3.3 Simple and Compound Interest

Introduction

What you’ll learn to do: Calculate interest rate, time, initial deposit, and total balance of both simple and compound interest situations

We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for retirement, or need a loan, we need to understand some mathematics.

Four credit cards sticking out of the back pocket of a pair of jeans

Learning Outcomes

  • Calculate one-time simple interest
  • Calculate simple interest over time
  • Determine APY given an interest scenario
  • Calculate compound interest given an interest scenario
  • Calculate the initial balance given an interest scenario

Simple 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.

recall converting percent to a decimal

To convert a percent to a decimal, remove the % symbol and move the decimal 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 => [latex]0.05\cdot100=5[/latex].

four rolled-up dollar bills seeming to grow out of dirt, with a miniature rake lying in between them

Simple One-time Interest

Begin by defining the following variables:

  • I is the interest
  • [latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] is the principal (starting amount)
  • r is the interest rate (written in decimal form. Example: 5% = 0.05)
  • A is the end amount: principal plus interest

Then, the formula we use to calculate the interest is given by:  [latex]I={{P}_{0}}r[/latex]

To calculate the final amount, we use the formula: [latex]A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}r={{P}_{0}}(1+r)[/latex]

Example

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 by defining the following variables:

  • I is the interest
  • [latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] is the principal (starting amount)
  • r is the interest rate (written in decimal form)
  • A is the end amount: principal plus interest
  • t is time

Then, the formula we use to calculate the interest is given by:  [latex]I={{P}_{0}}rt[/latex]

To calculate the final amount, we use the formula: [latex]A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)[/latex]

Note: The units of measurement (years, months, etc.) for the time should match the time period for the interest rate.

Important Idea: Observe that the formula given above to calculate the “Amount” when calculating simple interest over time is a linear equation. Consider the previous example exploring the value of a certain bond at maturity. Here is a graph of the data for that problem:

Annual Percentage Rate (APR)

Interest rates are often given in terms of the total interest that will be paid in the year.  The Annual Percentage Rate (APR) refers to the simple interest rate over a year’s time.

Note that, often, the interest is paid in smaller time increments (say quarters or months) and is these cases, the annual interest rates will be divided up.

 

Ex.:  A 6% APR paid monthly would be divided into twelve 0.5% payments because [latex]6\div{12}=0.5[/latex].

Ex.:  A 5% annual rate paid quarterly would be divided into four 1.25% payments because [latex]5\div{4}=1.25[/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 percentage rate, paid semi-annually, with a maturity in 42 months. How much interest will you earn?

 

This video below gives the solution to a similar example.

Example

A loan company charges $30 interest for a one month loan of $500. Find the annual rate they are charging.

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