# Compound **interest**

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**Compound interest** is the addition of

**interest**to the principal sum of a loan or deposit, or in other words,

**interest**on

**interest**. It is the result of reinvesting

**interest**, rather than paying it out, so that

**interest**in the next period is then earned on the principal sum plus previously accumulated

**interest**. Compound

**interest**is standard in finance and economics.

Compound **interest** is contrasted with *simple interest*, where previously accumulated

**interest**is not added to the principal amount of the current period, so there is no compounding. The

*simple annual*is the

**interest**rate**interest**amount per period, multiplied by the number of periods per year. The simple annual

**interest**rate is also known as the

**nominal**(not to be confused with the

**interest**rate**interest**rate not adjusted for inflation, which goes by the same name).

## Compounding frequency[edit]

The *compounding frequency* is the number of times per year (or rarely, another unit of time) the accumulated **interest** is paid out, or *capitalized* (credited to the account), on a regular basis. The frequency could be yearly, half-yearly, quarterly, monthly, weekly, daily, or continuously (or not at all, until maturity).

For example, monthly capitalization with **interest** expressed as an annual rate means that the compounding frequency is 12, with time periods measured in months.

The effect of compounding depends on:

- The nominal
**interest**rate which is applied and - The frequency
**interest**is compounded.

## Annual equivalent rate[edit]

The nominal rate cannot be directly compared between loans with different compounding frequencies. Both the nominal **interest** rate and the compounding frequency are required in order to compare **interest**-bearing financial instruments.

To help consumers compare retail financial products more fairly and easily, many countries require financial institutions to disclose the annual compound **interest** rate on deposits or advances on a comparable basis. The **interest** rate on an annual equivalent basis may be referred to variously in different markets as effective *annual percentage rate* (EAPR), *annual equivalent rate* (AER), *effective interest rate*,

*effective annual rate*,

*annual percentage yield*and other terms. The effective annual rate is the total accumulated

**interest**that would be payable up to the end of one year, divided by the principal sum.

There are usually two aspects to the rules defining these rates:

- The rate is the annualised compound
**interest**rate, and - There may be charges other than
**interest**. The effect of fees or taxes which the customer is charged, and which are directly related to the product, may be included. Exactly which fees and taxes are included or excluded varies by country, may or may not be comparable between different jurisdictions, because the use of such terms may be inconsistent, and vary according to local practice.

## Examples[edit]

- 1,000 Brazilian real (BRL) is deposited into a Brazilian savings account paying 20% per annum, compounded annually. At the end of one year, 1,000 x 20% = 200 BRL
**interest**is credited to the account. The account then earns 1,200 x 20% = 240 BRL in the second year. - A rate of 1% per month is equivalent to a simple annual
**interest**rate (nominal rate) of 12%, but allowing for the effect of compounding, the annual equivalent compound rate is 12.68% per annum (1.01^{12}− 1). - 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 and multiplied by the principal. The yearly compounded rate is higher than the disclosed rate. - Canadian mortgage loans are generally compounded semi-annually with monthly (or more frequent) payments.
^{[1]} - U.S. mortgages use an amortizing loan, not compound
**interest**. With these loans, an amortization schedule 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, for example, in the valuation of derivatives, to use
*continuous compounding*, which is the limit as the compounding period approaches zero. Continuous compounding in pricing these instruments is a natural consequence of Itō calculus, where financial derivatives are valued at ever increasing frequency, until the limit is approached and the derivative is valued in continuous time.

## Discount instruments[edit]

- US and Canadian T-Bills (short term Government debt) have a different convention. Their
**interest**is calculated on a discount basis as (100 −*P*)/*Pbnm*,^{[clarification needed]}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).

## Calculation[edit]

### Periodic compounding[edit]

The total accumulated value, including the principal sum plus compounded **interest** , is given by the formula:^{[2]}^{[3]}

where:

*P*is the original principal sum*P'*is the new principal sum*r*is the nominal annual**interest**rate*n*is the compounding frequency*t*is the overall length of time the**interest**is applied (expressed using the same time units as*r*, usually years).

The total compound **interest** generated is the final value minus the initial principal:^{[4]}

#### Example 1[edit]

Suppose a principal amount of $1,500 is deposited in a bank paying an annual **interest** rate of 4.3%, compounded quarterly.

Then the balance after 6 years is found by using the formula above, with *P* = 1500, *r* = 0.043 (4.3%), *n* = 4, and *t* = 6:

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

Subtracting the original principal from this amount gives the amount of **interest** received:

#### Example 2[edit]

Suppose the same amount of $1,500 is compounded biennially (every 2 years). (This is very unusual in practice.) Then the balance after 6 years is found by using the formula above, with *P* = 1500, *r* = 0.043 (4.3%), *n* = 1/2 (the **interest** is compounded every two years), and *t* = 6 :

So, the balance after 6 years is approximately $1,921.24.

The amount of **interest** received can be calculated by subtracting the principal from this amount.

The **interest** is less compared with the previous case, as a result of the lower compounding frequency.

### Accumulation function[edit]

Since the principal *P* is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used instead. The accumulation function shows what $1 grows to after any length of time. Accumulation functions for simple and compound **interest** are

### Continuous compounding[edit]

As *n*, the number of compounding periods per year, increases without limit, the case is known as continuous compounding, in which case the effective annual rate approaches an upper limit of e^{r} − 1, where e is a mathematical constant that is the base of the natural logarithm.

Continuous compounding can be thought of as making the compounding period infinitesimally small, achieved by taking the limit as *n* goes to infinity. See definitions of the exponential function for the mathematical proof of this limit. The amount after *t* periods of continuous compounding can be expressed in terms of the initial amount *P*_{0} as

### Force of **interest**[edit]

As the number of compounding periods reaches infinity in continuous compounding, the continuous compound **interest** rate is referred to as the force of **interest** .

In mathematics, the accumulation functions are often expressed in terms of *e*, the base of the natural logarithm. This facilitates the use of calculus to manipulate **interest** formulae.

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

This is the logarithmic derivative of the accumulation function.

Conversely:

- (since ; this can be viewed as a particular case of a product integral).

When the above formula is written in differential equation format, then 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*:

- or

The force of **interest** is less than the annual effective **interest** rate, but more than the annual effective discount rate. It is the reciprocal of the *e*-folding time. See also notation of **interest** rates.

A way of modeling the force of inflation is with Stoodley's formula: where *p*, *r* and *s* are estimated.

### Compounding basis[edit]

To convert an **interest** rate from one compounding basis to another compounding basis, use

where
*r*_{1} is the **interest** rate with compounding frequency *n*_{1}, and
*r*_{2} is the **interest** rate with compounding frequency *n*_{2}.

When **interest** is continuously compounded, use

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

### Monthly amortized loan or mortgage payments[edit]

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The **interest** on loans and mortgages that are amortized—that is, have a smooth monthly payment until the loan has been paid off—is often compounded monthly. The formula for payments is found from the following argument.

#### Exact formula for monthly payment[edit]

An exact formula for the monthly payment () is

or equivalently

where:

- = monthly payment
- = principal
- = monthly
**interest**rate - = number of payment periods

This can be derived by considering how much is left to be repaid after each month.

The Principal remaining after the first month is

that is, the initial amount plus **interest** less the payment.

If the whole loan is repaid after one month then

- , so

After the second month is left, so

If the whole loan was repaid after two months,

- , so

This equation generalises for a term of n months, . This is a geometric series which has the sum

which can be rearranged to give

- Spreadsheet formula

In spreadsheets, the **PMT()** function is used. The syntax is:

*PMT(***interest**_rate, number_payments, present_value, future_value,[Type] )

See Excel, Mac Numbers, LibreOffice, Open Office, Google Sheets for more details.

For example, for **interest** rate of 6% (0.06/12), 25 years * 12 p.a., PV of $150,000, FV of 0, type of 0 gives:

- = PMT(0.06/12, 25 * 12, -150000, 0, 0)
- = $966.45

#### Approximate formula for monthly payment[edit]

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

- .

Here 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 of mortgage payment[edit]

For a $10,000 mortgage with a term of 30 years and a note rate of 4.5%, payable yearly, we find:

which gives

so that

The exact payment amount is so the approximation is an overestimate of about a sixth of a percent.

## History[edit]

Compound **interest** was once regarded as the worst kind of usury and was severely condemned by Roman law and the common laws of many other countries.^{[5]}

The Florentine merchant Francesco Balducci Pegolotti provided a table of compound **interest** in his book *Pratica della mercatura* of about 1340. It gives the **interest** on 100 lire, for rates from 1% to 8%, for up to 20 years.^{[6]} The *Summa de arithmetica* of Luca Pacioli (1494) gives the Rule of 72, stating that to find the number of years for an investment at compound **interest** to double, one should divide the **interest** rate into 72.

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.^{[7]}^{[8]}

Jacob Bernoulli discovered the constant in 1683 by studying a question about compound **interest**.

In the 19th century, and possibly earlier, Persian merchants used a slightly modified linear Taylor approximation to the monthly payment formula that could be computed easily in their heads.^{[9]}

## See also[edit]

Wikiquote has quotations related to: Compound interest |

Look up in Wiktionary, the free dictionary.interest |

- Credit card
**interest** - Exponential growth
- Fisher equation
**Interest****Interest**rate- Rate of return
- Rate of return on investment
- Real versus nominal value (economics)
- Yield curve

## References[edit]

**^**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.**^**"Compound**Interest**Formula".*qrc.depaul.edu*. Retrieved 2018-12-05.**^**Staff, Investopedia (2003-11-19). "Continuous Compounding".*Investopedia*. Retrieved 2018-12-05.**^**"Compound**Interest**Formula - Explained".*www.thecalculatorsite.com*. Retrieved 2018-12-05.**^**This article incorporates text from a publication now in the public domain: Chambers, Ephraim, ed. (1728). "^{article name needed}".*Cyclopædia, or an Universal Dictionary of Arts and Sciences*(first ed.). James and John Knapton, et al.**^**Evans, Allan (1936).*Francesco Balducci Pegolotti, La Pratica della Mercatura*. Cambridge, Massachusetts. pp. 301–2.**^**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.**^**Milanfar, Peyman (1996). "A Persian Folk Method of Figuring**Interest**".*Mathematics Magazine*.**69**(5): 376.