# Lesson 6: Present Worth of $1 Per Period

(Assessors’ Handbook 505, Column 5)

This lesson discusses the Present Worth of $1 Per Period (*PW*$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
*PW*$1/*P*factors, and - Shows practical examples of how to apply the
*PW*$1/*P*factor.

*PW*$1/*P*: Meaning and Purpose

The *PW*$1/*P* is the present value of a series of future periodic payments of $1,
discounted at periodic interest rate *i* over *n* periods, assuming the payments occur at the end of each period.
The *PW*$1/*P* is typically used to discount a future level income stream to its present value.

Another way to conceptualize the *PW*$1/*P* is the amount that must be deposited today to fund withdrawals
of $1 at the end of each of the *n* periods at periodic interest rate *i*,
assuming a periodic rate *i* can be earned on the outstanding balance.

This compound interest function, together with the *PW*$1, is the basis of yield capitalization and its primary variant,
discounted cash flow analysis. The *PW*$1/*P* factors are in column 5 of AH 505.

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

The formula for the calculation of the *PW*$1/*P* factors is as follows:

Where:

*PW*$1/*P*= Present 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 *PW*$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 initial deposit of $3.465106 is shown as negative because from the point of view of a depositor it would be a cash outflow. The future values of $1 at the end of each year are shown as positive because they would be cash inflows.

To locate the *PW*$1/*P* factor, go to page 33 of AH 505, go down 4 years and across to column 5. The correct factor is 3.465106.

### Practical Applications of *PW*$1/*P*

**Example 1:**

You’ve admired your neighbor’s vintage car for years, and he’s finally agreed to sell it to you.
He offers you the following payment alternatives:

- Pay $20,000 now, or
- Pay $6,000 at the end of each of the next 4 years with an annual interest rate of 8%

Which is the better alternative?

**Solution:**

*PV*=*PMT*×*PW*$1/*P*(8%, 4*years*,*annual*)*PV*= $6,000 × 3.312127*PV*= $19,873

Calculate the present value of the 4-year payment plan (alternative 2) using the *PW*$1/*P*
factor and compare it to the immediate payment of $20,000 (alternative 1).

- Find the annual
*PW*$1/*P*factor (annual compounding) for 8% at a term of 4 years. In AH 505, page 41, go down 4 years and across to column 5 to find the correct factor of 3.312127. - The present value of $19,873 is equal to the periodic payment of $6,000 multiplied by the factor.
- You want to select the payment alternative with the lowest cost in present-value terms. Because the present value of the four payments ($19,873) is less than the immediate payment of $20,000 (no discounting of the immediate payment is required), the four-payment alternative is preferable after adjusting for the time value of money.

**Example 2:**

You will receive annual payments of $10,000 at the end of each year for the next 15 years with an annual interest rate of 5%. What is the present value of this stream of payments?

**Solution:**

*PV*=*PMT*×*PW*$1/*P*(5%, 15*years*,*annual*)*PV*= $10,000 × 10.379658*PV*= $103,796.58

- Find the annual
*PW*$1/*P*factor (annual compounding) for 5% and 15 years. In AH 505, page 29, go down 15 years and across to column 5 to find the correct factor of 10.379658. - The present value of $103,796.58 is equal to the periodic payment of $10,000 multiplied by the factor.

**Example 3:**

At the end of each year following your retirement, you want to withdraw $20,000 from your 401k retirement account.
You expect to live for 20 years after you retire. Assuming that you can earn an annual interest rate of 6%,
what balance will you need in your retirement account to fund your planned withdrawals?

**Solution:**

*PV*=*PMT*×*PW*$1/*P*(6%, 20*years*,*annual*)*PV*= $20,000 × 11.469921*PV*= $229,398

- Find the annual
*PW*$1/*P*factor (annual compounding) for 6% and 20 years. In AH 505, page 33, go down 20 years and across to column 5 to find the correct factor of 11.469921. - The present value of $229,398 is equal to the annual payment of $20,000 multiplied by the
*PW*$1/*P*factor. When you retire, the balance in your account must equal the present value of the 20 years of planned future withdrawals.

**Example 4:**

Mr. Fortunate has won the $64 million California lottery. He will receive 20 annual payments of
$3,200,000, with the first payment to be received immediately.
Acme Investment Company is offering Mr. Fortunate $30,000,000 for the right to receive his 20 payments.
If the annual interest rate is 8% with annual compounding, should he accept the offer?

**Solution:**

*PV*=*PMT*×*PW*$1/*P*(8%, 19*years*,*annual*)*PV*= $3,200,000 × 9.603599*PV*= $30,731,517- Total
*PV*= $30,731,517 + $3,200,000 = $33,931,517

- Use the
*PW*$1/*P*factor for 19 years to discount the future 19 payments of $3,200,000 (AH 505, page 41, column 5). - Add the initial payment of $3,200,000 (this occurs immediately and is not discounted) to calculate the total present value of the promised payments.
- Acme is offering $30,000,000 for a stream of cash flows valued at $33,931,517 (assuming an 8% discount rate). Mr. Fortunate should decline Acme’s offer.

**Example 5:**

The subscription to your favorite magazine is about to expire. The magazine company offers you three renewal options:

- Pay $100 now for a four-year subscription.
- Pay $32 per year at the end of each year for four years.
- Pay $54 today and another $54 two years from today.

Assuming you want to receive the magazine for at least four more years, if the annual interest rate is 10%, which option is the best deal?

**Solution:**

*PV*=*PMT*×*PW*$1/*P*(10%, 4*years*,*annual*)*PV*= $32 × 3.169865*PV*= $101.44*PV*=*FV*×*PW*$1 (10%, 2*years*,*annual*)*PV*= $54.00 × 0.826446*PV*= $44.63*Total PV*= $54.00 + $44.63 = $98.63

- Determine the present value of each renewal option and select the option with the lowest present value.
- The present value of option 1 is $100; payment is immediate and no discounting is required.
- The present value of option 2 is calculated using the
*PW*$1/*P*factor (AH 505, page 49, column 5). - The present value of option 3 is the initial payment of $54 (no discounting required)
plus the present value of the second payment of $54 discounted for 2 years using the
*PW$1*factor (AH 505, page 49, column 4). Option 3 has the lowest present value and is the best deal.