Survey of Math - MAT 140
Copyright 2007 by Tom Killoran
Page: 1
Compound Interest
Compound interest is how most things work with bank today (at least for savings and most loans,sortof). The good
thing about compound is that savings increase with interest much more quickly. The bad thing is the banks can reap
the same bene ts from a loan to us.
The idea behind compound interest is that the interest is computed at intervals throughout the life of the loan and
added to the principal. In this way we will be gaining interest on the interest, like putting grains of sand on top of one
another, it will eventually amount to a beach!
A new formula comes in handy here. If we just want the nal balance of the account after a period of time we get:
A D P C Pr t
A D P .1 C r t/
Thus if we have $150 in an account that earns simple interest for one month at 8% then:
A D 150 1 C 0:08
1
12
D $151: 00
this account would be worth $1 more at the end of the month.
Example 1 What is the compound interest earned on $P if the interest is computed monthly at r%
APR over the rst 4 months?
r
will equal how many months the money stays in
By the third month the pattern is apparent. The power on 1 C
12
the bank
To simplify this result we need to convert this to years. To convert from months to years we need to divide the number
of months in a year, 12. But this changes the value of the exponent. To balance this out we need to also multiply the
power by 12
4
r 12 12
A D P 1C
12
Survey of Math - MAT 140
Copyright 2007 by Tom Killoran
Page: 2
Replacing the pieces with "letter representations" , and letting t D year s then
A D P 1C
r
n
nt
where n is the number of computations for interest in a given year.
Compounded
yearly
semi anually
quar terly
monthly
weekly
daily
n=
1
2
4
12
52
365
Example 2 Mary deposited $550 into an account that pays 7% interest compounded quarterly. How much is the
account worth after 8 years?
Since this is compounded Quarterly, we know that n D 4 (four times a year)
A D P 1C
r
n
A D 550 1 C
nt
:07
4
4.8/
Calculator
550 .1 C :07=4/^.4 8/
A D $958: 22
She earned
958:22
550 D $408: 22 interest
Example 3 APY (Annual Percentage Yield). This is used by banks to advertise the one year effect of compound
interest, in terms of simple interest. Thus if a Bank has an account that earns 5.5% interest compounded monthly,
then if $1 is deposited for a year we would have:
A D1 1C
:055
12
12.1/
D 1: 056 4
In simple interest terms, we earned 5.64 cents for our one dollar, in other words 5.64% simple interest. Thus this
accounts APY would be given by:
5:64%A PY
Just remember that this gives the effects of compounding in simple interest terms for ONLY the rst year.
Survey of Math - MAT 140
Copyright 2007 by Tom Killoran
Page: 3
Example 4 Amy received a inheritance from a long lost uncle. He left her his entire savings account that he started
35 years ago, in an account that has earned 6.5% interest compounded daily. The account had a balance of $32,500
when her uncle died. How much did he start the account with?
***careful, here we know the ending balance, not the principal!!!
r nt
n
:065
P 1C
365
P .9: 726 /
P .9: 726 /
9:726
P
A D P 1C
32500 D
32500 D
32500
D
9:726
$3341: 56 D
365.35/
So he started the account with $3341.56, 35 years ago and it grew with just accumulating interest to $32,500.
1
Compound Interest Compared to Simple Interest
A little note on the power of compound interest.
Account A (blue, lower): Is a 6% simple interest loan on $100 for 18 years
Account B (red, upper): Is a 6% compounded monthly interest on $100 for 18 years
350
$A
300
250
200
150
100
50
-2
Years
2 4 6 8 10 12 14 16 18
Notice that the red graph (compound interest) matches the simple interest for at least the rst 2 years but distinctly
after year 6 the compounded interest becomes signi cantly more than the simple interest. And after 18 years it is
almost $100 in the lead. What happens is the graph keeps growing in a constantly increasing rate.
2
NOT IN TEXT BOOK, Systematic Savings
Most people do not save money in this way, it is usually done on a Monthly Deposit scheme. They decide on an
amount that they are willing to put away month to month for a certain time period. This is called systematic savings.
Survey of Math - MAT 140
Copyright 2007 by Tom Killoran
Page: 4
Now this is just a complicated compound interest problem. Lets say that you deposit $200 a month into an account
that earns 3% interest compounded monthly and you want to calculate the value of the account after 3 deposits.
Thus we deposited $600 by the end of the third cycle (or $800 by the beginning of the fourth) and earned $3.25 in
interest. The rst deposit has gained 3 months of interest, the second only received 2 months and the third on only
received 1 month.
There has to be an easier way to do this.... After a lot of mathematical searching and manipulation of the compounded
compound interest we (that is I can) get:
FD
12D
r
1C
r
12
12t
1
Where F is the Future Value of this Account
D is the constant monthly deposit
r is the interest rate (compounded monthly)
t is the years the savings is going to happen
Example 5 Misty is going to save $125 per month for 18 years, to save for her new babies college education. Her
bank has a education account that earns 6.5% interest compounded monthly. How much will she have for her babies
education fund at the end of 18 years?
F
D
r
t
12D
r
D?
D $125
D 0:065
D 18
r 12t
1C
1
12
"
#
12 .125/
:065 12.18/
D
1C
1
:065
12
#
"
12 .125/
:065 216
D
1C
1
:065
12
FD
Survey of Math - MAT 140
Copyright 2007 by Tom Killoran
calculator
12 125 :065 ..1 C :065=12/^216
F D $51; 04 2:37
Page: 5
1/
Now Misty deposited $125, 12 times a year, for 18 years
Deposits D 125 12 18
D 27 000
The difference between this amount and the future value will be the interest that was earned over 18 years
51; 04 2:37
27 000 D $24; 042:37 interest earned!!!!!