Statistics

Question 2

2a. Calculation of returns for the S&P, Yahoo and Google 

Formula: 100*Ln(current price/Price of previous month)

100*Ln(P_t-P_t-1)

Date	Monthly Return (S&P)	Monthly Return (Google)	Monthly Return (Yahoo)
1/6/2012	Â	Â	Â
1/7/2012	1.251895	8.727424	0.063151
1/8/2012	1.95706	7.912719	-7.80981
1/9/2012	2.394706	9.650506	8.68976
1/10/2012	-1.99878	-10.3521	5.241907
1/11/2012	0.284266	2.621531	10.85028
1/12/2012	0.704345	1.281881	5.845988
1/1/2013	4.919776	6.606337	-1.36607
1/2/2013	1.099988	5.847923	8.211744
1/3/2013	3.535537	-0.87879	9.909976
1/4/2013	1.792416	3.753931	4.974088
1/5/2013	2.055017	5.503249	6.155186
1/6/2013	-1.5113	1.044786	-4.55066
1/7/2013	4.827776	0.834778	11.13513
1/8/2013	-3.17983	-4.71075	-3.51422
1/9/2013	2.931554	3.368072	20.13744
1/10/2013	4.362998	16.26137	-0.69581
1/11/2013	2.766328	2.776029	11.56895
1/12/2013	2.328951	5.608037	8.944217
1/1/2014	-3.62314	5.237372	-11.6023
1/2/2014	4.221338	2.894278	7.126743
1/3/2014	0.690824	-8.68641	-7.43268
1/4/2014	0.618165	-4.19812	0.139179
1/5/2014	2.08122	-62.5662	-3.68314
1/6/2014	1.887898	2.252067	1.375774
1/7/2014	-1.51947	-0.87956	1.917172
1/8/2014	3.696367	0.483687	7.269076
1/9/2014	-1.56355	1.033517	5.653789
1/10/2014	2.293639	-3.55315	12.22719
1/11/2014	2.423747	-3.36484	11.65034
1/12/2014	-0.41973	-3.41226	-2.40598
1/1/2015	-3.15328	1.290026	-13.8209
1/2/2015	5.343887	4.560044	0.657077
1/3/2015	-1.75491	-1.41948	0.360686
1/4/2015	0.848467	-1.0748	-4.29902
1/5/2015	1.043679	-0.63066	0.865401
1/6/2015	-2.12357	-0.97296	-8.88338
1/7/2015	1.954969	19.68016	-6.90111
1/8/2015	-6.46247	-1.48319	-8.47579
1/9/2015	-2.67987	-1.46948	-15.3014
1/10/2015	7.971934	14.41989	20.87197
1/11/2015	0.050484	3.39445	-5.21507
1/12/2015	-1.76857	1.967796	-1.64011
1/1/2016	-5.20676	-2.16462	-11.9626
1/2/2016	-0.41369	-5.97105	7.442259
1/3/2016	6.390498	6.174433	14.66178
1/4/2016	0.269573	-7.48524	-0.57213
1/5/2016	1.520841	5.626411	3.595772
1/6/2016	0.091043	-6.24282	-1.00663
1/7/2016	3.499044	11.76172	1.663405
1/8/2016	-0.12199	-0.18847	11.27955
1/9/2016	-0.12352	1.783084	0.81538
1/10/2016	-1.96168	0.723689	-3.66255
1/11/2016	3.360347	-4.29129	-1.28378
1/12/2016	1.80371	4.235714	-5.89956
1/1/2017	1.77263	1.318102	13.07152

2b. Obtaining summary statistics and risk and average return relationship

Summary statistics of Google and Yahoo’s monthly return

Â	Google return	Yahoo return
Mean	0.629794972	1.861586191
Standard Error	1.409574475	1.109479885
Median	1.281881236	0.815380173
Mode	#N/A	#N/A
Standard Deviation	10.45368409	8.228123045
Sample Variance	109.279511	67.70200884
Kurtosis	24.98016236	-0.344618753
Skewness	-4.015978724	0.110839548
Range	82.24635155	36.17332232
Minimum	-62.56619623	-15.30135014
Maximum	19.68015532	20.87197218
Sum	34.63872348	102.3872405
Count	55	55

From the summary statistics, it is identified that average monthly return of Yahoo is comparatively greater to 1.86%, however, in Google, average return is derived to 0.63% respectively. The central tendency measure showcase that in comparison to Google, Yahoo is delivering high return to the investors on the capital invested in the business. For the risk, standard deviation presents the scatter and spread in the return over the period from the mean. It is founded greater for Google’s share as it reported a standard deviation of 10.45 whereas for Yahoo’s stock, it is computed to 8.22 which are comparatively lower. High value of standard deviation indicates high volatility in the average monthly return on such stock at a higher risk or vice-versa.

Correlation is a statistical tool that determines the level, direction and strength of relationship between two independent variables. In Google and Yahoo’s monthly stock return, very less correlation is determined to 0.14 that is below 0.25. Although positive relationship demonstrate that with the increase in either Google or Yahoo’s return, other stock return also changes in same direction but at very less percentage @ 14%. 

Jarque-Bera Test is a multiplier test that is used for the normality of the data set. Many of the statistical tests assumes normal distribution of the data, here, JB test can be run to confirm normality of the large data sets available for a given time series.

H₀: The data set is normally distributed.

H₁: The data set is not normally distributed.

Formula of JB test statistics: n[(√b₁)²/6 + (b₂-3)²/24]

Here: n- Sample size

√b₁ - Skewness coefficient

b₂ – Kurtosis coefficient

JB statistics: 55/(√-4.01)^ 2/6 + (24.98-3)²/24]

55/(√-4.01)^ 2/6 + (24.98-3)²/24]

Question 3

A) Sampling distribution of mean

σ_p = [ σ / sqrt(n) ] * sqrt[ (N - n ) / (N - 1) ]

[1.40/Sqrt(36)]*Sqrt[(55-36)/(55-1)]

= 0.0187

σ_p = [ σ / sqrt(n) ] * sqrt[ (N - n ) / (N - 1) ]

[1.20/Sqrt(36)]*Sqrt[(55-36)/(55-1)]

= 0.016

(b) Probability of return of 4%

Table 1Calculation of probability

Return on stock	4%
Average return on market index	0.935651
Probability	0.42751

© Likelihood of loss 

Table 2Likelihood of loss

Value at risk on Google	-9.49949
Value at risk on S&P	-3.70815
Probability	0.390353
Â	Â
Value at risk on Yahoo	14.49173
Value at risk on S&P	-3.70815
Probability	-0.25588

4 Creating excess return on preferred stock and excess market return

excess market return	excess return on preferred stock
Â	Google	Yahoo
-0.24011	7.235424174	-1.428848751
0.39506	6.350718884	-9.371805092
0.757706	8.013506171	7.052760488
-3.68478	-12.03814009	3.555906848
-1.32173	1.015531412	9.244284178
-1.05166	-0.474118764	4.089988113
2.934776	4.621336768	-3.351072344
-0.78801	3.959923229	6.323743782
1.683537	-2.730787646	8.057975663
0.117416	2.0789311	3.299087879
-0.10898	3.339248986	3.991185765
-3.9893	-1.433214096	-7.028658763
2.234776	-1.758221542	8.542128956
-5.92883	-7.45975479	-6.263217804
0.316554	0.7530722	17.52243899
1.820998	13.71936836	-3.237812821
0.025328	0.03502903	8.827950121
-0.69705	2.582037218	5.918216819
-6.29114	2.569372166	-14.27027165
1.563338	0.236277847	4.468742796
-2.03218	-11.40940667	-10.15568101
-2.02983	-6.846124709	-2.508821133
-0.37578	-65.02319623	-6.140135853
-0.6281	-0.263933049	-1.140225936
-4.07547	-3.435564517	-0.638828227
1.353367	-1.859312713	4.926076382
-4.07155	-1.474482938	3.145789186
-0.04136	-5.888146454	9.892192301
0.229747	-5.55883689	9.456341406
-2.58973	-5.582263554	-4.575984028
-4.82828	-0.384973591	-15.49590011
3.341887	2.558043956	-1.344922752
-3.68891	-3.353479143	-1.573314307
-1.19753	-3.120801687	-6.345018493
-1.05132	-2.725663188	-1.229598739
-4.45857	-3.307956743	-11.21837589
-0.25003	17.47515532	-9.106105149
-8.66247	-3.683188406	-10.67579247
-4.73987	-3.529482744	-17.36135014
5.820934	12.2688872	18.72097218
-2.16752	1.176449501	-7.433066114
-4.03757	-0.301204199	-3.909114296
-7.13776	-4.095620428	-13.89362848
-2.15369	-7.711054572	5.702258597
4.604498	4.388433353	12.87577725
-1.54943	-9.304237329	-2.391130698
-0.31316	3.792410864	1.76177241
-1.39696	-7.730817687	-2.494631017
2.041044	10.30371915	0.20540469
-1.68999	-1.756465705	9.711549415
-1.73152	0.175083925	-0.792619827
-3.79568	-1.110311001	-5.496547581
0.992347	-6.65929142	-3.651776866
-0.67529	1.756713564	-8.378564703
-0.67837	-1.132898395	10.62051732

After the analyzing the existing position of both the stocks such as Google as well as Yahoo stocks are properly evaluated by applying this particular technique in or order to determine their current position in the external business market. In terms of volume of both the stocks of Google and Yahoo, Google has higher volume size as compared to yahoo which is the main reason behind the increasing higher market risks as higher the share market higher will be its overall business risks. Excess return on preferred stock and excess market return of Google is higher in volume that increases overall risk of an entity which will directly affected all the shareholders who have invested in the business. The return of Google is decreasing on a constant basis as compared to the price of Yahoo.

which is increasing with the passage of time. Volume size of Yahoo is less that safeguards its entity from the external market changes in terms of risks incurred on the firm. The share value of Yahoo increases due to higher efforts applied by the entity owner on improving its existing business performance along with the considerations of each and every factors considered by the business in order to maintain their survival in the external environment as costs and risks are eliminated by the firm by focuses on its strength and the capabilities.

Question 5 and 6

6A. Estimating CAPM using linear regression

Summary statistics

Regression Statistics
Multiple R	0.231725
R Square	0.053696
Adjusted R Square	0.035842
Standard Error	10.32065
Observations	55

 

ANOVA	Â	Â	Â	Â	Â
Â	df	SS	MS	F	Significance F
Regression	1	320.335	320.335	3.007395	0.088698959
Residual	53	5645.336	106.5158	Â	Â
Total	54	5965.671	Â	Â	Â

 

Â	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-0.47396	1.513165	-0.31323	0.755339	-3.508985979	2.561064	-3.50899	2.561064
Google	0.859734	0.495757	1.734184	0.088699	-0.134628726	1.854097	-0.13463	1.854097

Linear regression equation (Google) Y = a+ bx

= -0.47396 + 0.859734 (S&P 500’s excess market return)

 Regression Statistics

Â
Multiple R	0.615822
R Square	0.379237
Adjusted R Square	0.367524
Standard Error	6.546886
Observations	55

 

ANOVA	Â	Â	Â	Â	Â
Â	df	SS	MS	F	Significance F
Regression	1	1387.811	1387.811	32.3788	5.61E-07
Residual	53	2271.671	42.86172	Â	Â
Total	54	3659.482	Â	Â	Â

 

Â	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	1.87211	0.959874	1.950371	0.056429	-0.05315	3.797373	-0.05315	3.797373
yahoo	1.789481	0.314483	5.690237	5.61E-07	1.158709	2.420254	1.158709	2.420254

 

Regression equation = 1.87211 + 1.789481(S&P 500’s excess market return)

6b. Interpretation of the coefficient

Slope presents the steepness of the line and presents the change in dependent variable with the change in independent variable. Linear regression equation for the Google return founded a slope coefficient of 0.859734 whereas for the Yahoo, it is predicted to 1.789481 which is very high indicates highly riskier security. It indicates with the increase or decrease in excess market return on S&P 500, Google’s and Yahoo’s excess portfolio return goes up by 0.85 and 1.78 respectively or vice-versa. High beta value above than 1 in Yahoo is a clear indication of excessive riskiness on this portfolio.

6c. interpreting the value of R²

R square is a statistical measure that measure that how closely the data are fitted on regression line, also known as coefficient of determination (Schroeder, Sjoquist and Stephan, 2016).

R-Square: Explained Variation/Total Variations

It ranges between 0% to 100% and for the made regression analysis, for the Google’s stock, R² is derived to^ 0.053696 whereas for the Yahoo’s portfolio, it is computed to 0.379237. Higher the value of R^{2 in Google} indicates better fitness to the model.

6d. Interpreting 95% confidence interval for the slope coefficient

Confidence interval provides greater information in comparison to the point estimates. With reference to the Google’s stock, at 95%, the CI is derived to -0.1346 to 1.8541. In contrast, for the Yahoo’s stock, lower and upper CI is derived to 1.1587 and 2.42054 respectively. It indicates that slope coefficient (beta) of both the stock lies in the given range and for the Yahoo, it is comparatively greater which shows high risk as well as volatility in the excess portfolio return with the change in excess market return on S&P 500 Index.  

7 Using confidence interval approach

Â	Lower 95%	Upper 95%	Lower 95.0%
Intercept	-3.508985979	2.561064	-3.50899
Google	-0.134628726	1.854097	-0.13463

 

Â	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-0.05315	3.797373	-0.05315	3.797373
yahoo	1.158709	2.420254	1.158709	2.420254

 

The efficiency of both the company’s stock price is based on the value of confidence interval approach at 95% is compared with the neutral stock that includes beta value of 1. The value of both the companies is higher as compared to the neutral stock beta value. The most preferred stock by an entity on the basis of above conditions suggest an entity in order to consider by the business entity for the future purpose.