This lesson discusses the Present Worth of $1 (PW$1); one of six compound interest functions presented in Assessors’ Handbook Section 505 (AH 505), Capitalization Formulas and Tables. The Lesson:
The PW$1 factor is the amount that must be deposited today to grow to $1 in the future, given periodic interest rate i and n periods.
The PW$1 factor is used to discount a single future amount to its present amount. The PW$1 factors are in column 4 of AH 505.
The PW$1 factor can be thought of as the opposite of the FW$1 factor; mathematically, the PW$1 and FW$1 factors are reciprocals:
Whereas the FW$1, discussed in Lesson 1, provides the future value of a single present amount, the PW$1 provides the present value of a single future amount.
The value of the PW$1 factor will always be less than $1, explicitly demonstrating that a dollar to be received in the future is worth less than a dollar today.
The formula for the calculation of the PW$1 factors is:
Where:
In order to calculate the PW$1 factor for 4 years at an annual interest rate of 6%, use the formula below:
Viewed on a timeline:
A depositor would be willing to give up $0.792094 today (shown as negative on the timeline) in order to receive $1 at the end of 4 years (shown as positive).
To locate the factor, go to page 33 of AH 505, go down 4 years, and then across to column 4. The correct PW$1 factor is 0.792094.
Example 1:
How much must be deposited today in order to have $15,000 at the end of 10 years,
assuming an annual interest rate of 7% with annual compounding?
Solution:
Example 2:
Someone promises to pay you $25,000 in 5 years. Given an annual interest rate of 6% with annual compounding,
how much should you pay for this promise today?
Solution:
Example 3:
If you want to have $10,000 after 3 years, and you can invest at an annual rate of 5% compounded annually,
how much should you invest today?
Solution:
Example 4:
Ten years from now, you will need to make a lump-sum payment of $500,000. Assuming annual compounding,
how much should you invest today in order to cover the future payment? The annual interest rate is 10%.
Solution:
Example 5:
Acme Enterprises promises to pay the holders of its most recent bond issue $1,000 per bond at the end of 25 years
(there are no annual or semi–annual interest payments; this is called a "zero coupon" bond).
If the annual interest rate is 8.50%, assuming annual compounding, how much should each bond sell for when issued?
Solution:
We have calculated the present value of single amounts or payments, using the PW$1 factors.
Many problems involve more than one payment, making it necessary to calculate the present value of multiple payments–that is, the present value of a stream of payments. Determining the present value of multiple payments is a straightforward extension of the single-payment situation.
When we calculated the present value of a single future payment, we multiplied the future payment by the appropriate PW$1 factor. This discounted the future payment to its present value.
If there is more than one future payment, multiple each payment by the appropriate PW$1 factor and add the present values. The sum of the present values is the total present value of the stream of future payments.
Example 1:
Consider the following 3 payments:
At an annual interest rate of 5%, what is the total present value of the 3 payments?
Solution:
Calculate the present value of each payment using the PW$1 factors and add those present values.
The sum is the present value of all 3 payments.
Thus:
Viewed on a timeline:
A person would be willing to pay $40,406 now (shown as negative on the timeline) in order to receive the three future payments of $10,000, and $15,000, and $20,000 (shown as positive).