AMERICAN UNIVERSITY IN DUBAI
FINANCIAL MODELLING AND EMPIRICAL ANALYSIS
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Irfan A. Khan - 0505009092
4/17/2010
Abstract: The paper contains the valid results on the returns of Goldman Sacks for two years and, to test whether it moves along with the S&P500 Index, a series of histograms, box-whisker plots, best-fit plots, confident intervals and summary statistics of both population and random sample has been used to analysis the validity and accuracy of the data.
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Collections of data (Prices and Return) for Goldman Sacks (Ticker GS) and S&P 500
The data of daily prices for GS which is traded in New York Stock Exchange and S&P 500 was
retrieved from Yahoo Finance for the range dated from 26th March 2008 to 26th March 2009, and
the daily return was calculated through Holding Period Returns [HPR] formula [(Price old – Price
new)/Price new].
Calculation of the mean, standard deviation, max, min, 1st quartile, median, 3rd
quartile for both the returns of the whole data (Population) for 2 years:
GS analysis:
Through Stat-Tools:
Through Excel Functions:
Pop. Mean Pop SD % Pop Mean % Pop SD
0.000955488 0.043325542 0.0955% 4.3326%
Quartile 3 Median Quartile 1 Max Min
0.0182 0.0000 -0.0181 0.2648 -0.1896
Interpretation of the analyzed data on daily return for GS stocks:
The average or mean return on GS stock is 0.096% daily for the past 2 years. This
entails that on average, if an investor buys a GS stock and sells it the next day, the investor will
receive 0.096% of the stock price. The standard deviation is a measure of variation and it verifies
the distribution of the stock prices around the mean. GS’s standard deviation is equals to 4.33%,
which means on average, the distance of the returns on GS stock from its mean is 4.33%. The
maximum daily return on the GS stock during the past 2 years was 26.48%. However, the minimum
daily return gained on the GS stock in the past 2 years was -18.96%. GS 1st quartile is -1.81% this
shows that 25% of the returns are under -1.81% and 75% of the returns are higher from -1.81%. GS
median is 0% which denotes that 50% of the daily returns are less than 0% and 50% of the daily
Return
One Variable Summary Goldman Sachs Population #1
Mean 0.00096 Std. Dev. 0.04333 Median 0.00000 Mean Abs. Dev. 0.02777 Minimum -0.18958 Maximum 0.26475 1st Quartile -0.01815 3rd Quartile 0.01814
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returns are higher than 0%. GS 3rd quartile is 1.82% which denotes that 25% of the daily return are
higher than 1.82% and 75% of the daily returns are less than 1.82%.
S&P 500 analysis:
Through Stat-Tools:
Through Excel Functions:
Pop. Mean Pop SD % Pop Mean % Pop SD
-0.000045999 0.021554822 -0.0046% 2.1555%
Quartile 3
Median Quartile 1
Max Min
0.0087 0.0013 -0.0094 0.1158 -0.0903
Interpretation of the analyzed data on daily return for S&P 500 Index:
The average or mean return on S&P 500 Index is -0.0046% daily for the past 2
years. This entails that on average, if an investor buys a S&P 500 Index and sells it the next day, the
investor will receive -0.0046% of the index price. The standard deviation is a measure of variation
and it verifies the distribution of the stock prices around the mean. S&P 500’s standard deviation is
equals to 2.155%, which means on average, the distance of the returns on S&P 500 Index from its
mean is 2.155%. The maximum daily return on the S&P 500 Index during the past 2 years was
11.58%. However, the minimum daily return gained on the S&P 500 Index in the past 2 years was –
0.0903%. GS 1st quartile is -0.0094% this shows that 25% of the returns are under -0.0094% and
75% of the returns are higher than -0.0094%. GS median is 0.13% which denotes that 50% of the
daily returns are less than 0.13% and 50% of the daily returns are higher than 0.13%. GS 3rd
quartile is 0.87% which denotes that 25% of the daily return are higher than 0.87% and 75% of the
daily returns are less than 0.87%.
Return
One Variable Summary S&P Population
Mean -0.00005
Std. Dev. 0.02155
Median 0.00127
Mean Abs. Dev. 0.01425
Minimum -0.09035
Maximum 0.11580
1st Quartile -0.00951
3rd Quartile 0.00864
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Histogram of Return / Goldman Sachs Population #1
0
50
100
150
200
250
300-0
.16687
-0.1
2143
-0.0
7600
-0.0
3057
0.0
1487
0.0
6030
0.1
0573
0.1
5117
0.1
9660
0.2
4203
Fre
quency
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
Plotting Histogram for GS and S&P 500 daily returns of the whole data (Population)
for 2 years:
GS Histogram
Return / Goldman Sachs Population #1
Histogram Bin Min Bin Max Bin Midpoint Freq. Rel. Freq. Prb. Density
Bin #1 -0.18958 -0.14415 -0.16687 2 0.0040 0.1 Bin #2 -0.14415 -0.09872 -0.12143 9 0.0179 0.4 Bin #3 -0.09872 -0.05328 -0.07600 22 0.0437 1.0 Bin #4 -0.05328 -0.00785 -0.03057 154 0.3056 6.7 Bin #5 -0.00785 0.03758 0.01487 256 0.5079 11.2 Bin #6 0.03758 0.08302 0.06030 47 0.0933 2.1 Bin #7 0.08302 0.12845 0.10573 6 0.0119 0.3 Bin #8 0.12845 0.17388 0.15117 4 0.0079 0.2 Bin #9 0.17388 0.21932 0.19660 2 0.0040 0.1 Bin #10 0.21932 0.26475 0.24203 2 0.0040 0.1
Interpretation of the GS Histogram;
GS return histogram is almost symmetric (bell-shaped).
The returns with the highest frequency of 256 lies between -0.00785 and 0.03758
with 50.79% probability of making between that range if an investors invests.
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Histogram of Return / Data Set #2
0
50
100
150
200
250
300
-0.0
8004
-0.0
5943
-0.0
3881
-0.0
1820
0.0
0242
0.0
2303
0.0
4365
0.0
6426
0.0
8488
0.1
0549
Fre
quency
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
StatTools Student VersionFor Academic Use Only
The returns with the lowest frequency of 2 lies between the ranges of -0.18958 to -
0.14415, 0.17388 to 0.21932 and 0.21932 to 0.26475 with 0.4% probability for each
range of making between that ranges if an investor invests.
S&P 500 Histogram
Return / S&P Population
Histogram Bin Min Bin Max Bin Midpoint Freq. Rel. Freq. Prb. Density
Bin #1 -0.09035 -0.06973 -0.08004 4 0.0079 0.4 Bin #2 -0.06973 -0.04912 -0.05943 9 0.0179 0.9 Bin #3 -0.04912 -0.02850 -0.03881 27 0.0536 2.6 Bin #4 -0.02850 -0.00789 -0.01820 103 0.2044 9.9 Bin #5 -0.00789 0.01273 0.00242 267 0.5298 25.7 Bin #6 0.01273 0.03334 0.02303 69 0.1369 6.6 Bin #7 0.03334 0.05396 0.04365 17 0.0337 1.6 Bin #8 0.05396 0.07457 0.06426 6 0.0119 0.6 Bin #9 0.07457 0.09519 0.08488 0 0.0000 0.0 Bin #10 0.09519 0.11580 0.10549 2 0.0040 0.2
Interpretation of the S&P 500 Histogram;
S&P 500 return histogram is almost symmetric (bell-shaped).
The returns with the highest frequency of 267 lies between -0.00789 and 0.01273
with 52.98% probability of making between that range if an investors invests.
The returns with the lowest frequency of 0 lies between of 0.07457 to 0.09519, with
0% probability of making between that range if an investor invests.
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Testing the Empirical Rule:
GS returns:
S&P 500 returns:
Category Upper limit Frequency
More than 3 stdev below mean -0.064710 5
Between 2 and 3 stdevs below mean
-0.043156 12
Between 1 and 2 stdevs below mean
-0.021601 39
Between mean and 1 stdev below mean
-0.000046 174
Between mean and 1 stdev above mean
0.021509 224
Between 1 and 2 stdevs above mean
0.043064 38
Between 2 and 3 stdevs above mean
0.064618 7
More than 3 stdevs above mean
5
Percentages within k stdevs of mean
Empirical rule 68% 95% 99.70% Actual data 78.97% 94.25% 98.02%
Category Upper limit Frequency
More than 3 stdev below mean -0.1290 3
Between 2 and 3 stdevs below mean
-0.0857 8
Between 1 and 2 stdevs below mean
-0.0424 37
Between mean and 1 stdev below mean
0.0010 210
Between mean and 1 stdev above mean
0.0443 196
Between 1 and 2 stdevs above mean
0.0876 38
Between 2 and 3 stdevs above mean
0.1309 4
More than 3 stdevs above mean
8
Percentages within k stdevs of mean
Empirical rule 68% 95% 99.70% Actual data 80.56% 95.44% 97.82%
Plotting Box-Plots for GS and S&P 500 daily
for 2 years:
GS returns:
1) Mild outliers in the right tail of dist.
mild outlier= q3+1.5IQR
0.0726 starting of the mild outlier
0.1270 ending of the mild outlier
We will consider a return as a mild outlier if it falls between
2) Return will be considered as exceptionally good return.
the interval is 0.12700 plus infinity
3) Mild outliers in the left tail of dist.
mild outlier= q1-1.5iqr
-0.0726 starting of the mild outlier
-0.1270 ending of the mild outlier
The return is extremely bad return if it falls below
Plots for GS and S&P 500 daily returns of the whole data
1) Mild outliers in the right tail of dist.
starting of the mild outlier
ending of the mild outlier
We will consider a return as a mild outlier if it falls
0.0726 and
2) Return will be considered as exceptionally good return.
plus infinity
3) Mild outliers in the left tail of dist.
starting of the mild outlier
ending of the mild outlier
The return is extremely bad return -0.12702
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returns of the whole data (Population)
0.1270
S&P 500 returns:
1) Mild outliers in the right tail of dist.
mild outlier= q3+1.5IQR
0.03587 starting of the mild outlier
0.06310 ending of the mild outlier
We will consider a return as a mild outlier if it between
2) Return will be considered as exceptionally good return.
the interval is 0.06310 plus infinity
3) Mild outliers in the left tail of dist.
mild outlier= q1-1.5iqr
-0.03674 starting of the
-0.06397 ending of the mild outlier
The return is extremely bad return if it falls below
1) Mild outliers in the right tail of dist.
starting of the mild outlier
ending of the mild outlier
We will consider a return as a mild outlier if it falls
0.03587 and
eturn will be considered as exceptionally good return.
plus infinity
ild outliers in the left tail of dist.
starting of the mild outlier
ending of the mild outlier
The return is extremely bad return if it falls -0.06397
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0.06310
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Calculation of the mean and standard deviation for both GS and S&P 500 returns of
the sampled data (from the Population of 2 years):
Samplings involve choosing a sample from a population. The sample should be randomly chosen
and should be a representative of the data. The method for obtaining the sampl is by using the
“=Rand ()” excel function to acquire random numbers. We afterward arrange that information into
ascending order and choose the first 100 observations as our sample data.
GS analysis:
Through Stat-Tools:
Return
One Variable Summary Goldman Sachs Sample 100
Mean 0.00057
Std. Dev. 0.04815
Median -0.00056
Mean Abs. Dev. 0.02965
Minimum -0.12130
Maximum 0.24997
Range 0.37128
Count 100
1st Quartile -0.01766
3rd Quartile 0.01610
Interquartile Range 0.03376
Through Excel Functions:
Sample Mean Sample SD % Sample Mean
% Sample SD
0.000568478 0.048153791 0.0568% 4.82%
Quartile 3 Median Quartile 1 Max Min
0.0162 0.0002 -0.0152 0.2500 -0.1213
Interpretation of the analyzed data on sample daily return for GS stocks:
The sample average or sample mean return on GS stock is 0.0568% daily. This
entails that on average, if an investor buys a GS stock and sells it the next day, the investor will
receive 0.0568% of the stock price. The standard deviation is a measure of variation and it verifies
the distribution of the stock prices around the mean. GS’s standard deviation is equals to 4.82%,
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which means on average, the distance of the returns on GS stock from its mean is 4.33%. The
maximum daily return on the GS stock was 25%. However, the minimum daily return gained on the
GS stock was -12.13%. GS 1st quartile is -1.52% this shows that 25% of the returns are under -
1.52% and 75% of the returns are higher from -1.52%. GS median is 0.02% which denotes that 50%
of the daily returns are less than 0.02% and 50% of the daily returns are higher than 0.02%. GS 3rd
quartile is 1.62% which denotes that 25% of the daily return are higher than 1.62% and 75% of the
daily returns are less than 1.62%.
S&P 500 analysis:
Through Stat-Tools:
Through Excel Functions:
Sample Mean Sample SD % Sample Mean % Sample SD
-0.000232742 0.021565491 -0.0233% 2.16%
Quartile 3 Median Quartile 1 Max Min
0.0071 0.0021 -0.0101 0.1079 -0.0881
Interpretation of the analyzed data on sample daily return for S&P 500 Index:
The sample average or sample mean return on S&P 500 Index is -0.0233% daily.
This entails that on average, if an investor buys a S&P 500 Index and sells it the next day, the
investor will receive -0.0233% of the index price. The standard deviation is a measure of variation
and it verifies the distribution of the stock prices around the mean. S&P 500’s standard deviation is
equals to 2.156%, which means on average, the distance of the returns on S&P 500 Index from its
mean is 2.156%. The maximum daily return on the S&P 500 Index was 10.79%. However, the
Return
One Variable Summary S&P Sample
100
Mean -0.00023
Std. Dev. 0.02157
Median 0.00198
Mean Abs. Dev. 0.01388
Minimum -0.08807
Maximum 0.10789
Range 0.19596
Count 100
1st Quartile -0.01007
3rd Quartile 0.00686
Interquartile Range 0.01693
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minimum daily return gained on the S&P 500 Index was –8.81%. GS 1st quartile is -1.01% this
shows that 25% of the returns are under -1.01% and 75% of the returns are higher than -1.01%. GS
median is 0.21% which denotes that 50% of the daily returns are less than 0.21% and 50% of the
daily returns are higher than 0.21%. GS 3rd quartile is 0.71% which denotes that 25% of the daily
return are higher than 0.71% and 75% of the daily returns are less than 0.71%.
Comparison and conclusion for the sample mean, sample standard deviation to the
population mean and population standard deviation.
The sample mean for GS is lower than the population mean, while the sample mean of S&P500
higher than the population mean. Also, the sample standard deviation for GS is higher than the
population standard deviation, while the sample standard deviation of S&P500 is same to the
population standard deviation.
Since the sample means and standard deviations differ from the population means and standard
deviations, they could be not good representatives for the population.
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Best distribution that fits the sample data
GS Sampled returns:
Logistic Normal
The best distribution that fits the GS sample data is the Logistics and the Normal comes number 7 in
the list of best distributions.
S&P Sampled returns:
LogLogistic Normal
The best distribution that fits the S&P 500 sample data is the LogLogistics and the Normal comes
number 4 in the list of best distributions.
Logistic(-0.00033128, 0.022046)
0
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-0.1
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-0.1
0
-0.0
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5
0.2
0
0.2
5
0.3
0
< >5.0% 5.0%90.0%
-0.0652 0.0646
BestFit Student VersionFor Academic Use Only
Normal(0.00056600, 0.048151)
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10
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-0.1
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-0.1
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-0.0
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0.2
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< >5.0% 5.0%90.0%
-0.0786 0.0798
BestFit Student VersionFor Academic Use Only
LogLogistic(-0.87524, 0.87499, 86.710)
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-0.1
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-0.0
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< >5.0% 5.0%90.0%
-0.0295 0.0300
BestFit Student VersionFor Academic Use Only
Normal(-0.00023274, 0.021565)
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-0.1
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-0.0
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< >5.0% 5.0%90.0%
-0.0357 0.0352
BestFit Student VersionFor Academic Use Only
Page 13 of 15
95% and 90% confidence intervals for the population mean
GS Returns:
Return
Conf. Intervals (One-Sample) Goldman Sachs Sample 100
Sample Size 100 Sample Mean 0.00057 Sample Std Dev 0.04815 Confidence Level (Mean) 95.0% Degrees of Freedom 99 Lower Limit -0.00899 Upper Limit 0.01012
Interpretation: We are 95% confident that the returns will lie between the limits of
-0.00899 and 0.01012.
Return
Conf. Intervals (One-Sample) Goldman Sachs Sample 100
Sample Size 100 Sample Mean 0.00057 Sample Std Dev 0.04815 Confidence Level (Mean) 90.0% Degrees of Freedom 99 Lower Limit -0.00743 Upper Limit 0.00856
Interpretation: We are 90% confident that the returns will lie between the limits of
-0.00743 and 0.00856.
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S&P 500 Returns:
Return
Conf. Intervals (One-Sample) S&P Sample 100
Sample Size 100 Sample Mean -0.00023 Sample Std Dev 0.02157 Confidence Level (Mean) 95.0% Degrees of Freedom 99 Lower Limit -0.00451 Upper Limit 0.00405
Interpretation: We are 95% confident that the returns will lie between the limits of
-0.00451 and 0.00405.
Return
Conf. Intervals (One-Sample) S&P Sample 100
Sample Size 100 Sample Mean -0.00023 Sample Std Dev 0.02157 Confidence Level (Mean) 90.0% Degrees of Freedom 99 Lower Limit -0.00381 Upper Limit 0.00335
Interpretation: We are 90% confident that the returns will lie between the limits of
-0.00381 and 0.00335.
Page 15 of 15
Simple regression conducted on the S&P 500 Index
Multiple R-Square
Adjusted StErr of Durbin
Summary R R-Square Estimate Watson
0.6795 0.4617 0.4562 0.030970615 2.0115
Degrees of Sum of Mean of F-Ratio p-Value ANOVA
Table Freedom Squares Squares
Explained 1 0.080633249 0.080633249 84.0649 < 0.0001
Unexplained 98 0.093999544 0.000959179
Coefficient Standard
t-Value p-Value Lower Upper
Regression Table Error Limit Limit
Constant 0.00320442 0.003102904 1.0327 0.3043 -
0.002953193 0.009362033 S&P-Return-risk free rate 1.196832687 0.130534767 9.1687 < 0.0001 0.93779069 1.455874684
R-Square
The r-square is 46.71%, therefore it’s not a very good regression and not good for prediction.
P-value
The p-value for Beta 1 or S&P500 index is less than 5%, therefore it’s significant.
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