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5.4 | Applications of Percentage in Banking. Simple and Compound Interest |

In these revision notes for Applications of Percentage in Banking. Simple and Compound Interest, we cover the following key points:

- How does a bank operate?
- What are the key terms used in banking?
- What is "interest applied" in banking?
- What is the difference between the terms "interest" and "interest rates"?
- How many types of interest are there?
- How is the simple interest calculated?
- How is compound interest calculated?
- What is a compound percentage change? How it is applied in banking?

A bank is an institution that **accepts customer deposits and offers loans to individuals and corporate clients**.

All banks share a common feature: they apply interest on clients who decide to have agreements with them. **Interest**, I represent the extra money one customer has to pay to a bank when he/she loans money from it or to receive when he/she decides to deposit money in that bank.

Banks make money by charging higher interest on loans than the interest they pay on customer deposits.

The original amount deposited or loaned is called the **principal**, P.

**Interest rates** (R) are expressed as a percentage (of principal) and usually they are calculated on a yearly base. Interest and interest rates are two different things. Interest (I) express the extra money you receive by or pay to the bank after making a deposit to or borrowing money from it, while interest rates (R) represent the percentage of the principal you earn or pay in a year when depositing or borrowing money from bank.

There are two types of interest: **simple** and **compound** (SI = simple interest; CI = compound interest).

Simple interest (SI) is the extra money you have to earn by or pay to the bank after a period T (in years) if the interest rate R given in a yearly base is expressed as a percentage of the principal. We calculate the simple interest using the formula

SI = *P × R × T**/**100*

From the concept of simple interest, it is clear that the formula which gives the total amount of money (A) a customer receives when he/she deposits a principal P to a bank during a period T if the value of simple interest SI is known, is

A = P + SI

When expressing the simple interest in terms of P, T and R, this formula becomes after a few rearrangements,

A = *P × (100 + R × T)**/**100*

Sometimes, the interest rates are not given per year (annually) but per month (monthly), per 3 months (quarterly), per 6 months (semi-annually), etc. In such cases, it is better to have the interest rate expressed per year first, and then apply the formulae of simple interest.

**Compound percentage change** encountered in various situations in daily life is different from the simple percentage change, as a compound percentage change involves a recurrent percentage change applied each time on the actual value rather than on the initial value. The general formula used when a compound percentage change is involved in calculations, is

A_{n} = A_{0} × (1 ± r)^{n}

where A_{n} is the amount accumulated after n compounds, r is the rate of growth or decay expressed as a decimal and A_{0} is the initial amount. The sign "plus" in the formula is for compound percentage growth while "minus" is used in compound percentage decays.

If we express the change r as a percentage instead as decimal, we obtain

A_{n} = A_{0} × (1 ± *r**/**100*)^{n}

**Compound interest**, otherwise known as "interest on interest" is an application in banking of the general concept of compound percentage change. In other words, Compound interest (or compounding interest) is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. It occurs when a bank calculates the rates based on the actual deposit and not on the principal. Compound interest is calculated together with the principal P (i.e. by calculating the total amount A) through the compound interest formula:

A_{n} = P × (1 + *r**/**n*)^{n ×t}

where n is the number of times the interest is compounded in a year (in our previous example n = 1, as the interest was compounded once in a year), r is the compound interest rate expressed as a decimal and t is the total period of deposit or loan. Then, the principal is subtracted from the amount A to give only the compound interest CI if required, i.e.

CI = A_{n} - P

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