NPV
A present value is the value now of a stream of future cash flows, negative or positive. The value of each cash flow needs to be adjusted for risk and the time value of money.
A net present value (NPV) includes all cash flows, such cost of acquisition of an asset, whereas a present value does not. So the net present value of a purchase being considered would include its purchase cost (as a negative cash flow), whereas the present value would not.
A discount rate needs to be used to adjust for risk and time value, and it is applied like this:
NPV = CF0 + CF1/(1+r) + CF2/(1+r)2 + CF3/(1+r)3 ...
where CF1 is the cash flow the investor receives in the first year, CF2 the cash flow the investor receives in the second year etc.
and r is the discount rate.
The series will usually end in a terminal value, which is a rough estimate of the value at that point. It is usual for this to be sufficiently far in the future to have only a minor effect on the NPV, so a rough estimate,usually based on a valuation ratio, is acceptable.
Periods other than an year could be used, but the discount rate needs to be adjusted. Assuming we start from an annual discount rate then to adjust to another period we would use, to get a rate i, given annual rate r, for a period x, where x is a fraction (e.g., six months = 0.5) or a multiple of the number of years:
i + 1 = (r + 1)x
To use discount rates that vary over time (so r1 is the rate in the first period, r2 = rate in the second period etc.) we would have to resort to a more basic form of the calculation:
NPV = CF0 + CF1/(1+r1) + CF2/((1+r1) ×(1+r2)) + CF3/((1+r1) ×(1+r2) ×(1+r3)) ...
This would be tedious to calculate by hand but is fairly easy to implement in a spreadsheet.