# Interest

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Interest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets. It is most commonly the price paid for the use of borrowed money,[1] or money earned by deposited funds.[2]

When money is borrowed, interest is typically paid to the lender as a percentage of the principal, the amount owed to the lender. The percentage of the principal that is paid as a fee over a certain period of time (typically one month or year) is called the interest rate. A bank deposit will earn interest because the bank is paying for the use of the deposited funds. Assets that are sometimes lent with interest include money, shares, consumer goods through hire purchase, major assets such as aircraft, and even entire factories in finance lease arrangements. The interest is calculated upon the value of the assets in the same manner as upon money.

Interest is compensation to the lender, for a) risk of principal loss, called credit risk; and b) forgoing other investments that could have been made with the loaned asset. These forgone investments are known as the opportunity cost. Instead of the lender using the assets directly, they are advanced to the borrower. The borrower then enjoys the benefit of using the assets ahead of the effort required to pay for them, while the lender enjoys the benefit of the fee paid by the borrower for the privilege. In economics, interest is considered the price of credit.

Interest is often compounded, which means that interest is earned on prior interest in addition to the principal. The total amount of debt grows exponentially, most notably when compounded at infinitesimally small intervals, and its mathematical study led to the discovery of the number e.[3] However, in practice, interest is most often calculated on a daily, monthly, or yearly basis, and its impact is influenced greatly by its compounding rate.

## History of interest

According to historian Paul Johnson, the lending of "food money" was commonplace in Middle East civilizations as far back as 5000 BC. They regarded interest as legitimate since acquired seeds and animals could "reproduce themselves"; whilst the ancient Jewish religious prohibitions against usury (נשך NeSheKh) were a "different view".[4] On this basis, the Laws of Eshnunna (early 2nd millennium BC) instituted a legal interest rate, specifically on deposits of dowry, since the silver being used in exchange for livestock or grain could not multiply of its own.

The First Council of Nicaea, in 325, forbade clergy from engaging in usury[5] which was defined as lending on interest above 1 percent per month (12.7% APR). Later ecumenical councils applied this regulation to the laity.[5][6] 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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[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.[7] For the same reason, interest has often been looked down upon in Islamic civilization, with most 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. Of Usury, from Brant's Stultifera Navis (the Ship of Fools); woodcut attributed to Albrecht Dürer 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 School of Salamanca elaborated on various reasons that justified the charging of interest: the person who received a loan benefited, and one could consider interest as a premium paid for the risk taken by the loaning party. There was also the question of opportunity cost, in that the loaning party lost other possibilities of using the loaned money. Finally and perhaps most originally was the consideration of money itself as merchandise, and the use of one's money as something for which one should receive a benefit in the form of interest. Martín de Azpilcueta also considered the effect of time. Other things being equal, one would prefer to receive a given good now rather than in the future. This preference indicates greater value. Interest, under this theory, is the payment for the time the loaning individual is deprived of the money. Economically, the interest rate is the cost of capital and is subject to the laws of supply and demand of the money supply. The first attempt to control interest rates through manipulation of the money supply was made by the French Central Bank in 1847. The first formal studies of interest rates and their impact on society were conducted by Adam Smith, Jeremy Bentham and Mirabeau during the birth of classic economic thought.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} In the late 19th century leading 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 early 20th century, Irving Fisher made a major breakthrough in the economic analysis of interest rates by distinguishing nominal interest from real interest. Several perspectives on the nature and impact of interest rates have arisen since then.

The latter half of the 20th century saw the rise of interest-free Islamic banking and finance, a movement that attempts to apply religious law developed in the medieval period to the modern economy. Some entire countries, including Iran, Sudan, and Pakistan, have taken steps to eradicate interest from their financial systems altogether.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[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 "fee" for the service of lending. ## Types of interest Simple interest is calculated only on the principal amount, or on that portion of the principal amount that remains. The amount of simple interest is calculated according to the following formula: ${\displaystyle I_{\text{simple}}=r\cdot B_{0}\cdot m_{t}}$ where r is the period interest rate (I/m), B0 the initial balance and mt the number of time periods elapsed. To calculate the period interest rate r, one divides the interest rate I by the number of periods mt. For example, imagine that a credit card holder has an outstanding balance of$2500 and that the simple interest rate is 12.99% per annum. The interest added at the end of 3 months would be,

${\displaystyle I_{\text{simple}}=\left({\frac {0.1299}{12}}\cdot \2500\right)\cdot 3=\81.19}$

and they would have to pay $2581.19 to pay off the balance at this point. If instead they make interest-only payments for each of those 3 months at the period rate r, the amount of interest paid would be, ${\displaystyle I=\left({\frac {0.1299}{12}}\cdot \2500\right)\cdot 3=(\27.0625/{\text{month}})\cdot 3=\81.19}$ Their balance at the end of 3 months would still be$2500.

In this case, the time value of money is not factored in. The steady payments have an additional cost that needs to be considered when comparing loans. For example, given a $100 principal: • Credit card debt where$1/day is charged: 1/100 = 1%/day = 7%/week = 365%/year.
• Corporate bond where the first $3 are due after six months, and the second$3 are due at the year's end: (3 + 3)/100 = 6%/year.
• Certificate of deposit (GIC) where $6 is paid at the year's end: 6/100 = 6%/year. There are two complications involved when comparing different simple interest bearing offers. 1. When rates are the same but the periods are different a direct comparison is inaccurate because of the time value of money. Paying$3 every six months costs more than $6 paid at year end so, the 6% bond cannot be 'equated' to the 6% GIC. 2. When interest is due, but not paid, does it remain 'interest payable', like the bond's$3 payment after six months or, will it be added to the balance due? In the latter case it is no longer simple interest, but compound interest.

A bank account that offers only simple interest, that money can freely be withdrawn from is unlikely, since withdrawing money and immediately depositing it again would be advantageous.

### Composition of interest rates

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 (i.e., 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. A simple formula for the nominal interest is:

${\displaystyle i=r+\pi }$

Where i is the nominal interest, r is the real interest and π is inflation.

This formula attempts to measure the value of the interest in units of stable purchasing power. However, if this statement were true, it would imply at least two misconceptions. First, that all interest rates within an area that shares the same inflation (that is, the same country) should be the same. Second, that the lenders know the inflation for the period of time that they are going to lend the money.

One reason behind the difference between the interest that yields a treasury bond and the interest that yields a mortgage loan is the risk that the lender takes from lending money to an economic agent. In this particular case, a government is more likely to pay than a private citizen. Therefore, the interest rate charged to a private citizen is larger than the rate charged to the government.

To take into account the information asymmetry aforementioned, both the value of inflation and the real price of money are changed to their expected values resulting in the following equation:

${\displaystyle i_{t}=r_{(t+1)}+\pi _{(t+1)}+\sigma }$

Here, it is the nominal interest at the time of the loan, r(t+1) is the real interest expected over the period of the loan, π(t+1) is the inflation expected over the period of the loan and ${\displaystyle \sigma }$ is the representative value for the risk engaged in the operation.

## Interest in mathematics

It is thought that Jacob Bernoulli discovered the mathematical constant e by studying a question about compound interest.[10] He realized that if an account that starts with $1.00 and pays say 100% interest per year, at the end of the year, the value is$2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding$1.00×1.52 = $2.25. Compounding quarterly yields$1.00×1.254 = \$2.4414…, and so on.

Bernoulli noticed that if the frequency of compounding is increased without limit, this sequence can be modeled as follows:

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

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

## Formulas

The balance of a loan with regular monthly payments is augmented by the monthly interest charge and decreased by the payment so

${\displaystyle B_{k+1}={\big (}1+r{\big )}B_{k}-p,}$

where

i = loan rate/100 = annual rate in decimal form (e.g. 10% = 0.10 The loan rate is the rate used to compute payments and balances.)
r = period rate = i/12 for monthly payments (customary usage for convenience)[1]
B0 = initial balance (loan principal)
Bk = balance after k payments
k = balance index
p = period (monthly) payment

By repeated substitution one obtains expressions for Bk, which are linearly proportional to B0 and p and use of the formula for the partial sum of a geometric series results in

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

A solution of this expression for p in terms of B0 and Bn 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 Bn = 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, IT, 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 rk and pk 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 Bk 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 Bk can be written as

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

Solving for rk we find a formula for rk involving known quantities and Bk, the balance after k periods,

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

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),

${\displaystyle \lambda _{k}={\frac {p_{k}}{p}}={\frac {r_{k}}{r}}={\frac {(1+r)^{k}-1}{r}}=k\left[1+{\frac {(k-1)r}{2}}+\cdots \right]}$ (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, 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 = rk B* into the equation for the Bk we get,

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

Since Bn = 0 we can solve for B*,

${\displaystyle B^{*}=B_{0}\left({\frac {1}{r_{n}}}+1\right).}$

Substituting back into the formula for the Bk shows that they are a linear function of the rk and therefore the λk,

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

This is the easiest way of estimating the balances if the λk are known. Substituting into the first formula for Bk above and solving for λk+1 we get,

${\displaystyle \lambda _{k+1}=1+(1+r)\lambda _{k}\;}$

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,

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

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.

## Notes

1. {{#invoke:citation/CS1|citation |CitationClass=book }}
2. {{#invoke:citation/CS1|citation |CitationClass=book }}
3. Template:Cite web
4. Johnson, Paul: A History of the Jews (New York: HarperCollins Publishers, 1987) ISBN 0-06-091533-1, pp. 172–73.
5. Moehlman, 1934, p. 6.
6. Noonan, John T., Jr. 1993. "Development of Moral Doctrine." 54 Theological Stud. 662.
7. No. 2547: Charging Interest
8. Rule of 78 - Watch out for this auto loan trick
9. Template:Usc
10. Template:Cite web

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