1

Let's say I have the following table representing 3 data points:

Year | Number of units
---- | ----
2002 | 1349  --> P1
2003 | 1391  --> P2
2004 | 1324  --> P3

My goal is to calculate the Number of unit percent change from 2002 to 2004. I know I can achieve this using the following equation:

(P3-P1)/P1 = (1324-1349)/1349 = -0.0185 = -1.85%

I assume adding the percent change from year to year would yield the same value, but it doesn't:

(P2-P1)/P1 = (1391-1349)/1349 = 0.0311 = 3.11%
(P3-P2)/P2 = (1324-1391)/1391 = -0.0482 = -4.82%
3.11% + (-4.82%) = -1.71%

My question is why are these not equal? Am I doing something wrong?

Johnny Metz
  • 111
  • 2
    Suppose you have 100 units, and increase it by 50% and then decrease it by 50%. You end up with 150 units, then 75 units in the end; this is because 50% of the larger number is more important than 50% of the original, smaller number. This is the same principle you're dealing with. –  Sep 06 '17 at 17:36
  • 2
    This is because $(1+0.0311)\cdot(1-0.0482)\ne(1+0.0311-0.0482)$ – Hagen von Eitzen Sep 06 '17 at 17:40
  • That makes sense. I've also failed to derive (P3-P1)/P1 from (P2-P1)/P1 + (P3-P2)/P2 – Johnny Metz Sep 06 '17 at 17:44
  • 2
    Here is a heuristic: If percentages come from fractions of the same whole (i.e. have the same denominator), you can add them. Otherwise, if the percentages are applied consecutively in the time line, so to speak, you multiply them (don't add them). The correct way to multiply them is as in the comment by HvE. Check compound interest anywhere online. – Antoni Parellada Sep 06 '17 at 18:15

1 Answers1

1

Let $$ \frac{p_i-p_j}{p_j}=d_{i,j} \qquad\Longrightarrow \qquad p_i=p_j (1+d_{i,j}) $$ So we have $$ p_3=p_2 (1+d_{3,2}) =p_1 (1+d_{2,1})\cdot (1+d_{3,2})=p_1 (1+d_{3,1}) $$ that is $$ (1+d_{2,1})\cdot (1+d_{3,2})= (1+d_{3,1}) $$ and then $$ d_{3,1}= (1+d_{3,2})\cdot (1+d_{2,1})-1 $$ In general $$ d_{n,1}=\left [\prod_{i=1}^n (1+d_{i+1,i})\right]-1 $$

alexjo
  • 14,976