Signal Theory and Earnings Surprise



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.

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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:

[pic] Peak Time: When the reaction reaches its peak value (time of maximum overshoot)

[pic] Maximum Overshoot: maximum amount by which the price exceeds its final value

[pic] 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.

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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. 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%)

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Figure 3 (Ticker: NWK, 4/25/90, Correlation: 63.0%)

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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.

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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.

3 themes ran throughout our hypotheses and tests.

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Theme 1 - Uncertainty about Company

How little do people know about the company and the earnings surprise? Is the surprise known by analysts before it hits the market (is it really a surprise at all)? Were expectations for the news actually opposite to what was announced? Is the company well known and well researched by the markets? Is it large or small?

We felt that if there is a great deal of uncertainty about the company, and its earnings we should see a greater magnitude on the day of the surprise. Further more the offset will be greater (or less negative, since in most instances offset is negative), in other words the surprise will not be known by some members of the market before the release date. It will be a true surprise in the sense that the stock price will jump at time zero.

The variables we used to gauge Uncertainty

- Surprise Dollar Amount

- Surprise %

- SUE - Standard Deviation of Analyst Estimates

- # of Estimates

- # of revisions

- Volume on the day of the surprise

- Market Cap

Theme 2 - Momentum

Is the company on a tear? Is it a growth company? Does the market have reason to be excited about the company?

We felt that stocks, which had significant momentum in their fundamental performance, would have increased magnitude, and perhaps reduced peak times. The stocks would be more exciting and enticing.

The variables we used to gauge Momentum

- Volume

- Price/Book

- Price/Earnings

- 1 Yr Sales Growth

Theme 3 - Fundamental/Financial Strength

How strong are the companies experiencing the surprise? Is the surprise due to improved operations and underlying business economics?

We felt that stronger companies would elicit stronger surprise reactions; greater magnitudes, persistent setting times, and shorten peak times.

The variables we used to gauge Fundamental/Financial Strength

- S&P Senior Bond Rating

- Debt/Equity

- Div Yield

- ROA

- 1 Yr Sales Growth

Predicting Magnitude

We tested Magnitude with greatest vigor. We believed Magnitude would be influenced by uncertainty, momentum, and Financial/Fundamental character.

The table outlines our predictions and their directionality.

We believed that uncertainty would yield greater magnitude as the surprise would be more genuine.

Greater momentum would yield greater magnitudes.

Finally sound Financial Fundamentals would yield favorable stock price reactions and greater magnitude.

Predicting Offset

Whispers in the market; a phrase coined by Zack’s Investment Research, describes the amount of information leakage before the announcement date. A positive earnings announcement might not be a surprise at all if indeed the news was leaked and traded upon prior to the earnings release date.

We observed that most reactions to positive or negative surprise occur 7 1/3 days prior to the earnings release date. This is the average Offset of over 600 observations. It is very rare that Offset is positive.

Our tests for Offset focused on the uncertainty theme. Is the company obscure or well covered? How many Analysts are following the stock, talking to management etc.?

Predicting Peak Time

We predicted Peak Time with variable that reflected uncertainty about the stock, as well as momentum variables.

We expected to see longer peak times the less known the company, as the market caught up with the surprise.

We expected to see shorter peak times for momentum stocks.

Predicting Maximum Overshoot Percentage

We predicted that maximum overshoot percentage would be a function of the uncertainty surrounding the event, and the subsequent momentum of the company.

Predicting Settling Time

Muck like Maximum Overshoot, we predicted Settling time with a predominant uncertainty theme.

The greater the uncertainty surrounding the event, the longer it would take for the reaction to settle.

Momentum also played a small role in our hypothesis. The greater the participation the longer it should take to settle.

Results of Statistical Analysis

Magnitude

Data Preparation:

1. We filtered the data for Magnitudes above and below 1 or -1. Such data points were small in number and usually the result of a fit error.

2. Several variables had incomplete and missing data. We filtered our sample set for these, no other adjustments were made to the data.

3. We analyzed positive and negative surprises separately.

Degrees of Freedom Positive Surprises Regression: 2426

Degrees of Freedom Negative Surprises Regression: 1844

Results: Please see Appendix I (positive surprise), & Appendix II (negative surprise) for our regression output

Results of interest:

1. Surprise $ amount, and Surprise % are not significant.

2. # of Estimates is borderline significant with a p-value of .16, the direction is in line with our hypothesis. The more Estimates the lower the expected Magnitude of the reaction.

3. Daily Volume was highly significant in both positive and negative cases. It exhibited a T-stat of 3.06, and a p-value of .0023 for positive surprises, directionality was in line with our hypothesis. The greater the participation the greater the Magnitude of the reaction. The Negative surprises actually had reversed directionality; greater participation meant lower magnitude in negative surprise situations.

4. # of Estimate Revisions Upward is significant with a p-value of 0.09. The directionality is negative and in line with our hypothesis. The larger the number of estimate revisions, the greater the whispers in the market betraying the surprise and reducing magnitude.

5. # of Estimate Revisions Downward was significant with a p-value of 0.04 for negative surprises; the directionality of this result was not in line with our hypothesis. We predicted that downward estimate revisions would dampen magnitude of negative surprises; instead it seems that downward estimate revisions by analysts actually increased the magnitude of the earnings surprise.

6. Price/Book is highly significant with a t-stat of -2.19, and a p-value of 0.03, the directionality is opposite to our hypothesis. We believed that higher Price/Book Ratios would signal growth stocks, and would increase Magnitude – a momentum story. This does not seem to be the case. Higher Price/Book ratios time and time again exhibit smaller magnitudes. 1-Year Sales growth was another variable we used to gauge momentum, and growth in companies. This variable did not fare as well in the regression (t-stat: -1.22, p-value: 0.22) however its direction told the same story. It is possible that growth companies do not exhibit greater magnitudes, or perhaps that 1-year Sales Growth, and Price/Book variables are not a good measure for true growth companies.

7. Debt to Equity was also significant with a t-stat of -1.88 and a p-value of 0.06, directionality suggested that higher debt reduces the magnitude - in line with our hypothesis. This can be telling a story of industry differences, as well as financial health.

8. Dividend Yield was highly significant with a t-stat of -3.89 and a p-value of 0.0001. Directionality suggested that higher dividend yielding companies had less magnitude. Our hypothesis was that Dividend Yield would signal financial strength and hence larger magnitudes on average. It is possible that Dividend is a better gauge for growth companies than the 1-year sales growth, and Price/Book ratio we saw in observation #6. If this is the case then the observed direction of Dividend yield would make sense

9. Return on Assets was significant for negative surprises only (t-stat: -3.4). The Directionality suggested that distressed companies (low return on assets) suffer from greater negative stock price reactions to negative earnings surprise - as predicted

Offset

Data Preparation:

1. A small number of data points were eliminated due to faulty curve fits

2. Several variables had incomplete and missing data. We filtered our sample set for these, no other adjustments were made to the data

3. We tested for Offset in Positive surprises, and Negative surprises separately

Degrees of Freedom Positive Surprises Regression: 2611

Degrees of Freedom Negative Surprises Regression: 2535

Results: Please see Appendix III (positive surprise), and Appendix IV (negative surprise) for our regression output

Results of interest:

Predictive tests for Offset returned several significant results however the directionality of the results befuddled us. We estimated that greater uncertainty about the company (less analyst attention) would yield higher Offset and vice versa. Instead we found that less uncertainty about the company yielded greater Offset values. Ie. The stock market reaction would come relatively late.

1. # of Analysts is a highly significant factor in determining Offset (t-stat:2.66, p-value: 0.007). However the directionality is not in line with our hypothesis under the tests for positive surprises. An increasing # of analysts yields a later stock price reaction. This is not to say that the market reacts after the earnings date (most offsets are less than 0), however on average the # of analysts does not seem to make the reaction come earlier. When we test for negative surprise 2 things happen. First, the # of analysts has a less significant impact and Second, the direction reverses and acts in accordance to our hypothesis – more analysts = earlier stock price reaction. This finding coincides with the findings of Mikhail, Walther and Willis (2002)

2. Daily Volume for positive surprise is a highly significant factor in determining Offset (t-stat: -3.19, p-value: 0.0014). The direction however is not in line with our hypothesis. The greater the daily volume, the less the offset is. We expected to see high volume on the surprise day for companies without ‘whisper’, for true surprises. Our findings show that the earlier the reaction the greater the volume on the day of surprise announcement.

3. Both negative and positive surprise intercepts have a t-stat in the upper 20s, and coefficients of approximately -7. This is good evidence that on average reaction to earnings surprises takes place approximately a week before the actual surprise date.

4. The # of downward revisions by Analysts for negative surprise is a significant factor in determining Offset (t-stat: 2.38, p-value: 0.02), the direction is not in line with our hypothesis.

5. Finally SUE – Deviation in Earnings Estimates is significant for negative surprises when predicting Offset (t-stat: -1.96, p-value: 0.05). The direction however is not in line with our hypothesis.

Peak Time

Data Preparation:

1. Some data points were eliminated due to faulty curve fits, or just fits that resulted in extremely small Wd, forcing peak time to misbehave.

2. Several variables had incomplete and missing data. We filtered our sample set for these, no other adjustments were made to the data

3. We tested for Peak Time in Positive surprises, and Negative surprises separately

Degrees of Freedom Positive Surprises Regression: 1485

Degrees of Freedom Negative Surprises Regression: 1181

Results: Please see Appendix V (positive surprise), the negative surprise tests did not return statistically reliable variables however the results are found in Appendix VI.

Results of interest:

1. Price to Book Ratio had the greatest statistical significance (t-stat: -2.49, p-value: 0.01), the directionality was in-line with our hypothesis. Price to Book may be regarded as a gauge for ‘hot’ or momentum stocks. We estimated that ‘hot’ stocks would experience a shorter time to their peak price. The direction suggests that greater surprise leads to longer peak times.

2. Surprise % and Surprise $ exhibited correct directionality according to our original hypothesis but their statistical significance was borderline (p-values of approx. 26)

3. Likewise SUE – a proxy for uncertainty – exhibited directionality in line with our hypothesis with a p-value of only 0.16. The direction suggests that greater uncertainty leads to longer peak times.

4. With borderline significance (p-value: 0.26) we find that larger market values experience longer peak times after positive surprise.

5. 1 Year Sales Growth Rate has borderline significance (p-value: 0.27). The direction suggests that higher growth rates yield shorter peak times. Very much in line with our hypotheses.

Maximum Overshoot Percentage

Data Preparation:

1. A small number of data points were eliminated due to faulty curve fits

2. Several variables had incomplete and missing data. We filtered our sample set for these

3. We removed instances where reactions had not settled before 30 days (the limit of our observation period)

4. We tested both negative and positive surprise in the same regression

Degrees of Freedom both Positive & Negative Surprise Regression: 1599

Results: Please see Appendix VII

Results of interest:

1. The intercept solidly confirms that an overshoot exists. (t-stat: 37), coefficient 63%

2. The absolute dollar value of surprise is significant (t-stat: -1.57, p-value: 0.12). The greater the Surprise dollar value the less the maximum overshoot. This directionality contradicts our initial hypothesis.

3. # of Analyst Estimates returns borderline statistical significance (t-stat: 1.19, p-value:0.23). The direction suggests that greater # of analysts, increases the overshoot. Perhaps Analysts tend to create momentum rather than certainty about the appropriate settling value.

Settling Time

Data Preparation:

1. A number of data points were eliminated due to faulty curve fits

2. Several variables had incomplete and missing data. We filtered our sample set for these, no other adjustments were made to the data

3. We removed instances that had not settled within the 30 day time horizon

4. We tested both negative and positive surprise in the same regression

Degrees of Freedom both Positive & Negative Surprise Regression: 1874

Results: Please see Appendix VIII

Results of interest:

1. Absolute Value of Surprise % was significant (t-stat: 1.63, p-value: 0.1), the direction was in line with our hypothesis; Greater surprises have longer settling times. It is also noteworthy that Surprise $ amount returned the same direction although it was borderline significant.

2. Volume on surprise day was also significant (t-stat: 1.76, p-value: 0.08), the direction was in line with our hypothesis; greater volume resulted in longer settling times.

3. Market Value was particularly strong (t-stat: -2.75, p-value: 0.01). The direction suggested that smaller stocks would take a longer time to settle, in-line with our hypothesis was. This finding is also in agreement with Bernhard & Thomas who found a size effect in the drift of reactions that was greater for smaller companies.

Extra Findings

Although this part of the analysis is not part of our main analysis, while we were collecting and working with the data, we found interesting patterns for earnings surprises. We have decided to look into these patterns, as they affect the key parameters that we are trying to analyze in our report. Therefore, in this section, we will discuss a few of our interesting findings regarding how stock prices react to earnings announcement in different industries and across different time periods, and how our reaction function looks across industries and through time.

Predicting Cross-Industry Effects

Hypothesis

| |Industry Effect |

| |Relevance? |Direction |

|Magnitude |Yes |+ |

|Offset |No |NA |

|Peak Time |No |NA |

|Max Overshoot % |Yes |+ |

|Settling Time |No |NA |

In the very early stage of the project, we stumbled upon some interesting trends about how industry affects price reaction in general. While looking at price history of stocks for general brainstorming purposes, we found that the “old economy”, traditional industries seems to have a small reaction to earnings surprises, while the new economy industries seems to have a bigger reaction to surprises.

We believed that market reaction for these new economy stocks on earning surprises is usually higher than that of the old economy stocks due to the more cyclical nature of these stocks. A steel company missing earnings estimate might just indicate a bad quarter of demand slowdown, while a computer communication company missing earnings estimate has a higher chance of denoting possible loss of competitiveness due to poor improvement in technological evolution. The positive overshoot comes directly from the high market reaction, as high magnitude of reaction tend to lead to higher overreaction.

Methodology and Data Preparation

|Steel |1 |

|Gas Distributors |2 |

|Pulp & Paper |3 |

|Trucks/Construction/Farm Machinery |4 |

|Homebuilding |5 |

|Electric Utilities |6 |

|Contract Drilling |7 |

|Finance/Rental/Leasing |8 |

|Electronic Equipment/Instruments |9 |

|Semiconductors |10 |

|Medical Specialties |11 |

|Packaged Software |12 |

|Telecommunications Equipment |13 |

|Computer Communications |14 |

|Biotechnology |15 |

Industry is our independent variable in this regression. The dependent variables are the reaction curve parameters. We aim to see how the general look of the curve changes across industries. Using the “industry grading” scale provided in the table to the right, we regressed the parameters of our reaction function to the corresponding industry number.

The industry number was arbitrarily assigned. Industries getting the lowest rating are the most stable, traditional industries. While industries getting the highest rating are the “new economy” industries that are technologically intensive.

We picked fifteen industries that represent the spectrum described above, using industries with more incidents of surprises while picking industries to fill the full spectrum of old to new economy industries.

We filtered out reactions with extreme magnitudes above 2 and below 2, and separated the regression into positive and negative surprises.

Results of interest

We discovered statistical evidence for what we had suspected all along – that the new economy industries tend to have a much higher magnitude and overshoot compared to the old economy industries. What was surprising was how strongly statistically significant was magnitude correlated with industry characteristic in positive surprises (t-stat of 6.29 and p-value of 5.56E-10). The negative surprise also showed good results for magnitude (t-stat of 2.03 and p-value of 0.04) and maximum overshoot % (t-stat of –2.63 and p-value of 0.008). This finding, however, informs us of the fact that there are some noises in our reaction that we did not account for. However, we believe that the dependent variable that we picked are capable of catching the same information. It is just really interesting to see this result in aggregate and in a more “primary” level.

Table - Effect of Industry on Reaction (Positive Surprises)

| |Coefficients |Standard Error|t Stat |P-value |Lower 95% |Upper 95% |

|Magnitude |3.689584 |0.586769 |6.28797 |5.56E-10 |2.537617 |4.841552 |

|Offset |-0.00285 |0.017156 |-0.16592 |0.868267 |-0.03653 |0.030835 |

|Peak Time |-2.5E-07 |6.35E-07 |-0.39844 |0.69042 |-1.5E-06 |9.93E-07 |

|Max Overshoot |0.443493 |0.264647 |1.675792 |0.09421 |-0.07607 |0.963059 |

|5% Settling Time |2.1E-05 |4.2E-05 |0.499815 |0.617357 |-6.1E-05 |0.000103 |

Please also see Appendix IX&X

The Effects of Time

Hypothesis

| |Time Effect |

| |Relevance? |Direction |

|Magnitude |Yes |+ |

|Offset |No |NA |

|Peak Time |No |NA |

|Max Overshoot % |Yes |+ |

|Settling Time |No |NA |

How earnings surprise reaction changed throughout our observation period was another interesting trend that we discovered during our project. Intuitively, with new economy stocks more dominant in the market recently, it makes sense for time to have some effect on the shape of the reaction curve. Indeed, while we were brainstorming using price history curves, we did seem to observe some sort of correlation between time and the magnitude of the reaction to earnings announcement.

We believe that time has a positive correlation with surprise magnitude. In fact, this is what we see in the market all the time now. The stock price of companies really suffer when they miss earnings estimates. The market knows that there are many legal accounting methods that can help companies arrive at their predicted forecast. In fact, there is a growing trend that companies play the “just-make” earnings strategy to maximum shareholder wealth. Therefore, when companies do significantly beat or miss the market estimate, it is believed by that market that there has to be enough fundamental reason priced beyond current market for that to happen, therefore causing larger-than-normal price reactions to earning surprise announcements.

Methodology and Data Preparation

We assigned ordinal number one to the first surprise in all our data, two to the next surprise, and so on. Again, we regressed positive and negative surprises separately. Data was filtered for extreme magnitudes, leaving us with 2004 negative and 3428 positive observations.

Results of interest

Regression result showed statistically significant result for positive correlation between magnitude and time, and negative correlation between offset and time. However, this result is only significant in the positive surprises (t-stats of 1.98 and p-value of 0.04 for magnitude, t-stats of –2.56 and p-value of 0.01 for offset). That means surprises tend to be bigger in recent periods. Surprisingly, another significant result is offset, showing decreased offset in more recent periods. That means there is more whisper in the market now. In other words, the efficiency of the market has been improving gradually. This is also an interesting finding to chew on. To sum things up, it did not surprise us to see how the market have changed in the past 15 years. For future studies, we should use regressions with a decay factor to account this changing market dynamics in order to put a heavier emphasis on more recent data. That way, we can get a more realistic view of what future reaction curves will be like.

Conclusion

Commentary on Regression Results

We had very strong results in some cases. In others, we found directionality to be counter-intuitive.

Offset especially returned statistically significant results with directions opposite from our hypothesis. Predicting Offset would be good grounds for further study.

Another puzzling result is the counter intuitive direction of the Price to Book ratio, and the 1-Year Sales Growth when predicting Magnitude. We estimated that High Price to Book and 1-Year Sales growth would signal growth companies and enhance Magnitude. This is not the case. It might be that a high 1-Year Sales Growth Figure lets the market know about surprise beforehand hence dampening the effect on the surprise date. High Price to Book values could also be evidence of market overvaluation dampening the effects of surprise.

Through cross-industry analysis we attempted to isolate growth vs. mature companies in another way. We had much better results, which were in line with our hypothesis about growth companies.

Also, our Dividend Yield variable supports the hypothesis that growth companies have a higher magnitude after a surprise date - Low Dividend Yielding companies had greater magnitudes.

Signs often change when we went from positive to negative surprise. For example: # of Analysts covering a stock, actually created a larger offset for positive surprises, but a smaller offset for negative surprise. Our original hypothesis was that the # of Estimates variable would signal whisper – greater whisper = early stock price reaction (smaller offset). We can not explain our results. This is a good de-mark for further study.

As for results in line with our hypothesis in significance and direction there are a number of great examples. Such examples are listed above in the Observations section of this paper. Many of these findings can be applied to trading strategies. Already, our results can provide a start point for estimations of magnitude and peak time for a given earning surprise - highly applicable to short-term long positions. Many other creative uses can be found.

Possible Implications of our Study

One of the biggest goals of our project is to be able to use our findings to get a better idea of asset pricing around abnormal events. From the beginning, we question how well Black-Scholes work around events, causing non-random price movement that the formula might be completely be able to capture. We feel that with the robust findings from our research, we are one step closer to seeing the light of using a different approach for options pricing around expected events.

Although our main motive is not to develop a trading strategy around our findings, we could barely resist the temptation to put our findings to work. Theories and statistical findings can be distant from real life situations. Scholes and Merton put their knowledge to use in the real world, and had quite a few years of success. Although we believe that our work lays sound fundamental background for a trading strategy that involves an adjusted Black-Scholes formula, the scope of this ambition is beyond the range of this project.

Besides option pricing, other transactions could be used to take advantage of our findings. The magnitude of reaction itself, although very interesting, can hardly be relied solely on when devising a trading strategy, because magnitude is a function of the direction of surprise, which is unknown until t=0. However, great opportunity exists after the reaction peaks. By marginally being able to predict the peaking time of the reactions, we can sell short at the proposed peaking time. To avoid taking direction, we can buy future call option at the proposed settling time, therefore taking advantage of the overshoot of the reaction. (Figure 4) Another interesting strategy might be to buy stock at t=0 and sell at peak time, therefore earning the some money before the stock peaks out. (Figure 5)

Figure 4

Figure 5

There are lots of interesting options for trading strategies out there that is applicable with our findings. The strategies listed here is just a couple that came to our mind immediately. Price movement around events is a really interesting topic. It follows its own path, and to put a generalized pricing formula for options around it can be hard. We believe that we have established a solid foundation for future work on this area.

Appendix – Regression Summaries

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Appendix IX – Cross-Industry Analysis Regression Results

Industry Regression (Positive Surprises)

|Regression Statistics |

|Multiple R |0.166145 |

|R Square |0.027604 |

|Adjusted R Square |0.016453 |

|Standard Error |1425.902 |

|Observations |442 |

|ANOVA | | | | | |

|　 |df |SS |MS |F |Significance F |

|Regression |5 |25165119 |5033024 |2.475425 |0.031561 |

|Residual |436 |8.86E+08 |2033196 | | |

|Total |441 |9.12E+08 |　 |　 |　 |

|　 |

|Multiple R |0.166145 |

|R Square |0.027604 |

|Adjusted R Square |0.016453 |

|Standard Error |1425.902 |

|Observations |442 |

|ANOVA | | | | | |

|　 |df |SS |MS |F |Significance F |

|Regression |5 |25165119 |5033024 |2.475425 |0.031561 |

|Residual |436 |8.86E+08 |2033196 | | |

|Total |441 |9.12E+08 |　 |　 |　 |

|　 |

|Multiple R |0.063898 |

|R Square |0.004083 |

|Adjusted R Square |0.002628 |

|Standard Error |562.0978 |

|Observations |3428 |

|ANOVA | | | | | |

|　 |df |SS |MS |F |Significance F |

|Regression |5 |4432570 |886513.9 |2.805833 |0.015567 |

|Residual |3422 |1.08E+09 |315953.9 | | |

|Total |3427 |1.09E+09 |　 |　 |　 |

|　 |

|Multiple R |0.041797 |

|R Square |0.001747 |

|Adjusted R Square |-0.00075 |

|Standard Error |584.6828 |

|Observations |2003 |

|ANOVA | | | | | |

|　 |df |SS |MS |F |Significance F |

|Regression |5 |1194711 |238942.1 |0.69896 |0.62424 |

|Residual |1997 |6.83E+08 |341854 | | |

|Total |2002 |6.84E+08 |　 |　 |　 |

　 |Coefficients |Standard Error |t Stat |P-value |Lower 95% |Upper 95% |Lower 95.0% |Upper 95.0% | |Intercept |1004.948 |23.71604 |42.3742 |1.9E-280 |958.4373 |1051.459 |958.4373 |1051.459 | |Magnitude |-5.27363 |7.661364 |-0.68834 |0.491318 |-20.2987 |9.751474 |-20.2987 |9.751474 | |Offset |-1.23728 |1.532354 |-0.80744 |0.41951 |-4.24246 |1.767899 |-4.24246 |1.767899 | |Peak Time |-1.8E-05 |2.19E-05 |-0.84124 |0.400312 |-6.1E-05 |2.45E-05 |-6.1E-05 |2.45E-05 | |Maximum Overshoot |-2.72966 |20.78278 |-0.13134 |0.895518 |-43.4879 |38.02855 |-43.4879 |38.02855 | |5% Settling Time |-0.00419 |0.003821 |-1.09738 |0.272608 |-0.01169 |0.0033 |-0.01169 |0.0033 | |

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