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# Simple Interest and Compound Interest

Interest is defined as the cost of borrowing money, as in the case of interest charged on a loan balance. Conversely, interest can also be the rate paid for money on deposit, as in the case of a certificate of deposit. Interest can be calculated in two ways: simple interest or compound interest.

• Simple interest is calculated on the principal, or original, amount of a loan.
• Compound interest is calculated on the principal amount and the accumulated interest of previous periods, and thus can be regarded as “interest on interest.”

There can be a big difference in the amount of interest payable on a loan if interest is calculated on a compound basis rather than on a simple basis. On the positive side, the magic of compounding can work to your advantage when it comes to your investments and can be a potent factor in wealth creation.

While simple interest and compound interest are basic financial concepts, becoming thoroughly familiar with them may help you make more informed decisions when taking out a loan or investing.

## Simple Interest Formula

The formula for calculating simple interest is:

\begin{aligned}&\text{Simple Interest} = P \times i \times n \\&\textbf{where:}\\&P = \text{Principal} \\&i = \text{Interest rate} \\&n = \text{Term of the loan} \\\end{aligned}

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## Compounding Periods

When calculating compound interest, the number of compounding periods makes a significant difference. Generally, the higher the number of compounding periods, the greater the amount of compound interest. So for every 100 of a loan over a certain period, the amount of interest accrued at 10% annually will be lower than the interest accrued at 5% semiannually, which will, in turn, be lower than the interest accrued at 2.5% quarterly. In the formula for calculating compound interest, the variables “i” and “n” have to be adjusted if the number of compounding periods is more than once a year. That is, within the parentheses, “i” or interest rate has to be divided by “n,” the number of compounding periods per year. Outside of the parentheses, “n” has to be multiplied by “t,” the total length of the investment. Therefore, for a 10-year loan at 10%, where interest is compounded semiannually (number of compounding periods = 2), i = 5% (i.e., 10% ÷ 2) and n = 20 (i.e., 10 x 2). To calculate the total value with compound interest, you would use this equation: \begin{aligned} &\text{Total Value with Compound Interest} = \Big( P \big ( \frac {1 + i}{n} \big ) ^ {nt} \Big ) - P \\ &\text{Compound Interest} = P \Big ( \big ( \frac {1 + i}{n} \big ) ^ {nt} - 1 \Big ) \\ &\textbf{where:} \\ &P = \text{Principal} \\ &i = \text{Interest rate in percentage terms} \\ &n = \text{Number of compounding periods per year} \\ &t = \text{Total number of years for the investment or loan} \\ \end{aligned} The following table demonstrates the difference that the number of compounding periods can make over time for a10,000 loan taken for a 10-year period.

## Other Compounding Interest Concepts

### Time Value of Money

Since money is not “free” but has a cost in terms of interest payable, it follows that a dollar today is worth more than a dollar in the future. This concept is known as the time value of money and forms the basis for relatively advanced techniques like discounted cash flow (DFC) analysis. The opposite of compounding is known as discounting. The discount factor can be thought of as the reciprocal of the interest rate and is the factor by which a future value must be multiplied to get the present value.

The formulas for obtaining the future value (FV) and present value (PV) are as follows:

### The Rule of 72

The Rule of 72 calculates the approximate time over which an investment will double at a given rate of return or interest “i” and is given by (72 ÷ i). It can only be used for annual compounding but can be very helpful in planning how much money you might expect to have in retirement.

For example, an investment that has a 6% annual rate of return will double in 12 years (72 ÷ 6%).

An investment with an 8% annual rate of return will double in nine years (72 ÷ 8%).

## Compound Annual Growth Rate (CAGR)

The compound annual growth rate (CAGR) is used for most financial applications that require the calculation of a single growth rate over a period.

For example, if your investment portfolio has grown from $10,000 to$16,000 over five years, then what is the CAGR? Essentially, this means that PV = $10,000, FV =$16,000, and nt = 5, so the variable “i” has to be calculated. Using a financial calculator or Excel spreadsheet, it can be shown that i = 9.86%.

## The Bottom Line

Get the magic of compounding working for you by investing regularly and increasing the frequency of your loan repayments. Familiarizing yourself with the basic concepts of simple interest and compound interest will help you make better financial decisions, saving you thousands of dollars and boosting your net worth over time.

Article Sources
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1. U.S. Securities and Exchange Commission. "Creating Choices."