# Understanding “interest”

“Interest” is an interesting word.

In common usage, it suggests that a person is at least curious about something that they take an “interest” in, like a hobby.  But in a political context, it can mean that someone is an advocate for a certain policy position that benefits them in a specific way, as in an “interest” group.

In a financial context, “interest” also has multiple meanings.  For example, it can mean that someone has an ownership stake or an “interest” in a company or an asset.  But in these videos from Khan Academy, interest is understood to be the cost of money — or the “rent” (per unit time) that a borrower must pay to a lender for the right to use the lenders money.

Understanding the units that are used to express interest rates (remember the importance of time) is very important.  Typically, interest rates are expressed as a dimensionless ratio, in percent.  For example, you might have to pay 13% per year on the balance of a credit card.

In this case, “per cent” means the amount of money that must be paid per 100 dollars borrowed in the time period specified.  The amount borrowed is the balance, or the principal.  So the ratio is actually \$i / \$b / time, or dollars of interest (\$i) per 100 dollars of balance (\$b) per unit time.

Where no unit of time is specified, then it is customary to assume that interest rates are expressed per year (sometimes called per annum).  But it is not so unusual to express interests rates in terms of months.  Unscrupulous lenders (called usurers, or loan sharks) may express interest rates on a per day basis (in which case the interest is called the vigorish, or “vig” for short).

In addition to the unit of time, it is important to understand the compounding period.  As interest accumulates, the balance of a loan increases.  When interest is compounded, the interest is added to the balance.  At the next compounding, the new balance on which the interest is computed will include the interest added at the last compounding.  So, unless payments are being made, the borrower now owes interest on the interest.  This way of computing an interest is called compound interest, which is different than simple interest.  This next video explains the difference between simple and compound interest.

The compounding period and the units of time can be different.  For example, the interest rate on a loan might be expressed as 12%/yr, compounded monthly.  In this case, there is a significant difference between the nominal interest rate and the effective interest rate.  The effective rate includes the interest paid on the interest after compounding.

By convention, an effective interest rate can be calculated from the nominal rate according to the formula:

effective = (1 + i/n)^n – 1

where i = the nominal interest rate, and  n = the number of compounding periods per unit of time expressed in the interest rate (in this case, 12 months per year).

Compounding periods can be yearly, monthly, daily, or continuous.  The formula for an effective interest rate in continuous compounding is a little different than the formula above, because it has to be evaluated at the limit of n as n -> infinity.  For continuous compounding:

effective (continuous) = e^(r) -1

where r = the nominal interest rate per year.