Stock Market Analysis and CAPM

This assignment explores the relationship between excess stock returns and market returns using the Capital Asset Pricing Model (CAPM). It involves conducting a regression analysis to determine the coefficients of the CAPM equation, calculating the coefficient of determination (R-squared), and constructing confidence intervals for average stock returns. The assignment also examines the normality of the residuals using Jarque-Berra test and graphical analysis. Ultimately, it aims to assess the predictive power of the CAPM model in explaining market return variations.

Running Head: STATISTICAL INFERENCE AND REGRESSION ANALYSIS
Statistical Inference and Regression Analysis
Name of the Student
Name of the University
Author Note

1STATISTICAL INFERENCE AND REGRESSION ANALYSIS
Table of Contents
1.0 Introduction................................................................................................................................2
2.0 Comparison of Stock Returns....................................................................................................2
2.1 Graphical Display..................................................................................................................2
2.2 Computation of Returns.........................................................................................................3
2.3 Summary of Returns..............................................................................................................5
2.4 Hypothesis Testing on Stock Returns....................................................................................6
2.5 Comparison of Risks..............................................................................................................7
2.6 Comparison of mean Return..................................................................................................7
2.7 Calculation of Excess Returns...............................................................................................8
2.8 Estimation of CAPM for IBM...............................................................................................9
2.9 Determination of Neutral Stock...........................................................................................11
Bibliography..................................................................................................................................13

2STATISTICAL INFERENCE AND REGRESSION ANALYSIS
1.0 Introduction
There is availability of two different stocks in the market. One is the stock prices of
Boeing company and the other is the stock prices of IBM (International Business Machines).
Historical data on the monthly stock prices of these two companies have been collected from
finance.yahoo.com for the time frame of 2nd Feb, 2010 to 31st July, 2015. The S&P price index of
this time frame and the interest rate of the 10 year US Treasury Note has been collected from the
same website. The client is interested to invest in the stocks of one of two available companies
Boeing or IBM. Thus, the main interest of this study is to identify which of the two stocks will
be better and safer to invest. For this, the risk on the stock prices and the returns from the stocks
are to be compared over the time frame to assess which stock would be better to invest. The
necessary analysis will be conducted using appropriate statistical techniques and on Microsoft
Excel.
2.0 Comparison of Stock Returns
2.1 Graphical Display
Comparison of the stock prices has been done in the following figures 2.1, 2.2 and 2.3.
The stock prices of S&P, Boeing and IBM have been plotted in the graph over the selected time
frame. It has been observed that the stock prices for both the companies Boeing and IBM follow
an increasing trend. The S&P stock prices also follow an increasing trend.
2/1/2010
6/1/2010
10/1/2010
2/1/2011
6/1/2011
10/1/2011
2/1/2012
6/1/2012
10/1/2012
2/1/2013
6/1/2013
10/1/2013
2/1/2014
6/1/2014
10/1/2014
2/1/2015
6/1/2015
0
500
1000
1500
2000
2500
Trend of S&P Stock Prices
Date
S&P Stock Prices

3STATISTICAL INFERENCE AND REGRESSION ANALYSIS
2/1/2010
6/1/2010
10/1/2010
2/1/2011
6/1/2011
10/1/2011
2/1/2012
6/1/2012
10/1/2012
2/1/2013
6/1/2013
10/1/2013
2/1/2014
6/1/2014
10/1/2014
2/1/2015
6/1/2015
0
40
80
120
160
Trend of Boeing Stock Prices
Date
Boeing Stock Prices
2/1/2010
6/1/2010
10/1/2010
2/1/2011
6/1/2011
10/1/2011
2/1/2012
6/1/2012
10/1/2012
2/1/2013
6/1/2013
10/1/2013
2/1/2014
6/1/2014
10/1/2014
2/1/2015
6/1/2015
0
50
100
150
200
250
Trend of IBM Stock Prices
Date
IBM Stock Prices
2.2 Computation of Returns
The returns on the closing stock prices for S&P, BE and IBM are given in the following
table:
Date
Price Series Returns
S&P Boeing IBM
US TN
(10 Year)
Return
S&P
Return
Boeing
Return
IBM
2/1/2010 1104.49 63.16 127.16 3.595 – – –
3/1/2010 1169.43 72.61 128.25 3.833 5.71 13.94 0.85
4/1/2010 1186.69 72.43 129 3.663 1.47 -0.25 0.58
5/1/2010 1089.41 64.18 125.26 3.301 -8.55 -12.09 -2.94
6/1/2010 1030.71 62.75 123.48 2.951 -5.54 -2.25 -1.43
7/1/2010 1101.6 68.14 128.4 2.907 6.65 8.24 3.91
8/1/2010 1049.33 61.13 123.13 2.477 -4.86 -10.86 -4.19
9/1/2010 1141.2 66.54 134.14 2.517 8.39 8.48 8.56

4STATISTICAL INFERENCE AND REGRESSION ANALYSIS
10/1/2010 1183.26 70.64 143.6 2.612 3.62 5.98 6.81
11/1/2010 1180.55 63.77 141.46 2.797 -0.23 -10.23 -1.50
12/1/2010 1257.64 65.26 146.76 3.305 6.33 2.31 3.68
1/1/2011 1286.12 69.48 162 3.378 2.24 6.27 9.88
2/1/2011 1327.22 72.01 161.88 3.414 3.15 3.58 -0.07
3/1/2011 1325.83 73.93 163.07 3.454 -0.10 2.63 0.73
4/1/2011 1363.61 79.78 170.58 3.296 2.81 7.62 4.50
5/1/2011 1345.2 78.03 168.93 3.05 -1.36 -2.22 -0.97
6/1/2011 1320.64 73.93 171.55 3.158 -1.84 -5.40 1.54
7/1/2011 1292.28 70.47 181.85 2.805 -2.17 -4.79 5.83
8/1/2011 1218.89 66.86 171.91 2.218 -5.85 -5.26 -5.62
9/1/2011 1131.42 60.51 174.87 1.924 -7.45 -9.98 1.71
10/1/2011 1253.3 65.79 184.63 2.175 10.23 8.37 5.43
11/1/2011 1246.96 68.69 188 2.068 -0.51 4.31 1.81
12/1/2011 1257.6 73.35 183.88 1.871 0.85 6.56 -2.22
1/1/2012 1312.41 74.18 192.6 1.799 4.27 1.13 4.63
2/1/2012 1365.68 74.95 196.73 1.977 3.98 1.03 2.12
3/1/2012 1408.47 74.37 208.65 2.216 3.09 -0.78 5.88
4/1/2012 1397.91 76.8 207.08 1.915 -0.75 3.22 -0.76
5/1/2012 1310.33 69.61 192.9 1.581 -6.47 -9.83 -7.09
6/1/2012 1362.16 74.3 195.58 1.659 3.88 6.52 1.38
7/1/2012 1379.32 73.91 195.98 1.492 1.25 -0.53 0.20
8/1/2012 1406.58 71.4 194.85 1.562 1.96 -3.46 -0.58
9/1/2012 1440.67 69.6 207.45 1.637 2.39 -2.55 6.27
10/1/2012 1412.16 70.44 194.53 1.686 -2.00 1.20 -6.43
11/1/2012 1416.18 74.28 190.07 1.606 0.28 5.31 -2.32
12/1/2012 1426.19 75.36 191.55 1.756 0.70 1.44 0.78
1/1/2013 1498.11 73.87 203.07 1.985 4.92 -2.00 5.84
2/1/2013 1514.68 76.9 200.83 1.888 1.10 4.02 -1.11
3/1/2013 1569.19 85.85 213.3 1.852 3.54 11.01 6.02
4/1/2013 1597.57 91.41 202.54 1.675 1.79 6.28 -5.18
5/1/2013 1630.74 99.02 208.02 2.164 2.06 8.00 2.67
6/1/2013 1606.28 102.44 191.11 2.478 -1.51 3.40 -8.48
7/1/2013 1685.73 105.1 195.04 2.593 4.83 2.56 2.04
8/1/2013 1632.97 103.92 182.27 2.749 -3.18 -1.13 -6.77
9/1/2013 1681.55 117.5 185.18 2.615 2.93 12.28 1.58
10/1/2013 1756.54 130.5 179.21 2.542 4.36 10.49 -3.28
11/1/2013 1805.81 134.25 179.68 2.741 2.77 2.83 0.26
12/1/2013 1848.36 136.49 187.57 3.026 2.33 1.65 4.30
1/1/2014 1782.59 125.26 176.68 2.668 -3.62 -8.59 -5.98
2/1/2014 1859.45 128.92 185.17 2.658 4.22 2.88 4.69
3/1/2014 1872.34 125.49 192.49 2.723 0.69 -2.70 3.88
4/1/2014 1883.95 129.02 196.47 2.648 0.62 2.77 2.05

5STATISTICAL INFERENCE AND REGRESSION ANALYSIS
5/1/2014 1923.57 135.25 184.36 2.457 2.08 4.72 -6.36
6/1/2014 1960.23 127.23 181.27 2.516 1.89 -6.11 -1.69
7/1/2014 1930.67 120.48 191.67 2.556 -1.52 -5.45 5.58
8/1/2014 2003.37 126.8 192.3 2.343 3.70 5.11 0.33
9/1/2014 1972.29 127.38 189.83 2.508 -1.56 0.46 -1.29
10/1/2014 2018.05 124.91 164.4 2.335 2.29 -1.96 -14.38
11/1/2014 2067.56 134.36 162.17 2.194 2.42 7.29 -1.37
12/1/2014 2058.9 129.98 160.44 2.17 -0.42 -3.31 -1.07
1/1/2015 1994.99 145.37 153.31 1.675 -3.15 11.19 -4.55
2/1/2015 2104.5 150.85 161.94 2.002 5.34 3.70 5.48
3/1/2015 2067.89 150.08 160.5 1.934 -1.75 -0.51 -0.89
4/1/2015 2085.51 143.34 171.29 2.046 0.85 -4.59 6.51
5/1/2015 2107.39 140.52 169.65 2.095 1.04 -1.99 -0.96
6/1/2015 2063.11 138.72 162.66 2.335 -2.12 -1.29 -4.21
7/1/2015 2103.84 144.17 161.99 2.205 1.95 3.85 -0.41
2.3 Summary of Returns
From the results of the summary statistics, it can be seen that BE provides higher returns
than IBM. It can also be seen that the returns provided by BE is less than S&P 500. On the other
hand, it can be said that BE experiences higher risks than IBM. The risks suffered by IBM has a
similarity with that of the S&P 500.
Summary Statistics Return S&P Return Boeing Return IBM
Mean 0.99 1.27 0.37
Standard Error 0.45 0.74 0.57
Median 1.47 1.65 0.33
Mode #N/A #N/A #N/A
Standard Deviation 3.65 5.97 4.57
Sample Variance 13.34 35.58 20.85
Kurtosis 0.46 -0.32 0.56
Skewness -0.34 -0.22 -0.47
Range 18.78 26.04 24.26
Minimum -8.55 -12.09 -14.38
Maximum 10.23 13.94 9.88
Sum 64.44 82.53 24.21
Count 65 65 65
From the Jarque-Berra test, it can be seen that the returns provided by BE and IBM are
normally distributed. Normality test is performed on the data because any test performed on a
data which is not normal might not give a proper result on performing any tests.
Jarque Berra Test

6STATISTICAL INFERENCE AND REGRESSION ANALYSIS
Return S&P Return Boeing Return IBM
Test Statistic 1.850 0.793 3.254
P-Value 0.396 0.673 0.197
2.4 Hypothesis Testing on Stock Returns
To test whether the returns on Boeing stocks exceeds 3 percent, z-test has to be
conducted. From the comparison of the returns on Boeing stock, it has been observed that the
returns on the Boeing stocks do not exceed 3 percent.
Hypothesis Test for μ
Hypotheses
Null Hypothesis H0: μ ≤ 3
Alternative Hypothesis HA: μ > 3
Test Type Upper
Level of significance
alpha α set to: 0.05
Critical Region
Degrees of Freedom 64
Critical Value 1.6690
Sample Data
Sample Standard Deviation s 5.97
Sample Mean x bar 1.27
Sample Size n 65
Is Pop StDev known? Y/N N
Standard Error of the Mean 0.7399
t Sample Statistic -2.3385
p-value from the t distribution 0.0112
Hypothesis test decision:
Reject the Null Hypothesis
2.5 Comparison of Risks
Decision has to be made on which stock would be better to invest. In order to do that, the
risks of the stocks have to be compared. From the results of the comparison of stock returns of
BE and IBM, it can be seen that the variation in the returns are not same. Moreover, the variation
is higher in case of BE. Thus, IBM would be a preferred choice for investment.
F-Test Two-Sample for Variances

7STATISTICAL INFERENCE AND REGRESSION ANALYSIS
Boeing IBM
Mean 94.97 174.97
Variance 905.08 582.33
Observations 66 66
df 65 65
F 1.554
P(F<=f) one-tail 0.039
F Critical one-tail 1.508
2.6 Comparison of mean Return
Further, it has to be compared whether both the stocks have the same average returns.
From the results of the analysis, it has been observed that the average stock returns for both the
companies BE and IBM are same. There is no significant difference between the average returns
on the stocks of BE as well as IBM.
t-Test: Two-Sample Assuming Equal Variances
Return Boeing Return IBM
Mean 1.27 0.37
Variance 35.58 20.85
Observations 65 65
Pooled Variance 28.22
Hypothesized Mean Difference 0
df 128
t Stat 0.963
P(T<=t) one-tail 0.169
t Critical one-tail 1.657
P(T<=t) two-tail 0.337
t Critical two-tail 1.979
Thus, from the analysis above it has been observed that the stocks of BE and IBM
provide same returns with lesser risk in case of IBM and higher risk in case of BE. Thus, the
preferred choice of investment will be in the stocks of BE.
2.7 Calculation of Excess Returns
Date US TN (10
Year) Return S&P Return IBM
Excess Return
(Yt)
Excess Market
Return (Xt)
2/1/2010 3.595
3/1/2010 3.833 5.7133 0.8535 -2.9795 1.8803
4/1/2010 3.663 1.4651 0.5831 -3.0799 -2.1979
5/1/2010 3.301 -8.5532 -2.9421 -6.2431 -11.8542
6/1/2010 2.951 -5.5388 -1.4312 -4.3822 -8.4898
7/1/2010 2.907 6.6516 3.9071 1.0001 3.7446

8STATISTICAL INFERENCE AND REGRESSION ANALYSIS
8/1/2010 2.477 -4.8612 -4.1910 -6.6680 -7.3382
9/1/2010 2.517 8.3928 8.5643 6.0473 5.8758
10/1/2010 2.612 3.6193 6.8148 4.2028 1.0073
11/1/2010 2.797 -0.2293 -1.5015 -4.2985 -3.0263
12/1/2010 3.305 6.3256 3.6782 0.3732 3.0206
1/1/2011 3.378 2.2393 9.8798 6.5018 -1.1387
2/1/2011 3.414 3.1457 -0.0741 -3.4881 -0.2683
3/1/2011 3.454 -0.1048 0.7324 -2.7216 -3.5588
4/1/2011 3.296 2.8097 4.5025 1.2065 -0.4863
5/1/2011 3.05 -1.3593 -0.9720 -4.0220 -4.4093
6/1/2011 3.158 -1.8426 1.5390 -1.6190 -5.0006
7/1/2011 2.805 -2.1708 5.8307 3.0257 -4.9758
8/1/2011 2.218 -5.8468 -5.6211 -7.8391 -8.0648
9/1/2011 1.924 -7.4467 1.7072 -0.2168 -9.3707
10/1/2011 2.175 10.2307 5.4311 3.2561 8.0557
11/1/2011 2.068 -0.5072 1.8088 -0.2592 -2.5752
12/1/2011 1.871 0.8497 -2.2159 -4.0869 -1.0213
1/1/2012 1.799 4.2660 4.6332 2.8342 2.4670
2/1/2012 1.977 3.9787 2.1217 0.1447 2.0017
3/1/2012 2.216 3.0851 5.8826 3.6666 0.8691
4/1/2012 1.915 -0.7526 -0.7553 -2.6703 -2.6676
5/1/2012 1.581 -6.4699 -7.0933 -8.6743 -8.0509
6/1/2012 1.659 3.8793 1.3798 -0.2792 2.2203
7/1/2012 1.492 1.2519 0.2043 -1.2877 -0.2401
8/1/2012 1.562 1.9571 -0.5783 -2.1403 0.3951
9/1/2012 1.637 2.3947 6.2660 4.6290 0.7577
10/1/2012 1.686 -1.9988 -6.4304 -8.1164 -3.6848
11/1/2012 1.606 0.2843 -2.3194 -3.9254 -1.3217
12/1/2012 1.756 0.7043 0.7756 -0.9804 -1.0517
1/1/2013 1.985 4.9198 5.8402 3.8552 2.9348
2/1/2013 1.888 1.1000 -1.1092 -2.9972 -0.7880
3/1/2013 1.852 3.5355 6.0241 4.1721 1.6835
4/1/2013 1.675 1.7924 -5.1762 -6.8512 0.1174
5/1/2013 2.164 2.0550 2.6697 0.5057 -0.1090
6/1/2013 2.478 -1.5113 -8.4785 -10.9565 -3.9893
7/1/2013 2.593 4.8278 2.0355 -0.5575 2.2348
8/1/2013 2.749 -3.1798 -6.7716 -9.5206 -5.9288
9/1/2013 2.615 2.9316 1.5839 -1.0311 0.3166
10/1/2013 2.542 4.3630 -3.2770 -5.8190 1.8210
11/1/2013 2.741 2.7663 0.2619 -2.4791 0.0253
12/1/2013 3.026 2.3289 4.2975 1.2715 -0.6971
1/1/2014 2.668 -3.6231 -5.9812 -8.6492 -6.2911
2/1/2014 2.658 4.2213 4.6934 2.0354 1.5633

9STATISTICAL INFERENCE AND REGRESSION ANALYSIS
3/1/2014 2.723 0.6908 3.8770 1.1540 -2.0322
4/1/2014 2.648 0.6182 2.0466 -0.6014 -2.0298
5/1/2014 2.457 2.0812 -6.3619 -8.8189 -0.3758
6/1/2014 2.516 1.8879 -1.6903 -4.2063 -0.6281
7/1/2014 2.556 -1.5195 5.5787 3.0227 -4.0755
8/1/2014 2.343 3.6964 0.3282 -2.0148 1.3534
9/1/2014 2.508 -1.5635 -1.2928 -3.8008 -4.0715
10/1/2014 2.335 2.2936 -14.3826 -16.7176 -0.0414
11/1/2014 2.194 2.4237 -1.3657 -3.5597 0.2297
12/1/2014 2.17 -0.4197 -1.0725 -3.2425 -2.5897
1/1/2015 1.675 -3.1533 -4.5458 -6.2208 -4.8283
2/1/2015 2.002 5.3439 5.4764 3.4744 3.3419
3/1/2015 1.934 -1.7549 -0.8932 -2.8272 -3.6889
4/1/2015 2.046 0.8485 6.5064 4.4604 -1.1975
5/1/2015 2.095 1.0437 -0.9621 -3.0571 -1.0513
6/1/2015 2.335 -2.1236 -4.2075 -6.5425 -4.4586
7/1/2015 2.205 1.9550 -0.4128 -2.6178 -0.2500
2.8 Estimation of CAPM for IBM
Date CAPM Date CAPM
2/1/2010 11/1/2012 -3.56455
3/1/2010 -2.98785 12/1/2012 -1.76906
4/1/2010 -3.04909 1/1/2013 1.17901
5/1/2010 -4.97757 2/1/2013 -2.99866
6/1/2010 -3.84307 3/1/2013 1.372211
7/1/2010 -0.56164 4/1/2013 -5.34834
8/1/2010 -5.23661 5/1/2013 -0.86307
9/1/2010 2.515487 6/1/2013 -7.85119
10/1/2010 1.390917 7/1/2013 -1.51123
11/1/2010 -3.792 8/1/2013 -6.97573
12/1/2010 -0.94387 9/1/2013 -1.79999
1/1/2011 2.792546 10/1/2013 -4.71902
2/1/2011 -3.29795 11/1/2013 -2.68279
3/1/2011 -2.83062 12/1/2013 -0.3962
4/1/2011 -0.43582 1/1/2014 -6.4445
5/1/2011 -3.62345 2/1/2014 0.069555
6/1/2011 -2.1584 3/1/2014 -0.46782
7/1/2011 0.673323 4/1/2014 -1.53805
8/1/2011 -5.95061 5/1/2014 -6.54798
9/1/2011 -1.30356 6/1/2014 -3.73579
10/1/2011 0.813767 7/1/2014 0.671498

10STATISTICAL INFERENCE AND REGRESSION ANALYSIS
11/1/2011 -1.32939 8/1/2014 -2.39976
12/1/2011 -3.66299 9/1/2014 -3.48857
1/1/2012 0.556555 10/1/2014 -11.3636
2/1/2012 -1.08317 11/1/2014 -3.34162
3/1/2012 1.064031 12/1/2014 -3.14822
4/1/2012 -2.79936 1/1/2015 -4.96399
5/1/2012 -6.45982 2/1/2015 0.94685
6/1/2012 -1.34161 3/1/2015 -2.89502
7/1/2012 -1.95643 4/1/2015 1.547989
8/1/2012 -2.47621 5/1/2015 -3.03515
9/1/2012 1.650793 6/1/2015 -5.16013
10/1/2012 -6.11967 7/1/2015 -2.76733
The CAPM is calculated from the regression analysis. The calculation of the CAPM is
given by the following equation:
CAPM = -1.17 + 0.61 * Excess Return
Regression Statistics
Multiple R 0.4951
R Square 0.2452
Adjusted R Square 0.23
Standard Error 3.96
Observations 65
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept -1.17 0.53 -2.225 0.03 -2.22 -0.12
Xt 0.61 0.13 4.524 0.000 0.34 0.88
The coefficients of the regression analysis show that, if the excess return on the stocks is
zero, then the market return of the stocks will be -1.17. With one-unit increase in the excess
return of the stocks, the market return of the stocks will increase by 0.61.
The coefficient of determination (R Square) value indicates that the excess return on the
stocks will be able to explain 24.51 percent of the variability in the market return of the stocks.
The average excess stock returns of IBM are supposed to lie between 0.34 and 0.88 with 95
percent confidence.
2.9 Determination of Neutral Stock
It can be seen clearly from the computations of the confidence interval that the average
return on the stocks of IBM lies between -0.76 and 1.50 with 95 percent confidence. There is

11STATISTICAL INFERENCE AND REGRESSION ANALYSIS
always a possibility that the stocks will not have any returns. Thus, this stock can be termed as a
neutral stock.
Confidence Interval for the mean
Data
Sample Standard Deviation 4.5661
Sample Mean 0.37
Sample Size 65
Confidence Level 95%
Is Pop StDev known? Y/N N
Intermediate Calculations
Standard Error of the Mean 0.5664
Degrees of Freedom 64
t Value 1.9977
Margin of Error 1.1314
Confidence Interval
Interval Lower Limit -0.76
Interval Upper Limit 1.50
From the results of the Jarque-Berra test, it can be said that the errors in the developed
model of prediction of CAPM is normally distributed. The following figure also shows that the
errors are normally distributed as they follow a linear trend.
Test for Normality of Residuals
Skewness -0.755
Kurtosis 2.915
Count 65
Jarque-Berra test Statistic 29.181
P-Value 0.000

12STATISTICAL INFERENCE AND REGRESSION ANALYSIS
0 20 40 60 80 100 120
-20
-15
-10
-5
0
5
10
Normal Probability Plot
Sample Percentile
Yt

13STATISTICAL INFERENCE AND REGRESSION ANALYSIS
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