# Net Present Value with investments in multiple phases

I am trying to calculate the Net Present Value for a theoretical project in which, unlike most examples in textbooks and on internet tutorials, investments do not only happen in t=0, but they also occur in later years.

My assumption is that investment expenditures occur at the beginning of each year, while operating expenditures (incomes minus costs) occur at the end of each year.

I have tried to illustrate that with the image below. The image shows the t=0 point, the 3 years considered in the analysis and the facts that the Investment expenditures (Inv) occur at the beginning of the year, while the operating expenditures (op) at the end.

My question then is, would the following formula be correct in order to calculate the total NPV of the project? Is it ok to discount the $$Inv_y$$ costs with the $$(1+d)^{y-1}$$ and the $$Op$$ with $$(1+d)^y$$?

$$NPV = - \sum_{y=1}^{y=3} \frac{Inv_y}{(1+d)^{y-1}} - \sum_{y=1}^{y=3} \frac{Op_y}{(1+d)^{y}}$$

For me, it makes sense since, for instance, the $$Inv_1$$ occurs "now", while the $$Op_1$$ occurs 1 year later. A similar thinking applies to the rest of the $$Inv$$ and $$Op$$ expenditures.

Any feedback will be greatly appreciated!

• If you only have investments and operating expenses, you're right, but you need to add a minus sign to indicate that the NPV is negative. Hopefully, the project will also generate revenue, which would have a positive sign and can result in the NPV to be positive. – Patricio Mar 7 at 14:01
• Hi! Thanks for your comment! Indeed, the negative sign should be added to indicate what you mention! I have added it now. – gmavrom Mar 7 at 16:43
• Yes, it looks correct. – BB King Mar 7 at 19:24
• Thanks BB King (cool nickname btw)! – gmavrom Mar 12 at 10:02