Dr. NICHOLAS WATSON

Time Value of Money (TVM)

 

Adopted from: http://accountingexplained.com/misc/tvm/

 

Time value of money is the concept that the value of a dollar to be received in future is less than the value of a dollar on hand today. One reason is that money received today can be invested thus generating more money. Another reason is that when a person opts to receive a sum of money in future rather than today, he is effectively lending the money and there are risks involved in lending such as default risk and inflation. Default risk arises when the borrower does not pay the money back to the lender. Inflation is the rise in general level of prices.

Time value of money principle also applies when comparing the worth of money to be received in future and the worth of money to be received in further future. In other words, TVM principle says that the value of given sum of money to be received on a particular date is more than same sum of money to be received on a later date.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Few of the basic terms used in time value of money calculations are:

 

Present Value

 

When a future payment or series of payments are discounted at the given rate of interest up to the present date to reflect the time value of money, the resulting value is called present value.

 

Present Value of a Single Sum of Money

 

Present value of a future single sum of money is the value that is obtained when the future value is discounted at a specific given rate of interest. In the other words present value of a single sum of money is the amount that, if invested on a given date at a specific rate of interest, will equate the sum of the amount invested and the compound interest earned on its investment with the face value of the future single sum of money.

 

Formula

 

The formula to calculate present value of a future single sum of money is:

 

Present Value (PV) =

Future Value (FV)

(1 + i)n

 

Where,

   i is the interest rate per compounding period; and
   n are the number of compounding periods.

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Present Value of an Annuity

 

An annuity is a series of evenly spaced equal payments made for a certain amount of time. There are two basic types of annuity known as ordinary annuity and annuity due. Ordinary annuity is one in which periodic payments are made at the end of each period. Annuity due is the one in which periodic payments are made at the beginning of each period.

 

The present value an annuity is the sum of the periodic payments each discounted at the given rate of interest to reflect the time value of money. Alternatively defined, the present value of an annuity is the amount which if invested at the start of first period at the given rate of interest will equate the sum of the amount invested and the compound interest earned on the investment with the product of number of the periodic payments and the face value of each payment.

 

Formula

 

Although the present value (PV) of an annuity can be calculated by discounting each periodic payment separately to the starting point and then adding up all the discounted figures, however, it is more convenient to use the 'one step' formulas given below.

 

PV of an Ordinary Annuity = R ×

1 − (1 + i)-n

i

 

PV of an Annuity Due = R ×

1 − (1 + i)-n

 × (1 + i)

i

 

Where,

   i is the interest rate per compounding period;
   n are the number of compounding periods; and
   R is the fixed periodic payment.

 

 

Future Value

 

Future value is amount that is obtained by enhancing the value of a present payment or a series of payments at the given rate of interest to reflect the time value of money.
 

Future Value of a Single Sum of Money

 

Future value of a present single sum of money is the amount that will be obtained in future if the present single sum of money is invested on a given date at the given rate of interest. The future value is the sum of present value and the compound interest.

 

Formula

 

The future value of a single sum of money is calculated by using the following formula.

Future Value (FV) = Present Value (PV) × (1 + i)n

 

Where,

   i is the interest rate per compounding period; and
   n are the number of compounding periods.

 

 

 

Future Value of an Annuity

 

The future value of an annuity is the value of its periodic payments each enhanced at a specific rate of interest for given number of periods to reflect the time value of money. In other words, future value of an annuity is equal to the sum of face value of periodic annuity payments and the total compound interest earned on all periodic payments till the future value point.

 

Formula

 

There are two types of annuity. The one in which payments occur at the end of each period is called ordinary annuity and the other in which payments occur at the beginning of each period is called annuity due. Both types have different formulas for future value calculation:

FV of Ordinary Annuity = R ×

(1 + i)n − 1

i

 

FV of Annuity Due = R ×

(1 + i)n − 1

 × (1 + i)

i

In the above formulas,

   i is the interest rate per compounding period;
   n are the number of compounding periods; and
   R is the fixed periodic payment.

 

 

 

Interest

 

Interest is charge against use of money paid by the borrower to the lender in addition to the actual money lent.
 

 

Simple vs. Compound Interest Calculation

 

Interest is the charge against the use of money by the borrower. The same is profit earned by the lender of money. The amount which is invested in a bank in order to earn interest is called principal. The interest rate is normally expressed in percentage and represents the dollar interest earned per $100 of principal in a specific time, usually a year. Simple interest and compound interest are the two types of interest based on the way they are calculated.

 

Simple Interest

 

Simple interest is charged only on the principal amount. The following formula can be used to calculate simple interest:

Simple Interest (Is) = P × i × t

Where,

   P is the principle amount;
   i is the interest rate per period;
   t is the time for which the money is borrowed or lent.

 

Compound Interest

 

Compound interest is charged on the principal plus any interest accrued till the point of time at which interest is being calculated. In other words, compound interest system works as follows:

• Interest for the first period charged on principle amount.

• For the second period, its charged on the sum of principle amount and interest charged during the first period.

• For the third period, it is charged on the sum of principle amount and interest charged during first and second period, and so on ...

It can be proved mathematically, that the interest calculated as per above procedure is given by the following formula:

Compound Interest (Ic) = P × (1 + i) n – P

Where,

   P is the principle amount;
   i is the compound interest rate per period;
   n are the number of periods.

 

 

 

Application of Time Value of Money Principle

 

There are many applications of time value of money principle. For example, we can use it to compare the worth of cash flows occurring at different times in future, to find the present worth of a series of payments to be received periodically in future, to find the required amount of current investment that must be made at a given interest rate to generate a required future cash flow, etc.