Interest is a fee, paid on borrowed capital. Assets lent include money, shares, consumer goods through hire purchase, major assets such as aircraft, and even entire factories in finance lease arrangements. In Business and Accounting, assets are everything owned by a person or company (all tangible and intangible property that can be converted into cash. Money is anything that is generally accepted as Payment for Goods and services and repayment of Debts. In financial markets, a share is a Unit of account for various financial instruments including Stocks Mutual funds Limited partnerships 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 Ireland Aircraft finance refers to financing for the purchase and operation of Aircraft. Finance lease is a type of Lease - the other being an Operating lease. The interest is calculated upon the value of the assets in the same manner as upon money. Interest can be thought of as "rent on money".

The fee is compensation to the lender for foregoing other useful investments that could have been made with the loaned money. Investment or investing is a term with several closely-related meanings in Business management, Finance and Economics, related to saving 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 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. Credit is the provision of resources (such as granting a Loan) by one party to another party where that second party does not reimburse the first party immediately thereby generating 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. Interest is a fee paid on borrowed capital Assets lent include Money, Shares, Consumer goods through Hire purchase, major assets

Contents 1 History of interest 2 Types of interest 2.1 Simple interest 2.2 Compound interest 2.2.1 Fixed and floating rates 2.3 Composition of interest rates 2.4 Cumulative interest or return 2.5 Other conventions and uses 3 Market interest rates 3.1 Open market operations in the United States 3.2 Interest rates and credit risk 3.3 Money and inflation 4 Interest in mathematics 5 Formulas and Worksheets 6 See also 7 References 7.1 Specific references 7.2 General references 8 External links

History of interest

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. Usury (ˈjuːʒəri comes from the Medieval Latin usuria, "interest" or "excessive interest" from the Latin usura "interest" 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. Usury (ˈjuːʒəri comes from the Medieval Latin usuria, "interest" or "excessive interest" from the Latin usura "interest" Some scholars think that banking started among Jewish families because of the restrictions of the church.

. . . financial oppression of Jews tended to occur in areas where they were most disliked, and if Jews reacted by concentrating on money lending 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. Excommunication is a religious Censure used to deprive or suspend membership in a religious community 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]

Usury has always been viewed negatively by the Roman Catholic Church. Usury (ˈjuːʒəri comes from the Medieval Latin usuria, "interest" or "excessive interest" from the Latin usura "interest" 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. The Second Lateran and tenth Ecumenical council was held by Pope Innocent II in April 1139, and was attended by close to a thousand clerics 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.

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. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere An entrepreneur is a person who has possession over a company enterprise, or Venture, and assumes significant accountability for the inherent risks and the outcome 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. Opportunity cost or economic opportunity loss is the value of a product forgone to produce or obtain 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. Martín de Azpilcueta ( 13 December 1491 &ndash 1 June 1586) was an important Spanish Canonist and Theologian in his time Other things being equal, one would prefer to receive a given good now rather than in the future. This preference indicates greater value. In Economics, time preference (or "discounting" pertains to how large a premium a consumer will place on enjoyment nearer in time over more remote enjoyment 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. Supply and demand is an Economic model describing effects on price and quantity in a Market. In Economics, money supply, or money stock, is the total amount of money available in an Economy at a particular point in time The first attempt to control interest rates through manipulation of the money supply was made by the French Central Bank in 1847. The Banque de France is the Central bank of France; it is linked to the European Central Bank (ECB

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. Adam Smith ( baptised 16 June 1723 – 17 July 1790) was a Scottish moral philosopher and a pioneer of Political economy. Jeremy Bentham ( IPA: or) (15 February 1748&ndash6 June 1832 was an English Jurist, Philosopher, and legal and Social reformer 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. Irving Fisher ( February 27 1867 Saugerties, New York &ndash April 29 1947, New York was an American economist 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. John Maynard Keynes 1st Baron Keynes CB (ˈkeɪnz "cains" (5 June 1883 &ndash 21 April 1946 was a British Economist whose ideas Milton Friedman (July 31 1912 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 ( Arabic: ربا rɪbæː means Usury and is forbidden in Islamic economic jurisprudence. Sharia ( Arabic: ar شريعة) 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:

where r is the period interest rate (I/n), B₀ the initial balance and n the number of time periods elapsed.

To calculate the period interest rate r, one divides the interest rate I by the number of periods n.

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,

and he would have to pay $2581. 19 to pay off the balance at this point.

If instead he makes interest-only payments for each of those 3 months at the period rate r, the amount of interest paid would be,

His balance at the end of 3 months would still be $2500.

In this case, the time value of money is not factored in. The time value of money is based on 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 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. A Guaranteed Investment Certificate or GIC is a Canadian investment that offers a guaranteed rate of return over a fixed period of time most commonly issued by trust companies

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. The time value of money is based on 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 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.

Compound interest

Main article: Compound interest

Compound interest is very similar to simple interest, however, as time goes on the difference becomes considerably larger. Compound interest is the concept of adding accumulated Interest back to the principal so that interest is earned on interest from that moment on The conceptual difference is that unpaid interest is added to the balance due. Put another way, the borrower is charged interest on previous interest charges. Assuming that no part of the principal or subsequent interest has been paid, the debt is calculated by the following formulas:

where I_comp is the compound interest, B₀ the initial balance, B_n 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

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 over short periods of time so, 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. The effective interest rate, effective annual interest rate, Annual Equivalent Rate (AER or simply effective rate is the Interest rate on a However, interest rates in lending are often quoted as nominal interest rates (i. A loan is a type of Debt. This article focuses exclusively on monetary loans although in practice any material object might be lent In finance and economics nominal interest rate or nominal rate of interest refers to the rate of Interest before adjustment for inflation (in contrast with the 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. Compound interest is the concept of adding accumulated Interest back to the principal so that interest is earned on interest from that moment on

Loans often include various non-interest charges and fees. One example are points on a mortgage loan in the United States. A point, sometimes also called a " discount point " is one of the important factors in the calculation of the Annual percentage rate for a Mortgage A mortgage loan is a Loan secured by Real property through the use of a Mortgage (a legal instrument 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). Annual percentage rate (APR is the simplified counterpart to the Effective interest rate that the borrower will pay on a loan 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.

In economics, continuous compounding is often used due to its particular mathematical properties. Compound interest is the concept of adding accumulated Interest back to the principal so that interest is earned on interest from that moment on

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. A fixed interest rate loan is a Loan where the Interest rate doesn't fluctuate during the fixed rate period of the loan 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. A reference rate is a rate that determines pay-offs in a financial contract and that is outside the control of the parties to the contract The London Interbank Offered Rate (or LIBOR, ˈlaɪbɔr is a daily Reference rate based on the Interest rates at which Banks offer to lend The Euro Interbank Offered Rate (or Euribor) is a daily Reference rate based on the averaged Interest rates at which Banks offer to lend unsecured These are known as floating rate, variable rate or adjustable rate loans. A floating interest rate, also known as a variable rate or adjustable rate, refers to any type Debt instrument such as a Loan, bond

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. In economics inflation or price inflation is a rise in the general level of prices of goods and services over a period of time 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. The " real interest rate " is approximately the Nominal interest rate minus the Inflation rate (see Fisher equation and below for exact equation A simple formula for the nominal interest is:

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. 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 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.

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:

i_t = r_{t + 1} + π_{t + 1} + σ

Where i_t 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 σ is the representative value for the risk engaged in the operation. In Economics and Contract theory, information asymmetry deals with the study of decisions in transactions where one party has more or better Information

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). 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 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. 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 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 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. Year 1992 ( MCMXCII) was a Leap year starting on Wednesday (link will display full 1992 Gregorian calendar) The United States of America —commonly referred to as the ^[2] Certain other jurisdictions have outlawed application of the Rule of 78s in certain types of loans, particularly consumer loans. [2]

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. In Finance, the rule of 72, the rule of 70 and the rule of 69 are methods for estimating an Investment 's doubling time 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%.

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 or economic opportunity loss is the value of a product forgone to produce or obtain

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. The " real interest rate " is approximately the Nominal interest rate minus the Inflation rate (see Fisher equation and below for exact equation
• 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.

Creditworthiness of businesses is measured by bond rating services and individual's credit scores by credit bureaus. A credit score is a numerical expression based on a statistical analysis of a person's credit files to represent the Creditworthiness of that person 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.

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. A central bank, reserve bank, or monetary authority is the entity responsible for the Monetary policy of a country or of a group of member states Monetary policy is the process by which the Government, Central bank, or monetary authority of a country controls (i the Supply of Money, 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.

Investment can change rapidly to changes in interest rates, affecting national income, and, through Okun's Law, changes in output affect unemployment. Economics, Okun's law, named after Economist Arthur Okun who proposed the relationship in 1962 (Prachowny 1993 describes an inverse relationship between Unemployment occurs when a person is available to work and currently seeking work but the person is without work.

Open market operations in the United States

The Federal Reserve (often referred to as 'The Fed') implements monetary policy largely by targeting the federal funds rate. In the United States, the federal funds rate is the Interest rate at which private Depository institutions (mostly banks lend balances ( Federal funds This is the rate that banks charge each other for overnight loans of federal funds. See also Federal Reserve System This article is about funds maintained by the U 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. Open market operations are the means of implementing Monetary policy by which a Central bank controls its national Money supply by buying and selling government 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. A security is a Fungible, Negotiable instrument representing financial value The Federal Reserve Bank of New York is the most important of the twelve Federal Reserve Banks of the United States. 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. See also Federal Reserve System This article is about funds maintained by the U

Interest rates and credit risk

It is increasingly recognized that the business cycle, interest rates and credit risk are tightly interrelated. Credit risk is the risk of loss due to a debtor's non-payment of a Loan or other line of credit (either the principal or Interest (coupon or both Faced The Jarrow-Turnbull model was the first model of credit risk which explicitly had random interest rates at its core. The Jarrow-Turnbull credit risk model was published by Robert A 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. James Darrell Duffie is a Canadian economist He is the Dean Witter Distinguished Professor of Finance at Stanford Graduate School of Business, and has been on the Finance

Money and inflation

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

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. In Economics, the quantity theory of money is a theory emphasizing the Positive relationship of overall prices or the nominal value of expenditures to the This means that interest rates can affect inflation in the future.

Interest in mathematics

Jacob Bernoulli discovered the mathematical constant e by studying a question about compound interest. For other family members named Jacob see Bernoulli family. Jacob Bernoulli (also known as James or Jacques) ( Basel The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line

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:

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

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)

B_k = balance after k payments

k = balance index

p = period (monthly) payment

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

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

To find the payment if the loan is to be paid off in n payments one sets B_n = 0. Debt is that which is owed usually referencing Assets owed but the term can cover other obligations In Mathematics, a geometric series is a series with a constant ratio between successive terms.

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

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

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

The formulas for a regular savings program are similar but the payments are added to the balances instead of being subtracted and the formula for the payment is the negative of the one above. A spreadsheet is a Computer application that simulates a paper worksheet 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)B_n-1.

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,

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

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

and B_k can be written,

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

Since B₀ could be any balance in the loan, the formula works for any two balances separate by k periods and can be used to compute a value for the annual interest rate.

B* is a scale invariant since it doesn't change with changes in the length of the period. In Physics and Mathematics, scale invariance is a feature of objects or laws that do not change if length scales (or energy scales are multiplied by a common factor

Rearranging the equation for B^* one gets a transformation coefficient (scale factor),

(see binomial theorem)

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

The change in the balance transforms likewise,

which gives an insight into the meaning of some of the coefficients found in the formulas above. A scale factor is a number which scales, or multiplies some quantity In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says 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 p_k = r_k B* into the equation for the B_k we get,

Since B_n = 0 we can solve for B*,

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,

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,

λ₀ and λ_n can be found using the formula for λ_k above or computing the λ_k recursively from λ₀ = 0 to λ_n.

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

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

which is less than r if n>1.

See also

References

Specific references

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.  

External links

• Simple Interest Calculator
• Compound Interest Calculator
• White Paper: More than Math, The Lost Art of Interest calculation
• Mortgages made clear Financial Services Authority (UK)

The Financial Services Authority ( "FSA") is an independent non-governmental body Quasi-judicial body and a company limited by guarantee that regulates

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