Quantile Forecasts
Asymmetric costs have an asymmetric answer. If being short costs three times as much as being long, the target is not the mean.
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Probabilistic Forecasts
A point forecast is one summary of this. Once you keep the whole distribution, the decision can pick the summary it actually needs.
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Quantile Regression
Swap squared error for an asymmetric loss and the fitted line becomes a quantile. Three fits give you an interval with no normality assumption.
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Quantile Level
A quantile level selects a position in the outcome distribution. Exact exceedance probabilities require continuity; discrete outcomes can have ties.
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Continuous Ranked Probability Score (CRPS)
CRPS reads like a weighted absolute error computed from the entire distribution, so it rewards being both accurate and appropriately confident.
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Pinball Loss (a.k.a. Quantile Loss)
At p = 0.5 it is just absolute error. Away from the median it weights misses unevenly, which is what makes a quantile the thing being estimated.