MSE versus MAE for noisy targets

For a heavy-tailed return target, how do squared and absolute prediction losses differ in their population-optimal predictions?

When the required expectations exist, squared loss is minimised by the conditional mean, while absolute loss is minimised by a conditional median. Squared loss emphasises large residuals more strongly. Choose the objective according to the decision being supported and compare performance under plausible tail events. A robust objective changes the estimated target functional; it is not automatically a better estimator of the conditional mean.