7.1: Simple Interest
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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.
\[ I = P_0r \nonumber \]
\[ A = P_0 + I = P_0 + P_0 r = P_0(1+r) \nonumber \]
\(I\) is the interest
\(A\) is the end amount: principal plus interest
\(P_0\) 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?
Solution
The principal:
\(P_0 = $300\)
3% rate:
\(r = 0.03\)
Therefore:
\(I = $300(0.03) = $9\)
You will earn $9 interest.
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.
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 $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.
\[ I = P_0rt \nonumber \]
\[ A = P_0 + I = P_0 + P_0 rt = P_0(1+rt) \nonumber \]
\(I\) is the interest
\(A\) is the end amount: principal plus interest
\(P_0\) is the principal (starting amount)
\(r\) is the interest rate in decimal form.
\(t\) is time
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.
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.
The principal:
\(P_0 = $1000\)
3% rate:
\(r = 0.02\)
4 years = 8 half-years
\(t = 8\)
Therefore:
\(I = $1000(0.02)(8) = $160\)
You will earn $160 interest total over the four years.
A loan company charges $30 interest for a one month loan of $500. Find the annual interest rate they are charging.