  The Az\'{e}ma-Yor solution (resp., the Perkins solution) of the Skorokhod
embedding problem has the property that it maximizes (resp., minimizes) the law
of the maximum of the stopped process. We show that these constructions have a
wider property in that they also maximize (and minimize) expected values for a
more general class of bivariate functions $F(W_{\tau},S_{\tau})$ depending on
the joint law of the stopped process and the maximum. Moreover, for monotonic
functions $g$, they also maximize and minimize $\mathbb
{E}[\int_0^{\tau}g(S_t)\,dt]$ amongst embeddings of $\mu$, although, perhaps
surprisingly, we show that for increasing $g$ the Az\'{e}ma-Yor embedding
minimizes this quantity, and the Perkins embedding maximizes it. For
$g(s)=s^{-2}$ we show how these results are useful in calculating model
independent bounds on the prices of variance swaps. Along the way we also
consider whether $\mu_n$ converges weakly to $\mu$ is a sufficient condition
for the associated Az\'{e}ma-Yor and Perkins stopping times to converge. In the
case of the Az\'{e}ma-Yor embedding, if the potentials at zero also converge,
then the stopping times converge almost surely, but for the Perkins embedding
this need not be the case. However, under a further condition on the
convergence of atoms at zero, the Perkins stopping times converge in
probability (and hence converge almost surely down a subsequence).
