why adding pct_change per unit doesn't equal total pct_change

Question

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?

Answer 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 $$ d_{n,1}=\left [\prod_{i=1}^n (1+d_{i+1,i})\right]-1 $$