The addition of interest to the principal sum of a loan or deposit is called compounding. Compound interest is interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the next period is then earned on the principal sum plus previously-accumulated interest. Compound interest is standard in finance and economics.

Compound interest may be contrasted with simple interest, where interest is not added to the principal, so there is no compounding. The simple annual interest rate is the interest amount per period, multiplied by the number of periods per year. The simple annual interest rate is also known as the nominal interest rate (not to be confused with nominal as opposed to real interest rates).


References

Example[edit]

1,000 Brazilian real (BRL) is deposited into a Brazilian savings account paying 20% per annum, compounded annually. At the end of one year, 1,000 x 20% = 200 BRL interest is credited to the account. The account then earns 1,200 x 20% = 240 BRL in the second year.

Compounding frequency[edit]

The compounding frequency is the number of times per year (or other unit of time) the accumulated interest is paid out, or capitalized (credited to the account), on a regular basis. The frequency could be yearly, half-yearly, quarterly, monthly, weekly, daily (or not at all, until maturity).

For example, monthly capitalization with annual rate of interest means that the compounding frequency is 12, with time periods measured in years.

The effect of compounding depends on:

  1. The nominal interest rate which is applied and
  2. The frequency interest is compounded.

Annual equivalent rate[edit]

The nominal rate cannot be directly compared between loans with different compounding frequencies. Both the nominal interest rate and the compounding frequency are required in order to compare interest-bearing financial instruments.

To assist consumers compare retail financial products more fairly and easily, many countries require financial institutions to disclose the annual compound interest rate on deposits or advances on a comparable basis. The interest rate on an annual equivalent basis may be referred to variously in different markets as annual percentage rate (APR), annual equivalent rate (AER), effective interest rate, effective annual rate, annual percentage yield and other terms. The effective annual rate is the total accumulated interest that would be payable up to the end of one year, divided by the principal sum.

There are usually two aspects to the rules defining these rates:

  1. The rate is the annualised compound interest rate, and
  2. There may be charges other than interest. The effect of fees or taxes which the customer is charged, and which are directly related to the product, may be included. Exactly which fees and taxes are included or excluded varies by country. may or may not be comparable between different jurisdictions, because the use of such terms may be inconsistent, and vary according to local practice.

Examples[edit]

  • A rate of 1% per month is equivalent to a simple annual interest rate (nominal rate) of 12%, but allowing for the effect of compounding, the annual equivalent compound rate is 12.68% per annum (1.0112 − 1).
  • The interest on corporate bonds and government bonds is usually payable twice yearly. The amount of interest paid (each six months) is the disclosed interest rate divided by two and multiplied by the principal. The yearly compounded rate is higher than the disclosed rate.
  • Canadian mortgage loans are generally compounded semi-annually with monthly (or more frequent) payments.[1]
  • U.S. mortgages use an amortizing loan, not compound interest. With these loans, an amortization schedule is used to determine how to apply payments toward principal and interest. Interest generated on these loans is not added to the principal, but rather is paid off monthly as the payments are applied.
  • It is sometimes mathematically simpler, e.g. in the valuation of derivatives, to use continuous compounding, which is the limit as the compounding period approaches zero. Continuous compounding in pricing these instruments is a natural consequence of Itō calculus, where financial derivatives are valued at ever increasing frequency, until the limit is approached and the derivative is valued in continuous time.

Discount instruments[edit]

  • US and Canadian T-Bills (short term Government debt) have a different convention. Their interest is calculated on a discount basis as (100 − P)/Pbnm,[clarification needed] where P is the price paid. Instead of normalizing it to a year, the interest is prorated by the number of days t: (365/t)×100. (See day count convention).


Albert Einstein is apocryphally quoted as saying "Compound interest is the eighth wonder of the world. He who understands it, earns it ... he who doesn't ... pays it.[6]

See also[edit]

Look up interest in Wiktionary, the free dictionary.

References[edit]

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    ^     http://laws.justice.gc.ca/en/showdoc/cs/I-15/bo-ga:s_6//en#anchorbo-ga:s_6 Interest Act (Canada), Department of Justice. The Interest Act specifies that interest is not recoverable unless the mortgage loan contains a statement showing the rate of interest chargeable, "calculated yearly or half-yearly, not in advance." In practice, banks use the half-yearly rate.
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    ^     Munshi, Jamal. "A New Discounting Model". ssrn.com.
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    ^      This article incorporates text from a publication now in the public domain: Chambers, Ephraim, ed. (1728). "article name needed". Cyclopædia, or an Universal Dictionary of Arts and Sciences (first ed.). James and John Knapton, et al.
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    ^     Lewin, C G (1970). "An Early Book on Compound Interest - Richard Witt's Arithmeticall Questions". Journal of the Institute of Actuaries. 96 (1): 121-132.
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    ^     Lewin, C G (1981). "Compound Interest in the Seventeenth Century". Journal of the Institute of Actuaries. 108 (3): 423-442.
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    ^     http://quoteinvestigator.com/2011/10/31/compound-interest/
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