How to compare -50% and +100% This might seem (and be) a really trivial question!
I have this table:
id until_2000 after_2000
1  1          1          1
2  2          1          2
3  3          2          1

Now I want to compare the relative change between the columns after_2000 and until_2000. Just dividing the after_2000 values by the until_2000 (and substract 1?!) values yields the colum rel_change
  id until_2000 after_2000 rel_change
1  1          1          1        0.0
2  2          1          2        1.0
3  3          2          1       -0.5

While the first row stays as is (has a change of 0%), the second row doubles (has a relative change of 100 %). Yet the third row is half the value and thus has a relative value of -50 %.
Now my trivial question is how to correctly compute and average these relative changes. In my line of thinking the values of 0, doubling, and "halfing" should result in an average change of 0.
Yet adding 0 + 1 - 0.5 and dividing it by three yields a value of 0.16666.
What am I conceptually understanding wrong here?

As good as is kqr answer, I would go for another method, which would work even if you have a 0 value somewhere and would keep his way of calculating the relative change
When you ask the question "how to correctly compute and average these relative changes" we could also first do the average then compute the relative change. You would then have
id    until_2000 after_2000 rel_change
total       4          4        0    

Then you can calculate the difference as you were calculating before
The average and the sum of the different rows lead to the same result here as your denominator is the same for both (3 rows)
$\frac{sumAfter/nRows - sumUntil/nRows}{sumUntil/nRows}  = \frac{sumAfter - sumUntil}{sumUntil}$
