## What Is an Annuity Table?

An annuity table is a tool for determining the present value of an annuity or other structured series of payments. Such a tool, used by accountants, actuaries, and other insurance personnel, takes into account how much money has been placed into an annuity and how long it has been there to determine how much money would be due to an annuity buyer or annuitant.

Figuring the present value of any future amount of an annuity may also be performed using a financial calculator or software built for such a purpose.

### Key Takeaways

- An annuity table is a tool used to determine the present value of an annuity.
- An annuity table calculates the present value of an annuity using a formula that applies a discount rate to future payments.
- An annuity table uses the discount rate and number of period for payment to give you an appropriate factor.
- Using an annuity table, you will multiply the dollar amount of your recurring payment by the given factor.

#### What Is An Annuity?

## How an Annuity Table Works

An annuity table provides a factor, based on time, and a discount rate (interest rate) by which an annuity payment can be multiplied to determine its present value. For example, an annuity table could be used to calculate the present value of an annuity that paid $10,000 a year for 15 years if the interest rate is expected to be 3%.

According to the concept of the time value of money, receiving a lump sum payment in the present is worth more than receiving the same sum in the future. As such, having $10,000 today is better than being given $1,000 per year for the next 10 years because the sum could be invested and earn interest over that decade. At the end of the 10-year period, the $10,000 lump sum would be worth more than the sum of the annual payments, even if invested at the same interest rate.

### Annuity Table Uses

A lottery winner could use an annuity table to determine whether it makes more financial sense to take his lottery winnings as a lump-sum payment today or as a series of payments over many years. Lottery winnings are a rare form of an annuity. More commonly, annuities are a type of investment used to provide individuals with a steady income in retirement.

## Annuity Table and the Present Value of an Annuity

### Present Value of an Annuity Formulas

The formula for the present value of an ordinary annuity, as opposed to an annuity due, is as follows:

$\begin{aligned}&\text{P} =\text{PMT}\times\frac{ 1 - (1 + r) ^ -n}{r}\\&\textbf{where:}\\&\text{P} = \text{Present value of an annuity stream}\\&\text{PMT} =\text{Dollar amount of each annuity payment}\\&r = \text{Interest rate (also known as the discount rate)}\\&n = \text{Number of periods in which payments will be made}\end{aligned}$

Assume an individual has an opportunity to receive an annuity that pays $50,000 per year for the next 25 years, with a discount rate of 6%, or a lump sum payment of $650,000. He needs to determine the more rational option. Using the above formula, the present value of this annuity is:

$\begin{aligned}&\text{PVA} = \$50,000 \times \frac{1 - (1 + 0.06) ^ -25}{0.06} = \$639,168\\&\textbf{where:}\\&\text{PVA}=\text{Present value of annuity}\end{aligned}$

Given this information, the annuity is worth $10,832 less on a time-adjusted basis, and the individual should choose the lump sum payment over the annuity.

Note, this formula is for an ordinary annuity where payments are made at the end of the period in question. In the above example, each $50,000 payment would occur at the end of the year, each year, for 25 years. With an annuity due, the payments are made at the beginning of the period in question. To find the value of an annuity due, simply multiply the above formula by a factor of (1 + r):

$\begin{aligned}&\text{P} = \text{PMT} \times\left(\frac{1 - (1 + r) ^ -n}{r}\right) \times (1 + r)\end{aligned}$

If the above example of an annuity due, its value would be:

$\begin{aligned}&\text{P}= \$50,000\\&\quad \times\left( \frac{1 - (1 + 0.06) ^ -25}{0.06}\right)\times (1 + 0.06) = \$677,518\end{aligned}$

In this case, the individual should choose the annuity due, because it is worth $27,518 more than the lump sum payment.

### Present Value of an Annuity Table

Rather than working through the formulas above, you could alternatively use an annuity table. An annuity table simplifies the math by automatically giving you a factor for the second half of the formula above. For example, the present value of an ordinary annuity table would give you one number (referred to as a factor) that is pre-calculated for the (1 - (1 + r) ^ - n) / r) portion of the formula.

The factor is determined by the the interest rate (r in the formula) and the number of periods in which payments will be made (n in the formula). In an annuity table, the number of periods is commonly depicted down the left column. The interest rate is commonly depicted across the top row. Simply select the correct interest rate and number of periods to find your factor in the intersecting cell. That factor is then multiplied by the dollar amount of the annuity payment to arrive at the present value of the ordinary annuity.

Below is an example of a present value of an ordinary annuity table:

n | 1% | 2% | 3% | 4% | 5% | 6% |

1 | 0.9901 | 0.9804 | 0.9709 | 0.9615 | 0.9524 | 0.9434 |

2 | 1.9704 | 1.9416 | 1.9135 | 1.8861 | 1.8594 | 1.8334 |

3 | 2.9410 | 2.8839 | 2.8286 | 2.7751 | 2.7233 | 2.6730 |

4 | 3.9020 | 3.8077 | 3.7171 | 3.6299 | 3.5460 | 3.4651 |

5 | 4.8534 | 4.7135 | 4.5797 | 4.4518 | 4.3295 | 4.2124 |

10 | 9.4713 | 8.9826 | 8.5302 | 8.1109 | 7.7217 | 7.3601 |

15 | 13.8651 | 12.8493 | 11.9380 | 11.1184 | 10.3797 | 9.7123 |

20 | 18.0456 | 16.3514 | 14.8775 | 13.5903 | 12.4622 | 11.4699 |

25 | 22.0232 | 19.5235 | 17.4132 | 15.6221 | 14.0939 | 12.7834 |

If we take the example above with a 6% interest rate and a 25 year period, you will find the factor = 12.7834. If you multiply this 12.7834 factor from the annuity table by the $50,000 payment amount, you will get $639,168. Notice, this is the same as the result of the formula above.

There is a separate table for the present value of an annuity due, and it will give you the correct factor based on the second formula.