We've seen how to calculate compound interest using the compound interest formula when the compounding period is a year. What do we do if the interest is being compounded more frequently; perhaps daily, weekly, monthly, quarterly or semi-annually?
The same formula applies but we may need to convert the rate and time to match the compounding period.
$A=P\times(1+i)^n$A=P×(1+i)n
Where:
$A$A is the final amount of money (principal and interest together)
$P$P is the principal (the initial amount of money invested)
$i$i is the interest rate per compounding period, expressed as a decimal
$n$n is the total number of compounding periods
From an annual rate, the rate per period can be found by dividing the annual rate by the number of compounds per year. For example, $4.5%$4.5% p.a. compounded monthly gives us a rate of $i=\frac{4.5%}{12}=0.375%$i=4.5%12=0.375% per month.
Worked example
Suppose $\$500$$500 is invested in a compound interest account with an interest rate of $10%$10% p.a. compounded semi-annually (that is, with a compounding period of $6$6 months) for $3$3 years.
Think: Since the interest is being compounded semi-annually, the number of compounding periods in a year is two. The interest rate is $10%$10% per year and so the interest rate per compounding period as a decimal is $i=\frac{0.01}{2}=0.005$i=0.012=0.005 . Moreover, in $3$3 years, there are a total of $n=2\times3=6$n=2×3=6 compounding periods.
Do: Now we can substitute into the formula:
| $A$A | $=$= | $P\left(1+i\right)^n$P(1+i)n |
| $=$= | $500\times\left(1+0.005\right)^6$500×(1+0.005)6 | |
| $\approx$≈ | $\$670.05$$670.05 |
Reflect: For comparison, if the $\$500$$500 is invested in a compound interest account with an interest rate of $10%$10% p.a. compounded annually for $3$3 years, then
| $A$A | $=$= | $P\left(1+i\right)^n$P(1+i)n |
| $=$= | $500\times\left(1+0.1\right)^3$500×(1+0.1)3 | |
| $\approx$≈ | $\$665.50$$665.50 |
Practice questions
Question 1
A $\$3400$$3400 investment earns interest at $3%$3% p.a. compounded quarterly over $19$19 years.
Use the compound interest formula to calculate the value of this investment in dollars.
Round your answer to the nearest cent.
Question 2
Katrina borrows $\$4000$$4000 at a rate of $6.6%$6.6% p.a. compounded semi-annually. If she pays off the loan in a lump sum at the end of $6$6 years, find how much interest she pays in dollars.
Round your answer to the nearest cent.
Question 3
Charlie is expecting a Christmas bonus of $\$2000$$2000 in $6$6 months time. What is the most he can borrow now, $x$x in dollars, at a rate of $3.9%$3.9% p.a. compounded daily, and still be able to pay off the loan with his bonus?
Assume there are $365$365 days in a year
Round your answer to the nearest cent.