Financial accounts appreciate in value according to an annual interest rate, which we typically refer to as APR (annual percentage rate). When using the APR to calculate account balances with interest, we always express the percentage as a decimal. For example, a 5% APR would be expressed as 0.05.
When multiplying an account balance by its growth factor according to interest, we can generally say that the growth factor is (1 + r), where r is the ARP expressed as a decimal. We add 1 to r, because the new account balance includes 100% of the original balance, plus any interest gained.
If interest is applied only once per year, we can multiply this growth factor of (1 + r) by the initial account balance after each year, leading to the expression:
Compound Interest Formula
Many accounts gather interest several times per year, not just once. This is compounding interest. When interest is compounded, a portion of the APR is applied throughout the year. For example, if interest is compounded twice a year, half of the APR is applied mid-year, and the other half is applied at the end of the year. If interest is compounded monthly, one-twelfth of the APR is applied each month, for a total of 12 times per year.
We need to adjust our interest formula to account for compounding interest. To do so, we introduce a new variable, n, which represents the number of times per year interest is compounded. How does n affect our formula? It divides r, the annual percentage rate, and it must also be multiplied by t, since it represents the number of times per year that interest is compounded.
Our formula for compounding interest is:
Solving for Account Balance
Now that we have a formula to work with, we can use this formula to answer questions about account balances, provided information about APR and how many times per year the interest is compounded. Consider the following scenario asking us to find the balance of account after a certain time period.
A checking account has an initial balance of $800.00. The account gains 3.5% interest, which is compounded monthly. What is the value of the account after 4 years, assuming no additional withdrawals or deposits are made?
First, we need to write down the values of as many variables as possible, based on the information provided to us. We know that the starting balance of the account is $800.00, so P = 800. Our annual percentage rate is 3.5%, which as a decimal is 0.035, our value for r. Now to interpret the compounding interest: interest is compounded monthly, so n = 12. What we need to find is the account balance after 4 years, so t = 4, and A remains unknown.
Fitting these into our formula, we get:
Follow the order of operations! Evaluate inside parenthesis first, then apply the exponent, and finally multiply by the principal value.
Solving for Growth Time
In this next sample, we are going to see how to apply logarithms to solve for the variable time, t. We will work with the same account, with an initial balance of $800.00 with 3.5% interest compounding monthly. We would like to know how many years it will take for the account balance to double.
When the account balance doubles, our A value will equal 1600. We still have 800 for P, 0.035 for r, and 12 for n. What we don't know is t, time:
We can simplify this formula a bit before we worry about solving for t. Let's clean up what is inside the parentheses first. Then, we want to divide by 800, because the exponent is attached to the fraction, not the 800. When we divide the equation by 800, we get 2 on the left side of the equation (which represents the account doubling in value).
To undo the variable exponent, we apply a logarithm to both sides. It doesn't matter in this case if you use the common log or natural log. Whichever logarithm you prefer will work just fine.
When we take the log of both sides, we need to apply the Power Property of Logs to isolate our variable t. This property allows us to move the variable from an exponent within the log to a coefficient outside of the log. Now we can isolate t by dividing by everything else on that side.
This means it will take almost 20 years for the account to double, given our assumptions of rate of growth.