# LO 8.2: Assess the Pearson correlation approach, Spearm ans rank correlation, and

LO 8.2: Assess the Pearson correlation approach, Spearm ans rank correlation, and Kendall s t , and evaluate their lim itations and usefulness in finance.
Pearson Correlation
The Pearson correlation coefficient is commonly used to measure the linear relationship between two variables. The Pearson correlation is defined by dividing covariance (covXY) by the product of the two assets standard deviations (oxa Y).
PXY
covXY ox a Y
Covariance is a measure of how two assets move with each other over time. The Pearson correlation coefficient standardizes covariance by dividing it by the standard deviations of each asset. This is very convenient because the correlation coefficient is always between 1 and 1.
Covariance is calculated by finding the product of each assets deviation from their respective mean return for each period. The products of the deviations for each period are then added together and divided by the number of observations less one for degrees of freedom.
( X t – n x )(Yt – j i Y)
c o v x y
n 1
There is a second methodology that is used for calculating the Pearson correlation coefficient if the data is drawn from random processes with unknown outcomes (e.g., rolling a die). The following equation defines covariance with expectation values. If E(X) and E(Y) are the expected values of variables X and Y, respectively, then the expected product of deviations from these expected values is computed as follows:
E{ [X – E(X)1 [Y – E(Y)]} or E(XY) – E(X)E(Y)
When using random sets of data, the correlation coefficient can be rewritten as:
PXY
E(XY) – E(X)E( Y)
Ve (X 2) (E(X))2 x Ve (Y2)-(E (Y ))2
Because many financial variables have nonlinear relationships, the Pearson correlation coefficient is only an approximation of the nonlinear relationship between financial
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Topic 8 Cross Reference to GARP Assigned Reading – Meissner, Chapter 3
variables. Thus, when applying the Pearson correlation coefficient in financial models, risk managers and investors need to be aware of the following five limitations:
1. The Pearson correlation coefficient measures the linear relationship between two
variables, but financial relationships are often nonlinear.
2. A Pearson correlation of zero does not imply independence between the two variables. It simply means there is not a linear relationship between the variables. For example, the parabola relationship defined as Y = X2 has a correlation coefficient of zero. There is, however, an obvious nonlinear relationship between variables Y and X.
3. When the joint distribution between variables is not elliptical, linear correlation
measures do not have meaningful interpretations. Examples of common elliptical joint distributions are the multivariate normal distribution and the multivariate Students ^-distribution.
4. The Pearson correlation coefficient requires that the variance calculations of the
variables X and Y are finite. In cases where kurtosis is very high, such as the Students ^-distribution, the variance could be infinite, so the Pearson correlation coefficient would be undefined.
3. The Pearson correlation coefficient is not meaningful if the data is transformed. For
example, the correlation coefficient between two variables X and Ywill be different than the correlation coefficient between ln(X) and ln(Y).
Spearm ans Rank Correlation
Ordinal measures are based on the order of elements in data sets. Two examples of ordinal correlation measures are the Spearman rank correlation and the Kendall t . The Spearman rank correlation is a nonparametric approach because no knowledge of the joint distribution of the variables is necessary. The calculation is based on the relationship of the ranked variables. The following equation defines the Spearman rank correlation coefficient where n is the number of observations for each variable, and d- is the difference between the ranking for period i.
The Spearman rank correlation coefficient is determined in three steps:
Step 1: Order the set pairs of variables X and Y with respect to the set X. Step 2: Determine the ranks of Xy and Y- for each time period i. Step 3: Calculate the difference of the variable rankings and square the difference.
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Topic 8 Cross Reference to GARP Assigned Reading – Meissner, Chapter 3
Example: Spearmans rank correlation
Calculate the Spearman rank correlation for the returns of stocks X and Yprovided in Figure 1.
Figure 1: Returns for Stocks X and Y
X
25.0% 60.0% -20.0% 40.0% -10.0% 19.0%
Y
-20.0% 40.0% 10.0% 20.0% 30.0% 16.0%
Year 2010 2011 2012 2013 2014 Average
The calculations for determining the Spearman rank correlation coefficient are shown in Figure 2. The first step involves ranking the returns for stock X from lowest to highest in the second column. The first column denotes the respective year for each return. The returns for stock Y are then listed for each respective year. The fourth and fifth columns rank the returns for variables X and Y The differences between the rankings for each year are listed in column six. Lastly, the sum of squared differences in rankings is determined in column 7.
Figure 2: Ranking Returns for Stocks X and Y
Year 2012 2014 2010 2013 2011
X
-20.0% -10.0% 25.0% 40.0% 60.0%
Y
10.0% 30.0% -20.0% 20.0% 40.0%
X Rank
Y Rank
1 2 3 4 5
2 4 1 3 5
d, -1 -2 2 1 0
Sum =
1 4 4 1 0 10
The Spearman rank correlation coefficient can then be determined as 0.3:
n
6Ed? i=l n(n2 – l )
Ps 1
1 –
6×10 5(23-1)
0.5
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Kendall S T
Topic 8 Cross Reference to GARP Assigned Reading – Meissner, Chapter 3 is also a
is another ordinal correlation measure that is becoming more widely applied Kendalls t in financial models for ordinal variables such as credit ratings. Kendalls t is also a nonparametric measure that does not require any assumptions regarding the joint probability distributions of variables. Both Spearmans rank correlation coefficient and Kendalls t perfectly correlated variables will have a coefficient of 1 . The Kendall t Y always increases with an increase in variable X. The numerical amount of the increase does not matter for two variables to be perfectly correlated. Therefore, for most cases, the Kendall t
are similar to the Pearson correlation coefficient for ranked variables because
and the Spearman rank correlation coefficients will be different.
will be 1 if variable
The mathematical definition of Kendalls t
is provided as follows:
nc ~ nd n(n 1)/ 2
In this equation, the number of concordant pairs is represented as nc, and the number of discordant pairs is represented as nd. A concordant pair of observations is when the rankings of two pairs are in agreement:
< Y and X ~ < Y* or X^ >
t*
t*
t
t
t
t
and X * > Y * and t ^ t
*
t*
t*
A discordant pair of observations is when the rankings of two pairs are not in agreement: