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# The Annuity Formula for the Present and Future Value of Annuities

July 31, 2024
Bill Kimball

A future value factor of 1.0 means the value of the series will be equal to the value today. An annuity is a series of payments made over a period of time, often for the same amount each period. Investors can determine the future value of their annuity by considering the annuity amount, projected rate of return, and number of periods. There are also implications as to whether the annuity payments are made at the beginning or at the end of a period.

## Continuous Compounding (m → ∞)

Or, put another way, it’s the sum that must be invested now to guarantee a desired payment in the future. In contrast to the future value calculation, a present value (PV) calculation tells you how much money would be required now to produce a series of payments in the future, again assuming a set interest rate. So, let’s assume that you invest \$1,000 every year for the next five years, at 5% interest. Future value (FV) is a measure of how much a series of regular payments will be worth at some point in the future, given a specified interest rate. So, for example, if you plan to invest a certain amount each month or year, it will tell you how much you’ll have accumulated as of a future date. If you are making regular payments on a loan, the future value is useful in determining the total cost of the loan.

## What Is a Future Value Factor?

John Egan is a veteran personal finance writer whose work has been published by outlets such as Bankrate, Experian, Newsweek Vault and Investopedia. “Essentially, a sum of money’s value depends on how long you must wait to use it; the sooner you can use it, the more valuable it is,” Harvard Business School says. Due to inflation, \$1,000 today is worth more than what that same \$1,000 will be worth in 10 years. Let’s say someone decides to invest \$125,000 per year for the next five years in an annuity that they expect to compound at 8% per year.

## Future Value of an Annuity: What It Is, Formula, and Calculation

All else being equal, the future value of an annuity due will be greater than the future value of an ordinary annuity because the money has had an extra period to accumulate compounded interest. In this example, the future value of the annuity due is \$58,666 more than that of the ordinary annuity. Continuously compounding interest will cause annuities to generate slightly more value—although this also creates some calculation challenges. When interest growth is continuous, the payment schedule relies on a logarithmic scale. This formula incorporates both the time value of money within the period and the additional interest earned due to earlier payments. Similarly, the formula for calculating the present value of an annuity due takes into account the fact that payments are made at the beginning rather than the end of each period.

## Future Value of Growing Annuities

1. These online calculators typically require the interest rate, payment amount and investment duration as inputs.
2. Though it may not seem like much of a distinction, there may be considerable differences between the two when considering what interest is accrued.
3. In contrast to the future value calculation, a present value (PV) calculation tells you how much money would be required now to produce a series of payments in the future, again assuming a set interest rate.
4. In this example, the future value of the annuity due is \$58,666 more than that of the ordinary annuity.

An annuity is a fixed sum of money that will be paid to a person or party in the future at regular intervals. In most cases, an annuity will be paid annually to the intended party for the rest of their life. Imagine you plan to invest a fixed amount, say \$1,000, every year for the next five years at a 5 percent interest rate. The first \$1,000 you invest earns interest for a longer period compared to subsequent contributions.

In this example, with a 5 percent interest rate, the present value might be around \$4,329.48. These recurring or ongoing payments are technically referred to as annuities (not to be confused with the financial product called an annuity, though the two are related). The purpose of this calculator is to compute the future value of a series of deposits.

Present value and future value indicate the value of an investment looking forward or looking back. The two concepts are directly related, as the future value of a series of cash flows also has a present value. For example, a present value of \$1,000 today may be equal to the future value of \$1,200 today. Determining the future value of an annuity is critical when deciding whether to invest.

It shows that \$4,329.48, invested at 5% interest, would be sufficient to produce those five \$1,000 payments. With this option, you can set the payment to be made at the end of the period (ordinary annuity) or the beginning of each period (annuity due). There are fixed annuities, where the payments are constant, but there are also variable annuities that allow you to accumulate the payments and then invest them on a tax-deferred basis. Most often, investors and analysts will know one value and try to solve for the other. For instance, if you buy a stock today for \$100 that awards a 2% dividend each year, you can calculate the future value of that stock. Alternatively, if you want to have \$10,000 of future value on hand for a down payment for a car next year, you can solve for the present value.

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Because there are two types of annuities (ordinary annuity and annuity due), there are two ways to calculate present value. An annuity’s value is the sum of money you’ll need to invest in the present to provide income payments down the road. Calculating an annuity’s future value will help you determine if investing in one makes sense for you. While annuities can be a great retirement-planning vehicle, we recommend exploring all your available investment options. These annuities also offer an immediate stream of income; however, the payments will be based on changing market conditions, and your annual payment may increase or decrease over time.

Besides, you can read about different types of annuities and get some insight into the analytical background. When calculating future values, one component of the calculation is called the future value factor. The future value factor is the aggregated growth that a lump sum or series of cash flow will entail. For example, if the future value of \$1,000 is \$1,100, the future value factor must have been 1.1.

The future value tells you how much a series of regular investments will be worth at a specific point in the future, considering the interest earned over time. An ordinary annuity is a series of recurring payments that are made at the end of a period, such as quarterly stock dividends. An annuity due, by contrast, is a series of recurring payments that are made at the beginning of a period. You may hear about a life annuity where payments are handed out for the rest of the purchaser’s (annuitant) life.

It calculates the current amount of money you’d need to invest today to generate a stream of future payments, considering a specific interest rate. Future value, on the other hand, is a measure of how much a series of regular payments will be worth at some point in the future, given a set interest rate. If you’re making regular payments on a mortgage, for example, calculating the future value can help you determine the total cost of the loan.

The formulas described above make it possible—and relatively easy, if you don’t mind the math—to determine the present or future value of either an ordinary annuity or an annuity due. Financial calculators also have the ability to calculate these for you with the correct inputs. The graph below shows the timelines of the two types of annuity with their future values. As you can see, in the case of an annuity due, each payment occurs a year before the payment at the ordinary annuity.

The reason the values are higher is that payments made at the beginning of the period have more time to earn interest. For example, if the \$1,000 was invested on January 1 rather than January 31, it would have an additional month to grow. ​An annuity due differs from an ordinary annuity in that the annuity due’s payments are made at the beginning, rather than the end, of each period. You can calculate the present or future value for an ordinary annuity or an annuity due using the following formulas.

The easiest way to understand the difference between these types of annuities is to consider a simple example. Let’s assume that you deposit 100 dollars annually for three years, and the interest rate is 5 percent; thus, you have a \$100, 3-year, 5% annuity. In this context, an “ordinary annuity” is the same as an immediate fixed annuity, meaning that the holder of the annuity will begin to immediately receive payments for the rest of their life. This annuity plan provides you with an annual stream of income at some predetermined point in the future, and the payment amount will not fluctuate. These annuities involve making a large lump sum payment and immediately gaining access to an annual payout for the rest of your life. These annuities will give you an income right away, although they require a larger initial payment and might not keep pace with inflation.