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In
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of fina ...
and
economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
, interest is payment from a
borrower A debtor or debitor is a legal entity (legal person) that owes a debt to another entity. The entity may be an individual, a firm, a government, a company or other legal person. The counterparty is called a creditor. When the counterpart of this ...
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 property ...
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 A fee is the price one pays as remuneration for rights or services. Fees usually allow for overhead, wages, costs, and markup. Traditionally, professionals in the United Kingdom (and previously the Republic of Ireland) receive a fee in contra ...
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-in ...
which is paid by a company to its shareholders (owners) from its
profit Profit may refer to: Business and law * Profit (accounting), the difference between the purchase price and the costs of bringing to market * Profit (economics), normal profit and economic profit * Profit (real property), a nonpossessory intere ...
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 vi ...
, 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 environme ...
taking entrepreneurs when the revenue earned exceeds the total costs. For example, a customer would usually pay interest to
borrow Borrow or borrowing can mean: to receive (something) from somebody temporarily, expecting to return it. *In finance, monetary debt *In language, the use of loanwords * In arithmetic, when a digit becomes less than zero and the deficiency is taken f ...
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 Profit may refer to: Business and law * Profit (accounting), the difference between the purchase price and the costs of bringing to market * Profit (economics), normal profit and economic profit * Profit (real property), a nonpossessory intere ...
, 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 accountancy, 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 ...
,
investment Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose of investing i ...
or
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 * Enterpris ...
. (Interest may be part or the whole of the profit on an
investment Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose of investing i ...
, but the two concepts are distinct from each other from an
accounting Accounting, also known as accountancy, is the measurement, processing, and communication of financial and non financial information about economic entities such as businesses and corporations. Accounting, which has been called the "languag ...
perspective.) The
rate of interest In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is distinct ...
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 ch ...
(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 The Laws of Eshnunna (abrv. LE) are inscribed on two cuneiform tablets discovered in Tell Abū Harmal, Baghdad, Iraq. The Iraqi Directorate of Antiquities headed by Taha Baqir unearthed two parallel sets of tablets in 1945 and 1947. The two tablets ...
instituted a legal interest rate, specifically on deposits of
dowry A dowry is a payment, such as property or money, paid by the bride's family to the groom or his family at the time of marriage. Dowry contrasts with the related concepts of bride price and dower. While bride price or bride service is a payment b ...
. Early Muslims called this ''riba'', translated today as the charging of interest. The
First Council of Nicaea The First Council of Nicaea (; grc, Νίκαια ) was a council of Christian bishops convened in the Bithynian city of Nicaea (now İznik, Turkey) by the Roman Emperor Constantine I in AD 325. This ecumenical council was the first effort ...
, 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 ch ...
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 are ...
s applied this regulation to the
laity In religious organizations, the laity () consists of all members who are not part of the clergy, usually including any non-ordained members of religious orders, e.g. a nun or a lay brother. In both religious and wider secular usage, a layperson ...
.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 largest Christian church, with 1.3 billion baptized Catholics worldwide . It is among the world's oldest and largest international institutions, and has played a ...
opposition to interest hardened in the era of scholastics, when even defending it was considered a
heresy Heresy is any belief or theory that is strongly at variance with established beliefs or customs, in particular the accepted beliefs of a church or religious organization. The term is usually used in reference to violations of important religi ...
. St.
Thomas Aquinas Thomas Aquinas, OP (; it, Tommaso d'Aquino, lit=Thomas of Aquino; 1225 – 7 March 1274) was an Italian Dominican friar and priest who was an influential philosopher, theologian and jurist in the tradition of scholasticism; he is known wi ...
, the leading theologian of the
Catholic Church The Catholic Church, also known as the Roman Catholic Church, is the largest Christian church, with 1.3 billion baptized Catholics worldwide . It is among the world's oldest and largest international institutions, and has played a ...
, 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, 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 ideas ...
era, greater mobility of people facilitated an increase in commerce and the appearance of appropriate conditions for
entrepreneur Entrepreneurship is the creation or extraction of economic value. With this definition, entrepreneurship is viewed as change, generally entailing risk beyond what is normally encountered in starting a business, which may include other values th ...
s 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 Circulation (curren ...
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 F ...
in 1847.


Islamic finance

The latter half of the 20th century saw the rise of interest-free
Islamic banking and finance Islamic banking, Islamic finance ( ar, مصرفية إسلامية), or Sharia-compliant finance is banking or financing activity that complies with Sharia (Islamic law) and its practical application through the development of Islamic economic ...
, 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 Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
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: : \lim_ \left( 1 + \dfrac \right)^n = e, 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 Credit (from Latin verb ''credit'', meaning "one believes") is the trust which allows one party to provide money or resources to another party wherein the second party does not reimburse the first party immediately (thereby generating a debt), ...
, and it plays the role of the
cost of capital In economics and accounting, the cost of capital is the cost of a company's funds (both debt and equity), or from an investor's point of view is "the required rate of return on a portfolio company's existing securities". It is used to evaluate new ...
. 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 o ...
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 Circulation (curren ...
, and one explanation of the tendency of interest rates to be generally greater than zero is the scarcity of
loanable funds In economics, the loanable funds doctrine is a theory of the market interest rate. According to this approach, the interest rate is determined by the demand for and supply of loanable funds. The term ''loanable funds'' includes all forms of credit, ...
. Over centuries, various schools of thought have developed explanations of interest and interest rates. The
School of Salamanca The School of Salamanca ( es, Escuela de Salamanca) is the Renaissance of thought in diverse intellectual areas by Spanish theologians, rooted in the intellectual and pedagogical work of Francisco de Vitoria. From the beginning of the 16th cen ...
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 Martín de Azpilcueta (Azpilkueta in Basque) (13 December 1491 – 1 June 1586), or Doctor Navarrus, was an important Spanish canonist and theologian in his time, and an early economist who independently formulated the quantity theory of mone ...
applied a
time preference In economics, time preference (or time discounting, delay discounting, temporal discounting, long-term orientation) is the current relative valuation placed on receiving a good or some cash at an earlier date compared with receiving it at a later ...
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 considere ...
. By applying an
opportunity cost In microeconomic theory, the opportunity cost of a particular activity is the value or benefit given up by engaging in that activity, relative to engaging in an alternative activity. More effective it means if you chose one activity (for example ...
argument, comparing the loan rate with the
rate of return In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, ca ...
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 conso ...
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 Adam Smith (baptized 1723 – 17 July 1790) was a Scottish economist and philosopher who was a pioneer in the thinking of political economy and key figure during the Scottish Enlightenment. Seen by some as "The Father of Economics"——— ...
,
Carl Menger Carl Menger von Wolfensgrün (; ; 28 February 1840 – 26 February 1921) was an Austrian economist and the founder of the Austrian School of economics. Menger contributed to the development of the theories of marginalism and marginal utility ...
, 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 th ...
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 Bertil Gotthard Ohlin () (23 April 1899 – 3 August 1979) was a Swedish economist and politician. He was a professor of economics at the Stockholm School of Economics from 1929 to 1965. He was also leader of the People's Party, a social-liberal ...
and Dennis Robertson and became known as the
loanable funds In economics, the loanable funds doctrine is a theory of the market interest rate. According to this approach, the interest rate is determined by the demand for and supply of loanable funds. The term ''loanable funds'' includes all forms of credit, ...
theory. Other notable interest rate theories of the period are those of
Irving Fisher Irving Fisher (February 27, 1867 – April 29, 1947) was an American economist, statistician, inventor, eugenicist and progressive social campaigner. He was one of the earliest American neoclassical economists, though his later work on debt def ...
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: :\frac 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, th ...
:''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, th ...
is 12.99% ''per annum'', applied monthly, so the frequency of applying interest is 12 per year. Over one month, :\frac = \$27.06 interest is due (rounded to the nearest cent). Simple interest applied over 3 months would be :\frac = \$81.19 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 :\frac \times 3 = \$27.06\text \times 3\text =\$81.18 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 In economics, time preference (or time discounting, delay discounting, temporal discounting, long-term orientation) is the current relative valuation placed on receiving a good or some cash at an earlier date compared with receiving it at a later ...
), 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: :\frac = \frac = \$300 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: :\$10\,000 + \$300 = \left(1 + \frac\right) \cdot B = \left(1 + \frac\right) \times \$10\,000 and so earns a coupon at the end of the next 6 months of: :\begin\frac &= \frac \\ &= \frac \\ &=\$309\end Assuming the bond remains priced at par, the investor accumulates at the end of a full 12 months a total value of: :\begin\$10,000 + \$300 + \$309 &= \$10\,000 + \frac + \frac \\ &= \$10\,000 \times \left(1 + \frac\right)^2\end and the investor earned in total: :\begin\$10\,000 \times \left(1 + \frac \right)^2 - \$10\,000\\ = \$10\,000 \times \left( \left( 1 + \frac \right)^2 - 1\right)\end The formula for the annual equivalent compound interest rate is: :\left(1 + \frac\right)^n - 1 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: :\left(1 + \frac\right)^2 - 1 = 1.03^2 - 1 = 6.09\%


Other formulations

The outstanding
balance Balance or balancing may refer to: Common meanings * Balance (ability) in biomechanics * Balance (accounting) * Balance or weighing scale * Balance as in equality or equilibrium Arts and entertainment Film * ''Balance'' (1983 film), a Bulgarian ...
''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: :B_ = \big( 1 + r \big) B_ - p, 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 In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each succ ...
results in :B_n = (1 + r)^n B_0 - \frac p A solution of this expression for ''p'' in terms of ''B''0 and ''B''''n'' reduces to :p = r \left \frac \right/math> To find the payment if the loan is to be finished in ''n'' payments, one sets ''B''''n'' = 0. The PMT function found in
spreadsheet A spreadsheet is a computer application for computation, organization, analysis and storage of data in tabular form. Spreadsheets were developed as computerized analogs of paper accounting worksheets. The program operates on data entered in cel ...
programs can be used to calculate the monthly payment of a loan: :p=\mathrm(\text,\text,\text,\text,) = \mathrm(r,n,-B_0,B_n,) An interest-only payment on the current balance would be :p_I= r B. The total interest, ''I''''T'', paid on the loan is :I_ = np - B_0. The formulas for a regular savings program are similar, but the payments are added to the balances instead of being subtracted, and the formula for the payment is the negative of the one above. These formulas are only approximate since actual loan balances are affected by rounding. To avoid an underpayment at the end of the loan, the payment must be rounded up to the next cent. Consider a similar loan but with a new period equal to ''k'' periods of the problem above. If ''r''''k'' and ''p''''k'' are the new rate and payment, we now have :B_k = B'_0 = (1 + r_k) B_0 - p_k. Comparing this with the expression for Bk above, we note that :r_k = (1 + r)^k - 1 and :p_k = \frac r_k. The last equation allows us to define a constant that is the same for both problems: :B^ = \frac = \frac and ''B''''k'' can be written as :B_k = (1 + r_k) B_0 - r_k B^*. Solving for ''r''''k'', we find a formula for ''r''''k'' involving known quantities and ''B''''k'', the balance after ''k'' periods: :r_k = \frac. Since ''B''0 could be any balance in the loan, the formula works for any two balances separate by ''k'' periods and can be used to compute a value for the annual interest rate. ''B''* is a
scale invariant In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term ...
, since it does not change with changes in the length of the period. Rearranging the equation for ''B''*, one obtains a transformation coefficient (
scale factor In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
): :\lambda_k = \frac = \frac = \frac = k\left + \frac + \cdots\right/math> (see
binomial theorem In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
) and we see that ''r'' and ''p'' transform in the same manner: :r_k=\lambda_k r :p_k=\lambda_k p. The change in the balance transforms likewise: :\Delta B_k=B'-B=(\lambda_k rB-\lambda_k p)=\lambda_k \, \Delta B , which gives an insight into the meaning of some of the coefficients found in the formulas above. The annual rate, ''r''12, assumes only one payment per year and is not an "effective" rate for monthly payments. With monthly payments, the monthly interest is paid out of each payment and so should not be compounded, and an annual rate of 12·''r'' would make more sense. If one just made interest-only payments, the amount paid for the year would be 12·''r''·''B''0. Substituting ''p''''k'' = ''r''''k'' ''B''* into the equation for the ''B''''k'', we obtain :B_k=B_0-r_k(B^*-B_0). Since ''B''''n'' = 0, we can solve for ''B''*: :B^ = B_0 \left(\frac + 1 \right). Substituting back into the formula for the ''B''''k'' shows that they are a linear function of the ''r''''k'' and therefore the ''λ''''k'': :B_k=B_0\left(1-\frac\right)=B_0\left(1-\frac\right). This is the easiest way of estimating the balances if the ''λ''''k'' are known. Substituting into the first formula for ''B''''k'' above and solving for ''λ''''k''+1, we obtain :\lambda_=1+(1+r)\lambda_k. ''λ''0 and ''λ''''n'' can be found using the formula for ''λ''''k'' above or computing the ''λ''''k'' recursively from ''λ''0 = 0 to ''λ''''n''. Since ''p'' = ''rB''*, the formula for the payment reduces to :p=\left(r+\frac\right)B_0 and the average interest rate over the period of the loan is :r_\text = \frac = r + \frac - \frac, which is less than ''r'' if ''n'' > 1.


Discount instruments

* US and Canadian T-Bills (short term Government debt) have a different calculation for interest. 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 also:
Day count convention In finance, a day count convention determines how interest accrues over time for a variety of investments, including bonds, notes, loans, mortgages, medium-term notes, swaps, and forward rate agreements (FRAs). This determines the number of days b ...
). The total calculation is ((100 − ''P'')/''P'')·((365/''t'')·100). This is equivalent to calculating the price by a process called discounting at a simple interest rate.


Rules of thumb


Rule of 78s

In the age before electronic computing power was widely available, flat rate consumer loans in the United States of America would be priced using the Rule of 78s, or "sum of digits" method. (The sum of the integers from 1 to 12 is 78.) The technique required only a simple calculation. Payments remain constant over the life of the loan; however, payments are allocated to interest in progressively smaller amounts. In a one-year loan, in the first month, 12/78 of all interest owed over the life of the loan is due; in the second month, 11/78; progressing to the twelfth month where only 1/78 of all interest is due. The practical effect of the Rule of 78s is to make early pay-offs of term loans more expensive. For a one-year loan, approximately 3/4 of all interest due is collected by the sixth month, and pay-off of the principal then will cause the effective interest rate to be much higher than the APR used to calculate the payments.Rule of 78 – Watch out for this auto loan trick
/ref> In 1992, the
United States The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
outlawed the use of "Rule of 78s" interest in connection with mortgage refinancing and other consumer loans over five years in term. Certain other jurisdictions have outlawed application of the Rule of 78s in certain types of loans, particularly consumer loans.


Rule of 72

To approximate how long it takes for money to double at a given interest rate, that is, for accumulated compound interest to reach or exceed the initial deposit, divide 72 by the percentage interest rate. For example, compounding at an annual interest rate of 6 percent, it will take 72/6 = 12 years for the money to double. The rule provides a good indication for interest rates up to 10%. In the case of an interest rate of 18 percent, the rule of 72 predicts that money will double after 72/18 = 4 years. :1.18^4 =1.9388 \text In the case of an interest rate of 24 percent, the rule predicts that money will double after 72/24 = 3 years. :1.24^3 = 1.9066 \text


Market interest rates

There are markets for investments (which include the money market, bond market, as well as retail financial institutions like banks) that set
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, th ...
s. Each specific debt takes into account the following factors in determining its interest rate:


Opportunity cost and deferred consumption

Opportunity cost In microeconomic theory, the opportunity cost of a particular activity is the value or benefit given up by engaging in that activity, relative to engaging in an alternative activity. More effective it means if you chose one activity (for example ...
encompasses any other use to which the money could be put, including lending to others, investing elsewhere, holding cash, or spending the funds. Charging interest equal to inflation preserves the lender's purchasing power, but does not compensate for the
time value of money The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference. The t ...
in
real terms In economics, nominal value is measured in terms of money, whereas real value is measured against goods or services. A real value is one which has been adjusted for inflation, enabling comparison of quantities as if the prices of goods had not c ...
. The lender may prefer to invest in another product rather than consume. The return they might obtain from competing investments is a factor in determining the interest rate they demand.


Inflation

Since the lender is deferring consumption, they will ''wish'', as a bare minimum, to recover enough to pay the increased cost of goods due to
inflation In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reductio ...
. Because future inflation is unknown, there are three ways this might be achieved: * Charge X% interest "plus inflation" Many governments issue "real-return" or "inflation indexed" bonds. The principal amount or the interest payments are continually increased by the rate of inflation. See the discussion at
real interest rate The real interest rate is the rate of interest an investor, saver or lender receives (or expects to receive) after allowing for inflation. It can be described more formally by the Fisher equation, which states that the real interest rate is approx ...
. * Decide on the "expected" inflation rate. This still leaves the lender exposed to the risk of "unexpected" inflation. * Allow the interest rate to be periodically changed. While a "fixed interest rate" remains the same throughout the life of the debt, "variable" or "floating" rates can be reset. There are derivative products that allow for hedging and swaps between the two. However interest rates are set by the market, and it happens frequently that they are insufficient to compensate for inflation: for example at times of high inflation during, for example, the oil crisis; and currently (2011) when real yields on many inflation-linked government stocks are negative.


Default

There is always the risk the borrower will become
bankrupt Bankruptcy is a legal process through which people or other entities who cannot repay debts to creditors may seek relief from some or all of their debts. In most jurisdictions, bankruptcy is imposed by a court order, often initiated by the debt ...
,
abscond Absconding may refer to: * Escaping from incarceration or bail * A form of swarming (honey bee) Swarming is a honey bee colony's natural means of reproduction. In the process of swarming, a single colony splits into two or more distinct colonies. ...
or otherwise default on the loan. The
risk premium A risk premium is a measure of excess return that is required by an individual to compensate being subjected to an increased level of risk. It is used widely in finance and economics, the general definition being the expected risky return less t ...
attempts to measure the integrity of the borrower, the risk of his enterprise succeeding and the security of any collateral pledged. For example, loans to developing countries have higher risk premiums than those to the US government due to the difference in creditworthiness. An operating line of credit to a business will have a higher rate than a
mortgage loan A mortgage loan or simply mortgage (), in civil law jurisdicions known also as a hypothec loan, is a loan used either by purchasers of real property to raise funds to buy real estate, or by existing property owners to raise funds for any p ...
. The
creditworthiness A credit risk is risk of default on a debt that may arise from a borrower failing to make required payments. In the first resort, the risk is that of the lender and includes lost principal and interest, disruption to cash flows, and increased c ...
of businesses is measured by bond rating services and individual's
credit score A credit score is a numerical expression based on a level analysis of a person's credit files, to represent the creditworthiness of an individual. A credit score is primarily based on a credit report, information typically sourced from credit bu ...
s by
credit bureau A credit bureau is a data collection agency that gathers account information from various creditors and provides that information to a consumer reporting agency in the United States, a credit reference agency in the United Kingdom, a credit repor ...
s. The risks of an individual debt may have a large standard deviation of possibilities. The lender may want to cover his maximum risk, but lenders with portfolios of debt can lower the risk premium to cover just the most probable outcome.


Composition of interest rates

In economics, interest is considered the price of credit, therefore, it is also subject to distortions due to
inflation In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reductio ...
. The nominal interest rate, which refers to the price before adjustment to inflation, is the one visible to the consumer (that is, the interest tagged in a loan contract, credit card statement, etc.). Nominal interest is composed of the
real interest rate The real interest rate is the rate of interest an investor, saver or lender receives (or expects to receive) after allowing for inflation. It can be described more formally by the Fisher equation, which states that the real interest rate is approx ...
plus inflation, among other factors. An approximate formula for the nominal interest is: : i= r + \pi Where :''i'' is the nominal interest rate :''r'' is the real interest rate :and is inflation. However, not all borrowers and lenders have access to the same interest rate, even if they are subject to the same inflation. Furthermore, expectations of future inflation vary, so a forward-looking interest rate cannot depend on a single real interest rate plus a single expected rate of inflation. Interest rates also depend on credit quality or risk of default.
Governments A government is the system or group of people governing an organized community, generally a state. In the case of its broad associative definition, government normally consists of legislature, executive, and judiciary. Government is a ...
are normally highly reliable
debtor A debtor or debitor is a legal entity (legal person) that owes a debt to another entity. The entity may be an individual, a firm, a government, a company or other legal person. The counterparty is called a creditor. When the counterpart of this ...
s, and the interest rate on government securities is normally lower than the interest rate available to other borrowers. The equation: : i = r + \pi + c relates expectations of inflation and credit risk to nominal and expected real interest rates, over the life of a loan, where :''i'' is the nominal interest applied :''r'' is the real interest expected : is the inflation expected and :''c'' is
yield spread Yield may refer to: Measures of output/function Computer science * Yield (multithreading) is an action that occurs in a computer program during multithreading * See generator (computer programming) Physics/chemistry * Yield (chemistry), the am ...
according to the perceived credit risk.


Default interest

Default interest is the rate of interest that a borrower must pay after material breach of a loan covenant. The default interest is usually much higher than the original interest rate since it is reflecting the aggravation in the financial risk of the borrower. Default interest compensates the lender for the added risk. From the borrower's perspective, this means failure to make their regular payment for one or two payment periods or failure to pay taxes or insurance premiums for the loan collateral will lead to substantially higher interest for the entire remaining term of the loan. Banks tend to add default interest to the loan agreements in order to separate between different scenarios. In some jurisdictions, default interest clauses are unenforceable as against public policy.


Term

Shorter terms often have less risk of default and exposure to inflation because the near future is easier to predict. In these circumstances, short-term interest rates are lower than longer-term interest rates (an upward sloping
yield curve In finance, the yield curve is a graph which depicts how the yields on debt instruments - such as bonds - vary as a function of their years remaining to maturity. Typically, the graph's horizontal or x-axis is a time line of months or ye ...
).


Government intervention

Interest rates are generally determined by the market, but government intervention - usually by a
central bank A central bank, reserve bank, or monetary authority is an institution that manages the currency and monetary policy of a country or monetary union, and oversees their commercial banking system. In contrast to a commercial bank, a central ba ...
- may strongly influence short-term interest rates, and is one of the main tools of
monetary policy Monetary policy is the policy adopted by the monetary authority of a nation to control either the interest rate payable for very short-term borrowing (borrowing by banks from each other to meet their short-term needs) or the money supply, often a ...
. The central bank offers to borrow (or lend) large quantities of money at a rate which they determine (sometimes this is money that they have created ''ex nihilo'', that is, printed) which has a major influence on supply and demand and hence on market interest rates.


Open market operations in the United States

The
Federal Reserve The Federal Reserve System (often shortened to the Federal Reserve, or simply the Fed) is the central banking system of the United States of America. It was created on December 23, 1913, with the enactment of the Federal Reserve Act, after a ...
(Fed) implements monetary policy largely by targeting the
federal funds rate In the United States, the federal funds rate is the interest rate at which depository institutions (banks and credit unions) lend reserve balances to other depository institutions overnight on an uncollateralized basis. Reserve balances a ...
. This is the rate that banks charge each other for overnight loans of
federal funds In the United States, federal funds are overnight borrowings between banks and other entities to maintain their bank reserves at the Federal Reserve. Banks keep reserves at Federal Reserve Banks to meet their reserve requirements and to clear ...
. Federal funds are the reserves held by banks at the Fed.
Open market operations In macroeconomics, an open market operation (OMO) is an activity by a central bank to give (or take) liquidity in its currency to (or from) a bank or a group of banks. The central bank can either buy or sell government bonds (or other financial as ...
are one tool within monetary policy implemented by the Federal Reserve to steer short-term interest rates. Using the power to buy and sell treasury
securities A security is a tradable financial asset. The term commonly refers to any form of financial instrument, but its legal definition varies by jurisdiction. In some countries and languages people commonly use the term "security" to refer to any for ...
, the Open Market Desk at the
Federal Reserve Bank of New York The Federal Reserve Bank of New York is one of the 12 Federal Reserve Banks of the United States. It is responsible for the Second District of the Federal Reserve System, which encompasses the State of New York, the 12 northern counties of New ...
can supply the market with dollars by purchasing U.S. Treasury notes, hence increasing the nation's money supply. By increasing the money supply or
Aggregate Supply of Funding Aggregate or aggregates may refer to: Computing and mathematics * collection of objects that are bound together by a root entity, otherwise known as an aggregate root. The aggregate root guarantees the consistency of changes being made within the ...
(ASF), interest rates will fall due to the excess of dollars banks will end up with in their reserves.
Excess reserves Excess reserves are bank reserves held by a bank in excess of a reserve requirement for it set by a central bank. In the United States, bank reserves for a commercial bank are represented by its cash holdings and any credit balance in an account ...
may be lent in the
Fed funds In the United States, federal funds are overnight borrowings between banks and other entities to maintain their bank reserves at the Federal Reserve. Banks keep reserves at Federal Reserve Banks to meet their reserve requirements and to clear ...
market to other banks, thus driving down rates.


Interest rates and credit risk

It is increasingly recognized that during the business cycle,
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, th ...
s and
credit risk A credit risk is risk of default on a debt that may arise from a borrower failing to make required payments. In the first resort, the risk is that of the lender and includes lost principal and interest, disruption to cash flows, and increased ...
are tightly interrelated. The Jarrow-Turnbull model was the first model of credit risk that explicitly had random interest rates at its core. Lando (2004),
Darrell Duffie James Darrell Duffie (born May 23, 1954) is a Canadian financial economist and is Dean Witter Distinguished Professor of Finance at Stanford Graduate School of Business. He is the author of numerous research articles, and several books, includi ...
and Singleton (2003), and van Deventer and Imai (2003) discuss interest rates when the issuer of the interest-bearing instrument can default.


Money and inflation

Loans and bonds have some of the characteristics of money and are included in the broad money supply. National governments (provided, of course, that the country has retained its own currency) can influence interest rates and thus the supply and demand for such loans, thus altering the total of loans and bonds issued. Generally speaking, a higher real interest rate reduces the broad money supply. Through the
quantity theory of money In monetary economics, the quantity theory of money (often abbreviated QTM) is one of the directions of Western economic thought that emerged in the 16th-17th centuries. The QTM states that the general price level of goods and services is directly ...
, increases in the money supply lead to inflation. This means that interest rates can affect inflation in the future.


Liquidity

Liquidity Liquidity is a concept in economics involving the convertibility of assets and obligations. It can include: * Market liquidity, the ease with which an asset can be sold * Accounting liquidity, the ability to meet cash obligations when due * Liqui ...
is the ability to quickly re-sell an asset for fair or near-fair value. All else equal, an investor will want a higher return on an illiquid asset than a liquid one, to compensate for the loss of the option to sell it at any time. U.S. Treasury bonds are highly liquid with an active secondary market, while some other debts are less liquid. In the
mortgage A mortgage loan or simply mortgage (), in civil law jurisdicions known also as a hypothec loan, is a loan used either by purchasers of real property to raise funds to buy real estate, or by existing property owners to raise funds for any pu ...
market, the lowest rates are often issued on loans that can be re-sold as securitized loans. Highly non-traditional loans such as seller financing often carry higher interest rates due to a lack of liquidity.


Theories of interest


Aristotle's view of interest

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
and the Scholastics held that it was unjust to claim payment except in compensation for one's own efforts and sacrifices, and that since money is by its nature sterile, there is no loss in being temporarily separated from it. Compensation for risk or for the trouble of setting up a loan was not necessarily impermissible on these grounds.


Development of the theory of interest during the seventeenth and eighteenth centuries

Nicholas Barbon Nicholas Barbon ( 1640 – 1698) was an English economist, physician, and financial speculator. Historians of mercantilism consider him to be one of the first proponents of the free market. In the aftermath of the Great Fire of London, he b ...
(c.1640–c.1698) described as a "mistake" the view that interest is a monetary value, arguing that because money is typically borrowed to buy assets (goods and stock), the interest that is charged on a loan is a type of rent – "a payment for the use of goods". According to Schumpeter, Barbon's theories were forgotten until similar views were put forward by Joseph Massie in 1750. In 1752
David Hume David Hume (; born David Home; 7 May 1711 NS (26 April 1711 OS) – 25 August 1776) Cranston, Maurice, and Thomas Edmund Jessop. 2020 999br>David Hume" ''Encyclopædia Britannica''. Retrieved 18 May 2020. was a Scottish Enlightenment philo ...
published his essay "Of money" which relates interest to the "demand for borrowing", the "riches available to supply that demand" and the "profits arising from commerce". Schumpeter considered Hume's theory superior to that of Ricardo and Mill, but the reference to profits concentrates to a surprising degree on 'commerce' rather than on industry.
Turgot Turgot may refer to: * Turgot of Durham ( – 1115), Prior of Durham and Bishop of St Andrews * Michel-Étienne Turgot (1690–1751), mayor of Paris * Anne Robert Jacques Turgot (1727–1781), French economist and statesman * Louis Félix Étienn ...
brought the theory of interest close to its classical form. Industrialists...
... share their profits with capitalists who supply the funds (''Réflexions'', LXXI). The share that goes to the latter is determined like all other prices (LXXV) by the play of supply and demand amongst borrowers and lenders, so that the analysis is from the outset firmly planted in the general theory of prices.


The classical theory of the interest rate

The classical theory was the work of a number of authors, including Turgot,
Ricardo Ricardo is the Spanish and Portuguese cognate of the name Richard. It derived from Proto-Germanic ''*rīks'' 'king, ruler' + ''*harduz'' 'hard, brave'. It may be a given name, or a surname. People Given name *Ricardo de Araújo Pereira, Portugu ...
,
Mountifort Longfield Samuel Mountifort Longfield (1802 – 21 November 1884) was an Irish lawyer, judge, mathematician, and academic. He was the first Professor of Political Economy at Trinity College, Dublin. Life He was son of Mountifort Longfield, vicar of Desert ...
,
J. S. Mill John Stuart Mill (20 May 1806 – 7 May 1873) was an English philosopher, political economist, Member of Parliament (MP) and civil servant. One of the most influential thinkers in the history of classical liberalism, he contributed widely to ...
, and
Irving Fisher Irving Fisher (February 27, 1867 – April 29, 1947) was an American economist, statistician, inventor, eugenicist and progressive social campaigner. He was one of the earliest American neoclassical economists, though his later work on debt def ...
. It was strongly criticised by
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 m ...
whose remarks nonetheless made a positive contribution to it. Mill's theory is set out the chapter "Of the rate of interest" in his "Principles of political economy". He says that the interest rate adjusts to maintain equilibrium between the demands for lending and borrowing. Individuals lend in order to defer consumption or for the sake of the greater quantity they will be able to consume at a later date owing to interest earned. They borrow in order to anticipate consumption (whose relative desirability is reflected by the
time value of money The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference. The t ...
), but entrepreneurs also borrow to fund investment and governments borrow for their own reasons. The three sources of demand compete for loans. For entrepreneurial borrowing to be in equilibrium with lending:
The interest for money... is... regulated... by the rate of profits which can be made by the employment of capital...
Ricardo's and Mill's 'profit' is made more precise by the concept of the marginal efficiency of capital (the expression, though not the concept, is due to Keynes), which may be defined as the annual revenue which will be yielded by an extra increment of capital as a proportion of its cost. So the interest rate ''r'' in equilibrium will be equal to the marginal efficiency of capital ''r''. Rather than work with ''r'' and ''r'' as separate variables, we can assume that they are equal and let the single variable ''r'' denote their common value. The investment schedule ''i'' (''r'') shows how much investment is possible with a return of at least ''r''. In a stationary economy it is likely to resemble the blue curve in the diagram, with a step shape arising from the assumption that opportunities to invest with yields greater than ''r̂'' have been largely exhausted while there is untapped scope to invest with a lower return.Mill §3; Longfield. Saving is the excess of deferred over anticipated consumption, and its dependence on income is much as described by Keynes (see
The General Theory ''The General Theory of Employment, Interest and Money'' is a book by English economist John Maynard Keynes published in February 1936. It caused a profound shift in economic thought, giving macroeconomics a central place in economic theory and ...
), but in classical theory definitely an increasing function of ''r''. (The dependence of ''s'' on income ''y'' was not relevant to classical concerns prior to the development of theories of
unemployment Unemployment, according to the OECD (Organisation for Economic Co-operation and Development), is people above a specified age (usually 15) not being in paid employment or self-employment but currently available for Work (human activity), w ...
.) The rate of interest is given by the intersection of the solid red saving curve with the blue investment schedule. But so long as the investment schedule is almost vertical, a change in income (leading in extreme cases to the broken red saving curve) will make little difference to the interest rate. In some cases the analysis will be less simple. The introduction of a new technique, leading to demand for new forms of capital, will shift the step to the right and reduce its steepness. Or a sudden increase in the desire to anticipate consumption (perhaps through military spending in time of war) will absorb most available loans; the interest rate will increase and investment will be reduced to the amount whose return exceeds it. This is illustrated by the dotted red saving curve.


Keynes's criticisms

In the case of extraordinary spending in time of war the government may wish to borrow more than the public would be willing to lend at a normal interest rate. If the dotted red curve started negative and showed no tendency to increase with ''r'', then the government would be trying to buy what the public was unwilling to sell at any price. Keynes mentions this possibility as a point "which might, perhaps, have warned the classical school that something was wrong" (p. 182). He also remarks (on the same page) that the classical theory does not explain the usual supposition that "an increase in the quantity of money has a tendency to reduce the rate of interest, at any rate in the first instance". Keynes's diagram of the investment schecule lacks the step shape which can be seen as part of the classical theory. He objects that
the functions used by classical theory... do not furnish material for a theory of the rate of interest; but they could be used to tell us... what the rate of interest will have to be, if the level of employment hich determines incomeis maintained at a given figure.
Later (p. 184) Keynes claims that "it involves a circular argument" to construct a theory of interest from the investment schedule since
the 'marginal efficiency of capital' partly depends on the scale of current investment, and we must already know the rate of interest before we can calculate what this scale will be.


Theories of exploitation, productivity and abstinence

The classical theory of interest explains it as the capitalist's share of business profits, but the pre-marginalist authors were unable to reconcile these profits with the
labor theory of value The labor theory of value (LTV) is a theory of value that argues that the economic value of a good or service is determined by the total amount of " socially necessary labor" required to produce it. The LTV is usually associated with Marxian e ...
(excluding Longfield, who was essentially a marginalist). Their responses often had a moral tone: Ricardo and Marx viewed profits as exploitation, and
McCulloch McCulloch is a Scottish surname. It's a variation of the Northern Irish surname McCullough. It's commonly found in Galloway. Notable people with the surname include: *Alan McCulloch (politician), New Zealand politician *Alan McLeod McCulloch ( ...
's productivity theory justified profits by portraying capital equipment as an embodiment of accumulated labor. The theory that interest is a payment for abstinence is attributed to
Nassau Senior Nassau William Senior (; 26 September 1790 – 4 June 1864), was an English lawyer known as an economist. He was also a government adviser over several decades on economic and social policy on which he wrote extensively. Early life He was born ...
, and according to Schumpeter was intended neutrally, but it can easily be understood as making a moral claim and was sharply criticised by Marx and Lassalle.


Wicksell's theory

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 th ...
published his "Interest and Prices" in 1898, elaborating a comprehensive theory of economic crises based upon a distinction between natural and nominal interest rates.
Wicksell's contribution, in fact, was twofold. First he separated the monetary rate of interest from the hypothetical "natural" rate that would have resulted from equilibrium of capital supply and demand in a barter economy, and he assumed that as a result of the presence of money alone, the effective market rate could fail to correspond to this ideal rate in actuality. Next he supposed that through the mechanism of credit, the rate of interest had an influence on prices; that a rise of the monetary rate above the "natural" level produced a fall, and a decline below that level a rise, in prices. But Wicksell went on to conclude that if the natural rate coincided with the monetary rate, stability of prices would follow.
In the 1930s Wicksell's approach was refined by
Bertil Ohlin Bertil Gotthard Ohlin () (23 April 1899 – 3 August 1979) was a Swedish economist and politician. He was a professor of economics at the Stockholm School of Economics from 1929 to 1965. He was also leader of the People's Party, a social-liberal ...
and Dennis Robertson and became known as the
loanable funds In economics, the loanable funds doctrine is a theory of the market interest rate. According to this approach, the interest rate is determined by the demand for and supply of loanable funds. The term ''loanable funds'' includes all forms of credit, ...
theory.


Austrian theories

Eugen Böhm von Bawerk Eugen is a masculine given name which may refer to: * Archduke Eugen of Austria (1863–1954), last Habsburg Grandmaster of the Teutonic Order from 1894 to 1923 * Prince Eugen, Duke of Närke (1865–1947), Swedish painter, art collector, and pat ...
and other members of the
Austrian School The Austrian School is a heterodox school of economic thought that advocates strict adherence to methodological individualism, the concept that social phenomena result exclusively from the motivations and actions of individuals. Austrian school ...
also put forward notable theories of the interest rate. The doyen of the Austrian school, Murray N. Rothbard, sees the emphasis on the loan market which makes up the general analysis on interest as a mistaken view to take. As he explains in his primary economic work, ''Man, Economy, and State'', the market rate of interest is but a ''manifestation'' of the natural phenomenon of time preference, which is to prefer present goods to future goods. To Rothbard, Interest is explainable by the rate of time preference among the people. To point to the loan market is insufficient at best. Rather, the rate of interest is what would be observed between the "stages of production", indeed a time market itself, where capital goods which are used to make consumers' goods are ordered out further in time away from the final consumers' goods stage of the economy where consumption takes place. It is ''this'' spread (between these various stages which will tend toward uniformity), with consumers' goods representing present goods and producers' goods representing future goods, that the real rate of interest is observed. Rothbard has said that Rothbard has furthermore criticized the Keynesian conception of interest, saying


Pareto's indifference

Pareto held that
The interest rate, being one of the many elements of the general system of equilibrium, was, of course, simultaneously determined with all of them so that there was no point at all in looking for any particular element that 'caused' interest.


Keynes's theory of the interest rate

Interest is one of the main components of the economic theories developed in
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 m ...
's 1936 ''General theory of employment, interest, and money''. In his initial account of
liquidity preference __NOTOC__ In macroeconomic theory, liquidity preference is the demand for money, considered as liquidity. The concept was first developed by John Maynard Keynes in his book ''The General Theory of Employment, Interest and Money'' (1936) to expl ...
(the demand for money) in Chapter 13, this demand is solely a function of the interest rate; and since the supply is given and equilibrium is assumed, the interest rate is determined by the money supply. In his later account (Chapter 15), interest cannot be separated from other economic variables and needs to be analysed together with them. See
The General Theory ''The General Theory of Employment, Interest and Money'' is a book by English economist John Maynard Keynes published in February 1936. It caused a profound shift in economic thought, giving macroeconomics a central place in economic theory and ...
for details.


In religious contexts


Judaism

Jews are forbidden from usury in dealing with fellow Jews, and this lending is to be considered
tzedakah ''Tzedakah'' or ''Ṣedaqah'' ( he, צדקה ) is a Hebrew word meaning "righteousness", but commonly used to signify ''charity''. This concept of "charity" differs from the modern Western understanding of "charity". The latter is typically un ...
, or charity. However, there are permissions to charge interest on loans to
non-Jews Gentile () is a word that usually means "someone who is not a Jews, Jew". Other Groups claiming affiliation with Israelites, groups that claim Israelite heritage, notably Mormons, sometimes use the term ''gentile'' to describe outsiders. More ra ...
. This is outlined in the
Jewish Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""The ...
scripture Religious texts, including scripture, are texts which various religions consider to be of central importance to their religious tradition. They differ from literature by being a compilation or discussion of beliefs, mythologies, ritual prac ...
s of the
Torah The Torah (; hbo, ''Tōrā'', "Instruction", "Teaching" or "Law") is the compilation of the first five books of the Hebrew Bible, namely the books of Genesis, Exodus, Leviticus, Numbers and Deuteronomy. In that sense, Torah means the s ...
, which
Christians Christians () are people who follow or adhere to Christianity, a monotheistic Abrahamic religion based on the life and teachings of Jesus Christ. The words ''Christ'' and ''Christian'' derive from the Koine Greek title ''Christós'' (Χρι ...
hold as part of the
Old Testament The Old Testament (often abbreviated OT) is the first division of the Christian biblical canon, which is based primarily upon the 24 books of the Hebrew Bible or Tanakh, a collection of ancient religious Hebrew writings by the Israelites. The ...
, and other books of the
Tanakh The Hebrew Bible or Tanakh (;"Tanach"
''
Jewish Publication Society The Jewish Publication Society (JPS), originally known as the Jewish Publication Society of America, is the oldest nonprofit, nondenominational publisher of Jewish works in English. Founded in Philadelphia in 1888, by reform Rabbi Joseph Krauskop ...
's 1917 Tanakh, with Christian verse numbers, where different, in parentheses: Several historical rulings in
Jewish law ''Halakha'' (; he, הֲלָכָה, ), also Romanization of Hebrew, transliterated as ''halacha'', ''halakhah'', and ''halocho'' ( ), is the collective body of Judaism, Jewish religious laws which is derived from the Torah, written and Oral Tora ...
have mitigated the allowances for usury toward non-Jews. For instance, the 15th-century commentator Rabbi Isaac Abrabanel specified that the rubric for allowing interest does not apply to Christians or Muslims, because their faith systems have a common ethical basis originating from Judaism. The medieval commentator Rabbi David Kimchi extended this principle to non-Jews who show consideration for Jews, saying they should be treated with the same consideration when they borrow.


Islam

The following quotations are English translations from the
Qur'an The Quran (, ; Standard Arabic: , Quranic Arabic: , , 'the recitation'), also romanized Qur'an or Koran, is the central religious text of Islam, believed by Muslims to be a revelation from God. It is organized in 114 chapters (pl.: , sing. ...
: The attitude of Muhammad to usury is articulated in his Last Sermon


Christianity

The
Old Testament The Old Testament (often abbreviated OT) is the first division of the Christian biblical canon, which is based primarily upon the 24 books of the Hebrew Bible or Tanakh, a collection of ancient religious Hebrew writings by the Israelites. The ...
"condemns the practice of charging interest because a loan should be an act of compassion and taking care of one's neighbor"; it teaches that "making a profit off a loan is exploiting that person and dishonoring God's covenant (Exodus 22:25–27)". The first of the scholastic Christian theologians,
Saint Anselm of Canterbury Anselm of Canterbury, OSB (; 1033/4–1109), also called ( it, Anselmo d'Aosta, link=no) after his birthplace and (french: Anselme du Bec, link=no) after his monastery, was an Italian Benedictine monk, abbot, philosopher and theologian of the ...
, led the shift in thought that labeled charging interest the same as theft. Previously usury had been seen as a lack of
charity Charity may refer to: Giving * Charitable organization or charity, a non-profit organization whose primary objectives are philanthropy and social well-being of persons * Charity (practice), the practice of being benevolent, giving and sharing * Ch ...
. St.
Thomas Aquinas Thomas Aquinas, OP (; it, Tommaso d'Aquino, lit=Thomas of Aquino; 1225 – 7 March 1274) was an Italian Dominican friar and priest who was an influential philosopher, theologian and jurist in the tradition of scholasticism; he is known wi ...
, the leading scholastic theologian of the
Roman Catholic Church The Catholic Church, also known as the Roman Catholic Church, is the largest Christian church, with 1.3 billion baptized Catholics worldwide . It is among the world's oldest and largest international institutions, and has played a ...
, argued charging of interest is wrong because it amounts to "double charging", charging for both the thing and the use of the thing. Aquinas said this would be morally wrong in the same way as if one sold a bottle of wine, charged for the bottle of wine, and then charged for the person using the wine to actually drink it. Similarly, one cannot charge for a piece of cake and for the eating of the piece of cake. Yet this, said Aquinas, is what usury does. Money is a medium of exchange, and is used up when it is spent. To charge for the money and for its use (by spending) is therefore to charge for the money twice. It is also to sell time since the usurer charges, in effect, for the time that the money is in the hands of the borrower. Time, however, is not a commodity that anyone can charge. In condemning usury Aquinas was much influenced by the recently rediscovered philosophical writings of
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of phil ...
and his desire to assimilate
Greek philosophy Ancient Greek philosophy arose in the 6th century BC, marking the end of the Greek Dark Ages. Greek philosophy continued throughout the Hellenistic period and the period in which Greece and most Greek-inhabited lands were part of the Roman Empir ...
with
Christian theology Christian theology is the theology of Christianity, Christian belief and practice. Such study concentrates primarily upon the texts of the Old Testament and of the New Testament, as well as on Christian tradition. Christian theology, theologian ...
. Aquinas argued that in the case of usury, as in other aspects of Christian revelation, Christian doctrine is reinforced by Aristotelian
natural law Natural law ( la, ius naturale, ''lex naturalis'') is a system of law based on a close observation of human nature, and based on values intrinsic to human nature that can be deduced and applied independently of positive law (the express enacte ...
rationalism. Aristotle's argument is that interest is unnatural, since money, as a sterile element, cannot naturally reproduce itself. Thus, usury conflicts with natural law just as it offends Christian revelation: see
Thought of Thomas Aquinas Thomas Aquinas, OP (; it, Tommaso d'Aquino, lit=Thomas of Aquino; 1225 – 7 March 1274) was an Italian Dominican friar and priest who was an influential philosopher, theologian and jurist in the tradition of scholasticism; he is known wit ...
. As such, Aquinas taught "that interest is inherently unjust and one who charges interest sins." Outlawing usury did not prevent investment, but stipulated that in order for the investor to share in the profit he must share the risk. In short he must be a joint-venturer. Simply to invest the money and expect it to be returned regardless of the success of the venture was to make money simply by having money and not by taking any risk or by doing any work or by any effort or sacrifice at all, which is usury. St Thomas quotes Aristotle as saying that "to live by usury is exceedingly unnatural". Islam likewise condemns usury but allowed commerce (Al-Baqarah 2:275) – an alternative that suggests investment and sharing of profit and loss instead of sharing only profit through interests. Judaism condemns usury towards Jews, but allows it towards non-Jews (Deut. 23:19–20). St Thomas allows, however, charges for actual services provided. Thus a banker or credit-lender could charge for such actual work or effort as he did carry out, for example, any fair administrative charges. The Catholic Church, in a decree of the
Fifth Council of the Lateran The Fifth Council of the Lateran, held between 1512 and 1517, was the eighteenth ecumenical council of the Catholic Church and was the last council before the Protestant Reformation and the Council of Trent. It was convoked by Pope Julius II to ...
, expressly allowed such charges in respect of credit-unions run for the benefit of the poor known as " montes pietatis". In the 13th century Cardinal Hostiensis enumerated thirteen situations in which charging interest was not immoral. The most important of these was ''lucrum cessans'' (profits given up) which allowed for the lender to charge interest "to compensate him for profit foregone in investing the money himself" . This idea is very similar to opportunity cost. Many scholastic thinkers who argued for a ban on interest charges also argued for the legitimacy of ''lucrum cessans'' profits (for example, Pierre Jean Olivi and
St. Bernardino of Siena Bernardino of Siena, OFM (8 September 138020 May 1444), also known as Bernardine, was an Italian priest and Franciscan missionary preacher in Italy. He was a systematizer of Scholastic economics. His preaching, his book burnings, and his " bon ...
). However, Hostiensis' exceptions, including for ''lucrum cessans'', were never accepted as official by the Roman Catholic Church. The
Westminster Confession of Faith The Westminster Confession of Faith is a Reformed confession of faith. Drawn up by the 1646 Westminster Assembly as part of the Westminster Standards to be a confession of the Church of England, it became and remains the " subordinate standard" ...
, a confession of faith upheld by the
Reformed Church Calvinism (also called the Reformed Tradition, Reformed Protestantism, Reformed Christianity, or simply Reformed) is a major branch of Protestantism that follows the theological tradition and forms of Christian practice set down by John Cal ...
es, teaches that usury--charging interest at any rate--is a
sin In a religious context, sin is a transgression against divine law. Each culture has its own interpretation of what it means to commit a sin. While sins are generally considered actions, any thought, word, or act considered immoral, selfish, s ...
prohibited by the eighth commandment. The Roman Catholic Church has always condemned usury, but in modern times, with the rise of capitalism and the disestablishment of the Catholic Church in majority Catholic countries, this prohibition on usury has not been enforced.
Pope Benedict XIV Pope Benedict XIV ( la, Benedictus XIV; it, Benedetto XIV; 31 March 1675 – 3 May 1758), born Prospero Lorenzo Lambertini, was head of the Catholic Church and ruler of the Papal States from 17 August 1740 to his death in May 1758.Antipope ...
's encyclica
''Vix Pervenit''
gives the reasons why usury is sinful:See also
''Church and the Usurers: Unprofitable Lending for the Modern Economy''
b
Dr. Brian McCall
o
''Interest and Usury''
b
Fr. Bernard W. Dempsey, S.J.
(1903–1960).)
The
Congregation of the Missionary Sons of the Immaculate Heart of Mary , image = Herb CMF.jpg , image_size = 175px , caption = Coat of arms of the Claretians , abbreviation = CMF , nickname = Claretians , formation = , founders = Anto ...
, a Catholic Christian
religious order A religious order is a lineage of communities and organizations of people who live in some way set apart from society in accordance with their specific religious devotion, usually characterized by the principles of its founder's religious practi ...
, thus teaches that:


See also

*
Actuarial notation Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables. Traditional notation uses a halo system where symbols are placed as superscript or subscript before or ...
*
Credit card interest Credit card interest is a way in which credit card issuers generate revenue. A card issuer is a bank or credit union that gives a consumer (the cardholder) a card or account number that can be used with various payees to make payments and borrow m ...
*
Credit rating agency A credit rating agency (CRA, also called a ratings service) is a company that assigns credit ratings, which rate a debtor's ability to pay back debt by making timely principal and interest payments and the likelihood of default. An agency may ra ...
* DIRTI 5 * Discount * Fisher equation *
Hire purchase A hire purchase (HP), also known as an installment plan, is an arrangement whereby a customer agrees to a contract to acquire an asset by paying an initial installment (e.g., 40% of the total) and repaying the balance of the price of the asset pl ...
*
Interest expense Interest expense relates to the cost of borrowing money. It is the price that a lender charges a borrower for the use of the lender's money. On the income statement, interest expense can represent the cost of borrowing money from banks, bond inve ...
*
Leasing A lease is a contractual arrangement calling for the user (referred to as the ''lessee'') to pay the owner (referred to as the ''lessor'') for the use of an asset. Property, buildings and vehicles are common assets that are leased. Industrial ...
*
Promissory note A promissory note, sometimes referred to as a note payable, is a legal instrument (more particularly, a financing instrument and a debt instrument), in which one party (the ''maker'' or ''issuer'') promises in writing to pay a determinate sum of ...
*
Risk-free interest rate The risk-free rate of return, usually shortened to the risk-free rate, is the rate of return of a hypothetical investment with scheduled payments over a fixed period of time that is assumed to meet all payment obligations. Since the risk-free ra ...


Notes


References

* * * * *


External links


White Paper: More than Math, The Lost Art of Interest calculation


Financial Services Authority The Financial Services Authority (FSA) was a quasi-judicial body accountable for the financial regulation, regulation of the financial services industry in the United Kingdom between 2001 and 2013. It was founded as the Securities and Investmen ...
(UK) *List of current interest rates:
World Interest Rates



"Which way to pay"

Deposit Rates in European Countries
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