Compound interest

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.

Effective interest rates

The effect of earning 20% annual interest on an initial $1,000 investment and various compounding frequencies

Compound interest is contrasted with 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, 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

 

Compound interest of 15% on initial $10,000 investment over 40 years

 

Annual dividend of 1.5% on initial $10,000 investment
$266,864 in total dividend payments over 40 years
Dividends were not reinvested in this scenario

 

Inflation compounded over 40 years at different rates

  8%

  7%

  6%

  5%

  4%

  3%

  2%

  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 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. Continuous compounding in pricing these instruments is a natural consequence of Itô calculus, where financial derivatives are valued at ever-increasing frequency, until the limit is approached and the derivative is valued in continuous time.

History

Further information: 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 laws of many other countries.^[2]

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.^[3] 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 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.^[4][5]

Jacob Bernoulli discovered the constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle e} 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.^[6] In modern times, Albert Einstein's supposed quote regarding compund interest rings true. "He who understands it earns it; he who doesn't pays it."^[7]

Calculation

See also: Time value of money and Interest § Calculation

Periodic compounding

The total accumulated value, including the principal sum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle P} plus compounded interest Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle I} , is given by the formula:^[8][9]Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle A=P\left(1+\frac{r}{n}\right)^{nt}}

where:

• A is the final amount
• P is the original 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:^[10]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle I=P\left(1+\frac{r}{n}\right)^{nt}-P}

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:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle a(t) = \left(1 + \frac {r} {n}\right) ^ {nt} }

Continuous compounding

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 e^r − 1. Continuous compounding can be regarded as letting the compounding period become infinitesimally small, achieved by taking the limit as n goes to infinity. The amount after t periods of continuous compounding can be expressed in terms of the initial amount P₀ as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle P(t)=P_0 e ^ {rt}.}

Force of interest

As the number of compounding periods Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle n} tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \delta} . 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 as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \delta_{t}=\frac{a'(t)}{a(t)}=\frac{d}{dt} \ln a(t)}

This is the logarithmic derivative of the accumulation function.

Conversely: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle a(t)=e^{\int_0^t \delta_s\, ds}\, ,} (Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle a(0) = 1} , 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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle da(t)=\delta_{t}a(t)\,dt}

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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \delta=\ln(1+r)} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle a(t)=e^{t\delta}}

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.

A way of modeling the force of inflation is with Stoodley's formula: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \delta_t = p + {s \over {1+rse^{st}}}} 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \left(1+\frac{r_1}{n_1}\right)^{n_1} = \left(1+\frac{r_2}{n_2}\right)^{n_2}}

use

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle r_2=\left[\left(1+\frac{r_1}{n_1}\right)^\frac{n_1}{n_2}-1\right]{n_2},}

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

When interest is continuously compounded, use

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \delta=n\ln{\left(1+\frac{r}{n}\right)},}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \delta} 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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle c} ) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle c = \frac{rP}{1-\frac{1}{(1+r)^n}} } or equivalently Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle c = \frac{rP}{1-e^{-n\ln(1+r)}}}

where:

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle c} = monthly payment
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle P} = principal
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle r} = monthly interest rate
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle n} = 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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle I<8\%} and terms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle T} =10–30 years), the monthly note rate is small compared to 1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle r << 1} so that the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \ln(1+r)\approx r} which yields the simplification:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle c\approx \frac{Pr}{1-e^{-nr}}= \frac{P}{n}\frac{nr}{1-e^{-nr}}}

which suggests defining auxiliary variables

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle Y\equiv n r = IT} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle c_0\equiv \frac{P}{n} .}

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle c_0} is the monthly payment required for a zero–interest loan paid off in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle n} installments. In terms of these variables the approximation can be written Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\textstyle c\approx c_0 \frac{Y}{1-e^{-Y}}} .

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\textstyle X = \frac{1}{2}Y} . The expansion Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\textstyle c\approx c_0 \left(1 + X + \frac{X^2}{3}\right)} is valid to better than 1% provided Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X\le 1 } .

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle T=30} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle I=0.045} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle c_0=\frac{$120,000}{360}=$333.33}

which gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X=\frac{1}{2}IT=.675}

so that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle c\approx c_0 \left(1 + X + \frac{1}{3}X^2 \right)=\$333.33 (1+.675+.675^2/3)=\$608.96}

The exact payment amount is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle c=\$608.02} 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).^[11]

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle P} = principal deposit
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle r} = rate of return (monthly)
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle M} = monthly deposit, and
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle t} = time, in months

The compound interest for each deposit is: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle M'=M(1+r)^{t}} 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): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle M'=\sum^{t-1}_{i=0}{M(1+r)^{t-i}}} Recognizing the geometric series: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle M'=M\sum^{t-1}_{i=0}(1+r)^{t}\frac{1}{(1+r)^{i}}} and applying the closed-form formula (common ratio :Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 1/(1+r)} ):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle P' = M\frac{(1+r)^{t}-1}{r}+P(1+r)^t}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \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}}

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:^[12]Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle r=\left(\frac{P'-P-\sum{M}}{P+\sum{M}/2}\right)^{1/t}} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle r=\left(\frac{P'-\sum{M}/2}{P+\sum{M}/2}\right)^{1/t}-1}

See also

 

Wikiquote has quotations related to Compound interest.

 

Look up interest in Wiktionary, the free dictionary.

• 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

1. http://laws.justice.gc.ca/en/showdoc/cs/I-15/bo-ga:s_6//en#anchorbo-ga:s_6^{[permanent dead link]} 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.
2.   This article incorporates text from a publication now in the public domain: Chambers, Ephraim, ed. (1728). "Interest". Cyclopædia, or an Universal Dictionary of Arts and Sciences (1st ed.). James and John Knapton, et al.
3. Evans, Allan (1936). Francesco Balducci Pegolotti, La Pratica della Mercatura. Cambridge, Massachusetts. pp. 301–2.{{cite book}}: CS1 maint: location missing publisher (link)
4. 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. doi:10.1017/S002026810001636X.
5. Lewin, C G (1981). "Compound Interest in the Seventeenth Century". Journal of the Institute of Actuaries. 108 (3): 423–442. doi:10.1017/S0020268100040865.
6. Milanfar, Peyman (1996). "A Persian Folk Method of Figuring Interest". Mathematics Magazine. 69 (5): 376. doi:10.1080/0025570X.1996.11996479.
7. {cite url=https://www.inc.com/jim-schleckser/why-einstein-considered-compound-interest-most-powerful-force-in-universe.html
8. "Compound Interest Formula". qrc.depaul.edu. Retrieved 2018-12-05.
9. Investopedia Staff (2003-11-19). "Continuous Compounding". Investopedia. Retrieved 2018-12-05.
10. "Compound Interest Formula - Explained". www.thecalculatorsite.com. Retrieved 2018-12-05.
11. "Using Compound Interest to Optimize Investment Spread".
12. http://moneychimp.com/features/portfolio_performance_calculator.htm "recommended by The Four Pillars of Investing and The Motley Fool"