Compound interest

Last updated October 31, 2024

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.

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

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

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 $e$ 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 compound interest rings true. "He who understands it earns it; he who doesn't pays it."^[7]

Calculation

Periodic compounding

The total accumulated value, including the principal sum $P$ plus compounded interest $I$ , is given by the formula:^[8]^[9] $A=P\left(1+{\frac {r}{n}}\right)^{tn}$

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]

$I=P\left(1+{\frac {r}{n}}\right)^{tn}-P$

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

Continuous compounding

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:

$P(t)=P_{0}e^{rt}.$

Force of interest

As the number of compounding periods $n$ tends to infinity in continuous compounding, the continuous compound interest rate is referred to as the force of interest $\delta$ . 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:

$\delta _{t}={\frac {a'(t)}{a(t)}}={\frac {d}{dt}}\ln a(t)$

This is the logarithmic derivative of the accumulation function.

Conversely: $a(t)=e^{\int _{0}^{t}\delta _{s}\,ds}\,,$ (Since $a(0)=1$ , 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: $da(t)=\delta _{t}a(t)\,dt$

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: $\delta =\ln(1+r)$ or $a(t)=e^{t\delta }$

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: $\delta _{t}=p+{s \over {1+rse^{st}}}$ where p, r and s are estimated.

Compounding basis

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

$\left(1+{\frac {r_{1}}{n_{1}}}\right)^{n_{1}}=\left(1+{\frac {r_{2}}{n_{2}}}\right)^{n_{2}}$

use

$r_{2}=\left[\left(1+{\frac {r_{1}}{n_{1}}}\right)^{\frac {n_{1}}{n_{2}}}-1\right]{n_{2}},$

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

$\delta =n\ln {\left(1+{\frac {r}{n}}\right)},$

where $\delta$ 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

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 ( $c$ ) is $c={\frac {rP}{1-{\frac {1}{(1+r)^{n}}}}}$ or equivalently $c={\frac {rP}{1-e^{-n\ln(1+r)}}}$

where:

$c$ = monthly payment
$P$ = principal
$r$ = monthly interest rate
$n$ = 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 ( $I<8\%$ and terms $T$ =10–30 years), the monthly note rate is small compared to 1. $r<<1$ so that the $\ln(1+r)\approx r$ which yields the simplification:

$c\approx {\frac {Pr}{1-e^{-nr}}}={\frac {P}{n}}{\frac {nr}{1-e^{-nr}}}$

which suggests defining auxiliary variables

$Y\equiv nr=IT$ $c_{0}\equiv {\frac {P}{n}}.$

Here $c_{0}$ is the monthly payment required for a zero–interest loan paid off in $n$ installments. In terms of these variables the approximation can be written ${\textstyle c\approx c_{0}{\frac {Y}{1-e^{-Y}}}}$ .

Let ${\textstyle X={\frac {1}{2}}Y}$ . The expansion ${\textstyle c\approx c_{0}\left(1+X+{\frac {X^{2}}{3}}\right)}$ is valid to better than 1% provided $X\leq 1$ .

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:

$T=30$ $I=0.045$ $c_{0}={\frac {\$120,000}{360}}=\$333.33$

which gives

$X={\frac {1}{2}}IT=.675$

so that

$c\approx c_{0}\left(1+X+{\frac {1}{3}}X^{2}\right)=\$333.33(1+.675+.675^{2}/3)=\$608.96$

The exact payment amount is $c=\$608.02$ 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]

$P$ = principal deposit
$r$ = rate of return (monthly)
$M$ = monthly deposit, and
$t$ = time, in months

The compound interest for each deposit is: $M'=M(1+r)^{t}$ 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): $M'=\sum _{i=0}^{t-1}{M(1+r)^{t-i}}$ Recognizing the geometric series: $M'=M\sum _{i=0}^{t-1}(1+r)^{t}{\frac {1}{(1+r)^{i}}}$ and applying the closed-form formula (common ratio : $1/(1+r)$ ):

$P'=M{\frac {(1+r)^{t}-1}{r}}+P(1+r)^{t}$

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

${\text{Value}}=M{\frac {(1+r)^{t}-1}{r}}+P(1+r)^{t}+k{\frac {(1+r)^{t-x}-1}{r}}+C(1+r)^{t-y}$

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] $r=\left({\frac {P'-P-\sum {M}}{P+\sum {M}/2}}\right)^{1/t}$ or $r=\left({\frac {P'-\sum {M}/2}{P+\sum {M}/2}}\right)^{1/t}-1$

Related Research Articles

The number $e$ is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted $. Alternatively, e can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest.$

<span class="mw-page-title-main">Exponential function</span> Mathematical function, denoted exp(x) or e^x

The exponential function is a mathematical function denoted by $or . Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number, but various modern definitions allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".$

In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has interest-earning potential, a characteristic referred to as the time value of money, except during times of negative interest rates, when the present value will be equal or more than the future value. Time value can be described with the simplified phrase, "A dollar today is worth more than a dollar tomorrow". Here, 'worth more' means that its value is greater than tomorrow. A dollar today is worth more than a dollar tomorrow because the dollar can be invested and earn a day's worth of interest, making the total accumulate to a value more than a dollar by tomorrow. Interest can be compared to rent. Just as rent is paid to a landlord by a tenant without the ownership of the asset being transferred, interest is paid to a lender by a borrower who gains access to the money for a time before paying it back. By letting the borrower have access to the money, the lender has sacrificed the exchange value of this money, and is compensated for it in the form of interest. The initial amount of borrowed funds is less than the total amount of money paid to the lender.

In finance and economics, interest is payment from a debtor or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum, at a particular rate. It is distinct from a fee which the borrower may pay to the lender or some third party. It is also distinct from dividend which is paid by a company to its shareholders (owners) from its profit or reserve, but not at a particular rate decided beforehand, rather on a pro rata basis as a share in the reward gained by risk taking entrepreneurs when the revenue earned exceeds the total costs.

A eutectic system or eutectic mixture is a type of a homogeneous mixture that has a melting point lower than those of the constituents. The lowest possible melting point over all of the mixing ratios of the constituents is called the eutectic temperature. On a phase diagram, the eutectic temperature is seen as the eutectic point.

The time value of money refers to the fact that there is normally a greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference.

In probability theory and statistics, the gamma distribution is a versatile two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-squared distribution are special cases of the gamma distribution. There are two equivalent parameterizations in common use:

With a shape parameter $k$ and a scale parameter $θ$
With a shape parameter $and an inverse scale parameter ⁠ ⁠, called a rate parameter.$

Future value is the value of an asset at a specific date. It measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return; it is the present value multiplied by the accumulation function. The value does not include corrections for inflation or other factors that affect the true value of money in the future. This is used in time value of money calculations.

The classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity and can thereby move due to the conservation of momentum. It is credited to Konstantin Tsiolkovsky, who independently derived it and published it in 1903, although it had been independently derived and published by William Moore in 1810, and later published in a separate book in 1813. Robert Goddard also developed it independently in 1912, and Hermann Oberth derived it independently about 1920.

In finance, the duration of a financial asset that consists of fixed cash flows, such as a bond, is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a function of yield, duration also measures the price sensitivity to yield, the rate of change of price with respect to yield, or the percentage change in price for a parallel shift in yields.

Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables.

In statistics, G-tests are likelihood-ratio or maximum likelihood statistical significance tests that are increasingly being used in situations where chi-squared tests were previously recommended.

The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change. For a given set of reaction conditions, the equilibrium constant is independent of the initial analytical concentrations of the reactant and product species in the mixture. Thus, given the initial composition of a system, known equilibrium constant values can be used to determine the composition of the system at equilibrium. However, reaction parameters like temperature, solvent, and ionic strength may all influence the value of the equilibrium constant.

An amortization calculator is used to determine the periodic payment amount due on a loan, based on the amortization process.

In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows which the investor receives from that investment over a specified time period, such as interest payments, coupons, cash dividends and stock dividends. It may be measured either in absolute terms or as a percentage of the amount invested. The latter is also called the holding period return.

Mortgage calculators are automated tools that enable users to determine the financial implications of changes in one or more variables in a mortgage financing arrangement. Mortgage calculators are used by consumers to determine monthly repayments, and by mortgage providers to determine the financial suitability of a home loan applicant. Mortgage calculators are frequently on for-profit websites, though the Consumer Financial Protection Bureau has launched its own public mortgage calculator.

Analogous to continuous compounding, a continuous annuity is an ordinary annuity in which the payment interval is narrowed indefinitely. A (theoretical) continuous repayment mortgage is a mortgage loan paid by means of a continuous annuity.

A recurring deposit is a special kind of term deposit in India that is offered by Indian banks and India Post which helps people with regular incomes to deposit a fixed amount every month into their recurring deposit account and earn interest at the rate applicable to fixed deposits.

The public market equivalent (PME) is a collection of performance measures developed to assess private equity funds and to overcome the limitations of the internal rate of return and multiple on invested capital measurements. While the calculations differ, they all attempt to measure the return from deploying a private equity fund's cash flows into a stock market index.

In investment, an annuity is a series of payments made at equal intervals. Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly insurance payments and pension payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time. Annuities may be calculated by mathematical functions known as "annuity functions".

References

↑ "Interest Act, R.S.C., 1985, c. I-15, s. 6: Interest on Moneys Secured by Mortgage on Real Property or Hypothec on Immovables". Justice Laws Website. Department of Justice (Canada). 2002-12-31. Archived from the original on 2022-09-18. Retrieved 2024-08-14.
↑ 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.
↑ Evans, Allan (1936). Francesco Balducci Pegolotti, La Pratica della Mercatura. Cambridge, Massachusetts. pp. 301–2.{{cite book}}: CS1 maint: location missing publisher (link)
↑ 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.
↑ Lewin, C G (1981). "Compound Interest in the Seventeenth Century". Journal of the Institute of Actuaries. 108 (3): 423–442. doi:10.1017/S0020268100040865.
↑ Milanfar, Peyman (1996). "A Persian Folk Method of Figuring Interest". Mathematics Magazine. 69 (5): 376. doi:10.1080/0025570X.1996.11996479.
↑ Schleckser, Jim (January 21, 2020). "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?". Inc.
↑ "Compound Interest Formula". qrc.depaul.edu. Retrieved 2018-12-05.
↑ Investopedia Staff (2003-11-19). "Continuous Compounding". Investopedia. Retrieved 2018-12-05.
↑ "Compound Interest Formula - Explained". www.thecalculatorsite.com. Retrieved 2018-12-05.
↑ "Using Compound Interest to Optimize Investment Spread".
↑ http://moneychimp.com/features/portfolio_performance_calculator.htm "recommended by The Four Pillars of Investing and The Motley Fool"

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] "Interest Act, R.S.C., 1985, c. I-15, s. 6: Interest on Moneys Secured by Mortgage on Real Property or Hypothec on Immovables". Justice Laws Website. Department of Justice (Canada). 2002-12-31. Archived from the original on 2022-09-18. Retrieved 2024-08-14.

[r1728-2] 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] Schleckser, Jim (January 21, 2020). "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?". Inc.

[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] ttp://moneychimp.com/features/portfolio_performance_calculator.htm "recommended by The Four Pillars of Investing and The Motley Fool"

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]