## ★ Interest

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, 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 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.

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.

The rate of interest is equal to the interest amount paid or received over a particular period divided by in principal sum borrowed or lent usually expressed as a percentage.

Compound interest means that interest is earned on prior interest in addition to the principal. Due to compounding, the total amount of debt grows exponentially, and its mathematical study led to the discovery of the number e. In practice, interest is most often calculated on a daily, monthly, or yearly basis, and its impact is influenced greatly by its compounding rate.

## 1. History. * (История) *

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".

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 (since the 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 usury which 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. 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 Quran 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.

### * 1.1. History. * Islamic finance. * (Исламские финансы) *

The latter half of the 20th century saw the rise of interest-free Islamic banking and finance, a movement that applies Islamic law to financial institutions and the economy. Some countries, including Iran, Sudan, and Pakistan, have taken steps to eradicate interest from their financial systems. Rather than charging interest, the interest-free lender shares the risk by investing as a partner in profit loss sharing scheme, because predetermined loan repayment as interest is prohibited, as well as making money out of money is unacceptable (is ending). All financial transactions must be asset-backed and it does not charge any interest or fee for the service of lending.

### * 1.2. History. * In the history of mathematics. * (В истории математики) *

It is thought that Jacob Bernoulli discovered the mathematical constant e by studying a question about compound interest. He realized that if an account that starts with n.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 n is multiplied by 1.5 twice, yielding (Kareem kniesel) N.00×1.5 2 = $2.25. Compounding quarterly yields N.00×1.25 4 = $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 n → ∞ 1 + 1 n = e, {\displaystyle \lim _{n\rightarrow \infty }\left1+{\dfrac {1}{n}}\right^{n}=e,}where n is the number of times the interest this to be compounded in a year.

## 2. Economics. * (Экономика) *

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. In the sixteenth century, Martin de Azpilcueta applied a time preference argument: it is preferable for 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 Frederic 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, Wicksells 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.

### * 3.1. Calculation. * Simple interest. * (Простые проценты) *

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:

r ⋅ B ⋅ m n {\displaystyle {\frac {r\cdot B\cdot m}{n}}}where

r is the simple annual interest rate 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 is 12.99% per annum, applied monthly, so the frequency of applying interest is 12 per year. Over one month,

0.1299 × $ 2500 12 = $ 27.06 {\displaystyle {\frac {0.1299\times \$2500}{12}}=\$27.06}interest is due rounded to the nearest cent.

Simple interest applied over 3 months would be

0.1299 × $ 2500 × 3 12 = $ 81.19 (12 3 = $ 81.19) {\displaystyle {\frac {0.1299\times \$2500\times 3}{12}}=\$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

0.1299 × $ 2500 12 × 3 = $ 27.06 per month × 3 months = $ 81.18 {\displaystyle {\frac {0.1299\times \$2500}{12}}\times 3=\$27.06{\text{ per month}}\times 3{\text{ months}}=\$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.

### * 3.2. Calculation. * Compound interest. * (Сложные проценты) *

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 n00 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 n0.000 par value of a US dollar bond, which pays coupons twice a year, and that the bonds 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 the value. At the end of 6 months, the issuer pays the holder:

r ⋅ B ⋅ m n = 6 % × $ 10 000 × 1 2 = $ 300 {\displaystyle {\frac {r\cdot B\cdot m}{n}}={\frac {6\%\times \n0\,000\times 1}{2}}=\$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 = 1 + r n ⋅ B = 1 + 6 % 2 × $ 10 000 {\displaystyle \n0\,000+\$300=\left1+{\frac {r}{n}}\right\cdot B=\left1+{\frac {6\%}{2}}\right\times \n0\,000}and so earns a coupon at the end of the next 6 months from:

r ⋅ B ⋅ m n = 6 % × $ 10 000 + $ 300 2 = 6 % × 1 + 6 % 2 × $ 10 000 2 = $ 309 {\displaystyle {\begin{aligned}{\frac {r\cdot B\cdot m}{n}}&,={\frac {6\%\times \left\n0\,000+\$300\right}{2}}\\&,={\frac {6\%\times \left1+{\frac {6\%}{2}}\right\times \n0\,000}{2}}\\&,=\$309\end{aligned}}}Assuming the bond remains priced at par, the investor accumulates at the end of a full 12 months a total value of:

$ 10, 000 + $ 300 + $ 309 = $ 10 000 + 6 % × $ 10, 000 2 + 6 % × 1 + 6 % 2 × $ 10 000 2 = $ 10 000 (A $ 10.000 2 = $ 10 000) × 1 + 6 % 2 {\displaystyle {\begin{aligned}\n0.000+\$300+\$309&,=\n0\,000+{\frac {6\%\times \n0.000}{2}}+{\frac {6\%\times \left1+{\frac {6\%}{2}}\right\times \n0\,000}{2}}\\&,=\n0\,000\times \left1+{\frac {6\%}{2}}\right^{2}\end{aligned}}}and the investor earned in total:

$ 10 000 × 1 + 6 % 2 − $ 10 000 = $ 10 000 × 1 + 6 % 2 − 1) {\displaystyle {\begin{aligned}\n0\,000\times \left1+{\frac {6\%}{2}}\right^{2}-\n0\,000\\=\n0\,000\times \left\left1+{\frac {6\%}{2}}\right^{2}-1\right)\end{aligned}}}The formula for the annual equivalent compound interest rate is:

1 + r n − 1 {\displaystyle \left1+{\frac {r}{n}}\right^{n}-1}where

r is the simple annual rate of interest n is the frequency of applying interestFor example, in the case of a 6% simple annual rate, the annual equivalent compound rate is:

1 + 6 % 2 − 1 = 1.03 2 − 1 = 6.09 % {\displaystyle \left1+{\frac {6\%}{2}}\right^{2}-1=1.03^{2}-1=6.09\%}### * 3.3. Calculation. * Other formulations. * (Другие формулировки) *

The outstanding balance B n 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 n = 1 + r B n − 1 − p, {\displaystyle B_{n}={\big }1+r{\big}B_{n-1}-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 } see binomial theoremand we see that r and p transform in the same manner,

r k = λ k r {\displaystyle r_{k}=\lambda _{k}r} p k = λ k p {\displaystyle p_{k}=\lambda _{k}p}The change in the balance transforms likewise,

Δ B k = B ′ − B = λ k r B − λ k p = λ k Δ B {\displaystyle \Delta B_{k}=B-B=\lambda _{k}rB-\lambda _{k}p=\lambda _{k}\,\Delta B}which gives an insight into the meaning of some of the coefficients found in the formulas above. The annual rate, r 12, assumes only one payment per year and is not an "effective" rate for monthly payments. With monthly payments the monthly interest is paid out of each payment and so should not be compounded and an annual rate of 12 r would make more sense. If one just made interest-only payments the amount paid for the year would be 12 r B 0.

Substituting p k = r k B * into the equation for the B k we get,

B k = B 0 − r k B ∗ − B 0 {\displaystyle B_{k}=B_{0}-r_{k}B^{*}-B_{0}}Since B n = 0 we can solve for B *,

B ∗ = B 0 1 r n + 1. {\displaystyle B^{*}=B_{0}\left{\frac {1}{r_{n}}}+1\right.}Substituting back into the formula for the B k shows that they are a linear function from the r k and therefore the λ k,

B k = B 0 1 − r k r n = B 0 1 − λ k λ n {\displaystyle B_{k}=B_{0}\left1-{\frac {r_{k}}{r_{n}}}\right=B_{0}\left1-{\frac {\lambda _{k}}{\lambda _{n}}}\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 get,

λ k + 1 = 1 + 1 + r λ k {\displaystyle \lambda _{k+1}=1+1+r\lambda _{k}}λ 0 and λ n can be found using the formula for λ k above or computing the λ k recursively from λ 0 = 0 to λ n.

Since p = rB * the formula for the payment reduces to,

p = r + 1 λ n B 0 {\displaystyle p=\leftr+{\frac {1}{\lambda _{n}}}\rightB_{0}}and the average interest rate over the period of the loan is

r loan = I T n B 0 = r (loan r = I T n B 0 = r) + 1 λ n − 1 n, {\displaystyle r_{\text{loan}}={\frac {I_{T}}{nB_{0}}}=r+{\frac {1}{\lambda _{n}}}-{\frac {1}{n}},}which is less than r if n > 1.

## 4. Discount instruments. * (Скидки инструментов) *

- 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 days: 365) t 100. See also: Day count convention. 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.

### * 5.1. Rules of thumb. * Rule of 78s. * (Правило 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 credit, 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 from the loan is because, 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 (and pay off of the principal then will cause) in 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.

### * 5.2. Rules of thumb. * Rule of 72. * (Правило 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 4 d.p. {\displaystyle 1.18^{4}=1.9388{\text{ 4 d.p.}}}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 4 d.p. {\displaystyle 1.24^{3}=1.9066{\text{ 4 d.p.}}}## 6. Market interest rates. * (Процентных ставок на рынке) *

There are markets for investments set interest rates. Each specific debt takes into account the following factors in determining its interest rate:

### * 6.1. Market interest rates. * Opportunity cost and deferred consumption. * (Возможность стоимость и отложенным потреблением) *

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 lenders 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.

### * 6.2. Market interest rates. * Inflation. * (Инфляция) *

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:

- 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.
- Decide on the "expected" inflation rate. This still leaves the lender exposed to the risk of "unexpected" inflation.
- 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.

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 growth government stocks are negative.

### * 6.3. Market interest rates. * Default. * (По умолчанию) *

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 individuals 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.

### * 6.4. Market interest rates. * Composition of interest rates. * (Состав процентных ставок) *

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. Nominal interest is composed of the real interest rate plus inflation, among other factors. An approximate formula for the nominal interest is:

i = r + π {\displaystyle i=r+\pi }Where

i is the nominal interest rate r is the real interest rate and π 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 depends 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.

The equation:

i = r + π + c {\displaystyle 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 π is the inflation expected and c is yield spread according to the perceived credit risk.### * 6.5. Market interest rates. * Default interest. * (Интерес по умолчанию) *

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 borrowers 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.

### * 6.6. Market interest rates. * Term. * (Термин) *

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 long term interest rates an upward sloping yield curve.

### * 6.7. Market interest rates. * Government intervention. * (Вмешательство правительства) *

Interest rates are generally determined by the market, but government intervention - usually by a central bank - may strongly effect 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 which has a major influence on supply and demand and hence on market interest rates.

### * 6.8. Market interest rates. * Open market operations in the United States. * (Операции на открытом рынке в США) *

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 nations 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.

### * 6.9. Market interest rates. * Interest rates and credit risk. * (Процентные ставки и кредитный риск) *

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 from 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.

### * 6.10. Market interest rates. * Money and inflation. * (Деньги и инфляция) *

Loans and bonds have some of the characteristics of money and are included in the broad money supply.

National governments 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 wide 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.

### * 6.11. Market interest rates. * Liquidity. * (Ликвидность) *

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 funding often carry higher interest rates due to lack of liquidity.

### * 7.1. Theories of interest. * Aristotles view of interest. * (Аристотеля представление интересов) *

Aristotle and the Scholastics held that it was unjust to claim payment except in compensation for ones 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 for the trouble of setting up a loan was not necessarily unacceptable on these grounds.

### * 7.2. Theories of interest. * Development of the theory of interest during the seventeenth and eighteenth centuries. * (Развитие теории интереса в течение семнадцатого и восемнадцатого веков) *

Nicholas Barbon S. 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 (buy goods and assets) stock, the interest that is charged on a loan is a type of rent – "a payment for the use of goods". According to Schumpeter, Barbons 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 (which relates to interest) the "demand for borrowing", the "riches available to supply that demand" and the "profits arising from commerce". Schumpeter considered Humes 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 Reflexions, 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.

### * 7.3. Theories of interest. * The classical theory of the interest rate. * (Классическая теория процентной ставки) *

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 he.

Mills theory is set out the chapter "Of the rate of interest" in his "Principles from 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 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.

Ricardos and Mills profit is made more precise by the concept of the marginal efficiency of capital, 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 limit 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 remind the blue curve in the diagram, with a step shape arising from the assumption that opportunities to invest with yields greater than R have been largely exhausted while there is untapped scope to invest with a lower return.

Saving is the excess of deferred over anticipated (of deferred over the 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 (on ys income 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 for 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.

### * 7.4. Theories of interest. * Keyness criticisms. * (Критика Keyness) *

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 doesnt explain the usual supposition that "an increase in the quantity of money there a tendency to reduce the rate of interest, at any rate in the first instance".

Keyness 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 to 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 is maintained at a given figure.

Later p. 184 Keynes claims that "it involves to 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.

### * 7.5. Theories of interest. * Theories of exploitation, productivity and abstinence. * (Теории эксплуатации, производительности и воздержания) *

The classical theory of interest explains it as the capitalists 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 profit as exploitation, and McCullochs 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.

### * 7.6. Theories of interest. * Wicksells theory. * (Wicksells теории) *

Knut Wicksell published his "Interest and Prices" in 1898, elaborating a comprehensive theory of economic crises based upon a distinction between natural and rated interest rates.

Wicksells contribution, in fact, was twofold. First he separated the monetary rate of interest from the hypothetical "natural" rate that would have resulted from balance 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 Wicksells approach was refined at Bertil Ohlin and Dennis Robertson and became known as the loanable funds theory.

### * 7.7. Theories of interest. * Austrian theories. * (Австрийской теории) *

Eugen Bohm von Bawerk (Eugen von Bohm 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 from 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

### * 7.8. Theories of interest. * Paretos indifference. * (Paretos безразличие) *

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.

### * 7.9. Theories of interest. * Keyness theory of the interest rate. * (Теория Keyness процентной ставки) *

Interest is one of the main components of the economic theories developed in Keyness 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.

### * 8.1. In religious contexts. * Judaism. * (Иудаизм) *

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 Societys 1917 Tanakh, with Christian verse numbers, where different, in parentheses:

If thou lend money to any of My people, even to the poor with thee, thou shalt not be to him as a the lender, neither shall ye lay upon him interest.

Take thou no interest of him or to increase, but fear thy God, that thy brother may live with thee.

Thou shalt not give him thy money upon interest, nor give him thy victuals for increase.

Thou shalt not lend upon interest to thy brother: interest of money, interest of victuals, interest of any thing that is lent upon interest.

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.

. that hath withdrawn his hand from the poor, that hath not received interest nor increase, hath executed Mine ordinances, hath walked in My Charter, he shall not die for the iniquity of his father, he shall surely live.

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.

### * 8.2. In religious contexts. * Islam. * (Ислам) *

The following quotations are English translations from the Quran:

Those who charge usury are in the same position as those controlled by the devils 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.

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.

O you who believe, you shall not take usury, compounded over and over. Observe God, that you may succeed.

And for practicing usury, which was forbidden, and for consuming the peoples money illicitly. We have prepared for the disbelievers among them painful retribution.

The usury that is practiced to increase some peoples wealth, does not gain anything at God. But if people give to charity, seeking Gods pleasure, these are the ones who receive their reward many fold.

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 Abdal Muttalib Prophets uncle shall henceforth be waived.

### * 8.3. In religious contexts. * Christianity. * (Христианство) *

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. Aristotles 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. 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 lists 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" Rothbard 1995, p. 46. This idea is very similar to opportunity cost. Many scholastic thinkers who argued for a ban on interest charges also claimed 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 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 XIVs 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. 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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