1. Indian Institute of Management, Bangalore
Dharmesh Gandhi
PGSEM – Section ‘A’
Quantitative Methods 1
Final Assignment
Stocks Analyzed : RIL, SBI, Dr Reddy’s, Hindalco, Satyam
and Nifty
2. QM1 Assignment
All the data are adjusted for stock/splits and bonuses .Dividends are ignored
The returns are in %
1. Descriptive Statistics
1.1 Daily returns
Stock RIL SBI Dr Reddy Hindalco Satyam Nifty
Mean 0.099013838 0.069136782 0.120480846 0.0345841 0.28548121 0.033745968
Median 0.005703856 0 0 0 0 0.046533869
Mode 0 0 0 0 0 0
Skewness 0.444522438 0.175873627 0.075526213 -0.65485711 0.201215992 -0.12423968
Kurtosis 2.854520157 2.547972933 2.890787744 14.94130293 0.994890493 5.049793853
Max 15.1026393 16.81338028 12.69599441 16.56072265 16.4893617 10.91974165
Min -15.352349 -14.7669895 -18.1323334 -31.1499631 -15.9923237 -12.2377401
SD 2.723 2.704 2.9 2.43 4.17 1.691
Analysis
RIL: Mean, median and mode are all quite close to zero. Min, max and mean values suggest
symmetricity. However Range is approx 30 which is much more than 6σ .It is not a normal distribution.
Kurtosis is more than 0 implying a leptokurtic curve. It is a positively skewed curve.
SBI: Mean, median and mode are all quite close to zero. Min, max and mean values suggest
symmetricity. However Range is approx 30 which is much more than 6σ . It is not a normal distribution.
Kurtosis is more than 0 implying a leptokurtic curve. It is a positively skewed curve.
Dr Reddy’s : Mean is not equal to median and mode.Min and max values suggest a non-symmetric
curve.Range is approx 30 which is much more than 6σ. It is not a normal distribution. Kurtosis is more
than 0 implying a leptokurtic curve. It is a positively skewed curve.
Hindalco: Mean, median and mode are quite close. Min and max values suggest a non-symmetric
curve.Range is approx 47 which is much more than 6σ. It is not a normal distribution. Kurtosis is much
more than 0 implying a leptokurtic curve. It is a negatively skewed curve.
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3. QM1 Assignment
Satyam: Mean, median and mode are not quite close. Min and max values suggest a symmetric
curve.Range is approx 32 which is more than 6σ. It is not a normal distribution. Kurtosis is much more
than 0 implying a leptokurtic curve. It is a positively skewed curve.
Nifty: Mean, median and mode are quite close. Min and max values suggest a symmetric curve. Range is
approx 33 which is more than 6σ. It is not a normal distribution. Kurtosis is much more than 0 implying
a leptokurtic curve. It is a positively skewed curve.
1.2 Monthly return
Stock RIL SBI Dr Reddy Hindalco Satyam Nifty
Mean 2.259506206 1.743758789 2.861548397 1.033867437 7.12144637 0.87903147
Median 1.114649682 1.037089119 1.453488372 0.706190061 3.035032952 0.589599845
Mode 0 0 0 0 0 #N/A
Skewness 0.544226315 0.639824424 0.343847796 0.224917217 1.349531713 0.028199808
Kurtosis 1.217994303 1.727850253 0.057784501 0.614954091 3.802653999 -0.313170521
Max 57.98462852 72.26070529 46.57008948 52.23140496 160.4017217 26.07249791
Min -44.13177 -33.15266486 -36.78387097 -41.17212509 -49.3184466 -26.0693657
SD 13.03 13.42 13.56 12.82 24.88 8.32
Analysis
RIL: Mean, median and mode are not close. Min and max values suggest a non-symmetric curve. Range
is approx 102 which is more than 6σ. It is not a normal distribution. Kurtosis is much more than 0
implying a leptokurtic curve. It is a positively skewed curve.
SBI: Mean, median and mode are not close. Min and max values suggest a highly non-symmetric curve.
Range is approx 105 which is more than 6σ. It is not a normal distribution. Kurtosis is much more than 0
implying a leptokurtic curve. It is a positively skewed curve.
Dr Reddy: Mean, median and mode are not close. Min and max values suggest a non-symmetric curve.
Range is approx 83 which is more than 6σ. It is not a normal distribution. Kurtosis is slightly more than 0
implying a slight leptokurtic curve. It is a positively skewed curve.
Hindalco: Mean, median and mode are not close. Min and max values suggest a non-symmetric curve.
Range is approx 93 which is more than 6σ. It is not a normal distribution. Kurtosis is more than 0
implying a leptokurtic curve. It is a positively skewed curve.
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4. QM1 Assignment
Satyam: Mean, median and mode are quite far apart. Min and max values suggest a highly non-
symmetric curve. Range is approx 210 which is more than 6σ. It is not a normal distribution. Kurtosis is
more than 0 implying a leptokurtic curve. It is a positively skewed curve.
Nifty: Mean, median and mode are reasonably close. Min and max values suggest a symmetric curve.
Range is approx 52 which is quite close to 6σ. Kurtosis is slightly less than 0 implying a slightly
platykurtic curve. However, the skewness is almost 0. The curve has lot of properties of a normal
distribution.
1.3 Yearly return
Stock RIL SBI Dr Reddy Hindalco Satyam Nifty
Mean 29.65791997 15.48411764 39.76150302 9.914655201 139.7434961 9.968663069
Median 21.22872801 10.13151486 37.64470783 -7.594285714 54.15122313 -0.950419
Mode 100 5.535714286 -27.76548673 73.91304348 300 #N/A
Skewness 0.619341948 0.87772208 0.991949901 1.187115272 0.988186367 1.026198069
Kurtosis -0.421263142 0.536904004 0.897045857 0.522072956 0.188012776 0.043321
Max 202.5 132.3058378 237.9045529 146.7416667 827.5221652 102.184
Min -48.62697448 -49.07088782 -39.48145025 -59.31397096 -80.76086957 -34.18092156
SD 47.78 37.7 53.82 46.13 190.77 30.77
Analysis
RIL: Mean, median and mode are quite far apart. Min and max values suggest a highly non-symmetric
curve. Range is approx 250 which is less than 6σ(288). It is not a normal distribution. Kurtosis is less
than 0 implying a platykurtic curve. It is a positively skewed curve.
SBI: Mean, median and mode are quite far apart. Min and max values suggest a highly non-symmetric
curve. Range is approx 172 which is less than 6σ(228). It is not a normal distribution. Kurtosis is more
than 0 implying a leptokurtic curve. It is a positively skewed curve.
Dr Reddy: Mean, median and mode are quite far apart. Min and max values suggest a highly non-
symmetric curve. Range is approx 277 which is less than 6σ(324). It is not a normal distribution. Kurtosis
is more than 0 implying a leptokurtic curve. It is a positively skewed curve.
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5. QM1 Assignment
Hindalco: Mean, median and mode are quite far apart. Min and max values suggest a highly non-
symmetric curve. Range is approx 206 which is less than 6σ(276). It is not a normal distribution. Kurtosis
is more than 0 implying a leptokurtic curve. It is a positively skewed curve.
Satyam: Mean, median and mode are quite far apart. Min and max values suggest a highly non-
symmetric curve. Range is approx 908 which is less than 6σ(1146). It is not a normal distribution.
Kurtosis is more than 0 implying a leptokurtic curve. It is a positively skewed curve.
Nifty: : Mean, median and mode are quite far apart. Min and max values suggest a highly non-symmetric
curve. Range is approx 136 which is less than 6σ(180). It is not a normal distribution. Kurtosis is slightly
more than 0 implying a slightly-leptokurtic curve. It is a positively skewed curve.
2 Frequency distributions and histograms
2.1 Daily returns
2.1.1 RIL daily returns
Bin Freq
-9 4
-7.5 15
-6 13
-4.5 39
-3 144
-1.5 301
0 559
1.5 552
3 296
4.5 107
6 48
7.5 32
9 29
10.5 6
More 5
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30. QM1 Assignment
2.3.6 Nifty Yearly returns
Bin Frequency
-30 5
-20 203
-10 365
0 417
10 217
20 183
30 107
40 43
50 66
60 98
70 91
80 55
90 42
100 15
110 4
More 0
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31. QM1 Assignment
3 Using the frequency distribution of daily/monthly/yearly returns,
answer the following questions:
3.1 Probability of a positive return
Stock RIL SBI Dr Reddy Hindalco Satyam Nifty
Daily 0.509302326 0.50744186 0.510232558 0.505116279 0.521860465 0.519534884
Monthly 0.540629403 0.535932363 0.55284171 0.527477689 0.56552372 0.528417097
Yearly 0.631083203 0.64678179 0.700156986 0.407639979 0.714809001 0.481946625
3.2 Probability of a negative return
RIL SBI Dr Reddy Hindalco Satyam Nifty
Daily 0.490697674 0.49255814 0.489767442 0.494883721 0.478139535 0.480465116
Monthly 0.459370597 0.464067637 0.44715829 0.472522311 0.43447628 0.471582903
Yearly 0.368916797 0.35321821 0.299843014 0.592360021 0.285190999 0.518053375
3.3 Probability of a loss of more than 10%
RIL SBI Dr Reddy Hindalco Satyam Nifty
Monthly 0.14372945 0.173790512 0.155941757 0.186002818 0.218882104 0.096289338
Yearly 0.258503401 0.244897959 0.203558346 0.46310832 0.252223967 0.299843014
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32. QM1 Assignment
3.4 Probability of a gain of more than 15%
RIL SBI Dr Reddy Hindalco Satyam Nifty
Monthly 0.14372945 0.124001879 0.179426961 0.117895726 0.291686238 0.048379521
Yearly 0.545787546 0.423338566 0.610675039 0.312401884 0.639455782 0.321821036
4 Conditional probability if today’s return is between 5-10%
A= event that return after one month is
B= event that today’s return is between 5-10%
P(A/B)=P(A ∩Β)/ P(B)
4.1 If today’s return is 5%-10%, what is the probability that return of after
one month
EventAStock RIL SBI Dr Reddy Hindalco Satyam Nifty
5%-10% 0.101123596 0.023529412 0.076923077 0 0.131147541 0
>5% 0.101123596 0.023529412 0.076923077 0 0.135245902 0
>10% 0 0 0 0 0.004098361 0
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33. QM1 Assignment
4.2 If today’s return is 5%-10%, what is the probability that return of after
one year
RIL SBI Dr Reddy Hindalco Satyam Nifty
5%-10% 0.056818182 0.024691358 0.061403509 0 0.140350877 0
>5% 0.056818182 0.024691358 0.061403509 0 0.162280702 0
>10% 0 0 0 0 0.021929825 0
5 Assuming the distribution of daily return to be normal answer the
above questions and compare and comment
X= return after one month/year
B=event that today’s return is between 5-10%
Assume the conditional variable X/B follows a normal distribution
5.1 If today’s return is 5%-10%, what is the probability that return of after
one month
EventAStock RIL SBI Dr Reddy Hindalco Satyam Nifty
Mean(X/B) -0.16674872 0.148229898 -0.08257506 -0.5691103 -0.11021055 0.339731985
SD(X/B) 3.295984188 2.735541665 3.671949127 2.318709615 4.617763138 1.924326132
5%-10% 0.057469358 0.037906007 0.080136485 0.00815441 0.119941113 0.007722383
>5% 0.058488535 0.038064263 0.083154319 0.00815699 0.134224375 0.007722642
>10% 0.001019176 0.000158256 0.003017834 2.57986E-06 0.014283261 2.58252E-07
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34. QM1 Assignment
Comparison:
RIL: The conditional probabilities are about half of those derived from the data. The assumption of
normality needs to be checked.
SBI: The conditional probabilities are quite close to those derived from the data.
Dr Reddy: The conditional probabilities are quite close to those derived from the data.
Hindalco: The conditional probabilities are quite close to those derived from the data.
Satyam: The conditional probabilities are quite close to those derived from the data.
Nifty: The conditional probabilities are quite close to those derived from the data.
5.2 If today’s return is 5%-10%, what is the probability that return of after
one year
EventAStock RIL SBI Dr Reddy Hindalco Satyam Nifty
Mean(X/B) 3.242826551 3.602205133 7.305029634 0.463840804 9.90210935 1.683467465
SD(X/B) 10.98286795 14.43921291 12.57186648 11.57493635 24.6384195 8.060838659
5%-10% 0.167247701 0.132588216 0.15760646 0.142559148 0.080438796 0.18927429
>5% 0.436443601 0.461440438 0.572737608 0.347567914 0.578853767 0.340375825
>10% 0.2691959 0.328852222 0.415131148 0.205008767 0.498414971 0.151101534
Comparison:
RIL: The conditional probabilities are quite different. The assumption of normality needs to be checked.
SBI: The conditional probabilities are quite different. The assumption of normality needs to be checked.
Dr Reddy: The conditional probabilities are quite different. The assumption of normality needs to be
checked.
Hindalco: The conditional probabilities are quite different. The assumption of normality needs to be
checked.
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35. QM1 Assignment
Satyam: The conditional probabilities are quite different. The assumption of normality needs to be
checked.
Nifty: The conditional probabilities are quite different. The assumption of normality needs to be
checked.
6 Estimate a 95% confidence interval for the average
daily/monthly/yearly return.
6.1 RIL
Average daily return Average monthly return Average yearly return
K1 -0.01610043 1.7055041 27.51544905
K2 0.214128104 2.813508313 31.80039089
6.2 SBI
Average daily return Average monthly return Average yearly return
K1 -0.04517166 1.173506538 13.79382365
K2 0.183445227 2.314011041 17.17441162
6.3 Dr. Reddy’s
Average daily return Average monthly return Average yearly return
K1 -0.0021716 2.285435212 37.34823327
K2 0.243133288 3.437661581 42.17477278
6.4 Hindalco
Average daily return Average monthly return Average yearly return
K1 -0.0680646 0.489090237 7.846038599
K2 0.137232799 1.578644638 11.9832718
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36. QM1 Assignment
6.5 Satyam
Average daily return Average monthly return Average yearly return
K1 0.10936845 6.064716894 131.1902405
K2 0.46159397 8.178175845 148.2967518
6.6 Nifty
Average daily return Average monthly return Average yearly return
K1 -0.03774256 0.525498641 8.589232903
K2 0.105234498 1.232564299 11.34809323
7 Test the hypothesis that the average daily/monthly/yearly return is
more than 10%.
Test hypothesis
H0: µ >= 10% v/s Ha: µ< 10%
Reject H0 if sample mean < critical value
7.1 RIL
Daily Monthly Yearly
Sample Mean 0.099013838 2.259506206 29.65791997
Critical Value 9.903393063 9.535066776 8.201981723
Decision Reject H0 Reject H0 Accept H0
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37. QM1 Assignment
7.2 SBI
Daily Monthly Yearly
Sample Mean 0.069136782 1.743758789 15.48411764
Critical Value 9.904069329 9.521429224 8.581460573
Decision Reject H0 Reject H0 Accept H0
7.3 Dr Reddy’s
Daily Monthly Yearly
Sample Mean 0.120480846 2.861548397 39.76150302
Critical Value 9.897066826 9.516510575 7.974720179
Decision Reject H0 Reject H0 Accept H0
7.4 Hindalco
Daily Monthly Yearly
Sample Mean 0.0345841 1.033867437 9.914655201
Critical Value 9.913854496 9.542808561 8.263962223
Decision Reject H0 Reject H0 Accept H0
7.5 Satyam
Daily Monthly Yearly
Sample Mean 0.28548121 7.12144637 139.7434961
Critical Value 9.852201513 9.113164668 2.821881622
Decision Reject H0 Reject H0 Accept H0
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38. QM1 Assignment
7.6 Nifty
Daily Monthly Yearly
Sample Mean 0.033745968 0.87903147 9.968663069
Critical Value 9.940004935 9.703305897 8.842345712
Decision Reject H0 Reject H0 Accept H0
8 Estimate a 95% confidence interval for the proportion of time when
the daily/monthly/yearly return exceeds 10%.
P = (Number of days return > 10%)/Sample Size
Confidence Interval = P^ +/- Zα/2× √ (p (1-p)/n); where n = number of samples
n = 2150 for daily returns, 2129 for monthly returns, 1911 for yearly returns
8.1 RIL
Daily Monthly Yearly
Sample proportion 0.002790698 0.231564115 0.574045003
K1 0.000560833 0.213645695 0.551874651
K2 0.005020562 0.249482534 0.596215354
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39. QM1 Assignment
8.2 SBI
Daily Monthly Yearly
Sample proportion 0.000465116 0.235791451 0.500784929
K1 -0.00044628 0.217760019 0.478367427
K2 0.001376515 0.253822884 0.523202431
K1 Proportion cannot be less than 0 .This
result can come because sample
proportion is very small
8.3 Dr Reddy
Daily Monthly Yearly
Sample proportion 0.002325581 0.274307186 0.636839351
K1 0.000289529 0.255355177 0.615277697
K2 0.004361634 0.293259196 0.658401006
8.4 Hindalco
Daily Monthly Yearly
Sample proportion 0.000930233 0.225927666 0.340136054
K1 -0.00035838 0.208163871 0.318895234
K2 0.002218846 0.24369146 0.361376875
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40. QM1 Assignment
8.5 Satyam
Daily Monthly Yearly
Sample proportion 0.011162791 0.374823861 0.669806384
K1 0.006721823 0.354261401 0.648721233
K2 0.015603759 0.395386321 0.690891536
8.6 Nifty
Daily Monthly Yearly
Sample proportion 0.000465116 0.150305308 0.368393511
K1 -0.00044628 0.135125059 0.346766474
K2 0.001376515 0.165485556 0.390020548
9 Test the hypothesis that 90% of the time the daily/monthly/yearly
returns exceeds 10%.
p= (number of times the return exceed 10%)/(sample size)
H0: P=0.9 v/s Ha : P≠0.9
Reject H0 if p<k1 or p>k2; wherein k1 and k2 are the critical values
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42. QM1 Assignment
K2 0.912680901 0.912743288 0.913450518
Decision Reject H0 Reject H0 Reject H0
9.5 Satyam
Daily Monthly Yearly
Sample proportion 0.011162791 0.374823861 0.669806384
K1 0.887319099 0.887256712 0.886549482
K2 0.912680901 0.912743288 0.913450518
Decision Reject H0 Reject H0 Reject H0
9.6 Nifty
Daily Monthly Yearly
Sample proportion 0.000465116 0.150305308 0.368393511
K1 0.887319099 0.887256712 0.886549482
K2 0.912680901 0.912743288 0.913450518
Decision Reject H0 Reject H0 Reject H0
10 Assuming standard deviation to be a measure of risk, compute a 95%
confidence interval for the risk measure for daily/monthly/yearly
returns.
√((n-1)s2/χ2α/2) < σ < √((n-1)s2/χ21-α/2)
χ2 has degrees of freedom n-1.Since n is large χ2 will be a normal distribution
χ2 ~N ( n-1, 2n-2 )
n = 2150 for daily returns, 2129 for monthly returns, 1911 for yearly returns
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44. QM1 Assignment
10.5 Satyam
Daily Monthly Yearly
Sample SD 4.166409587 24.8773182 190.7715916
K1 4.047171356 24.16200104 184.9952978
K2 4.296848565 25.66017591 197.1251088
10.6 Nifty
Daily Monthly Yearly
Sample SD 1.691248809 8.322800569 30.76677472
K1 1.642847058 8.083488518 29.83520032
K2 1.74419722 8.584708566 31.79144108
11 Compute a 95% VaR for the top three high-risk stocks assuming the
daily/monthly/yearly return distribution to be normal
Compute the mean and variance of negative daily returns .VaR(Value at risk) here would be the
maximum loss faced by the investor 95% of the time given the fact that he faces a loss. Hence we pick all
the negative returns as our data set. 5th percentile of this distribution (assuming it to be normal) would
be the VaR.
Here, a negative value would imply a LOSS.
The 3 most volatile stocks are RIL, Satyam and Dr Reddy’s (decided by looking at the SD of yearly return).
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45. QM1 Assignment
11.1 RIL
Daily Monthly Yearly
Mean -1.89724528 -8.354053438 -17.47098572
Variance 2.819324919 46.83624966 115.151983
95% VaR -4.65909359 -19.61094061 -35.12172774
Number of days in last 6
months when |VaR|
was exceeded 5 8 0
11.2 Dr Reddy’s
Daily Monthly Yearly
Mean -1.91531609 -8.811026718 -15.90522586
Variance 4.01711585 46.42409842 89.45323824
95% VaR -5.2120541 -20.01827507 -31.46220575
Number of days in last 6
months when |VaR|
was exceeded 4 11 20
11.3 Satyam
Daily Monthly Yearly
Mean -2.94826329 -12.57627678 -37.74318809
Variance 6.42648655 100.4819388 529.7815985
95% VaR -7.11805209 -29.06440135 -75.60275935
Number of days in last 6
0 0 0
months when |VaR|
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46. QM1 Assignment
was exceeded
12 Count the number of days when VaR is exceeded during the last six
months in the data set and comment on the effectiveness of the VaR
estimate
12.1 RIL
The VaR is exceeded 5 times for the daily returns which is 3.79% < 5% which is good.
It is exceeded 8 times for the monthly returns which is 6 % > 5% ; not very good.
It is never exceeded for the yearly returns. Excellent.
12.2 Dr Reddy’s
The VaR is exceeded 4 times for the daily returns which is 3% < 5% which is good.
It is exceeded 11 times for the monthly returns which is 8.33 % > 5% ; not very good.
It is exceeded 20 times for the yearly returns which is 15.15% which is really bad.
12.3 Satyam
The VaR estimate is never exceeded for Satyam which is excellent.
13 Test if the assumption of normality of the daily/monthly/yearly
return distribution is valid.
Goodness of fit tests
H0: distribution is Normal v/s Ha : distribution is not normal
χ2calc =∑(Oi – Ei)2 /Ei
2 2
Reject H0 if χ calc > χ 0.05,n-1
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