| Simple and Compound Interest | | Print | |
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This topic explains simple interest and compound interest through a series of problems and examples. Compounding continuously and the annual percentage rate is also worked on. Attention is given to the problem of finding the doubling time for an investment. Definition (Simple Interest and Future Value) If
a sum of money (called the principal) is invested for a period of time Example (Future Value for Simple Interest) If
$21,200 is invested at an annual simple interest rate of 5%, what is the future
value of the investment after 2 years? Example (Interest for Simple Interest) If
$7,700 is invested for 5 years at an annual simple interest rate of 15%, how
much interest is earned? Example (Principal for Simple Interest) A
firm buys 15 file cabinets at $166.23 each, with the bill due in 90 days. How
much must the firm deposit now to have enough to pay the bill if money is worth
6% per year? Use 360 days in a year. Example (Doubling Time for Simple Interest) If
$5000 is invested at 8% annual simple interest, how long does it take to double
to $10,000? Definition (Periodic Compounding Interest) If Example (Future Value for Compounding Periocially) Find
the future value if $3500 is invested for 6 years at 8% compounded quarterly. Example (Interest for Compounding Periocially) Find
the interest that will be earned if $5000 is invested for 3 years at 10% compounded
semiannually. Example (Principal for Compounding Periocially) What
present value amounts to $100,000 if it is invested for 10 years at 8% compounded
quarterly? Example (Doubling Time for Componding Periocially) How
long in years would $700 have to be invested at 11.9% compounded monthly to
have $1,400? Definition (Continuous Compounding Interest) If Example (Future Value for Compounding Continuously) What
lump sum do parents need to deposit in an account earning 9%, compounded continuously,
so that it will grow to $40,000 for their daughter's college tuition in 18 years? Example (Interest for Compounding Continuously) Which
investment will earn more money, a $1000 investment for 6 years at 8% componded
annually, or a $1000 investment for 6 years compounded continuously? Example (Principal for Compounding Continuously) What
present value needs to be deposited to have $20,000 in 3 years with an investment
that is compounded continuously at 4%? Example (Doubling Time for Compounding Continuously)
(a) How long in years would $700 have to be invested at 12.3%, componded
continuously, to have Definition (Annual Percentage Yield) If Example (Annual Percentage Yield) Suppose
there are three investements to invest in (a) one at 10% compounded
annually, (b) another at 9.8% compounded quarterly, and (c) a
third investment at 9.65% compounded continuously. Which investment is best? Example (Interest Problems) (a) What
is the present value of an investment at 6% annual simple interest if it is
worth $832 in 8 months? |
at
an interest rate 
per
period, the simple interest is given by the formula: 
and
the future value of the investment is 
and
since 
and 
we
have
where 
and 
so
we have
We
are looking for the principal, 
and 
We
use the formula 
and
we have 
and
solving for 
we
get
and
we are given a value of 
We
are asked to find 
when 
and 
We
have 
years.
dollars
is invested for 
years
at a nominal interest rate 
componded 
times
per year, then the total number of compounded periods is 
and
the interest rate per period is 
and
the future value is 
or 
where 
and 
so
we have 
where 
and 
so
we have
where
the future value 
and 
so
we have
and
can be found using the formula 
where 
and 
so
we have
years.
dollars
is invested for 
years
at an interest rate 
compounded
continuously, then the future value is given by 
where 
and 
and
so we have
where 
and 
which
is 
The
investment that is compounding continuously will have future value 
where 
and 
which
is 
Thus,
the investment which is compounding continuously is the better investment. 
where 
and 
and
so we have
and
is given by the formula 
where 
and 
and
so we have
years
and
principal 
where 
is
in years.
is
the present value, 
is
the interest rate, and 
is
the time in years, so we have
is
the number of compounding periods per year, then 
is
the interest rate per period and if 
is
the annual interest rate for an investment, then the annual percentage yield
is defined by the formula
and 
and
so will have A.P.Y. 
For
the second investment we have 
and 
and
so we have A.P.Y. 
For
the last investment we have A.P.Y. 
and
so the best investment is the second. 
where 
and 
and
so we have
where 
and 
and
so we have 
Thus
the interest earned is 
If
we use compounding quarterly then the future value is given by 
where 
and 
and
so we have future value of 
Thus
for compounding quarterly we have interest earned as 
Therefore,
the first investment is better by 
