Compound Interest. What is it. Introduction. How is it calculated. Formula. Examples

When someone lends money to someone else, the borrower usually pays a fee to the lender. So the due interest is a sum paid or charged for the use of money or for borrowing it.

Number 4 in the list above, the frequency of the compounding interest, also makes the difference between compound and simple interest.
1. Simple Interest...
... is calculated only at the end of the loan period as a fixed percentage of the borrowed amount.
2. Compound Interest...
... is calculated not only at the end of the loan period, but periodically, over shorter periods of time, for example daily, weekly, monthly, ... These shorter periods are called compounding periods.
The interest compounding frequency shows us how often is the interest calculated over the loan period. If the interest is calculated daily - then we have a daily compounding frequency, and the interest is said to be compounded daily. If it is calculated weekly - then we have a weekly compounding frequency, and the interest is said to be compounded weekly. And so on, monthly, quarterly, half-yearly, annually, ...
The amount that results as an interest at the end of such a shorter compounding period is immediately added to the principal amount, so that the interest accrued up to that point contributes to the calculation of the interest for the next compounding period, and not only the principal amount, as it was in the case of simple interest.
Thus, in the end, the interest also earns interest and not just the principal. It follows that the compound interest will produce a slightly higher amount in the form of interest than the simple interest.

1. If the annual interest rate is r% then it means that for a principal amount of P, an interest of (r% of P) is to be paid, in a year:
I₁ = r% × P
The total amount accumulated in one year is:
A₁ = P + I₁ = P + r% × P = P × (1 + r%)
2. If the period for which the interest is being calculated is of two years, the accumulated interest over the entire period is:
I₂ = I₁ + r% × P = r% × P + r% × P = 2 × r% × P
And the total amount is:
A₂ = P + I₂ = P + 2 × r% × P = P × (1 + 2 × r%)
n. If the period for which the simple interest is being calculated is of n years, a deposit of P earns by the end of the n years period a cumulative interest of:
I_n = n × r% × P
And the total amount is:
A_n = P + I_n = P + n × r% × P = P × (1 + n × r%).

If the annual interest rate is r% and the interest compounding frequency is annual, then it means that the compounding period is one year long and the interest is calculated once a year.
1. In one year, for the amount of P, an interest of (r% of P) is earned:
C₁ = r% × P
The total amount accumulated after one year is:
A₁ = P + C₁ = P + r% × P = P × (1 + r%)

2. After the second year, the interest is:
C₂ = r% × A₁ = r% × P × (1 + r%)
And the total amount is:
A₂ = A₁ + C₂ = P × (1 + r%) + r% × P × (1 + r%) = P × (1 + r%) × (1 + r%) = P × (1 + r%)²
3. After three years, the accrued interest is:
C₃ = r% × A₂ = r% × P × (1 + r%)²
And the total amount is:
A₃ = A₂ + C₃ = P × (1 + r%)² + r% × P × (1 + r%)² = P × (1 + r%) × (1 + r%)² = P × (1 + r%)³
n. After n years, the interest is:
C_n = r% × A_n-1 = r% × P × (1 + r%)^n-1
And the total amount is:
A_n = A_n-1 + C_n = P × (1 + r%)^n-1 + r% × P × (1 + r%)^n-1 = P × (1 + r%) × (1 + r%)^n-1 = P × (1 + r%)ⁿ

What is the simple interest, I, earned in n = 5 years by a principal amount of P = 20,000 if the annual simple interest rate is r% = 3.5%?
Answer:
I = n × r% × P = 5 × 3.5% × 20,000 = 5 × (3.5 ÷ 100) × 20,000 = 1,000 × 3.5 = 3,500

What is the compound interest, C, earned in n = 5 years by a principal amount of P = 20,000 if the annual compound interest rate is r% = 3.5%?
Answer:
F = P × (1 + r%)ⁿ = 20,000 × (1 + 3.5%)⁵ = 20,000 × (1 + 3.5 ÷ 100)⁵ = 20,000 × (1 + 0.035)⁵ = 20,000 × 1.035⁵ = 20,000 × 1.187686305647 = 23,753.7261129375 ≈ 23,753.73 lei.
C = F - P = 23,753.73 - 20,000 = 3,753.73

It can be easily seen, for the identical data sets above - the same P = 20,000, r% = 3.5% and n = 5 years - the difference between the compound interest, C, and the simple interest, I, is:
C - I = 3,753.73 - 3,500 = 253.73 lei.
The compound interest earns 253.73 more than the simple interest does.