Aggregating exponential growth rates

Question

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.

Answer 1

The mix of the 3 lines, at the end of 2014, produces a total of 3200 and that means an average growth on the total equal to $100\%$. So the projected values of the 3 lines will produce a total of 6,869.62 and that means an average growth on the total equal to $\frac{6,869.62}{3200}=114.68\%$. The row of total is just a result! So you can plan the growth of the line and then calculate the average growth of the total.

Otherwise you can plan the total growth as the previous average CAGR=$100\%$ and the distribute the growth over the lines using the weight of each line over the total. So for example line A has weight $400/3200=13\%$ and if the new total is $3200\times (1+100\%)=6400$, you will have that line A must produce $6400\times 13\%=800$ that is a growth of $100\%$ instead of the previous $59\%$. Nevertheless this approach is mathematically correct but practically inconsistent.