# 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:
$(X_T + Y_T) (1 + R_{X+Y})=X_T(1+f(R_X))+Y_T(1+f(R_Y))$