Compound interest, previously called anatocism, is interest which is regularly added to the debt (compounded). Interest is then calculated not only over the principal (as it is done in the case of simple interest), but also over the interest that has been added to the debt before – in other words, it is calculated over the total amount owed. With compound interest, the frequency of compounding influences the total amount of interest paid over the life of the loan.

Many banks advertise an annual percentage yield (APY) which is the return on the principal over an entire year. For example, a 5% rate compounded monthly would have an approximate APY of 5.12%.

Compound interest was once regarded as the worst kind of usury, and was severely condemned by Roman law, as well as the common laws of many other countries.

A popular example

---

If the Native American tribe that accepted goods worth 60 guilders for the sale of Manhattan in 1626 had invested the money in a Dutch bank at 6 1/2 % interest, compounded annually, then in 2005 their investment would be worth over € 700 billion (around US$ 820 billion), more than the assessed value of the real estate in all five boroughs of New York City.

Mathematics of interest rates

---

The amount function for compound interest is an exponential function in terms of time.

$A(t)\; =\; A\_0\; \backslash left(1\; +\; \backslash frac\; \{r\}\; \{n\}\backslash right)\; ^\; \{n\; \backslash cdot\; t\}$

• $n$ = Number of compounding periods per year (note that the total number of compounding periods is $n\; \backslash cdot\; t$)

As $n$ increases the rate approaches an upper limit of $e\; ^\; r$. This rate is called continuous compounding, see below.

Since the principal A(0) is simply a coefficient, it is often dropped for simplicity, and the resulting accumulation function is used in interest theory instead. Accumulation functions for simple and compound interest are listed below:

$a(t)=1+t\; r\backslash ,$

$a(t)=(1+\backslash frac\{r\}\{n\})^\{n\; \backslash cdot\; t\}\backslash ,$

Note: A(t) is the amount function and a(t) is the accumulation function.

Force of interest

In mathematics, the accumulation function are often expressed in terms of e, the base of the natural logarithm. This facilitates the use of calculus methods in manipulation of interest formulas. This is called the force of interest.

The force of interest is defined as the following:

$\backslash delta\_\{t\}=\backslash frac\{a\text{'}(t)\}\{a(t)\}\backslash ,$

$a(n)=e^\{\backslash int\_0^n\; \backslash delta\_t\backslash ,\; dt\}\backslash ,$

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

$da(t)=\backslash delta\_\{t\}a(t)\backslash ,dt\backslash ,$

The force of interest for compound interest is a constant for a given r, and the accumulation function of compounding interest in terms of force of interest is a simple power of e:

$\backslash delta=\backslash ln(1+r)\backslash ,$

$a(t)=e^\{t\backslash delta\}\backslash ,$

Continuous compounding

For interest compounded a certain number of times, n, per year, such as monthly or quarterly, the formula is:

$a(t)=\backslash left(1+\backslash frac\{r\}\{n\}\backslash right)^\{n\; \backslash cdot\; t\}\backslash ,$

Continuous compounding can be thought as making the compounding period infinitely small; therefore achieved by taking the limit of n to infinity. One should consult definitions of the exponential function for the mathematical proof of this limit.

$a(t)=\backslash lim\_\{n\backslash to\backslash infty\}\backslash left(1+\backslash frac\{r\}\{n\}\backslash right)^\{n\; \backslash cdot\; t\}$

$a(t)=e^\{r\; \backslash cdot\; t\}$

The amount function is simply

$A(t)=A\_0\; e^\{r\; \backslash cdot\; t\}$

Compounding bases

See Day count convention

To convert an interest rate from one compounding basis to another compounding basis, the following formula applies:

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

When interest is continuously compounded:

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

Another formula to calculate compound interest is y=a x b^x (y is equal to a times b to the x power) where y is the money after interest is calculated a is the initial deposite b is the interest rate in a decimal form (divided by the number of compoundings if applicable) x is the exponent of b which represents the time (multiplied by the number of compoundings if applicable)

See also

---

• Compound annual growth rate (CAGR)
• Credit card interest
• Fisher equation
• Term Structure of Interest Rates

References

---

Basic financial concepts | Exponentials | Mathematical finance | Actuarial science

Zinseszins | Procent składany | 复利