# Present value of an annuity

## What is the present value of an annuity?

The present value of a series of future promises to pay or receive an annuity at a specified interest rate. 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.

## Explanation

As in the case of a 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 payment 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-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 4 years. The appropriate interest or discount rate is 12%. To solve this, we can construct a table that determines the present values of each of the receipts:

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Thus, 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 (Table 2). For example, the present value of the dollar received at the end of year 4 when discounted back 4 years is \$0.63552. It must be discounted back 4 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 3 periods; the dollar received at the end of year 2 must be discounted back 2 periods; and so forth.

Table 2

As with the calculation of the future value of an annuity, we can use prepared tables. The following table (Table 4) is the example of such a table.

Table 4

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 \$ I .00 to be received at the end of each of the next 4 years when discounted back at 12% is 3.03735, the value we 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:

### Formula

Present value of an annuity = Factor x Amount of the annuity

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

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### Determining the Present Value

To demonstrate how to calculate the present value of’ an annuity, assume that you were offered an investment that paid \$2,000 a year at the end of each of the next 10 years. How much would you pay for it if you wanted 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 discountéd back at 8%. This amount is \$13,420.16, determined as follows:

Present value of an annuity = Factor x Amount of the annuity

= 6.71008 x \$2,000

= \$13,420.16

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

### 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; we will assume that you will make 30 yearly payments at the end of each of the next 30 years. (Most require monthly payments.) How much will your yearly-payments be?

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

Present value of an annuity = Factor x Amount of the annuity

Amount of the annuity = Present value of an annuity / Factor

= \$80,000 / 9.42691

= \$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 rate is 12% annually, or 3% per quarter. How many payments will be required to pay off the loan? In this case, 40 payments are required, determined as follows:

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Present value of an annuity = Factor x Amount of the annuity

Factor = Present value of an annuity / Amount of the annuity

= \$100,000 / \$4,326.24

= 23.11477

Looking down the 3% column in Table 2 we find the factor 23.11477 at the fortieth-period row. Thus 40 quarterly payments are needed to pay off the loan.

### 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 2 cash flows discounted at 10%. That value is \$122,296, as determined below:

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