Discussing interest starts with the principal, or amount your account starts with. This could be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: \(\$ 100(0.05)=\$ 5\). The total amount you would repay would be $105, the original principal plus the interest.
Simple One-time Interest
\[I=P_{0} r \nonumber \]
\[A=P_{0}+I=P_{0}+P_{0} r=P_{0}(1+r) \nonumber \]
where
- \(I\) is the interest
- \(A\) is the end amount: principal plus interest
- \(P_0\) is the principal (starting amount)
- \(r\) is the interest rate (in decimal form. Example: \(5\% = 0.05\))
Example 1
A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?
Solution
\(\begin{array}{ll} P_{0}=\$ 300 & \text{the principal } \\ r=0.03 & 3 \%\text{ rate} \\
I=\$ 300(0.03)=\$ 9. & \text{You will earn }\$ 9 \text{ interest.}\end{array}\)
One-time simple interest is only common for extremely short-term loans. For longer term loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In that case, interest would be earned regularly. For example, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bond holder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which time the issuer pays back the original bond value.
Example 2
Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn?
Solution
Each year, you would earn 5% interest: \(\$ 1000(0.05)=\$ 50\) in interest. So over the course of five years, you would earn a total of $250 in interest. When the bond matures, you would receive back the $1,000 you originally paid, leaving you with a total of $1,250.
We can generalize this idea of simple interest over time.
Simple Interest over Time
\(I=P_{0} r t\)
\(A=P_{0}+I=P_{0}+P_{0} r t=P_{0}(1+r t)\)
where
- \(I\) is the interest
- \(A\) is the end amount: principal plus interest
- \(P_0\) is the principal (starting amount)
- \(r\) is the interest rate in decimal form
- \(t\) is time
The units of measurement (years, months, etc.) for the time should match the time period for the interest rate.
APR – Annual Percentage Rate
Interest rates are usually given as an annual percentage rate (APR) – the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up.
For example, a \(6 \%\) APR paid monthly would be divided into twelve \(0.5 \%\) payments.
A \(4 \%\) annual rate paid quarterly would be divided into four \(1 \%\) payments.
Example 3: Treasury Notes
Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?
Solution
Since interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2% payments.
\(\begin{array}{ll} P_{0}=\$ 1000 & \text{the principal } \\ r=0.02 & 2 \%\text{ rate per half-year} \\ t = 8 & \text{4 years = 8 half-years} \\
I=\$ 1000(0.02)(8)=\$ 160. & \text{You will earn }\$ 160 \text{ interest total over the four years.}\end{array}\)
Try it Now 1
A loan company charges $30 interest for a one month loan of $500. Find the annual interest rate they are charging.
- Answer
-
\(I=\$ 30\) of interest
\(P_{0}=\$ 500\) principal
\(r= \frac{30}{500} = 0.06\) per month
\((0.06)(12) = 0.72\) per year
They are charging an annual interest rate of 72%.
