Simple Interest is calculated only on the principal, or on that portion of the principal which remains unpaid. The amount of simple interest is calculated according to the following formula:
where A is the amount of interest, P the principal, r the interest rate as a percentage, and n the number of time periods elapsed since the loan was taken.
For example, imagine Jim borrows $23,000 to buy a car, and simple interest is charged at a rate of 5.5% per annum. After five years, and assuming none of the loan has been paid off, Jim owes:
At this point, Jim owes a total of $29,325 (principal plus interest).
To calculate the simple interest rate r, add together all interest paid, or payable, in a period. Divide the result by the principal at the beginning of the period. The result is the simple interest rate. For example, given a $100 principal:
Credit card debt where $1/day is charged: 1/100 = 1%/day.
Corporate bond where the first $3 are due after six months, and the second $3 are due at the year's end: (3+3)/100 = 6%/year.
Certificate of deposit (GIC) where $6 is paid at the year's end: 6/100 = 6%/year.
There are three problems with simple interest.
The time periods used for measurement can be different, making comparisons wrong. One cannot claim that 1%/day of credit card interest is 'equal' to a 365%/year GIC.
The time value of money means that $3 paid every six months costs more than $6 paid only at year end. So the 6% bond cannot be 'equated' to the 6% GIC.
When interest is due, but not paid, the consequences are unclear. For example, does it remain 'interest payable', like the bond's $3 payment after six months? Alternatively, will it be added to the original principal, as would typically be the case in the 1%/day borrowed via the credit card? In the latter case, it is no longer simple interest, but compound interest.
Compund Interest
In the short run, compound Interest is very similar to simple interest, however, as time continues the difference becomes considerably larger. The conceptual difference is that the principal changes with every time period, as any interest incurred over the period is added to the principal. Put another way, the lender is charging interest on the interest.
Assuming that no part of the principal or subsequent interest has been paid, the amount of compound interest incurred is calculated by the following formula:
where A, P, r and n have the same meanings as before.
For example, if the 5.5% interest on Jim's car were calculated as compound interest, he would end up owing, in addition to the $23,000 principal, the following interest:
= 7060
In this case, then, Jim would owe a total of $30,060.
A problem with compound interest is that the resulting obligation can be difficult to interpret. To simplify this problem, a common convention in economics is to disclose the interest rate as though the term were one year, with annual compounding, yielding the effective interest rate. However, interest rates in lending are often quoted as nominal interest rates, i.e., compounding interest uncorrected for the frequency of compounding. The discussion at compound interest shows how to convert to and from the different measures of interest.
Loans often include various non-interest charges and fees. One example are points on a mortgage loan in the United States. When such fees are present, lenders are regularly required to provide information on the 'true' cost of finance, often expressed as an annual percentage rate (APR). The APR attempts to express the total cost of a loan as an interest rate after including the additional fees and expenses, although details may vary by jurisdiction.
Fixed vs. Floating Interest Rate
Commercial loans generally use compound interest, but they may not always have a single interest rate over the life of the loan. Loans for which the interest rate does not change are referred to as fixed rate loans. Loans may also have a changeable rate over the life of the loan based on some reference rate (such as LIBOR), usually plus (or minus) a fixed margin. These are known as floating rate, variable rate or adjustable rate loans.
Combinations of fixed-rate and floating-rate loans are possible and frequently used. Less frequently, loans may have different interest rates applied over the life of the loan, where the changes to the interest rate are governed by specific criteria other than an underlying interest rate. An example would be a loan that uses specific periods of time to dictate specific changes in the rate, such as a rate of 5% in the first year, 6% in the second, and 7% in the third.
