The risk of an estimator $\delta$ is defined as $$E_\theta[L(\theta,\delta(X))],$$ where, say, $L(\theta,\delta(X)) = (\theta-\delta(X))^2$, and $E_\theta(X)$ is defined as $\int XdP_\theta$, namely the expectation of random variable $X$ when the parameter is $\theta$.
I wonder why people never considered $$Var_\theta[L(\theta,\delta(X))]$$ in addition to the risk as a way to evaluate an estimator given a predefined loss function?
Say $\delta_1$ and $\delta_2$ have similar risk, but different variance of loss. Intuitively I would choose the one with a smaller variance of loss. However, I've never seen people ever talked about it. Have they?