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.

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## Simple Interest

Simple interest is calculated using the following formula:

﻿\begin{aligned} &\text{Simple Interest} = P \times r \times n \\ &\textbf{where:} \\ &P = \text{Principal amount} \\ &r = \text{Annual interest rate} \\ &n = \text{Term of loan, in years} \\ \end{aligned}﻿

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 their college tuition, which costs 18,000, and the annual interest rate on their loan is 6%. They repay their loan over three years. The amount of simple interest they pay is: ﻿\begin{aligned} &\3,240 = \18,000 \times 0.06 \times 3 \\ \end{aligned}﻿ and the total amount paid is: ﻿\begin{aligned} &\21,240 = \18,000 + \3,240 \\ \end{aligned}﻿ ## Real-Life Simple Interest Loans Two good examples of simple interest loans are auto loans and the interest owed on lines of credit such as credit cards. A person could take out a simple interest car loan, for example. If the car cost a total of100, to finance it the buyer would need to take out a loan with a 100 principal, and the stipulation could be that the loan has an annual interest rate of 5% and must be paid back in one year. ## Compound Interest Compound interest accrues and is added to the accumulated interest of previous periods; it includes interest on interest, in other words. The formula for compound interest is: ﻿\begin{aligned} &\text{Compound Interest} = P \times \left ( 1 + r \right )^t - P \\ &\textbf{where:} \\ &P = \text{Principal amount} \\ &r = \text{Annual interest rate} \\ &t = \text{Number of years interest is applied} \\ \end{aligned}﻿ It is calculated by multiplying the principal amount by one plus the annual interest rate raised to the number of compound periods, and then minus the reduction in the principal for that year. ## Examples of Simple and Compound Interest Below are some examples of simple and compound interest. Example 1: Suppose you plunk5,000 into a one-year certificate of deposit (CD) that pays simple interest at 3% per annum. The interest you earn after one year would be 150: ﻿\begin{aligned} &\5,000 \times 3\% \times 1 \\ \end{aligned}﻿ Example 2: Continuing with the above example, suppose your certificate of deposit is cashable at any time, with interest payable to you on a pro-rated basis. If you cash the CD after four months, how much would you earn in interest? You would earn50:

﻿\begin{aligned} &\5,000 \times 3\% \times \frac{ 4 }{ 12 } \\ \end{aligned}﻿

Example 3: Suppose Bob the Builder borrows $500,000 for three years from his rich uncle, who agrees to charge Bob simple interest at 5% annually. How much would Bob have to pay in interest charges every year, and what would his total interest charges be after three years? (Assume the principal amount remains the same throughout the three-year period, i.e., the full loan amount is repaid after three years.) Bob would have to pay$25,000 in interest charges every year:

﻿\begin{aligned} &\500,000 \times 5\% \times 1 \\ \end{aligned}﻿

or 75,000 in total interest charges after three years: ﻿\begin{aligned} &\25,000 \times 3 \\ \end{aligned}﻿ Example 4: Continuing with the above example, Bob the Builder needs to borrow an additional500,000 for three years. But as his rich uncle is tapped out, he takes a loan from Acme Borrowing Corporation at an interest rate of 5% per annum compounded annually, with the full loan amount and interest payable after three years. What would be the total interest paid by Bob?

Since compound interest is calculated on the principal and accumulated interest, here's how it adds up:

﻿\begin{aligned} &\text{After Year One, Interest Payable} = \25,000 \text{,} \\ &\text{or } \500,000 \text{ (Loan Principal)} \times 5\% \times 1 \\ &\text{After Year Two, Interest Payable} = \26,250 \text{,} \\ &\text{or } \525,000 \text{ (Loan Principal + Year One Interest)} \\ &\times 5\% \times 1 \\ &\text{After Year Three, Interest Payable} = \27,562.50 \text{,} \\ &\text{or } \551,250 \text{ Loan Principal + Interest for Years One} \\ &\text{and Two)} \times 5\% \times 1 \\ &\text{Total Interest Payable After Three Years} = \78,812.50 \text{,} \\ &\text{or } \25,000 + \26,250 + \27,562.50 \\ \end{aligned}﻿

This can also be determined using the compound interest formula from above:

﻿\begin{aligned} &\text{Total Interest Payable After Three Years} = \78,812.50 \text{,} \\ &\text{or } \500,000 \text{ (Loan Principal)} \times (1 + 0.05)^3 - \500,000 \\ \end{aligned}﻿

## The Bottom Line

In real life situations, compound interest is often a factor in business transactions, investments, and financial products intended to extend for multiple periods or years. Simple interest is mainly used for easy calculations: those generally for a single period or less than a year, though they also apply to open-ended situations, such as credit card balances.