Picking a Regression Metric for Demand Forecasting

You forecast next-week unit demand for 10,000 SKUs. Many SKUs sell zero units in a typical week; a few sell thousands. Stockouts cost roughly 4× what overstock costs per unit. A colleague proposes MAPE as the single offline metric and trains a model to minimise MSE.

1. Explain two concrete problems with MAPE on this data.
2. Is MSE the right training loss given the cost asymmetry? What loss and what evaluation metric would you use instead, and what quantile?
3. For actuals [0, 5, 50, 500] and forecasts [2, 4, 60, 450], compute MAE, RMSE, and the pinball loss at τ = 0.8. Comment on which errors dominate each metric.