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?
$A=P\left(1+\frac{r}{n}\right)^{nt}$A=P(1+rn)nt
where:
$A$A is the final amount of money (principal and interest together)
$P$P is the principal (the initial amount of money invested)
$r$r is the interest rate per year, expressed as a decimal or fraction
$n$n is the number of compounding periods in a year
$t$t is the number of years
Notice that since $n$n is the number of compounding periods in a year, $\frac{r}{n}$rn is the interest rate per compounding period, and $nt$nt is the total number of compounding periods.
Worked example
Suppose $\$500$$500 is invested in a compound interest account with an interest rate of $10%$10% p.a. compounded semi-annually (that is, with a compounding period of $6$6 months) for $3$3 years.
Since the interest is being compounded semi-annually, the number of compounding periods in a year is $n=2.$n=2. The interest rate is $10%$10% per year and so the interest rate per compounding period as a decimal is $\frac{r}{n}=\frac{0.01}{2}=0.005$rn=0.012=0.005 . Moreover, in $3$3 years, there are a total of $nt=2\times3=6$nt=2×3=6 compounding periods. Now we can substitute into the formula:
| $A$A | $=$= | $P\left(1+\frac{r}{n}\right)^{nt}$P(1+rn)nt |
| $=$= | $500\times\left(1+\frac{0.01}{2}\right)^{2\times3}$500×(1+0.012)2×3 | |
| $=$= | $500\times\left(1+0.005\right)^6$500×(1+0.005)6 | |
| $\approx$≈ | $\$670.05$$670.05 |
For comparison, if the $\$500$$500 is invested in a compound interest account with an interest rate of $10%$10% p.a. compounded annually for $3$3 years, then
| $A$A | $=$= | $P\left(1+r\right)^t$P(1+r)t |
| $=$= | $500\times\left(1+0.1\right)^3$500×(1+0.1)3 | |
| $\approx$≈ | $\$665.50$$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.