Finance

What Is Time Value Of Money(Definition & Formula)

Time value of money formula

Time value of money definition: It is a known fact of economic life that money is not free and can be downright expensive. Money has earning power therefore the user must pay for using it. The cost of using money can be said to involve an explicit cost and implicit cost

A very practical benefit to be gained from studying the time value of money formula is that you will show how to apply them to your personal financial transactions which will enable you to understand how payments are determined for installment debt contract, such as mortgages and car loans etc.

What is time value of money

Time value of money definition and meaning: 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.

The principle of capital market efficiency states that current market price of financial securities reflect all available information. That is, financial securities are fairly priced. Another way to state this principle is to say that the NPV from investing in financial securities is zero. At this point, most people ask: if the NPV is zero, why would anyone purchase a financial security? The is to earn profit. A zero NPV implies that the investor will earn an appropriate for the investment risk; not a zero return. The principle of risk return trade ooff implies that investors who take more risk will earn a large profit, on average. Of course, with risky investment, the outcome may be extremely good, extremely bad, or anywhere in between. At the time of investment, however, the investor’s expectation must be favorable, or he would not be willing to make the investment. So the decision to purchase a risky financial security(NPV=0) is a decision to take risk in order to have a chance at a higher return than would be earned by a safe investment, such as depositing the money in an insured savings account.

Time value of money definition

It is almost impossible to overstate the importance of the net present value concept. NPV appears in connection with virtually the time value of money, and virtually all financial decisions in some way or another involve the assessment of the net present value associated with each of several courses of action.

The time value of money is the greater benefit of receiving money now rather than an identical sum later. It is founded on time preference.

The time value of money explains why interest is paid or earned: Interest, whether it is on a bank deposit or debt, compensates the depositor or lender for the time value of money.

It also underlies investment. Investors are willing to forgo spending their money now only if they expect a favorable return on their investment in the future, such that the increased value to be available later is sufficiently high to offset the preference to have money now.

Time Value of Money Formula

Below are the time value of money formula;

  • 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 formula for the time value of money; 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 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 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 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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I hope you now understand what is times values of money, it’s definition and time value of money formula 

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