Learning Outcomes
- Calculate one-time simple interest, and simple interest over time
- Determine APY given an interest scenario
- Calculate compound interest
Principal and 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.
Simple One-time Interest
[latex]\begin{align}&I={{P}_{0}}r\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}r={{P}_{0}}(1+r)\\\end{align}[/latex]
- I is the interest
- A is the end amount: principal plus interest
- [latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] is the principal (starting amount)
- r is the interest rate (in decimal form. Example: 5% = 0.05)
Examples
A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?
The following video works through this example in detail.
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.
Exercises
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?
Further explanation about solving this example can be seen here.
We can generalize this idea of simple interest over time.
Simple Interest over Time
[latex]\begin{align}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align}[/latex]
- I is the interest
- A is the end amount: principal plus interest
- [latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] 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.
[latex]6\div{12}=0.5[/latex]
A 4% annual rate paid quarterly would be divided into four 1% payments.
[latex]4\div{4}=1[/latex]
Example
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?
This video explains the solution.
Try It
Try It
A loan company charges $30 interest for a one month loan of $500. Find the annual interest rate they are charging.
