This lesson discusses the Future Worth of $1 per Period (FW$1/P); one of six compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The lesson:
The FW$1/P factor is the amount to which a series of periodic payments of $1 will compound at periodic interest rate i over n periods, assuming payments occur at the end of each period.
The FW$1/P factor is used to compound a series of periodic equal payments to their future value. The FW$1/P factors are in column 2 of AH 505.
FW$1/P factors are applicable to ordinary annuity problems. An annuity may be defined as a series of periodic payments, usually equal in amount, and payable at the end of the period. (See Lesson 10 for further discussion of annuities.)
The formula for the calculation of the FW$1/P factors is
Where:
In order to calculate the annual FW$1/P factor for 4 years at an annual interest rate of 6%, use the formula below:
Viewed on a timeline:
On the timeline, the deposits of $1 are shown as negative because from the point of view of a depositor they would be cash outflows. The future values are shown as positive because they would be cash inflows. The depositor gives up money at the end of each year in order to receive money at the end of year 4.
To locate the FW$1/P factor, go to page 33 of AH 505, go down 4 years and across to column 2. The correct factor is 4.374616.
Example 1:
What is the future value of 3 payments of $1,000 with the payments made at the end of each of the next 3 years?
(Assume an annual interest rate of 10%.)
Solution:
Viewed on a timeline:
The problem could have been solved by using the FW$1 factor applicable to each payment, but it would have taken 4 calculations.
Using the FW$1/P annuity factor simplifies the calculation. Annuity factors are essentially shortcuts that can be used when cash flows or payments are equal and at regular intervals.
Example 2:
You deposit $13,000 at the end of each year for 23 years. If the account earns an annual rate of 7.50%,
compounded annually, how much will be in the account after 23 years?
Solution:
Example 3:
Mr. Foresight deposits $1,500 at the end of each month into a retirement account that returns an annual rate of 6%,
compounded monthly. How much will he have after 10 years? After 30 years?
Solution:
After 10 years:
After 30 years:
Example 4:
Mrs. Foresight invests $20,000 in a 401k account at the end of each year for 10 years, earning an annual rate of 7%, compounded annually.
At the end of 10 years, she invests the lump-sum balance for another 10 years, earning an annual rate of 8%, compounded annually.
How much will Mrs. Foresight have at the end of 20 years?
(Hint: This problem combines the FW$1/P and the FW$1)
Solution:
This is a two-part problem.
Part I: First 10 years
Part II: End of 20 years (final answer)
Example 5:
You want to save $8,000 to buy a car. You will deposit $185.71 at the end of every month.
Your first deposit will be a month from today. If your account pays an annual interest rate of 12%,
compounded monthly, approximately how many months will it take to save $8,000?
Solution: