Interest rate

In financial transactions, interest is the charge a lender imposes for the use of money by a borrower. Interest exists because (almost) all people have time-preferences for money and goods; all else being equal, people will prefer to receive a fixed sum of money now rather than a somewhat larger sum of money later. The amount of money which a person requires to overcome their preference for immediate possession of the money is the interest.

In most transactions, interest is charged over time at a uniform rate; for example, the sum of money a borrower owes a lender might increase at 1% per month, thus the interest rate is 1% per month. Interest is called simple interest if the interest (or finance) charge does not have interest charged on it; if the interest charge is added to the original amount borrowed for future calculations of interest, the loan has compound interest. Most lending transactions in modern economies are compound interest transactions. (People who deposit money in interest-bearing accounts at banks are lenders to the bank, who is a borrower in that transaction.)

The difference between simple and compound interest can amount to a large difference. For an interest rate of 10% per year, an original loan amount (principal) of $100 will grow by $10 per year in a simple-interest loan, while the amount of the loan will be $110, $121, $133.10, $146.41, $161.05 and $177.16 in each of the following six years in a compound-interest loan. (Both examples assume that none of the loan is paid off.) At higher interest rates, compounding works more quickly: a $100 loan at 20% per year, in six years of simple interest, becomes $220, while six years of compound interest will increase it to $298.60.

Individual lenders will charge interest based on their time preferences for money and the perceived risk of the loan, while borrowers will pay interest based on their own time preferences and their anticipated use of the money.

Interest payment formulae

Calculation of interest payments appears to have driven adoption of Arabic numerals and widespread knowledge of arithmetic in the developing capitalist society of Renaissance northern Italy.

A variety of formulae are available for calculation of specific payment scenarios. In the following equations, the following symbols are used:

• P = principal; the amount of money loaned
• F = final value; the amount owed at the end of the loan period
• p = periodic payment; the amount of money paid in each period
• i = interest rate; the rate per period
• n = the number of periods in the life of the loan

For a loan where the entire amount owed is paid back after n periods, and no payments are made in the interim: $\scriptstyle F=P\times (1+i)^{n}$

For a loan where equal periodic payments are made over a fixed period, paying off the loan entirely: $\scriptstyle p=P{\frac {i(1+i)^{n}}{(1+i)^{n}-1}}$

Note: Both formulae are for compounding interest.