The Power of Compounding

Principal = balance upon which interest is being paid
simple interest = interest only paid on principal, not on any interest already accumulated
compound interest =  interest paid on principal aswell as on already accumulated interest

A = P * (1 + i * Y)

where     A = accumulated balance after Y years
              P = principal
              i = interest
             Y = number of years

• A savings bond pays 6% interest on a principal of $100. What is the accumulation after 8 years?

                A = P * (1 + i)^Y

Use the y^x button on the calculator.

• A savings account earns 6% interest compounded annually on a principal of $ 100. What is the accumulation after 8 years?

• invest  $100 in a savings account which earns 4.5% compounded annually for 10 years.
• invest  $100 in a savings account which earns 4.75% compounded annually for 10 years.

        A = P * (1 + APR/n)^nY

where     A = accumulated balance after Y years
              P = principal
              APR = interest
              n = number of times the interest is compounded per year
             Y = number of years
 
• A savings account earns 6% interest compounded monthly on a principal of $ 100. What is the accumulation after 8 years?

annual percentage yield (APY) = actual percentage by which a balance increases in 1 year (this is greater than APR if the interest compounds more than once a year.)

      APY =  (1 + APR/n)ⁿ -1
 

• A bank offers an APR of 4.5% compounded daily. Find the APY.
• A bank offers an APR of 4.5% compounded monthly. Find the APY.

growth of compounding:

• invest  $ 1 in year 0 at 5% annual compound interest. Compute the accumulated balance in the years 100, 500, 1000, 1500, 2000, 2002.

• Is there a limit, when you compound more and more frequently over a fixed time? Invest $1 for one year at an APR of 100 %, compounding n times over the course of the year. What is the accumulation for n = 1, 5, 10, 50, 100, 1000, 10 000?

Compound interest formula for continuous compounding:

            A = P * e^{(APR * Y)}

where     A = accumulated balance after Y years
              P = principal
              APR = interest
             Y = number of years
             e = 2.71828... (= Euler number, irrational)
 
• deposit  $10,000 into an account with an APR of 6% continuously compounded.
• College Fund: you want to establish a  $100,000 college fund in 18 years. How much will you have to deposit now under these scenarios
1. APR of 6% compounded daily
2. APR of 6% compounded continuously
3. APR of 6% compounded monthly