At some point in your life, you may have had to make a series of fixed payments over a period of time - such as rent or car payments - or have received a series of payments over a period of time, such as bond coupons. These are called annuities. If you understand the time value of money, you're ready to learn about annuities and how their present and future values are calculated.
What Are Annuities?
Annuities are essentially a series of fixed payments required from you or paid to you at a specified frequency over the course of a fixed time period. The most common payment frequencies are yearly, semi-annually (twice a year), quarterly and monthly. There are two basic types of annuities: ordinary annuities and annuities due.
- Ordinary Annuity: Payments are required at the end of each period. For example, straight bonds usually pay coupon payments at the end of every six months until the bond's maturity date.
- Annuity Due: Payments are required at the beginning of each period. Rent is an example of annuity due. You are usually required to pay rent when you first move in at the beginning of the month, and then on the first of each month thereafter.
Since the present and future value calculations for ordinary annuities and annuities due are slightly different, we will first discuss the present and future value calculation for ordinary annuities.
Calculating the Future Value of an Ordinary Annuity
If you know how much you can invest per period for a certain time period, the future value of an ordinary annuity formula is useful for finding out how much you would have in the future by investing at your given interest rate. If you are making payments on a loan, the future value is useful in determining the total cost of the loan.
Let's now run through Example 1. Consider the following annuity cash flow schedule:
To calculate the future value of the annuity, we have to calculate the future value of each cash flow. Let's assume that you are receiving $1,000 every year for the next five years, and you invested each payment at 5%. The following diagram shows how much you would have at the end of the five-year period:
Since we have to add the future value of each payment, you may have noticed that if you have an ordinary annuity with many cash flows, it would take a long time to calculate all the future values and then add them together. Fortunately, mathematics provides a formula that serves as a shortcut for finding the accumulated value of all cash flows received from an ordinary annuity:
C = Cash flow per period
i = interest rate
n = number of payments
Using the above formula for Example 1 above, this is the result:
Note that the 1 cent difference between $5,525.64 and $5,525.63 is due to a rounding error in the first calculation. Each value of the first calculation must be rounded to the nearest penny - the more you have to round numbers in a calculation, the more likely rounding errors will occur. So, the above formula not only provides a shortcut to finding FV of an ordinary annuity but also gives a more accurate result.
Calculating the Present Value of an Ordinary Annuity
If you would like to determine today's value of a future payment series, you need to use the formula that calculates the present value of an ordinary annuity. This is the formula you would use as part of a bond pricing calculation. The PV of an ordinary annuity calculates the present value of the coupon payments that you will receive in the future.
For Example 2, we'll use the same annuity cash flow schedule as we did in Example 1. To obtain the total discounted value, we need to take the present value of each future payment and, as we did in Example 1, add the cash flows together.
Again, calculating and adding all these values will take a considerable amount of time, especially if we expect many future payments. As such, we can use a mathematical shortcut for PV of an ordinary annuity.
C = Cash flow per period
i = interest rate
n = number of payments
The formula provides us with the PV in a few easy steps. Here is the calculation of the annuity represented in the diagram for Example 2:
Calculating the Future Value of an Annuity Due
When you are receiving or paying cash flows for an annuity due, your cash flow schedule would appear as follows:
Since each payment in the series is made one period sooner, we need to discount the formula one period back. A slight modification to the FV-of-an-ordinary-annuity formula accounts for payments occurring at the beginning of each period. In Example 3, let's illustrate why this modification is needed when each $1,000 payment is made at the beginning of the period rather than at the end (interest rate is still 5%):
Notice that when payments are made at the beginning of the period, each amount is held longer at the end of the period. For example, if the $1,000 was invested on January 1 rather than December 31 each year, the last payment before we value our investment at the end of five years (on December 31) would have been made a year prior (January 1) rather than the same day on which it is valued. The future value of annuity formula would then read:
Calculating the Present Value of an Annuity Due
For the present value of an annuity due formula, we need to discount the formula one period forward as the payments are held for a lesser amount of time. When calculating the present value, we assume that the first payment was made today.
We could use this formula for calculating the present value of your future rent payments as specified in a lease you sign with your landlord. Let's say for Example 4 that you make your first rent payment at the beginning of the month and are evaluating the present value of your five-month lease on that same day. Your present value calculation would work as follows:
Of course, we can use a formula shortcut to calculate the present value of an annuity due:
Recall that the present value of an ordinary annuity returned a value of $4,329.48. The present value of an ordinary annuity is less than that of an annuity due because the further back we discount a future payment, the lower its present value: each payment or cash flow in an ordinary annuity occurs one period further into the future.
Now you can see how annuity affects how you calculate the present and future value of any amount of money. Remember that the payment frequencies, or number of payments, and the time at which these payments are made (whether at the beginning or end of each payment period) are all variables you need to account for in your calculations.