Consider the problem of drawing random variates (X
1, …, X
n
) from a distribution where the marginal of each X
i
is specified, as well as the correlation between every pair X
i
and X
j
. For given marginals, the Fréchet-Hoeffding bounds put a lower and upper bound on the correlation between X
i
and X
j
. Any achievable correlation between X
i
and X
j
is a convex combination of these bounds. We call the value λ(X
i
, X
j
) ∈ [0, 1] of this convex combination the convexity parameter of (X
i
, X
j
) with λ(X
i
, X
j
) = 1 corresponding to the upper bound and maximal correlation. For given marginal distributions functions F
1, …, F
n
of (X
1, …, X
n
), we show that λ(X
i
, X
j
) = λ
ij
if and only if there exist symmetric Bernoulli random variables (B
1, …, B
n
) (that is {0, 1} random variables with mean ½) such that λ(B
i
, B
j
) = λ
ij
. In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three, and four dimensions.