An Empirical Study for Testing the Stock Market Efficiency and Identifying Abnormal Return Opportunities

Transcription

1 An Empirical Study for Testing the Stock Market Efficiency and Identifying Abnormal Return Opportunities Merve Artman* Central Bank of Turkey, Ankara, Turkey Murat Artman Central Bank of Turkey, Ankara, Turkey Abstract The efficient market theory states that active management in the long term is a waste of money and that an investor is better off placing assets into every type on index fund and should take a passive strategy approach to investing. However, investors can outperform the market and identify abnormalities that give them a trading advantage. This paper studies the stock return data of Herbalife Ltd., a NASDAQ Company, from including the unexpected event day that caused the share price to fall down 19.94%. The contribution of this paper is to show normality, event study, monthly and January effects on stock return performance with using econometrical and statistical tests. First we ask, does the information arrive linearly to the market or do investors react linearly to its arrival? Our results indicate that the stock return data of the company is not normally distributed and there is a possibility of earning abnormal returns by investors. The second question we ask, does the investor s reaction to the market last longer than the event day itself? Our results suggest that the event effect did not absorb on the same day of event. Our evidence suggests that a trader can profit from shorting a stock several days after a major negative event has occurred. We also ask if there are monthly or January effects on the performance of stock returns of the company. We observed that there are significant gains to be had from moving into the stock in the beginning of the month and moving out of the stock by the end of the month and repeating that process over and over. However, there is no statistically significant evidence that unusually high returns amass in the first couple of days of January, while the return for the rest of the year statistically indistinguishable from zero. Key Words: event study, january effect, market anomaly, normality, stock market efficiency

2 1. Introduction About the company Herbalife Ltd. (HLF), incorporated on April 04, 2002, is a global nutrition company. The Company sells weight management, healthy meals and snacks, sports and fitness, energy and targeted nutritional products as well as personal care products. It distributes and sells its products through a network of independent distributors by leveraging direct selling channels. Herbalife categorizes its sciencebased products into four principal groups: weight management, targeted nutrition, energy, sports & fitness and Outer Nutrition. It sells its products in 88 countries to and through a network of approximately 3.2 million independent distributors. 1 Event Herbalife Ltd. s first-quarter conference call on May 01, 2012 is the event we will be analyzing. During the call, activist hedge-fund manager David Einhorn asked some skeptical questions about the company's revenue structure. The share price fell down 19.94% on that day Testing For Normality in Stock Return Data In theoretical finance, the assumption that stock returns are normally distributed is common. This may be because of the fact that if stock prices follow a random walk, then stock returns should be independent and identically distributed. According to the central limit theorem if we can collect enough independent and identically distributed stock return data, than the limiting distribution of these returns should be normal. We have collected 1,000 daily stock returns of Herbalife Ltd. This data includes the day of the event we are studying, May 01, We are interested in testing whether actual stock returns for Herbalife appear to be drawn from a normal distribution. We are trying to prove that our data is normally distributed. Therefore, we will conduct the Kolmogorov-Smirnov test to prove this. Our null hypothesis is that our data is normally distributed. We will test the actual frequency distribution of our sample compared to theoretical probability distribution frequency. Model & Data We choose 1,000 trading days of Herbalife s adjusted closing stock price data such that we would have 899 days of trading days before the event and 100 days after the event. Therefore, the first day of our data is October 6, 2008; and the last date is September 21, We have made October 3, 2008, the trading date before October 6, 2008, date zero for calculating stock return for day 1. We computed the daily returns using this formula: eno64- wsj&url=http%3a%2f%2fonline.wsj.com%2farticle%2fsb html

3 We used the adjusted price and this incorporated the effects of dividends. Therefore, the Ending Price t + Dividend t is captured in the Adjusted Price. This ensures that we don t need to handle dividends separately. We choose the S&P market index data for the general market. We computed the daily return data for the market index as well. Since there are no dividends, we used this formula: (Ending Price- Beginning Price) / Beginning Price. After calculating daily stock returns, we sorted our 1,000 observations from the lowest daily return to highest daily return. The next step we did was to calculate the actual distribution and the theoretical distribution. We used normal distribution as the theoretical distribution. For actual distribution, we expected that each daily return would have a 1/1000 or a 0.10% probability of occurring. Theoretical cumulative probability is the cumulative probability in the standard normal tables. For our 1,000 daily stock returns; the mean was and the standard deviation was By using these, we calculated the Z score for each day and found the related cumulative probability. Then we found the absolute differences between the actual probability and the theoretical probability for each the 1,000 observations. Our aim was to find the maximum difference. To test our hypothesis, we compared actual difference ( ) with the critical D from the KS table at the 5% significance level for 1,000 observations (0.043). Results Before we came to a conclusion with statistical tests for the normality in the Herbalife stock returns; we constructed the histogram of the 1,000 daily returns to give us general impression of the data:

4 When one looks at the graph, one can see that there is symmetry on both sides of the peak. Although the peak is closer to the right side (positively skewed). The tails seem a lot flatter than those of a normal distribution but, in general one might say that our stock return data is normally distributed by quickly looking at this graph. When we conducted the statistical tests we got a better understanding of the data. The statistics below summarize the data for the 1,000 observations. Actual D and Critical D can be seen at the end of the table. Since our observed value of is greater than from the table we reject the null hypothesis that the 1,000 daily returns are normally distributed. So, the 1,000 returns do not appear to be normally distributed based on the KS test. Computation Area Daily Annualized Sample Average % % Variance % % Standard Deviation % % Std Error of Mean % Confidence Interval for For Mean - Data From 1 to 1000 Lower Limit % % Mean % % Upper Limit % % Actual D Critical D Mean Variance MAX D We conducted the same test for the 10 sub samples of 100 observations each. Our Analysis has led us to the following conclusion; with the exception of days and ; we cannot reject the null hypothesis. Therefore, we can say that other sub-samples appear to be normally distributed because actual D values are smaller than critical D value of It is a surprising result when 1,000 observations don t appear to be normally distributed. Output Table 1: Actual and Critical D, Mean and Variance of Samples From To Actual D Critical D Mean Variance

5 We also looked at the mean and variance values for each of the samples because if we can prove that the largest mean is not statistically different than the smallest mean, then we can prove the normality as well. This is a one-tailed test because the alternative hypothesis is that larger mean> smaller mean. From the table one can see that the t statistics for the difference between two means is greater than the critical t value. Therefore we reject the null hypothesis. Also the F statistic for the difference between two variances is greater than the critical F value. Again, we reject the null hypothesis that highest variance and lowest variance are equal. These two test results show us that these two samples are not coming from the same distribution. In other words, this tests support our KS test about the normality of the stock returns. Output Table 2: Difference Between Two Means and Variance Mean Variance Sample Size Ref Variance Highest Mean N30 Lowest Mean N29 t Statistic Degrees of Freedom Approx One Tailed or Two Tailed Test 1 Confidence Ineterval 95.00% Critical t Reject the Null Difference Between Two Variances - Calculated from Above Summary Highest Variance Lowest Variance F Statistic Critical F Reject the Null Conclusion According to the KS test result for Herbalife s 1, 000 stock observation, we conclude that returns don t come from a normal distribution. There is no overlap in the mean results between the lower limit and the mean and the upper limit when we look at 95% confidence interval for mean data from 1 to 1000 observations. Moreover, our data demonstrates that the largest mean is significantly different from the smallest mean. Our event date is in the lower limit data, in 1 to 100 observations in KS test order. This subinterval is normally distributed according to KS test result.

6 When we look at the subintervals ( and ) which are not normally distributed, we couldn t catch any common trend about years, months or days. Our overall test about the normality of 1,000 stock returns doesn t surprise us a lot. Because, we think that normality of the stock returns is questionable if information doesn t arrive linearly to the market. Alternatively, if we assume information arrives in the market linearly, investors may not react linearly to its arrival. What we find in the histogram of the 1,000 daily returns support this argument because our stock distribution has fatter tails than expected under the Normal distribution. We think that normal distribution assumption can underestimate the risk in investing the Herbalife stock data. Normal distribution assumption provides us the unbiased estimates of risky securities which result in eliminating the possibilities of earning abnormal return under the condition of certainty. However, our study proves the possibility of earning abnormal returns by the investors. Summary We conclude based on our observation that stock returns for Herbalife data are not normally distributed. We worked with 1,000 observations which may be relatively small sample. If we worked with a larger size of data, the results may in fact be a normal distribution. From the histogram, we can see a somewhat skewed distribution with fat tails and a high peak. We may want to think of alternative distributions for our stock returns. The first one can be the logistic distribution which is also similar to normal distribution but has thicker tails. The second one can be the exponential power distribution, which includes high peak and exponential rate at fat tails. From our data distribution, we can think this model may fit better than logistic distribution. Alternatively, our subinterval analysis of 1,000 stock returns shows us that we can use normal distribution with an adjustment such that we can generate stock returns with the mixture of continuous changes in prices and discontinuous jumps. We can assign the probability of occurrence to the each group. The mixture of two normal distributions will allow us to deal with the normality problem in our data. 3. Empirical Study of Market Anomaly (Monthly Effect) Just over 40 years ago Burton Malkiel s classic book A Random Walk Down Wall Street hit the bookshelves. The thesis of his book is that the efficient market theory does exist. However, Malkiel believes that a weaker version of the EMT. While Malkiel makes it clear in his book that occasionally certain market conditions exist that allow for active investors to outperform the general market indexes, those occasions are rare. Overtime Malkiel believes that active management is not as successful as indexing and even if it can be achieved the costs typically outweigh the returns. Malkiel s book and the Efficient Market Theory are widely debated topics on Wall Street. While some industry leaders, such as Vanguard, believe that markets are efficient there are plenty of asset managers pitching their abilities to generate alpha via active trading, complex trading rules, and the use of technical

7 trading. This ability to beat the market over long time provides a strong counter point to the EMT. Additionally, a study done by Robert Ariel in the late 1980 s around a turn of the month effect provides evidence that an investor can consistently beat the market by investing in stocks during the first half of the month and selling out of them and sitting on cash during the middle and end of the month. The investor would then buy the securities back at the start of the next month and repeat the process. Our goal is to test this Monthly Effect using data from Herbalife and the general market index (S&P 500). We are going to examine 1000 days of returns (Herbalife and S&P 500) from Specifically we are going to be looking at returns from the beginning of the month and end of the month. From there we are going to perform tests to determine if the two means are different. Model & Data The data set we used was Herbalife s stock return from October 3 rd 2008 to September 10 th Our benchmark was S&P 500 s returns from that same time period. The stock s closing price and the index s closing price can be viewed on the data input sheet in the excel model. That sheet also contains columns that include the daily returns for both Herbalife and the index. Additionally, the model identifies each day as beginning, middle, or end. In order to come to a conclusion on the validity of a turn of a monthly effect we leveraged the results of three models, each of them increasing in complexity. To build the models we broke the S&P s returns and Herbalife s returns into two samples; beginning of the month and end of the month. The first ten days represent the beginning of the month and the last ten days represent the end of the month. The first model tests the difference between the mean of the beginning of the month and the mean of the end of the month using the arithmetic average approach. The second model uses regression, factors in compounding, and uses the F Test to identify any differences. The third model that we used in order to examine the data is the Chow Test, which will allow us to run regression on both sets of data (Herbalife and S&P) without designating a period in the month. The third model will allow us to test if the data was structurally changed, which would alter our conclusion. Results The First Model we ran was a simple test of Arithmetic return differences. Our null hypothesis was that the means from both data sets are not different. Therefore we believed that the results will be in line with what Arial found. The results are pasted in below:

8 Difference Between Two Means - Stock Difference Between Two Means - Index Sample Mean Variance Sample Sample Mean Variance Sample Beg Of Month Beg Of Month End of Month End of Month t Statistic t Statistic ( ) ( ) Degrees of Freedom 705 Degrees of Freedom 705 Computed t Statistic Computed t Statistic ( ) Critical t Critical t Cannot Reject Null Hypothesis Cannot Reject Null Hypothesis We calculated the means and variances for the beginning and end of the month for both the S&P and Herbalife. We then computed a T Statistic and found the Critical T value for both data sets. Examining the model s output we can see that we don t get the same results as Ariel. Both the S&P and Herbalife show no difference in returns from the beginning to end of the month. Phase two of the first model was conducting a test to determine if there was a difference between the variances of Herbalife s stock and the S&P during the month. We tested if the stock or S&P was less risky at the start of the month or end of the month. Variances measure risk. It is a good metric to examine when looking at investments because two investments that have the same expected return might not have the same actually return because of the risk. The model is pasted in below: Difference Between Two Variances - Stock Difference Between Two Variances - Index Highest Variance Highest Variance Lowest Variance Lowest Variance F Statistic F Statistic Critical F Critical F Can Reject Null Hypothesis Cannot Reject Null Hypothesis It is interesting to see that Herbalife s stock doesn t have the same risk level at the beginning of the month and the end of the month. The S&P on the other hand doesn t show any evidence that the level of risk is different. This result shows that towards the end of the month Herbalife s stock becomes less risky. This has tremendous implications for a trader and portfolio manager. For example if a portfolio manager wants to add Herbalife to the portfolio and sends the order down to the trader the trader may wait until the end of the month to execute the order because he or she knows that the stock is less risky at the end of the month. This is a strategy that could generate alpha for the portfolio thus beating the market and disproving the EMT. The second model that we used to determine if there was in fact a monthly effect on stocks was a regression based model that was able to take into account the compounding effect of stocks. We altered the data by compounding the stock and index on a daily basis. We also took the log of each series in the hope of getting a better understanding of the data. This regression analysis will give us the ability to run an F Test to determine significance. The model has been pasted in below:

9 SUMMARY Regression OUTPUT for Company - Beginning of Month Stock SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 329 ANOVA df SS MS F ignificance F Regression E-87 Residual Total Coefficients tandard Erro t Stat P-value Lower 95% Upper 95% ower 95.0%er 95 Intercept X Variable E SUMMARY Regression OUTPUT for Company - End of Month Stock SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Standard E Observation 379 ANOVA df SS MS F Significance F Regression E-86 Residual Total % CoefficientsStandard Error t Stat P-value Lower 95% Upper 95% Lower 95.0%Upper 95.0% Intercept X Variable E E Summary Statistics Stock Daily Annualized Std Error Beg of Month % % % End of Month % % % Confidence Interval Beg Month Upper Limit % Mean % Lower Limit % End Month Upper Limit % Mean % Lower Limit % The results from the third chart show that for Herbalife s stock; owning it in the beginning of the month has a much larger daily and annualized compounded return then owning it towards the end of the month. One realizes a 163% annualized compounding return by owning the stock during the first ten days of the month and only a 53.9% annualized compounded return by owning it in the end of the month. This is a significant difference between the beginning and end of

10 the month for Herbalife s stock. While we have not performed a significance test on these results this conclusion provides a counter point to the EMT. While this conclusion can t be used to build out a trading-strategy one could begin to hypothesis that if more test like this were done on all stocks then one could begin to build a strategy to capitalize off that trend. SUMMARY Regression OUTPUT for Market - Beginning of Month Stock SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 329 ANOVA df SS MS F ignificance F Regression E-51 Residual Total Coefficients tandard Erro t Stat P-value Lower 95% Upper 95% ower 95.0%er 95 Intercept X Variable E E SUMMARY Regression OUTPUT for Market - End of Month Stock SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Standard E Observation 379 ANOVA df SS MS F Significance F Regression E-08 Residual Total % CoefficientsStandard Error t Stat P-value Lower 95% Upper 95% Lower 95.0%Upper 95.0% Intercept X Variable E E Summary Statistics Index Daily Annualized Std Error % % % % % % % % % % % % The results from the third chart show that for the index; owning it in the beginning of the month has a much larger daily and annualized compounded return then owning it towards the end of the month. One realizes a 32.15% annualized compounding return by owning the market during the first ten days of the month and only a 5.11% annualized compounded return by owning it in the end of the

11 month. This is a significant difference between the beginning and end of the month for the overall market. While we have not performed a significance test on these results this conclusion provides a great counter point to the EMT. Based off this conclusion and the evidence we saw with Herbalife s stock a trading strategy be built. The strategy would buy into the overall market at the start of the month and sell out of the market by the end of the month. An investor could simply buy ETF s during the first ten days of the month and sell them once the 10 th trading day came and buy them again the next month. We now tested the significance of the above results with an F test. F Test Stock Index Beg of Month Critical F Value End of Month Critical F Value The output of the F Test shows us that in all cases the Calculated F value is much greater than the Critical F value. This leads us to the conclusion that all the regression equations are highly significant. The third model we leveraged in this research is the Chow Test. This model allowed us to compare and test if the end and beginning of month returns are statistically different. The two samples are statistically different than two different regression models describe two different data series. Stock Index Beg Month SSR SSE SST End Month SSR SSE SST Combined SSR SSE SST k Total - 2k Num Den Chow Test - Result F Test Critical F Conclusion Can Reject NuCan Reject Null Hypothesis The F Test results for both Herbalife and the S&P show that in both cases the F Test is much larger than the Critical F. The two models are statistically different. This means that returns generated in the start of the month are statistically different then returns generated in the beginning of the month. This leads us to conclude that there is in fact a monthly effect and that the EMT may

12 exist but it is much weaker than Malkiel thinks. It may not be that much of a random walk. Conclusion The efficient market theory states that active management in the long term is a waste of money and that an investor is better off placing assets into every type on index fund and should take a passive strategy approach to investing. We have examined Herbalife s stock from During that time span our research demonstrates that there is in fact a monthly effect. Actively trading Herbalife stock can allow an investor to beat the market. We observed that there are significant gains to be had from moving into the stock in the beginning of the month and moving out of the stock by the end of the month and repeating that process over and over. While we have completely ignored trading costs and tax implications of moving in and out of a position there is evidence that active management will in fact generate alpha. With regression analysis and taking into account compounding we found that there is a significant difference between monthly returns for both Herbalife and the index. Therefore, an investor can benefit greatly from owing a stock or index fund in the beginning of the month and selling out of it by the end of the month. This regression analysis was supported by an F Test and a Chow test that support our outcomes. While our early research found that there is no evidence of different returns between the start and end of the month we were only using a simply t test and were using the arithmetic mean. Using three years of data we have found evidence suggestion that stocks may unexpectedly and unexplainable generate large returns in the beginning of the month while lagging towards the end of the month. This conclusion leads us to believe that actively trading into the market at the start of the month and trading out of it after the 10 th trading day will ensure above average returns. Summary Our data suggests that there is in fact a turn of the month effect in the stock market. We used a number of tests that support our conclusion. However, there are several areas that we believe need to be investigate more in order to determine if the turn of the month effect is in fact true. Our data set only covers three years. Additionally, those three years of data are taken from one of the most volatile times in the stock markets history. October of 2008, when our data begins, was a month after the largest bankruptcy in history took place. When Lehman failed the market got crushed and the United States economy dipped into a recession. We did not control for this once in a life time event and the sudden plunge of the market and volatility of the market could distort our results.

13 Moreover, as time pasted and the market moved on from Lehman, we witnessed an incredible bull market that we have never seen before. This market growth benefited every stock. Therefore during this time period trends may have formed that is purely a function of a red hot bull market. Therefore the very high annualized compounded gains might be a function of the market and not a potential trading pattern. Another issue that could cause this monthly effect we are realizing are inflows into the market. For example at the end of the month people are paid. If those people have an employer sponsored 401k plan then money is taken out of their pay checks and sent to the firm the managed the 401k. Once that asset manager has the funds it invests them into the market place. This inflow of new capital spikes prices and increases returns and once time passes the market adjusts. This could be the monthly effect that we are witnessing. 4. Event Study Event studies are very important research subjects in corporate finance. The reaction of stock prices and their related returns according to significant news or event are subject to the interest. Events are expected to generate statistically abnormal performance, so we will test to see if the return on the actual event day is significantly different than that expected. Event studies assume that markets are efficient. Stock prices are expected to fully reflect all available information, so the only factor that will change prices is new information. This fact is a fundamental principal in the random walk thesis and the efficient market theory. We want to measure the impact of the announcement on the value of Herbalife stock and determine if the effect is greater than that normally forecasted or expected. Our event occurred in May 01, 2012 in Herbalife Ltd. s first-quarter conference call when famous hedge-fund manager David Einhorn joined the call and asked some skeptical questions about the company's revenue structure. The share price fell down 19.94% on that day. We want to compare the forecasted return that we would expect under normal circumstances to the actual return on the event day. Therefore, we can see if the market reacted positively or negatively with this event and if there is an impact on Herbalife stock price. We will test the market efficiency as well, because we will see if our event has instant effect in the price of stock. If the market is efficient, investors should reevaluate the riskiness of the Herbalife stock and this will be reflected immediately in the price of the stocks. Model & Data We define the event date as May 01, That was the day of conference call when famous hedge-fund manager David Einhorn joined the call. Then, we centered an event window around the event date such that we had the 20 days before the event day, the 20 days including the day, and the 20 days after the event date. Our event window begins on April 02, 2012 and ends on May 29, By creating an event window, we wanted to see if there was any abnormal price reaction leading up to or following shortly after the event date. After defining our event window, we constructed the estimation window. We made May 01, the event date-as t, our estimation window is from t-140 to t-21. Therefore, the estimation window starts at day 760 (October 10, 2011) and ends at day 879

14 (March 30, 2012). After we listed stock returns of Herbalife and the S&P 500 index returns for this period, we ran an OLS regression on this data using the regression model: We ran the regression to see abnormal returns which are the difference between the actual security return and the predicted from the estimated market model. We found the Standard Error of the Forecast to calculate Standardized Abnormal Returns (Abnormal return/standard error of the forecast). We also calculated cumulative results over the event window: Cumulative Abnormal Returns and Standardized Cumulative Abnormal Returns. We also used the Chow test to see the stability of the market model around the event. We took data from 760 to 879 as pre event and from 920 to 1000 as past event. We ran OLS regression for two data sets separately on the above market model using stock returns and index returns. We took SSE for each regression and add them for unrestricted model. Then we ran OLS regression for combined sample as restricted model for Chow test. Results We estimated the pre-event market model from days 760 to 879, that is a 120 day period before the event window. The OLS results for our market model are: SUMMARY OUTPUT (Pre Event - Market Model) Regression Statistics Multiple R R Square Adjusted R Standard E Observation 120 ANOVA df SS MS F ignificance F Regression E-13 Residual Total Coefficientstandard Erro t Stat P-value Lower 95% Upper 95% Intercept X Variable E The R squared of 36.28% and the F test of shows that the regression is significant. Also the t statistic on the slope which is indicates that the slope is significantly different than zero. We also estimated the post-event market model from days , that is 81 day period after the event window. The OLS results for our market model are:

15 SUMMARY OUTPUT (Post Event - Market Model) Regression Statistics Multiple R R Square Adjusted R Standard E Observation 81 ANOVA df SS MS F ignificance F Regression E-06 Residual Total Coefficientstandard Erro t Stat P-value Lower 95% Upper 95% Intercept X Variable E The R squared of 23.33% and the F test of shows that the regression is significant. Also the t statistic on the slope which is indicates that the slope is significantly different than zero. We used the pre-event market model to construct the abnormal returns and the cumulative abnormal returns for the event window. For illustration purposes, here we show the results for five days before the event day, the event day and five days after the event day. Event Window Cummulative Date Stock Index Predicted Abnormal Std Error Standard Abnormal Standardized Count Num Return Return Return Return (AR) Forecast AR (SAR) Return (CAR) CAR (SCAR) % % % % (3.5064) % % % % % % % % % % % % % % % (1.7952) % % % % % % % % % % ( ) % (2.7394) % % % % (4.0782) % (3.5458) % % % % (7.6096) % (5.0546) % % % % % (4.5483) % % % % (0.0224) % (4.4609) % % % % (1.7879) % (4.7249) The actual return on the event day, day 900, is % and the predicted return is %. Therefore, the abnormal return is [ % %] = %. The Standard Error of the Forecast is as seen on the table. So, the Standardized Abnormal Return is [ / ]= The critical t value for 118 degrees of freedom at the 95% confidence interval is Therefore, we can conclude that on the event day the abnormal return observed is significantly different from that expected. The questions asked by David Einhorn on the conference call had a significant impact on the stock return when compared to how it normally would have been expected to perform. When we look at the chart of Standardized Abnormal Returns, we can see that our event day is well beyond the 95% confidence interval for the critical t factor which is ± However, there are seven other days for the stock returns of

16 Herbalife behaved abnormally when compared to how it normally would have been expected to perform. We show them in the table below. It is interesting to see that these days are very close to the event date. This situation can be a clue about market efficiency. We can see that the situation appeared on the event day has not been absorbed on the same day. We can see significant abnormal returns on the day after and two days after the announcement. This is inconsistent with the market efficiency. Standardized Cumulative Abnormal Return (SCAR) on day 902 is which indicates significant abnormal return over two trading days. On day 909, SCAR is that is smaller but still shows significant abnormal return. On the other hand, when we look at the four days just before the event day from the table above, Standardized Abnormal Returns are below the critical value of ± So, we can say that event was not anticipated before it happened. Finally, we look at the SCAR for the day before, the day of and the day after the event day. The calculation is as follows: [ ]/ 3 = For three days period, the result is also significant. Event Window Cummulative Date Stock Index Predicted Abnormal Std Error Standard Abnormal Standardized Count Num Return Return Return Return (AR) Forecast AR (SAR) Return (CAR) CAR (SCAR) % % % % (3.5064) % % % % % ( ) % (2.7394) % % % % (4.0782) % (3.5458) % % % % (7.6096) % (5.0546) % % % % % (4.1229) % % % % (2.7065) % (4.8083) % % % % % (2.8031) % % % % (5.5937) % (3.7340)

17 On the entire event window the Cumulative Abnormal Return is %. The Standardized Cumulative Abnormal Return is % indicating that the abnormal returns for the entire 40 day event window are significant % % % % % (3.3937) We also tested if the risk or the return of the stock had changed significantly after the event. We ran a Chow test using the returns from days and from days The OLS results from pre and post event market model are shown above. SSE from the pre-event model is and SSE from the post event model is Therefore our total unrestricted SSE is = We also ran regression for the combined sample. The OLS results are as follows: SUMMARY OUTPUT (Pre and Post Event) Regression Statistics Multiple R R Square Adjusted R Standard E Observation 201 ANOVA df SS MS F ignificance F Regression E-17 Residual Total Coefficientstandard Erro t Stat P-value Lower 95% Upper 95% Intercept X Variable E From the ANOVA results above, our restricted SSE is So, our Chow test result is: Sample SSE n k Pre Event Post Event Total Unrestricted Restricted (Combined Sample) Chow Test Results Numerator Denominator Chow Test Critical F Test Fail to reject the Null Hypothesis

18 The difference in the market model between the pre and post event periods is insignificant. Our market model is stable across these periods and as result the event did not change the risk or return characteristics of Herbalife stock. Conclusion The questions asked by famous hedge-fund manager David Einhorn in Herbalife Ltd. s first-quarter conference call about the company's revenue structure created significant abnormal return on Herbalife stock on that day. It was a sudden and unexpected event so the days leading up the event don t show significant abnormal returns. However, the skeptical questions about company may take the attention of investors and the effect of event continuous after the event day of May 01, The event effect didn t absorb on the same day of event. Since, when we take the returns from days 900, 901 and 902 together; we can still see the significant abnormal returns. This can be evaluated as a sign of inefficient market. Our results show that investors take notice of negative point of view of Mr.Einhorn. Their reaction to the market lasts longer than the event day itself. If there is an efficient market, we expect that information be absorbed into the stock price right away. However, we can see that the biggest reaction was given on the event day, and the effect of the event decreased gradually. Even on May16, 2012, stock return is 16.66%. This is probably related to the Herbalife announcement about the reports on net income and sales after the event. The reports on increase in net income and sales relative to previous year calm the market. Investors may think as an opportunity to buy Herbalife stocks when prices are decreased. The abnormal returns for entire 40 days even periods are still significant. However, the stability of the market model pre and post the event indicates that our event didn t in any meaningful change the returns generating behavior or the risk of the stock. Summary Event studies are very important for evaluating the market efficiency and market manipulation and insider trading at the same time. In our case, we have doubts about market efficiency and also we think that the aim of Mr.Einhorn may bet against the Herbalife through market manipulation. We used daily data as advised to eliminate the bias and make the model more robust. We also used 120 days for estimation period which is commonly used for event studies. We are very certain about our event day and our stock return data prove our argument. In our situation, the market price impact of our event is absorbed over a period of time but in a decreasing basis. The event occurred in the conference call; Mr.Einhorn attended the conference call with analyst reporters. The event is instantly announced to the public. Therefore, we don t think any delay for reporting the event. But we are not sure about the exact last trade of Herbalife stock in a day. On some days, stocks may be traded thinly and the last trade may be early in the day. This may affect the estimates we get from the market model. An additional concern that we may have is how we define an event. Many events such as a miss in earnings or a surprise uptick in earnings are more typical then an activist investor calling into an investor conference call and basically accusing a firm of a type of fraud. We stand by the results that we found however, we realize that

19 certain events may in fact be so great and the implications might take some time for the market to understand that the market may be acting efficiently but it may take some time. 5. Is There a January Effect? Several studies in finance have shown the calendar effects in stocks returns. More specific, empirical research has shown that returns during the first days of January are significantly different than zero. This has been so well document that is known as the January Effect or January Anomaly. This effect was first documented by Rozeff and Kinney in 1976 in the paper published in the Journal of Financial Economics titled Capital Market Seasonality: The Case for Stock Returns. Rozeff and Kinney documented that unusually high returns where amass in the first couple of days of January, while the return for the rest of the year where statistically indistinguishable from zero. This effect is still studied, as a recent paper by Li, Jing in 2013 titled Testing for January Effect in Canada Finance Industry. Other studies have shown that the January effect is limited to small stocks, rather than large capitalized stocks. The paper mentioned before ( Testing for January Effect in Canada Finance Industry ) arrived to the same conclusion. Studies have also tried to discover the reason of the January Effect. Some argue tax loss selling, that artificially push the prices down in December, while creating a buying opportunity in January. Other attribute the January Effect to the Bid-Ask spread. The difference between the bid (price at which investor sell and market maker buy) and ask (price at which investor buy and market maker sell) price is usually 0.25cents per share. Considering the absolute value of the spread, this will have a higher effect on lower priced stocks (0.25cent represent a larger percentage of the total price). Model & Data To test the January Effect on Herbalife, we introduced a Dummy Variable for the first days of January (from 1-4). The regression model was in the form of: Results The results of the regression are presented below:

20 SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 1000 ANOVA df SS MS F ignificance F Regression E-80 Residual Total Coefficients Standard Erro t Stat P-value Lower 95% Upper 95% Intercept Market Index E D_Jan The overall model is statistically significant, with an F value of 220, resulting in a p-value of almost zero. Additionally, the R 2 is 30.62%, while the Adjusted R 2 is 30.49%. In our regression, we found that our stock does not present a January Effect. Our Dummy variable not only is not significant, but also its coefficient is negative. As mentioned is our introduction for Assignment 4, this can be due the size of Herbalife (large capitalized stock). In order to prove any problems with serial correlation, we ran a Durbin Watson test. The results are shown below: Durbin Watson 2.06 Appears to be no problems with serial correlation d u d l As presented above, there is no apparent problem with serial correlation, since our Durbin Watson coefficient is: We also tested if this new model (including the possible January Effect expanded model) represent an improvement over our simple market model (without the January Effect reduce model). The results are presented below:

21 SSEr Wald test SSEur At 95% significance level with 1 and 997 degrees of m 1 freedom the critical F value is Therefor, the n 1,000 expand model with the introduction of the Dummy k 1 variable does not add any significant explanatory power over the simple Market Index Model The addition of the Dummy variable does not add any significant explanatory power to our regression over the simple model. This result is also consistent with the fact that our Dummy variable was not significant in our expanded model. In order to review for other types of calendar effect, we extended our analysis to calendar months. Analysis of Calendar Month Effect As we did previously we wanted to test for the calendar month. In order to do so, we create Dummy variables for each one of the months (our January Dummy Variable from days 1-4 stayed the same). Our new regression model will be: The results of our models are presented below: SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Sq Standard Error Observations 1000 ANOVA df SS MS F ignificance F Regression E-70 Residual Total CoefficientsStandard Error t Stat P-value Lower 95% Upper 95% Intercept Market Index E D_Jan D_Feb D_Mar D_Apr D_May D_Jun D_Jul D_Aug D_Sep D_Oct D_Nov D_Dec

22 The overall model is statistically significant, with an F value of 33.95, resulting in a p-value of almost zero. Additionally, the R 2 is 30.92%, while the Adjusted R 2 is 30.01%. In our regression, we found that our stock does not present any calendar effects. All of our Dummy variables were not significant at 95%. Again, the only variable that helps predict Herbalife returns is the Market Index. We also tested any problems with serial correlation by running a Durbin Watson test. The results are shown below: Durbin Watson Appears to be no problems with serial correlation d u d l As presented above, there is no apparent problem with serial correlation, since our Durbin Watson coefficient is: We also tested if this new model (including the Calendar Effect expanded model) represent an improvement over our simple market model (Simple Market index reduce model). The results are presented below: SSEr Wald test SSEur At 95% significance level with 11 and 986 degrees of m 12 freedom the critical F value is Therefor, the n 1,000 expand model with the introduction of several Dummy k 13 variables does not add any significant explanatory power over the simple Market Index Model As we can observe, the addition of the Dummy variables does not add any significant explanatory power to our regression over the simple model. Conclusions As we presented with our analysis, Herbalife stock return does not contain any type of Calendar effect. We can then assume that the market is behaving in an efficient way, including all the information related to calendar month on its pricing, and hence, its return. We could extend this test to several other calendar variables, such as season, day of the week or even in more detail, time of day. However, in an article published in 2001 in the Journals of Economics title Dangers of Data Mining: The Case of Calendar Effects in Stock Returns, the author (Ryan Sullivan) argue that there is no statistically significant evidence for calendar effects in the stock market, and that all such patterns are the result of data dredging. Our stock analysis agrees with that.