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This Lesson (PAYMENT FOR A FUTURE VALUE) was created by by Theo(13342)
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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 FUTURE VALUE PMT = Payment per Time Period FV = Future 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 future value. Unless otherwise specified, payments are assumed to be at the end of the time period specified. 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 that requires you to make yearly payments that will earn the lender a total of $10,000 by the end of the loan period of 5 years at the rate of 20% per 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 rate of 20% 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). FV = $10,000 (take as given except you need to remove the $ sign and the commas before entering into the formula). Formula becomes: EXAMPLE 2 You have taken out a personal loan that requires you to make monthly payments that will earn the lender a total of $10,000 by the end of the loan period of 5 years at the rate of 20% per 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 12 to get monthly time periods). i = 20% / 100% / 12 = .0166666666 per month (percent interest rate of 20% 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: WHAT YOU PAY VERSUS WHAT THE LENDER MAKES In example 1 your payments were $1,343.80 per year for 5 years. The total of all your payments is therefore 5 * $1,343.80 = $6,719.00 approximately (some rounding error in there). The future value of the loan, however, is $10,000. The difference is what the lender makes by re-investing the payments at the rate of return of the investment. Similarly, in example 2 your payments were $98.27 per month. The total of all your payments is therefore 5 * 12 * $98.27 = $5,892 approximately (some rounding error in there). The future value of the loan, however, is $10,000. The difference is what the lender makes by re-investing the payments at the rate of return of the investment. 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 future value formula needs to be adjusted as follows: PAYMENT FOR A FUTURE VALUE WHEN THE PAYMENT IS MADE AT THE END OF EACH TIME PERIOD. PAYMENT FOR A FUTURE VALUE WHEN THE PAYMENT IS MADE AT THE BEGINNING OF EACH TIME PERIOD. As you can see, the end of time period payments for a future value formula is divided by (1+i) to get the beginning of time period payments for a future value. An example of what happens is shown below: Payment for a future value of $10,000 at 20% interest per time period for 2 time periods assuming end of time period payments (payment calculated to be $4,545.45 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 4545.45 = remaining balance 4545.45 end of time period 2 principal = 4545.45 * 1.2 = 545454 + payment 4545.45 = remaining balance 9999.99 Future Value of payments is $9,999.00 with rounding = $10,000 without rounding. Payment for a future value of $10,000 at 20% interest per time period for 2 time periods assuming beginning of time period payments (payment calculated to be 3787.88 per time period). start of time period 1 principal = 0000.00 + payment 3787.88 = remaining balance 3787.88 end of time period 1 principal = 3787.88 * 1.2 = 4545.46 + payment 3787.88 = remaining balance 8333.34 end of time period 2 principal = 8333.34 * 1.2 = 1000.01+ payment 0000.00 = remaining balance 10000.01 Future Value of payments is $10,000.01 with rounding = $10,000.00 without rounding. Note: I rounded the payment to the nearest penny. This caused an insignificant error of 1 penny in the future value of payments. If I had not rounded the payments to the nearest penny, the future value of payments would have been exactly $10,000. The actual end of time period payment was 4545.4545454545 without rounding. The actual beginning of time period payment was 3787.87878787 without rounding. Since the interest rate was 20% per year which translates to an interest rate of .2 per year, if I divide 4545.45454545… by 1.2 I get 3787.8768787…. which shows that Beginning of Time Period Payment for a Future Value is the same as End of Time Period Payment for a Future Value divided by (1 + the interest rate) per time period. This lesson has been accessed 10887 times. |
