What Is the Annuity Formula?

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An annuity is an insurance contract you purchase to receive payments for a specific period, such as 30 years, or for the rest of your life. By applying a mathematical formula consisting of variables such as payment amount, the discount or interest rate and the number of payment periods, it’s possible to know at the beginning of the annuity how much it’ll be worth at the end.

Types of Annuities

Insurance companies offer two main categories of annuity contracts: immediate and deferred.

Immediate Annuities

An immediate annuity begins paying out anywhere from 30 days to one year after you purchase it. You purchase an immediate annuity with a single lump-sum payment; you can choose how often you want to receive your annuity payments and the number of years you want the payments to last.

Immediate annuities often appeal to retirees and those within a year of retirement. Although they have to tap into their savings to fund the annuity, the annuity assures them a certain level of retirement income that can begin almost immediately and last their entire lives.

Deferred Annuities

A deferred annuity begins paying out at a future date. You purchase it with either a single lump-sum payment or a series of payments and designate a future date to begin receiving your payments. The time before your payments are due to begin is called the accumulation phase of the annuity. This phase gives your payments into the annuity time to earn interest.

Deferred annuities might appeal to individuals who are a number of years from retirement and who have maxed out their 401(k) and individual retirement accounts. Investors also use deferred annuities strategically, to diversify their portfolios and mitigate risk affecting stocks, bonds and other assets.

Components of the Annuity Formula

Two different but related measures provide insights on an annuity’s value.

Present Value

An annuity’s present value is the value of its future payments to you in today’s dollars. It relies on a principle called the “time value of money” that says money you receive today is worth more than money you receive in the future.

Say, for example, you’re given a choice between receiving a dollar today and waiting a year to receive it. The better choice would likely be to receive it now. Not only would you get the dollar before a year’s worth of inflation had diminished its value; but you could also save or invest the money and reap a year’s worth of appreciation.

In the case of an annuity, the present value considers the amount of each payment you receive, the number of payment periods and the interest rate. The interest rate is the rate at which the money is discounted, and this accounts for the time value of money. Think of it as negative interest.

Future Value

An annuity’s future value looks ahead to tell you how much that future value will be. The calculation uses the same variables as the present value calculation, but it flips them:

• Instead of payments you receive, it uses the amount you pay in.
• Instead of the number of payments you receive, it uses the number of payments you make.
• Instead of using the interest rate at which the annuity’s outgoing payments are discounted, it uses the rate earned on the money you contribute.

Knowing the future value benefits you in two ways. First, it tells you how much you’ll receive from the annuity based on the three variables. It can help you plan for the future by setting a particular goal for the future value and determining the amount and number of payments you’ll have to make to reach that goal.

Used together, present value and future value help you compare your options when deciding whether to take a lump-sum payout from an annuity or annuitize the funds to spread payments out over several years.

The Present Value of Annuity Formula

The present value calculation has three variables: payment amounts, number of payment periods and interest rate, which is the rate at which the payments are discounted.

Formula Breakdown

Here’s the formula and what it means.

PV = PMT x ((1 – (1 + r) ^ -n ) / r)

• PV: Present value
• PMT: Payment amount
• r: Interest (discount) rate
• n: Number of payment periods

Step-by-Step Example Calculation

Imagine you have a $10,000 savings account balance, and you want to draw $1,000 per year for 10 years. Here are the variables you use for the calculation.

PV = 1000 x [(1- (1 + .03) ^ -10)/.03])

The Future Value of an Annuity Formula

The future value calculation also has three variables: payment amounts, number of payment periods and interest rate.

Formula Breakdown

Here’s the formula and what it means.

 FV = PMT * (((1 + i)^n – 1) / i):

• FV: Future value
• PMT: Payment amount
• r: Interest rate
• n: Number of payment periods

Step-by-Step Example Calculation

For this calculation imagine you’re saving for a vacation you want to take three years from now, and you plan to pay for it by depositing your next three $2,000 annual bonuses into a savings account earning 3% interest. Here’s the formula with those variables.

FV = 2000 * (((1 + .03)^3 – 1) / .03)

This example uses payment amount, number of payments and interest rate to figure out the final balance after five years. If you know the future value needed, you could use it as a variable for figuring out how many $2000 deposits you’d have to make to meet the goal. Or how much you’d need to deposit if you knew you wanted to make three payments.

Differences Between PV and FV of Annuities

PV, or present value, is the value of future annuity payments you’ll receive, in today’s dollars. FV, or future value, is what your annuity will be worth after you’ve made your payments. Whereas PV discounts the payments received to account for the time value of money, FV compounds interest on your payments.

Here’s a look at the differences between PV and FV.

How To Use a Financial Calculator or Spreadsheet for Annuities

Although you can simply input the variables into any number of annuity value calculators and let the calculators do the math for you, you can also do the calculations yourself using a calculator or spreadsheet.

Using a Financial Calculator

A financial calculation has features standard calculators don’t have, and they make it easier to do calculations based on complex formulas. For example, it has parenthese buttons to ensure that you follow the correct order of operations. In addition, you can use the STO and RCL buttons to store numbers and then recall them later. Remember to clear your work from the calculator’s memory before moving on to new calculations.

Using Excel or Google Sheets

Spreadsheets have an FV function for calculating future annuity value.

To calculate future value using Excel, use the following syntax:

FV (rate, nper, pmt, [pv], [type])

• FV: Future value
• rate: Interest rate
• npr: Number of payment periods
• pmt: Payment amount
• pv: Present value (optional)
• type: 0 for payments due at end of period, 1 for payments due at beginning (optional)

The syntax for Google Sheets is similar but more descriptive:

FV(rate, number_of_periods, payment_amount, [present_value], [end_or_beginning])

As with Excel, present value and “end_or_beginning” (i.e., type) are optional.

Karen Doyle contributed to the reporting for this article.

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