question archive Perform a Regression on this data

Perform a Regression on this data

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Perform a Regression on this data. 

Year Return on Stock Z Return on S&P 500
1 10% 5%
2 -15 -10
3 15 10
4 5 0
5 -5 -10
     


Follow these steps with the data you entered above: 
? Enter returns on stock Z as y variable range.
? Enter returns on S&P 500 as x variable range.
? Plot a scatter diagram from the returns given.
? Estimate the beta for stock Z.
? Provide a linear equation relating stock Z returns to returns on the S&P 500 index.
? What type of relationship between the two variables is predicted from your equation?
? If the return on the S&P 500 index is expected to be 10% over the next year, using your regression equation what
is the expected return on stock Z over the same period?

pur-new-sol

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  • Scatter Diagram can be found below.
  • Beta for Stock Z = 1.28125 or 1.28
  • Linear equation:
    • 2 decimal places: yi = 0.03 + 1.28xi + ε
    • Exact: yi = 0.0328125 + 1.28125xi + ε
  • positive relationship can be predicted from the equation
  • Expected return on Stock Z if return on S&P 500 will be 10% = 0.1609375

Step-by-step explanation

Scatter Diagram:

Scatter Diagram on the Returns Given 0.2 0.15 0.1 0.05 O Return on Stock Z 0.15 -0.1 -0.05 O 0.05
0.15 0.05 -0.1 0. 15 -0.2 Return on S&P 500

After using data analysis (regression tool) in Excel using the given data from Stock Z and S&P 500 returns, it returned the following data:

SUMMARY OUTPUT                
                 
Regression Statistics              
Multiple R 0.951688608              
R Square 0.905711207              
Adjusted R Square 0.874281609              
Standard Error 0.042695628              
Observations 5              
                 
ANOVA                
  df SS MS F Significance F      
Regression 1 0.052531 0.052531 28.81714 0.012654      
Residual 3 0.005469 0.001823          
Total 4 0.058            
                 
  Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 0.0328125 0.019243 1.705196 0.186703 -0.02843 0.094051 -0.02843 0.094051
X Variable 1 1.28125 0.238676 5.36816 0.012654 0.521677 2.040823 0.521677 2.040823
                 
                 
                 

From here, it can be seen that the beta for Stock Z = 1.28125 or 1.28.

The y-intercept, on the other hand, is 0.0328125 or 0.03. Having these 2 figures, we can come up with the following linear equation:

  • 2 decimal places: yi = 0.03 + 1.28xi + ε
  • OR Exact: yi = 0.0328125 + 1.28125xi + ε

A positive relationship can be predicted from the equation, because the 2 figures (y-intercept and slope/beta) are both positive. This can also be seen in the scatter diagram, where the points are not far off a straight line and indicates that these 2 have a positive relationship. This signifies that as S&P 500 returns increase, Stock Z returns will also increase.

 

Lastly, using the linear equation from above, we can compute for the expected return on Stock Z if return on S&P 500 will be 10%. We will simply substitute x with 10% or 0.10 and multiply it with our beta of 1.28125 and add it with the y-intercept of 0.0328125:

yi = 0.0328125 + 1.28125xi + ε

= 0.0328125 + (1.28125 x .10) + ε

= 0.1609375

Please see attached file