Interest, in finance and economics, 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 from a fee which the borrower may pay the lender or some third party. It is also distinct from dividend which is paid by a company to its shareholders (owners) from its profit 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 taking entrepreneurs when the revenue earned exceeds the total costs.
Finance is a field that is concerned with the allocation (investment) of assets and liabilities over space and time, often under conditions of risk or uncertainty. Finance can also be defined as the art of money management. Participants in the market aim to price assets based on their risk level, fundamental value, and their expected rate of return. Finance can be split into three sub-categories: public finance, corporate finance and personal finance.
Economics is the social science that studies the production, distribution, and consumption of goods and services.
A debtor is an entity 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 debt arrangement is a bank, the debtor is more often referred to as a borrower.
For example, a customer would usually pay interest to borrow from a bank, so they pay the bank an amount which is more than the amount they borrowed; or a customer may earn interest on their savings, and so they may withdraw more than they originally deposited. In the case of savings, the customer is the lender, and the bank plays the role of the borrower.
Interest differs from profit, in that interest is received by a lender, whereas profit is received by the owner of an asset, investment or enterprise. (Interest may be part or the whole of the profit on an investment, but the two concepts are distinct from each other from an accounting perspective.)
Profit, in accounting, is an income distributed to the owner in a profitable market production process (business). Profit is a measure of profitability which is the owner’s major interest in income formation process of market production. There are several profit measures in common use.
Ownership is the state or fact of exclusive rights and control over property, which may be an object, land/real estate or intellectual property. Ownership involves multiple rights, collectively referred to as title, which may be separated and held by different parties.
In financial accounting, an asset is any resource owned by the business. Anything tangible or intangible that can be owned or controlled to produce value and that is held by a company to produce positive economic value is an asset. Simply stated, assets represent value of ownership that can be converted into cash. The balance sheet of a firm records the monetary value of the assets owned by that firm. It covers money and other valuables belonging to an individual or to a business.
The rate of interest is equal to the interest amount paid or received over a particular period divided by the principal sum borrowed or lent (usually expressed as a percentage).
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited or borrowed. The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, the compounding frequency, and the length of time over which it is lent, deposited or borrowed.
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.
Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the next period is then earned on the principal sum plus previously accumulated interest. Compound interest is standard in finance and economics.
The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828, and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series
According to historian Paul Johnson, the lending of "food money" was commonplace in Middle Eastern 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 (נשך NeSheKh) represented a "different view".
Paul Bede Johnson is an English journalist, popular historian, speechwriter, and author. While associated with the political left in his early career, he is now a conservative popular historian.
The Middle East is a transcontinental region centered on Western Asia, Turkey, and Egypt. Saudi Arabia is geographically the largest Middle Eastern nation while Bahrain is the smallest. The corresponding adjective is Middle Eastern and the derived noun is Middle Easterner. The term has come into wider usage as a replacement of the term Near East beginning in the early 20th century.
Usury is the practice of making unethical or immoral monetary loans that unfairly enrich the lender. Originally, usury meant interest of any kind. A loan may be considered usurious because of excessive or abusive interest rates or other factors. Historically, in some Christian societies, and in many Islamic societies even today, charging any interest at all would be considered usury. Someone who practices usury can be called a usurer, but a more common term in contemporary English is loan shark.
The first written evidence of compound interest dates roughly 2400 BC.The annual interest rate was roughly 20%. Compound interest was necessary for the development of agriculture and important for urbanization.
While the traditional Middle Eastern views on interest was the result of the urbanized, economically developed character of the societies that produced them, the new Jewish prohibition on interest showed a pastoral, tribal influence.In the early 2nd millennium BC, since silver used in exchange for livestock or grain could not multiply of its own, the Laws of Eshnunna instituted a legal interest rate, specifically on deposits of dowry. Early Muslims called this riba, translated today as the charging of interest.
The First Council of Nicaea, in 325, forbade clergy from engaging in usurywhich was defined as lending on interest above 1 percent per month (12.7% APR). Ninth century ecumenical councils applied this regulation to the laity. Catholic Church opposition to interest hardened in the era of scholastics, when even defending it was considered a heresy. St. Thomas Aquinas, the leading theologian of the Catholic Church, 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.[ citation needed ] 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.
In the Renaissance era, greater mobility of people facilitated an increase in commerce and the appearance of appropriate conditions for entrepreneurs to start new, lucrative businesses. Given that borrowed money was no longer strictly for consumption but for production as well, interest was no longer viewed in the same manner.
The first attempt to control interest rates through manipulation of the money supply was made by the Banque de France in 1847.[ citation needed ]
The latter half of the 20th century saw the rise of interest-free Islamic banking and finance, a movement that applies Islamic law to financial institutions and the economy. Some countries, including Iran, Sudan, and Pakistan, have taken steps to eradicate interest from their financial systems.[ citation needed ] 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 economics, the rate of interest is the price of credit, and it plays the role of the cost of capital. In a free market economy, interest rates are subject to the law of supply and demand of the money supply, and one explanation of the tendency of interest rates to be generally greater than zero is the scarcity of loanable funds.
Over centuries, various schools of thought have developed explanations of interest and interest rates. The School of Salamanca justified paying interest in terms of the benefit to the borrower, and interest received by the lender in terms of a premium for the risk of default.[ citation needed ] In the sixteenth century, Martín de Azpilcueta applied a time preference argument: it is preferable to receive a given good now rather than in the future. Accordingly, interest is compensation for the time the lender forgoes the benefit of spending the money.
On the question of why interest rates are normally greater than zero, in 1770, French economist Anne-Robert-Jacques Turgot, Baron de Laune proposed the theory of fructification. By applying an opportunity cost argument, comparing the loan rate with the rate of return on agricultural land, and a mathematical argument, applying the formula for the value of a perpetuity to a plantation, he argued that the land value would rise without limit, as the interest rate approached zero. For the land value to remain positive and finite keeps the interest rate above zero.
Adam Smith, Carl Menger, and Frédéric Bastiat also propounded theories of interest rates.In the late 19th century, Swedish economist Knut Wicksell in his 1898 Interest and Prices elaborated a comprehensive theory of economic crises based upon a distinction between natural and nominal interest rates. In the 1930s, Wicksell's approach was refined by Bertil Ohlin and Dennis Robertson and became known as the loanable funds theory. Other notable interest rate theories of the period are those of Irving Fisher and John Maynard Keynes.
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. 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:
For example, imagine that a credit card holder has an outstanding balance of $2500 and that the simple annual interest rate is 12.99% per annum, applied monthly, so the frequency of applying interest is 12 per year. Over one month,
interest is due (rounded to the nearest cent).
Simple interest applied over 3 months would be
If the card holder pays off only interest at the end of each of the 3 months, the total amount of interest paid would be
which is the simple interest applied over 3 months, as calculated above. (The one cent difference arises due to rounding to the nearest cent.)
Compound interest includes interest earned on the interest which was previously accumulated.
Compare for example a bond paying 6 percent biannually (that is, coupons of 3 percent twice a year) with a certificate of deposit (GIC) which 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 biannual bond receives half the $6 per year after only 6 months (time preference), and so has the opportunity to reinvest the first $3 coupon payment after the first 6 months, and earn additional interest.
For example, suppose an investor buys $10,000 par value of a US dollar bond, which pays coupons twice a year, and that the bond's simple annual coupon rate is 6 percent per year. This means that every 6 months, the issuer pays the holder of the bond a coupon of 3 dollars per 100 dollars par value. At the end of 6 months, the issuer pays the holder:
Assuming the market price of the bond is 100, so it is trading at par value, suppose further that the holder immediately reinvests the coupon by spending it on another $300 par value of the bond. In total, the investor therefore now holds:
and so earns a coupon at the end of the next 6 months of:
Assuming the bond remains priced at par, the investor accumulates at the end of a full 12 months a total value of:
and the investor earned in total:
The formula for the annual equivalent compound interest rate is:
For example, in the case of a 6% simple annual rate, the annual equivalent compound rate is:
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.
In 1992, the United States 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.
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.
In the case of an interest rate of 24 percent, the rule predicts that money will double after 72/24 = 3 years.
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There are markets for investments (which include the money market, bond market, as well as retail financial institutions like banks) set interest rates. Each specific debt takes into account the following factors in determining its interest rate:
Opportunity cost 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 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.
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. Because future inflation is unknown, there are three ways this might be achieved:
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.
There is always the risk the borrower will become bankrupt, abscond or otherwise default on the loan. The risk premium 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.
The creditworthiness of businesses is measured by bond rating services and individual's credit scores by credit bureaus. 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.
In economics, interest is considered the price of credit, therefore, it is also subject to distortions due to inflation. 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 plus inflation, among other factors. An approximate formula for the nominal interest is:
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 are normally highly reliable debtors, and the interest rate on government securities is normally lower than the interest rate available to other borrowers.
relates expectations of inflation and credit risk to nominal and expected real interest rates, over the life of a loan, where
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.
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).
Interest rates are generally determined by the market, but government intervention - usually by a central bank - may strongly influence short-term interest rates, and is one of the main tools of monetary policy. 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.
The Federal Reserve (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. Federal funds are the reserves held by banks at the Fed.
Open market operations 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, the Open Market Desk at the Federal Reserve Bank of New York 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 may be lent in the Fed funds market to other banks, thus driving down rates.
It is increasingly recognized that during the business cycle, interest rates and credit risk 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.
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, increases in the money supply lead to inflation. This means that interest rates can affect inflation in the future.[ citation needed ]
Liquidity 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 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.
Aristotle 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.
Nicholas Barbon (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 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”. Schumpeterconsidered 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 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 was the work of a number of authors, including Turgot, Ricardo,Mountifort Longfield, J. S. Mill, and Irving Fisher. It was strongly criticised by 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), 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 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.), 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
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.
The solid red curve in the diagram shows the desired level of saving s as a function of r for the current income ŷ. Saving is the excess of deferred over anticipated consumption, and its dependence on income is much as described by Keynes (see 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.
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.
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 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, 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.
Knut Wicksell 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 and Dennis Robertson and became known as the loanable funds theory.
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,
Too many writers consider the rate of interest as only the price of loans on the loan market. In reality...the rate of interest pervades all time markets, and the productive loan market is a strictly subsidiary time market of only derivative importance. 371:
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
Interest rate is equal to the rate of price spread in the various stages. 371:
Rothbard has furthermore criticized the Keynesian conception of interest, saying
One grave and fundamental Keynesian error is to persist in regarding the interest rate as a contract rate on loans, instead of the price spreads between stages of production. 789:
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.
Even if Pareto was right, the equations might have a structure which made it possible to say more about interest than that it satisfies a complicated set of conditions.
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 for details.
It is thought that Jacob Bernoulli discovered the mathematical constant e by studying a question about compound interest. ×1.52 = $2.25. Compounding quarterly yields $1.00×1.254 = $2.4414..., and so on.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
Bernoulli noticed that if the frequency of compounding is increased without limit, this sequence can be modeled as follows:
where n is the number of times the interest is to be compounded in a year.
The outstanding balance Bn of a loan after n regular payments increases each period by a growth factor according to the periodic interest, and then decreases by the amount paid p at the end of each period:
By repeated substitution one obtains expressions for Bn, which are linearly proportional to B0 and p and use of the formula for the partial sum of a geometric series results in
A solution of this expression for p in terms of B0 and Bn reduces to
To find the payment if the loan is to be finished in n payments one sets Bn = 0.
The PMT function found in spreadsheet programs can be used to calculate the monthly payment of a loan:
An interest-only payment on the current balance would be
The total interest, IT, paid on the loan is
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 rk and pk are the new rate and payment, we now have
Comparing this with the expression for Bk above we note that
The last equation allows us to define a constant that is the same for both problems,
and Bk can be written as
Solving for rk we find a formula for rk involving known quantities and Bk, the balance after k periods,
Since B0 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 since it does not change with changes in the length of the period.
Rearranging the equation for B* one gets a transformation coefficient (scale factor),
and we see that r and p transform in the same manner,
The change in the balance transforms likewise,
which gives an insight into the meaning of some of the coefficients found in the formulas above. The annual rate, r12, 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·B0.
Substituting pk = rkB* into the equation for the Bk we get,
Since Bn = 0 we can solve for B*,
Substituting back into the formula for the Bk shows that they are a linear function of the rk and therefore the λk,
This is the easiest way of estimating the balances if the λk are known. Substituting into the first formula for Bk above and solving for λk+1 we get,
λ0 and λn can be found using the formula for λk above or computing the λk recursively from λ0 = 0 to λn.
Since p = rB* the formula for the payment reduces to,
and the average interest rate over the period of the loan is
which is less than r if n > 1.
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 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:
Exodus 22:24 (25)—If thou lend money to any of My people, even to the poor with thee, thou shalt not be to him as a creditor; neither shall ye lay upon him interest.
Leviticus 25:36— Take thou no interest of him or increase; but fear thy God; that thy brother may live with thee.
Leviticus 25:37— Thou shalt not give him thy money upon interest, nor give him thy victuals for increase.
Deuteronomy 23:20 (19)—Thou shalt not lend upon interest to thy brother: interest of money, interest of victuals, interest of any thing that is lent upon interest.
Deuteronomy 23:21 (20)—Unto a foreigner thou mayest lend upon interest; but unto thy brother thou shalt not lend upon interest; that the LORD thy God may bless thee in all that thou puttest thy hand unto, in the land whither thou goest in to possess it.
Ezekiel 18:17—that hath withdrawn his hand from the poor, that hath not received interest nor increase, hath executed Mine ordinances, hath walked in My statutes; he shall not die for the iniquity of his father, he shall surely live.
Psalm 15:5—He that putteth not out his money on interest, nor taketh a bribe against the innocent. He that doeth these things shall never be moved.
Several historical rulings in Jewish law 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.
The following quotations are English translations from the Qur'an:
Those who charge usury are in the same position as those controlled by the devil's influence. This is because they claim that usury is the same as commerce. However, God permits commerce, and prohibits usury. Thus, whoever heeds this commandment from his Lord, and refrains from usury, he may keep his past earnings, and his judgment rests with God. As for those who persist in usury, they incur Hell, wherein they abide forever (Al-Baqarah 2:275)
God condemns usury, and blesses charities. God dislikes every sinning disbeliever. Those who believe and do good works and establish worship and pay the poor-due, their reward is with their Lord and there shall no fear come upon them neither shall they grieve. O you who believe, you shall observe God and refrain from all kinds of usury, if you are believers. If you do not, then expect a war from God and His messenger. But if you repent, you may keep your capitals, without inflicting injustice, or incurring injustice. If the debtor is unable to pay, wait for a better time. If you give up the loan as a charity, it would be better for you, if you only knew. (Al-Baqarah 2:276-280)
O you who believe, you shall not take usury, compounded over and over. Observe God, that you may succeed. (Al-'Imran 3:130)
And for practicing usury, which was forbidden, and for consuming the people's money illicitly. We have prepared for the disbelievers among them painful retribution. (Al-Nisa 4:161)
The usury that is practiced to increase some people's wealth, does not gain anything at God. But if people give to charity, seeking God's pleasure, these are the ones who receive their reward many fold. (Ar-Rum 30:39)
The attitude of Muhammad to usury is articulated in his Last Sermon
O People, just as you regard this month, this day, this city as Sacred, so regard the life and property of every Muslim as a sacred trust. Return the goods entrusted to you to their rightful owners. Hurt no one so that no one may hurt you. Remember that you will indeed meet your LORD, and that HE will indeed reckon your deeds. ALLAH has forbidden you to take usury , therefore all usurious obligation shall henceforth be waived. Your capital, however, is yours to keep. You will neither inflict nor suffer any inequity. Allah has Judged that there shall be no usury and that all the usury due to Abbas ibn 'Abd'al Muttalib (Prophet's uncle) shall henceforth be waived...
[ unreliable source? ][ better source needed ]
The first of the 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.
St. Thomas Aquinas, 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 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.
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) [ citation needed ] 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 Cardinal Hostiensis enumerated thirteen situations in which charging interest was not immoral. ( Rothbard 1995 , p. 46) 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 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."
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 encyclical Vix Pervenit gives the reasons why usury is sinful:
The nature of the sin called usury has its proper place and origin in a loan contract… [which] demands, by its very nature, that one return to another only as much as he has received. The sin rests on the fact that sometimes the creditor desires more than he has given…, but any gain which exceeds the amount he gave is illicit and usurious.
One cannot condone the sin of usury by arguing that the gain is not great or excessive, but rather moderate or small; neither can it be condoned by arguing that the borrower is rich; nor even by arguing that the money borrowed is not left idle, but is spent usefully…
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Money paid regularly at a particular rate for the use of money lent, or for delaying the repayment of a debt.
a share in a pro rata distribution (as of profits) to stockholders.
Discounting is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee. Essentially, the party that owes money in the present purchases the right to delay the payment until some future date. The discount, or charge, is the difference between the original amount owed in the present and the amount that has to be paid in the future to settle the debt.
In finance, the net present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount rate. NPV accounts for the time value of money. It provides a method for evaluating and comparing capital projects or financial products with cash flows spread over time, as in loans, investments, payouts from insurance contracts plus many other applications.
In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is always less than or equal to the future value because money has interest-earning potential, a characteristic referred to as the time value of money, except during times of negative interest rates, when the present value will be more than the future value. Time value can be described with the simplified phrase, "A dollar today is worth more than a dollar tomorrow". Here, 'worth more' means that its value is greater. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Interest can be compared to rent. Just as rent is paid to a landlord by a tenant without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the money for a time before paying it back. By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest. The initial amount of the borrowed funds is less than the total amount of money paid to the lender.
The time value of money is the greater benefit of receiving money now rather than an identical sum later. It is founded on time preference.
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 incurs a debt, and is usually liable to pay interest on that debt until it is repaid, and also to repay the principal amount borrowed.
In finance and economics, the nominal interest rate or nominal rate of interest is either of two distinct things:
A fixed-rate mortgage (FRM), often referred to as a "vanilla wafer" mortgage loan, is a fully amortizing mortgage loan where the interest rate on the note remains the same through the term of the loan, as opposed to loans where the interest rate may adjust or "float". As a result, payment amounts and the duration of the loan are fixed and the person who is responsible for paying back the loan benefits from a consistent, single payment and the ability to plan a budget based on this fixed cost.
Rational pricing is the assumption in financial economics that asset prices will reflect the arbitrage-free price of the asset as any deviation from this price will be "arbitraged away". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.
The term annual percentage rate of charge (APR), corresponding sometimes to a nominal APR and sometimes to an effective APR (EAPR), is the interest rate for a whole year (annualized), rather than just a monthly fee/rate, as applied on a loan, mortgage loan, credit card, etc. It is a finance charge expressed as an annual rate. Those terms have formal, legal definitions in some countries or legal jurisdictions, but in general:
Credit 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, but promises either to repay or return those resources at a later date. In other words, credit is a method of making reciprocity formal, legally enforceable, and extensible to a large group of unrelated people.
The real interest rate is the rate of interest an investor, saver or lender receives after allowing for inflation. It can be described more formally by the Fisher equation, which states that the real interest rate is approximately the nominal interest rate minus the inflation rate.
Credit card interest is the principal way in which credit card issuers generate revenue. A card issuer is a bank or credit union that gives a consumer a card or account number that can be used with various payees to make payments and borrow money from the bank simultaneously. The bank pays the payee and then charges the cardholder interest over the time the money remains borrowed. Banks suffer losses when cardholders do not pay back the borrowed money as agreed. As a result, optimal calculation of interest based on any information they have about the cardholder's credit risk is key to a card issuer's profitability. Before determining what interest rate to offer, banks typically check national, and international, credit bureau reports to identify the borrowing history of the card holder applicant with other banks and conduct detailed interviews and documentation of the applicant's finances.
Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. Mortgage calculators are used by consumers to determine monthly repayments, and by mortgage providers to determine the financial suitability of a home loan applicant.
Analogous to continuous compounding, a continuous annuity is an ordinary annuity in which the payment interval is narrowed indefinitely. A (theoretical) continuous repayment mortgage is a mortgage loan paid by means of a continuous annuity.
An EMI or Equated Monthly Installment is defined by Investopedia as "A fixed payment amount made by a borrower to a lender at a specified date each calendar month. Equated monthly installments are used to pay off both interest and principal each month, so that over a specified number of years, the loan is fully paid off along with interest."
An annuity is a series of payments made at equal intervals. Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time.
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