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    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.

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    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

    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

    © 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

     

     

    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

    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

     

     

     

     

     

     

    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)

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    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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