Time Value of Money - Six Functions of a Dollar
Using Assessors’ Handbook Section 505 (Capitalization Formulas and Tables)
Appraisal Training: Self-Paced Online Learning Session

Lesson 4: Future Worth of $1 Per Period
(Assessors’ Handbook 505, Column 2)

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:

  • Explains the function’s meaning and purpose,
  • Provides the formula for the calculation of FW$1/P factor, and
  • Contains practical examples of how to apply the FW$1/P factor.

FW$1/P: Meaning and Purpose

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.)


Formula for Calculating FW$1/P

The formula for the calculation of the FW$1/P factors is

Image of an equation showing that the future 
								worth of one dollar per period factor is equal to the quantity 1 plus i raised to the power n minus 1 over i

Where:

  • FW$1/P = Future Worth of $1 per Period Factor
  • i = Periodic Interest Rate, often expressed as an annual percentage rate
  • n = Number of Periods, often expressed in years


In order to calculate the annual FW$1/P factor for 4 years at an annual interest rate of 6%, use the formula below:

Image of an equation showing that the future worth of one dollar per period factor 
								is equal to the quantity 1 plus i raised to the power n minus 1 over i. The value for i is 0.06 (six percent, the annual periodic rate), 
								the value for n is 4 (four years) and the final result is 4.374617

Viewed on a timeline:

Image of a timeline showing how deposits of $1 at the end of each year for 4 years 
								would compound at an annual interest of 6 percent with annual compounding to $4.374616 at the end of year 4

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.


Image of a compound interest table (AH 505, page 33) highlighting the future worth 
								of one dollar per period factor for 4 years with annual compounding at an annual interest rate of 6 percent. The highlighted factor is 4.374616

Link to AH 505, page 33



Practical Applications of FW$1/P

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:

  • FV = PMT × FW$1/P (10%, 3 yrs, annual)
  • FV = $1,000 × 3.310000
  • FV = $3,310
  • Find the annual FW$1/P factor (annual compounding) for 10% and a term of 3 years. In AH 505, page 49, go down 3 years and across to column 2 to find the correct factor of 3.310000.
  • The future value of $3,310 is equal to the periodic payment of $1,000 multiplied by the factor.
Image of a compound interest table (AH 505, page 49) highlighting the future 
								worth of one dollar per period factor for 3 years with annual compounding at an annual interest rate of 10 percent. The highlighted factor is 3.310000

Link to AH 505, page 49

Viewed on a timeline:

Image of a timeline showing three payments of $1,000 at the end of years 1, 2 and 3, 
								with each payment multiplied by the appropriate future worth of one dollar factor to arrive at its future value at the end of year 3. 
								The three future values are then summed to show the total future value of the three payments at the end of year 3.
								A payment of $1,000 at the end of year 1 multiplied by 1.210000, the future worth of one dollar factor for 2 years at an annual interest rate of 10 percent, 
								given annual compounding.  The resulting future value is equal to $1,210.
								A payment of $1,000 at the end of year 2 multiplied by 1.100000, the future worth of one dollar factor for 1 year at an annual interest rate of 10 percent, 
								given annual compounding. The resulting future value is equal to $1,100.
								A payment of $1,000 at the end of year 3 multiplied by 1, indicating no compounding, because the payment already occurs at the end of year 3. 
								The resulting future value is simply $1,000.
								Adding the three future values, the total future value of the three payments is equal to $3,310

The problem could have been solved by using the FW$1 factor applicable to each payment, but it would have taken 4 calculations.

  • End-of-year 1 payment FW = $1,000 × 1.210000* = $1,210
    End-of-year 2 payment FW = $1,000 × 1.100000** = $1,100
    End-of-year 3 payment FW = $1,000 × 1.000000 (no compounding) = $1,000
    Total value at end of year 3 = $3,310

    * 1.210000 = FW$1 (10%, 2 years, annual) (AH 505, page 49, column 1)
    ** 1.100000 = FW$1 (10%, 1 year, annual) (AH 505, page 49, column 1)

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:

  • FV = PMT × FW$1/P (7.50%, 23 yrs, annual)
  • FV = $13,000 × 57.027895
  • FV = $741,363
  • Find the annual FW$1/P factor (annual compounding) for 7.50% and a term of 23 years. In AH 505, page 39, go down 23 years and across to column 2 to find the correct factor of 57.027895.
  • The future value of $741,363 is equal to the payment of $13,000 multiplied by the factor.
Image of a compound interest table (AH 505, page 39) highlighting the future worth 
								of one dollar factor for 23 years with annual compounding at an annual interest rate of 7.5 percent. The highlighted factor is 57.027895.

Link to AH 505, page 39



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:

  • FV = PMT × FW$1/P (6%, 10 yrs, monthly)
  • FV = $1,500 × 163.879347
  • FV = $245,819

After 30 years:

  • FV = PMT × FW$1/P (6%, 30 yrs, monthly)
  • FV = $1,500 × 1,004.515042
  • FV = $1,506,773
  • Find the monthly FW$1/P factors (monthly compounding) for 6% at terms of 10 and 30 years, respectively. In AH 505, go to page 32, column 2, to find the correct factors of 163.879347 and 1,004.515042.
  • The future values of $245,819 and $1,506,773, respectively, are equal to the payment of $1,500 multiplied by the appropriate factors. What a difference 20 years makes.
Image of a compound interest table (AH 505, page 32) highlighting the future worth 
								of one dollar per period factor for 10 and 30 years with annual compounding at an annual interest rate of 7.5 percent. 
								The highlighted factors are 163.879347 and 1004.515042 respectively

Link to AH 505, page 32



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

  • FV = PMT × FW$1/P (7%, 10 yrs, annual)
  • FV = $20,000 × 13.816448
  • FV = $276,329
  • Find the correct FW$1/P factor (AH 505, page 37, column 2) to calculate the future value of the payments after the first 10 years; the correct factor is 13.816448.
  • After the first 10 years the future value is $276,329.
Image of a compound interest table (AH 505, page 37) highlighting the future worth 
								of one dollar per period factor for 10 years with annual compounding at an annual interest rate of 7 percent. The highlighted factor is 13.816448

Link to AH 505, page 37

Part II: End of 20 years (final answer)

  • FV = PMT × FW$1/P (8%, 10 yrs, annual)
  • FV = $276,329 × 2.158925
  • FV = $596,574
  • Use the future value calculated in Part I ($276,329) as the present value in the next calculation.
  • Find the FW$1 factor (AH 505, page 41, column 1); the correct factor is 2.158925.
  • Calculate the final future value (at the end of 20 years) using the formula:
Image of a compound interest table (AH 505, page 41) highlighting the future worth 
								of one dollar per period factor for 10 years with annual compounding at an annual interest rate of 8 percent. The highlighted factor is 2.158925

Link to AH 505, page 41



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:
Image of an equation showing that the future value is equal to the payment multiplied 
								by the future worth of one dollar per period factor at an annual interest rate of 12 percent with monthly compounding. 
								The equation is rearranged to show that the desired future worth of one dollar per period factor would be equal to the future value divided by the payment amount.
								Given a future value of $8,000 and a payment amount of $185.71, the value of the future worth of one dollar per period factor is calculated as 43.077917

  • In AH 505, page 56, column 2, find the FW$1/P factor that is closest to 43.077917; the correct factor is 43.076878.
  • In order to save $8,000, you will need to deposit $185.71 for approximately 3 years or 36 months.
Image of a compound interest table (AH 505, page 56) highlighting the future worth 
								of one dollar per period factor for 3 years with monthly compounding at an annual interest rate of 12 percent. The highlighted factor is 43.076878

Link to AH 505, page 56