An annuity is the annual amortization of capital to which interest is added.

The term also refers to the annual payment of a sum of money for investment purposes.

Calculations for an annuity, for a loan, depend on the rule used to apply the interest owed. The most common rules are **constant amortization** (same reimbursement each year) and l’**constant annuity** (same amount to be paid each year, including applicable interest).

For an amount \(C_{0}\) borrowed for a period of * n* years at an annual interest rate of

*(expressed as a decimal), the annuity \(A\) is calculated as follows for a constant annuity :*

**t**\(A\space=\space\frac{C_{0}\cdot t}{1-(1+t)^{-n}}\)

In the case of constant annual amortization, the interest decreases yearly, and the annuities decrease accordingly, like a mathematical sequence. For an amount \(C_{0}\) borrowed for a period of * n* years at an annual interest rate of

*(expressed as a decimal), the annuity \(A_{i}\) to be paid after \(i\) years is calculated as follows for constant amortization :*

**t**\(A_{i} = A_{1}-\left ( \frac{C_{0}\cdot t}{n} \right )\cdot \left ( i-1 \right )\)

where *A*_{1} represents the amount to be paid after the first year, which is \(\frac{C_{0}}{n}+C_{0}\cdot t\).

### Examples

If an individual borrows an amount \(C_{0}\) of $1000 at an annual interest rate *t* of 8 % to be amortized over a period *n* of 5 years, the amount of a **constant annuity** is equal to :

\(A=\dfrac {1000 \times 0.08} {1-\left( 1+0.08\right) ^{-5}}=\dfrac {80} {1-\left( 1.08\right) ^{-5}}=\dfrac {80} {0.3194168}\)

And : *A* ≈ 250.46

If the individual chooses **constant annual amortization**, reimbursement, annuity amounts for the 5-year period are given in the following table :

Year i |
Initial Capital |
Interestt |
AmortizationM / n |
AnnuityA_{i} |
Non-amortized capital |

1 | 1000 | 80 | 200 | 280 | 800 |

2 | 800 | 64 | 200 | 264 | 600 |

3 | 600 | 48 | 200 | 248 | 400 |

4 | 400 | 32 | 200 | 232 | 200 |

5 | 200 | 16 | 200 | 216 | 0 |

### Educational note

The term *annuity* is sometimes confused with related terms such as *monthly installments, semi-annual installments*, etc. The term *annuity* should only be used for a sum paid annually, whereas the term * monthly installment* should be used for a sum paid every month, and the term

*, for a sum paid every six months.*

**semi-annual installment**