Assignment 1: Second-order Systems and Earnings Surprises
Midas Capital Management
matt mcConnell
David Nabwangu
Johnson Yeh
Prof: Campbell Harvey
2/26/04
TABLE OF CONTENTS
Abstract 4
Introduction 4
Signal Theory 6
Data Collection & Preparation 7
Data Filtering 8
Cumulative Abnormal Return Calculation 8
Statistical Analysis 12
Hypotheses 12
Theme 1 - Uncertainty about Company 13
Theme 2 - Momentum 13
Theme 3 - Fundamental/Financial Strength 14
Predicting Magnitude 15
Predicting Offset 16
Predicting Peak Time 17
Predicting Maximum Overshoot Percentage 17
Predicting Settling Time 18
Results of Statistical Analysis 19
Magnitude 19
Offset 20
Peak Time 21
Maximum Overshoot Percentage 22
Settling Time 22
Extra Findings 23
Predicting Cross-Industry Effects 23
Hypothesis 23
Methodology and Data Preparation 24
Results of interest 24
The Effects of Time 25
Hypothesis 25
Magnitude 25
Methodology and Data Preparation 25
Results of interest 25
Conclusion 26
Commentary on Regression Results 26
Possible Implications of our Study 26
Abstract
Our paper takes inspiration from the study of control system engineering to model and explain the shape of short term stock price reaction curves to news events..Earnings surprises were chosen as an easily quantifiable and readily available news event.
After fitting a flexible reaction curve to thousands of earnings surprise events, we performed statistical regressions to explain the shape of the market reaction curve. We modeled the size of the reaction, the time it took to settle to a new price, the amount of time by which the market pre-empted the earnings announcement, the time it took the price to reach its peak-level and the amount of overestimation made relative to the settling price. We then used statistical tests to examine the impact of certain variables on the shape of a surprise reaction.
Earnings surprise is just one news event that hits the market and produces a reaction. It is expectecd that our methods can be applied to any event that hits the market.
Introduction
There is a large body of research on the return pattern around and after earnings announcement. Most studies conclude that a surprise in the earnings announcement leads to abnormal returns in the period following the announcement. This period is most often the trading days from the day after the announcement up to a couple of months after the announcement. In the academic literature the abnormal return pattern is called post-earnings-announcement drift. The drift is in general positive for positive earning announcement surprises and negative for negative earnings announcement surprises. Therefore, studies claim that investors under-react to the information embedded in the earnings surprise. If investors would incorporate the information fully at the time of the announcement, no post announcement drift pattern should appear.
From earliest works of Ball and Brown (1968), numerous studies have been done on delays on a firm’s price responses to earnings announcement. However, specific studies on investor’s reaction appear in Bernard and Thomas’s seminal study (1989), which provides some explanation to the problem. Investors fail to understand the characteristics of the serial correlation in earnings. They prove that investor reliance on a naïve seasonal random walk earnings forecasting model provides some explanation for the post announcement drift. For the decile of stocks with the most negative surprise, they find an abnormal return of -2% up to 60 days after the announcement.
Mikhail, Walther and Willis (2002) find that the significance of the drift decreases with the aggregate experience level of the security analysts that cover the stock. Relating this to Bernard and Thomas, it could be argued that more experienced analysts understand the implication of the current earning on future earning to a higher degree than less experienced analysts.
Bernhard and Thomas also found a size effect in the drift. A smaller market capitalization leads to more significant drift. Controlling for this and other effects, Mikhail, Walther and Willis found that the effect from analyst experience was persistent.
Research has also been conducted on other announcement patterns. Bulkley and Herrias (2002) investigated the returns subsequent to profit warnings. They found no evidence of abnormal returns after the initial reaction.
Kvist and Åberg (2002) make use of the reliability theorem to explain and predict stock price reactions around profit warnings. The theorem states that investors will overstate the value of information with relatively low reliability while understating information with relatively high reliability. In their sample of Swedish stocks they find that most profit warnings leads to investor under reactions. They also find that proxies for information reliability can predict the return pattern subsequent to a profit warning to some extent. Due to the similarity between profit warnings and earnings announcement surprises, their findings is in line with the research on post-earnings-announcement drift.
The post-earnings-announcement drift is intriguing since it is one of the few persistent market anomalies that researchers have not been able to attribute to solely inadequate risk adjustment. In fact, Fama called the post-earnings-announcement drift the “Granddaddy” of market anomalies.
To our knowledge, there has been no academic study that investigates the entire shape of the return pattern around earnings announcements. Making use of control systems engineering, we can characterize the return pattern around earnings announcement surprises by six parameters, and analyze these.
Background on Control Systems
Stock prices react continuously to a stream of news about the company and external economic conditions. Each individual piece of news may be seen as a shock to what would otherwise be a steady value for the stock. The market might be modeled as a system that determines stock prices based on a series of shocks. The input to the system is a stream of news items; the output is a stream of stock prices. The figure below is a schematic representation of a second order control system.
Figure 1: Schematic Representation of a Second Order System
When an abrupt change enters the system (called a step function), the output moves to a newly determined level. However, it does not do so immediately. It takes some time to find the correct level. If the output shots upwards very quickly, it is likely to overshoot the correct level and oscillate a bit as it settles down. The output of the system is fed back and compared to the input, generating an error signal that indicates in which direction the output move move to reach the correct level. A clear analogy to the market can be drawn: traders may initially over- or under-react to the news and then take some time to reach a consensus as to a new stock price. In the interim, the stock price might swing above and below its new market value.
The output response of the second system to a step function follows the form of the equation
The parameters ωd, ωn and ζ, termed the natural frequency, the damped natural frequency, and the dampening ration, determine the shape of the curve. In examining the shape of a stock price reaction curve the an earnings announcement, we adjusted these three variables, plus the addition of three variables specific to the curve fitting procedure, to determine the reaction curve that best fit each surprise. The three additional variables are initial level, magnitude, and offset. Initial level is the non-zero initial level of the stock price. Magnitude is the relative size of the price reaction; a greater magnitude indicates a greater reaction. Offset is the distance from time zero at which the reaction begins. The more negative the offset value is, the earlier the reaction started prior to the recorded announcement date.
For interpretation and relation to tangible factors, we transformed ωd, ωn and ζ in the following ways:
Peak Time: When the reaction reaches its peak value (time of maximum overshoot)
Maximum Overshoot: maximum amount by which the price exceeds its final value
Settling Time: Time it takes for price to be within 5% of its final value
These transformed variables are used in the regression analysis which relates the curve models to observed market data. These transformed variables contain all of the information necessary to define a unique reaction curve.
Data Collection & Preparation
We collected earnings surprise data dating back to 1990. Each instance of surprise was accompanied by a company name and variables listed in the table below.
By using the FACTSET database we were able to compile a list of multiple surprises for over 1000 companies across time, and collect data specific to each company at each instance in time. This enabled us to get an accurate picture of what was happening to a specific company at a specific time.
Data Filtering
1. We examined stocks in the S&P500, and the S&P small cap universe. Most of our stocks were therefore US companies
2. We examined earnings surprise from 1990 onward. The data was best in this time frame, and the time frame had both economic booms and busts. It also restricted our sample size
3. We filtered out surprises less than 30% (both negative and positive). Primarily to restrict the amount of data
4. We filtered out surprises less than 2 cents (both negative and positive). This was to eliminate companies that had little to no earnings to speak of before the surprise. Eg. Amazon.com has earnings estimates of 1 cent, and the reported earnings amount is 2 cents, creating a 50% surprise (passes filter #3). Reaction to this event may not be the shock we are interested in
5. We filtered out surprises in which the original Earnings Estimate was zero. This was to avoid Dividing by zero errors in our statistical analysis
Filtering reduced our sample size to 6060 instances of earning surprise from 1,100 companies, dating back to 1990.
Cumulative Abnormal Return Calculation
Cumulative returns were calculated for each earnings surprise instance beginning from 31 days prior to the earnings announcement to 30 days after the earnings announcement. In order to calculate the cumulative surprise for each company we did the following:
1. Gathered Price data surrounding the surprise using the Datastream database
2. Calculated Daily returns for each company in the days surrounding the surprise
3. Subtracted the relevant S&P500 daily return times the S&P500 Beta at the time of the surprise from the daily returns of each company
4. Calculated abnormal cumulative returns from t = -31 days to t = 30 days
Right away we spotted trends in our data that indicated the existence of abnormal returns around surprise times. The ‘heat map’ depicted below is a snap shot of our cumulative return data.
Exert From Heat Map - Illustrates Stock Price Reaction to Earnings Surprise (positive surprise sample)
We colored orange those days in our surprise window (-31 days from reaction to + 30days) above the average abnormal cumulative return we colored red the maximum observed cumulative return. By surveying the heat map it is plain to see that an abnormal reaction is prominent and forces the latter days in our surprise window to have a higher cumulative abnormal return than the earlier days (for positive surprises). And vice versa for negative surprises.
Curve Fitting Methodology – Application of Signal Theory
With such a large data set to deal with, we have decided to use SAS instead of Microsoft Excel to handle the curve fitting part of our project. We used Excel for the curve fitting for our initial project, but we found it to be extremely time consuming. Besides, Excel only enabled us to perform ordinary least square (OLS) optimization for our curve fits. With a powerful statistical package like SAS, we had a lot more choice in what kind of optimization we used. Furthermore, We were able to perform our analysis much faster.
We chose to utilize the optimization capability in the SAS/OR software. Within SAS/OR, we employed the non-linear programming technique, due to the complicated form of our reaction function. Specifically, we used PROC NLP for curve fits. After, we ended up using its trust region optimization technique for our least square optimization because of the relative low degree of freedom of data that this technique requires. We performed unconstrained least square optimization to begin with, and we very happy with the resulting fits that we got. Therefore, we did not attempt UN-constrained least square optimization. On average, our fits have a 79% correlation with the actual price curve, which is very satisfactory to us. The low correlation fits usually come from the curves with extraordinary numbers in one of the parameters - then our equation fails to capture the real movement of the price. A lot of other low correlation comes from high price movement before the reaction occurred, which we did not attempt to capture in our equation.
Although we have a very complicated reaction function, it does have its limitations. Some yielded extremely nice fits (figure 2), while others produced only marginally satisfactory results (figure 3). Overall, our curve fits were very good, and we feel, resulted in minimal distortion in our final regression results.
Figure 2 (Ticker: APOG, 12/17/97, Correlation: 98.8%)
Figure 3 (Ticker: NWK, 4/25/90, Correlation: 63.0%)
Statistical Analysis
The curve fitting returned 5 variables that were descriptive of our stock price reactions:
1. Magnitude: The size of the initial stock price reaction
2. Offset: The delay, or pre-emption of the reaction relative to the surprise date
3. Peak Time: The time required for the response to reach the first peak of the overshoot. It is not part of our curve fitting function. Rather, this variable is calculated using Wd and Offset numbers.
4. Maximum Overshoot Percentage: The maximum peak value of the response curves less the level at which the curve settles. The maximum overshoot indicates the relative stability of the system. This variable is also not part of the actual curve fitting equation. It was calculated from Zeta.
5. Settling Time: The time required for the response curve to reach and stay within a settling range. This can be compared to the post-announcement drift of Mikhail, Walther and Willis (2002) & Bernard and Thomas’s seminal study (1989) Again, this variable did not result from the actual curve fitting, but rather was calculated using Zeta, Wn, and Offset numbers.
Our regression analysis focused on estimating these 5 parameters.
We also conducted 2 interesting experiments analyzing how earnings surprise reactions have changed over time, and across industry.
Hypotheses
We created hypotheses for each parameter. We want to be in the position where we can estimate each parameter and hence the stock reaction shapes at the time of the surprise.