Editing Net present value (section)

== Example ==
<!-- People like to mess with this example. Please be 100% sure you are correct before you make a change to it!-->
A corporation must decide whether to introduce a new product line. The company will have immediate costs of 100,000 at&nbsp;{{math|1=''t'' = 0}}. Recall, a cost is a negative for outgoing cash flow, thus this cash flow is represented as −100,000. The company assumes the product will provide equal benefits of 10,000 for each of 12 years beginning at&nbsp;{{math|1=''t'' = 1}}. For simplicity, assume the company will have no outgoing cash flows after the initial 100,000 cost. This also makes the simplifying assumption that the net cash received or paid is lumped into a single transaction occurring ''on the last day'' of each year. At the end of the 12 years the product no longer provides any cash flow and is discontinued without any additional costs. Assume that the effective annual discount rate is 10%.

The present value (value at&nbsp;{{math|1=''t'' = 0}}) can be calculated for each year:
{| class="wikitable" align="center" font-size="120%" border:"2px solid black;"
!Year!!Cash flow!!Present value
|-
|''T'' = 0||<math>\frac{-100,000}{(1+0.10)^0}</math>||−100,000
|-
|''T'' = 1||<math>\frac{10,000}{(1+0.10)^1}</math>||9,090.91
|-
|''T'' = 2||<math>\frac{10,000}{(1+0.10)^2}</math>||8,264.46
|-
|''T'' = 3||<math>\frac{10,000}{(1+0.10)^3}</math>||7,513.15
|-
|''T'' = 4||<math>\frac{10,000}{(1+0.10)^4}</math>||6,830.13
|-
|''T'' = 5||<math>\frac{10,000}{(1+0.10)^5}</math>||6,209.21
|-
|''T'' = 6||<math>\frac{10,000}{(1+0.10)^6}</math>||5,644.74
|-
|''T'' = 7||<math>\frac{10,000}{(1+0.10)^7}</math>||5,131.58
|-
|''T'' = 8||<math>\frac{10,000}{(1+0.10)^8}</math>||4,665.07
|-
|''T'' = 9||<math>\frac{10,000}{(1+0.10)^9}</math>||4,240.98
|-
|''T'' = 10||<math>\frac{10,000}{(1+0.10)^{10}}</math>||3,855.43
|-
|''T'' = 11||<math>\frac{10,000}{(1+0.10)^{11}}</math>||3,504.94
|-
|''T'' = 12||<math>\frac{10,000}{(1+0.10)^{12}}</math>||3,186.31
|}

The total present value of the incoming cash flows is 68,136.91.  The total present value of the outgoing cash flows is simply the 100,000 at time&nbsp;{{math|1=''t'' = 0}}.
Thus:

: <math>\mathrm{NPV} = PV(\text{benefits}) - PV(\text{costs})</math>

In this example:

: <math>\begin{align}
\mathrm{NPV} &=  68,136.91 - 100,000
\\
& = -31,863.09
\end{align}</math>

Observe that as ''t'' increases the present value of each cash flow at ''t'' decreases. For example, the final incoming cash flow has a future value of 10,000 at {{math|1=''t'' = 12}} but has a present value (at&nbsp;{{math|1=''t'' = 0}}) of 3,186.31. The opposite of discounting is compounding. Taking the example in reverse, it is the equivalent of investing 3,186.31 at {{math|1=''t'' = 0}} (the present value) at an interest rate of 10% compounded for 12 years, which results in a cash flow of 10,000 at {{math|1=''t'' = 12}} (the future value).

The importance of NPV becomes clear in this instance. Although the incoming cash flows ({{math|1=10,000 × 12 = 120,000}}) appear to exceed the outgoing cash flow (100,000), the future cash flows are not adjusted using the discount rate. Thus, the project appears misleadingly profitable. When the cash flows are discounted however,  it indicates the project would result in a net loss of 31,863.09.  Thus, the NPV calculation indicates that this project should be disregarded because investing in this project is the equivalent of a loss of 31,863.09 at&nbsp;{{math|1=''t'' = 0}}. The concept of time value of money indicates that cash flows in different periods of time cannot be accurately compared unless they have been adjusted to reflect their value at the same period of time (in this instance,&nbsp;{{math|1=''t'' = 0}}).<ref name="Berk, DeMarzo p. 94"/> It is the present value of each future cash flow that must be determined in order to provide any meaningful comparison between cash flows at different periods of time.  There are a few inherent assumptions in this type of analysis:

# The ''investment horizon'' of all possible investment projects considered are equally acceptable to the investor (e.g. a 3-year project is not necessarily preferable vs. a 20-year project.)
# The 10% discount rate is the appropriate (and stable) rate to discount the expected cash flows from each project being considered. Each project is assumed equally speculative.
# The shareholders cannot get above a 10% return on their money if they were to directly assume an equivalent level of risk. (If the investor could do better elsewhere, no projects should be undertaken by the firm, and the excess capital should be turned over to the shareholder through dividends and stock repurchases.)

More realistic problems would also need to consider other factors, generally including: smaller time buckets, the calculation of taxes (including the cash flow timing), inflation, currency exchange fluctuations, hedged or unhedged commodity costs, risks of technical obsolescence, potential future competitive factors, uneven or unpredictable [[cash flow]]s, and a more realistic [[salvage value]] assumption, as well as many others.

A more simple example of the net present value of incoming cash flow over a set period of time, would be winning a Powerball lottery of {{US$|long=no|500 million}}.  If one does not select the "CASH" option they will be paid {{US$|long=no|25000000}} per year for 20 years, a total of {{US$|long=no|500000000}}, however, if one does select the "CASH" option, they will receive a one-time lump sum payment of approximately {{US$|long=no|285 million}}, the NPV of {{US$|long=no|500000000}} paid over time.  See "other factors" above that could affect the payment amount.  Both scenarios are before taxes.