# Annuity Formulas

An ordinary annuity is a series of annual payments made at the end of each year for a fixed number of years. Annuities with payments at the end of each year are sometimes referred to as regular annuities or simply annuity. Annuity formulas are used to calculate annuity values. The formula to use will depend on which components of the annuity are already known.

## Annuity Formulas Examples

The listing below summarizes the various formulas to use for annuity calculations.

In all annuity formulas the following symbols are used.

• FV = Future value
• PV = Present value
• i = Periodic rate
• Pmt = Periodic payment
• n = Number of years
• LN is a natural logarithm

The example used below for each annuity formula is based on the following information.

• Future value = FV = 7,335.93
• Present value = PV = 4,622.88
• Periodic rate = i = 8%
• Periodic payment = Pmt = 1,000
• number of years = n = 6

### Future Value of an Annuity

```Formula
FV = Pmt x ((1+i)n-1)/i

Excel Formula
FV = -FV(i%,n,Pmt)

Example
FV = 1000 x ((1+8%)6-1)/8% = 7,335.93
```

### Present Value of Annuity

```Formula
PV = Pmt x ((1-1/(1+i)n)/i)

Excel Formula
PV = -PV(i%,n,Pmt)

Example
PV = 1000*((1-1/(1+8%)6)/8%) = 4,622.88
```

### Calculate Annuity Payments based on Present Value

```Formula
Pmt = PV/((1-1/(1+i)n)/i)

Excel Formula
Pmt = PMT(i%,n,-PV)

Example
Pmt = 4,622.88/((1-1/(1+8%)6)/8%) = 1,000
```

### Calculate Annuity Payment based on Future Value

```Formula
Pmt = FV/(((1+i)n-1)/i)

Excel Formula
Pmt = PMT(i%,n,,-FV)

Example
Pmt = 7,335.93 /(((1+8%)6-1)/8%) = 1,000
```

### Number of years based on Future Value

```Formula
n = LN(1+FV/Pmt x i)/LN(1+i)

Excel Formula
n = nPER(i%,Pmt,,-FV)

Example
n = LN(1+7,335.93/1000 x 8%)/LN(1+8%) = 6
```

### Number of years based on Present Value

```Formula
n = -LN(1-PV/Pmt x i)/LN(1+i)

Excel Formula
n = nPER(i%,Pmt,-PV)

Example
n = -LN(1-4,622.88/1000 x 8%)/LN(1+8%) = 6
```

The formulas are summarized below. The other type of annuity is the annuity due where payments are made at the beginning of each year.

Last modified December 11th, 2019

## About the Author

Chartered accountant Michael Brown is the founder and CEO of Double Entry Bookkeeping. He has worked as an accountant and consultant for more than 25 years and has built financial models for all types of industries. He has been the CFO or controller of both small and medium sized companies and has run small businesses of his own. He has been a manager and an auditor with Deloitte, a big 4 accountancy firm, and holds a degree from Loughborough University.