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Reference

Interest

Concept

Simple Interest

The interest that is applied only to an initial amount of money is called simple interest. The initial amount is known as the principal. Simple interest is calculated as a product of principal, annual interest rate, and the time in years.

An interest rate is a percent used to calculate the interest on the principal. It may be easier to write it in decimal form to make the calculations easier. For instance, assume that a savings account earns

3 %

simple interest per year on a deposit of

$ 1000 .

1000 \cdot 0.03 \cdot 1 = $ 30

This means that the simple interest earned on

$ 1000

in one year is

$ 30 .

The final amount of money in the account is called the balance. The following table shows the balance over five years of an account that earns

3 %

simple interest each year.

Years	Amount of Simple Interest	Balance
$1$	$1000 \cdot 0.03 \cdot 1 = $ 30$	$1000 + 30 = $ 1030$
$2$	$1000 \cdot 0.03 \cdot 2 = $ 60$	$1000 + 60 = $ 1060$
$3$	$1000 \cdot 0.03 \cdot 3 = $ 90$	$1000 + 90 = $ 1090$
$4$	$1000 \cdot 0.03 \cdot 4 = $ 120$	$1000 + 120 = $ 1120$
$5$	$1000 \cdot 0.03 \cdot 5 = $ 150$	$1000 + 150 = $ 1150$

Notice that the balance at the end of each period is calculated by adding the principal and simple interest earned during that period.

Concept

Compound Interest

Compound interest is the interest earned depending on both the initial investment and previously earned interest. To find the balance

A

of an account that earns compound interest, an exponential growth function can be used.

In this function,

P

stands for the principal, or the initial amount of money,

r

is the interest rate in decimal form, and

n

is the number of times the interest is compounded per year. For an account with the principal

$ 100

and an annual interest of

18 %

compounded twice a year, the balance in the account after

t

years is shown in the graph.

Notice that the function grows continuously, whereas, in reality, the account balance only increases at the times of compound. When calculating compound interest, the number of compounding periods

n

creates a difference. That is, the higher the number of compounding periods, the greater the amount of compound interest.

Rule

Continuously Compounded Interest

When interest is compounded infinitely many times, it is said to be continuously compounded. Let $A$ be the balance of an account that is continuously compounded, $P$ the initial amount, $r$ the interest rate, and $t$ the time. These values are connected by the following formula.

$A = P e^{r t}$

Keep in mind that, in this formula, the value of $r$ must be written as a decimal and the time $t$ must be in years. Also, the initial amount $P$ is usually called principal.

Proof

Recall the compound interest formula.

A = P (1 + \frac{r}{n})^{n t}

In the formula, there is the account's balance

A

, the initial amount

P,

the time in years

t,

and the number of times

n

the interest is compounded per year. The formula can be rewritten by using the Power of a Power Property.

A = P (1 + \frac{r}{n})^{n t} ⇕ A = P [(1 + \frac{r}{n})^{n}]^{t}

If the interest rate is

100 %,

r = 1,

and the interest is continuously compounded, the formula can be written in terms of

P,

t,

and

e .

This is because as

n

goes to infinity, the value of

(1 + \frac{1}{n})^{n}

approaches

e .

A = P [(1 + \frac{1}{n})^{n}]^{t} n \to \infty A = P e^{t}

The value of the expression

(1 + \frac{r}{n})^{n}

, on the other hand, approaches

e^{r} .

Examine the following interactive graph to better grasp how this is possible.

Using this last approximation, the final form of the formula can be obtained.

A = P [(1 + \frac{r}{n})^{n}]^{t} n \to \infty A = P e^{r t}

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Proof