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The data set consists of 5 minute ticks for 28 stocks in the DAX. The missing stocks are TUI and Hypo Real Estate. With a time frame of 4 months between April 2001 and Juli 2001, the data set comprises 10.000 data points in total.
Out of the 28 available stock data, we chose 3 stocks corresponding to a small, medium and large capitalization. That is, we chose Altana as the stock with the smallest capitalization in the DAX, the Deutsche Börse with a medium capizalization and Allianz with the largest capizalization. The weighing file for the DAX in November 2006 can be found here: [1]
With the 28 available stock data, we calculated the average stock price as a market index as
with as the market at time and the prices of each stock at time .
Getting to know the data set
Datei:Xpl plot stocks and market.gifFigure: Timeplots of the stock price of three stocks over the period of April 2001 to Juli 2001 as well as of the market index.
We first explored the data set with basic XploRe commands such as 'dimensions', descriptive and 'countNotNumbers'. The commands gave us the feedback that the 10.000 data points of 28 stocks were correctly read in and that the number of {-inf, inf, NaN} is zero. Therefore we concluded that our data quality is quite good for further analysis and no further modifications are necessary. The data source is Landesbank Berlin, so that we can be sure that the data was assessed in a professional manner.
We can see from the plot that the Altana and Dt. Börse stock decreased over the time frame, while the stocks of Allianz increased in price. The overall market decreased as well, which is in accordance with the global market trend in the Q2-Q3 2001.
It might be interesting to mention, that the sudden decrease in stockprice, as well as the high volatility at around ???? that we see in Altana and Deutsche Boerse Group is due to ???.
The questions
Does any of the three selected stocks perform better than the market?
How much better?
How does the distribution of returns vary within different time-horizons (5min, 1h, 1day)?
Stock returns
The return can be calculated as relative return
or logarithmic return
, which we decided for.
The actual calculation is done with the function "vEarnings5min":
proc(return)=vEarnings5min(x)
return = (1:9999)*0
i = 1
while(i<9998)
i = i + 1
return[i] = log(x[i,2]/x[i-1,2])
endo
endp
which receives as input the timeseries 'x' and computes the output 'return'. The computation is done for every time step 'i', beginning from i = 2 until i = 9999. The length of the output is decreased by one data point, because the return cannot be calculated for the fist data point.
We calculated the log. return for each stock price and then averaged over all 28 stocks with
with stock price p of stock i at time t. Given this market return, we could then calculate the log. return of a stock above the market with .
Time horizons
Secondly, we computed the return above the market also for different time horizons of 5 min, 1 hour and 1 day, which equals 1 tick, 12 ticks and 108 ticks respectively. Therefore we calculated not only the difference between following data points but also over 12 and 108 data points. We found that the time horizon influences the smoothness of the curves quite effectively. The results also show an increase of stock price above market for Altana and Allianz, while Dt. Börse decreased compared to the index in the time frame.
Datei:Xpl timeplot time.gifFigure: Cumulated returns for 5 minutes, 1 hour and 1 day for each stock over market and the market.
One commonly observed phenomenon for stock market returns is leptokurtosis, also known as fat tails. It means that extremely low (negative) returns occur more often than extremely high returns. At the same time it is more peaked than a normal distribution, which has a kurtosis of around 3. If it is higher than 3 a distribution is called leptokurtotic, if it is smaller than 3 playtkurtotic.
As can be seen from the plots, all examined stock returns possess the feature of leptokurtosis.
Datei:Xpl histograms2.gifFigure: Histogramms of cumulated returns over market for our three stocks and the market.
Kurtosis and skewness are both used in the Jarques-Bera-Test on normality of a distribution.
positive correlation between index stocks
DAX-stocks are said to be positively correlated, so that the following observation is rather surprising:
Correlation matrix
Altana
Dt.Börse
Allianz
Altana
1
-0.04214
-0.060218
Dt. Börse
-0.04214
1
-0.1533
Allianz
-0.060218
-0.1533
1
Key figures and findings
Altana
Dt. Börse
Allianz
Mean
0.000580483
-0.000551137
0.00502454
Median
0.000392772
-0.000899603
0.00333357
Variance
0.000136357
0.000141299
0.000631811
Skewness
0.147285
0.354333
1.24129
Kurtosis
3.20132
4.97229
7.98768
Altana stock
5 min
1 hour
1 day
Mean
4.17631e-06
4.68395e-05
0.000580483
Median
0
0
0.000392772
Variance
2.48647e-06
1.54758e-05
0.000136357
Skewness
-0.730336
-0.350057
0.147285
Kurtosis
22.2724
8.28639
3.20132
Outlook
data, short period, only dax-companies (2001,black friday after 9/11), etc. different
effect of industry and world economy is stronger that that of market capitalization (share buy-back) sample is too small and randomly chosen, why should mcap be a distinct measure?
Literature
J.Franke, W.Härdle, C.Hafner: „Einführung in die Statistik der Finanzmärkte“, 2001, Springer-Verlag, Heidelberg
![{\displaystyle kurt(X)={\frac {E[\left(X-\mu )^{4}\right]}{\sigma ^{4}}}}](/media/api/rest_v1/media/math/render/svg/cb5e4ba047b7b1dfd8254fe7d958ab8626de27f7)
