Account of politics derived from but not limited to Marxism.

As members of groups, classes, and so on, people have interests which can be identified by the outside observer. These interests are real whether individuals or whole groups are aware of them or not, and enable commentators to detect success, exploitation, and so on, in the relations between one group and another. An obvious difficulty arises with attributing to a group an interest of which none of its members appear to be aware.

Source:

Geoffrey Roberts and Alistair Edwards, A New Dictionary of Political Analysis (London, 1991)

## History

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”.^{[5]}

The first written evidence of compound interest dates roughly 2400 BC.^{[6]} The annual interest rate was roughly 20%. Compound interest was necessary for the development of agriculture and important for urbanization.^{[7]}^{[dubious – discuss]}

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.^{[8]} 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.^{[9]}

The First Council of Nicaea, in 325, forbade clergy from engaging in usury^{[10]} which was defined as lending on interest above 1 percent per month (12.7% AER). Ninth century ecumenical councils applied this regulation to the laity.^{[10]}^{[11]} 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.^{[12]} 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]}

### Islamic finance

The latter half of the 20th century saw the rise of interest-free Islamic banking and finance, a movement that applies Islamic law to financial institutions and the economy. Some countries, including Iran, Sudan, and Pakistan, have taken steps to eradicate interest from their financial systems.^{[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 the history of mathematics

It is thought that Jacob Bernoulli discovered the mathematical constant e by studying a question about compound interest.^{[13]} He realized that if an account that starts with $1.00 and pays say 100% interest per year, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.5^{2} = $2.25. Compounding quarterly yields $1.00×1.25^{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:

- {\displaystyle \lim _{n\rightarrow \infty }\left(1+{\dfrac {1}{n}}\right)^{n}=e,}

where *n* is the number of times the interest is to be compounded in a year.

## Economics

In economics, the rate of interest is the price of credit, 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.^{[14]} 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.

## Calculation

### Simple interest

**Simple interest** is calculated only on the principal amount, or on that portion of the principal amount that remains. It excludes the effect of compounding. 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:

- {\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,

- {\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

- {\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

- {\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.)

### Compound interest

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:

- {\displaystyle {\frac {r\cdot B\cdot m}{n}}={\frac {6\%\times \$10\,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:

- {\displaystyle \$10\,000+\$300=\left(1+{\frac {r}{n}}\right)\cdot B=\left(1+{\frac {6\%}{2}}\right)\times \$10\,000}

and so earns a coupon at the end of the next 6 months of:

- {\displaystyle {\begin{aligned}{\frac {r\cdot B\cdot m}{n}}&={\frac {6\%\times \left(\$10\,000+\$300\right)}{2}}\\&={\frac {6\%\times \left(1+{\frac {6\%}{2}}\right)\times \$10\,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:

- {\displaystyle {\begin{aligned}\$10,000+\$300+\$309&=\$10\,000+{\frac {6\%\times \$10,000}{2}}+{\frac {6\%\times \left(1+{\frac {6\%}{2}}\right)\times \$10\,000}{2}}\\&=\$10\,000\times \left(1+{\frac {6\%}{2}}\right)^{2}\end{aligned}}}

and the investor earned in total:

- {\displaystyle {\begin{aligned}\$10\,000\times \left(1+{\frac {6\%}{2}}\right)^{2}-\$10\,000\\=\$10\,000\times \left(\left(1+{\frac {6\%}{2}}\right)^{2}-1\right)\end{aligned}}}

The formula for the **annual equivalent compound interest rate** is:

- {\displaystyle \left(1+{\frac {r}{n}}\right)^{n}-1}

where

- r is the simple annual rate of interest
- n is the frequency of applying interest

For example, in the case of a 6% simple annual rate, the annual equivalent compound rate is:

- {\displaystyle \left(1+{\frac {6\%}{2}}\right)^{2}-1=1.03^{2}-1=6.09\%}

### Other formulations

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:

- {\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) [2]*B*_{0}= initial balance, which equals the principal sum

By repeated substitution one obtains expressions for *B*_{n}, which are linearly proportional to *B*_{0} and p and use of the formula for the partial sum of a geometric series results in

- {\displaystyle B_{n}=(1+r)^{n}B_{0}-{\frac {(1+r)^{n}-1}{r}}p}

A solution of this expression for *p* in terms of *B*_{0} and *B*_{n} reduces to

- {\displaystyle p=r\left[{\frac {(1+r)^{n}B_{0}-B_{n}}{(1+r)^{n}-1}}\right]}

To find the payment if the loan is to be finished in *n* payments one sets *B*_{n} = 0.

The PMT function found in spreadsheet programs can be used to calculate the monthly payment of a loan:

- {\displaystyle p=\mathrm {PMT} ({\text{rate}},{\text{num}},{\text{PV}},{\text{FV}},)=\mathrm {PMT} (r,n,-B_{0},B_{n},)}

An interest-only payment on the current balance would be

- {\displaystyle p_{I}=rB.}

The total interest, *I*_{T}, paid on the loan is

- {\displaystyle I_{T}=np-B_{0}.}

The formulas for a regular savings program are similar but the payments are added to the balances instead of being subtracted and the formula for the payment is the negative of the one above. These formulas are only approximate since actual loan balances are affected by rounding. To avoid an underpayment at the end of the loan, the payment must be rounded up to the next cent.

Consider a similar loan but with a new period equal to *k* periods of the problem above. If *r*_{k} and *p*_{k} are the new rate and payment, we now have

- {\displaystyle B_{k}=B’_{0}=(1+r_{k})B_{0}-p_{k}.}

Comparing this with the expression for B_{k} above we note that

- {\displaystyle r_{k}=(1+r)^{k}-1}

and

- {\displaystyle p_{k}={\frac {p}{r}}r_{k}.}

The last equation allows us to define a constant that is the same for both problems,

- {\displaystyle B^{*}={\frac {p}{r}}={\frac {p_{k}}{r_{k}}}}

and *B*_{k} can be written as

- {\displaystyle B_{k}=(1+r_{k})B_{0}-r_{k}B^{*}.}

Solving for *r*_{k} we find a formula for *r*_{k} involving known quantities and *B*_{k}, the balance after *k* periods,

- {\displaystyle r_{k}={\frac {B_{0}-B_{k}}{B^{*}-B_{0}}}}

Since B_{0} could be any balance in the loan, the formula works for any two balances separate by *k* periods and can be used to compute a value for the annual interest rate.

*B** is a scale invariant 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),

- {\displaystyle \lambda _{k}={\frac {p_{k}}{p}}={\frac {r_{k}}{r}}={\frac {(1+r)^{k}-1}{r}}=k\left[1+{\frac {(k-1)r}{2}}+\cdots \right]} (see binomial theorem)

and we see that *r* and *p* transform in the same manner,

- {\displaystyle r_{k}=\lambda _{k}r}
- {\displaystyle p_{k}=\lambda _{k}p}

The change in the balance transforms likewise,

- {\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,

- {\displaystyle B_{k}=B_{0}-r_{k}(B^{*}-B_{0})}

Since *B*_{n} = 0 we can solve for *B**,

- {\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 of the *r*_{k} and therefore the *λ*_{k},

- {\displaystyle B_{k}=B_{0}\left(1-{\frac {r_{k}}{r_{n}}}\right)=B_{0}\left(1-{\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,

- {\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,

- {\displaystyle p=\left(r+{\frac {1}{\lambda _{n}}}\right)B_{0}}

and the average interest rate over the period of the loan is

- {\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.

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