Summation of growth problem

[Boaman888]

I have a business that did $X in sales in September, 2008. I would like to do $Y in annual revenue in 2009. Now I want to calculate the necessary month-on-month compound growth rate to get me there, R%.

October 08 sales = X * (1 + R%)
November 08 sales = X * (1 + R%)^2
December 08 sales = X * (1 + R%)^3

January 09 sales = X * (1 + R%)^4
Etc.
December 09 sales = X * (1 + R%)^16

Summation of all these 12 months for 2009 = $Y. In other words, summation from n=4 to n=16 of X * (1 + R%)^n equals Y. How do I express R% as a function of X and of Y?

Notes of things I've thought about so far:
(1) Is there any way to calculate from n=4 to n=infinity and then subtract off the sum from n=17 to infinity. that would give n=4 up to n=16. I don't think this is possible because this series diverges . . .

(2) I can do an approximation by truncating the series to only quadratic terms:
The summation becomes X * [556 R^2 + 114R + 12 ] = Y, which I can then solve using the quadratic equation (556R^2+114R+12 - Y/X = 0]. Only problem is the error is too big for my purposes. Is there a way to solve using the R^3 or R^4 terms? i.e. is there a solution to:
X * [1819R^3+556R^2+114R+12] = Y, OR even better,
X * [4368R^4+1819R^3+556R^2+114R+12] = Y

Any help much appreciated!

 

[Denis]

Note: your "December 09 sales = X * (1 + R%)^16" should be "December 09 sales = X * (1 + R%)^15"

No matter how you cook it, cannot be solved directly; iteration must be used.

The Dec/08 sales will be x(1 + r)^3.
The 2009 sales will then be: [x(1 + r)^3] * (1 + r)^1 + [x(1 + r)^3] * (1 + r)^2 + .... + [x(1 + r)^3] * (1 + r)^12 = y

The summation of the above flow is:
y = {x(1 + r)^3 * (1 + r) * [(1 + r)^12 - 1]} / r : if you ask me for the steps involved in getting this, answer is NO!
Simplified a bit:
y = {x(1 + r)^4 * [(1 + r)^12 - 1]} / r ************
To make it a little more "manageable", let f = 1 + r; then:
y = xf^4(f^12 - 1) / (f - 1) or y = x(f^16 - f^4) / (f - 1)
Re-arranging:
(f^16 - f^4) / (f - 1) = y / x

As example, Sep/08 sales = 20,000, and you want sales for 2009 to be 300,000:
(f^16 - f^4) / (f - 1) = 300000 / 20000
(f^16 - f^4) / (f - 1) = 15

A little iteration gave me f = 1.0235, left side computing to 15.0110....

So r = .0235 or ~2.35% (that's per month).

**********may be a good idea to change that slightly to:
y = {x(1 + r)^(m + 1) * [(1 + r)^12 - 1]} / r
m being the number of months left in previous year (in your case m = 3),
since you may not always want Sept to be the base.

 

[Boaman888]

Thanks Denis! That makes sense; I will close in on the answer using a numerical approximation.