# Statistics

## Question 1

### 1. Creating line charts for closing prices of S&P, Yahoo and Google series

From the presented line chart, it can be seen that S&P’s closing price is highest than Google and Yahoo’s stock prices. Till the end of 2012, all the three stock’s closing prices shows a stable trend, afterwards, S&P’s price gone up in 2013 to 1498.11, Google & Yahoo’s price also goes increase to 755.69 and 19.63 respectively. Following 2013, in 2014, S&P and Yahoo’s share prices has been increased, however, in contrast, Google share price dropped down. S&P’s price regularly shows a increasing trend over the duration of 2012 to 2016. Out of all the three stock, Yahoo’s share price is lowest, on the other side, after decline in 2014, in the next two years, Google’s share price shows rising trend which is good.

## Question 2

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

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

100*Ln(Pt-Pt-1)

 Date Monthly Return (S&P) Monthly Return (Google) Monthly Return (Yahoo) 1/6/2012 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

 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]

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## Question 3

### A) Sampling distribution of mean

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

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

= 0.0187

σp = [ 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

Table 2Likelihood of loss

 Value at risk on Google -9.49949 Value at risk on S&P -3.70815 Probability 0.390353 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 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

#### Summary statistics

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

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

 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

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

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

 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 R2 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

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

 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.

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