# Aggregating growth rates

I'm working on a simple forecast model that uses Cumulative Annual Growth Rate (CAGR) to project future growth, and I've run into an apparent paradox.

The model includes multiple lines of business that change at different rates. I'm ultimately concerned with the projected total of all the lines combined. However I'd also like to project the growth of the individual lines to show how they contribute to the total.

Issue: the sum of individual line projections does not equal the projection of the total.

Example:


2011   2012   2013   2014 |   CAGR    2015(P)
Line A  100    200    300    400 |    59%     634.96
Line B  100    100    200    300 |    44%     432.67
Line C  200    800   1500   2500 |   132%   5,801.99
TOTAL   400   1100   2000   3200 |   100%   6,400.00


CAGR is 2014/2011^(1/3). Projected values 2015(P) are found by multiplying the previous year by 1+CAGR.

• 2015(P) Line A + Line B + Line C = 6,869.62

• 2015(P) TOTAL = 3200 * (1 + 100%) = 6400

What accounts for the difference? Is there a way to reconcile the growth rates of the individual lines and the total, or do I just need to pick a level of detail and stick with it? Been banging my head against a wall on this one for a while and any help is appreciated.

This is just basic mathematics. Simplify to two business lines just to make the point more transparent. Suppose you have two business lines with initial values $X_0, Y_0$ and terminal values $X_T, Y_T$. Then the sum of the initial values are $X_0 + Y_0$ and the terminal values are $X_T + Y_T$. Let the projected $T+1$ values be $X_{T+1}, Y_{T+1}, (X+Y)_{T+1}$. Then the $T$ period CAGR's are simply \begin{align} CAGR^X &= \left(\frac{X_T}{X_0}\right)^{(1/T)} \\ CAGR^Y &= \left(\frac{Y_T}{Y_0}\right)^{(1/T)} \\ CAGR^{X+Y} &= \left(\frac{X_T + Y_T}{X_0 + Y_0}\right)^{(1/T)} \\ \end{align}
By your "projection formula", then \begin{align} X_{T+1} &= (1 + CAGR^X) X_T \\ Y_{T+1} &= (1 + CAGR^Y) Y_T \\ (X+Y)_{T+1} &= (1 + CAGR^{X+Y}) (X_T + Y_T) \\ \end{align}
Clearly, then, $X_{T+1} + Y_{T+1} \neq (X+Y)_{T+1}$ simply because the CAGR rate computation is not linear in the values argument.