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Compound interest

Overview

Compound interest arises when interest

Interest

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

is added to the principal, so that from that moment on, the interest that has been added also itself earns interest. This addition of interest to the principal is called compounding. A bank account, for example, may have its interest compounded every year: in this case, an account with $1000 initial principal and 20% interest per year would have a balance of $1200 at the end of the first year, $1440 at the end of the second year, and so on.

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Encyclopedia

Compound interest arises when interest

Interest

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

is added to the principal, so that from that moment on, the interest that has been added also itself earns interest. This addition of interest to the principal is called compounding. A bank account, for example, may have its interest compounded every year: in this case, an account with $1000 initial principal and 20% interest per year would have a balance of $1200 at the end of the first year, $1440 at the end of the second year, and so on.

In order to define an interest rate fully, and enable one to compare it with other interest rates, the interest rate and the compounding frequency must be disclosed. Since most people prefer to think of rates as a yearly percentage, many governments require financial institutions to disclose the equivalent yearly compounded interest rate on deposits or advances. For instance, the yearly rate for a loan with 1% interest per month is approximately 12.68% per annum (1.01¹² − 1). This equivalent yearly rate may be referred to as annual percentage rate
Annual percentage rate
The term annual percentage rate , also called nominal APR, and the term effective APR, also called EAR, describe the interest rate for a whole year , rather than just a monthly fee/rate, as applied on a loan, mortgage loan, credit card, etc. It is a finance charge expressed as an annual rate...

(APR), annual equivalent rate (AER), annual percentage yield
Annual Percentage Yield
Annual percentage yield is a normalized representation of an interest rate, based on a compounding period of one year. APY figures allow for a reasonable, single-point comparison of different offerings with varying compounding schedules...

, effective interest rate
Effective interest rate
The effective interest rate, effective annual interest rate, annual equivalent rate or simply effective rate is the interest rate on a loan or financial product restated from the nominal interest rate as an interest rate with annual compound interest payable in arrears.It is used to compare the...

, 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 compare the actual costs of borrowing more easily.

For any given interest rate and compounding frequency, an "equivalent" rate for any 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 is standard in finance and economics, and simple interest is used infrequently (although certain financial products may contain elements of simple interest).

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, divided by the amount of the principal. The periodic rate is used primarily for calculations, and is rarely used for comparison. The nominal annual rate or nominal interest rate

Nominal interest rate

In finance and economics nominal interest rate or nominal rate of interest refers to the rate of interest before adjustment for inflation ; or, for interest rates "as stated" without adjustment for the full effect of compounding...

is defined as the periodic rate multiplied by the number of compounding periods per year. For example, a monthly rate of 1% is equivalent to an annual nominal interest of 12%.

Effective annual rate: this reflects the effective rate as if annual compounding were applied: in other words it is the total accumulated interest that would be payable up to the end of one year, divided by the principal.

Economists generally prefer to use effective annual rates to allow for comparability. In finance and commerce, the nominal annual rate may however be the one quoted instead. When quoted together 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 the nominal rate cannot be directly compared with loans that have a 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

Annual percentage rate

The term annual percentage rate , also called nominal APR, and the term effective APR, also called EAR, describe the interest rate for a whole year , rather than just a monthly fee/rate, as applied on a loan, mortgage loan, credit card, etc. It is a finance charge expressed as an annual rate...

and annual percentage yield

Annual Percentage Yield

Annual percentage yield is a normalized representation of an interest rate, based on a compounding period of one year. APY figures allow for a reasonable, single-point comparison of different offerings with varying compounding schedules...

may have specific legal definitions and may or may not be comparable, depending on the jurisdiction.

The use of the terms above (and other similar terms) may be inconsistent, and vary according to local custom or marketing demands, for 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)/Pbnm, 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
Day count convention
In finance, a day count convention determines how interest accrues over time for a variety of investments, including bonds, notes, loans, mortgages, medium-term notes, swaps, and forward rate agreements . This determines the amount transferred on interest payment dates, and also the calculation of...

).
The interest on corporate bonds and government bonds is usually 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 loan
Mortgage loan
A mortgage loan is a loan secured by real property through the use of a mortgage note which evidences the existence of the loan and the encumbrance of that realty through the granting of a mortgage which secures the loan...

s are generally semi-annual compounding with monthly (or more frequent) payments.
U.S. mortgages use an amortizing loan
Amortizing loan
In banking and finance, an amortizing loan is a loan where the principal of the loan is paid down over the life of the loan, according to some amortization schedule, typically through equal payments....

, not compound interest. With these loans, an amortization schedule
Amortization schedule
An amortization schedule is a table detailing each periodic payment on an amortizing loan , as generated by an amortization calculator. Amortization refers to the process of paying off a debt over time through regular payments...

is used to determine how to apply payments toward principal and interest. Interest generated on these loans is not added to the principal, but rather is paid off monthly as the payments are applied.
It is sometimes mathematically simpler, e.g. in the valuation of derivatives
Derivative (finance)
A derivative instrument is a contract between two parties that specifies conditions—in particular, dates and the resulting values of the underlying variables—under which payments, or payoffs, are to be made between the parties.Under U.S...

to use continuous compounding, which is the limit
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....

as the compounding period approaches zero. Continuous compounding in pricing these instruments is a natural consequence of Itō calculus
Ito calculus
Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to stochastic processes such as Brownian motion . It has important applications in mathematical finance and stochastic differential equations....

, where derivatives
Derivative (finance)
A derivative instrument is a contract between two parties that specifies conditions—in particular, dates and the resulting values of the underlying variables—under which payments, or payoffs, are to be made between the parties.Under U.S...

are valued at ever increasing frequency, until the limit is approached and the derivative is valued in continuous time.

Simplified calculation

Formulae are presented in greater detail at time value of money

Time value of money

The time value of money is the value of money figuring in a given amount of interest earned over a given amount of time. The time value of money is the central concept in finance theory....

.

In the formula below, i is the effective interest rate per period. FV and PV represent the future and present value of a sum. n represents the number of periods.

These are the most basic formulas:

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

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

The above calculates the compound interest rate achieved if an initial investment of PV returns a value of FV after n accrual periods.

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), as long as consistent bases are used all throughout calculation.

Compound

A formula for calculating annual compound interest is

Where,

A = final amount
P = principal amount (initial investment)
r = annual nominal interest rate (as a decimal)

(it should not be in percentage)

n = number of times the interest is compounded per year
t = number of years

Example usage: An amount of $1500.00 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:

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

Periodic compounding

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

= Total time in years

= Number of compounding periods per year (note that the total number of compounding periods is )

= Nominal annual interest rate
Nominal interest rate
In finance and economics nominal interest rate or nominal rate of interest refers to the rate of interest before adjustment for inflation ; or, for interest rates "as stated" without adjustment for the full effect of compounding...

expressed as a decimal. e.g.: 6% = 0.06

As

increases, the rate approaches an upper limit of

. This rate is called continuous compounding, see below.

Since the principal A(0) is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function

Accumulation function

The accumulation function a is a function defined in terms of time t expressing the ratio of the value at time t and the initial investment...

is used in interest theory instead. Accumulation functions for simple and compound interest are listed below:

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

Continuous compounding

Continuous compounding can be thought of as making the compounding period infinitesimally small; therefore achieved by taking the limit

Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

of n to infinity

Infinity

Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

. One should consult definitions of the exponential function for the mathematical proof of this limit.

take t as 1 so

)

where R is simple return and r is called log return because it is the logarithm of normal return.

The amount function is simply

The interest rate expressed as a continuously compounded rate is called the force of interest. The annual force of interest is simply 12 times the monthly force of interest.

The effective interest rate

Effective interest rate

The effective interest rate, effective annual interest rate, annual equivalent rate or simply effective rate is the interest rate on a loan or financial product restated from the nominal interest rate as an interest rate with annual compound interest payable in arrears.It is used to compare the...

per year is

Using this i the amount function can be written as:

or

See also logarithmic or continuously compounded return.

Force of interest

In mathematics, the accumulation functions are often expressed in terms of e
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

, the base of the natural logarithm

Natural logarithm

The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

. This facilitates the use of calculus methods in manipulation of interest formulae.

For any continuously differentiable accumulation function

Accumulation function

The accumulation function a is a function defined in terms of time t expressing the ratio of the value at time t and the initial investment...

a(t) the force of interest, or more generally the logarithmic or continuously compounded return is a function of time defined as follows:

which is the rate of change with time of the natural logarithm of the accumulation function.

Conversely:

(since

)

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

For compound interest with a constant annual interest rate r the force of interest is a constant, and the accumulation function of compounding interest in terms of force of interest is a simple power of e:

The force of interest is less than the annual effective interest rate

Effective interest rate

The effective interest rate, effective annual interest rate, annual equivalent rate or simply effective rate is the interest rate on a loan or financial product restated from the nominal interest rate as an interest rate with annual compound interest payable in arrears.It is used to compare the...

, but more than the annual effective discount rate. It is the reciprocal of the e-folding

E-folding

In science, e-folding is the time interval in which an exponentially growing quantity increases by a factor of e. This term is often used in theoretical physics, especially when cosmic inflation is investigated...

time. See also notation of interest rates.

Compounding basis

See Day count convention
Day count convention
In finance, a day count convention determines how interest accrues over time for a variety of investments, including bonds, notes, loans, mortgages, medium-term notes, swaps, and forward rate agreements . This determines the amount transferred on interest payment dates, and also the calculation of...

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

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:

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

Monthly mortgage payments

The interest on mortgages is often compounded monthly. The formula for payments is found from the following argument.

Notation

I = Note percentage rate

i = Monthly percentage rate = I/12 (so that the APR = (1+i)^12 - 1)

T = Term in years

Y= I•T

X = ½ I•T = ½ Y

n = 12•T = term in months

L = Principal or amount of loan

P = monthly payment

Exact formula for P

If the term were only one month then clearly

so that

.
If the term were two months then

so that

.
For a term of n months then

.

This can be simplified by noting that

and taking the difference:

so that

This formula for the monthly payment on a U.S. mortgage is exact and is what banks use.

Approximate formula for P

A formula that is accurate to within a few percent can be found by
noting that for typical U.S. note rates (

and terms T=10-30 years), the monthly note rate is small compared to 1:

so that the

which yields
a simplification so that

which suggests defining auxiliary variables

.

is the monthly payment required for a zero
interest loan paid off in

installments. In terms of these variables the
approximation can be written

The function

is even:

implying that it can be expanded in even powers of

.

It follows immediately that

can be expanded in even powers
of

plus the single term:

It will prove convenient then to define

so that

which can be expanded:

where the ellipses indicate terms that are higher order in even powers of

. The expansion

is valid to better than 1% provided

.

Example

For a mortgage with a term of 30 years and a note rate of 4.5% we find:

which suggests that the approximation

is accurate to better than one percent for typical U.S. mortgage terms in January 2009.
The formula becomes less accurate for higher rates and longer terms.

For a 30-year term on a loan of $120,000 and a 4.5% note rate we find:

so that

The exact payment amount is

so the approximation is an overestimate of about a sixth of a percent.

Other approximations

The approximate formula

yields

which is a slight underestimate of the exact result. This underestimate results from
the approximation

. Keeping the next correction in the expansion
of

results in an approximate formula

which is off by two tenths of a cent.
The simplest approximation discussed

is good to within better than a percent for typical US mortgages in early 2009. The approximation

is an underestimate of around 10% for such mortgage payments.

History

Compound interest was once regarded as the worst kind of usury

Usury

Usury Originally, when the charging of interest was still banned by Christian churches, usury simply meant the charging of interest at any rate . In countries where the charging of interest became acceptable, the term came to be used for interest above the rate allowed by law...

, and was severely condemned by Roman law

Roman law

Roman law is the legal system of ancient Rome, and the legal developments which occurred before the 7th century AD — when the Roman–Byzantine state adopted Greek as the language of government. The development of Roman law comprises more than a thousand years of jurisprudence — from the Twelve...

, as well as the common law

Common law

Common law is law developed by judges through decisions of courts and similar tribunals rather than through legislative statutes or executive branch action...

s of many other countries.

In one passage, the Bible

Bible

The Bible refers to any one of the collections of the primary religious texts of Judaism and Christianity. There is no common version of the Bible, as the individual books , their contents and their order vary among denominations...

addresses the charging of interest in the following manner:
The Qur'an

Qur'an

The Quran , also transliterated Qur'an, Koran, Alcoran, Qur’ān, Coran, Kuran, and al-Qur’ān, is the central religious text of Islam, which Muslims consider the verbatim word of God . It is regarded widely as the finest piece of literature in the Arabic language...

explicitly mentions compound interest as a great sin. Usury

Usury

Usury Originally, when the charging of interest was still banned by Christian churches, usury simply meant the charging of interest at any rate . In countries where the charging of interest became acceptable, the term came to be used for interest above the rate allowed by law...

(oppressive interest), known in Arabic as "riba

Riba

Riba means one of the senses of "usury" . Riba is forbidden in Islamic economic jurisprudence fiqh and considered as a major sin...

", is considered wrong:
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.