Maximum negative corrleation for $n$ identically distributed normal variables

Suppose you have $n$ identically and symmetrically distributed normal random variables. What is the most negative common correlation coefficient possible? I believe the answer is $\frac{-1}{n-1}$, but I cannot prove it.

Solution 1:

$\newcommand{Cov}{\operatorname{Cov}}$ $\newcommand{Var}{\operatorname{Var}}$ $\newcommand{corr}{\operatorname{corr}}$

Assume WLOG $\Var(X_i)=1$ for all $i$, so $\Cov(X_i,X_j)=\corr(X_i,X_j)=\rho$ for $i\neq j$. Then $$0\leq\Var\left(\sum X_i\right)=\sum\Var(X_i)+\sum_{i\neq j}\Cov(X_i,X_j) =n+n(n-1)\rho,$$ and so $\rho\geq-\frac{1}{n-1}$ as desired.