View source for Compound interest

{{short description|Compounding sum paid for the use of money}}
[[File:Compound interest (English).gif|thumb|300px|Effective interest rates]]
[[File:Compound Interest with Varying Frequencies.svg|thumb|right|310px|The effect of earning 20% annual interest on an initial $1,000 investment and various compounding frequencies]]
{{E (mathematical constant)}}

'''Compound interest''' is [[interest]] accumulated from a principal sum and previously accumulated interest. It is the result of reinvesting or retaining interest that would otherwise be paid out, or of the accumulation of debts from a borrower. 

Compound interest is contrasted with [[Interest#Calculation|simple interest]], where previously accumulated interest is not added to the principal amount of the current period. Compounded interest depends on the simple interest rate applied and the frequency at which the interest is compounded. 

==Compounding frequency==
The ''compounding frequency'' is the number of times per given unit of time the accumulated interest is capitalized, on a regular basis. The frequency could be yearly, half-yearly, quarterly, monthly, weekly, daily, [[continuous compounding|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.

==Annual equivalent rate==
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. These rates are usually the annualised compound interest rate alongside charges other than interest, such as taxes and other fees.

==Examples==

[[File:Compound interest.webp|thumb|300px|Compound interest of 15% on initial $10,000 investment over 40 years]]
[[File:Annual dividend.webp|thumb|300px|Annual [[dividend]] of 1.5% on initial $10,000 investment <br> $266,864 in total dividend payments over 40 years <br> Dividends were not reinvested in this scenario]]
[[File:Inflation compounded over 40 years.webp|thumb|300px|Inflation compounded over 40 years at different rates
{{legend-line|#000000 solid 3px|8%}}
{{legend-line|#FF95CA solid 3px|7%}}
{{legend-line|#929292 solid 3px|6%}}
{{legend-line|#017100 solid 3px|5%}}
{{legend-line|#FF2600 solid 3px|4%}}
{{legend-line|#D41876 solid 3px|3%}}
{{legend-line|#FEAE00 solid 3px|2%}}
{{legend-line|#00A2FF solid 3px|1%}}
]]
* The interest on corporate bonds and government bonds is usually payable twice yearly. The amount of interest paid every 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 loan]]s are generally compounded semi-annually with monthly or more frequent payments.<ref>{{Cite web |url=https://laws.justice.gc.ca/eng/acts/I-15/page-1.html#h-270681 |title=Interest Act, R.S.C., 1985, c. I-15, s. 6: Interest on Moneys Secured by Mortgage on Real Property or Hypothec on Immovables |date=2002-12-31 |access-date=2024-08-14 |website=Justice Laws Website |archive-url=https://web.archive.org/web/20220918222651/https://laws.justice.gc.ca/eng/acts/I-15/page-1.html#h-270681 |archive-date=2022-09-18 |url-status=live |publisher=[[Department of Justice (Canada)]]}}</ref>
* 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 [[Derivative (finance)|derivatives]], to use continuous compounding. Continuous compounding in pricing these instruments is a natural consequence of [[Itô calculus]], where [[Derivative (finance)|financial derivatives]] are valued at ever-increasing frequency, until the limit is approached and the derivative is valued in continuous time.

==History==
{{further|Interest#History}}

Compound interest when charged by lenders was once regarded as the worst kind of [[usury]] and was severely condemned by [[Roman law]] and the [[common law]]s of many other countries.<ref name="r1728">{{1728|title=Interest}}</ref>

The Florentine merchant [[Francesco Balducci Pegolotti]] provided a [http://www.medievalacademy.org/resource/resmgr/maa_books_online/evans_0024.htm#hd_ma0024_head_755 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.<ref>{{cite book | last = Evans |first = Allan | title = Francesco Balducci Pegolotti, La Pratica della Mercatura | place = Cambridge, Massachusetts | year = 1936 |pages = 301–2}}</ref> 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 [[wikt:anatocism#Etymology|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 maximum rate of interest allowable on loans) and 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.<ref>{{cite journal | last = Lewin | first = C G | year = 1970 | title = An Early Book on Compound Interest - Richard Witt's Arithmeticall Questions| journal = Journal of the Institute of Actuaries | volume = 96 | issue = 1 | pages = 121–132 | doi = 10.1017/S002026810001636X }}</ref><ref>{{cite journal | last = Lewin | first = C G | year = 1981 | title = Compound Interest in the Seventeenth Century | journal = Journal of the Institute of Actuaries | volume = 108 | issue = 3 | pages = 423–442 | doi = 10.1017/S0020268100040865 }}</ref>

[[Jacob Bernoulli]] discovered the constant [[e (mathematical constant)#Compound interest|<math>e</math>]] 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.<ref>{{cite journal | last = Milanfar | first = Peyman | year = 1996 | title = A Persian Folk Method of Figuring Interest | journal = Mathematics Magazine | volume = 69 | issue = 5 | pages = 376 | doi = 10.1080/0025570X.1996.11996479 }}</ref>
In modern times, Albert Einstein's supposed quote regarding compound interest rings true. "He who understands it earns it; he who doesn't pays it."<ref>{{cite magazine|url=https://www.inc.com/jim-schleckser/why-einstein-considered-compound-interest-most-powerful-force-in-universe.html|title=Why Einstein Considered Compound Interest the Most Powerful Force in the Universe: Is the power of compound interest really the 8th Wonder of the World?|magazine=Inc.|first=Jim|last=Schleckser|date=January 21, 2020}}</ref>

==Calculation==
{{See also|Time value of money|Interest#Calculation}}{{more citations needed|section|date=June 2019}}
===Periodic compounding===
The total accumulated value, including the principal sum <math>P</math> plus compounded interest <math>I</math>, is given by the formula:<ref>{{Cite web|url=https://qrc.depaul.edu/StudyGuide2009/Notes/Savings%20Accounts/Compound%20Interest.htm|title=Compound Interest Formula|website=qrc.depaul.edu|access-date=2018-12-05}}</ref><ref>{{Cite web|url=https://www.investopedia.com/terms/c/continuouscompounding.asp|title=Continuous Compounding|author=Investopedia Staff| date=2003-11-19|website=Investopedia| language=en|access-date=2018-12-05}}</ref>
<math display="block">A=P\left(1+\frac{r}{n}\right)^{tn}</math>

where:
*''A'' is the final amount
*''P'' is the original principal sum
*''r'' is the [[nominal annual interest rate]]
*''n'' is the compounding frequency (1: annually, 12: monthly, 52: weekly, 365: daily)<ref>{{Cite web|url=https://www.investopedia.com/terms/c/compounding.asp|title=Compounding Interest: Formulas and Examples|author=JAMES CHEN|date=2024-08-01|website=Investopedia|language=en|access-date=2024-12-26}}</ref>
*''t'' is the overall length of time the interest is applied (expressed using the same time units as ''n'', usually years).

The total compound interest generated is the final amount minus the initial principal, since the final amount is equal to principal plus interest:<ref>{{Cite web| url=https://www.thecalculatorsite.com/articles/finance/compound-interest-formula.php|title=Compound Interest Formula - Explained| website=www.thecalculatorsite.com|access-date=2018-12-05}}</ref>

<math display="block">I=P\left(1+\frac{r}{n}\right)^{tn}-P</math>

===Accumulation function===

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. The accumulation function for compound interest is:<math display="block">a(t) = \left(1 + \frac {r} {n}\right) ^ {tn} </math>

===Continuous compounding===
<!-- This section is linked from [[Interest]] and from [[Continuously compounded interest]] -->
{{see also|Logarithmic return}}

When the number of compounding periods per year increases without limit, continuous compounding occurs, in which case the effective annual rate approaches an upper limit of {{math|''e''<sup>''r''</sup> − 1}}. Continuous compounding can be regarded as letting the compounding period become infinitesimally small, achieved by taking the [[Limit (mathematics)|limit]] as ''n'' goes to [[infinity]]. The amount after ''t'' periods of continuous compounding can be expressed in terms of the initial amount ''P''<sub>0</sub> as:

<math display="block">P(t)=P_0 e ^ {rt}.</math>

===Force of interest===
As the number of compounding periods <math>n</math> tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest <math>\delta</math>. For any continuously differentiable [[accumulation function]] a(t), the force of interest, or more generally the [[Rate of return#Logarithmic or continuously compounded return|logarithmic or continuously compounded return]], is a function of time as follows:

<math display="block">\delta_{t}=\frac{a'(t)}{a(t)}=\frac{d}{dt} \ln a(t)</math>

This is the [[logarithmic derivative]] of the accumulation function.

Conversely:
<math display="block">a(t)=e^{\int_0^t \delta_s\, ds}\, ,</math> (Since <math>a(0) = 1</math>, 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:
<math display="block">da(t)=\delta_{t}a(t)\,dt</math>

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'':
<math display="block">\delta=\ln(1+r)</math> or
<math display="block">a(t)=e^{t\delta}</math>

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|''e''-folding]] time. 

A way of modeling the force of inflation is with Stoodley's formula: <math>\delta_t = p + {s \over {1+rse^{st}}}</math> where ''p'', ''r'' and ''s'' are estimated.

===Compounding basis===
{{See also|Day count convention}}

To convert an interest rate from one compounding basis to another compounding basis, so that

<math display="block">\left(1+\frac{r_1}{n_1}\right)^{n_1} = \left(1+\frac{r_2}{n_2}\right)^{n_2}</math>

use

<math display="block">r_2=\left[\left(1+\frac{r_1}{n_1}\right)^\frac{n_1}{n_2}-1\right]{n_2},</math>

where ''r''<sub>1</sub> is the interest rate with compounding frequency ''n''<sub>1</sub>, and ''r''<sub>2</sub> is the interest rate with compounding frequency ''n''<sub>2</sub>.

When interest is [[#Continuous compounding|continuously compounded]], use

<math display="block">\delta=n\ln{\left(1+\frac{r}{n}\right)},</math>

where <math>\delta</math> 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===
{{See also|Mortgage calculator#Monthly payment formula}}
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====
An exact formula for the monthly payment (<math>c</math>) is
<math display="block"> c = \frac{rP}{1-\frac{1}{(1+r)^n}} </math>
or equivalently
<math display="block"> c = \frac{rP}{1-e^{-n\ln(1+r)}}</math>

where:
* <math>c</math> = monthly payment
* <math>P</math> = principal
* <math>r</math> = monthly interest rate
* <math>n</math> = number of payment periods

===== Spreadsheet formula =====
In spreadsheets, the '''PMT()''' function is used.  The syntax is:

 PMT(interest_rate, number_payments, present_value, future_value, [Type])
====Approximate formula for monthly payment====
A formula that is accurate to within a few percent can be found by noting that for typical U.S. note rates (<math>I<8\%</math> and terms <math>T</math>=10–30 years), the monthly note rate is small compared to 1. <math>r << 1</math> so that the <math>\ln(1+r)\approx r</math> which yields the simplification:

<math display="block">c\approx \frac{Pr}{1-e^{-nr}}= \frac{P}{n}\frac{nr}{1-e^{-nr}}</math>

which suggests defining auxiliary variables

<math display="block">Y\equiv n r = IT</math><math display="block">c_0\equiv \frac{P}{n} .</math>

Here <math>c_0</math> is the monthly payment required for a zero–interest loan paid off in <math>n</math> installments. In terms of these variables the approximation can be written <math display="inline">c\approx c_0 \frac{Y}{1-e^{-Y}}</math>.

Let <math display="inline">X = \frac{1}{2}Y</math>.
The expansion <math display="inline"> c\approx c_0 \left(1 + X + \frac{X^2}{3}\right)</math> is valid to better than 1% provided <math>X\le 1 </math>.

====Example of mortgage payment====
For a $120,000 mortgage with a term of 30 years and a note rate of 4.5%, payable monthly, we find:

<math display="block">T=30</math><math display="block">I=0.045</math><math display="block">c_0=\frac{$120,000}{360}=$333.33</math>

which gives

<math display="block">X=\frac{1}{2}IT=.675</math>

so that

<math display="block">c\approx c_0 \left(1 + X + \frac{1}{3}X^2 \right)=\$333.33 (1+.675+.675^2/3)=\$608.96</math>

The exact payment amount is <math>c=\$608.02</math> so the approximation is an overestimate of about a sixth of a percent.

===Monthly deposits===
Given a principal deposit and a recurring deposit, the total return of an investment can be calculated via the compound interest gained per unit of time. If required, the interest on additional non-recurring and recurring deposits can also be defined within the same formula (see below).<ref>{{Cite web|url=https://www.wavesofpaper.com/finance/Using-Compound-Interest-to-Optimize-Investment-Spread|title=Using Compound Interest to Optimize Investment Spread}}</ref>

* <math>P</math> = principal deposit
* <math>r</math> = rate of return (monthly)
* <math>M</math> = monthly deposit, and
* <math>t</math> = time, in months

The compound interest for each deposit is:
<math display="block">M'=M(1+r)^{t}</math>
Adding all recurring deposits over the total period t, (i starts at 0 if deposits begin with the investment of principal; i starts at 1 if deposits begin the next month):
<math display="block">M'=\sum^{t-1}_{i=0}{M(1+r)^{t-i}}</math>
Recognizing the [[geometric series]]: <math>M'=M\sum^{t-1}_{i=0}(1+r)^{t}\frac{1}{(1+r)^{i}}</math> and applying the [[Geometric series#Closed-form formula|closed-form formula]] (common ratio :<math>1/(1+r)</math>):

<math display="block">P' = M\frac{(1+r)^{t}-1}{r}+P(1+r)^t</math>

If two or more types of deposits occur (either recurring or non-recurring), the compound value earned can be represented as

<math display="block">\text{Value}=M\frac{(1+r)^{t}-1}{r}+P(1+r)^t+k\frac{(1+r)^{t-x}-1}{r}+C(1+r)^{t-y}</math>

where C is each lump sum and k are non-monthly recurring deposits, respectively, and x and y are the differences in time between a new deposit and the total period t is modeling.

A practical estimate for reverse calculation of the [[rate of return]] when the exact date and amount of each recurring deposit is not known, a formula that assumes a uniform recurring monthly deposit over the period, is:<ref>http://moneychimp.com/features/portfolio_performance_calculator.htm  "recommended by The Four Pillars of Investing and The Motley Fool"</ref>
<math display="block">r=\left(\frac{P'-P-\sum{M}}{P+\sum{M}/2}\right)^{1/t}</math> or <math display="block">r=\left(\frac{P'-\sum{M}/2}{P+\sum{M}/2}\right)^{1/t}-1</math>

==See also==
{{wikiquote}}
{{wiktionary|interest}}
* [[Credit card interest]]
* [[Exponential growth]]
* [[Fisher equation]]
* [[Interest]]
* [[Interest rate]]
* [[Rate of return]]
* [[Rate of return on investment]]
* [[Real versus nominal value (economics)]]
* [[Usury]]
* [[Yield curve]]

==References==
{{Reflist}}

{{Authority control}}

[[Category:Interest]]
[[Category:Exponentials]]
[[Category:Mathematical finance]]
[[Category:Actuarial science]]

[[it:Anatocismo]]