Thursday, April 19, 2012

Correlation Analysis in Investment Analysis

Problem on Hand
A portfolio manager collected five years of return data on both ABC shares and the S&P 500 index. These return pairs are presented in the following table:


Year                      Return on ABC shares              Return on S&P


1                                     10%                                         5%

2                                     -15                                           -10

3                                     15                                            10

4                                     5                                              0

5                                     -5                                             -10


Analyze the relationship between ABC  common share returns and the returns generated by the S&P 500 index.
First you have to verify that there is a relationship between Y and X. This is the role played by Correlation Analysis.

1.     Scatter Plot

One simple way of determining whether any plausible relationship exists between two variables is to draw a scatter plot, which is a two-dimensional diagram depicting the paired sets of data.


A visual inspection of the scatter plot can be used to determine whether or not the two variables seem to be related and, if they are, what the probable mathematical form of the relationship might be.


2.     The Correlation Coefficient

a.     Correlation Coefficient between Y and X = rYX. rYX = +1 indicates that Y and X are perfectly and positively related in a linear manner.

b.     rYX = 0 indicates Y and X are not related.

c.     rYX = -1 indicates that the two variables, X and Y are perfectly and negatively related in a linear manner. The data point lie on a down-ward sloping straight line in this case.


3.     Calculating Covariance




                   COVYX  = Sum of (Y – Y)(X – X)/(n – 1)


                   It is the average value of the cross-product of the deviations from the observations of two random variables from their sample means.



4.     Coefficient of Determination


The square of the correlation coefficient is called the coefficient of determination. The coefficient of determination measures the percentage of the total variation in the dependent variable (Y) that is explained by the variation in independent variable (X).


5.     Testing the Significance of a Correlation Coefficient


H0: rYX = 0

H1: rYX = 0


Test Statistic  = t(calc) = (rYX*Sqrt(n-2))/Sqrt(1- rYX 2)

D of f = n – 2 (for simple linear regressions)

If t(calc) is less than t(critical) do not reject H0.


Ex: Determine whether or not the correlation coefficient between ABC and S&P (.952) is statistically significant at the 5% level of significance.


t(calc) = (rYX*Sqrt(n-2))/Sqrt(1- rYX 2)

            = .952*(Sqrt(5-2))/Sqrt(1- (.952)2)

             = 5.39

T(critical) = t 0.025,3   =  3.182

T(calc) is greater than t(critcal)


6.     Limitations of Correlation Analysis

a.     The correlation coefficient assumes that the relationship is linear. There can be stronger non-linear relationship.

b.     The presence of outliers can distort the correlation coefficient. Calculate the coefficient without outliers also.

c.     There can be spurious correlation. Two rules to avoid spurious correlation trap.


·        remember that correlation does not imply causation.

·        Correlations without theoretical basis should be suspect.

Original knol - 488

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