Finance

Time Value of Money Formula With Definition

Time value of money definition

You will get all the time value of money formula on this with its definition and also how to calculate it.

Time Value of money Definition

This is the value of money recieved today is different from the value of money recieved after sometime in the future. The value of money is time dependent, this is a very important financial principle that is based on the following four factors. This is just a simple summary of time value of definition, to know more about it, go through here.

Now let’s head over to the time value of money formula.

Time value of money formula

Time value of money formula

With these Time value of money formula, you can easily get your calculation right especially when followed carefully.

  • PV is the value at time=0 (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.

  FV   \ = \  PV \cdot (1+i)^n

Present value of a future sum

The present value formula is the core for the time value of money formula ; each of the other formulae 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:

  PV \ = \ \frac{FV}{(1+i)^n}

The cumulative present value of future cash flows can be calculated by summing the contributions of FVt, the value of cash flow at time t:

\PV\ =\ \sum _{{t=1}}^{{n}}{\frac  {FV_{{t}}}{(1+i)^{t}}}

Note that this series can be summed for a given value of n, or when n is ∞. This is a very general time value of money formula, which leads to several important special cases given below.

Present value of an annuity for n payment period

In this case the cash flow values remain the same throughout the n periods. The present value of an annuity (PVA) formula has four variables, each of which can be solved for by numerical methods:

PV(A) \,=\,\frac{A}{i} \cdot \left[ {1-\frac{1}{\left(1+i\right)^n}} \right]

To get the PV of an 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 gas 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 :

{\displaystyle PV(A)\,=\,{A \over (i-g)}\left[1-\left({1+g \over 1+i}\right)^{n}\right]}

Where i = g :

{\displaystyle PV(A)\,=\,{A\times n \over 1+i}}

To get the PV of a growing 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.

PV(P) \ = \ { A \over i }

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 time value of money formula for a growing perpetuity:

{\displaystyle PV(A)\,=\,{A \over i-g}}

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 Growth modelused 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:

FV(A) \,=\,A\cdot\frac{\left(1+i\right)^n-1}{i}

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 :

FV(A) \,=\,A\cdot\frac{\left(1+i\right)^n-\left(1+g\right)^n}{i-g}

Where i = g :

FV(A) \,=\,A\cdot n(1+i)^{n-1}

Formula table

The following table summarizes the different time value of money formulas commonly used in calculating the time value of money. These values are often displayed in tables where the interest rate and time are specified.

FindGivenFormula
Future value (F)Present value (P){\displaystyle F=P\cdot (1+i)^{n}}F=P\cdot (1+i)^n
Present value (P)Future value (F){\displaystyle P=F\cdot (1+i)^{-n}}P=F\cdot (1+i)^{-n}
Repeating payment (A)Future value (F){\displaystyle A=F\cdot {\frac {i}{(1+i)^{n}-1}}}A=F\cdot \frac{i}{(1+i)^n-1}
Repeating payment (A)Present value (P){\displaystyle A=P\cdot {\frac {i(1+i)^{n}}{(1+i)^{n}-1}}}A=P\cdot \frac{i(1+i)^n}{(1+i)^n-1}
Future value (F)Repeating payment (A){\displaystyle F=A\cdot {\frac {(1+i)^{n}-1}{i}}}F=A\cdot \frac{(1+i)^n-1}{i}
Present value (P)Repeating payment (A){\displaystyle P=A\cdot {\frac {(1+i)^{n}-1}{i(1+i)^{n}}}}P=A\cdot \frac{(1+i)^n-1}{i(1+i)^n}
Future value (F)Initial gradient payment (G){\displaystyle F=G\cdot {\frac {(1+i)^{n}-in-1}{i^{2}}}}F=G\cdot \frac{(1+i)^n-in-1}{i^2}
Present value (P)Initial gradient payment (G){\displaystyle P=G\cdot {\frac {(1+i)^{n}-in-1}{i^{2}(1+i)^{n}}}}P=G\cdot \frac{(1+i)^n-in-1}{i^2(1+i)^n}
Fixed payment (A)Initial gradient payment (G){\displaystyle A=G\cdot \left[{\frac {1}{i}}-{\frac {n}{(1+i)^{n}-1}}\right]}A=G\cdot \left[\frac{1}{i}-\frac{n}{(1+i)^n-1}\right]
Future value (F)Initial exponentially increasing payment (D)Increasing percentage (g){\displaystyle F=D\cdot {\frac {(1+g)^{n}-(1+i)^{n}}{g-i}}}F=D\cdot \frac{(1+g)^n-(1+i)^n}{g-i}  (for i ≠ g){\displaystyle F=D\cdot {\frac {n(1+i)^{n}}{1+g}}}F=D\cdot \frac{n(1+i)^n}{1+g}   (for i = g)
Present value (P)Initial exponentially increasing payment (D)Increasing percentage (g){\displaystyle P=D\cdot {\frac {\left({1+g \over 1+i}\right)^{n}-1}{g-i}}}P=D\cdot \frac{\left({1+g \over 1+i}\right)^n-1}{g-i}   (for i ≠ g){\displaystyle P=D\cdot {\frac {n}{1+g}}}P=D\cdot \frac{n}{1+g}   (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 Gand 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.

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Above are all the time value of money formula and its definition.

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