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P2.T5.504. Rank correlations: Spearman's and Kendall's

Nicole Seaman

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Learning outcomes: Evaluate the limitations of financial modeling with respect to the model itself, calibration of the model, and the model’s output. Assess the Pearson correlation approach, Spearman’s rank correlation, and Kendall’s τ, and evaluate their limitations and usefulness in finance.

Questions:

504.1. The annual returns of two assets, X(i) and Y(i), are shown below for the five years from 2010 to 2014, inclusive. The returns have been sorted with respect to X(i); for example, in 2010 X(i) returned 4.3% which ranked 3rd among its annual returns and in the same year Y(i) returned 6.0% which ranked 4th among its annual returns (ranking is from worst to best).



The Pearson correlation coefficient, taken from the actual return pairs--for example, (X,Y) = (-12.5%, 4.3%)--is about 0.756. But we are interested instead in a rank correlation. Which is nearest to the Spearman's rank correlation?

a. -0.25
b. 0.33
c. 0.60
d. 0.85


504.2. The annual returns of two assets, X(i) and Y(i), are shown below for the five years from 2010 to 2014, inclusive. The returns have been sorted with respect to X(i); for example, in 2010 X(i) returned 1.0% which ranked 3rd among its annual returns and in the same year Y(i) returned -3.3% which ranked 1st among its annual returns (ranking is from worst to best). The final two columns compute the number of concordant pairs, which is six, and the number of discordant pairs, which is four.



The Pearson correlation coefficient, taken from the actual return pairs--for example, (X,Y) = (-8.8%, 0.8%)--is 0.5490. But we are interested instead in a rank correlation. Which is nearest to the Kendall's tau?

a. -0.15
b. 0.20
c. 0.50
d. 0.67


504.3. About correlation measures including Pearson's, Spearman's and Kendall's tau, each of the following is true EXCEPT which is false?

a. Pearson is a cardinal correlation measure while Spearman's and Kendall's tau are ordinal correlation measures
b. The problem with applying ordinal rank correlations to cardinal observations is that ordinal correlation are less sensitive to outliers (an unwelcome property in risk management)
c. An advantage of Pearson's correlation coefficient is that it is invariance to transformations; e.g., Pearson's correlation between pairs [x,y] will equal Pearson's correlation between[ln(x), ln(y)]
d. Pearson's correlation coefficient is a natural (good) dependence measure when variables are distributed as multivariate elliptical (e.g., normal, student's t); however, we know many financial variables are not elliptically distributed

Answers here:
 
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#2
I feel small amendment is required. It should be "X(i) and Y(i), are shown below for the five years from 2010 to 2014" instead of "X(i) and Y(i), are shown below for the five years from 2010 to 2103" in questions 504.1 & 504.2.
 

David Harper CFA FRM

David Harper CFA FRM
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#5
Hi @Merlinius The LO verb is "assess" such that your inference is correct: probably a calculation (of the rank correlation metrics) will not be required on the exam. You know how we roll here: we are sincere about understanding the concepts. Can somebody who has not actually calculated Spearman’s rank correlation and Kendall’s tau (τ) at least once--on a super-simple dataset as above--explain these measures, much less tell us "their limitations and usefulness in finance"?! ;)
 
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