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1 - )
In the first question, the answers are based on Sabancı Holding (SAHOL) stock price series between 01.10.2001 and 08.10.2007. a - ) Time Plot of the price series of SAHOL.

b - ) Descriptive statistics and histogram of simple returns of SAHOL are below.

c -) Descriptive statistics and histogram of log returns of SAHOL are below.

d )- In the simple return statistics, skewness is 0.4490 which is positive and meaning that the histogram is right skewed and kurtosis is 5.35 which means that it has fatter tails than the standard normal distribution. In the log returns statistics, skewness is 0.28 which is right skewed still but is more close to a standard normal distribution statistic. Kurtosis of log returns is 5.03 which are close to standard normal distribution statistic number of 3 kurtosis in comparison to simple returns. Simply it can be observable that log returns have a more close distribution to standard normal distribution in comparison to simple returns.

e ) –
RI is simple return of ISE.
RS is the simple return of SAHOL stock.
Covariance matrix of ISE simple returns and SAHOL stock simple returns is below:

Correlation matrix of ISE simple returns and SAHOL stock simple returns is below:

f ) –
RI is simple return of ISE.
RS is the simple return of SAHOL stock.
β = cov (RS, RI) / var (RI)

Cov(RS, RI) = 0.000506
var(RI)=0.000468

β = 0.000506 / 0.000468
β = 1.082205
Since β is greater than 1 which means that stock is trading aggressively. From covariance it can be understood that stock is moving in the same direction with ISE but it is moving with a strong covariance that the variance of ISE.

g ) –
CVs= St.Dev of SAHOL / Mean of SAHOL ( SAHOL’s CV)

CVS= 0.027445/ 0.001528 = 1796 %
CVI= St.Dev of ISE / Mean of ISE ( ISE’s CV)

CVI= 0.021630/ 0.001550 = 1395%
They both have almost the same standard deviation but ISE is safer since it is less varied comparison to SAHOL stock.
h ) -
DS is calculated in E-Views as the difference between SAHOL stock return and market interest rate.
Sharp ratio of SAHOL stock is the division of mean of DS by standard deviation of DS.

S (SAHOL) = DS mean / DS standard deviation = 0.000588 / 0.027436 = 0.2143
DI is calculated in E-Views as the difference between ISE index return and market interest rate.
Sharp ratio of ISE index is the division of mean of DI by standard deviation of DI.

S (ISE) = DI mean / DI standard deviation = 0.000610 / 0.021625 = 0.2820
ISE index is preferable against SAHOL in terms of Sharp ratios, as sharpe ratio measures the excess return for each unit of risk taken. In the comparison of the above 2 sharp ratios, ISE is performing more return for each unit of risk taken than SAHOL stock does. i )-

Date observed between period 01.10.2001 and 01.10.2007.
Below are the founded statistics for the date observed.
Number of returns| 1508| | |
Number of months| 73| | |
Average number of returns per month (µ)| 21| | |
| | | |
Mean of returns| 0.15%| | |
Stdev of Returns| 2.74%| | |
| | | Number of Jumps|
Jump| greater than| 5.64174%| 53.00|
| lower than| -5.33611%| 30.00|
Total Jumps| | | 83.00|
| | | |
Average number of jumps per month| | | 1.1370|

According to the above date, jump occurs at an average rate of 1 ( rounded) in every month.

P( 1 jump per month) = (e^( -1) * 1^1)/ 1! = 36.79 %
P( 2 jumps per month) = (e^( -1) * 1^2)/ 2! = 18.739 %
j ) –
In SAHOL stock returns I have observed 1508 data and this data series have a mean of 0.15% and a standard deviation of 2.74%. So when the data is tested for Tchebychev’s rule for m=2, I number of 5.49% obtained for a r.v to be greater than in the probability. According to this, the proof is gone be this ; P( [X-µ] >=5.49%) <= 25% ( 1/m^2=1/4) Below is shown the proof of this probability in xls for the returns greater than...
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