INTEREST
By Richard Gooden
-Principal: The amount of money initially invested.
-Accumulated Value: Denoted by A(t), the
amount of
money that the principal has grown to after any given time
period.
– -Interest: the money that has
been gained on the
principal or the difference between the accumulated
value and the principal.
-Formula: Interest=Accumulated Value – Principal.
-i is defined as the effective rate of interest.
-We define effective rate of interest in the nth year as:
in= [A(n)-A(n-1)]/A(n-1).
-There are two types of interest:
(1) Simple Interest and (2) Compound Interest.
--Simple
Interest: 1 +it (where t>=0)
***This is just something to be aware of:
-Example- If we were to borrow 1000
dollars at fifteen percent simple interest
for 16 days, we
would assume exact
simple interest, which means at the end
of the 16 days, we would owe:
1000[1 + (16/365)(.15)]=1006.58.
-However, under the Banker’s rule, we
would owe:
1000[1 + (16/360)(.15)]=1006.67
because they use 360 days instead of
365 days.
-Compound Interest- This is when you earn interest on
interest. Your accumulated value is re-invested.
Formula: A(t)= (1 + i)t (where t >=0)
-Present Value and Future Value:
In life, compound interest is assumed
unless directly stated.
In mathematics, one dollar now is not
worth one
dollar in three years. Similarly, one
dollar now is not worth one dollar a year ago.
Example 1: How much is 2000 dollars
worth
in four years from now
assuming an effective interest rate of
three percent?
Solution: 2000(1 + .03)4=2251.02.
Example 2: How much is 2000 dollars
worth four years ago, assuming an
effective rate of interest of three percent?
Solution: 2000(1.03)-4=1776.97.
So as we can see from the two examples,
when moving forward in time, (1 + i) is
raised to the positive power of time.
However, when moving back in time, (1
+ i) is raised to
the negative power
of time.
-Nominal Rates:
Definition: It is an interest rate that is
convertible over a period other than a year.
i(m) is a nominal rate that is converted m times a
year, which would make the effective rate of
interest, i={[1 + (i(m)/m)]m-1}.
Example: If you have a credit card that
charges eighteen percent compounded
monthly, they are charging more than
eighteen percent, though the credit card
holder may not know it. Let’s see!
Solution: i(12)=.18, because it is eighteen
percent compounded monthly. So, when
we calculate the
effective rate of
interest, i, we have
[1 + (.18/12)]12-1=.1956 or 19.56% (not
18% as
stated).
Therefore a general rule is the more times
your rate is compounded, the more debt you will
be in at the end of the period.