In finance, discounted cash flow (DCF) analysis is a method of valuing a project, company, or asset using the concepts of the time value of money. All future cash flows are estimated and discounted to give their present values (PVs) — the sum of all future cash flows, both incoming and outgoing, is the net present value (NPV), which is taken as the value or price of the cash flows in question. Using DCF analysis to compute the NPV takes as input cash flows and a discount rate and gives as output a price; the opposite process — taking cash flows and a price and inferring a discount rate, is called the yield. Discounted cash flow analysis is widely used in investment finance, real estate development, and corporate financial management. Discount rate
Main article: Discounting
The most widely used method of discounting is exponential discounting, which values future cash flows as "how much money would have to be invested currently, at a given rate of return, to yield the cash flow in future." Other methods of discounting, such as hyperbolic discounting, are studied in academia and said to reflect intuitive decision-making, but are not generally used in industry. The discount rate used is generally the appropriate Weighted average cost of capital (WACC), that reflects the risk of the cashflows. The discount rate reflects two things: 1. The time value of money (risk-free rate) – according to the theory of time preference, investors would rather have cash immediately than having to wait and must therefore be compensated by paying for the delay. 2. A risk premium – reflects the extra return investors demand because they want to be compensated for the risk that the cash flow might not materialize after all. An alternative to including the risk in the discount rate is to use the risk free rate, but multiply the future cash flows by the estimated probability that they will occur (the success rate). This method, widely used in drug development, is referred to as rNPV (risk-adjusted NPV), and similar methods are used to incorporate credit risk in the probability model of CDS valuation. Oxera (2011)  reviews the selection of a discount rate suitable for the assessment of new and emerging energy technologies.  History
Discounted cash flow calculations have been used in some form since money was first lent at interest in ancient times. As a method of asset valuation it has often been opposed to accounting book value, which is based on the amount paid for the asset. Following the stock market crash of 1929, discounted cash flow analysis gained popularity as a valuation method for stocks. Irving Fisher in his 1930 book "The Theory of Interest" and John Burr Williams's 1938 text 'The Theory of Investment Value' first formally expressed the DCF method in modern economic terms.  Mathematics
 Discounted cash flows
The discounted cash flow formula is derived from the future value formula for calculating the time value of money and compounding returns.
Thus the discounted present value (for one cash flow in one future period) is expressed as:
* DPV is the discounted present value of the future cash flow (FV), or FV adjusted for the delay in receipt; * FV is the nominal value of a cash flow amount in a future period; * i is the interest rate, which reflects the cost of tying up capital and may also allow for the risk that the payment may not be received in full; * d is the discount rate, which is i/(1+i), i.e. the interest rate expressed as a deduction at the beginning of the year instead of an addition at the end of the year; * n is the time in years before the future cash flow occurs. Where multiple cash flows in multiple time periods are discounted, it is necessary to sum them as follows:
for each future cash flow (FV) at any time period (t) in years from the present time, summed over all time periods. The sum can then be used as a net present value figure....