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Introduction

Interest is the cost of borrowing money, where the borrower pays a fee to the lender for using the latter's money. The interest, typically expressed as a percentage, can be either simple or compounded. Simple interest is based on the principal amount of a loan or deposit, while compound interest is based on the principal amount and the interest that accumulates on it in every period. Since simple interest is calculated only on the principal amount of a loan or deposit, it's easier to determine than compound interest.

Simple interest

Simple interest is a easy tool for estimating the interest earned or paid on a certain balance in one period. However, it does not take into account the effects of compounding, which is the process of earning interest on principal plus interest that was earned earlier. This means it can dramatically understate the amount of interest earned or paid over time.

For a borrower, simple interest is advantageous, since the total interest expense will be less without the effect of compounding.

Simple interest is calculated using the following formula:

Simple Interest, I = P x R x T

Where:

P = Principal Amount

R = Interest Rate

T = # of Periods

The period must be expressed for the same time span as the rate. If, for example, the interest is expressed in a yearly rate, such as in a 5% per annum (yearly) interest rate loan, then the number of periods must also be expressed in years. Note that sometimes changes to interest rates may be expressed in basis points (BPS). It may be worth your while, as a financial professional, to learn how to convert BPS into interest rates. If the interest rate is expressed as an annual figure, but the relevant time period is less than a year, than the interest rate must be prorated for one year. For example, if the interest rate is 8% per year, but the calculation in question calls for a quarterly interest rate, then the relevant interest rate is 2% per quarter. This 2% per quarter is equivalent to a simple interest rate of 8% per year. This is not the same, however, in the case of compounded interest.

Generally, simple interest paid or received over a certain period is a fixed percentage of the principal amount that was borrowed or lent.

For example, say a student obtains a simple-interest loan to pay one year of her college tuition, which costs $18,000, and the annual interest rate on her loan is 6%. She repays her loan over three years.

The amount of simple interest she pays $3,240 ($18,000 x 0.06 x 3).

The total amount she repays is $21,240 ($18,000 + $3,240).

Compound Interest

Compound interest is interest calculated on the initial principal and which also includes all of the accumulated interest of previous periods of a deposit or loan.

Originated in 17th century Italy, compound interest can be thought of as “interest on interest,” and will make a sum grow at a faster rate than simple interest, which is calculated only on the principal amount. The rate at which compound interest accrues depends on the frequency of compounding such that the higher the number of compounding periods, the greater the compound interest. Thus, the amount of compound interest accrued on $100 compounded at 10% annually will be lower than that on $100 compounded at 5% semi-annually over the same time period.

Compound interest is calculated by multiplying the initial principal amount by one plus the annual interest rate raised to the number of compound periods minus one. The total initial amount of the loan is then subtracted from the resulting value.

The formula for calculating compound interest is:

Compound Interest = Total amount of Principal and Interest in future (or Future Value) less Principal amount at present (or Present Value)

= [P (1 + i)n] – P

= P [(1 + i)n – 1]

(Where P = Principal, i = nominal annual interest rate in percentage terms, and n = number of compounding periods.)

Example:

Take a three-year loan of $10,000 at an interest rate of 5% that compounds annually.

What would be the amount of interest? 

In this case, it would be: $10,000 [(1 + 0.05)3 – 1] = $10,000 [1.157625 – 1] = $1,576.25.

Real Life Applications of Simple and Compound interests

Car Loans

Since car loans are amortized monthly, part of the loan is allocated to paying the outstanding monthly loan balance. The remainder goes to the interest payment. The interest payable reduces as the outstanding balance lowers, allowing a greater portion of the monthly payment to be placed towards principal payment.

For instance, Mar has a car loan of $30,000 and the interest is payable at 4%. Using an auto loan calculator, the monthly payment is $552.50 over 60 months (that’s within a five-year period). Now, in the first month, the interest is $100. It means the principal repayment is $452.50. The principal amount, at the end of the first month, is $29,547.50, and the interest payable is $100. Assuming the loan is paid on time, the loan amount outstanding will be zero by the end of the 60th month.

Certificates of Deposits

We are talking about certificates of deposits for periods of one year or less. For example, Dan invests $100,000 in a one-year CD which pays interest at 3% per year. He would earn $2,000 in interest income after a year. However, if it pays the same interest but only for six months, he will gain $1,500 in interest income.

Consumer Loans

Frequently, malls offer certain appliances on a simple interest basis for periods of up to 12 months (or 24 months in some cases). If you want to purchase a microwave oven, which costs $3,000 and has an annual rate of 8% in monthly installments, the payment would be $270 / month. Multiply the amount by 12 months, you would end up paying $3,240 for buying a microwave oven.

Discounts on Early Payments

Suppliers normally offer a discount to buyers who pay their invoices early. Let’s say establishment XYZ offers a 0.10% discount to entities that settle their payments within a month. If Company ABC has an invoice of $150,000 and pays it ahead of the scheduled payment, that company will receive a 0.10%, which is equivalent to $15,000.

References.

Evans, Allan (1936). Francesco Balducci Pegolotti, La Pratica della Mercatura. Cambridge, Massachusetts. pp. 301–2.

Lewin, C G (1970). "An Early Book on Compound Interest - Richard Witt's Arithmeticall Questions". Journal of the Institute of Actuaries. 96 (1): 121–132.

Lewin, C G (1981). "Compound Interest in the Seventeenth Century". Journal of the Institute of Actuaries. 108 (3): 423–442.

Duffie, Darrell and Kenneth J. Singleton (2003). Credit Risk: Pricing, Measurement, and Management. Princeton University Press.