = \mathbb{E} \bigg{[} \frac{\Delta P_{t,T}}{(E_{act,t}-E_{exp,t})} \bigg{]}$$ And then my formula simplifies: $$\mathbb{E}[m] = \frac{1}{P_t} \bigg{(} \frac{dP}{dE}(E_{act,t}-E_{exp,t}) - \Delta P_{t,T} \bigg{)}$$ where now, $m$ is the percentage excess drift. But I am still not sure how to estimate price sensitivity to a change in earnings. How much depends on whether this surprise is seen as a one time perturbation versus an impetus to revise future expectations? – David Addison May 17 '17 at 18:04

As pure speculative commentary and a non-quantitative answer, you may want to look at the short-term options volatility as a factor for possibly fine tuning your model.

Your goal seems to be to find the elasticity or 'beta' factor of the price movement. The price drift you are looking at implies some form of information dissemination or perception change in the market. I would expect that a large part of calculating your beta factor would be determining how much the market's opinions changed. Was this surprise far outside of expectations or was it within an expected range.

The options implied volatility on the short term expirations that expire immediately after the release would give you a measurement of the market's perception of risk before the announcement. If you use this to determine the shock factor of the earnings data it could be related to the elasticity of the movement.

In simple terms, the IV of the short-term options represents the market pricing of the predicted risks associated with the earnings announcement. Think of it as a measurement of Fear, Uncertainty, and Doubt. If IV is extremely high, the market is pricing in a broad range of possible earnings results. Even a large surprise may not be a significant market mover in this case because the risks are priced in and there was little shock from the event.

On the other hand, if the options showed an abnormally low IV prior to the announcement you could imply a high level of confidence in the expected earnings results. The market is pricing in a smaller range of possibilities, and a significant deviation could be interpreted as a large information "Shock" to the market. Theoretically, this would be realized as a greater beta to your expected price movement.

Please understand that this is all speculative and I have no study to back this idea up. On the other hand, it shouldn't be that tough to at least do a sniff test on the concept.

A simple approach would be to take some historical data and run a test. Get a handful of companies' historical earnings data and associated options implied volatilities from just prior to the announcement. Segment your data into two sets. I would normally go with 2/3rd in sample data and 1/3rd out of sample. In this case you may want to just do 50/50.

Run your current model on the out of sample data and figure out the MSE of that test run.

Then use the in sample data to calibrate a simple kernel regression using your model's output as one input and the IV factor as the other. Aim for the smallest predictive MSE you can get during your calibration. Then run the new combined model on the out of sample data you used before. Calculate the new MSE and compare it to the origional test of your raw model.

This isn't the most analytically pure approach, but it should give you a bit of insight. If your error significantly drops in the combined model then there might be some level of information content in the IV that may be useful in a model. It could be worth analyzing further.

If the difference in error rates between the tests isn't significant then there are two possibilities. The first is that the relationship between IV and your beta factor exists, but it is too complex to suss out with a simple kernel regression.

The second and far more likely answer is that no information exists in the relationship and the idea is a dead end. At least it is an easy test, a fast failure is always good.

If you have any interest in the concept send me a comment. I have a pretty good set of historical options data I can contribute to a test. I do a lot of work with volatility trades around earnings, and the results would be interesting to me.

• Using implied volatility as a proxy for expectations is a good idea. Like you, I would assume that IV should be inversely related to price elasticity to an earnings surprise. If I get any latitude in my personal time, I will definitely let you know if/when I can take on this research project. It seems like a low-hanging fruit where a) the mechanism is facile; c) persistence is robust; and, c) the data is readily available. – David Addison May 25 '17 at 23:48
• @DavidAddison I agree. This concept should be fairly simple to prove/disprove. With a fairly simple test, we should be able to either see a reduction of prediction error or not. Right now I am engaged in a project that has most of this data readily available. 10yrs of detailed options history and a reasonably robust earnings history. It wouldn't be a high priority, but I would be interested in the results if you feel like collaborating. – drobertson May 26 '17 at 3:21