When taking the difference between two numbers, with the second number being negative, you have to add a -1 to the normal method to produce a sensible result. Conversely, when moving from negative to positive territory and taking a difference you have to add a 1 to account for the movement back to zero. In both cases it is advisable to add an indicator to the result to show that there is a sign change in the calculation.
Let’s say that I anticipated a profit of 3 million for the year, but profits came in at 4 million, how much better did I do? Well, I can either divide 4 by 3 and subtract 1 (33%) or subtract 3 from 4 and divide by 3 (33%). Either way I can say that I did 33% better than expected. It works out when calculating the movement too, if I add 33% of 3 to my base of 3 I get 4.
What happens if my profits come in at -1 million? Now the results are somewhat meaningless. Negative one divided by three is -33%. I didn’t do 33% worse than I expected, that would suggest that I had profits of 2 million, I actually did 133% worse than expected. I lost all the profit I expected (3) plus a third more.
Therein lies the answer, I have to account for the fact that I lost all positive profits at all first and then add the difference. Similarly, the old method works when you add a negative sign to account for the change from positive to negative. The difference between 3 and -1 is 4, making this negative and dividing by 3 gives the same answer as the logic above, -133%.
Similarly, when moving from a negative to a positive number you have to account for a 100% gain to get back to zero then take the difference. If I had anticipated profits of -5 but they came in at 2, the difference between -5 and 2 is 7, divided by 5 and flipping the sign leads to an increase of 140%. This makes sense when thinking of the number line, the -5 increased by 100% to get back to zero, then moved up by an additional 2 units or 2 divided by 5 or 40%, for a total move of 140%.
In either case, to take into account a sign change when calculating percent changes it may be helpful to your audience to add an indicator to the end of your number indicating what happened. For instance in my first example I could give an answer of -133%L, adding the L at the end to indicate that I’ve dropped from a positive to a negative number. Similarly the second example could be listed as 140%U, though there is no mathematical convention currently in place for the L or U suffixes and they will have to be explained to your audience.