  We investigate how well the graph of a bilinear function
$b:[0,1]^n\to\mathbb{R}$ can be approximated by its McCormick relaxation. In
particular, we are interested in the smallest number $c$ such that the
difference between the concave upper bounding and convex lower bounding
functions obtained from the McCormick relaxation approach is at most $c$ times
the difference between the concave and convex envelopes. Answering a question
of Luedtke, Namazifar and Linderoth, we show that this factor $c$ cannot be
bounded by a constant independent of $n$. More precisely, we show that for a
random bilinear function $b$ we have asymptotically almost surely
$c\geqslant\sqrt n/4$. On the other hand, we prove that $c\leqslant
600\sqrt{n}$, which improves the linear upper bound proved by Luedtke,
Namazifar and Linderoth. In addition, we present an alternative proof for a
result of Misener, Smadbeck and Floudas characterizing functions $b$ for which
the McCormick relaxation is equal to the convex hull.
