Compound interest

From Wikicpa

Compound interest. When interest is periodically added to the principal and this new amount is used as a the principal for the following time period and this procedure is repeated over a set number of periods, the final amount is called the compounded amount. The difference between the compounded amount and the original principal is called the compounded interest.

The interest rate is usually quoted on a yearly basis and must be changed to the interest rate per interest period for computational purposes.

1 Example & Solution
2 Mathematics of interest rates
- 2.1 Force of interest
- 2.2 Continuous compounding
3 Also see

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Example & Solution

Example: Find the compounded amount at the end of 1 year on $1,000 if the rate is 6% compounded quarterly.

Solution: The phrase "compounded quarterly" means 1.5% per 3 months. Thus, at the end of the first 3 months, $15 would be added to the original $1,000 principal (1.5% x $1,000) for a new principal of $1,015. At the end of the second 3 month period $15.23 would be added (1.5% x $1,015) resulting in a new principal of $1,030.23. At the end of the third 3 month period, $15.45 would be added (1.5% x 1,030.23) resulting in $1,045.68. And finally at the end of the fourth 3 month period $15.69 (1.5% x $1,045.68) would be added resulting in the 1 year compouned amount of $1,061.36. In this example the 1 year compounded interest is $61.36 ($1,061.36 less the original $1,000)

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Mathematics of interest rates

Source:wikipedia.org
The amount function for compound interest is an exponential function in terms of time.

$A(t) = A_0 \left(1 + \frac {r} {n}\right) ^ {n \cdot t}$

$n$ = Number of compounding periods per each $t$ (time in years) (note that the total number of compounding periods is $n \cdot t$ )

$r$ = Interest rate expressed as a decimal. eg: 6% = .06

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:

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

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

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

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Force of interest

In mathematics, the accumulation function are often expressed in terms of E (mathematical constant)|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:

$\delta_{t}=\frac{a'(t)}{a(t)}\,$

$a(n)=e^{\int_0^n \delta_t\, dt}\,$

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

$da(t)=\delta_{t}a(t)\,dt\,$

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:

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

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

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Continuous compounding

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

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

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.

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