Present Value of an Annuity

The value today of a series of equal payments or receipts to be made or received on specified future dates is called the present value of an annuity. As in the case of the future value of an annuity, the receipts or payments are made in the future. Present value is the value today, and future value relates to accumulated future value. Furthermore, the present value of a series of payments or receipts will be less than the total of the same payments or receipts, because cash received in the future is not as valuable as cash received today. On the other hand, the future value of an annuity will be greater than the sum of the individual payments or receipts, because interest is accumulated on the payments. It is important to distinguish between the future value and the present value of an annuity. Again, timelines are helpful in this respect.

Mortgages and certain notes payable in equal installments are examples of present value of an annuity problems. For example, assume that a bank lends you $60,000 today, to be repaid in equal monthly installments over 30 years. The bank is interested in knowing what series of monthly payments, when discounted back at the agreed-upon interest rate, is equal to the present value today of the amount of the loan, or $60,000.

Determining the Present Value of an Annuity

Assume that you want to determine the value today of receiving $1 at the end of each of the next four years. The appropriate interest or discount rate is 12%. To solve this, construct a table that determines the present values of each of the receipts, as shown in Figure 11.4. The exhibit shows that the present value of receiving the four $1.00 payments is $3.03735 when discounted at 12%. Each of the individual dollars was discounted by using the factors in the present value of a single amount table in Table 11.4. For example, the present value of the dollar received at the end of year 4, when discounted back four years, is $0.63552. It must be discounted back four years because the present, or today, is the beginning of year 1. The dollar received at the end of year 3 must be discounted back three periods; the dollar received at the end of year 2 must be discounted back two periods; and so forth.

As with the calculation of the future value of an annuity, you can use prepared tables. Table II in Appendix B is such a table. It is constructed by summing the individual present values of $1 at set interest rates and periods. Thus the factor for the present value of four $1.00 payments to be received at the end of each of the next four years, when discounted back at 12%, is 3.03735, the value that was determined independently.

Problems Involving the Present Value of an Annuity

Problems involving the present value of an annuity can be solved by using the following general formula:

$\text{Present value of an annuity}=\text{Factor}\times \text{Amount of the annuity}$

As long as you know two of the three variables, you can solve for the third. Thus, you can determine the present value of the annuity, the interest rate, the number of periods, or the amount of the annuity.

Determining the Present Value

To demonstrate how to calculate the present value of an annuity, assume that you are offered an investment that pays $2,000 a year at the end of each of the next 10 years. How much would you pay for it if you want to earn a rate of return of 8%? This is a present value problem, because you would pay the value today of this stream of payments discounted back at 8%. This amount is $13,420.16, determined as follows:

$\text{Present value of an annuity}=\text{Factor}\times \text{Amount of the annuity}$

$\mathrm{\$}\mathrm{13,420.16}=6.71008\times \mathrm{\$}\mathrm{2,000}$

Another way to interpret this problem is to say that if you want to earn 8%, it makes no difference whether you keep $13,420.16 today or receive $2,000 a year for 10 years.

A business calculator can be used to compute the present value of an annuity as follows:

Hewlett-Packard Keystrokes:

1. CLEAR ALL.

2. Set P/YR to 1.

3. 2,000 Press PMT

4. 10 Press N

5. 8 Press I/YR

6. Press PV for the answer of $13,420.16.

Determining the Annuity Payment

A common variation of present value problems requires computing the annuity payment. In many cases, these are loan or mortgage problems. For example, assume that you purchase a house for $100,000 and make a 20% down payment. You will borrow the rest of the money from the bank at 10% interest. To make the problem easier, assume that you will make a payment at the end of each of the next 30 years. (Most mortgages require monthly payments.) How much will your yearly payments be?

In this case, you are going to borrow $80,000 ($100,000 × 80%). The yearly payment would be $8,486.34, determined as follows:

$\text{Present value of an annuity}=\text{Factor}\times \text{Amount of the annuity}$

$\text{Amount of the annuity}=\frac{\text{Present value of an annuity}}{\text{Factor}}$

$\mathrm{\$}8,486.34=\frac{\mathrm{\$}80,000}{9.42691}$

The necessary keystrokes to compute the annual payment amount are as follows:

Hewlett-Packard Keystrokes:

1. CLEAR ALL.

2. Set P/YR to 1.

3. 80,000 Press PV

4. 30 Press N

5. 10 Press I/YR

6. Press PMT for the answer of $8,486.34.

Determining the Number of Payments

Assume that the Black Lighting Co. purchased a new printing press for $100,000. The quarterly payments are $4,326.24 and the interest rate is 12% annually, or 3% a quarter. How many payments will be required to pay off the loan? In this case, 40 payments are required, determined as follows:

$\text{Present value of an annuity}=\text{Factor}\times \text{Amount of the annuity}$

$\text{Factor}=\frac{\text{Present value of an annuity}}{\text{Amount of the annuity}}$

$23.11477=\frac{\mathrm{\$}100,000}{\mathrm{\$}4,326.24}$

Looking down the 3% column in Table II (see Appendix B), you find the factor 23.11477 at the fortieth-period row. Thus, 40 quarterly payments are needed to pay off the loan.

The necessary number of payments can also be computed as follows:

Hewlett-Packard Keystrokes:

1. CLEAR ALL.

2. Set P/YR to 1.

3. –100,000 Press PV

4. 4,326.24 Press PMT

5. 3 Press I/YR

6. Press N for the answer of 40 quarters, or 10 years.

Solving Combination Problems

Many accounting applications related to the time value of money involve both single amounts and annuities. For example, say that you are considering purchasing an apartment house. After much analysis, you determine that you will receive net yearly cash flows of $10,000 from rental revenue, less rental expenses, from the apartment. To make the analysis easier, assume that the cash flows are generated at the end of each year. These cash flows will continue for 20 years, at which time you estimate that you can sell the apartment building for $250,000. How much should you pay for the building, assuming that you want to earn a rate of return of 10%?

This problem involves an annuity—the yearly net cash flows of $10,000—and a single amount—the $250,000 to be received once at the end of the twentieth year. As a rational person, the maximum that you would be willing to pay is the value today of these two cash flows discounted at 10%. That value is $122,296, as determined below.

The necessary business calculator keystrokes are as follows:

Hewlett-Packard Keystrokes:

1. CLEAR ALL.

2. Set P/YR to 1.

3. 10,000 Press PMT

4. 250,000 Press FV

5. 20 Press N

6. 10 Press I/YR

7. Press PV for the answer of $122,296.54.

Exercises