an empirical study for testing the stock market …...an empirical study for testing the stock...

An Empirical Study for Testing the Stock Market Efficiency and Identifying

Abnormal Return Opportunities

Merve Artman*

Central Bank of Turkey, Ankara, Turkey [email protected]

Murat Artman

Central Bank of Turkey, Ankara, Turkey [email protected]

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 2008-2012 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

mailto:[email protected]

mailto:[email protected]

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 science-

based 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.2

2. 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, 2012. 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, 2012. 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:

1 http://www.reuters.com/finance/stocks/companyProfile?symbol=HLF

2

http://online.wsj.com/news/articles/SB10000872396390444450004578002783686418120?mg=reno64-wsj&url=http%3A%2F%2Fonline.wsj.com%2Farticle%2FSB10000872396390444450004578002783686418120.html

http://www.reuters.com/finance/stocks/companyProfile?symbol=HLF





We used the adjusted price and this incorporated the effects of dividends.

Therefore, the Ending Pricet + Dividendt 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 0.001524 and the

standard deviation was 0.031385. 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 (0.106303) 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:

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 0.10633 is greater than 0.043 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.

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 601-700 and 801-900; 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 0.136000. It is a surprising result when

1,000 observations don’t appear to be normally distributed.

Computation Area

Daily Annualized

Sample Average 0.152373% 46.322275%

Variance 0.098539% 24.634741%

Standard Deviation 3.139092% 49.633397%

Std Error of Mean 0.099267%

Confidence Interval for For Mean - Data From 1 to 1000

Lower Limit -0.042190% -10.012396%

Mean 0.152373% 46.322275%

Upper Limit 0.346935% 137.699598%

Actual D Critical D Mean Variance

MAX D 0.106303 0.043000 0.001524 0.000985

Output Table 1: Actual and Critical D, Mean and Variance of Samples

From To Actual D Critical D Mean Variance

1 1000 0.106303 0.043000 0.001524 0.000985

1 100 0.086245 0.136000 -0.008001 0.003420

101 200 0.045971 0.136000 0.009249 0.001480

201 300 0.074625 0.136000 0.003096 0.000687

301 400 0.066927 0.136000 0.001359 0.000367

401 500 0.106942 0.136000 0.002851 0.000503

501 600 0.065865 0.136000 0.001774 0.000186

601 700 0.145167 0.136000 0.005323 0.000532

701 800 0.065426 0.136000 0.000239 0.000916

801 900 0.165801 0.136000 0.000476 0.000689

901 1000 0.125930 0.136000 -0.001131 0.000982

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.

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.

Output Table 2: Difference Between Two Means and Variance

Mean Variance Sample Size Ref Variance

Highest Mean 0.009249 0.001480 100 N30

Lowest Mean -0.008001 0.003420 100 N29

t Statistic 0.017250 2.464124

0.007000

Degrees of Freedom 0.000000 172.905761

0.000000

Approx 198.000000

One Tailed or Two Tailed Test 1

Confidence Ineterval 95.00%

Critical t 1.652586

Reject the Null

Difference Between Two Variances - Calculated from Above Summary

Highest Variance 0.003420

Lowest Variance 0.000186

F Statistic 18.396863

Critical F 1.394061

Reject the Null

When we look at the subintervals (601-700 and 801-900) 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

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 2008-2012. 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 3rd

2008 to

September 10th

2012. 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:

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:

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:

Difference Between Two Means - Stock Difference Between Two Means - Index

Sample Mean Variance Sample Sample Mean Variance Sample

Beg Of Month 0.001882 0.001234 328 Beg Of Month 0.000333 0.000318 328

End of Month 0.001554 0.000984 379 End of Month 0.000655 0.000271 379

t Statistic 0.000328 0.130080 t Statistic (0.000323) (0.248524)

0.002521 0.001298

Degrees of Freedom 705 Degrees of Freedom 705

Computed t Statistic 0.130080 Computed t Statistic (0.248524)

Critical t 1.963335 Critical t 1.963335

Cannot Reject Null Hypothesis Cannot Reject Null Hypothesis

Difference Between Two Variances - Stock Difference Between Two Variances - Index

Highest Variance 0.001234 Highest Variance 0.000318

Lowest Variance 0.000984 Lowest Variance 0.000271

F Statistic 1.253478 F Statistic 1.175352

Critical F 1.191270 Critical F 1.191270

Can Reject Null Hypothesis Cannot Reject Null Hypothesis

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

SUMMARY Regression OUTPUT for Company - Beginning of Month Stock

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.8353005

R Square 0.6977269

Adjusted R Square 0.6968025

Standard Error 0.2427853

Observations 329

ANOVA

df SS MS F Significance F

Regression 1 44.49165 44.49165 754.80323 5.85E-87

Residual 327 19.27492 0.0589447

Total 328 63.76657

CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Lower 95.0%Upper 95.0%

Intercept 4.3202131 0.026832 161.01264 0 4.267429 4.3729973 4.267429 4

X Variable 1 0.003872 0.000141 27.473683 5.851E-87 0.003595 0.0041493 0.003595 0

SUMMARY Regression OUTPUT for Company - End of Month Stock

SUMMARY OUTPUT


Multiple R 0.802157

R Square 0.643456

Adjusted R Square0.64251

Standard Error0.14097

Observations 379

ANOVA


Regression 1 13.52083682 13.5208368 680.37307 1.9139E-86

Residual 377 7.492000605 0.01987268

Total 378 21.01283742

Upper 95.0% CoefficientsStandard Error t Stat P-value Lower 95% Upper 95% Lower 95.0%Upper 95.0%

Intercept 4.383131 0.014511042 302.054853 0 4.35459803 4.411663471 4.354598 4.411663

X Variable 1 0.001726 6.61853E-05 26.083962 1.914E-86 0.00159624 0.001856513 0.0015962 0.001857

Summary Statistics

Stock Index

Daily Annualized Std Error

Beg of Month 0.387202% 162.777306% 0.014094%

End of Month 0.172637% 53.913658% 0.006619%

Confidence Interval

Beg Month Upper Limit 0.359579%

Mean 0.387202%

Lower Limit 0.414826%

End Month Upper Limit 0.159665%

Mean 0.172637%

Lower Limit 0.185610%

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.

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

SUMMARY Regression OUTPUT for Market - Beginning of Month Stock

SUMMARY OUTPUT



R Square 0.4987888



Observations 329

ANOVA


Regression 1 3.694131 3.6941309 325.41962 5.59E-51

Residual 327 3.712071 0.0113519

Total 328 7.406202

CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Lower 95.0%Upper 95.0%

Intercept 4.423335 0.011775 375.65807 0 4.400171 4.4464991 4.400171 4

X Variable 1 0.0011157 6.18E-05 18.039391 5.59E-51 0.000994 0.0012374 0.000994 0

SUMMARY Regression OUTPUT for Market - End of Month Stock

SUMMARY OUTPUT


Multiple R 0.271564

R Square 0.073747



Observations 379

ANOVA


Regression 1 0.180396877 0.18039688 30.016301 7.8481E-08

Residual 377 2.265756312 0.00600996

Total 378 2.446153189

Upper 95.0% CoefficientsStandard Error t Stat P-value Lower 95% Upper 95% Lower 95.0%Upper 95.0%

Intercept 4.643554 0.007980063 581.894442 0 4.62786313 4.659245152 4.6278631 4.659245

X Variable 1 0.000199 3.63973E-05 5.47871341 7.848E-08 0.00012784 0.000270978 0.0001278 0.000271

Summary Statistics

Index

Daily Annualized Std Error

0.111572% 32.150908% 0.006185%

0.019941% 5.111093% 0.003640%

0.099450%

0.111572%

0.123694%

0.012807%

0.019941%

0.027075%

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 10th

trading

day came and buy them again the next month.

We now tested the significance of the above results with an F test.

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.

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

F Test

Stock Index

Beg of Month 754.8032329 325.4196209

Critical F Value 3.870053307 3.870053307

End of Month 680.3730737 30.01630067

Critical F Value 3.866242833 3.866242833

Stock Index

Beg Month SSR 44.4916 3.6941

SSE 19.2749 3.7121

SST 63.7666 7.4062

End Month SSR 13.5208 0.1804

SSE 7.4920 2.2658

SST 21.0128 2.4462

Combined SSR 28.8237 5.1569

SSE 60.3734 7.6932

SST 89.1971 12.8501

k 2.0000 2.0000

Total - 2k 704.0000 704.0000

Num 16.8033 0.8577

Den 0.0380 0.0085

Chow Test - Result F Test 441.9444 101.0097

Critical F 3.0085 3.0085

Conclusion Can Reject Null HypothesisCan Reject Null Hypothesis

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 2008-2012. 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 10th

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.

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, 2012. 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, 2012. 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, 2012-

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

(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:

The R squared of 36.28% and the F test of 67.1646 shows that the

regression is significant. Also the t statistic on the slope which is 8.195401

indicates that the slope is significantly different than zero.

We also estimated the post-event market model from days 920-1000, that is

81 day period after the event window. The OLS results for our market model are:

SUMMARY OUTPUT (Pre Event - Market Model)


Multiple R 0.60227

R Square 0.362729



Observations 120

ANOVA


Regression 1 0.015682 0.015682 67.1646 3.42E-13

Residual 118 0.027551 0.000233

Total 119 0.043232

CoefficientsStandard Error t Stat P-value Lower 95%Upper 95%

Intercept 0.000908 0.001409 0.644318 0.520619 -0.00188 0.003697

X Variable 10.936555 0.114278 8.195401 3.42E-13 0.710253 1.162856

The R squared of 23.33% and the F test of 24.035 shows that the regression

is significant. Also the t statistic on the slope which is 4.902532 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.

The actual return on the event day, day 900, is -19.9404% and the

predicted return is 0.6207%. Therefore, the abnormal return is [-19.9404%-

0.6207%] = -20.5611%. The Standard Error of the Forecast is 0.015350 as seen on

the table. So, the Standardized Abnormal Return is [-0.205611/0.015350]= -

13.3949. The critical t value for 118 degrees of freedom at the 95% confidence

interval is 1.98. 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 ± 1.98. However, there are seven other days for the stock returns of

SUMMARY OUTPUT (Post Event - Market Model)


Multiple R 0.482979

R Square 0.233269



Observations 81

ANOVA


Regression 1 0.007607 0.007607 24.03482 4.96E-06

Residual 79 0.025002 0.000316

Total 80 0.032609


Intercept -0.00056 0.001993 -0.28038 0.779916 -0.00453 0.003409

X Variable 11.073456 0.21896 4.902532 4.96E-06 0.637628 1.509284

Event WindowCummulative

Date Stock Index Predicted Abnormal Std Error Standard Abnormal Standardized

Count Num Return Return Return Return (AR) Forecast AR (SAR) Return (CAR) CAR (SCAR)

16 895 -4.9452% 0.3680% 0.4354% -5.3806% 0.015345 (3.5064) 1.5933% 0.2571

17 896 1.5168% 1.3645% 1.3687% 0.1481% 0.015404 0.0961 1.7414% 0.2727

18 897 2.9733% 0.6680% 0.7164% 2.2569% 0.015354 1.4699 3.9983% 0.6115

19 898 -2.4376% 0.2414% 0.3169% -2.7545% 0.015344 (1.7952) 1.2438% 0.1833

20 899 -0.2082% -0.3884% -0.2729% 0.0647% 0.015357 0.0422 1.3085% 0.1881

21 900 -19.9404% 0.5658% 0.6207% -20.5611% 0.015350 (13.3947) -19.2526% (2.7394)

22 901 -6.4036% -0.2497% -0.1431% -6.2605% 0.015351 (4.0782) -25.5131% (3.5458)

23 902 -12.3309% -0.7659% -0.6265% -11.7044% 0.015381 (7.6096) -37.2175% (5.0546)

24 903 1.6107% -1.6147% -1.4215% 3.0322% 0.015479 1.9589 -34.1853% (4.5483)

25 904 0.0893% 0.0351% 0.1236% -0.0343% 0.015344 (0.0224) -34.2196% (4.4609)

26 905 -3.0560% -0.4279% -0.3100% -2.7460% 0.015359 (1.7879) -36.9656% (4.7249)

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 -5.0546 which indicates significant abnormal return over two trading days. On

day 909, SCAR is -4.8083 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 ± 1.98. 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:

[0.0422-13.3942-4.0782]/√3 = -10.0633. For three days period, the result is also

significant.

Event WindowCummulative

Date Stock Index Predicted Abnormal Std Error Standard Abnormal Standardized

Count Num Return Return Return Return (AR) Forecast AR (SAR) Return (CAR) CAR (SCAR)

16 895 -4.9452% 0.3680% 0.4354% -5.3806% 0.015345 (3.5064) 1.5933% 0.2571

21 900 -19.9404% 0.5658% 0.6207% -20.5611% 0.015350 (13.3947) -19.2526% (2.7394)

22 901 -6.4036% -0.2497% -0.1431% -6.2605% 0.015351 (4.0782) -25.5131% (3.5458)

23 902 -12.3309% -0.7659% -0.6265% -11.7044% 0.015381 (7.6096) -37.2175% (5.0546)

27 906 3.5665% -0.6702% -0.5369% 4.1034% 0.015374 2.6691 -32.8622% (4.1229)

30 909 -5.1217% -1.1113% -0.9500% -4.1717% 0.015414 (2.7065) -40.4197% (4.8083)

32 911 16.6585% -0.4404% -0.3217% 16.9802% 0.015360 11.0551 -24.3236% (2.8031)

33 912 -9.9685% -1.5051% -1.3189% -8.6496% 0.015463 (5.5937) -32.9732% (3.7340)

On the entire event window the Cumulative Abnormal Return is -

32.9839%. The Standardized Cumulative Abnormal Return is -3.3937% indicating

that the abnormal returns for the entire 40 day event window are significant.

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 760-

879 and from days 920-1000. The OLS results from pre and post event market

model are shown above. SSE from the pre-event model is 0.027551 and SSE from

the post event model is 0.025002.

Therefore our total unrestricted SSE is 0.027551+0.025002 = 0.052553.

We also ran regression for the combined sample. The OLS results are as follows:

From the ANOVA results above, our restricted SSE is 0.052723.So, our

Chow test result is:

40 919 1.9271% 1.1079% 1.1284% 0.7987% 0.015381 0.5193 -32.9839% (3.3937)

SUMMARY OUTPUT (Pre and Post Event)


Multiple R 0.553439

R Square 0.306295



Observations 201

ANOVA


Regression 1 0.023279 0.023279 87.8655467 1.59E-17

Residual 199 0.052723 0.000265

Total 200 0.076002


Intercept 0.000324 0.001159 0.279403 0.78022573 -0.00196 0.002609

X Variable 10.974879 0.104002 9.373662 1.5886E-17 0.769791 1.179966

Sample SSE n k

Pre Event 0.027551 120 1

Post Event 0.025002 81 1

Total Unrestricted 0.052553 201 1

Restricted (Combined Sample) 0.052723 201 1

Chow Test Results

Numerator 0.0001

Denominator 0.0003

Chow Test 0.3191

Critical F Test 3.0418

Fail to reject the Null Hypothesis

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, 2012. 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

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:

The overall model is statistically significant, with an F value of 220,

resulting in a p-value of almost zero. Additionally, the R2

is 30.62%, while the

Adjusted R2 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:

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:

SUMMARY OUTPUT



R Square 0.306275558



Observations 1000

ANOVA


Regression 2 0.301499 0.150749 220.085031 6.79E-80

Residual 997 0.6829053 0.000685

Total 999 0.9844043

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

Intercept 0.001140777 0.0008303 1.373932 0.16977149 -0.00049 0.00277

Market Index 1.029320642 0.0490654 20.97855 3.168E-81 0.933037 1.125604

D_Jan1-4 -0.009370084 0.0107326 -0.87305 0.38284516 -0.03043 0.011691

1.4094 Durbin Watson 2.06

Appears to be no problems with serial correlation

0.6829 du 1.748

dl 1.789

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:

SSEr 0.68343 Wald test 0.7630

SSEur 0.68291 At 95% significance level with 1 and 997 degrees of

m 1 freedom the critical F value is 3.85. 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

SUMMARY OUTPUT


Multiple R 0.55609

R Square 0.309236



Observations 1000

ANOVA


Regression 13 0.30441283 0.023416372 33.95417 2.31E-70

Residual 986 0.67999143 0.000689646

Total 999 0.98440426


Intercept 0.001263 0.00307484 0.410601005 0.681454 -0.00477 0.007297

Market Index 1.029063 0.04934498 20.8544701 2.67E-80 0.93223 1.125897

D_Jan1-4 -0.00949 0.01117423 -0.849159052 0.395999 -0.03142 0.012439

D_Feb -0.00107 0.00429069 -0.250007265 0.802634 -0.00949 0.007347

D_Mar 0.000548 0.00414058 0.132453014 0.894653 -0.00758 0.008674

D_Apr 0.002934 0.00422904 0.693864197 0.487931 -0.00536 0.011233

D_May 0.002478 0.00421388 0.587949884 0.556701 -0.00579 0.010747

D_Jun 0.001587 0.00416884 0.380754563 0.703467 -0.00659 0.009768

D_Jul 0.000369 0.0042053 0.087691147 0.93014 -0.00788 0.008621

D_Aug -0.00161 0.00414729 -0.388095118 0.698029 -0.00975 0.006529

D_Sep -0.00091 0.00429232 -0.212664732 0.831632 -0.00934 0.00751

D_Oct -0.00288 0.00420329 -0.684381573 0.493895 -0.01113 0.005372

D_Nov -0.00174 0.0042389 -0.411369079 0.680891 -0.01006 0.006575

D_Dec -0.00123 0.00417088 -0.293784225 0.768985 -0.00941 0.006959

The overall model is statistically significant, with an F value of 33.95,

resulting in a p-value of almost zero. Additionally, the R2

is 30.92%, while the

Adjusted R2 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:

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:

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.

1.4079 Durbin Watson 2.0705

Appears to be no problems with serial correlation

0.6800

du 1.632

dl 1.908

SSEr 0.6834 Wald test 0.4152

SSEur 0.6800 At 95% significance level with 11 and 986 degrees of

m 12 freedom the critical F value is 1.79. 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

Summary

Our analysis leads us to believe that the market is in fact efficient or at

least the January effect does not exist in Herbalife’s stock. That being said, a better

test of this thesis would include a much larger sample size of Herbalife’s stock.

Additionally our sample may in fact be distorted because of the remarkable events

that were taking place during 2008 – 2012. Furthermore from the start we were

not sure that Herbalife’s stock would show signs of a January trading effect

because it is a highly liquid large cap security.

6. Conclusion

Our analysis of Herbalife’s data has presented us with some conflicting

data points. We discovered that the returns for Herbalife are not normally

distributed; however we believe that the incredible market events that were going

on between 2008-2012 played a major role in shaping the data that we looked at.

We found that there is evidence of a “turn of the month effect” which does not

support the efficient market theory and implies that a trader could make money

buying into a security at the start of the month and quickly selling out of it by the

end of the month (assuming no trading costs). Additionally, after studying the

return data for the days preceding and following the conference call we found

evidence of abnormal returns in several days following the event, this goes against

the EMT. Our evidence suggests that a trader can profit from shorting a stock

several days after a major negative event has occurred. While two of our studies

found evidence that the markets may not be as efficient as many people believe

them to be our analysis around the January Effect supports the efficient market

hypothesis. We found that Herbalife’s stock returns show no signs of a January

effect or a calendar month effect. Not finding a calendar month effect is in

contradiction with the results we got from assignment two. While the tests were

very different we believed that the conclusions would be somewhat it line.

Our conclusion is very similar to that of Burton Malkiel’s in A Random

Walk Down Wall Street. Our data suggests that the markets are not perfectly

efficient. Our analysis shows that opportunities do exist where active investors can

gain an upper hand on passive investors by capitalizing on short lived

inefficiencies. However, capitalizing off those inefficiencies is expensive and

difficult to replicate over time. Our study of the turn of the month effect and the

event date show there are opportunities to generate alpha but our analysis of the

January effect show that the markets are very efficient and some “trading

strategies” may simply just be random occurrences.

We believe that as technology plays a large role in the capital markets the

markets will become more efficient. As the markets become more efficient it will

be harder for investors to generate alpha. However, we believe that there will

always be investors who outperform the market and identify abnormalities that

give them a trading advantage.

an empirical study for testing the stock market …...an empirical study for testing the stock...

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