# Constant annuity

The constant annuity is a type of repayment of a loan by a constant amount, which is a function of the interest rate and the duration of the loan. Annuity is a linear function of the amount borrowed.

## Decomposition of the calculation of the constant annuity

### Simple interest

The calculation of a simple interest is expressed by the formula {\ displaystyle I = {C_ {0}} \ times {i} \ times {n}}

with:

• {\ displaystyle {C_ {n}}} is the value of the capital at maturity (earned value),
• {\ displaystyle C_ {0}} is the value of capital,
• {\ displaystyle i} is the interest rate
• n is the number of periods.

example: {\ displaystyle C_ {0} = 10000; i = 5 \%; n = 1}

So : {\ displaystyle I =}10,000 * 0.05 * 1 = 500 €

The calculation of the acquired value is expressed by the formula {\ displaystyle C_ {n} = C_ {0} + C_ {0} * i * n = C_ {0} (1 + i * n)}

example of calculation of the acquired value with the same data as before:

{\ displaystyle C_ {n}}= 10 000 (1 + 0.05 * 1) = 10 500 €

### Compound interest

If several capitalization periods take place, the interest is composed of as many periods, being added to the capital plus past interests:

{\ displaystyle C_ {n} = {C_ {0}} \ times {(1 + i)} \ times {(1 + i)} \ times {(1 + i)} \ times {(1 + i)} \ times … \ times {(1 + i)}} (n times)

or :

{\ displaystyle C_ {n} = {C_ {0}} \ times {(1 + i)} ^ {n}}

• It is the value of the capital at maturity,
• Co is the value of the initial capital,
• i is the interest rate
• n is the number of periods

### Geometric progression

The geometric progression is the product of a sequence of numbers whose ratio is constant.

The product P is calculated by the formula {\ displaystyle P = {\ frac {(1-r ^ {n})} {(1-r)}}}