economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
, interest is payment from a borrower or deposit-taking financial institution to a
lender
A creditor or lender is a party (e.g., person, organization, company, or government) that has a claim on the services of a second party. It is a person or institution to whom money is owed. The first party, in general, has provided some propert ...
or depositor of an amount above repayment of the
principal sum
Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to another party, the creditor. Debt is a deferred payment, or series of payments, which differentiates it from an immediate purchase. The ...
(that is, the amount borrowed), at a particular rate. It is distinct from a fee which the borrower may pay the lender or some third party. It is also distinct from
dividend
A dividend is a distribution of profits by a corporation to its shareholders. When a corporation earns a profit or surplus, it is able to pay a portion of the profit as a dividend to shareholders. Any amount not distributed is taken to be re-i ...
which is paid by a company to its shareholders (owners) from its profit or
reserve
Reserve or reserves may refer to:
Places
* Reserve, Kansas, a US city
* Reserve, Louisiana, a census-designated place in St. John the Baptist Parish
* Reserve, Montana, a census-designated place in Sheridan County
* Reserve, New Mexico, a US ...
, but not at a particular rate decided beforehand, rather on a pro rata basis as a share in the reward gained by
risk
In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environm ...
taking entrepreneurs when the revenue earned exceeds the total costs.
For example, a customer would usually pay interest to borrow from a bank, so they pay the bank an amount which is more than the amount they borrowed; or a customer may earn interest on their savings, and so they may withdraw more than they originally deposited. In the case of savings, the customer is the lender, and the bank plays the role of the borrower.
Interest differs from profit, in that interest is received by a lender, whereas profit is received by the
owner
Ownership is the state or fact of legal possession and control over property, which may be any asset, tangible or intangible. Ownership can involve multiple rights, collectively referred to as title, which may be separated and held by different ...
of an
asset
In financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) that can be used to produce positive economic value. Assets represent value of ownership that c ...
enterprise
Enterprise (or the archaic spelling Enterprize) may refer to:
Business and economics
Brands and enterprises
* Enterprise GP Holdings, an energy holding company
* Enterprise plc, a UK civil engineering and maintenance company
* Enterprise ...
. (Interest may be part or the whole of the profit on an investment, but the two concepts are distinct from each other from an accounting perspective.)
The rate of interest is equal to the interest amount paid or received over a particular period divided by the
principal sum
Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to another party, the creditor. Debt is a deferred payment, or series of payments, which differentiates it from an immediate purchase. The ...
borrowed or lent (usually expressed as a percentage).
Compound interest means that interest is earned on prior interest in addition to the principal. Due to compounding, the total amount of debt grows exponentially, and its mathematical study led to the discovery of the number '' e''. In practice, interest is most often calculated on a daily, monthly, or yearly basis, and its impact is influenced greatly by its compounding rate.
History
Credit is thought to have preceded the existence of coinage by several thousands of years. The first recorded instance of credit is a collection of old Sumerian documents from 3000 BC that show systematic use of credit to loan both grain and metals. The rise of interest as a concept is unknown, though its use in Sumeria argue that it was well established as a concept by 3000BC if not earlier, with historians believing that the concept in its modern sense may have arisen from the lease of animal or seeds for productive purposes. The argument that acquired seeds and animals could reproduce themselves was used to justify interest, but ancient Jewish religious prohibitions against
usury
Usury () is the practice of making unethical or immoral monetary loans that unfairly enrich the lender. The term may be used in a moral sense—condemning taking advantage of others' misfortunes—or in a legal sense, where an interest rate is c ...
(נשך ''NeSheKh'') represented a "different view".
The first written evidence of compound interest dates roughly 2400 BC. The annual interest rate was roughly 20%. Compound interest was necessary for the development of agriculture and important for urbanization.
While the traditional Middle Eastern views on interest was the result of the urbanized, economically developed character of the societies that produced them, the new Jewish prohibition on interest showed a pastoral, tribal influence. In the early 2nd millennium BC, since silver used in exchange for livestock or grain could not multiply of its own, the Laws of Eshnunna instituted a legal interest rate, specifically on deposits of dowry. Early Muslims called this ''riba'', translated today as the charging of interest.
The First Council of Nicaea, in 325, forbade
clergy
Clergy are formal leaders within established religions. Their roles and functions vary in different religious traditions, but usually involve presiding over specific rituals and teaching their religion's doctrines and practices. Some of the ter ...
from engaging in
usury
Usury () is the practice of making unethical or immoral monetary loans that unfairly enrich the lender. The term may be used in a moral sense—condemning taking advantage of others' misfortunes—or in a legal sense, where an interest rate is c ...
Conrad Henry Moehlman (1934). The Christianization of Interest. Church History, 3, p 6. doi:10.2307/3161033. which was defined as lending on interest above 1 percent per month (12.7% AER). Ninth century
ecumenical council
An ecumenical council, also called general council, is a meeting of bishops and other church authorities to consider and rule on questions of Christian doctrine, administration, discipline, and other matters in which those entitled to vote ar ...
s applied this regulation to the laity.Noonan, John T., Jr. 1993. "Development of Moral Doctrine." 54 Theological Stud. 662.
Catholic Church
The Catholic Church, also known as the Roman Catholic Church, is the List of Christian denominations by number of members, largest Christian church, with 1.3 billion baptized Catholics Catholic Church by country, worldwide . It is am ...
opposition to interest hardened in the era of scholastics, when even defending it was considered a heresy. St.
Thomas Aquinas
Thomas Aquinas, Dominican Order, OP (; it, Tommaso d'Aquino, lit=Thomas of Aquino, Italy, Aquino; 1225 – 7 March 1274) was an Italian Dominican Order, Dominican friar and Catholic priest, priest who was an influential List of Catholic philo ...
, the leading theologian of the
Catholic Church
The Catholic Church, also known as the Roman Catholic Church, is the List of Christian denominations by number of members, largest Christian church, with 1.3 billion baptized Catholics Catholic Church by country, worldwide . It is am ...
, argued that the charging of interest is wrong because it amounts to " double charging", charging for both the thing and the use of the thing.
In the
medieval economy
It has been estimated that throughout prehistory, the world average GDP per capita was about $158 per annum (adjusted to 2013 dollars), and did not rise much until the Industrial Revolution. The first object or physical thing specifically used i ...
, loans were entirely a consequence of necessity (bad harvests, fire in a workplace) and, under those conditions, it was considered morally reproachable to charge interest. It was also considered morally dubious, since no goods were produced through the lending of money, and thus it should not be compensated, unlike other activities with direct physical output such as blacksmithing or farming. For the same reason, interest has often been looked down upon in Islamic civilization, with almost all scholars agreeing that the Qur'an explicitly forbids charging interest.
Medieval jurists developed several financial instruments to encourage responsible lending and circumvent prohibitions on usury, such as the
Contractum trinius
{{unreferenced, date=August 2013
A ''contractum trinius'' was a set of contracts devised by European bankers and merchants in the Middle Ages as a method of circumventing canonical laws prohibiting usury as a part of Christian finance. At the ...
.
In the
Renaissance
The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ide ...
era, greater mobility of people facilitated an increase in commerce and the appearance of appropriate conditions for entrepreneurs to start new, lucrative businesses. Given that borrowed money was no longer strictly for consumption but for production as well, interest was no longer viewed in the same manner.
The first attempt to control interest rates through manipulation of the
money supply
In macroeconomics, the money supply (or money stock) refers to the total volume of currency held by the public at a particular point in time. There are several ways to define "money", but standard measures usually include currency in circu ...
was made by the
Banque de France
The Bank of France (French: ''Banque de France''), headquartered in Paris, is the central bank of France. Founded in 1800, it began as a private institution for managing state debts and issuing notes. It is responsible for the accounts of the ...
in 1847.
Islamic finance
The latter half of the 20th century saw the rise of interest-free Islamic banking and finance, a movement that applies Islamic law to financial institutions and the economy. Some countries, including Iran, Sudan, and Pakistan, have taken steps to eradicate interest from their financial systems. Rather than charging interest, the interest-free lender shares the risk by investing as a partner in profit loss sharing scheme, because predetermined loan repayment as interest is prohibited, as well as making money out of money is unacceptable. All financial transactions must be asset-backed and it does not charge any interest or fee for the service of lending.
In the history of mathematics
It is thought that Jacob Bernoulli discovered the mathematical constant e by studying a question about compound interest. He realized that if an account that starts with $1.00 and pays say 100% interest per year, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.52 = $2.25. Compounding quarterly yields $1.00×1.254 = $2.4414..., and so on.
Bernoulli noticed that if the frequency of compounding is increased without limit, this sequence can be modeled as follows:
:
where ''n'' is the number of times the interest is to be compounded in a year.
Economics
In economics, the rate of interest is the price of credit, and it plays the role of the cost of capital. In a
free market
In economics, a free market is an economic system in which the prices of goods and services are determined by supply and demand expressed by sellers and buyers. Such markets, as modeled, operate without the intervention of government or any ot ...
economy, interest rates are subject to the law of
supply and demand
In microeconomics, supply and demand is an economic model of price determination in a Market (economics), market. It postulates that, Ceteris paribus, holding all else equal, in a perfect competition, competitive market, the unit price for a ...
of the
money supply
In macroeconomics, the money supply (or money stock) refers to the total volume of currency held by the public at a particular point in time. There are several ways to define "money", but standard measures usually include currency in circu ...
, and one explanation of the tendency of interest rates to be generally greater than zero is the scarcity of loanable funds.
Over centuries, various schools of thought have developed explanations of interest and interest rates. The School of Salamanca justified paying interest in terms of the benefit to the borrower, and interest received by the lender in terms of a premium for the risk of default. In the sixteenth century, Martín de Azpilcueta applied a time preference argument: it is preferable to receive a given good now rather than in the future. Accordingly, interest is compensation for the time the lender forgoes the benefit of spending the money.
On the question of why interest rates are normally greater than zero, in 1770, French economist Anne-Robert-Jacques Turgot, Baron de Laune proposed the
theory of fructification
In economics, the theory of fructification is a theory of the interest rate which was proposed by French economist and finance minister Anne Robert Jacques Turgot. The term ''theory of fructification'' is due to Eugen von Böhm-Bawerk who consider ...
. By applying an opportunity cost argument, comparing the loan rate with the rate of return on agricultural land, and a mathematical argument, applying the formula for the value of a
perpetuity
A perpetuity is an annuity that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence. For example, the United Kingdom (UK) government issued them in the past; these were known as cons ...
to a plantation, he argued that the land value would rise without limit, as the interest rate approached zero. For the land value to remain positive and finite keeps the interest rate above zero.
Adam Smith, Carl Menger, and Frédéric Bastiat also propounded theories of interest rates. In the late 19th century, Swedish economist
Knut Wicksell
Johan Gustaf Knut Wicksell (December 20, 1851 – May 3, 1926) was a leading Swedish economist of the Stockholm school. His economic contributions would influence both the Keynesian and Austrian schools of economic thought. He was married to t ...
in his 1898 ''Interest and Prices'' elaborated a comprehensive theory of economic crises based upon a distinction between natural and nominal interest rates. In the 1930s, Wicksell's approach was refined by Bertil Ohlin and Dennis Robertson and became known as the loanable funds theory. Other notable interest rate theories of the period are those of Irving Fisher and
John Maynard Keynes
John Maynard Keynes, 1st Baron Keynes, ( ; 5 June 1883 – 21 April 1946), was an English economist whose ideas fundamentally changed the theory and practice of macroeconomics and the economic policies of governments. Originally trained in ...
.
Calculation
Simple interest
Simple interest is calculated only on the principal amount, or on that portion of the principal amount that remains. It excludes the effect of
compounding
In the field of pharmacy, compounding (performed in compounding pharmacies) is preparation of a custom formulation of a medication to fit a unique need of a patient that cannot be met with commercially available products. This may be done for me ...
. Simple interest can be applied over a time period other than a year, for example, every month.
Simple interest is calculated according to the following formula:
:
where
:''r'' is the simple annual
interest rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, t ...
:''B'' is the initial balance
:''m'' is the number of time periods elapsed and
:''n'' is the frequency of applying interest.
For example, imagine that a credit card holder has an outstanding balance of $2500 and that the simple annual
interest rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, t ...
is 12.99% ''per annum'', applied monthly, so the frequency of applying interest is 12 per year. Over one month,
:
interest is due (rounded to the nearest cent).
Simple interest applied over 3 months would be
:
If the card holder pays off only interest at the end of each of the 3 months, the total amount of interest paid would be
:
which is the simple interest applied over 3 months, as calculated above. (The one cent difference arises due to rounding to the nearest cent.)
Compound interest
Compound interest includes interest earned on the interest that was previously accumulated.
Compare, for example, a bond paying 6 percent semiannually (that is, coupons of 3 percent twice a year) with a certificate of deposit ( GIC) that pays 6 percent interest once a year. The total interest payment is $6 per $100 par value in both cases, but the holder of the semiannual bond receives half the $6 per year after only 6 months ( time preference), and so has the opportunity to reinvest the first $3 coupon payment after the first 6 months, and earn additional interest.
For example, suppose an investor buys $10,000 par value of a US dollar bond, which pays coupons twice a year, and that the bond's simple annual coupon rate is 6 percent per year. This means that every 6 months, the issuer pays the holder of the bond a coupon of 3 dollars per 100 dollars par value. At the end of 6 months, the issuer pays the holder:
:
Assuming the market price of the bond is 100, so it is trading at par value, suppose further that the holder immediately reinvests the coupon by spending it on another $300 par value of the bond. In total, the investor therefore now holds:
:
and so earns a coupon at the end of the next 6 months of:
:
Assuming the bond remains priced at par, the investor accumulates at the end of a full 12 months a total value of:
:
and the investor earned in total:
:
The formula for the annual equivalent compound interest rate is:
:
where
:r is the simple annual rate of interest
:n is the frequency of applying interest
For example, in the case of a 6% simple annual rate, the annual equivalent compound rate is:
:
Other formulations
The outstanding balance ''Bn'' of a loan after ''n'' regular payments increases each period by a growth factor according to the periodic interest, and then decreases by the amount paid ''p'' at the end of each period:
:
where
:''i'' = simple annual loan rate in decimal form (for example, 10% = 0.10. The loan rate is the rate used to compute payments and balances.)
:''r'' = period interest rate (for example, ''i''/12 for monthly payments :''B''0 = initial balance, which equals the
principal sum
Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to another party, the creditor. Debt is a deferred payment, or series of payments, which differentiates it from an immediate purchase. The ...
By repeated substitution, one obtains expressions for ''B''''n'', which are linearly proportional to ''B''0 and ''p'', and use of the formula for the partial sum of a geometric series results in
:
A solution of this expression for ''p'' in terms of ''B''0 and ''B''''n'' reduces to
: