Algebraically this works, but I'm looking to understand (1) why it works and (2) if there is a simpler formula.
Problem: I want to get total % change over time.
Facts:
Example Data -
Jan 2013 - 14.29%
May 2013 - 25%
Oct 2013 - 20%
**Solution Methodology:**
1. Take a random baseline number to plugin: 100
2. Calculate the actual numbers:
Jan 2013 - 14.29% Increase = 114.29
May 2013 - 25% Increase = 142.8625
Oct 2013 - 20% Increase = 171.435
3. Calculate Total % Difference:
(171.435 - 100) / 100 = 71.435%
4. Plug in variables to determine formula to do this:
Baseline Number = X
A = Jan 2013 Increase % = (x)(A + 1) = (xA + x)
B = May 2013 Increase % = (xA + x)(B + 1) = (xAB + xA + xB + x)
C = Oct 2013 Increase % = (xAB + xA + xB + x)(C + 1) = (xABC + xAC + xBC + xC + xAB + xA + xB + x) = x(ABC + AC + BC + C + AB + A + B + 1)
Calculate % Change =
(x(ABC + AC + BC + C + AB + A + B + 1) - x) / x
Final Algebraic Formula = ABC + AC + BC + C + AB + A + B
Other Notes Interestingly enough if I only looked to get the Total % Difference of May 2013 and Jan 2013 the algebraic formula becomes: AB + A + B
There is a chance I'm overcomplicating this or am calling it the wrong thing.
Appreciate your help!
It may be easier to understand why this works if you consider the percentage increases as decimal multiplications.
Example: starting with $100$, a $A=10\%$ increase is equivalent to $100*1.10$, where the percentage $10\%$ is divided by $100$ and added to $1$.
We add $1$ to include the initial amount.
When multiple interests are compounded (applied sequentially), you are simply multiplying the respective decimal values together. Your final algebraic formula works as a result of multiplication's distributive property, but it can be written more generally as $$(\prod_{i=1}^n (A_i+1)) - 1$$ Where $A$ represents the interest rate, and $\prod$ means multiply all of the following together, for $i$ starting at $1$, and incrementing by $1$ until it reaches $n$.