Interest

Pronunciation: /ˈɪn trɪst/ ?

Interest is an amount paid on the principal of loans for the use of money. When one borrows money, one is expected to pay it back to the lender, adding on interest.

The amount of money borrowed is called principal. If one borrows $100, the principal is $100.

The interest rate is the portion of the principle that is added as interest. For example, if the interest rate is 10% and the loan amount is £140, the interest amount is £14 (£140·.10 = £14). Interest rates are typically given as an nominal interest rate per year.

If the interest is compounded more than once a year, the actual interest rate is more than the nominal interest rate. This rate is called the annual interest rate. To convert a nominal interest rate to an annual interest rate, use the formula ia = (1+i/n)^n - 1 where ia is the annualized interest rate, i is the nominal interest rate, and n is the number of times per year the interest is compounded. The part of the equation (1+i/n) is called the accumulation factor. The accumulation factor is a measure of how fast the principle grows. The greater the accumulation factor, the faster the principle grows.

The interest on first mortgages on homes usually runs from about 5% to 18%. Interest on credit cards can run from 10% to 36%. Interest on payday loans and title loans often runs as high as 1500%.

Variables

The variables that are used for calculation of interest in this article are:

VariableDescription
IInterest
rRate of interest
P0Initial principal
tNumber of periods elapsed (time)
Table 1: Descriptions of variables

Simple Interest

In simple interest, the interest is not added into the principal each period, but tracked separately. The formula for simple interest for a single period is I = P0·r. The formula for simple interest for multiple periods is I = (P0·r)·t. Typically, interest is rounded to the nearest two decimal digits.

Example 1: What is the interest on a loan of £120 at a rate of 7%?
I = P0·r, P0 = £120, r = 7%
I = £120·0.07 (7% = 0.07. See Percent.)
I = £8.40

Example 2: What is the interest on $1,423.67 at a rate of 8.7%?
I = P0·r, P0 = $1,423.67, r = 8.7%
I = $1,423.67·.087 = $123.85929
I = 123.86 (Round to two decimal places.)

Compound Interest

Compound interest is interest calculated using both principal and accrued interest. Table 1 show the compounding of interest on $100 at 1% per month.

MonthStarting
Principal
InterestEnding
Principal
1100.001.00101.00
2101.001.01102.01
3102.011.02103.03
4103.031.03104.06
5104.061.04105.10
6105.101.05106.15
7106.151.06107.21
8107.211.07108.28
9108.281.08109.36
10109.361.09110.45
11110.451.10111.55
12111.551.12112.67
Table 2 - Compound interest

The simplest equation for compound interest is P=P0*(1+i)^n where P0 is the initial principal, i is the interest rate per period, and n is the number of periods. Plugging the example from table 1 into the equation gives

P=100*(1+0.01)^12=100*1.01^12 which is approximately 100*1.1268=112.68}
Note that the result from using the table 1 is 112.67 and the result from using the formula is 112.68. The equation is more accurate than the table.

Another equation used with compound interest is P=P0*(1+i/12)^(12n). This equation is used if the interest is given as an annual interest, but compounds monthly. Most home mortgages in the United States use this interest formula.

In order to make interest rates comparable, interest rates can be annualized. The formula for annualizing interest rates is i_a=(1+i/n)^n-1, where ia is the annualized interest rate, i is the interest rate and n is the number of times per year interest is compounded.

Graphs of Compound Interest

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Manipulative 1 - Compound interest. Click on the blue point on the slider to change the figure. Created with GeoGebra.

Click on the blue point on the slider in manipulative 1 to change the interest rate.

Discovery

  • Move the blue point on the slider until the value at year 10 is $200. Since $200 is twice $100, the starting value, the money has doubled in 10 years. At what interest rate does the money double in 10 years?
  • At what interest rate does the money triple in 10 years?
  • At what interest rate does the money quadruple in 10 years?
  • At what interest rate does the money increase 5 times in 10 years?
  • At what interest rate does the money increase 6 times in 10 years?

Continuous Compounding

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Manipulative 1 - Compound interest. Created with GeoGebra.

Continuous compounding assumes interest is compounded continuously. This can be calculated using the equation A=P*e^(it). In this equation A represents the accumulated principal and interest, P represents the initial principal, e is Euler's number, i represents the interest rate, and t represents the time in years. The graph in manipulative 2 shows continuous compounding. Click on the slider labeled i and drag it to change the interest rate.

Cite this article as:


Interest. 2010-06-13. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/i/interest.html.

Translations

Image Credits

Revision History


2010-06-13: Added accumulation factor. (McAdams, David.)
2010-05-21: Rewrote first few paragraphs for clarification. (McAdams, David.)
2010-02-18: Rewrote article and included article 'Compound Interest' (McAdams, David.)
2008-11-29: Initial version (McAdams, David.)

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