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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

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1 Answer 1

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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 $\color{blue}{(1)}$.

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  • $\begingroup$ Should 'discount' be a negative? $\endgroup$
    – Mike
    Sep 3, 2015 at 1:48
  • $\begingroup$ @Mike The discount rate ($\delta$) itself can be negative. Suppose $\delta=r+\pi$. r is the real interest rate and $\pi$ is the inflation rate. If the inflation rate is higher than the real interest rate, then the discount rate is negative. The discount rate must not be smaller than -1. Otherwise $1+\delta$ itself would be negative. I have made an edit, which is referring to your penultimate comment. $\endgroup$ Sep 3, 2015 at 8:16

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