Compound
Interest
With simple
interest, the amount of interest is found once. In many investments, loans, and savings accounts, the amount
of interest is found every year, month, day, or some other period of time. If the amount of interest is found
periodically, this is compound interest. Each time the interest is calculated is called a compounding. The formula for compound interest is:
A = P*(1
+ r ¸
n)nt
Where A =
total amount at the end of the term of an investment or total amount to be paid
back for a loan (Principal + interest), P = Principal, r = annual rate, t =
time in years, and n = the number of periods or compoundings in a year.
Some common
values for n are:
|
n
= 1
|
Compounded
annually
|
|
n
= 4
|
Compounded
quarterly
|
|
n
= 12
|
Compounded
monthly
|
|
n
= 52
|
Compounded
weekly
|
|
n
= 365
|
Compounded
daily
|
- What is the total amount
if $500 is invested for 4 years at 5% compounded monthly?
Here,
P = $500, r = .05, n = 12, and t = 4
A = P*(1 + r ¸ n)nt
A = $500*(1 + .05 ¸ 12)(12*4)
A = $610.44767…
A» $610.45 (rounded)
Thus the
total amount is $610.45.Notice that this includes the $500 principal. The interest made is $610.45 - $500.00 = $110.45.
- What
if the previous example was compounded daily?
The change
here is that n = 365.
A = P*(1 + r ¸ n)nt
A = $500*(1 + .05 ¸ 365)(365*4)
A = $610.69301…
A» $610.69 (rounded)
Notice that the
total amount here is 24 cents higher than in the previous example. The more often the compounding, the
more money there is in interest. It
may not be a lot, but it is more.
- What
if the previous example had a term of 30 months
Remember
that in this formula the time is expressed in years.
A = P*(1 + r ¸ n)nt
A = $500*(1 + .05 ¸ 365)(365*30/12)
A = $566.56937…
A» $566.57 (rounded)
- What
is the amount of interest if $2500 is invested for 3 years at 10%
compounded weekly?
Here, P =
$2500, r = .10, n = 52, and t = 3
A = P*(1 + r ¸ n)nt
A = $2500*(1 + .10 ¸ 52)(52*3)
A = $3373.67494…
A» $3373.67 (rounded)
I = $3373.67 - $2500 = $873.67
Using a little algebra, one can also
find the principal needed to obtain an amount in the future. One must solve for P by dividing to
obtain
A = P*(1 + r ¸ n)nt
divide both sides by (1 +
r ¸
n)nt
P = A ¸(1
+ r ¸
n)nt
- How
much money do you need to put in a 4% compounded monthly savings account
to have $5000 in 15 years?
In this
problem we are looking for the Principal (the initial amount).
A = P*(1 + r ¸ n)nt
$5000 = P*(1 + .04 ¸ 12)(12*15)
P = $5000 ¸(1 + .04 ¸ 12)(12*15)
P = $2746.79752…
P» $2746.80 (rounded)
Thus if you
invested $2746.80 in a savings account with these conditions now, you would have
$5000 in 15 years.
There is one other type of
compounding, continuous compounding, which means that instead of
compounding every day, month, etc, compounding is done every instant. The formula for continuous compounding
is:
A = P*ert
where e =
2.71828…. The number e is found in most calculators. Here is the
first example with a slight change.
- What is the total amount
if $500 is invested for 4 years at 5% compounded continuously?
A = P*ert
A = $500*e(.05*4)
A = $610.70137…
A» $610.70 (rounded)
Notice this
total is only 1¢ higher than the compounded daily example
(example 2).This is probably why continuous compounding is not very widely used.
Exercises
- What
is the interest on a $650 compound interest loan with a 10% compounded
quarterly rate for 18 months? (Round answer to whole cents)
- What
is the total amount made on a $1000 compound interest investment if the
rate is 8% compounded monthly and the term is 3 years? (Round answer to
whole cents)
- What
is the Principal of a 2-year loan if the total amount paid is $6000 with a
rate of 6% compounded daily? (Round answer to whole cents)
- How
much money is needed now if one wants $25,000 in 10 years and an 8%
compounded weekly rate is found? (Round answer to whole cents)
- What
is the interest on a $4500 compounded continuously investment if the rate
is 9% and the term is 7 years? (Round answer to whole cents)
- What
would the answer to #5 be if the compounding was daily?
Solutions:
- $103.80
- $1270.24
- $5321.58
- $11240.13
- $8449.25
- $8448.59
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