Isolate Costs in NPV equation

Question

Hey can anyone help with this? This is the classic NPV equation:

NPV = -CapEx + ∑ (Revenue − Costs) / (1+Discount)^i

The partial sum is from i = 0 to n years.

For my purposes all the elements are know except costs.

I need to isolate costs in this equation. Is this possible?

Thanks,

Mike

Answer 1

Let $\frac{1}{\texttt{1+discount}}=a$. The equation becomes

$NVP=-CapEx+\sum_{i=0}^n\texttt{(Revenue-Costs)}\cdot a^i $

$NVP=-CapEx+\texttt{(Revenue-Costs)}\cdot\sum_{i=0}^n a^i $

$\sum_{i=0}^n a^i $ is the partial sum of a geometric series.

$\sum_{i=0}^n a^i =\frac{1-a^{n+1}}{1-a}$ Therefore

$NVP=-CapEx+\texttt{(Revenue-Costs)}\cdot\frac{1-a^{n+1}}{1-a} $

Adding CapEx on both sides of the equation and after that dividing the equation by $\frac{1-a^{n+1}}{1-a} $

$(NVP+CapEx)\cdot \frac{1-a}{1-a^{n+1}}=\texttt{Revenue-Costs} \quad \color{blue}{(1)}$

Now it is just one step to isolate the costs.

Further transformations: Multiplying the equation by (-1). The signs are going to be the opposite.

$-(NVP+CapEx)\cdot \frac{1-a}{1-a^{n+1}}=\texttt{-Revenue+Costs}\quad | +\texttt{Revenue} $

$\texttt{Revenue}-(NVP+CapEx)\cdot \frac{1-a}{1-a^{n+1}}=\texttt{Costs}\quad $

Pay attention on the negative sign on the RHS. The costs can be higher or lower than the revenue. The (constant) revenue must be higher than the (constant) costs, if you want a positive NVP. But this is not sufficient (only necessary condition), because of the CapEx. This can be seen in $\color{blue}{(1)}$.