# Annuities

## Annuities

Sum paid annually to meet a financial goal, such as the reimbursement of a loan and the interest owed, or a regular investment to reach a given capitalization..

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

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

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

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

where A1 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 Interest t Amortization M / n Annuity Ai 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 semi-annual installment, for a sum paid every six months.