STATGB 3302 : Statistical Inference And Regression Analysis

Answer:

1.Comparison of Stock Returns

1.1 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.

1.2 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
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
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

1.3 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
Return S&P	Return Boeing	Return IBM
Test Statistic	1.850	0.793	3.254
P-Value	0.396	0.673	0.197

1.4 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 H_A:	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

1.5 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
	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

1.6 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.

1.7 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
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
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

1..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
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.

1.9 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 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

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 2^nd Feb, 2010 to 31^st 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.

Referance:

Anderson, D.R., Sweeney, D.J., Williams, T.A., Camm, J.D. and Cochran, J.J., 2016. Statistics for business & economics. Nelson Education.

Siegel, A., 2016. Practical business statistics. Academic Press.

Ruppert, D., 2014. Statistics and finance: an introduction. Springer.

Swift, L. and Piff, S., 2014. Quantitative methods: for business, management and finance. Palgrave Macmillan.

Asongu, S.A. and Moulin, B., 2016. Research in International Business and Finance.

Sullivan III, M., 2015. Fundamentals of statistics. Pearson.

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