Simple interest is calculated only on the principal (that is, the initial amount) so the amount of interest being added to a loan or investment remains constant or fixed. However, most of the time when banks and financial institutions calculate interest, they are using compound interest.

Ideas

Compound interest
Compound interest formula

Compound interest

Compound interest is calculated at the end of each compounding period, which is typically a day, month, quarter, or year. At the end of each compounding period, the total amount (principal plus interest) from previous compounding periods is used to calculate the new quantity of interest.

Examples

Example 1

\$8000 is invested for 3 years at a rate of 3\% p.a. compounded annually.

	Interest ($)	Balance ($)
\text{After }0\text{ years}	-	8000
\text{After }1\text{ year}
\text{After }2\text{ years}
\text{After }3\text{ years}

Complete the above table.

Worked Solution

Create a strategy

For each row, we want to apply the interest to the previous balance, then we write this into the "Interest" column.

The new balance will be the previous balance plus the interest, which is then entered into the "Balance" column.

Apply the idea

For the first year:

\displaystyle \text{Interest}	\displaystyle =	\displaystyle 8000 \times 0.03	Multiply the balance by the interest rate
	\displaystyle =	\displaystyle \$240	Evaluate
\displaystyle \text{Balance}	\displaystyle =	\displaystyle 8000+240	Add the balance and interest
	\displaystyle =	\displaystyle \$8240	Evaluate

We can use the above method to complete the rest of the table:

	Interest ($)	Balance ($)
\text{After }0\text{ years}	-	8000
\text{After }1\text{ year}	240	8240
\text{After }2\text{ years}	247.20	8487.20
\text{After }3\text{ years}	254.62	8741.82

Calculate the total interest accumulated over 3 years.

Worked Solution

Create a strategy

Add the values in the Interest column from the table in part (a).

Apply the idea

\displaystyle \text{Interest}	\displaystyle =	\displaystyle 240+247.20+254.62	Add the interest for each year
	\displaystyle =	\displaystyle \$741.82	Evaluate

Reflect and check

We could have also subtracted the initial investment of \$8000 from the final balance of \$8741.82:

\displaystyle \text{Interest}	\displaystyle =	\displaystyle 8741.82-8000	Subtract the investment from the final balance
	\displaystyle =	\displaystyle \$741.82	Evaluate

Idea summary

Compound interest is calculated at the end of each compounding period, on the total amount (principal plus interest) from the previous compounding period.

Compound interest formula

Why does money grow faster with compound interest than simple interest?

The image shows a graph with a curve and straight line. Ask your teacher for more information.

As seen before, simple interest is interest earned on the principal invested amount only, whereas compound interest is interest earned on the principal amount plus interest on the interest already earned. So instead of the value of your investment increasing it a straight line as with simple interest, it will exponentially grow something like this.

For example, consider a deposit of \$1000 into an online account for 2 years that pays 10\% pa simple interest. The interest earned in the 2 years is \$1000 \times 10\% \times 2 =\$200.

But suppose that, instead of simple interest, the paid interest compounded annually. In this case, the interest earned in the first year would be \$1000 \times 10\% \times 1 = \$100.

The new principal at the end of the first year would be \$1000 + \$100 = \$1100.

The interest earned in the second year would then be \$1100 \times 10\% \times 1 = \$110.

So the total compound interest earned over the two years would be \$100 + \$110 = \$210 which is \$10 more than what was earned with simple interest. Although a \$10 difference may not seem like much, think of how much the difference would have been if a million dollars was invested instead of a thousand, or if the investment was made for twenty years instead of two.

Notice that in the above example, at the end of each compounding period there is a two step process: calculate the interest and then add it to the account balance. We could treat this as a percentage increase and combine these two steps as follows:

\displaystyle \text{Balance after }1\text{ year} =	\displaystyle =	\displaystyle 500+500 \times 0.1
	\displaystyle =	\displaystyle 500 \times (1+0.1)
	\displaystyle =	\displaystyle 550

This suggests a rule:\text{New balance } = \text{ Previous balance } \times (1+0.1) In other words, we can find the balance at the end of each year by repeatedly multiplying by (1+0.1):

\displaystyle \text{Balance after }1\text{ year}	\displaystyle =	\displaystyle 500 \times (1+0.1)
\displaystyle \text{Balance after }2\text{ years}	\displaystyle =	\displaystyle 500 \times (1+0.1)\times (1+0.1)
	\displaystyle =	\displaystyle 500 \times (1+0.1)^2
\displaystyle \text{Balance after }3\text{ years}	\displaystyle =	\displaystyle 500 \times (1+0.1)\times (1+0.1) \times (1+0.1)
	\displaystyle =	\displaystyle 500 \times (1+0.1)^3

This leads us to the compound interest formula: A=P\left(1+\dfrac{r}{100}\right)^n where A is the final amount of money , P is the principal , r is the interest rate per compounding period, expressed as a whole number, and n is the number of compounding time periods.

For interest rates r that are expressed as a decimal or a fraction, this formula can be used instead: A=P(1+r)^n

This formula gives us the total amount (ie. the principal and interest together). To calculate how much of the final amount is interest I, this can be done by subtracting the principal P from the total amount of the investment / loan A, I=A-P

Examples

Example 2

William's investment of \$2000 earns interest at a rate of 6\% p.a, compounded annually over 4 years. What is the future value of the investment to the nearest cent?

Worked Solution

Create a strategy

We can use the formula: A=P(1+r)^n

Apply the idea

\displaystyle A	\displaystyle =	\displaystyle 2000(1+0.06)^4	Substitute P=2000, \, r=0.06, \, n=4
	\displaystyle =	\displaystyle \$2524.95	Evaluate

Example 3

Tom wants to put a deposit on a house in4 years. In order to finance the \$12\,000 deposit, he decides to put some money into a high interest savings account that pays 5\% p.a. interest compounded monthly. If P is the amount of money that he must put into his account now to accumulate enough for the deposit, find P to the nearest cent.

Worked Solution

Create a strategy

We can use the formula: A=P(1+r)^n

Apply the idea

\displaystyle 12\,000	\displaystyle =	\displaystyle P \times \left(1+\dfrac{0.05}{12}\right)^{12\times 4}	Substitute the values
\displaystyle P	\displaystyle =	\displaystyle \dfrac{12\,000}{\left(1+\dfrac{0.05}{12}\right)^{12\times 4}}	Make P the subject of the equation
\displaystyle P	\displaystyle =	\displaystyle \$9828.85	Evaluate

Idea summary

We can find the final value of an investment or loan using the formula:

\displaystyle A=P\left(1+\dfrac{r}{100}\right)^n

\bm{A}

is the final amount of money

\bm{P}

is the principal

\bm{r}

is the interest rate per year

\bm{n}

is the number of compounding time periods

For interest rates r that are expressed as a decimal or a fraction, this formula can be used instead: A=P(1+r)^n

We can find the interest earned using the formula:

\displaystyle I=A-P

\bm{I}

is the interest earned

\bm{A}

is the final amount of money

\bm{P}

is the principal

Outcomes

U2.AoS4.8

scientific notation, exact and approximate answers, significant figures and rounding

4.07 Compound interest

Introduction

Ideas

Compound interest

Examples

Example 1

Compound interest formula

Examples

Example 2

Example 3

Outcomes

U2.AoS4.8

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