### 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.