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.

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.

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.