Reconciling Component and Aggregate CAGR Forecasts

Summary

The document describes a forecasting problem: applying compound annual growth rates separately to business lines does not necessarily produce the same next-period total as applying a CAGR to their combined value. The reason is that aggregate growth depends on the starting weights of the components, while each line can have a different growth rate. The author wants to preserve the aggregate forecast while explaining how individual components contribute to it.

The question asks whether a function can transform each line’s rate so that the transformed component projections sum to the aggregate projection. No answer or evidence is provided. The setup also appears to mix CAGR as a growth factor with a percentage rate: the displayed ratio power is a factor, while the projection multiplies by one plus that quantity. This distinction matters when formulating a reconciliation. The document does not specify assumptions about negative or zero values, changing component definitions, or the desired attribution method.

Key ideas

• The CAGR of a sum generally differs from the individual CAGRs of its components.
• An aggregate growth rate reflects the starting-value weights and component growth rates.
• A reconciliation method is needed when component projections must add to a fixed aggregate forecast.
• The displayed formulas may confuse a growth factor with a percentage growth rate.
• The document poses the reconciliation question but provides no proposed solution or validation.

Tags

# Reconciling forecasted growth of components and sum


# Reconciling forecasted growth of components and sum












I'm working with a very basic basic forecast model using Compound Annual Growth Rate and I need to reconcile the forecasts at different levels of detail.

Suppose I 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 their projected $T+1$ values be $X_{T+1},Y_{T+1},(X+Y)_{T+1}$.

I find the Compound Annual Growth Rate $R$ for each line and the sum of all lines: \begin{align} R_X &= \left(\frac{X_T}{X_0}\right)^{(1/T)} \\ R_Y &= \left(\frac{Y_T}{Y_0}\right)^{(1/T)} \\ R_{X+Y} &= \left(\frac{X_T + Y_T}{X_0 + Y_0}\right)^{(1/T)} \\ \end{align} To project forward one period, I multiply the terminal value of each line and the sum of all lines by their respective $R$ values: \begin{align} X_{T+1} &= X_T(1+R_X)\\ Y_{T+1} &= X_T(1+R_Y)\\ (X+Y)_{T+1} &= (X_T + Y_T) (1 + R_{X+Y}) \\ \end{align} However, I find that $X_{T+1}+Y_{T+1}\neq(X+Y)_{T+1}$ because the rate computation is not linear in the values argument. I need to work with $(X+Y)_{T+1}$ as is, but I also need to discuss how $X_{T+1}$ and $Y_{T+1}$ individually contribute to the total.

Is there a function $f$ such that: $(X_T + Y_T) (1 + R_{X+Y})=X_T(1+f(R_X))+Y_T(1+f(R_Y))$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.