History

Credit for the invention of the electric telephone is frequently disputed, and new controversies over the issue have arisen from time-to-time. As with other great inventions such as radio, television, light bulb, and computer, there were several inventors who did pioneering experimental work on voice transmission over a wire and improved on each other's ideas. Innocenzo Manzetti, Antonio Meucci, Johann Philipp Reis, Elisha Gray, Alexander Graham Bell, and Thomas Edison, among others, have all been credited with pioneering work on the telephone. An undisputed fact is that Alexander Graham Bell was the first to be awarded a patent for the electric telephone by the United States Patent and Trademark Office (USPTO) in March 1876.^[2] That first patent by Bell was the master patent of the telephone, from which all other patents for electric telephone devices and features flowed.

The early history of the telephone became and still remains a confusing morass of claims and counterclaims, which were not clarified by the huge mass of lawsuits that hoped to resolve the patent claims of many individuals and commercial competitors. The Bell and Edison patents, however, were forensically victorious and commercially decisive.

A Hungarian engineer, Tivadar Puskás quickly invented the telephone switchboard in 1876, which allowed for the formation of telephone exchanges, and eventually networks. ^[3]

Basic principles

A traditional landline telephone system, also known as "plain old telephone service" (POTS), commonly handles both signaling and audio information on the same twisted pair of insulated wires: the telephone line. Although originally designed for voice communication, the system has been adapted for data communication such as Telex, Fax and dial-up Internet communication. The signaling equipment consists of a bell, beeper, light or other device to alert the user to incoming calls, and number buttons or a rotary dial to enter a telephone number for outgoing calls. A twisted pair line is preferred as it is more effective at rejecting electromagnetic interference (EMI) and crosstalk than an untwisted pair.

The telephone consists of an alerting device, usually a ringer, that remains connected to the phone line whenever the phone is "on hook", and other components which are connected when the phone is "off hook". These include a transmitter (microphone), a receiver (speaker) and other circuits for dialing, filtering, and amplification. A calling party wishing to speak to another party will pick up the telephone's handset, thus operating a button switch or "switchhook", which puts the telephone into an active (off hook) state by connecting the transmitter (microphone), receiver (speaker) and related audio components to the line. This circuitry has a low resistance (less than 300 ohms) which causes DC current from the telephone exchange to flow through the line. The exchange detects this current, attaches a digit receiver circuit to the line, and sends a dial tone to indicate readiness. On a modern push-button telephone, the calling party then presses the number buttons in a sequence corresponding to the telephone number of the called party. The buttons are connected to a tone generator circuit that produces DTMF tones which end up at a circuit at the exchange. A rotary dial telephone employs pulse dialing, sending electrical pulses corresponding to the telephone number to the exchange. (Most exchanges are still equipped to handle pulse dialing.) Provided the called party's line is not already active or "busy", the exchange sends an intermittent ringing signal (about 90 volts AC in North America and UK and 60 volts in Germany) to alert the called party to an incoming call. If the called party's line is active, the exchange sends a busy signal to the calling party. However, if the called party's line is active but has call waiting installed, the exchange sends an intermittent audible tone to the called party to indicate an incoming call.

The phone's ringer is connected to the line through a capacitor, a device which blocks the flow of DC current but permits AC current. This constitutes a mechanism whereby the phone draws no current when it is on hook, but exchange circuitry can send an AC voltage down the line to activate the ringer for an incoming call. When a landline phone is inactive or "on hook", the circuitry at the telephone exchange detects the absence of DC current flow and therefore "knows" that the phone is on hook with only the alerting device electrically connected to the line. When a party initiates a call to this line, and the ringing signal is transmitted. When the called party picks up the handset, they actuate a double-circuit switchhook which simultaneously

disconnects the alerting device and connects the audio circuitry to the line. This, in turn, draws DC current through the line, confirming that the called phone is now active. The exchange circuitry turns off the ring signal, and both phones are now active and connected through the exchange. The parties may now converse as long as both phones remain off hook. When a party "hangs up", placing the handset back on the cradle or hook, DC current ceases to flow in that line, signaling the exchange to disconnect the call.

Calls to parties beyond the local exchange are carried over "trunk" lines which establish connections between exchanges. In modern telephone networks, fiber-optic cable and digital technology are often employed in such connections. Satellite technology may be used for communication over very long distances.

In most telephones, the transmitter and receiver (microphone and speaker) are located in the handset, although in a speakerphone these components may be located in the base or in a separate enclosure. Powered by the line, the transmitter produces an electric current whose voltage varies in response to the sound waves arriving at its diaphragm. The resulting current is transmitted along the telephone line to the local exchange then on to the other phone (via the local exchange or a larger network), where it passes through the coil of the receiver. The varying voltage in the coil produces a corresponding movement of the receiver's diaphragm, reproducing the sound waves present at the transmitter.

A Lineman's handset is a telephone designed for testing the telephone network, and may be attached directly to aerial lines and other infrastructure components.

Early development

• 1844 — Innocenzo Manzetti first mooted the idea of a “speaking telegraph” (telephone).
• 26 August 1854 — Charles Bourseul publishes an article in a magazine L'Illustration (Paris) : "Transmission électrique de la parole" [electric transmission of speech].
• 26 October 1861 — Johann Philipp Reis (1834–1874) publicly demonstrated the Reis telephone before the Physical Society of Frankfurt
• 22 August 1865, La Feuille d'Aoste reported “It is rumored that English technicians to whom Mr. Manzetti illustrated his method for transmitting spoken words on the telegraph wire intend to apply said invention in England on several private telegraph lines.”
• 28 December 1871 — Antonio Meucci files a patent caveat (n.3335) in the U.S. Patent Office titled "Sound Telegraph", describing communication of voice between two people by wire.
• 1874 — Meucci, after having renewed the caveat for two years, fails to find the money to renew it. The caveat lapses.
• 6 April 1875 — Bell's U.S. Patent 161,739 "Transmitters and Receivers for Electric Telegraphs" is granted. This uses multiple vibrating steel reeds in make-break circuits.
• 11 February 1876 — Gray invents a liquid transmitter for use with a telephone but does not build one.
• 14 February 1876 — Elisha Gray files a patent caveat for transmitting the human voice through a telegraphic circuit.
• 14 February 1876 — Alexander Bell applies for the patent "Improvements in Telegraphy", for electromagnetic telephones using undulating currents.
• 19 February 1876 — Gray is notified by the U.S. Patent Office of an interference between his caveat and Bell's patent application. Gray decides to abandon his caveat.
• 7 March 1876 — Bell's U.S. patent 174,465 "Improvement in Telegraphy" is granted, covering "the method of, and apparatus for, transmitting vocal or other sounds telegraphically … by causing electrical undulations, similar in form to the vibrations of the air accompanying the said vocal or other sound."
• 10 March 1876 — The first successful telephone transmission of clear speech using a liquid transmitter when Bell spoke into his device, “Mr. Watson, come here, I want to see you.” and Watson heard each word distinctly.
• 30 January 1877 — Bell's U.S. patent 186,787 is granted for an electromagnetic telephone using permanent magnets, iron diaphragms, and a call bell.
• 27 April 1877 — Edison files for a patent on a carbon (graphite) transmitter. The patent 474,230 was granted 3 May 1892, after a 15 year delay because of litigation. Edison was granted patent 222,390 for a carbon granules transmitter in 1879.

Early commercial instruments

Early telephones were technically diverse. Some used a liquid transmitter, some had a metal diaphragm that induced current in an electromagnet wound around a permanent magnet, and some were "dynamic" - their diaphragm vibrated a coil of wire in the field of a permanent magnet or the coil vibrated the diaphragm. The dynamic kind survived in small numbers through the 20th century in military and maritime applications where its ability to create its own electrical power was crucial. Most, however, used the Edison/Berliner carbon transmitter, which was much louder than the other kinds, even though it required an induction coil, actually acting as an impedance matching transformer to make it compatible to the impedance of the line. The Edison patents kept the Bell monopoly viable into the 20th century, by which time the network was more important than the instrument.

Early telephones were locally powered, using either a dynamic transmitter or by the powering of a transmitter with a local battery. One of the jobs of outside plant personnel was to visit each telephone periodically to inspect the battery. During the 20th century, "common battery" operation came to dominate, powered by "talk battery" from the telephone exchange over the same wires that carried the voice signals.

Early telephones used a single wire for the subscriber's line, with ground return used to complete the circuit (as used in telegraphs). The earliest dynamic telephones also had only one port opening for sound, with the user alternately listening and speaking (or rather, shouting) into the same hole. Sometimes the instruments were operated in pairs at each end, making conversation more convenient but also more expensive.

At first, the benefits of a telephone exchange were not exploited. Instead telephones were leased in pairs to a subscriber, who had to arrange for a telegraph contractor to construct a line between them, for example between a home and a shop. Users who wanted the ability to speak to several different locations would need to obtain and set up three or four pairs of telephones. Western Union, already using telegraph exchanges, quickly extended the principle to its telephones in New York City and San Francisco, and Bell was not slow in appreciating the potential.

Signalling began in an appropriately primitive manner. The user alerted the other end, or the exchange operator, by whistling into the transmitter. Exchange operation soon resulted in telephones being equipped with a bell, first operated over a second wire, and later over the same wire, but with a condenser (capacitor) in series with the bell coil to allow the AC ringer signal through while still blocking DC (keeping the phone "on hook"). Telephones connected to the earliest Strowger automatic exchanges had seven wires, one for the knife switch, one for each telegraph key, one for the bell, one for the push-button and two for speaking.

Rural and other telephones that were not on a common battery exchange had a magneto or hand-cranked generator to produce a high voltage alternating signal to ring the bells of other telephones on the line and to alert the operator.

In the 1890s a new smaller style of telephone was introduced, packaged in three parts. The transmitter stood on a stand, known as a "candlestick" for its shape. When not in use, the receiver hung on a hook with a switch in it, known as a "switchhook." Previous telephones required the user to operate a separate switch to connect either the voice or the bell. With the new kind, the user was less likely to leave the phone "off the hook". In phones connected to magneto exchanges, the bell, induction coil, battery and magneto were in a separate bell box called a "ringer box". ^[4] In phones connected to common battery exchanges, the ringer box was installed under a desk, or other out of the way place, since it did not need a battery or magneto.

Cradle designs were also used at this time, having a handle with the receiver and transmitter attached, separate from the cradle base that housed the magneto crank and other parts. They were larger than the "candlestick" and more popular.

Disadvantages of single wire operation such as crosstalk and hum from nearby AC power wires had already led to the use of twisted pairs and, for long distance telephones, four-wire circuits. Users at the beginning of the 20th century did not place long distance calls from their own telephones but made an appointment to use a special sound proofed long distance telephone booth furnished with the latest technology.

What turned out to be the most popular and longest lasting physical style of telephone was introduced in the early 20th century, including Bell's Model 102. A carbon granule transmitter and electromagnetic receiver were united in a single molded plastic handle, which when not in use sat in a cradle in the base unit. The circuit diagram of the Model 102 shows the direct connection of the receiver to the line, while the transmitter was induction coupled, with energy supplied by a local battery. The coupling transformer, battery, and ringer were in a separate enclosure. The dial switch in the base interrupted the line current by repeatedly but very briefly disconnecting the line 1-10 times for each digit, and the hook switch (in the center of the circuit diagram) disconnected the line and the transmitter battery while the handset was on the cradle.

After the 1930s, the base also enclosed the bell and induction coil, obviating the old separate ringer box. Power was supplied to each subscriber line by central office batteries instead of a local battery, which required periodic service. For the next half century, the network behind the telephone became progressively larger and much more efficient, but after the dial was added the instrument itself changed little until American Telephone & Telegraph Company (AT&T) introduced Touch-Tone dialing in the 1960s.

Digital telephony

The Public Switched Telephone Network (PSTN) has gradually evolved towards digital telephony which has improved the capacity and quality of the network. End-to-end analog telephone networks were first modified in the early 1960s by upgrading transmission networks with T1 carrier systems, designed to support the basic 3 kHz voice channel by sampling the bandwidth-limited analog voice signal and encoding using PCM. While digitization allows wideband voice on the same channel, the improved quality of a wider analog voice channel did not find a large market in the PSTN.

Later transmission methods such as SONET and fiber optic transmission further advanced digital transmission. Although analog carrier systems existed that multiplexed multiple analog voice channels onto a single transmission medium, digital transmission allowed lower cost and more channels multiplexed on the transmission medium. Today the end instrument often remains analog but the analog signals are typically converted to digital signals at the (Serving Area Interface (SAI), central office (CO), or other aggregation point. Digital loop carriers (DLC) place the digital network ever closer to the customer premises, relegating the analog local loop to legacy status.

IP telephony

Internet Protocol (IP) telephony (also known as Voice over Internet Protocol, VoIP), is a disruptive technology that is rapidly gaining ground against traditional telephone network technologies. As of January 2005, up to 10% of telephone subscribers in Japan and South Korea have switched to this digital telephone service. A January 2005 Newsweek article suggested that Internet telephony may be "the next big thing."^[5] As of 2006 many VoIP companies offer service to consumers and businesses.

IP telephony uses an Internet connection and hardware IP Phones or softphones installed on personal computers to transmit conversations encoded as data packets. In addition to replacing POTS (plain old telephone service), IP telephony services are also competing with mobile phone services by offering free or lower cost connections via WiFi hotspots. VoIP is also used on private networks which may or may not have a connection to the global telephone network.

IP telephones have two notable disadvantages compared to traditional telephones. Unless the IP telephone's components are backed up with an uninterruptible power supply or other emergency power source, the phone will cease to function during a power outage as can occur during an emergency or disaster, exactly when the phone is most needed. Traditional phones connected to the older PSTN network do not experience that problem since they are powered by the telephone company's battery supply, which will continue to function even if there's a prolonged power black-out. A second distinct problem for an IP phone is the lack of a 'fixed address' which can impact the provision of emergency services such as police, fire or ambulance, should someone call for them. Unless the registered user updates the IP phone's physical address location after moving to a new residence, emergency services can be, and have been, dispatched to the wrong location.

Usage

By the end of 2009, there were a total of nearly 6 billion mobile and fixed-line subscribers worldwide. This included 1.26 billion fixed-line subscribers and 4.6 billion mobile subscribers. ^[6]

Telephone operating companies

In some countries, many telephone operating companies (commonly abbreviated to telco in American English) are in competition to provide telephone services. The above Main article lists only facilities based providers and not companies which lease services from facilities based providers in order to serve their customers.

History of interest

In ancient biblical Israel, it was against the Law of Moses to charge interest on private loans^[3].During the Middle Ages, time was considered to be property of God. Therefore, to charge interest was considered to be commerce with God's property.^{[citation needed]} 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.^{[citation needed]}

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.^{[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.^[4]

As Jewish citizens were ostracized from most professions by local rulers, the church and the guilds, they were pushed into marginal occupations considered socially inferior, such as tax and rent collecting and moneylending. Natural tensions between creditors and debtors were added to social, political, religious, and economic strains. ...financial oppression of Jews tended to occur in areas where they were most disliked, and if Jews reacted by concentrating on moneylending to non-Jews, the unpopularity — and so, of course, the pressure — would increase. Thus 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. Catholic autocrats frequently 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 became identified with the hated trade of moneylending.

Interest has often been looked down upon in Islamic civilization as well for the same reason for which usury was forbidden by the Catholic Church, with most scholars agreeing that the Qur'an explicitly forbids charging interest. Medieval jurists therefore developed several financial instruments to encourage responsible lending.

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.^{[citation needed]} 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 entirely.^{[citation needed]} Rather than charging interest, the interest-free lender charges a "fee" for the service of lending. As any such fee can be shown to be mathematically identical to an interest charge, the distinction between "interest-free" banking and "for-interest" banking is merely one of semantics

Types of interest

Simple interest

Simple interest is calculated only on the principal amount, or on that portion of the principal amount that remains unpaid.

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

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

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

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 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.

Compound interest

Compound interest is very similar to simple interest; however, with time, the difference becomes considerably larger. This difference is because unpaid interest is added to the balance due. Put another way, the borrower is charged interest on previous interest. 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_m the balance after m periods (where m is not necessarily an integer) 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 frequent payments are the least expensive repayment 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).^{[citation needed]}

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.

In economics, continuous compounding is often used due to its particular mathematical properties.^{[citation needed]}

Fixed and floating rates

Commercial loans generally use simple 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.

Combinations of fixed-rate and floating-rate loans are possible and frequently used. Loans may also 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.^{[citation needed]}

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:

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 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:

i_t = r_{(t + 1)} + π_{(t + 1)} + σ

Here, 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.

Cumulative interest or return

This section requires expansion.

The calculation for cumulative interest 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.^{[citation needed]}

Other conventions and uses

Exceptions:

• US and Canadian T-Bills (short term Government debt) have a different calculation for interest. Their interest is calculated as (100-P)/P where 'P' is the price paid. Instead of normalizing it to a year, the interest is prorated by the number of days 't': (365/t)*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. ^[5]

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.^[6] Certain other jurisdictions have outlawed application of the Rule of 78s in certain types of loans, particularly consumer loans. ^[5]

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%.

Market interest rates

This section does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (January 2009)

There are markets for investments (which include the money market, bond market, as well as retail financial institutions like banks) 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.

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 or the interest payments are continually increased by the rate of inflation. 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 loan.

The 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 sooner rather than later. There will be an interest premium of the delay. 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

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.

Investment can change rapidly in response to changes in interest rates, affecting national income, and, through Okun's Law, changes in output affect unemployment.{Fact|date=January 2009}

Open market operations in the United States

The Federal Reserve (Fed) implements monetary policy largely by targeting the federal funds rate. This is the rate that banks charge each other for overnight loans of federal funds. Federal funds are the reserves held by banks at the Fed.

Open market operations are one tool within monetary policy implemented by the Federal Reserve to steer short-term interest rates. Using the power to buy and sell treasury securities, the Open Market Desk at the Federal Reserve Bank of New York can supply the market with dollars by purchasing 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.

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 that explicitly had random interest rates at its core. Lando (2004), Darrell Duffie and Singleton (2003), and van Deventer and Imai (2003) discuss interest rates when the issuer of the interest-bearing instrument can default.

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. This means that interest rates can affect inflation in the future.^{[citation needed]}

Interest in mathematics

It is thought that Jacob Bernoulli discovered the mathematical constant e by studying a question about compound interest.^{[citation needed]}

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.

Formulae

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

.

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)[2]

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.

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. 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. 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 that 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 does not change with changes in the length of the period.

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. 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.