First time posting here, so please be gentle
Suppose I have a numerical value which has sustained linear growth over 2 years
Lets say in Jan 2013 the figure is 22,000, and in Jan 2015 it's 33,000. I don't know the value for the year in between, just the number of years and the knowledge that its a linear growth.
22,000 > 33,000 is a 50% linear growth rate over two years. However, I want to get the annual linear growth rate. I can't just half that, because a 25% growth rate would lead to 34,375 by Jan 2015.
So how do I calculate annual percentage? It's probably fairly simple but my brain is a bit slow this week.
Best Answer
What a "$50\%$ linear growth rate over two years" means is as follows: if the amount we start with is $P$, then the amount we have two years later will be $$ A = P(1 + 0.50) = 1.5P $$ what we want is a one-year growth rate that yields the same two year effect. If $r$ (our rate as a decimal) is our yearly growth rate, then the amount after one year will be $P(1 + r)$, and the amount after 2 will be $[P(1+r)](1+r) = P(1+r)^2$.
So, we want to find a value $r$ so that $$ P(1+ r)^2 = 1.5 P \implies\\ (1 + r)^2 = 1.5 $$ solving for $r$, we have $$ r = \sqrt{1.5} - 1 \approx 0.225 $$ so, the corresponding yearly growth rate is $22.5\%$.