TheInfoList

In
finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available which could ...

and
economics Economics () is a social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant life and a bran ...

, interest is payment from a
borrower A debtor or debitor is a legal entity (legal person) that owes a debt Debt is an obligation that requires one party, the debtor A debtor or debitor is a legal entity (legal person) that owes a debt Debt is an obligation that re ...
or deposit-taking financial institution to a
lender A creditor or lender is a party 300px, '' Hip, Hip, Hurrah!'' (1888) by Peder Severin Krøyer, a painting portraying an artists' party in 19th century Denmark A party is a gathering of people who have been invited by a host A host is ...

or depositor of an amount above repayment of the
principal sum Debt is an obligation that requires one party, the debtor A debtor or debitor is a legal entity (legal person) that owes a debt Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to ...
(that is, the amount borrowed), at a particular rate. It is distinct from a
fee A fee is the price A price is the (usually not negative) quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "l ...
which the borrower may pay the lender or some third party. It is also distinct from
dividend A dividend is a distribution of 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 ...

which is paid by a company to its shareholders (owners) from its
profit Profit may refer to: Business and law * Profit (accounting) Profit, in accounting Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information about economic entity, economic en ...
or reserve, 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 Uncertainty refers to Epistemology, epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to ...

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 wikt:return, return it. *In finance, monetary debt *In language, the use of loanwords *Carry (arithmetic), In arithmetic, when a digit becomes less than zer ...

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) Profit, in accounting Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information about economic entity, economic en ...
, in that interest is received by a lender, whereas profit is received by the
owner Ownership is the state or fact of exclusive right In Anglo-Saxon law Anglo-Saxon law (Old English Old English (, ), or Anglo-Saxon, is the earliest recorded form of the English language English is a West Germanic languages, West Germa ...
of an
asset In financial accounting Financial accounting is the field of accounting Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information about economic entity, economic entities such a ...
,
investment Investment is the dedication 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 , the purpose of investing is to generate a from the inve ...

or
enterprise Enterprise (or the archaic spelling Enterprize) may refer to: Business and economics Brands and enterprises * Enterprise GP Holdings Enterprise GP Holdings was a midstream energy holding company based in Houston, Texas Houston ( ) is ...

. (Interest may be part or the whole of the profit on an
investment Investment is the dedication 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 , the purpose of investing is to generate a from the inve ...

, but the two concepts are distinct from each other from an
accounting Accounting or Accountancy is the measurement, processing, and communication of financial and non financial information about economic entity, economic entities such as businesses and corporations. Accounting, which has been called the "languag ...
perspective.) The
rate of interest Interest, in finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have mon ...
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 A debtor or debitor is a legal entity (legal person) that owes a debt Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to ...
borrowed or lent (usually expressed as a percentage).
Compound interest Compound interest is the addition of interest Interest, in finance and economics, is payment from a debtor, borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, ...

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

According to historian
Paul JohnsonPaul Johnson may refer to: Musicians *Paul Johnson (producer) (1971–2021), American producer and DJ *Paul Johnson (singer), British soul singer of the 1980s *Paul Johnson (guitarist), American *Paul Francis Johnson, Australian bassist, frontman o ...
, the lending of "food money" was commonplace in
Middle East The Middle East ( ar, الشرق الأوسط, ISO 233 The international standard An international standard is a technical standard A technical standard is an established norm (social), norm or requirement for a repeatable technical task whi ...

ern civilizations as early as 5000 BC. 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 s 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 charg ...
(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 EshnunnaThe Laws of Eshnunna (abrv. LE) are inscribed on two cuneiform Cuneiform is a Logogram, logo-Syllabary, syllabic writing system, script that was used to write several languages of the Ancient Near East. The script was in active use from the earl ...
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 Bride price, bridewealth, or bride token, is money, prop ...
. 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 Bithynia Bithynia (; Koine Greek Koine Greek (, , Greek approximately ;. , , , lit. "Common Greek"), also known as Alexandrian dialec ...
clergy Clergy are formal leaders within established s. Their roles and functions vary in different religious traditions, but usually involve presiding over specific rituals and teaching their religion's s and practices. Some of the terms used for ind ...
from engaging in
usury Usury () is the practice of making unethical or immoral monetary s 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 charg ...
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 (or oecumenical council; also general council) is a conference of ecclesiastical dignitaries and theological experts convened to discuss and settle matters of Church doctrine and practice in which those entitled to vote a ...
s applied this regulation to the
laity In religious organizations, the laity consists of all members who are not part of the clergy Clergy are formal leaders within established religions. Their roles and functions vary in different religious traditions, but usually involve presidin ...
.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 baptised Catholics Catholic Church by country, worldwide . As the wo ...

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 (; it, Tommaso d'Aquino, lit=Thomas of Aquino; 1225 – 7 March 1274) was an Italian Dominican Dominican may refer to: * Someone or something from or related to the Dominican Republic The Dominican Republic ( ; es, ...

, 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 baptised Catholics Catholic Church by country, worldwide . As the wo ...

, 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 , 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. In the
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in ...

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 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 than simply ...

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 money held by the public at a particular point in time in an economy. There are several ways to define "money", but standard measures usually include Circulati ...
Banque de France The Bank of France ( French: ''Banque de France''), headquartered in Paris Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, most populous city of France, with an estimated population of 2,17 ...

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 A bank is a financial institution that accepts Deposit account, deposits from the public and creates a demand deposit while simul ...
, 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 Leibn ...
discovered the mathematical constant e by studying a question about
compound interest Compound interest is the addition of interest Interest, in finance and economics, is payment from a debtor, borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, ...

. 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: : $\lim_ \left\left( 1 + \dfrac \right\right)^n = e,$ 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 px, Domestic credit to private sector in 2005 Credit (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, k ... , and it plays the role of the cost of capital In economics Economics () is a social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of plant l ... . In a free market In economics Economics () is a social science Social science is the branch A branch ( or , ) or tree branch (sometimes referred to in botany Botany, also called , plant biology or phytology, is the science of pl ... economy, interest rates are subject to the law of supply and demand In microeconomics Microeconomics is a branch of that studies the behavior of individuals and in making decisions regarding the allocation of and the interactions among these individuals and firms. Microeconomics focuses on the study ... of the money supply In macroeconomics, the money supply (or money stock) refers to the total volume of money held by the public at a particular point in time in an economy. There are several ways to define "money", but standard measures usually include Circulati ... , and one explanation of the tendency of interest rates to be generally greater than zero is the scarcity of loanable fundsIn economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and ... . Over centuries, various schools of thought have developed explanations of interest and interest rates. The School of Salamanca The School of Salamanca, also called Iberian School of Peace, ( es, Escuela de Salamanca) is the Renaissance The Renaissance ( , ) , from , with the same meanings. was a period in European history marking the transition from the Middle ... 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, 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 In economics Economics () is the social sci ... 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 Anne Robert Jacques Turgot, Baron de l'Aulne ( ; ; 10 May 172718 March 1781), commonly known as Turgot, was a French economist and statesman. Originally considered a Physiocracy, physiocrat, he is today best remembered as an early advocate for e ... proposed the theory of fructification. By applying an opportunity cost In microeconomic theory Microeconomics (from Greek prefix ''mikro-'' meaning "small" + ''economics'') is a branch of economics Economics () is the social science that studies how people interact with value; in particular, the Produ ... argument, comparing the loan rate with the rate of return In finance Finance is a term for the management, creation, and study of money In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in t ... on agricultural land, and a mathematical argument, applying the formula for the value of a perpetuity A perpetuity is an annuity An annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings ... 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 ( 1723 – 17 July 1790) was a Scottish economist, philosopher as well as a moral philosopher Ethics or moral philosophy is a branch of philosophy that "involves systematizing, defending, and recommending concepts of right and ... , Carl Menger Carl Menger (; ; 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 theory of marginalism (marginal utility), which rejected the co ... , and Frédéric Bastiat Claude-Frédéric Bastiat (; ; 30 June 1801 – 24 December 1850) was a French economist, writer and a prominent member of the French Liberal School. A Freemason Freemasonry or Masonry consists of fraternal organisations that trace th ... 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 Swedish or ' may refer to: * Anything from or related to Sweden, a country in Northern Europe * Swedish language, a North Germanic language spoken primarily i ... 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 Swedish or ' may refer to: * Anything from or related to Sweden, a country in Northern Europe * Swedish language, a North Germanic language spoken primarily in Sweden and ... and Dennis Robertson and became known as the loanable fundsIn economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and ... 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 An economist is a practitioner in the social sciences, social science discipline of economics. The individual may also study, develop, and apply theories and concep ... and John Maynard Keynes John Maynard Keynes, 1st Baron Keynes, ( ; 5 June 1883 – 21 April 1946) was an English economist An economist is a professional and practitioner in the social science Social science is the branch The branches and leaves of a ... . # 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 med ... . 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 In and , interest is payment from a or deposit-taking financial institution to a or depositor of an amount above repayment of the (that is, the amount borrowed), at a particular rate. It is disti ... :''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$2500 and that the simple annual
interest rate An interest rate is the amount of interest In and , interest is payment from a or deposit-taking financial institution to a or depositor of an amount above repayment of the (that is, the amount borrowed), at a particular rate. It is disti ...
is 12.99% ''per annum'', applied monthly, so the frequency of applying interest is 12 per year. Over one month, :$\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 In economics Economics () is the social sci ... ), 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\left(1 + \frac\right\right) \cdot B = \left\left(1 + \frac\right\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: :$\begin\10,000 + \300 + \309 &= \10\,000 + \frac + \frac \\ &= \10\,000 \times \left\left(1 + \frac\right\right)^2\end$ and the investor earned in total: :$\begin\10\,000 \times \left\left(1 + \frac \right\right)^2 - \10\,000\\ = \10\,000 \times \left\left( \left\left( 1 + \frac \right\right)^2 - 1\right\right)\end$ The formula for the annual equivalent compound interest rate is: :$\left\left(1 + \frac\right\right)^n - 1$ 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: :$\left\left(1 + \frac\right\right)^2 - 1 = 1.03^2 - 1 = 6.09\%$

## Other formulations

The outstanding
balance Balance may refer to: Common meanings * Balance (ability) in biomechanics * Balance (accounting) * Balance or weighing scale Arts and entertainment Film * Balance (1983 film), ''Balance'' (1983 film), a Bulgarian film * Balance (1989 film), ''Bal ...
''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: :$B_ = \big\left( 1 + r \big\right) B_ - p,$ 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 A debtor or debitor is a legal entity (legal person) that owes a debt Debt is an obligation that requires one party, the debtor, to pay money or other agreed-upon value to ...
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 (mathematics), series is the sum of an infinite number of Summand, terms that have a constant ratio between successive terms. For example, 1/2 + 1/4 + 1/8 + 1/16 + · · ·, the series :\frac \,+\, \frac \,+\, ...
results in :$B_n = \left(1 + r\right)^n B_0 - \frac p$ A solution of this expression for ''p'' in terms of ''B''0 and ''B''''n'' reduces to :''n'' = 0. The PMT function found in
spreadsheet A spreadsheet is a computer application for organization, analysis, and storage of data in table (information), tabular form. Spreadsheets were developed as computerized analogs of paper accounting Worksheet#Accounting, worksheets. The program op ...

programs can be used to calculate the monthly payment of a loan: :$p=\mathrm\left(\text,\text,\text,\text,\right) = \mathrm\left(r,n,-B_0,B_n,\right)$ 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 :$B_k = B\text{'}_0 = \left(1 + r_k\right) B_0 - p_k.$ Comparing this with the expression for Bk above, we note that :$r_k = \left(1 + r\right)^k - 1$ 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 :$B_k = \left(1 + r_k\right) B_0 - r_k B^*.$ 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 A scale factor is usually a decimal which scales Scale or scales may refer to: Mathematics * Scale (descriptive set theory)In the mathematical discipline of descriptive set theory, a scale is a certain kind of object defined on a set (mathemat ...
): :
binomial theorem In elementary algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas a ...
) 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: :$\Delta B_k=B\text{'}-B=\left(\lambda_k rB-\lambda_k p\right)=\lambda_k \, \Delta B$, 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 :$B_k=B_0-r_k\left(B^*-B_0\right)$. Since ''B''''n'' = 0, we can solve for ''B''*: :$B^ = B_0 \left\left(\frac + 1 \right\right).$ 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'': :$B_k=B_0\left\left(1-\frac\right\right)=B_0\left\left(1-\frac\right\right)$. 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 :$\lambda_=1+\left(1+r\right)\lambda_k$. ''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 :$p=\left\left(r+\frac\right\right)B_0$ 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 conventionIn finance, a day count convention determines how interest accrues over time for a variety of investments, including bond (finance), bonds, notes, loans, Mortgage loan, mortgages, medium-term notes, Swap (finance), swaps, and forward rate agreements ...
). 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 APY 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 Continental United States, primarily located in North America. It consists of 50 U.S. state, states, a Washington, D.C., ...

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 In and , interest is payment from a or deposit-taking financial institution to a or depositor of an amount above repayment of the (that is, the amount borrowed), at a particular rate. It is disti ...
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 Microeconomics (from Greek prefix ''mikro-'' meaning "small" + ''economics'') is a branch of economics Economics () is the social science that studies how people interact with value; in particular, the Produ ...
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 Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, ...
in real terms. 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 refers to a general progressive increase in prices 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 r ...

. 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 In financial mathematicsMathematical finance, also ...
. * 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 debtor. ...

, or otherwise
default Default may refer to: Law * Default (law), the failure to do something required by law ** Default (finance) In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It ...
on the loan. The
risk premium Overview A risk premium is a measure of excess return that is required by an individual to compensate them for being subjected to an increased level of risk. It is used widely in finance and economics with the general definition being the expect ...
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 () is a loan In finance Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money ...
. The
creditworthiness A credit risk is risk of default Default may refer to: Law * Default (law), the failure to do something required by law ** Default (finance) In finance Finance is the study of financial institutions, financial markets and how they ope ...
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 A credit risk is risk of default (finance), default on a debt that may arise from a borrower failing to make requi ...
s by
credit bureau A credit bureau is a data collection Data collection is the process of gathering and measuring Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. The scope an ...
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 refers to a general progressive increase in prices 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 r ...

. 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 In financial mathematicsMathematical finance, also ...
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 A legislature is a deliberative assemb ...
are normally highly reliable
debtor A debtor or debitor is a legal entity (legal person) that owes a debt Debt is an obligation that requires one party, the debtor A debtor or debitor is a legal entity (legal person) that owes a debt Debt is an obligation that ...
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) In ch ...
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 Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money avai ...

).

## 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 A currency, "in circulation", from la, currens, -entis, literally meaning "running" or "traversing" in the most specific sense is money ...

- 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 In finance and economics, a monetary authority is the entity that manages a country’s currency and money supply, often with the objective of controlling inflation targeting, infla ...

. 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 (also known as the Federal Reserve or simply the Fed) is the central bank A central bank, reserve bank, or monetary authority is an institution that manages the and of a or formal monetary union, and ove ...

(Fed) implements monetary policy largely by targeting the federal funds rate. This is the rate that banks charge each other for overnight loans of
federal funds In the United States The United States of America (USA), commonly known as the United States (U.S. or US), or America, is a country Contiguous United States, primarily located in North America. It consists of 50 U.S. state, states, a Wa ...
. Federal funds are the reserves held by banks at the Fed.
Open market operations In macroeconomics Macroeconomics (from the Greek prefix ''makro-'' meaning "large" + ''economics'') is a branch of economics dealing with performance, structure, behavior, and decision-making of an economy as a whole. For example, using interest ...
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 A financial asset is a non-physical asset In financial accounting Financial accounting is the field of accounting Accounting or Accountancy is the measurement, processing, and communication o ...
, 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 The United States of America (USA), commonly known as the United States (U.S. or US), or America, is a country Contiguous United States, prim ...
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 (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 The reserve requirement (or cash reserve ratio) is a central bank A central bank, reserve bank, or monetary authority is an institution that manages th ...
may be lent in the Fed funds 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 In and , interest is payment from a or deposit-taking financial institution to a or depositor of an amount above repayment of the (that is, the amount borrowed), at a particular rate. It is disti ...
s and
credit risk A credit risk is risk of default Default may refer to: Law * Default (law), the failure to do something required by law ** Default (finance) In finance Finance is the study of financial institutions, financial markets and how they ope ...
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 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#REDIRECT Quantity theory of money In monetary economics, the quantity theory of money (QTM) states that the general price level of goods and services is directly proportional to the amount of money in circulation, or money supply In macroeco ...
, 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 In business Business is the activity of making one's living or making money by producing or buying and selling ...
is the ability to quickly resell 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 () is a loan In finance, a loan is the lending of money by one or more individuals, organizations, or other entities to other individuals, organizations etc. The recipient (i.e., the borrower) incurs a ...
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 lack of liquidity.

# Theories of interest

## Aristotle's view of interest

Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questio ...

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 Speculation, 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 ...

(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 999999 or triple nine most often refers to: * 999 (emergency telephone number) 250px, A sign on a beach ...

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. 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 language, Spanish and Portuguese language, Portuguese cognate of the name Richard. It derived from Proto-Germanic language, Proto-Germanic ''*rīks'' 'king, ruler' + ''*harduz'' ''hard, brave'. It may be a given name, or a ...

, Mountifort Longfield, John Stuart Mill, J. S. Mill, and
Irving Fisher Irving Fisher (February 27, 1867 – April 29, 1947) was an American economist An economist is a practitioner in the social sciences, social science discipline of economics. The individual may also study, develop, and apply theories and concep ...

. It was strongly criticised by John Maynard Keynes, Keynes 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 Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, ...
), 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 of Employment, Interest and Money#bookiii, The General Theory), 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.) 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 doesn't 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 [which determines income] is 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 (excluding Longfield, who was essentially a marginalist). Their responses often had a moral tone: Ricardo and Marx viewed profits as exploitation, and John Ramsay McCulloch, 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 William Senior, Nassau Senior, 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 Ferdinand Lassalle, Lassalle.

## Wicksell's theory

Knut Wicksell Johan Gustaf Knut Wicksell (December 20, 1851 – May 3, 1926) was a leading Swedish Swedish or ' may refer to: * Anything from or related to Sweden, a country in Northern Europe * Swedish language, a North Germanic language spoken primarily i ...

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 Swedish or ' may refer to: * Anything from or related to Sweden, a country in Northern Europe * Swedish language, a North Germanic language spoken primarily in Sweden and ...

and Dennis Robertson and became known as the
loanable fundsIn economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and ...
theory.

## Austrian theories

Eugen Böhm von Bawerk and other members of the 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

Vilfredo Pareto, 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's 1936 ''General theory of employment, interest, and money''. In his initial account of liquidity preference (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 of Employment, Interest and Money#interestliquidity, The General Theory for details.

# In religious contexts

## Judaism

Jews are forbidden from usury in dealing with fellow Jews, and this lending is to be considered tzedakah, or charity. However, there are permissions to charge interest on loans to Gentile, non-Jews. This is outlined in the Jewish scriptures of the Torah, which Christians hold as part of the Old Testament, and other books of the Tanakh. From the Jewish Publication Society's 1917 Tanakh, with Christian verse numbers, where different, in parentheses: Several historical rulings in Halakha, Jewish law have mitigated the allowances for usury toward non-Jews. For instance, the 15th-century commentator Isaac Abarbanel, 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 David Kimhi, 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 attitude of Muhammad to usury is articulated in his Last Sermon

## Christianity

The Old Testament "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 theology, scholastic Christian theologians, Saint Anselm of Canterbury, led the shift in thought that labeled charging interest the same as theft. Previously usury had been seen as a lack of Charity (virtue), charity. St.
Thomas Aquinas Thomas Aquinas (; it, Tommaso d'Aquino, lit=Thomas of Aquino; 1225 – 7 March 1274) was an Italian Dominican Dominican may refer to: * Someone or something from or related to the Dominican Republic The Dominican Republic ( ; es, ...

, the leading scholastic theologian of the Roman Catholic Church, 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 A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questio ...

and his desire to assimilate Greek philosophy with Christian theology. Aquinas argued that in the case of usury, as in other aspects of Christian revelation, Christian doctrine is reinforced by Aristotelian natural law 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. 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, expressly allowed such charges in respect of credit-unions run for the benefit of the poor known as "montes pietatis". In the 13th century Henry of Segusio, 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). However, Hostiensis' exceptions, including for ''lucrum cessans'', were never accepted as official by the Roman Catholic Church. The Westminster Confession of Faith, a confession of faith upheld by the Reformed Churches, teaches that usury--charging interest at any rate--is a sin in Christianity, sin prohibited by the Ten Commandments, 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's encyclica
''Vix Pervenit''
''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, a Catholic Christian religious order, thus teaches that:

* Actuarial notation * Credit card interest * Credit rating agency * DIRTI 5 * Discounting, Discount * Fisher equation * Hire purchase * Interest expense * Leasing * Promissory note * Risk-free interest rate

# References

* * * * *

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

Financial Services Authority (UK)
OECD interest rate statistics
*You can see a list of current interest rates at these sites:
World Interest Rates

"Which way to pay"

Deposit Rates in European Countries
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