Compound 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.

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.

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 effective annual rate is the total accumulated interest that would be payable up to the end of one year, divided by the principal sum.

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.

[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.

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.

where: The total compound interest generated is the final amount minus the initial principal, since the final amount is equal to principal plus interest:[11]

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.

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 er − 1.

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:

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

A way of modeling the force of inflation is with Stoodley's formula:

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

=10–30 years), the monthly note rate is small compared to 1.

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

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

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).

and applying the closed-form formula (common ratio :

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

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:[13]