Which equal correlations of three random variables are possible? 3 random variables have equal pairwise correlation, what are the possible values of correlation?
I think 1 and 0 are the obvious answers, but are there any more, and how could I find them?
 A: Just a side note for the otherwise correct answers already given (+1 both). The correlation/covariance matrix described is compound symmetric. This has some rather important theoretical implication on how one interpreters a model; in particular one would assume that the co-variance of the variables examined can be perfectly partitioned in a  "shared" component and the "unshared" component between your variables. 
A (somewhat) common setting for such structures to be used is when the assumption for equal correlation of residuals is plausible; for example when one deals with repeated trials under the same condition in an experiment.
The CV link: "What is compound symmetry in plain english?" gives a more in-depth presentation of compound symmetry.
A: Let $X_1$, $X_2$, $X_3$ be three random variables with common pairwise correlation coefficient $\rho$, that is $\mbox{corr}(X_i, X_j)= \rho$ for $i \neq j$ with $|\rho|\leq 1$. So, the correlation matrix of $X = (X_1, X_2, X_3)$ is
$$
\left( \begin{array}{ccc}
1    & \rho & \rho \\
\rho & 1    & \rho \\
\rho & \rho & 1 \end{array} \right) .
$$
Correlation matrices need to be positive-semidefinite, which implies that their leading principal minors are all positive. So $\rho$ must satisfy the following two conditions
$$
\begin{cases}
1 - \rho^2 &\geq 0,\\
1 - 3\rho^2 + 2\rho^3 &\geq 0 .
\end{cases}
$$
The first condition is always satisfied, and the second condition implies that $\rho \geq -0.5$.
