Quantitative Financial Risk Management by Michael B. Miller

Author:Michael B. Miller [Miller, Michael B.]
Language: eng
Format: epub, pdf
ISBN: 9781119522263
Publisher: Wiley
Published: 2018-11-13T00:00:00+00:00

Data Set #1 Data Set #2

X Y Rank[X] Rank[Y] X Y Rank[X] Rank[Y]

A 1 5 3 3 A 1 5 3 3

B 2 10 2 2 B 2 10 2 2

C 3 15 1 1 C 3 18 1 1

Within both data sets, notice how if the X value of one point is greater than the X value of another point, then the Y value is also greater. Likewise, if the X value of one point is less than the X value of another point, then the Y value is also less. When the X and Y values of one point are both greater than or both less than the X and Y values of another point, we say that the two points are concordant. If two points are not concordant, we say they are discordant. More formally, for two distinct points i and j we have

(6.22)

Kendall's tau is defined as the probability of concordance minus the probability of discordance.

(6.23)

If P[concordance] is 100%, then P[discordance] must be 0%. Similarly, if P[discordance] is 100%, P[concordance] must be 0%. Because of this, like our standard correlation measure, Kendall's tau must vary between −100% and +100%.

To measure Kendall's tau in a given data set, we simply need to compare every possible pair of points and count how many are concordant and how many are discordant. In a data set with n points, the number of unique combinations of two points is

(6.24)

For example, in Table 6.9 each data set has three points, A, B, and C, and there are three possible combinations of two points: A and B, A and C, and B and C. For a given data set, then, Kendall's tau is

(6.25)

A potentially interesting feature of Kendall's tau is that it is less sensitive to outliers than Pearson's correlation. In Table 6.9, as long as the Y value of point C is greater than 10, all of the points will be concordant, and Kendall's tau will be unchanged. Point C could be 11 or 11,000. In this way, measures of dependence based on rank are analogous to the median when measuring averages.

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