Simple Interest
Simple interest is calculated using the following formula:
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).
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 of $100, 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. Therefore, the simple interest on the car loan can be calculated as follows:
($100) x (5%) x (1) = $5
Alternately, say an individual has a credit card with a $1,000 limit and a 20% APR. If, for example, he buys $1,000 worth of goods. He pays only the minimum the next month, $100. So the next month, with a $900 balance remaining, he would owe the following:
($900) x (20%) x (1) = $180
Compound Interest
Conversely, compound interest accrues on the principal amount and the accumulated interest of previous periods; it includes interest on interest, in other words. It is calculated by multiplying the principal amount by the annual interest rate raised to the number of compound periods, and then minus the reduction in the principal for that year. Or as a formula:
Compound Interest = Total amount of Principal and Interest in future less Principal amount at present
To demonstrate, let's go back to our student in the first example: She's borrowing the same amount ($18,000) for college and repaying over the same three years, only this time it's a compound-interest loan. The amount of compound interest that would be paid is $18,000 x (1.06)3 - 1) = $3,438.29 – obviously, higher than the simple interest of $3,240.
Examples of Simple and Compound Interest
Let's run through a few examples to demonstrate the formulas for both type of interest.
Example 1: Suppose you plunk $5,000 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: $5,000 x 3% x 1.
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 earn $50: $5,000 x 3% x (4 ÷ 12).
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 ($500,000 x 5% x 1), or $75,000 ($25,000 x 3) in total interest charges after three years.
Example 4: Continuing with the above example, Bob the Builder needs to borrow additional $500,000 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:
After year one, interest payable = $25,000 ($500,000 (loan principal) x 5% x 1).
After year two, interest payable = $26,250 ($525,000 (loan principal + year one interest) x 5% x 1).
After year three, interest payable = $27,562.50 ($551,250 (loan principal + interest for year one & year two) x 5% x 1).
Total interest payable after three years = $78,812.50 ($25,000 + $26,250 + $27,562.50).
Of course, rather than calculating interest payable for each year separately, one could easily compute the total interest payable by using the compound interest formula, which you'll recall is:
Compound Interest = Total amount of Principal and Interest in future, less Principal amount at present
= [P (1 + i)n] – P
= P [(1 + i)n – 1]
P = Principal, while i = annual interest rate expressed in percentage terms, and n = number of compounding periods.
Plugging the above numbers into the formula, we have P = $500,000, i = 0.05, and n = 3. Thus, compound interest = $500,000 [(1+0.05)3 - 1] = $500,000 [1.157625 - 1] = $78,812.50.
However you calculate it, the point is that by being charged compound interest rather than simple interest, Bob has to pay an additional $3,812.50 ($78,812.50 - $75,000) in interest over the three-year.
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