The Kendall tau distance was originally defined as a correlation coefficient. It seems clear to me that every metric function $d$ that is bounded by $b$, induces a correlation coefficient. That is:

Let $d$ be a metric. Then $-d(x,y)/b+1$ is a correlation coefficient.

I wonder if the converse is true, too:

Let $c(x,y)$ be a correlation coefficient. Then $-c(x,y) + 1$ is a pseudometric.

Is the latter statement true or false?