# E Net present value of an investment

Back to chapter 4: “The effect of the interest rate on investment demand”

The net present value of an investment can be calculated by subtracting the acquisition cost from the **present value** of the sum of the expected future cash flows. The periodic cash flows are the difference between the cash inflows and the cash outflows associated with the investment project in each period.

\[\text{Net present value} = \text{Present value of cash flows} - \text{Acquisition cost}\]

The present value refers to the current value of the sum of the cash flows. The present value of a future payment surplus can be obtained by calculating how much we would have to save today at a given interest rate in order to have saved a value of exactly the same amount at the time of the potential payment surplus. This value is called the present value.

The present value (\(V_t^E\)) of the expected cash flow in the future time periods can be calculated as follows, where \(\Pi_t^E\), \(\Pi_{t+1}^E\), … stand for the expected cash surplus in period \(t\), \(t+1\), etc. For simplicity, we assume that the real interest rate (\(r\)) remains constant over all periods:

\[\begin{equation} V_t^E = \Pi_t^E+\frac{\Pi_{t+1}^E}{\left(1+r\right)}+\frac{\Pi_{t+2}^E}{\left(1+r\right)^2}+...\ =\sum_{i=0}^T\frac{1}{\left(1+r\right)^i}\Pi_{t+i}^E \end{equation}\]

If we now compare the cash value of the surplus payments expected from the investment with the acquisition costs of the investment, we obtain the net present value of the investment.

\[\text{Net present value} = V_t^E - \text{Acquisition cost}\]

If the net present value is positive, the investment is profitable. If the net present value is negative, the investment represents a loss.