9.2: Simple Interest
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.
\[I=P_{0} r \nonumber \]
\[A=P_{0}+I=P_{0}+P_{0} r=P_{0}(1+r) \nonumber \]
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: \(5\% = 0.05\))
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.
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.
\(I=P_{0} r t\)
\(A=P_{0}+I=P_{0}+P_{0} r t=P_{0}(1+r t)\)
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.
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.
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.
\(\begin{array}{ll} P_{0}=\$ 1000 & \text{the principal } \\ r=0.02 & 2 \%\text{ rate per half-year} \\ t = 8 & \text{4 years = 8 half-years} \\
I=\$ 1000(0.02)(8)=\$ 160. & \text{You will earn }\$ 160 \text{ interest total over the four years.}\end{array}\)
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%.