NationMaster - Encyclopedia: Compound 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. The act of declaring interest to be principal is called compounding (i.e. interest is compounded). A loan, for example, may have its interest compounded every month: in this case, a loan with $1000 principal and 1% interest per month would have a balance of $1010 at the end of the first month.

Interest rates must be comparable in order to be useful, and in order to be comparable, the interest rate and the compounding frequency must be disclosed. Since most people think of rates as a yearly percentage, many governments require financial institutions to disclose a (notionally) comparable yearly interest rate on deposits or advances. Compound interest rates may be referred to as Annual Percentage Rate, Effective interest rate, Effective Annual Rate, and by other terms. When a fee is charged up front to obtain a loan, APR usually counts that cost as well as the compound interest in converting to the equivalent rate. These government requirements assist consumers to more easily compare the actual cost of borrowing. Annual percentage rate (APR) is an expression of the effective interest rate the borrower will pay on a loan, taking into account one-time fees and standardizing the way the rate is expressed. ... 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). ...

Compound interest rates may be converted to allow for comparison: for any given interest rate and compounding frequency, an "equivalent" rate for a different compounding frequency exists.

Compound interest may be contrasted with simple interest, where interest is not added to the principal (there is no compounding). Compound interest predominates in finance and economics, and simple interest is used infrequently (although certain financial products may contain elements of simple interest). In finance, interest has three general definitions. ...

1 Terminology
- 1.1 Exceptions
2 Mathematics of interest rates
3 History
4 See also
5 References
6 External links

Terminology

The effect of compounding depends on the frequency with which interest is compounded and the periodic interest rate which is applied. Therefore, in order to define accurately the amount to be paid under a legal contract with interest, the frequency of compounding (yearly, half-yearly, quarterly, monthly, daily, etc.) and the interest rate must be specified. Different conventions may be used from country to country, but in finance and economics the following usages are common:

Periodic rate: the interest that is charged (and subsequently compounded) for each period. The periodic rate is used primarily for calculations, and is rarely used for comparison. The periodic rate is defined as the annual nominal rate divided by the number of compounding periods per year.

Nominal interest rate or nominal annual rate: the annual rate, unadjusted for compounding. For example, 12% annual nominal interest compounded monthly has a periodic (monthly) rate of 1%. A nominal interest rate is the interest rate as stated - that is, not adjusted for compounding. ...

Effective annual rate: the nominal annual rate "adjusted" to allow comparisons; the nominal rate is restated to reflect the effective rate as if annual compounding were applied. In relation to the US: The Effective Annual Rate (EAR) is the interest rate that is annualized using compound interest, as opposed to using simple interest in the case of the Annual percentage rate (APR). ...

Economists generally prefer to use effective annual rates to allow for comparability. In finance and commerce, the nominal annual rate may be the most frequently used. When quoted with the compounding frequency, a loan with a given nominal annual rate is fully specified (the effect of interest for a given loan scenario can be precisely determined), but cannot be compared to loans with different compounding frequency.

Loans and finance may have other "non-interest" charges, and the terms above do not attempt to capture these differences. Other terms such as annual percentage rate and annual percentage yield may have specific legal definitions and may or may not be comparable, depending on the jurisdiction. Annual percentage rate (APR) is an expression of the effective interest rate the borrower will pay on a loan, taking into account one-time fees and standardizing the way the rate is expressed. ... The annual percentage yield provides a means for estimating the return on an investment with compound interest in terms of its effective annual yield. ...

The use of the terms above (and other similar terms) may be inconsistent, and vary according to local custom, marketing demands, simplicity or for other reasons.

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 day count convention).
Corporate Bonds are most frequently payable twice yearly. The amount of interest paid (each six months) is the disclosed interest rate divided by two (multiplied by the principal). The yearly compounded rate is higher than the disclosed rate.
Canadian mortgage loans are generally semi-annual compounding with monthly (or more frequent) payments.^[1]
U.S. mortgages generally use monthly compounding (with corresponding payment periods).
Certain techniques for, e.g., valuation of derivatives may use continuous compounding, which is the limit as the compounding period approaches zero. Continuous compounding in pricing these instruments is a natural consequence of Ito Calculus, where derivatives are valued at ever increasing frequency, until the limit is approached and the derivative is valued in continuous time.

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. ... A mortgage loan is a loan secured by real property through the use of a mortgage (a legal instrument). ... For other uses, see Derivative (disambiguation). ... In mathematics, the limit of a function is a fundamental concept in analysis. ... ItÃ´ calculus, named after Kiyoshi ItÃ´, treats mathematical operations on stochastic processes. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

Mathematics of interest rates

Simple

Formulae are presented in greater detail at time value of money. 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. ...

In the formulae below, i or r are the interest rate, expressed as a true percentage (i.e. 10% = 10/100 = 0.10). FV and PV represent the future and present value of a sum. n represents the number of periods.

These are the most basic formulae:

$FV = PV ( 1+i )^n,$

The above calculates the future value of FV of an investment's present value of PV accruing at a fixed interest rate of i for n periods.

$PV = frac {FV} {left( 1+i right)^n},$

The above calculates what present value of PV would be needed to produce a certain future value of FV if interest of i accrues for n periods.

$i = sqrt[n]{left( frac {FV} {PV} right)} -1 ,$

$i = left( frac {FV} {PV} right)^left(frac {1} {n} right)- 1$

The above two formulae are the same and calculate the compound interest rate achieved if an initial investment of PV returns a value of FV after n accrual periods.

$n = frac {log(FV) - log(PV)} {log(1 + i)}$

The above formula calculates the number of periods required to get FV given the PV and the interest rate i. The log function can be in any base, e.g. natural log (ln)

Compound

Formula for calculating compound interest:

$A = Pleft(1 + frac{r}{n}right)^{nt}$

Where,

P = principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
A = amount after time t

Example usage: An amount of $1,500.00 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Find the balance after 6 years.

A. Using the formula above, with P = 1500, r = 4.3/100 = 0.043, n = 4, and t = 6:

$A=1500(1 + frac{0.043}{4})^{4*6} =1938.84$

So, the balance after 6 years is approximately $1,938.84.

Translating different compounding periods

Each time unpaid interest is compounded and added to the principal, the resulting principal is grossed up to equal P(1+i%).

A) You are told the interest rate is 8% per year, compounded quarterly. What is the equivalent effective annual rate?

The 8% is a nominal rate. It implies an effective quarterly interest rate of 8%/4 = 2%. Start with $100. At the end of one year it will have accumulated to:
$100 (1+ .02) (1+ .02) (1+ .02) (1+ .02) = $108.24
We know that $100 invested at 8.24% will give you $108.24 at year end. So the equivalent rate is 8.24%. Using a financial calculator or a table is simpler still. Using the Future Value of a currency function, input

PV = 100
n = 4
i = .02
solve for FV = 108.24

B) You know the equivalent annual interest rate is 4%, but it will be compounded quarterly. You need to find the interest rate that will be applied each quarter.

$sqrt[4]{1+.04}-1 = .00985341$

$100 (1+ .009853) (1+ .009853) (1+ .009853) (1+ .009853) = $104
The mathematics to find the 0.9853% is discussed at Time value of money, but using a financial calculator or table is easier. Input 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. ...

PV = 100
n = 4
FV = 104
solve for interest = 0.9853%

C) You sold your house for a 60% profit. What was the annual return? You owned the house for 4 years, paid $100,000 originally, and sold it for $160,000.
$100,000 (1+ .1247) (1+ .1247) (1+ .1247) (1+ .1247) = $160,000
Find the 12.47% annual rate the same way as B.) above, using a financial calculator or table. Input

PV = 100,000
n = 4
FV = 160,000
solve for interest = 12.47%

Example question:

In January 1970 the S&P 500 index stood at 92.06 and in January 2006 the index stood at 1248.29. What has been the annual rate of return achieved? (ignoring dividends). The S&P 500 is an index containing the stocks of 500 Large-Cap corporations, most of which are American. ...

$PV = 92.06 ,$

$FV = 1248.29 ,$

$n = 36 (years) ,$

Answer:

$i = sqrt[36]{left( frac {1248.29} {92.06} right)} -1 = 7.51% ,$

The Rule of 72

The Rule of 72 is a very simple way of illustrating the growth potential of compound interest. The rule says simply this: In finance, the rule of 72, the rule of 71, the rule of 70 and the rule of 69. ...

$frac {72} {i} approx n$ , where $i$ is the interest rate in percentage ( i.e, $100 i$ ) and $n$ is the number of time periods needed to double the principal.

For example, say a mutual fund grows at 12% average interest rate. According to the rule of 72, if money were invested in this mutual fund, then it would double every 6 $left(frac {72} {12}=6right)$ years. This calculation deals only with the gross amount, taxes must be factored into growth if taxable vehicles (such as CDs, mutual funds, etc) are used.

However, the above Rule of 72 merely gives an approximation of the time needed to retain an investment before it doubles in value. The accurate calculation is as follows: In finance, the rule of 72, the rule of 71, the rule of 70 and the rule of 69. ...

$n = frac{ln 2}{ln(1+frac{i}{100})}$

Periodic compounding

The amount function for compound interest is an exponential function in terms of time.

$A(t) = A_0 left(1 + frac {r} {n}right) ^ {n cdot t}$

$t$ = Total time in years

$n$ = Number of compounding periods per year (note that the total number of compounding periods is $n cdot t$ )

$r$ = Nominal annual interest rate expressed as a decimal. e.g.: 6% = 0.06

As $n$ increases, the rate approaches an upper limit of $e r$ . This rate is called continuous compounding, see below. A nominal interest rate is the interest rate as stated - that is, not adjusted for compounding. ...

Since the principal A(0) is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used in interest theory instead. Accumulation functions for simple and compound interest are listed below: The accumulation function a(t) is a function defined in terms of time t expressing the time value of money. ... In finance, interest has three general definitions. ...

$a(t)=1+t r,$

$a(t) = left(1 + frac {r} {n}right) ^ {n cdot t}$

Note: A(t) is the amount function and a(t) is the accumulation function.

Force of interest

Part of a series of articles on
The mathematical constant, e 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. ...

Natural logarithm Image File history File links Euler's_formula. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ...

Applications in: compound interest · Euler's identity & Euler's formula · half-lives & exponential growth/decay For other uses, see List of topics named after Leonhard Euler. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... Half-Life For a quantity subject to exponential decay, the half-life is the time required for the quantity to fall to half of its initial value. ... In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...

Defining e: proof that e is irrational · representations of e · Lindemann–Weierstrass theorem In mathematics, the series expansion of the number e can be used to prove that e is irrational. ... The mathematical constant e can be represented in a variety of ways as a real number. ... In mathematics, the Lindemannâ€“Weierstrass theorem states that if Î±1,...,Î±n are algebraic numbers which are linearly independent over the rational numbers, then are algebraically independent over the algebraic numbers; in other words the set has transcendence degree n over . ...

People John Napier · Leonhard Euler
For other people with the same name, see John Napier (disambiguation). ... Euler redirects here. ...

Schanuel's conjecture
Schanuels conjecture is that given any set of n complex numbers which have linear independence over the rational numbers, the set (up to twice the size) has transcendence degree of at least n over the rationals. ...

In mathematics, the accumulation functions are often expressed in terms of e, the base of the natural logarithm. This facilitates the use of calculus methods in manipulation of interest formulae. This is called the force of interest. 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. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ...

The force of interest is defined as the following:

$delta_{t}=frac{a'(t)}{a(t)},$

$a(n)=e^{int_0^n delta_t, dt} ,$ (since

a (0) = 1

)

When the above formula is written in differential equation format, the force of interest is simply the coefficient of amount of change.

$da(t)=delta_{t}a(t),dt,$

The force of interest for compound interest is a constant for a given r, and the accumulation function of compounding interest in terms of force of interest is a simple power of e:

$delta=ln(1+r),$

$a(t)=e^{tdelta},$

Continuous compounding

For interest compounded a certain number of times, n, per year, such as monthly or quarterly, the formula is:

$a(t)=Pleft(1+frac{r}{n}right)^{n cdot t},$

Continuous compounding can be thought as making the compounding period infinitely small; therefore achieved by taking the limit of n to infinity. One should consult definitions of the exponential function for the mathematical proof of this limit. Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... For other uses, see Infinity (disambiguation). ... In mathematics, the exponential function can be characterized in many ways. ...

$a(t)=lim_{ntoinfty}left(1+frac{r}{n}right)^{n cdot t}$

$a(t)=e^{r cdot t}$

The amount function is simply

$A(t)=A_0 e^{r cdot t}$

A common mnemonic device considers the equation in the form

$A = P e^{r cdot t}$

called 'PERT' where P is the principal amount, e is the base of the natural log, R is the rate per period, and T is the time (in the same units as the rate's period), and A is the final amount. 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. ...

Compounding bases

See 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. ...

To convert an interest rate from one compounding basis to another compounding basis, the following formula applies:

$r_2=left[left(1+frac{r_1}{n_1}right)^frac{n_1}{n_2}-1right]{times}n_2$

where r₁ is the stated interest rate with compounding frequency n₁ and r₂ is the stated interest rate with compounding frequency n₂.

When interest is continuously compounded:

$R=n{times}ln{left(1+frac{r}{n}right)}$

where R is the interest rate on a continuous compounding basis and r is the stated interest rate with a compounding frequency n.

History

If the Native American tribe that accepted goods worth 60 guilders for the sale of Manhattan in 1626 had invested the money in a Dutch bank at 6.5% interest, compounded annually, then in 2005 their investment would be worth over €700 billion (around USD $1,000 billion), more than the assessed value of the real estate in all five boroughs of New York City. With a 6.0% interest however, the value of their investment today would have been €100 billion (7 times less!). This article is about the people indigenous to the United States and their history after European contact, chiefly in what is now the United States. ... http://www. ... Guilder is the English translation of gulden, (old) Dutch for golden. The gulden originated as a gold coin (hence the name) but has been a common name for a silver or base metal coin for some centuries. ... This article is about the borough of New York City. ... Events September 30 - Nurhaci, chieftain of the Jurchens and founder of the Qing Dynasty dies and is succeeded by his son Hong Taiji. ... For other uses, see Bank (disambiguation). ... The United States dollar is the official currency of the United States. ... New York, New York and NYC redirect here. ...

Compound interest was once regarded as the worst kind of usury, and was severely condemned by Roman law, as well as the common laws of many other countries. ^[2] Of Usury, from Brants Stultifera Navis (the Ship of Fools); woodcut attributed to Albrecht DÃ¼rer Usury (//,comes from the Medieval Latin usuria, interest or excessive interest, from the Latin usura interest) originally meant the charging of interest on loans. ... Using the term Roman law in a broader sense, one may say that Roman law is not only the legal system of ancient Rome but the law that was applied throughout most of Europe until the end of the 18th century. ... This article concerns the common-law legal system, as contrasted with the civil law legal system; for other meanings of the term, within the field of law, see common law (disambiguation). ...

Richard Witt's book Arithmeticall Questions, published in 1613, was a landmark in the history of compound interest. It was wholly devoted to the subject (previously called anatocism), whereas previous writers had usually treated compound interest briefly in just one chapter in a mathematical textbook. Witt's book gave tables based on 10% (the then maximum rate of interest allowable on loans) and on other rates for different purposes, such as the valuation of property leases. Witt was a London mathematical practitioner and his book is notable for its clarity of expression, depth of insight and accuracy of calculation, with 124 worked examples.^[3]^[4]

The Qu'ran, revealed over 1400 years ago, explicitly mentions compound interest as a great sin. Interest known in arabic as riba is considered wrong. The Quran ( Arabic al-qurʾān أَلْقُرآن; its literal meaning is the recitation and is often called Al Quran Al Karim: The Noble Quran, also transliterated as Quran, Koran, and less commonly Alcoran) is the holy book of Islam. ... 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...

Oh you who believe, you shall not take riba, compounded over and over. Observe God, that you may succeed. ( Qur'an 3:130) The Qurâ€™Än [1] (Arabic: , literally the recitation; also sometimes transliterated as Quran, Koran, or Al-Quran) is the central religious text of Islam. ...

References

^ http://laws.justice.gc.ca/en/showdoc/cs/I-15/bo-ga:s_6//en#anchorbo-ga:s_6 Interest Act (Canada), Department of Justice. The Interest Act specifies that interest is not recoverable unless the mortgage loan contains a statement showing the rate of interest chargeable, "calculated yearly or half-yearly, not in advance." In practice, banks use the half-yearly rate.
^ This article incorporates content from the 1728 Cyclopaedia, a publication in the public domain.
^ Lewin, C G (1970). "An Early Book on Compound Interest - Richard Witt's Arithmeticall Questions". Journal of the Institute of Actuaries 96 (1): 121–132.
^ Lewin, C G (1981). "Compound Interest in the Seventeenth Century". Journal of the Institute of Actuaries 108 (3): 423–442.

Table of Trigonometry, 1728 Cyclopaedia Cyclopaedia, or, A Universal Dictionary of Arts and Sciences (folio, 2 vols. ... The public domain comprises the body of all creative works and other knowledge—writing, artwork, music, science, inventions, and others—in which no person or organization has any proprietary interest. ...

External links

A Simple Introduction to Compound Interest
An Online Compound Interest Calculator
Practice using Compound Interest Formula
Compound interest, what it is and why you want it on your side
Continuously compounded interest formula and calculator

Results from FactBites:

Compound interest - Wikipedia, the free encyclopedia (906 words)
	Compound interest, is interest which is added to the original principal.
	Yearly compounded interest is considered the norm unless it is specified to be otherwise.
	Compound interest was once regarded as the worst kind of usury, and was severely condemned by Roman law, as well as the common laws of many other countries.

Simple and Compound Interest Rates (838 words)
	Compound interest is calculated each period on the original principal and all interest accumulated during past periods.
	The interest earned in each period is added to the principal of the previous period to become the principal for the next period.
	You must adjust the interest rate and the number of periods to be consistent with compounding periods.

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