Tag Archives: Kendall

The Pay-for-Performance Myth

Last week, Eric Chemi and Ariana Giorgi published an interesting article on “The Pay-for-Performance Myth

With all the public chatter about exorbitant executive compensation and income inequality, it’s useful to look at the relationship between chief executive officer pay and corporate performance. Typically, when the subject of their big pay packages arises, CEOs—usually through their spokespeople—say they are paid for performance. Does data back that up?

An analysis of compensation data publicly released by Equilar shows little correlation between CEO pay and company performance. Equilar ranked the salaries of 200 highly paid CEOs. When compared to metrics such as revenue, profitability, and stock return, the scattering of data looks pretty random, as though performance doesn’t matter. The comparison makes it look as if there is zero relationship between pay and performance.

In the article, they produce a copula-type plot (since ranks – only – are considered). Ariana kindly sent me the dataset (that was used in The Pay at the Top) to play with it

> base=read.table("ceo.csv",sep=";",header=TRUE)

Here I normalize (dividing by the size of the dataset) to have uniform distribution on the unit interval (instead of working with ranks, i.e. integers). If we remove that scaling factor, the scatterplot is that same as the one mentioned in  the Pay-for-performance myth.

> n=nrow(base)
> U=rank(base[,1])/(n+1)
> V=rank(base[,2])/(n+1)
> plot(U,V,xlab="Rank CEO Pay",
+ ylab="Rank Stock Return")

This is the copula type representation.

If we visualize the density of the copula (using the algorithm described in the joint paper with Gery and Davy), we get either

> library("copula")
> library("ks")
> library("MASS")
> library("locfit")
> n.res=32
> ctilde1=probtranscopkde(UVs,p=1,
+ u.out=seq(1/(2*n.res+1),1-1/(2*n.res+1),

Continue reading The Pay-for-Performance Myth

Kendall’s function for copulas

As mentioned in the course on copulas, a nice tool to describe dependence it Kendall’s cumulative function. Given a random pair http://freakonometrics.hypotheses.org/files/2015/12/conc-19.gif with distribution  http://freakonometrics.hypotheses.org/files/2015/12/conc-17.gif, define random variable http://freakonometrics.hypotheses.org/files/2015/12/conc-30.gif. Then Kendall’s cumulative function is


Genest and Rivest (1993) introduced that function to choose among Archimedean copulas (we’ll get back to this point below).

From a computational point of view, computing such a function can be done as follows,

  • for all http://freakonometrics.hypotheses.org/files/2015/12/kendall-02.gif, compute http://freakonometrics.hypotheses.org/files/2015/12/kendall-03.gif as the proportion of observation in the lower quadrant, with upper corner http://freakonometrics.hypotheses.org/files/2015/12/kendall-4.gif, i.e.


  • then compute the cumulative distribution function of http://freakonometrics.hypotheses.org/files/2015/12/kendall-03.gif‘s.

To visualize the construction of that cumulative distribution function, consider the following animation

Thus, here the code to compute simply that cumulative distribution function is


The graph can be obtain either using




The interesting point is that for an Archimedean copula with generator http://freakonometrics.hypotheses.org/files/2015/12/kendall-7.gif, then Kendall’s function is simply

http://freakonometrics.hypotheses.org/files/2015/12/kendall-8.gifIf we’re too lazy to do the maths, at least, it is possible to compute those functions numerically. For instance, for Clayton copula,


Similarly, let us consider Gumbel copula,


If we plot the empirical Kendall’s function (obtained from the sample), with different theoretical ones, derived from Clayton copulas (on the left, in blue) or Gumbel copula (on the right, in purple), we have the following,


Note that the different curves were obtained when Clayton copula has Kendall’s tau equal to 0, .1, .2, .3, …, .9, 1, and similarly for Gumbel copula (so that Figures can be compared). The following table gives a correspondence, from Kendall’s tau to the underlying parameter of a copula (for different families)

as well as Spearman’s rho,

To conclude, observe that there are two important particular cases that can be identified here: the case of perfect dependent, on the first diagonal when http://freakonometrics.hypotheses.org/files/2015/12/kennnn-04.gif, and the case of independence, the upper green curve, http://freakonometrics.hypotheses.org/files/2016/10/kennnnn-05.gif. It should also be mentioned that it is also common to plot not function http://freakonometrics.hypotheses.org/files/2015/12/kennnn-01.gif, but function http://freakonometrics.hypotheses.org/files/2015/12/kennnn-02.gif, defined as http://freakonometrics.hypotheses.org/files/2015/12/kennnn-03.gif,