Range of the Correlation Coefficient
What is the full range of possible values for the correlation coefficient $\rho$ between the returns of two assets? Justify why those are the limits.
Hints
- Correlation is covariance divided by the product of standard deviations -- the bound must come from that normalization.
- Apply Cauchy-Schwarz to the centered returns, or use that the variance of $X/\sigma_X \pm Y/\sigma_Y$ cannot be negative.
- $\operatorname{Var}(X/\sigma_X \pm Y/\sigma_Y) = 2(1 \pm \rho) \geq 0$ gives $-1 \leq \rho \leq 1$ directly.
Worked Solution
How to Think About It: Correlation is a normalized covariance, and the normalization is exactly what bounds it. The cleanest argument is Cauchy-Schwarz: covariance is an inner product on mean-zero random variables, and an inner product is always bounded by the product of the norms.
Derivation: By definition, for assets with returns $X$ and $Y$, $\rho = \frac{\operatorname{Cov}(X, Y)}{\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}}.$ By the Cauchy-Schwarz inequality applied to the centered variables, $|\operatorname{Cov}(X,Y)| \leq \sqrt{\operatorname{Var}(X)\,\operatorname{Var}(Y)},$ so $|\rho| \leq 1$, i.e. $-1 \leq \rho \leq 1$. Equality $\rho = \pm 1$ holds exactly when $Y$ is an affine function of $X$ ($Y = a + bX$): $\rho = +1$ when $b > 0$ and $\rho = -1$ when $b < 0$. Another way to see it: $\operatorname{Var}(X/\sigma_X \pm Y/\sigma_Y) = 2(1 \pm \rho) \geq 0$ since variances are non-negative, which forces $\rho \in [-1, 1]$.
Answer: $\rho \in [-1, 1]$. The extremes are attained only when the two return series are perfectly linearly related (positively at $+1$, negatively at $-1$).
Intuition
Correlation lives in $[-1, 1]$ because it is the cosine of the angle between two vectors in the space of mean-zero random variables, and a cosine is bounded by one. The endpoints correspond to perfect collinearity. In portfolio construction this bound is load-bearing: the variance of a two-asset portfolio depends on $\rho$, and the fact that $\rho$ cannot fall below $-1$ caps how much diversification can reduce risk. A frequent mistake is to think any correlation matrix you write down is valid -- in higher dimensions the pairwise correlations must also be jointly consistent (the matrix must be positive semidefinite), which is a stronger condition than each entry lying in $[-1, 1]$.