## Simple Interest vs. Compound Interest: An Overview

Interest is the cost of borrowing money, where the borrower pays a fee to the lender for the loan. 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. In contrast, compound interest is based on the principal amount and the interest that accumulates on it in every period. Simple interest is calculated only on the principal amount of a loan or deposit, so it is easier to determine than compound interest.

### Key Takeaways

- Interest is the cost of borrowing money, where the borrower pays a fee to the lender for the loan.
- Generally, simple interest paid or received over a certain period is a fixed percentage of the principal amount that was borrowed or lent.
- Compound interest accrues and is added to the accumulated interest of previous periods, so borrowers must pay interest on interest as well as principal.

## 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 college tuition, which costs $18,000, and the annual interest rate on the loan is 6%. The student repays the loan over three years. The amount of simple interest paid is:

$\begin{aligned} &\$3,240 = \$18,000 \times 0.06 \times 3 \\ \end{aligned}$

and the total amount paid is:

$\begin{aligned} &\$21,240 = \$18,000 + \$3,240 \\ \end{aligned}$

## 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. With compound interest, borrowers must pay interest on the interest as well as the principal.

## Simple Interest vs. Compound Interest Examples

Below are some examples of simple and compound 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:

$\begin{aligned} &\$5,000 \times 3\% \times 1 \\ \end{aligned}$

### Example 2

Continuing with the above example, suppose your certificate of deposit is cashable at any time, with interest payable to you on a prorated basis. If you cash the CD after four months, how much would you earn in interest? You would receive $50:$\begin{aligned} &\$5,000 \times 3\% \times \frac{ 4 }{ 12 } \\ \end{aligned}$

### Example 3

Suppose Bob 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 years, 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 needs to borrow an additional $500,000 for three years. Unfortunately, his rich uncle is tapped out. So, he takes a loan from the bank at an interest rate of 5% per year 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:

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

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

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

This example shows how the formula for compound interest arises from paying interest on interest as well as principal.