So far we have looked at investments that compound annually. How does the future value of an investment change if the interest is compounded more frequently, and how would we use the formula $FV=PV\left(1+r\right)^n$FV=PV(1+r)n to work it out?
We first need to adjust the interest rate to reflect the length of the compounding period we are working with, and determine the correct number of periods to use as the index in the formula.
For example, if we are looking at an investment with an annual interest rate of $3%$3% that is compounded monthly, then when calculating its future value in $3$3 years time:
- the number of compounding periods is $n=3\times12=36$n=3×12=36 months
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the interest rate must be expressed in terms of the same time period, so $r=3%$r=3% per year $=$=$\frac{3%}{12}=0.25%$3%12=0.25% per month
Once everything is expressed correctly for our particular compounding period, we can use these values with the compound interest formula as usual.
You'll notice that because the interest is being calculated and added to the balance more frequently, the total amount of the investment will increase faster than if interest is only calculated annually.
Here is a visual representation of the same investment with the same interest rate, with monthly (top), quarterly (middle), and annual (bottom) compounding over two years:
Though the effects of the three different kinds of compounding are only small at the beginning, monthly compounding ends up growing the investment significantly more over time.
It is most common for banks to apply compound interest on a daily basis, which is good for anyone with an investment!
Practice questions
Question 1
A $\$2620$$2620 investment earns interest at $3%$3% p.a. compounded over $16$16 years.
Use the compound interest formula to calculate the value of this investment to the nearest cent, if the interest is compounded annually.
Use the compound interest formula to calculate the value of this investment to the nearest cent, if the interest is compounded quarterly.
Use the compound interest formula to calculate the value of this investment to the nearest cent, if the interest is compounded daily.
Calculate the total interest earned in each of the above investments, correct to the nearest dollar.
Interest earned compounding yearly: $$\editable{}$
Interest earned compounding quarterly: $$\editable{}$
Interest earned compounding daily: $$\editable{}$
Question 2
Valentina's investment of $\$8000$$8000 earns interest at $2%$2% p.a., compounded semiannually over $2$2 years. Answer the following questions by repeated multiplication.
What is the value of the investment after $2$2 years, to the nearest cent?
What is the amount of interest earned to the nearest cent?