Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions | |
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5.4 | Applications of Percentage in Banking. Simple and Compound Interest |

In this Math tutorial, you will learn:

- 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?

In this tutorial, we will focus on banking terminology and focus on the key activity that makes a bank operate: interest. Banks work by collecting deposits from customers and using those deposits to fund investments and loans. The profits from those loans and investments are used to pay for the banks employees, offices and equipment etc. A portion of those profits is also used to provide extra money back to the customers who deposited the original money (the customers capital) with the bank. This extra money is called interest. Everybody, at some point in their life, will be in the situation where he/she is in doubt about which bank to choose for depositing his/her money. Therefore it is important to understand how banks work. As banks method of operation is based on percentages - a concept we cover in this chapter, we will now look at banking in more detail.

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

In simple terms, a bank is an institution that **accepts customer deposits and offers loans to individuals and corporate clients**. Banks make money by charging higher interest on loans than the interest they pay on customer deposits.

There are some banks which do not accept customer deposits and only offer loans to clients (investment funds). Their owners have enough money, so they don't need money from customers for their investments. They only lend money to their customers and the latter repay the money received in instalments based on the rates determined through mutual agreement. However, most banks work in both ways mentioned earlier, i.e. they accept customer deposits but also offer loans for those who need money. Then, they may or may not invest the money a customer deposits in other sectors of economy to have extra earnings, depending on the strategy they apply to extract profits from their activity.

For example, a customer deposits $10,000 in a bank and he receives an extra $1,200 as interest for the period of deposit. Hence, the customer receives, in total, $10,000 + $1,200 = $11,200 from the bank. During the same period, the bank invests the money in profitable activities or lends them to other customers and receives let's say an extra $2,000 from this activity. Thus, the bank receives in total $10,000 + $2,000 = $12,000 from this activity. The difference between these two amounts, i.e. $12,000 - $11,200 = $800 represents the bank's profit. This is a simple example that illustrates the financial logic of a banks operation.

The original amount deposited or loaned is called the **principal**. In our example, the principal (in short P) is $10,000.

Basically, when you deposit money in a bank, you are lending your money to the bank, which on the other hand uses your money to repay other customers or to invest in other assets. In this way, you indirectly become an investor in the economy through your bank. On the other hand, when you borrow money from a bank, they gain profit from your money through interest you have to pay in addition to the principal borrowed.

**Interest rates** (R) are expressed in percentage (of principal) and usually they are calculated on a yearly base (per annum). Thus, when you read the interest rates offered by a bank, you are reading the percentage of profit you earn in a year by loaning your money to the bank.

Interest and interest rates are two different things. Interest (I) express the extra money you receive from 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, we will discuss these in the upcoming paragraphs.

Simple interest (SI) is the extra money you earn from or pay to the bank after a deposit period T (in years) and 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*

For example, if you make a $20,000 deposit (P = 20,000) in a bank which offers 4% (yearly) interest rates (R = 4), the extra money you will receive after 5 years (T = 5) is

SI = *P × R × T**/**100*

=*$20,000 × 4 × 5**/**100*

= $4,000

=

= $4,000

A customer earned an extra $3,600 from depositing $24,000 in a bank for three years. The bank applies simple interest rates. What was the interest rate offered by the bank?

We have the following clues:

SI = 3,600

P = 24,000

T = 3

R = ?

Using the formula of simple interest

SI = *P × R × T**/**100*

we obtain for the interest rate after rearranging the formula above:

R = *SI × 100**/**P × T*

=*3,600 × 100**/**24,000 × 3*

=*360,000**/**72,000*

= 5

=

=

= 5

This means the bank offered a 5% interest rate to the given customer.

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, as we did in the previous paragraph, we obtain

A = P + *P × R × T**/**100*

=*P × 100 + P × R × T**/**100*

=

Thus,

A = *P × (100 + R × T)**/**100*

For example, if a customer deposits $8,700 to a bank (P = 8,700) that offers 2.6% yearly (annual) interest rates (R = 2.6), after a period of 3.5 years of deposit (T = 3.5) he will receive in total:

A = P + SI

=*P × (100 + R × T)**/**100*

=*8,700 × (100 + 2.6 × 3.5)**/**100*

=*8,700 × (100 + 9.1)**/**100*

=*8,700 × 109.1**/**100*

= 87 × 109.1

= $9,491.7

=

=

=

=

= 87 × 109.1

= $9,491.7

Obviously, the value of simple interest earned by the customer during this period is the difference between the total amount received and the principal deposited, i.e.

SI = A - P

= $9,491.7 - $8,700

= $791.7

= $9,491.7 - $8,700

= $791.7

a value that represents 9.1% of the principal ( ** $791.7/$8700** = 0.091 = 9.1%).

A customer needs to borrow $15,000 from a bank which offers loans with a simple interest rate of 3.8% per annum. The customer hopes to sell property in the future to settle the loan. The property is estimated to cost $18,500. How long after the loan agreement will the customer have to sell the property in order to settle the loan on time?

Clues:

P = 15,000

R = 3.8

A = 18,500

T = ?

Using the combined formula found earlier

A = *P × (100 + R × T)**/**100*

we obtain for the time T (in years) needed to settle the loan:

100 + R × T = *100 × A**/**P*

R × T =*100 × A**/**P* - 100

T =*100 × A**/**P* - 100*/**R*

=*100 × 18,500**/**15,000* - 100*/**3.8*

=*123.3 - 100**/**3.8*

=*23.3**/**3.8*

= 6.13 years

R × T =

T =

=

=

=

= 6.13 years

Sometimes, the interest rates are not given per year (annually [per annum]) but per month (monthly [per month]), per 3 months (quarterly), per 6 months (semi-annually), etc. In these situations it is better to have the interest rate expressed per year first, and then apply the formulae we explained above.

A bank offers 0.22% monthly interest rates for loans. How much will a customer receive after three years if he deposits $20,000 now?

We have the following clues:

R (monthly) = 0.22

P = 20,000

T = 3

A = ?

First, we have to calculate the yearly interest rates by multiplying the monthly ones by 12 (the number of months in a year). Thus,

R(yearly) = 12 × R(monthly)

= 12 × 0.22

= 2.64

= 12 × 0.22

= 2.64

This means the yearly interest is 2.64%.

In this way, we obtain for the money the customer will receive in three years:

A = *P × (100 + R × T)**/**100*

=*20,000 × (100 + 2.64 × 3)**/**100*

=*20,000 × (100 + 7.92)**/**100*

=*20,000 × 107.92**/**100*

= $21,584

=

=

=

= $21,584

Before explaining the other type of interest applied in banking (compound interest), it would be appropriate to explain the general concept of **compound percentage change**. It 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.

For example, if we have a compound percentage increase (growth) of the number of bacteria in a sample by 20% each hour and the original number of bacteria is 4,000 (A_{0} = 4,000), the number of bacteria in the sample after 3 hours (A_{3}) will be calculated through repeated calculations based on the actual value, i.e.

A_{1} = A_{0} + 20% × A_{0}

= A_{0} + 0.2 × A_{0}

= 1.2 × A_{0}

= 1.2 × 4,000

= 4,800

A_{2} = A_{1} + 20% × A_{1}

= A_{1} + 0.2 × A_{1}

= 1.2 × A_{1}

= 1.2 × 4,800

= 5,760

A_3 = A_{2} + 20% × A_{2}

= A_{2} + 0.2 × A_{2}

= 1.2 × A_{2}

= 1.2 × 5,760

= 6,912

= A

= 1.2 × A

= 1.2 × 4,000

= 4,800

A

= A

= 1.2 × A

= 1.2 × 4,800

= 5,760

A_3 = A

= A

= 1.2 × A

= 1.2 × 5,760

= 6,912

As you see, if we continue using this method, it would be a time-consuming procedure, so it is necessary to find an easier and shorter way to calculate compound percentage change. Thus, if we express the percentage change by r (as decimal), we can write

A_{1} = A_{0} + r × A_{0}

A_{2} = A_{1} + r × A_{1}

= A_{1} × (1 + r)

= (A_{0} + r × A_{0} ) × (1 + r)

= A_{0} × (1 + r) × (1 + r)

= A_{0} × (1 + r)^{2}

A

= A

= (A

= A

= A

Obviously, our example involved a compound percentage increase, but this method can be used for the percentage decrease (decay) as well. We just put a minus before r instead of plus. Therefore, using this procedure n times, we obtain the general formula for the compound percentage change:

A_{n} = A_{0} × (1 ± r)^{n}

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

A_{n} = A_{0} × (*1 ± r**/**100*)^{n}

The number of radioactive particles in a sample decreases by 5% every day. If the original number of radioactive particles was 30,000, how many particles are left in the sample after one week?

Since the recurrence occurs 7 times in total (once in a week) have the following clues:

A_{0} = 30,000

r = 5

n = 7

A_{n} = ?

Applying the equation of recurring percentage change

A_{n} = A_{0} × (*1 ± r**/**100*)^{n}

where the sign before n must be minus as we have a percentage decay involved in this situation, we obtain the number of radioactive particles left in the sample after the given recurrence:

A_{n}=A_{0} × (*1 - r**/**100*)^{n}

A_{7} = 30,000 × (*1 - 5**/**100*)^{7}

= 30,000 × (*100 - 5**/**100*)^{7}

= 30,000 × (*95**/**100*)^{7}

= 20,950

A

= 30,000 × (

= 30,000 × (

= 20,950

Now that you know what a **compound percentage change** represents, it is much easier to understand the concept of compound interest and the corresponding interest rates applied in banking.

**Compound interest**, otherwise known as "interest on interest" occurs when a bank calculates the rates based on the actual deposit and not on the principal. 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. 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

Let's get a better understanding of this through an example.

A customer deposits $3,600 in a bank that applies 1.2% compound interest rates every six months. Calculate:

- The amount the customer will receive after five years of deposit
- The compound interest earned by the client during this period

We have the following clues:

P = $3,600

r = 1.2% = 0.012

n = 2 (every six months means twice a year)

t = 5

A_{n} = ?

CI = ?

- Using the formula of compound interest A
_{n}= P × (1 +)*r**/**n*^{n × t}

= $3,600 × (1 +)*0.012**/**2*^{2 × 5}

= $3,600 × 0.006^{10}

= $3,821.9 - The compound interest is calculated by subtracting the principal from the total amount earned. Thus, we have CI = A
_{n}- P

= $3,821.9 - $3,600

= $221.9

You can also calculate other variables involved in the compound interest formula. For example, we can calculate the principal if the total amount after a certain period is given when the interest rate is known, etc. Let's consider another example.

A customer has $37,821 in his savings account. The bank offers an interest of 1.8% compounded thrice in a year. What was her balance four years ago?

The quantity to be calculated in this problem is the principal P. We have the following clues:

A_{n} = $37,821

r = 1.6% = 0.018

n = 3

t = 4

P = ?

Using the formula of compound interest

A_{n} = P × (1 + *r**/**n*)^{n × t}

we obtain after substitutions

P = *A*_{n}*/**(1 + **r**/**n*)^{n × t}

=*$37,821**/**(1 + **0.018**/**3*)^{3 × 4}

=*$37,821**/**(1 + 0.006)*^{12}

=*$37,821**/**1.006*^{12}

=*$37,821**/**1.0744*

= $35,202

=

=

=

=

= $35,202

If we are asked to calculate the value of compound interest CI from this example, we can write

CI = A_{n} - P

= $37,821 - $35,202

= $2,619

= $37,821 - $35,202

= $2,619

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