Why Component CAGRs Do Not Reconcile to Aggregate Growth
Summary
The document addresses why forecasting each business line with its own compound annual growth rate can produce a different next-period total from forecasting the combined business with its aggregate CAGR. It defines each line’s CAGR from its starting and ending values over the observation period, then applies that rate to the line’s latest value. The aggregate calculation instead uses the starting and ending sums to calculate one rate and applies it to the current total.
The discrepancy is mathematical: the CAGR calculation, which includes a ratio and exponentiation, is nonlinear in the values being combined. Therefore, adding projections based on individual rates generally does not equal the projection based on the total’s rate. The example illustrates the mismatch, and the answer formalizes it for two lines. It does not propose a reconciliation rule or address which forecast is more appropriate; that depends on whether the goal is to model component contributions or project the aggregate directly.
Key ideas
- A line’s CAGR is calculated from that line’s own beginning and ending values.
- The aggregate CAGR uses the sums of line values at the beginning and end of the period.
- CAGR is nonlinear in the values being combined, so component-based forecasts generally will not sum to the aggregate-based forecast.
- Choose the forecast level according to whether the goal is to project individual components or the total.
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Full text
# Aggregating growth rates
# 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
```
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 by user32416 (score 1, accepted)
https://quant.stackexchange.com/a/20768
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.