Editing Time value of money (section)

==Formula==
The following formula use these common variables:
* ''PV'' is the value at time zero (present value)
* ''FV'' is the value at time ''n'' (future value)
* ''A'' is the value of the individual payments in each compounding period
* ''n'' is the number of periods (not necessarily an integer)
* ''i'' is the [[interest rate]] at which the amount compounds each period
* ''g'' is the growing rate of payments over each time period

===Future value of a present sum===
The [[future value]]  (''FV'')  formula is similar and uses the same variables.
:<math>  FV   \ = \  PV \cdot (1+i)^n </math>

===Present value of a future sum===
The present value formula is the core formula for the time value of money; each of the other formulas is derived from this formula. For example, the annuity formula is the sum of a series of present value calculations.

The [[present value]] (''PV'') formula has four variables, each of which can be solved for by [[numerical methods]]:
:<math>  PV \ = \ \frac{FV}{(1+i)^n} </math>

The cumulative present value of future cash flows can be calculated by summing the contributions of ''FV<sub>t</sub>'', the value of cash flow at time ''t'':
:<math>  PV \ = \ \sum_{t=1}^{n} \frac{FV_{t}}{(1+i)^t} </math>

Note that this series can be summed for a given value of ''n'', or when ''n'' is ∞.<ref>http://mathworld.wolfram.com/GeometricSeries.html Geometric Series</ref> This is a very general formula, which leads to several important special cases given below.

===Present value of an annuity for n payment periods===
In this case the cash flow values remain the same throughout the ''n'' periods. The present value of an [[Annuity (finance theory)|annuity]] (PVA) formula has four variables, each of which can be solved for by numerical methods:
:<math>PV(A) \,=\,\frac{A}{i} \cdot \left[ {1-\frac{1}{\left(1+i\right)^n}} \right] </math>

To get the PV of an [[Annuity (finance theory)#Annuity-due|annuity due]], multiply the above equation by (1 + ''i'').

===Present value of a growing annuity===
In this case, each cash flow grows by a factor of (1 + ''g''). Similar to the formula for an annuity, the present value of a growing annuity (PVGA) uses the same variables with the addition of ''g'' as the rate of growth of the annuity (A is the annuity payment in the first period). This is a calculation that is rarely provided for on financial calculators.

'''Where ''i'' ≠ ''g'' :'''
:<math>PV(A)\,=\,{A \over (i-g)}\left[ 1- \left({1+g \over 1+i}\right)^n \right] </math>
'''Where ''i'' = ''g'' :'''
:<math>PV(A)\,=\,{A \times n \over 1+i} </math>

To get the PV of a growing [[Annuity (finance theory)#Annuity-due|annuity due]], multiply the above equation by (1 + ''i'').

===Present value of a perpetuity===
A [[perpetuity]] is payments of a set amount of money that occur on a routine basis and continue forever. When ''n'' → ∞, the ''PV'' of a perpetuity (a perpetual annuity) formula becomes a simple division.
:<math>PV(P) \ = \ { A \over i } </math>

===Present value of a growing perpetuity===
When the perpetual annuity payment grows at a fixed rate (''g'', with ''g'' < ''i'') the value is determined according to the following formula, obtained by setting ''n'' to infinity in the earlier formula for a growing perpetuity:

:<math>PV(A)\,=\,{A \over i-g}</math>

In practice, there are few securities with precise characteristics, and the application of this valuation approach is subject to various qualifications and modifications. Most importantly, it is rare to find a growing perpetual annuity with fixed rates of growth and true perpetual cash flow generation. Despite these qualifications, the general approach may be used in valuations of real estate, equities, and other assets.

This is the well known [[Gordon model|Gordon growth model]] used for [[stock valuation]].

===Future value of an annuity===
The future value (after ''n'' periods) of an annuity (FVA) formula has four variables, each of which can be solved for by numerical methods:
:<math>FV(A) \,=\,A\cdot\frac{\left(1+i\right)^n-1}{i}</math>

To get the FV of an annuity due, multiply the above equation by (1 + ''i'').

===Future value of a growing annuity===
The future value (after ''n'' periods) of a growing annuity (FVA) formula has five variables, each of which can be solved for by numerical methods:

'''Where i ≠ g :'''
:<math>FV(A) \,=\,A\cdot\frac{\left(1+i\right)^n-\left(1+g\right)^n}{i-g}</math>
'''Where i = g :'''
:<math>FV(A) \,=\,A\cdot n(1+i)^{n-1}</math>

===Formula table===
The following table summarizes the different formulas commonly used in calculating the time value of money.<ref>{{Cite web|url=http://ncees.org/exams/study-materials/download-fe-supplied-reference-handbook/|title=NCEES FE exam|work=NCEES }}</ref> These values are often displayed in tables where the interest rate and time are specified.

{| class=wikitable
!Find!!Given!!Formula
|-
|Future value (F)
|Present value (P)
|<math>F=P\cdot (1+i)^n</math>
|-
|Present value (P)
|Future value (F)
|<math>P=F\cdot (1+i)^{-n}</math>
|-
|Repeating payment (A)
|Future value (F)
|<math>A=F\cdot \frac{i}{(1+i)^n-1}</math>
|-
|Repeating payment (A)
|Present value (P)
|<math>A=P\cdot \frac{i(1+i)^n}{(1+i)^n-1}</math>
|-
|Future value (F)
|Repeating payment (A)
|<math>F=A\cdot \frac{(1+i)^n-1}{i}</math>
|-
|Present value (P)
|Repeating payment (A)
|<math>P=A\cdot \frac{(1+i)^n-1}{i(1+i)^n}</math>
|-
|Future value (F)
|Initial gradient payment (G)
|<math>F=G\cdot \frac{(1+i)^n-in-1}{i^2}</math>
|-
|Present value (P)
|Initial gradient payment (G)
|<math>P=G\cdot \frac{(1+i)^n-in-1}{i^2(1+i)^n}</math>
|-
|Fixed payment (A)
|Initial gradient payment (G)
|<math>A=G\cdot \left[\frac{1}{i}-\frac{n}{(1+i)^n-1}\right]</math>
|-
|Future value (F)
|Initial exponentially increasing payment (D)

Increasing percentage (g)
|<math>F=D\cdot \frac{(1+g)^n-(1+i)^n}{g-i}</math> &nbsp; (for ''i'' ≠ ''g'')

<math>F=D\cdot \frac{n(1+i)^n}{1+g}</math> &nbsp; (for ''i'' = ''g'')
|-
|Present value (P)
|Initial exponentially increasing payment (D)

Increasing percentage (g)
|<math>P=D\cdot \frac{\left({1+g \over 1+i}\right)^n-1}{g-i}</math> &nbsp; (for ''i'' ≠ ''g'')

<math>P=D\cdot \frac{n}{1+g}</math> &nbsp; (for ''i'' = ''g'')
|}
Notes:
*''A'' is a fixed payment amount, every period
*''G'' is the initial payment amount of an increasing payment amount, that starts at ''G'' and increases by ''G'' for each subsequent period.
*''D'' is the initial payment amount of an exponentially (geometrically) increasing payment amount, that starts at ''D'' and increases by a factor of (1 + ''g'') each subsequent period.