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 interest from bonds or CDs. These are called annuities (a more generic use of the word—not to be confused with the specific financial product called an annuity, though the two are related). If you understand the time value of money, you are 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. Payment frequencies can be 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 make 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 an 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 discuss them separately.

### Ordinary Annuities

#### Calculating the Future Value

If you know how much you can invest per period for a certain time period, the future value (FV) of an ordinary annuity formula is useful for finding out how much you would have in the future. If you are making payments on a loan, the future value is useful in determining the total cost of the loan. If you know how much you plan to invest each year and the fixed rate of return your annuity guarantees—or, for loans, the amount of your payments and the given interest rate—you can easily determine the value of your account at any point in the future.

Let us now run through Example 1. Consider the following annuity cash flow schedule:

### The Time Value of Money

The future value calculation is based on the concept of the time value of money. This simply means a dollar earned today is worth more than a dollar earned tomorrow because funds you control now can be invested and earn interest over time. Therefore, the future value of an annuity is greater than the sum of all your investments because those contributions have been earning interest over time. For example, the future value of $1,000 invested today at 10% interest is$1,100 one year from now. A single dollar today is worth $1.10 in a year because of the time value of money. Assume you make annual payments of$5,000 to your ordinary annuity for 15 years. It earns 9% interest, compounded annually.

﻿\begin{aligned} FV &= \5,000 \times \{(((1 + 0.09)^{15}) - 1) \div 0.09\}\\ &= \5,000 \times \{((1.09^{15}) - 1) \div 0.09\}\\ &= \5,000 \times 2.642 \div 0.09\\ &= \5,000 \times \146,804.58 \end{aligned}﻿

Without the power of interest compounding, your series of $5,000 contributions is only worth$75,000 at the end of 15 years. Instead, with compound interest, the future value of your annuity is almost twice that at $146,804.58. To calculate the future value of an annuity due, simply multiply the ordinary future value by 1+ i (the interest rate). In the above example, the future value of an annuity due with the same parameters is simply$146,804.58 x (1+0.09), or $160,016.99. ### Present Value Considerations When calculating the present value of an annuity, it is important that all variables are consistent. If the annuity generates annual payments, for example, the interest rate must also be expressed as an annual rate. If the annuity generates monthly payments, for example, the interest rate must also be expressed as a monthly rate. Assume an annuity has a 10% interest rate that generates annual payments of$3,000 for the next 15 years. The present value of this annuity is:

﻿\begin{aligned} &= \3,000 \times (((1 - (1 + 0.1)^{-15})) \div 0.1)\\ &= \3,000 \times ((1 - .239392) \div 0.1)\\ &= \3,000 \times (0.760608 \div 0.1)\\ &= \3,000 \times 7.60608\\ &= \22,818 \end{aligned}﻿

1:08

### The Bottom Line

Now you can see how annuities affect 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.

When planning for retirement, it is important to have a good idea of how much income you can rely on each year. While it may be relatively easy to keep track of how much you put into employer-sponsored retirement plans, individual retirement accounts (IRAs), and annuities, it is not always so easy to know how much you will get out. Luckily, when it comes to fixed-rate annuities or plans invested in fixed-rate securities, there is a simple way to calculate how much money you can expect to have available after retirement based on how much you put into the account during your working years.