# Mean absolute percentage error with respect to predictions

A friend of mine has suggested that instead of using mean absolute percentage error, i.e.

$$\frac{1}{N}\sum_{i=0}^N \left| \frac{A_i - P_i}{A_i} \right|$$

(where $$A_i$$ denotes an actual value, $$P_i$$ denotes a predicted value, and $$N$$ the number of observations), to modify it so that the predicted amount is in the denominator, i.e.

$$\frac{1}{N}\sum_{i=0}^N \left| \frac{A_i - P_i}{P_i} \right|$$

They've said that like this we penalise underforecasting more than overforecasting.

This seems strange to me, though I'm having trouble articulating why. What problems might there be with this latter error metric (as compared with regular MAPE on top)?

• Which one is "such a"? The first one looks more problematic because in case $A_i=0$ you're in trouble, whereas any decent prediction procedure in the second case will avoid the possibility of predicting $P_i=0.$ (This is part of an argument for using the absolute logarithm of $A_i/P_i$ as an improvement over either formula.)
– whuber
Feb 9, 2021 at 17:45
• @whuber the latter. Thanks, have edited to clarify Feb 9, 2021 at 18:03