Truncated distribution and the property of its expected value

I have a complicated random variable $Y$ where the support is on $\mathbb{R}$ (real value). Now, we construct a truncated random variable $\tilde{Y}$ such that$\tilde{Y}=Y$ if $-C_0\leq Y\leq C_1$, $\tilde{Y}=C_1$ if $Y\ge C_1$, and $\tilde{Y}=-C_0$ if $Y\le -C_0$, where $C_1$ and $C_0$ are positive real number. We firstly consider the case $E_{Y}[\tilde{Y}]>0$ and $E_Y[Y]>0$.

My question is what is the ratio $\frac{E_{Y}[Y]}{E_{Y}[\tilde{Y}]}$? Is there any theorem talking about this?