Finance Unit

1.7 Simple and Compound Interest

You can earn interest by saving or investing money at a bank or with a financial company. Simple interest is paid only on your initial investment. Compound interest is paid both on the original investment and on all interest that has been added to the original investment.

The simple interest formula is I=PRT

where I is the interest earned,

P is the starting principal,

R is the annual percentage interest rate written as a decimal,

and T is the number of years.

The compound interest formula is \( \begin{equation} A=P\left(1+\frac{R}{n}\right)^{n T} \end{equation} \)

Where A is the accumulated balance after T years,

P is the starting principal,

R is the annual percentage interest rate written as a decimal,

n is the number of compounding periods per year,

and T is the number of years.

  1. You deposit $5000 in a bank account that pays an APR of 3% and compounds interest monthly. How much money will you have after 5 years?

    2. \(\begin{equation} A=5000\left(1+\frac{.03}{12}\right)^{12 \cdot 5} \end{equation} \)

    \(\begin{equation} A=\$ 5808.08 \end{equation} \)

  2. Compare this amount to the amount you’d have if interest were compounded daily.

    3. \(\begin{equation} A=5000\left(1+\frac{.03}{365}\right)^{365 \cdot 5} \end{equation}\)

    \(\begin{equation} A=\$ 5809.14 \end{equation}\)

  3. Compare these amounts to the amount you’d have if the bank account paid simple interest.

    4. \(\begin{equation} I=5000 \cdot .03 \cdot 5 \end{equation} \)

    \(\begin{equation} I= \$ 750 \end{equation} \)

    \(\begin{equation} A=5000+750= \$ 5750 \end{equation} \)

  4. At your birth $10,000 was deposited in an account paying 5% interest compounded quarterly. What is the value of the account at your 21st birthday?

    5. \(\begin{equation} A=10000\left(1+\frac{.05}{4}\right)^{4 \cdot 21} \end{equation} \)

    \(\begin{equation} A= \$ 28391.13 \end{equation} \)

  5. How much money would you need to deposit at your child’s birth in an account paying 6.5% interest compounded monthly to have $20,000 when your child turns 18?

    6. \(\begin{equation} 20000=P\left(1+\frac{.065}{12}\right)^{12 \cdot 18} \end{equation} \)

    \(\begin{equation} 20000=3.211835712 P \end{equation} \)

    \(\begin{equation} P=\$ 6226.97 \end{equation} \)

  6. How much would you have to invest now at 0.25% interest compounded monthly to have $500 for Christmas in 11 months?

    7. \(\begin{equation} 500=P\left(1+\frac{.0025}{12}\right)^{12 \cdot \frac{11}{12}} \end{equation} \)

    \(\begin{equation} 500=1.002294055 P \end{equation} \)

    \(\begin{equation} P= \$ 498.86 \end{equation} \)