8.1: Simple Interest

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Learning Objectives

Learn to recognize and use the basic terminology for financial formulas: principal, interest rates, future value.
Be able to calculate simple interest.
Be able to find the time or interest rate of an investment given principal and future value.

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 borrow $100 from a friend and agree to repay it with 5% interest, then the amount of interest you pay is 5% of 100: $\$ 100(0.05)=\$ 5$. The total amount you would repay would be $105, the original principal ($100) plus the interest ($5).

Simple One-time Interest

\[I=P_{0} r\]

\[A=P_{0}+I=P_{0}+P_{0} r=P_{0}(1+r)\]

where

$I$ is the interest
$A$ is the future value (end amount): principal plus interest
$P_0$ is the principal (starting amount)
$r$ is the interest rate (in decimal form. Example: $5\% = 0.05$)

Example 1

A friend asks to borrow $300 and agrees to repay it in 30 days with 3% simple interest. How much interest will you earn?

Solution

$\begin{array}{ll} P_{0}=\$ 300 & \text{the principal } \\ r=0.03 & 3 \%\text{ rate} \\
I=\$ 300(0.03)=\$ 9. & \text{You will earn }\$ 9 \text{ in interest.}\end{array}$

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.

Example 2

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 $\$50(5) = \$250$ 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 in the following way.

Simple Interest over Time

$I=P_{0} r t$

$P_t=P_{0}+I=P_{0}+P_{0} r t=P_{0}(1+r t)$

where

$I$ is the interest
$P_t$ is the future value (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.

Annual Percentage Rate (APR)

Sometimes the duration of a loan or investment is not measured in years. In such cases, it can be helpful to convert the information into annual percentage rate (APR). This is especially helpful when trying to compare investment opportunities or loan terms.

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

Example 3: Treasury Notes

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.

$\begin{array}{ll} P_{0}=\$ 1000 & \text{the principal } \\ r=0.02 & 2 \%\text{ rate} \\ t = 8 & \text{4 years = 8 half-years} \\
I=\$ 1000(0.02)(8)=\$ 160. & \text{You will earn }\$ 160 \text{ interest total over the four years.}\end{array}$

Using our formula for simple interest over time, we can solve for other variables as well.

Example 4: Solving for time

Suppose you invest $2,500 that earns 4.5% simple interest. How long will it take your investment to be worth $3,287.50?

Solution

$\begin{array}{ll} P_{0}=\$ 2500 & \text{the principal } \\ r=0.045 & 4.5 \%\text{ rate} \\ t = ? & \\
I=\$ 3,287.50 - \$2,500 = \$ 787.50 \\ \\ \text{Using our formula } I = P_{0} r t \text{ we get} \\ \$787.50 = \$ 2500(0.045)t & \text{Multiply } 2500(0.045) \\ \$ 787.50 = \$ 112.50t & \text{Divide both sides by } 112.50 \\ 7 = t & \text{It will take 7 years for your investment to be worth } \$3,287.50 \end{array}$

Example 5: Solving for interest rate

Suppose you take out a 6-month loan of $1,200 loan that charges $80 in interest. What annual interest rate are you paying?

Solution

Since you are paying $80 for 6 months, that would be $\$ 80(2) = \$ 160$ for a full year. So,

$\begin{array}{ll} P_{0}=\$ 1200 & \text{the principal } \\ r=? & \\ t = 1 & \text{we need time in years to find annual interest} \\
I=\$ 160 & \text{one year of interest} \\ \\ \text{Using our formula } I = P_{0} r t \text{ we get} \\ \$160 = \$ 1200(r)(1) & \text{Multiply by 1} \\ \$ 160 = \$ 1200r & \text{Divide both sides by } 1200 \\ 0.1333 = r & \text{Your annual interest rate is 13.33% (rounded to two decimal places)} \end{array}$

Try it Now 1

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

$r=$ ?

$t=\frac{1}{12}$ (1 month is $\frac{1}{12}$ of a year and we want annual interest rate)

So, using our formula $I= P_{0} r t$ we get,

$\$30 = \$ 500(r)(\frac{1}{12}) $	Multiply by $\frac{1}{12}$
$\$ 30 = \$ \frac{500}{12}r $	Multiply both sides by $\frac{12}{500}$
$0.72 = r $	Your annual interest rate is 72%.

\(\$30 = \$ 500(r)(\frac{1}{12}) \)	Multiply by \(\frac{1}{12}\)
\(\$ 30 = \$ \frac{500}{12}r \)	Multiply both sides by \(\frac{12}{500}\)
\(0.72 = r \)	Your annual interest rate is 72%.

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