Learning Outcome
- Solve compound interest applications of exponential functions
The annual percentage rate (APR) of an account, also called the nominal rate, is the interest rate earned by an investment account per year. The term nominal is used when the compounding occurs a number of times other than once per year. When interest is compounded more than once a year, the effective interest rate ends up being greater than the nominal rate! This is a powerful tool for investing.
We can calculate the balance in an account with compounding interest using the following compound interest formula.
The Compound Interest Formula
Compound interest can be calculated using the formula
where
- [latex] A(T) [/latex] is the final account value, often called the future value
- [latex] t [/latex] is time measured in years
- [latex] P [/latex] is the starting value of the account, often called the principal or the present value
- [latex] r [/latex] is the annual percentage rate (APR) expressed as a decimal
- [latex] n [/latex] is the number of compounding periods in one year.
The variable [latex] n [/latex] appearing in the formula is usually expressed as how often the account generates interest. For example, an account with interest compounding monthly will use [latex] n=12 [/latex] since there are [latex] 12 [/latex] months in a year.
Note that this formula does NOT mean that the full APR is paid out [latex] n [/latex] times per year. The APR is split between the number of times interest is compounded. For example, if the APR is [latex] 6\% [/latex] and interest is compounded monthly, then the actual interest received per month is [latex] \dfrac{6\%}{12} [/latex] or [latex] 0.5\%. [/latex]
In our next example, we will calculate the value of an account after [latex]10[/latex] years of interest compounded quarterly.
Example
If we invest [latex]$3,000[/latex] in an investment account paying [latex]3\%[/latex] interest compounded quarterly, how much will the account be worth in [latex]10[/latex] years?
The following video shows an example of using exponential growth to calculate interest compounded quarterly.
In our next example, we will use the compound interest formula to solve for the principal.
Example
A [latex]529[/latex] Plan is a college-savings plan that allows relatives to invest money to pay for a child’s future college tuition. The account grows tax-free. Lily wants to set up a [latex]529[/latex] account for her new granddaughter and wants the account to grow to [latex]$40,000[/latex] over [latex]18[/latex] years. She believes the account will earn [latex]6\%[/latex] compounded semi-annually (twice a year). To the nearest dollar, how much will Lily need to invest in the account now?
In the following video, we show another example of finding the principal necessary to obtain a future value from compounded interest.
Investigating Continuous Compounding
So far, the examples we have used had interest paid at long and regular intervals. There are some accounts that compound interest much more frequently though. Let’s investigate what happens if the frequency of compounding is increased.
Suppose [latex]\$1,000[/latex] is invested in an account paying [latex]10\%[/latex] interest for [latex] 3 [/latex] years. What will the final balance be with different values of [latex] n [/latex]?
Annual ([latex] n=1 [/latex]): [latex]A(3)=1000\left(1+\dfrac{0.10}{1}\right)^{(1\cdot 3)}=\$1331[/latex]
Semi-annual ([latex] n=2 [/latex]): [latex]A(3)=1000\left(1+\dfrac{0.10}{2}\right)^{(2\cdot 3)}\approx\$1340.10[/latex]
Quarterly ([latex] n=4 [/latex]): [latex]A(3)=1000\left(1+\dfrac{0.10}{4}\right)^{(4\cdot 3)}\approx\$1344.89[/latex]
Monthly ([latex] n=12 [/latex]): [latex]A(3)=1000\left(1+\dfrac{0.10}{12}\right)^{(12\cdot 3)}\approx\$1348.18[/latex]
Weekly ([latex] n=52 [/latex]): [latex]A(3)=1000\left(1+\dfrac{0.10}{52}\right)^{(52\cdot 3)}\approx\$1349.47[/latex]
Daily ([latex] n=365 [/latex]): [latex]A(3)=1000\left(1+\dfrac{0.10}{365}\right)^{(365\cdot 3)}\approx\$1349.80[/latex]
Notice that the final balance is larger in each case, but the gains are getting smaller and it seems there is a limit to how much money we can make just by increasing compounding frequency. This limiting process is called continuous compounding, which you can think of as compounding millions of times per second. There is a formula for this type of compounding.
THE CONTINUOUS COMPOUNDING FORMULA
Continuously compounding interest, which can be though of as interest compounding millions of times per second, can be calculated using the formula
where
- [latex] A(T) [/latex] is the final account value
- [latex] t [/latex] is time measured in years
- [latex] P [/latex] is the starting value of the account, or the principal
- [latex] r [/latex] is the annual percentage rate (APR) expressed as a decimal
Note that this formula is actually very generic and can be applied in any real word application of exponential growth, as long as the growth is happening continuously (for example, population growth).
If we revisit our previous example with [latex]\$1,000[/latex] now invested with continuous compounding for three years, the final balance is [latex]A(3)=1000e^{0.1\cdot 3}\approx \$1,349.86.[/latex] The implication is that no matter how often interest is compounded in this example, the final balance after three years will never exceed [latex]\$1,349.86.[/latex]. This extreme amount of compounding only earned us [latex] 6 [/latex] more cents than the daily compounding.
In our next example, we will calculate continuous growth of an account. It is important to note the language that is used in the instructions for interest rate problems. You will know to use the continuous compounding when it uses the word continuous to describe the compounding instead of a frequency like monthly or quarterly.
Example
Credit card interest is often compounded continuously. Suppose a balance of [latex]$2,500[/latex] is left on a credit card with an APR of [latex]26.9\%[/latex] compounded continuously and no additional charges or payments are made to the card. What will the balance be after [latex] 3 [/latex] years?
In the following video, we show another example of interest compounded continuously.
Summary
The Compound Interest Formula can calculate the balance in an account which is gaining interest at regular intervals over time. It grows exponentially because the interest is calculated based on the current account balance rather than the principal, which means it generates interest from the previous interest that has already been earned. If interest is compounded more and more frequently, the limit of this process is called Continuous Compounding and uses a base of [latex] e [/latex] in the exponential growth.
