# Future Value of Annuities

An **annuity** is a series of equal payments or receipts that occur at evenly spaced intervals. Leases and rental payments are examples. The payments or receipts occur at the **end** of each period for an **ordinary annuity** while they occur at the **beginning** of each period.for an **annuity due**.

**Future Value of an Ordinary Annuity**

The** Future Value of an Ordinary Annuity** (FVoa) is the value that a stream of expected or promised future payments will grow to after a given number of periods at a specific compounded interest.

The Future Value of an Ordinary Annuity could** **be solved by calculating the future value of each individual payment in the series using the future value formula and then summing the results. A more direct formula is:

FVoa = PMT [((1 + i)^{n} - 1) / i] |

**Where:**

- FVoa = Future Value of an Ordinary Annuity
- PMT = Amount of each payment
- i = Interest Rate Per Period
- n = Number of Periods

**Example:** What amount will accumulate if we deposit $5,000 at the **end** of each year for the next 5 years? Assume an interest of 6% compounded annually.

PV = 5,000 i = .06 n = 5

**FVoa = 5,000 [ (1.3382255776 - 1) /.06 ] = 5,000 (5.637092) = 28,185.46**

Year | 1 | 2 | 3 | 4 | 5 |

Begin | 0 | 5,000.00 | 10,300.00 | 15,918.00 | 21,873.08 |

Interest | 0 | 300.00 | 618.00 | 955.08 | 1,312.38 |

Deposit | 5,000.00 | 5,000.00 | 5,000.00 | 5,000.00 | 5,000.00 |

End | 5,000.00 | 10,300.00 | 15,918.00 | 21,873.08 | 28,185.46 |

**Example 2: **In practical problems, you may need to calculate both the future value of an annuity (a stream of future periodic payments) and the future value of a single amount that you have today:

For example, you are 40 years old and have accumulated $50,000 in your savings account. You can add $100 at the **end** of each month to your

account which pays an interest rate of 6% per year. Will you have enough money to retire in 20 years?

You can treat this as the sum of two separate calculations:

- the future value of 240 monthly payments of $100
**Plus** - the future value of the $50,000 now in your account.

PMT = $100 per period i = .06 /12 = .005 Interest per period (6% annual rate / 12 payments per year)

n = 240 periods

**FVoa =** **100 [ (3.3102 - 1) /.005 ] = 46,204**

** +**

PV = 50,000 Present value (the amount you have today) i = .005 Interest per period

n = 240 Number of periods

**FV = PV (1+i) ^{240} = 50,000 (1.005)^{240} = 165,510.22**

After 20 years you will have accumulated $211,714.22 (46,204.00 + 165,510.22).

**Future Value of an Annuity Due (FVad)**

The** Future Value of an Annuity Due** is identical to an ordinary annuity except that each payment occurs at the beginning of a period rather than at the end. Since each payment occurs one period earlier, we can calculate the present value of an **ordinary annuity** and then multiply the result by (1 + i).

**Where:**

- FVad = Future Value of an Annuity Due
- FVoa = Future Value of an Ordinary Annuity
- i = Interest Rate Per Period

**Example:** What amount will accumulate if we deposit $5,000 at the **beginning** of each year for the next 5 years? Assume an interest of 6% compounded annually.

PV = 5,000 i = .06 n = 5

**FVoa = 28,185.46 (1.06) = 29,876.59**

Year | 1 | 2 | 3 | 4 | 5 |

Begin | 0 | 5,300.00 | 10,918.00 | 16,873.08 | 23,185.46 |

Deposit | 5,000.00 | 5,000.00 | 5,000.00 | 5,000.00 | 5,000.00 |

Interest | 300.00 | 618.00 | 955.08 | 1,312.38 | 1,691.13 |

End | 5,300.00 | 10,918.00 | 16,873.08 | 23,185.46 | 29,876.59 |

## Combined Formula

You can also combine these formulas and the future value of a single amount formula into one.Category: Annuity

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The Special Retirement Supplement Formula