Is this altered formula for correlation still bounded by $-1$ and $1$? Recall that
$$
‐1 \le \text{corr}(X,Y) = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \le 1
$$
The proof for this bound uses the Cauchy Schwarz inequality, and I've been trying to wrap my head around why this bound holds intuitively.
Question: Does this bound still hold for the altered formula
$$
‐1 \le \frac{\text{Cov}(X,Y)}{d_X d_Y} \le 1
$$
where
$$
d_X = E[|X - E[X]|]
$$
and
$$
d_Y = E[|Y - E[X]|]?
$$
I know that 
$$
d_X d_Y \le \sigma_X \sigma_Y
$$
so that, if it did hold, this would be a tighter bound than the original.

Let $X=-1$ with probability $2/3$, and $2$ with probability $1/3$, and let $Y=X$.
Compute. We have $d_X=d_Y=4/3$, and $\text{Cov}(X,Y)=2$. The ratio is $\frac{18}{16}$.
