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The fee is compensation to the lender for foregoing other useful investments that could have been made with the loaned money. Instead of the lender using the assets directly, they are advanced to the borrower. The borrower then enjoys the benefit of the use of the assets ahead of the effort required to obtain them, while the lender enjoys the benefit of the fee paid by the borrower for the privilege. The amount lent, or the value of the assets lent, is called the principal. This principal value is held by the borrower on credit. Interest is therefore the price of credit, not the price of money as it is commonly - and mistakenly - believed to be. The percentage of the principal that is paid as a fee (the interest), over a certain period of time, is called the interest rate. Investment is a term with several closely related meanings in finance and economics. ... Credit as a financial term, used in such terms as credit card, refers to the granting of a loan and the creation of debt. ...

The charge of interest dates back to 1500 B.C.[citation needed] among the Sumerian and Egyptian cultures. References to the concept can be found in the religious text of the Abrahamic religions such as the counsel against excessive interest. Image File history File links Emblem-important. ... The history of Sumer, taken to include the prehistoric Ubaid and Uruk periods, spans the 5th to 3rd millennia BC, ending with the downfall of the Third Dynasty of Ur around 2004 BC, followed by a transition period of Amorite states before the rise of Babylonia in the 18th century... map showing the prevalence of Abrahamic (purple) and Dharmic (yellow) religions in each country. ... Look up usury in Wiktionary, the free dictionary. ...

Interest is the price paid for the use of savings over a given period of time. In the Middle Ages, time was considered to be property of God. Therefore, to charge interest was considered to commerce with God's property. Also, 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. The church regarded this as a sin of usury; nevertheless, this rule was never strictly obeyed and eroded gradually until it disappeared during the industrial revolution. Some scholars think that banking started among Jewish families because of the restrictions of the church. The Middle Ages formed the middle period in a traditional schematic division of European history into three ages: the classical civilization of Antiquity, the Middle Ages, and modern times, beginning with the Renaissance. ... Saint Thomas Aquinas, O.P.(also Thomas of Aquin, or Aquino; c. ... The name Catholic Church can mean a visible organization that refers to itself as Catholic, or the invisible Christian Church, viz. ... Look up usury in Wiktionary, the free dictionary. ...

... financial oppression of Jews tended to occur in areas where they were most disliked, and if Jews reacted by concentrating on moneylending to gentiles, the unpopularity - and so, of course, the pressure - would increase. This is that the Jews became an element in a vicious circle. The Christians, on the basis of the Biblical rulings, condemned interest-taking absolutely, and from 1179 those who practiced it were excommunicated. But the Christians also imposed the harshest financial burdens on the Jews. The Jews reacted by engaging in the one business where Christian laws actually discriminated in their favor, and so became identified with the hated trade of moneylending. [1] Events Third Council of the Lateran condemned Waldensians and Cathars as heretics, institutes a reformation of clerical life, and creates the first ghettos for Jews Afonso I is recognized as the true King of Portugal by Portugal the protection of the Catholic Church against the Castillian monarchy Philip II is... Excommunication is a religious censure used to deprive or suspend membership in a religious community. ...

Usury has always been viewed negatively by the Roman Catholic Church. The Second Lateran Council condemned any repayment of a debt with more money than was originally loaned, the Council of Vienna explicitly prohibited usury and declared any legislation tolerant of usury to be heretical, and the first scholastics reproved the charging of interest. 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. Look up usury in Wiktionary, the free dictionary. ... The Second Lateran Council was called by Pope Innocent II in 1139 as an attempt to reunify the church after the two papacies. ...

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, it could not be viewed in the same manner. The School of Salamanca elaborated 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 utilizing 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. This article is about the European Renaissance of the 14th-17th centuries. ... An entrepreneur (a loanword from French introduced and first defined by the Irish economist Richard Cantillon) is a person who operates a new enterprise or venture and assumes some accountability for the inherent risks. ... Opportunity cost is a central concept of microeconomics. ...

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. MartÃ­n de Azpilcueta[1] (b. ... Time preference is the economists assumption that a consumer will place a premium on enjoyment nearer in time over more remote enjoyment. ...

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 until 1847. The supply and demand model describes how prices vary as a result of a balance between product availability at each price (supply) and the desires of those with purchasing power at each price (demand). ... In macroeconomics, money supply (monetary aggregates, money stock) is the quantity of currency and money in bank accounts in the hands of the non-bank public available within the economy to purchase goods, services, and securities. ... In macroeconomics, money supply (monetary aggregates, money stock) is the quantity of currency and money in bank accounts in the hands of the non-bank public available within the economy to purchase goods, services, and securities. ... One of the Banque de Frances offices in Paris. ...

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. 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. Among academics, the more modern views of John Maynard Keynes and Milton Friedman are widely accepted. For other persons named Adam Smith, see Adam Smith (disambiguation). ... Jeremy Bentham (IPA: ) (26 February [O.S. 15 February 15] 1748) â€“ June 6, 1832) was an English jurist, philosopher, and legal and social reformer. ... Mirabeau can refer to: Honoré Mirabeau Mirabeau, a commune of the Alpes-de-Haute-Provence département, in southeastern France Mirabeau, a commune of the Vaucluse département, in southern France Les Pennes-Mirabeau, a commune of the Bouches-du-Rhône département, in southern France the Cours Mirabeau is a famous street... Irving Fisher, 1927. ... Keynes redirects here. ... Milton Friedman (July 31, 1912 â€“ November 16, 2006) was an American Nobel Laureate economist and public intellectual. ...

Former Central President of the JUP Sahibzada Fazal Karim MNA has stated that the Council of Islamic ideology feels that Islamic banking ought to be interest-free by law. Islamic banking refers to a system of banking or banking activity that is consistent with Islamic law (Sharia) principles and guided by Islamic economics. ... Riba is the (Arabic: Ø±Ø¨Ø§ ) term for intrest, the charging of which is forbidden by the Quran here, among other places: And that which you give in gift (loan) (to others), in order that it may increase (your wealth by expecting to get a better one in return) from other... Sharia (Arabic: transliteration: ) is the body of Islamic religious law. ...

## Types of interest

### Simple interest

Simple Interest is calculated only on the principal, or on that portion of the principal which remains unpaid.

The amount of simple interest is calculated according to the following formula:

$I_{simp} = (r cdot B_0) cdot n$

where r is the period interest rate (10%=0.10), B0 the initial balance and n the number of time periods elapsed.

For example, imagine that a credit card holder has an outstanding balance of \$2500 and that the interest rate is 12.99% per annum. If simple interest were charged on the balance the interest added at the end of 3 months would be,

$I_{simp} = bigg(frac{0.1299}{12}cdot 2500bigg) cdot 3=81.19$

and he would have to pay \$2581.19 to pay off the balance at this time.

If instead he makes interest-only payments on the balance for 3 months at the same period rate the amount of interest paid would be,

$I = bigg(frac{0.1299}{12}cdot 2500bigg) cdot 3= (27.0625) cdot 3=81.19$

The balance at the end of 3 months would still be \$2500. So, interest-only payments result in amounts similar to simple interest but the difference is that the time value of money is not factored in and the steady payments have an additional cost which needs to be considered when comparing loans. The time value of money is the premise that an investor prefers to receive a payment of a fixed amount of money today, rather than an equal amount in the future, all else being equal. ...

To calculate the simple interest rate r one divides the interest paid by the number of periods and then divides the result by the initial balance. For example, given a \$100 principal:

• Credit card debt where \$1/day is charged: 1/100 = 1%/day.
• 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 in using the simple interest rate. A Guaranteed Investment Certificate is a Canadian investment that offers a guaranteed rate of return over a fixed period of time, most commonly issued by trust companies or banks. ...

1. If you want to compare two different interest bearing offers and the time periods for each offer are different, a direct comparison is wrong because of the time value of money. For example 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, the consequences are unclear. For example, does it remain 'interest payable', like the bond's \$3 payment after six months? Alternatively, will it be added to the original principal, as would typically be the case in the 1%/day borrowed via the credit card? In the latter case, it is no longer simple interest, but compound interest.

The time value of money is the premise that an investor prefers to receive a payment of a fixed amount of money today, rather than an equal amount in the future, all else being equal. ...

### Compound interest

Main article: Compound interest

In the short run, compound Interest is very similar to Simple Interest, however, as time goes on difference becomes considerably larger. The conceptual difference is that the principal changes with every time period, as any interest incurred over the period is added to the principal. Put another way, the lender is charging interest on the interest. Compound interest refers to the fact that whenever interest is calculated, it is based not only on the original principal, but also on any unpaid interest that has been added to the principal. ...

Assuming that no part of the principal or subsequent interest has been paid, the amount of the debt, including principal and compound interest incurred, is calculated by the following formulas:

begin{align} &I_{comp}=B_0cdotbig[left(1+rright)^n-1big] &B_n=B_0+I_{comp} end{align}

where Icomp is the compound interest, B0 the initial balance, Bn the balance after n months and r the period rate.

For example, if the credit card holder above chose not to make any payments the interest would accumulate,

begin{align} &mbox{Calculation for Compound Interest}: I_{comp}&=2500cdotbigg[bigg(1+frac{0.1299}{12}bigg)^3-1bigg] &=2500cdotleft(1.010825^3-1right) &=82.07 end{align}
begin{align} B_3&=B_0+I_{comp} &=2500+82.07 &=2582.07 end{align}

So, at the end of 3 months the credit card holder's balance would be \$2582.07 and he would now have to pay \$82.07 to get it down to the initial balance. Simple interest is approximately the same as compound interest for short periods of time. The interest payments indicate that more frequent payments is the better payment strategy.

A problem with compound interest is that the resulting obligation can be difficult to interpret. To simplify this problem, a common convention in economics is to disclose the interest rate as though the term were one year, with annual compounding, yielding the effective interest rate. However, interest rates in lending are often quoted as nominal interest rates (i.e., compounding interest uncorrected for the frequency of compounding). The discussion at compound interest shows how to convert to and from the different measures of interest. In contrast to a nominal interest rate, the period of time after that the interest is credited coincides with the basic time unit (normally one year). ... For other uses, see Loan (disambiguation). ... A nominal interest rate is the interest rate as stated - that is, not adjusted for compounding. ... Compound interest refers to the fact that whenever interest is calculated, it is based not only on the original principal, but also on any unpaid interest that has been added to the principal. ...

Loans often include various non-interest charges and fees. One example are points on a mortgage loan in the United States. When such fees are present, lenders are regularly required to provide information on the 'true' cost of finance, often expressed as an annual percentage rate (APR). The APR attempts to express the total cost of a loan as an interest rate after including the additional fees and expenses, although details may vary by jurisdiction. This article or section is in need of attention from an expert on the subject. ... A mortgage loan is a loan secured by real property through the use of a mortgage (a legal instrument). ... Annual Percentage Rate (APR) is an expression of the effective interest rate that the borrower will pay on a loan, taking into account one-time fees and standardizing the way the rate is expressed. ...

In economics, continuous compounding is often used due to its particular mathematical properties. Compound interest refers to the fact that whenever interest is calculated, it is based not only on the original principal, but also on any unpaid interest that has been added to the principal. ...

#### Fixed and floating rates

Commercial loans generally use compound interest, but they may not always have a single interest rate over the life of the loan. Loans for which the interest rate does not change are referred to as fixed rate loans. Loans may also have a changeable rate over the life of the loan based on some reference rate (such as LIBOR and EURIBOR), usually plus (or minus) a fixed margin. These are known as floating rate, variable rate or adjustable rate loans. A fixed interest rate loan is a loan where the interest rate doesnt not fluctuate over the life of the loan. ... A reference rate is any publicly available quoted number or value that is used by the parties to a financial contract. ... LIBOR stands for the London Interbank Offered Rate and is a daily reference rate based on the interest rates at which banks offer to lend unsecured funds to other banks in the London wholesale (or interbank) money market. ... Euribor-12m value between years 2001 and 2006 Euribor (Euro Interbank Offered Rate) is a daily reference rate based on the averaged interest rates at which banks offer to lend unsecured funds to other banks in the euro wholesale money market (or interbank market). ... This article does not cite any references or sources. ...

Combinations of fixed-rate and floating-rate loans are possible and frequently used. Less frequently, loans may have different interest rates applied over the life of the loan, where the changes to the interest rate are governed by specific criteria other than an underlying interest rate. An example would be a loan that uses specific periods of time to dictate specific changes in the rate, such as a rate of 5% in the first year, 6% in the second, and 7% in the third.

### Composition of interest rates

In economics, interest is considered the price of money, 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 by the real interest rate plus inflation, among other factors. A simple formula for the nominal interest is: A nominal interest rate is the interest rate as stated - that is, not adjusted for compounding. ... The real interest rate is the interest rate charged to a risk free borrower, minus the inflation rate. ...

i = r + π

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 was true, it would imply at least two misconceptions. First, that all interest rates within an area that shares the same inflation (i.e. the same country) should be the same. Second, that the lender knows the inflation for the period of time that he/she is 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, the US 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 US government. Treasury securities are government bonds issued by the United States Department of the Treasury through the Bureau of the Public Debt. ... A mortgage loan is a loan secured by real property through the use of a mortgage (a legal instrument). ...

To take into account the information asymmetry aforementioned, both the value of inflation and the real price of money is changed to their expected values resulting in the following equation: In probability (and especially gambling), the expected value (or expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical odds are...

it = rt + 1 + πt + 1 + σ

Where it is the nominal interest at the time of the loan, rt + 1 is the real interest expected over the period of the loan, πt + 1 is the inflation expected over the period of the loan and σ is the representative value for the risk engaged in the operation.

### Cumulative interest or return

Cumulative interest/return: This calculation is (FV/PV)-1. It ignores the 'per year' convention and assumes compounding at every payment date. It is usually used to compare two long term opportunities. Since the difference in rates gets magnified by time, so the speaker's point is more clearly made.

### Other conventions and uses

Other exceptions:

• US and Canadian T-Bills (short term Government debt) have a different convention. Their interest is calculated as (100-P)/P where 'P' is the price paid. Instead of normalizing it to a year, the interest is prorated by the number of days 't': (365/t)*100. (See also: Day count 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.
• Corporate Bonds are most frequently payable twice yearly. The amount of interest paid is the simple interest disclosed divided by two (multiplied by the face value of debt).

Flat Rate Loans and the Rule of 78s: Some consumer loans have been structured as flat rate loans, with the loan outstanding determined by allocating the total interest across the term of the loan by using the "Rule of 78s" or "Sum of digits" method. Seventy-eight is the sum of the numbers 1 through 12, inclusive. The practice enabled quick calculations of interest in the pre-computer days. In a loan with interest calculated per the Rule of 78s, the total interest over the life of the loan is calculated as either simple or compound interest and amounts to the same as either of the above methods. 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. [1] In finance, a day count convention determines how interest accrues over time for a variety of investments, including bonds, notes, loans, medium-term notes, swaps, and FRAs. ... Also known as the sum-of-the-digits method, the Rule of 78s is a term used in lending that refers to a method of yearly interest calculation. ...

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.[2] Certain other jurisdictions have outlawed application of the Rule of 78s in certain types of loans, particularly consumer loans. [2] Year 1992 (MCMXCII) was a leap year starting on Wednesday (link will display full 1992 Gregorian calendar). ...

Rule of 72: The "Rule of 72" is a "quick and dirty" method for finding out how fast money doubles for a given interest rate. For example, if you have an interest rate of 6%, it will take 72/6 or 12 years for your money to double, compounding at 6%. This is an approximation that starts to break down above 10%. In finance, the rule of 72, the rule of 71, the rule of 70 and the rule of 69. ...

## Market interest rates

There are markets for investments which include the money market, bond market, as well as retail financial institutions like banks, which set interest rates. Each specific debt takes into account the following factors in determining its interest rate:

Opportunity cost: This encompasses any other use to which the money could be put, including lending to others, investing elsewhere, holding cash (for safety, for example), and simply spending the funds. Opportunity cost is a central concept of microeconomics. ...

Inflation: Since the lender is deferring his consumption, he will at a bare minimum, want to recover enough to pay the increased cost of goods due to inflation. Because future inflation is unknown, there are three tactics.

• Charge X% interest 'plus inflation'. Many governments issue 'real-return' or 'inflation indexed' bonds. The principal amount and the interest payments are continually increased by the rate of inflations. See the discussion at real interest rate.
• Decide on the 'expected' inflation rate. This still leaves both parties exposed to the risk of 'unexpected' inflation.
• Allow the interest rate to be periodically changed. While a 'fixed interest rate' remains the same throughout the life of the debt, 'variable' or 'floating' rates can be reset. There are derivative products that allow for hedging and swaps between the two.

Default: 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. The real interest rate is the interest rate charged to a risk free borrower, minus the inflation rate. ...

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.

Deferred consumption: Charging interest equal only to inflation will leave the lender with the same purchasing power, but he would prefer his own consumption NOW rather than later. There will be an interest premium of the delay. See the discussion at time value of money. He may not want to consume, but instead would invest in another product. The possible return he could realize in competing investments will determine what interest he charges. The time value of money is the premise that an investor prefers to receive a payment of a fixed amount of money today, rather than an equal amount in the future, all else being equal. ...

Length of time: Time has two effects.

• Shorter terms have less risk of default and inflation because the near future is easier to predict. Broadly speaking, if interest rates increase, then investment decreases due to the higher cost of borrowing (all else being equal).

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 used as the main tool of monetary policy. The central bank offers to buy or sell money at the desired rate and, due to their control of certain tools (such as, in many countries, the ability to print money) they are able to influence overall market interest rates. Tax rates around the world Tax revenue as % of GDP Economic policy Monetary policy Central bank   Money supply Fiscal policy Spending   Deficit   Debt Trade policy Tariff   Trade agreement Finance Financial market Financial market participants Corporate   Personal Public   Banking   Regulation        Monetary policy is the process by which the government, central bank...

Investment can change rapidly to changes in interest rates, affecting national income, and, through Okun's Law, changes in output affect unemployment. Graph of US quarterly data (not annualized) from 1947 through 2002 produces the equation: %Change GNP = .856 - 1. ... CIA figures for world unemployment rates, 2006 Unemployment is the state in which a person is without work, available to work, and is currently seeking work. ...

### Open market operations in the United States

The effective federal funds rate charted over fifty years

The Federal Reserve (often referred to as 'The 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. Tax rates around the world Tax revenue as % of GDP Economic policy Monetary policy Central bank   Money supply Fiscal policy Spending   Deficit   Debt Trade policy Tariff   Trade agreement Finance Financial market Financial market participants Corporate   Personal Public   Banking   Regulation        Monetary policy is the process by which the government, central bank... The federal funds rate is the interest rate at which private depository institutions lend balances (federal funds) at the Federal Reserve to other depository institutions overnight. ... Federal Funds transactions redistribute bank reserves. ...

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 T-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. Open Market Operations are the means by which central banks control the liquidity of the national currency. ... Securities are tradeable interests representing financial value. ... The Federal Reserve Bank of New York is the most important of the twelve Federal Reserve Banks of the United States. ... Federal Funds transactions redistribute bank reserves. ...

### Interest rates and credit risk

It is increasingly recognized that the business cycle, interest rates and credit risk are tightly interrelated. The Jarrow-Turnbull model was the first model of credit risk which 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. Credit risk is the risk of loss due to a debtors non-payment of a loan or other line of credit (either the principal or interest (coupon) or both). ... The Jarrow-Turnbull credit risk model was published by Robert A. Jarrow of Kamakura Corporation and Cornell University and Stuart Turnbull, currently at the University of Houston, in March, 1995. ... J. Darrell Duffie is super cool, amazing, and generally awesome. ...

### Money and inflation

Loans, bonds, and shares have some of the characteristics of money and are included in the broad money supply. For other uses, see Money (disambiguation). ... In macroeconomics, money supply (monetary aggregates, money stock) is the quantity of currency and money in bank accounts in the hands of the non-bank public available within the economy to purchase goods, services, and securities. ...

By setting i*n, the government institution can affect the markets to alter the total of loans, bonds and shares 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. In economics, the velocity of money refers to a key term in the quantity theory of money, which centers on the equation of exchange: where is the total amount of money in circulation in an economy at any one time (say, on average during a month). ...

## Interest in mathematics

Jacob Bernoulli discovered the mathematical constant e by studying a question about compound interest. Jakob Bernoulli. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ...

He realized that if an account that starts with \$1.00 and pays 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 this sequence can be modeled as follows:

$lim_{nrightarrowinfty} left(1+dfrac{1}{n}right)^n=e,$

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

## Formulas and Worksheets

The balance of a loan with regular monthly payments is augmented by the monthly interest charge and decreased by the payment so, In banking and accountancy, the outstanding balance is the amount of money owned, (or due), that remains in a deposit account (or a loan account) at a given date, after all past remittances, payments and withdrawal have been accounted for. ...

$B_{k+1}=big(1+rbig)B_k-p$.

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)[3]
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, For other uses, see Debt (disambiguation). ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...

$B_k=(1+r)^k B_0 - frac{{(1+r)^k-1}}{r} p$

A solution of this expression for p in terms of B0 and Bn reduces to,

$p=rBigg[frac{B_0-B_n}{({1+r})^n-1}+B_0Bigg]$

To find the payment if the loan is to be paid off 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: Screenshot of a spreadsheet under OpenOffice A spreadsheet is a rectangular table (or grid) of information, often financial information. ...

$p=PMT(rate,num,PV,FV,) = PMT(r,n,-B_0,B_n,);$

An interest-only payment on the current balance would be,

$p_I=r B;$

The total interest, IT, paid on the loan is,

$I_T=np-B_0;$

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. In order to avoid an underpayment at the end of the loan the payment needs to be rounded up to the next cent. The final payment would then be (1+r)Bn-1.

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,

$B_k=B'_0=(1+r_k)B_0-p_k;$

Comparing this with the expression for Bk above we note that,

$r_k=(1+r)^k-1;$
$p_k=frac{p}{r} r_k$

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

$B^*=frac{p}{r}=frac{p_k}{r_k}$

and B_k can be written,

$B_k=(1+r_k)B_0-r_k B^*;$

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

$r_k=frac{B_0-B_k}{B^*-B_0}$

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 doesn't change with changes in the length of the period. In physics, scale invariance is the feature of physical objects of laws that do not change if the space is magnified, i. ...

Rearranging the equation for B* one gets a transformation coefficient (scale factor), lol rofl taco hahaThere is also a nscale factor for the expansion of the Universe Scale factors are used in computer science when certain real world numbers need to be represented on a different scale in order to fit a required number format. ...

$lambda_k=frac{p_k}{p}=frac{r_k}{r}=frac{(1+r)^k-1}{r}=k[1+frac{(k-1)r}{2}+cdots]$ (see binomial theorem)

and we see that r and p transform in the same manner, In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ...

$r_k=lambda_k r;$
$p_k=lambda_k p;$

The change in the balance transforms likewise,

$Delta B_k=B'-B=(lambda_k rB-lambda_k p)=lambda_k Delta B ;$

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,

$B_k=B_0-r_k(B^*-B_0);$

Since Bn = 0 we can solve for B*,

$B^*=B_0bigg(frac{1}{r_n}+1bigg)$

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

$B_k=B_0bigg(1-frac{r_k}{r_n}bigg)=B_0bigg(1-frac{lambda_k}{lambda_n}bigg)$

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,

$lambda_{k+1}=1+(1+r)lambda_k;$

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,

$p=bigg(r+frac{1}{lambda_n}bigg)B_0$

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

$r_{loan}=frac{I_T}{nB_0}=r+frac{1}{lambda_n}-frac{1}{n}$

which is less than r if n>1.

Look up interest in Wiktionary, the free dictionary.

Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 151 languages. ... In economics the rate of return on investment refers to the benefits to an investor (the profit) relative to the cost of the initial investment. ... The cash accumulation equation is an equation which calculates how much money will be in a bank account, at any point in time. ... A credit rating agency (CRA) is a company that assigns credit ratings for issuers of certain types of debt obligations. ... Credit card interest is the principal way in which card issuers generate revenue. ... In finance, discounting is the process of finding the current value of an amount of cash at some future date, and along with compounding cash from the basis of time value of money calculations. ... NOTE: this is not Fishers equation in differential equations The Fisher equation in financial mathematics and economics estimates the relationship between nominal and real interest rates under inflation. ... Hire purchase (frequently abbreviated to HP) is the legal term for a contract developed in the United Kingdom, and now found in India, Australia, New Zealand, and other states which have adopted the English law concept. ... This article is about a property agreement in private law. ... Tax rates around the world Tax revenue as % of GDP Economic policy Monetary policy Central bank   Money supply Fiscal policy Spending   Deficit   Debt Trade policy Tariff   Trade agreement Finance Financial market Financial market participants Corporate   Personal Public   Banking   Regulation        Monetary policy is the process by which the government, central bank... A mortgage loan is a loan secured by real property through the use of a mortgage (a legal instrument). ... The risk-free interest rate is the interest rate that it is assumed can be obtained by investing in financial instruments with no risk. ... The US dollar yield curve as of 9 February 2005. ... The time value of money is the premise that an investor prefers to receive a payment of a fixed amount of money today, rather than an equal amount in the future, all else being equal. ... Look up usury in Wiktionary, the free dictionary. ... In finance, interest has three general definitions. ...

## References

### Specific references

1. ^ Johnson, Paul: A History of the Jews (New York: HarperCollins Publishers, 1987) ISBN 0-06-091533-1. p.174
2. ^ 15 U.S.C. § 1615

Paul Johnson (born Paul Bede Johnson on 2 November 1928 in Manchester, England) is a British Roman Catholic journalist, historian, speechwriter and author. ...

### General references

• Duffie, Darrell and Kenneth J. Singleton (2003). Credit Risk: Pricing, Measurement, and Management. Princeton University Press. ISBN13 978-0691090467.
• Kellison, Stephen G. (1970). The Theory of Interest. Richard D. Irwin, Inc.. Library of Congress Catalog Card No. 79-98251.
• Lando, David (2004). Credit Risk Modeling: Theory and Applications. Princeton University Press. ISBN13 978-0691089294.
• van Deventer, Donald R. and Kenji Imai (2003). Credit Risk Models and the Basel Accords. John Wiley & Sons. ISBN13 978-0470820919.

Results from FactBites:

 Mortgage calculator | Mortgage rates | Compare interest rates for home loans in any state. Interest.com (662 words) Compare interest rates for home loans in any state. It favors loans with low interest rates and high fees, which are not the right choice for most buyers. I owe \$109,000 on my mortgage but my house is only worth \$80,000.
 Interest, by Paul Heyne: The Concise Encyclopedia of Economics: Library of Economics and Liberty (1387 words) Interest is conventionally expressed as a percentage rate for a period of one year. The standard procedure for calculating compound interest, under which the interest at the end of each year is added to the principal (the amount borrowed), requires borrowers who want to retain command for two years to repay 106.09 percent of the principal, assuming a 3 percent annual rate of interest. The interest rate is determined by demand and supply: the demand for present control of resources by those who do not have it, and the supply from those who do have control and are willing to surrender it for a price.
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