Lesson PAYMENT FOR A PRESENT VALUE

Source code of 'PAYMENT FOR A PRESENT VALUE'

This Lesson (PAYMENT FOR A PRESENT VALUE) was created by by Theo(13342)

See BASIC FORMULAS AND ASSUMPTIONS USED IN FINANCIAL FORMULAS for an explanation of interest rates, compound interest versus simple interest, number of time periods, definition of time periods and time points, and any other topic basic to the understanding of these financial formulas and how to use them.

PAYMENT FOR A PRESENT VALUE

{{{ PMT(PV) = (PV / ((((1 - (1 / ((1+i)^n))))/i))) }}}

PMT = Payment per Time Period	
PV = Present Value
i = Interest Rate per Time Period
n = Number of Time Periods

This formula deals with calculating the value of the payment used in a series of payments for a present value.

If the series of payments are specified as being made at the beginning of the time period, then a special adjustment to this formula is used that will be shown at the end of this presentation.

EXAMPLE 1

You have taken out a personal loan for $10,000 that will require yearly payments for a period of 5 years at the interest rate of 20% a year.
How much is each payment?

n = number of time periods = 5 (no adjustment is necessary since the number of time periods and the number of years is the same)
i = 20% / 100% = .2 per year (percent interest per year is divided by 100% to get the interest rate per year.  Since time periods and years are the same, no further adjustment is necessary).  
PV = $10,000 (take as given except you need to remove the $ sign and the commas before entering into the formula).
Formula becomes:

{{{ PMT(PV) = (10000 / ((((1 - (1 / ((1.2)^5))))/.2))) }}} = 3343.797033 = $3,343.80

EXAMPLE 2

You have taken out a personal loan for $10,000 that will require monthly payments for a period of 5 years at the interest rate of 20% a year.
How much is each payment?

n = number of time periods = 5 * 12 = 60 (you are given number of years but you want number of months so you need to multiply by 12 to get monthly time periods).
i = 20% / 100% / 12 = .0166666666 per month (percent interest rate per year is divided by 100% to get the interest rate per year and then divided by 12 to get the interest rate per month).  
FV = $10,000 (take as given except you need to remove the $ sign and the commas before entering into the formula).
Formula becomes:

{{{ PMT(PV) = (10000 / ((((1 - (1 / ((1.0166666666)^60))))/.0166666666))) }}} = 264.9388327 = $264.94

BEGINNING OF TIME PERIOD PAYMENT VERSUS END OF TIME PERIOD PAYMENT

The basic assumption is that payments are made at the end of the time period.
If you are given that the payments are being made at the beginning of the time period, then the payment for a present value formula needs to be adjusted as follows:

PAYMENT FOR A PRESENT VALUE WHEN THE PAYMENT IS MADE AT THE END OF EACH TIME PERIOD.

{{{ PMT(PV) = (PV / ((((1 - (1 / ((1+i)^n))))/i))) }}}

PAYMENT FOR A PRESENT VALUE WHEN THE PAYMENT IS MADE AT THE BEGINNING OF EACH TIME PERIOD.

{{{ PMT(PV) = ((PV / ((((1 - (1 / ((1+i)^n))))/i))))/(1+i) }}}

As you can see, the end of time period payments for a present value formula is divided by (1+i) to get the beginning of time period payments for a present value.

An example of what happens is shown below:

Payment for a present value of $10,000 at 20% interest per time period for 2 time periods assuming end of time period payments (payment calculated to be 6545.45 per time period):
start of time period 1 principal = 0000.00 + payment 0000.00 = remaining balance 0000.00
end of time period 1 principal = 0000.00 * 1.2 = 0000.00 + payment 6545.45 = remaining balance 6545.45
end of time period 2 principal = 6545.45 * 1.2 = 7854.54 + payment 6545.45 = remaining balance 14399.99
Future Value of payments is $14,399.99 with rounding = $14,400.00 without rounding.
Present Value of payments is $9,999.99 with rounding = $10,000 without rounding.

Payment for a present value of $10,000 at 20% interest per time period for 2 time periods assuming beginning of time period payments (payment calculated to be 5454.55 per time period.
start of time period 1 principal = 0000.00 + payment 5454.55 = remaining balance 5454.55
end of time period 1 principal = 5454.55 * 1.2 = 6545.46 + payment 5454.55 = remaining balance 12000.01
end of time period 2 principal = 12000.01 * 1.2 = 14400.01 + payment 0000.00 = remaining balance 14400.01
Future Value of payments is $14,400.01 with rounding = $14,400.00 without rounding.
Present Value of payments = $10,000.00 with rounding = $10,000.00 without rounding.

Note:  I rounded the payment to the nearest penny.  In this particuar case it didn't make any difference since the present value of payments with rounding was equal to the present value of payments without rounding ($10,000).
The actual end of time period payment was 6545.45454545
The actual beginning of time period payment was 5454.545454.

Since the interest rate was 20% per year which translates to an interest rate of .2 per year, if I divide 6545.45454545� by 1.2 I get 5454.54545454�. which shows that Beginning of Time Period Payment for a Present Value is the same as End of Time Period Payment for a Present Value divided by (1 + the interest rate) per time period.