Editing Discounted cash flow (section)

==Mathematics==
===Discounted cash flows===
The discounted cash flow formula is derived from the [[Time_value_of_money#Present_value_of_a_future_sum|present value formula for calculating the time value of money]]

:<math>DCF = \frac{CF_1}{(1+r)^1} + \frac{CF_2}{(1+r)^2} + \dotsb +
\frac{CF_n}{(1+r)^n}</math>

and [[Compound interest|compounding]] returns:

:<math>FV = DCF \cdot (1+r)^n</math>.
Thus the discounted present value (for one cash flow in one future period) is expressed as:

:<math>DPV =  \frac{FV}{(1+r)^n}</math>

where
* ''DPV'' is the discounted present value of the future cash flow (''FV''), or ''FV'' adjusted for the delay in receipt;
* ''FV'' is the [[Real versus nominal value (economics)|nominal value]] of a cash flow amount in a future period (see [[Mid-year adjustment]]);
* ''r'' is the [[interest rate]] or discount rate, which reflects the cost of tying up [[Capital (economics)|capital]] and may also allow for the risk that the payment may not be received in full;<ref>{{cite web |url=http://data.gov.uk/sib_knowledge_box/discount-rates-and-net-present-value |title=Discount rates and net present value |publisher=Centre for Social Impact Bonds |access-date=28 February 2014 |archive-url=https://web.archive.org/web/20140304094708/http://data.gov.uk/sib_knowledge_box/discount-rates-and-net-present-value |archive-date=4 March 2014 |url-status=dead }}</ref>
* ''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:

:<math>DPV = \sum_{t=0}^{N} \frac{FV_t}{(1+r)^{t}}</math>

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. If the amount to be paid at time&nbsp;0 (now) for all the future cash flows is known, then that amount can be substituted for ''DPV'' and the equation can be solved for ''r'', that is the [[internal rate of return]].

All the above assumes that the interest rate remains constant throughout the whole period.

If the cash flow stream is assumed to continue indefinitely, the finite forecast is usually combined with the assumption of constant cash flow growth beyond the discrete projection period. The total value of such cash flow stream is the sum of the finite discounted cash flow forecast and the [[Terminal value (finance)]].

===Continuous cash flows===
For continuous cash flows, the summation in the above formula is replaced by an integration:

:<math>DPV= \int_0^T  FV(t) \, e^{-\lambda t} dt = \int_0^T \frac{FV(t)}{(1 + r)^t} \, dt\,,</math>

where <math>FV(t)</math> is now the ''rate'' of cash flow, and <math>\lambda = \ln(1+r)</math>.