Editing Net present value (section)

== Formula ==
Each cash inflow/outflow is [[discounted]] back to its present value (PV). Then all are summed such that NPV is the sum of all terms:
<math display="block">\mathrm{PV} = \frac{R_t}{(1+i)^t}</math>
where:
* {{mvar|t}} is the time of the cash flow
* {{mvar|i}} is the discount rate, i.e. the [[rate of return|return]] that could be earned per unit of time on an [[opportunity cost|investment with similar risk]]
* <math>R_t</math> is the net cash flow i.e. cash inflow&nbsp;− cash outflow, at time ''t''. For educational purposes, <math>R_0</math> is commonly placed to the left of the sum to emphasize its role as (minus) the investment.
* <math>1/(1+i)^t</math> is the discount factor, also known as the present value factor.

The result of this formula is multiplied with the Annual Net cash in-flows and reduced by Initial Cash outlay the present value, but in cases where the cash flows are not equal in amount, the previous formula will be used to determine the present value of each cash flow separately. Any cash flow within 12 months will not be discounted for NPV purpose, nevertheless the usual initial investments during the first year ''R''<sub>0</sub> are summed up a negative cash flow.<ref>{{cite book | last = Khan | first = M.Y. | title = Theory & Problems in Financial Management | publisher = McGraw Hill Higher Education | location = Boston | year = 1993| isbn = 978-0-07-463683-1 }}</ref>

The NPV can also be thought of as the difference between the discounted benefits and costs over time. As such, the NPV can also be written as:

:<math>\mathrm{NPV} = \mathrm{PV}(B) - \mathrm{PV}(C)</math>

where:
* {{mvar|B}} are the benefits or cash inflows
* {{mvar|C}} are the costs or cash outflows

Given the (period, cash inflows, cash outflows) shown by ({{mvar|t}}, <math>B_t</math>, <math>C_t</math>) where {{mvar|N}} is the total number of periods, the net present value <math>\mathrm{NPV}</math> is given by:

:<math>\mathrm{NPV}(i, N) = \sum_{t=0}^N \frac{B_t}{(1+i)^t} - \sum_{t=0}^N \frac{C_t}{(1+i)^t}</math>

where:
* <math>B_t</math> are the benefits or cash inflows at time {{mvar|t}}.
* <math>C_t</math> are the costs or cash outflows at time {{mvar|t}}.
The NPV can be rewritten using the net cash flow <math>(R_t)</math> in each time period as:<math display="block">\mathrm{NPV}(i, N) = \sum_{t=0}^N \frac{R_t}{(1+i)^t}</math>By convention, the initial period occurs at time <math>t = 0</math>, where cash flows in successive periods are then discounted from <math>t = 1, 2, 3 ...</math> and so on. Furthermore, all future cash flows during a period are assumed to be at the end of each period.<ref>{{Cite web |last=Javed |first=Rashid |date=2016-12-28 |title=Net present value (NPV) method - explanation, example, assumptions, advantages, disadvantages |url=https://www.accountingformanagement.org/net-present-value-method/ |access-date=2023-04-21 |website=Accounting For Management |language=en-US}}</ref> For constant cash flow {{mvar|R}}, the net present value <math>\mathrm{NPV}</math> is a finite [[geometric series]] and is given by:

<math display="block">\mathrm{NPV}(i, N, R) = R \left( \frac{ 1 -\left( \frac{1}{1 + i} \right)^{N+1}} { 1 - \left( \frac{1}{1+i} \right) } \right), \quad i \ne 0 </math>

Inclusion of the <math>R_0</math> term is important in the above formulae. A typical capital project involves a large negative <math>R_0</math> cashflow (the initial investment) with positive future cashflows (the return on the investment). A key assessment is whether, for a given discount rate, the NPV is positive (profitable) or negative (loss-making). The IRR is the discount rate for which the NPV is exactly 0.