Compound Interest

 

 

Recall that with simple interest, any accrued interest on a principal amount P is paid once at the end of the compounding period. The accumulated amount available at this time would be

 

 

 

As an example, suppose that a bank pays simple interest at the rate of 8% per year. If a customer deposits $1000 and makes no withdrawals for three years, what is the total amount on deposit at the end of three years? What is the interest earned in that period of time?

 

 

To begin to understand compound interest, investigate the annual compounding of $100 over a three-year period with an interest rate of 10% by filling in the following chart.

 

           

 

It seems that after t years, the accumulated amount is given by the formula

 

 

 

With compound interest, earned interest is periodically added to the principal to earn interest itself at the same rate (called the nominal rate).

 

 

If the interest is compounded more than once a year (which is typically how things are done), say m times a year, then the rate per period becomes  and there are a total of mt conversion periods in t years. The accumulated amount after t years is now given by

 

 

 

As an example, find out how our $100 grows (over a three-year period) at 10% annual interest using different compounding frequencies.

 

 

Effective Rate of Interest

 

 

The effective rate (or effective annual yield) is the simple interest rate that would produce the same accumulated amount in one year as the nominal rate compounded m times a year. It is given by the formula

 

 

 

Knowing the effective rate of interest allows us to more easily compute the accumulated amount after t years on an investment of P dollars.

 

 

Present Value

 

 

The principal P and accumulated amount A are often called the present value and future value, respectively. A common problem is to determine the amount to invest now in order to have a certain amount available at a later date.

 

 

Find how much money should be deposited in a bank paying interest at the rate of 6% per year compounded monthly so that at the end of three years the accumulated amount will be $20000.

 

Find the present value of $49,158.60 due in five years at an interest rate of 10% per year compounded quarterly.

 

Jane has narrowed her investment options down to two:

·        Purchase a CD that matures in 12 years and pays interest upon maturity at the rate of 10% per year, compounded daily.

·        Purchase a zero coupon CD that will triple her investment in the same period.

Which option will optimize her investment?

 

Money is invested in a mutual fund that pays interest daily. Over a 2-year period in which no deposits or withdrawals were made, the account grew from $4500 to $5268.24. Find the effective rate of interest.