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), +length=n.res),plots=TRUE)

with distribution
, define random variable
. Then Kendall’s cumulative function is
, compute
as the proportion of observation in the lower quadrant, with upper corner
, i.e.


, then Kendall’s function is simply
If we’re too lazy to do the maths, at least, it is possible to compute those functions numerically. For instance, for Clayton copula,


, and the case of independence, the upper green curve,
. It should also be mentioned that it is also common to plot not function
, but function
, defined as
,