# Lesson 3: Present Worth of $1

(Assessors’ Handbook 505, Column 4)

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:

- Explains the function’s meaning and purpose,
- Provides the formula for the calculation of
*PW*$1 factor, - Shows how to calculate the present value of multiple payments, and
- Contains practical examples of how to apply the
*PW*$1 factor.

*PW*$1: Meaning and Purpose

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.

### Formula for Calculating *PW*$1 Factors

The formula for the calculation of the *PW*$1 factors is:

Where:

*PW*$1 = Present Worth of $1 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 *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.

### Practical Applications of *PW*$1

**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:**

*PV*=*FV*×*PW*$1 (7%, 10*yrs*,*annual*)*PV*= $15,000 × 0.508349*PV*= $7,625

- Find the annual
*PW*$1 factor (annual compounding) for 7% at a term of 10 years. In AH 505, page 37, go down 10 years and across to column 4 to find the correct factor of 0.508349. - The present value of $7,625 is equal to the future value of $15,000 multiplied by the factor.

**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:**

*PV*=*FV*×*PW*$1 (6%, 5*yrs*,*annual*)*PV*= $25,000 × 0.747258*PV*= $18,681

- Find the annual
*PW*$1 factor (annual compounding) for 6% at a term of 5 years. In AH 505, page 33, go down 5 years and across to column 4 to find the correct factor of 0.747258. - The present value of $18,681 is equal to the future value of $25,000 multiplied by the factor.

**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:**

*PV*=*FV*×*PW*$1 (5%, 3*yrs*,*annual*)*PV*= $10,000 × 0.863838*PV*= $8,638

- Find the annual
*PW*$1 factor (annual compounding) for 5% at a term of 3 years. In AH 505, page 29, go down 3 years and across to column 4 to find the correct factor of 0.863838. - The present value of $8,638 is equal to the future value of $10,000 multiplied by the factor.

**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:**

*PV*=*FV*×*PW*$1 (10%, 10*yrs*,*annual*)*PV*= $500,000 × 0.285543*PV*= $192,772

- Find the annual
*PW*$1 factor (annual compounding) for 10% at a term of 10 years. In AH 505, page 49, go down 10 years and across to column 4 to find the correct factor of 0.385543. - The present value of $192,772 is equal to the future value of $500,000 multiplied by the factor.

**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:**

*PV*=*FV*×*PW*$1 (8.50%, 25*yrs*,*annual*)*PV*= $1,000 × 0.130094*PV*= $130.09

- Find the annual
*PW*$1 factor (annual compounding) for 8.50% at a term of 25 years. In AH 505, page 43, go down 25 years and across to column 4 to find the correct factor of 0.130094. - The present value of $130.09 is equal to the future value of $1,000 multiplied by the factor. The bond should sell for $130.09.

### Multiple Payments and *PW*$1

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.

### Practical Applications of *PW*$1 with Multiple Payments

**Example 1:**

Consider the following 3 payments:

- $10,000 at the end of the first year
- $15,000 at the end of the second year
- $20,000 at the end of the third year

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).