Simple Interest

Learning Outcomes

• Calculate one-time simple interest
• Calculate simple interest over time
• Determine APY given an interest scenario

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.

recall converting percent to a decimal

To convert a percent to a decimal, remove the % symbol and move the decimal place two places to the left.

Ex. 5% = 0.05,  25% = 0.25, and 100% = 1.0

To take 5% of $100 as in the paragraph above, write the percent as a decimal translate the word of as multiplication. Ex. 5% of$100 => $0.5\cdot100=5$.

Simple One-time Interest

\begin{align}&I={{P}_{0}}r\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}r={{P}_{0}}(1+r)\\\end{align}

• I is the interest
• A is the end amount: principal plus interest
• \begin{align}{{P}_{0}}\\\end{align} 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? 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 and 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

\begin{align}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align}

• I is the interest
• A is the end amount: principal plus interest
• \begin{align}{{P}_{0}}\\\end{align} 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

APR is for interest paid by consumer on loans, APY is for interest paid to consumer on savings

APY – Annual Percentage Yield

Interest rates are usually given as an annual percentage yield (APY) – the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APY will be divided up.

For example, a 6% APY paid monthly would be divided into twelve 0.5% payments.
$6\div{12}=0.5$

A 4% annual rate paid quarterly would be divided into four 1% payments.
$4\div{4}=1$