Everything about Compound Interest totally explained
Compound interest is the concept of adding accumulated interest back to the principal, so that interest is earned on interest from that moment on. The act of declaring interest to be principal is called compounding (for example interest is compounded). A loan, for example, may have its interest compounded every month: in this case, a loan with $1000 principal and 1% interest per month would have a balance of $1010 at the end of the first month.
Interest rates must be comparable in order to be useful, and in order to be comparable, the interest rate
and the compounding frequency must be disclosed. Since most people think of rates as a yearly percentage, many governments require financial institutions to disclose a (notionally) comparable yearly interest rate on deposits or advances. Compound interest rates may be referred to as
Annual Percentage Rate,
Effective interest rate,
Effective Annual Rate, and by other terms. When a fee is charged up front to obtain a loan, APR usually counts that cost as well as the compound interest in converting to the equivalent rate. These government requirements assist consumers to more easily compare the actual cost of borrowing.
Compound interest rates may be converted to allow for comparison: for any given interest rate and compounding frequency, an "equivalent" rate for a different compounding frequency exists.
Compound interest may be contrasted with
simple interest, where interest isn't added to the principal (there is no compounding). Compound interest predominates in finance and economics, and simple interest is used infrequently (although certain financial products may contain elements of simple interest).
Terminology
The effect of compounding depends on the frequency with which interest is compounded and the periodic interest rate which is applied. Therefore, in order to define accurately the amount to be paid under a legal contract with interest, the frequency of compounding (yearly, half-yearly, quarterly, monthly, daily, etc.)
and the interest rate must be specified. Different conventions may be used from country to country, but in finance and economics the following usages are common:
Periodic rate: the interest that's charged (and subsequently compounded) for each period. The periodic rate is used primarily for calculations, and is rarely used for comparison. The periodic rate is defined as the annual nominal rate divided by the number of compounding periods per year.
Nominal interest rate or nominal annual rate: the annual rate, unadjusted for compounding. For example, 12% annual nominal interest compounded monthly has a periodic (monthly) rate of 1%.
Effective annual rate: the nominal annual rate "adjusted" to allow comparisons; the nominal rate is restated to reflect the effective rate as if annual compounding were applied.
Economists generally prefer to use effective annual rates to allow for comparability. In finance and commerce, the nominal annual rate may be the most frequently used. When quoted with the compounding frequency, a loan with a given nominal annual rate is fully specified (the effect of interest for a given loan scenario can be precisely determined), but can't be compared to loans with different compounding frequency.
Loans and finance may have other "non-interest" charges, and the terms above don't attempt to capture these differences. Other terms such as
annual percentage rate and
annual percentage yield may have specific legal definitions and may or may not be comparable, depending on the jurisdiction.
The use of the terms above (and other similar terms) may be inconsistent, and vary according to local custom, marketing demands, simplicity or for other reasons.
Exceptions
- US and Canadian T-Bills (short term Government debt) have a different convention. Their interest is calculated as (100-P)/P 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).
- Corporate Bonds are most frequently payable twice yearly. The amount of interest paid (each six months) is the disclosed interest rate divided by two (multiplied by the principal). The yearly compounded rate is higher than the disclosed rate.
- Canadian mortgage loans are generally semi-annual compounding with monthly (or more frequent) payments.
- U.S. mortgages generally use monthly compounding (with corresponding payment periods).
- Certain techniques for, for example, valuation of derivatives may use continuous compounding, which is the limit as the compounding period approaches zero. Continuous compounding in pricing these instruments is a natural consequence of Ito Calculus, where derivatives are valued at ever increasing frequency, until the limit is approached and the derivative is valued in continuous time.
Mathematics of interest rates
Simple
Formulae are presented in greater detail at
time value of money.
In the formulae below,
i or
r are the interest rate, expressed as a true percentage (for example 10% = 10/100 = 0.10).
FV and
PV represent the future and present value of a sum.
n represents the number of periods.
These are the most basic formulae:
»
The above calculates the future value of
FV of an investment's present value of
PV accruing at a fixed interest rate of
i for
n periods.
»
where
R is the interest rate on a continuous compounding basis and
r is the stated interest rate with a compounding frequency
n.
History
If the
Native American tribe that accepted goods worth 60
guilders for the sale of
Manhattan in
1626 had invested the money in a
Dutch bank at 6.5% interest, compounded annually, then in 2005 their investment would be worth over €700 billion (around
USD $1,000 billion), more than the assessed value of the real estate in all five boroughs of
New York City. With a 6.0% interest however, the value of their investment today would have been €100 billion (7 times less!).
Compound interest was once regarded as the worst kind of
usury, and was severely condemned by
Roman law, as well as the
common laws of many other countries.
Richard Witt's book
Arithmeticall Questions, published in 1613, was a landmark in the history of compound interest. It was wholly devoted to the subject (previously called
anatocism), whereas previous writers had usually treated compound interest briefly in just one chapter in a mathematical textbook. Witt's book gave tables based on 10% (the then maximum rate of interest allowable on loans) and on other rates for different purposes, such as the valuation of property leases. Witt was a London mathematical practitioner and his book is notable for its clarity of expression, depth of insight and accuracy of calculation, with 124 worked examples.
Further Information
Get more info on 'Compound Interest'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://compound_interest.totallyexplained.com">Compound interest Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
