Chapter 4
Mathematics of Finance
D. S. Malik
Creighton University, Omaha, NE
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Interest
Two types of Interests
Simple
Compound
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Simple Interest
Interest is …xed for the duration of the deposit or investment.
Example
Lisa wants to open a new account and deposit $1000. The bank agrees to
pay her the simple interest at the rate of 8% per year. Suppose no more
deposits are made in the account. Also suppose I denotes the interest
amount at the end of the period.
What is the interest amount at the end of one year? How much
money will Lisa have at the end of one year?
Because the interest is simple, the bank pays 8% of the amount in the
account. So the interest the bank pays is:
I = 1000 0.08 = 80.
The total amount at the end of one year is 1000 + 80 = $1080.
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Example
What is the interest amount at the end of …ve years? How much
money will Lisa have at the end of …ve years?
As in previous part, the interest is simple. Here we need to calculate
the interest for …ve years. Thus,
I = 1000 (0.08) 5 = 400.
The total interest at the end of …ve years is $400. The total amount
at the end of …ve years is $1000 + $400 = $1400.
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Remark
When the interest rate is speci…ed, it is usually speci…ed as an annual
interest rate, i.e., an interest rate per year.
For example, in the preceding example, the interest rate is 8% per
year.
Note that we multiplied 1000 with 0.08 rather than 8.
This is for the following reasons:
The interest rate is 8%. This is read as 8 percent, which means 8 for
every 100.
8 = 0.08 for every 1.
This is equivalent to 100
This can be interpreted as 0.08 of 1.
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Formula to Compute Simple Interest
Suppose that the amount P is invested at the simple interest rate of
r percent per year for t years.
Let I denote the total interest and A be the total amount after t
years. Then
I = Prt,
and
A = P + I = P + Prt = P (1 + rt ).
P is called the principal amount
r is called the interest rate per year or annual interest rate
t is the number of years the amount P is invested.
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Remark
In a problem, we typically specify interest rate as a percent, such as
7.25%.
During calculations, we convert a percent into a decimal number.
For example, 7.25% is converted into 0.0725.
We therefore say that as a percent the interest rate is 7.25 and as a
decimal number the interest rate is 0.0725.
When we give the description of a formula, we typically say that the
interest rate is r percent per year or r % per year.
When we write the formula, r is expressed as a decimal number.
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Example
An amount of $1500 is invested at the simple interest rate of 7.5%
annually.
1
What is the total amount of interest after 12 year? What is the total
money at the end of 12 year.
Here P = 1500, r = 0.075, and t = 12 . Let I denote the interest
amount and A be the total amount at the end of 12 year. Then
I = Prt = 1500 (0.075)
1
= 56.25.
2
Also,
A = P + Prt = 1500 + 56.25 = 1556.25.
Hence, the interest is $56.25 and the total amount is $1, 556.25.
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Example
2. What is the total amount of interest after 6 years? What is the total
money at the end of 6 years?
Here P = 1500, r = 0.075, and t = 6. Let I denote the interest
amount and A be the total amount at the end of 6 years. Then
I = Prt = 1500 (0.075) 6 = 675.
Also,
A = P + Prt = 1500 + 675 = 2175.
Hence, the interest is $675 and the total amount is $2, 175.
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Compound Interest
De…nition
Suppose that P dollars is invested at the interest rate of r percent per
year for t years and the interest is calculated m times a year.
Suppose that the interest is added to the balance at the end of each
period and the interest for the next period is calculated on the
balance at the end of the previous period.
In this case, we say that the interest is compounded periodically.
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Suppose that the interest is compounded m times a year. Then
If the interest is compounded monthly, then m = 12.
If the interest is compounded quarterly, then m = 4.
If the interest is compounded semiannually, then m = 2.
If the interest is compounded daily, then m = 365 or 366, depending
on whether the year is a leap year.
If the interest is compounded annually, then m = 1.
If the interest is compounded weekly, then m = 52.
If the interest is compounded biweekly (twice a month), then m = 26.
If the interest is compounded bimonthly (every other month), then
m = 6.
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In order to calculate the compound interest, the …rst thing that we
need to do is to …nd the interest rate per period.
Note that the interest rate r is for the year, not necessarily for each
period.
Suppose that i denotes the interest per period.
If the interest is calculated m times a year, then
i=
r
.
m
Moreover, if the interest is calculated m times a year for t years, then
the number of times the interest is calculated is mt.
Let n denote the number of interest periods, then
n = mt.
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Example
Suppose that the interest is compounded monthly at the rate of 8.4%
per year for 5 years.
Then r = 0.084, m = 12, and t = 5. Hence,
i=
r
0.084
=
= 0.007
m
12
and
n = mt = 12 5 = 60.
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Theorem
Suppose that P dollars is invested at the interest rate of r percent per year
for t years. Suppose that the interest is compounded m times a year and
A denotes the total amount at the end of t years. Then
A = P (1 + i )n ,
where
i = mr , is the interest per period, and
n = mt is the number of interest periods.
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Example
Suppose $1, 500 is deposited in an account that pays 6% interest per
year compounded quarterly.
Let us determine the total amount at the end of the sixth year.
Here P = 1500, r = 0.06, m = 4, and t = 6.
= 0.06
4 = 0.015 and n = mt = 4 6 = 24.
Therefore, the total amount, A, at the end of the sixth year is
Thus, i =
r
m
A = P (1 + i )n = 1500(1 + 0.015)24 = 1500 1.01524 = 2144.25.
Hence, the total amount is $2, 144.25.
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Example
Ravi won a jackpot worth $100, 000. He wants to buy a home after 8 years
and would like to make a down payment of $80, 000. His bank o¤ers an
account paying 7.25% interest per year compounded monthly. Ravi wants
to know how much money, out of $100, 000, should be deposited in that
account so that he will have $80, 000 at the end of 8 years.
Note that Ravi wants $80, 000 at the end of 8 years. That is, the total
amount, A, to be accumulated is given. So we want to determine the
amount, P, that is needed to be deposited now.
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Example
Here A = 80, 000, r = 0.0725, m = 12, and t = 8. Thus, i =
and n = mt = 12 8 = 96. We want to …nd P. Now
r
m
=
0.0725
12
A = P (1 + i )n .
This implies that
P=
80000
A
=
n
(1 + i )
1 + 0.0725
12
96
= 44870.10.
Hence, Ravi needs to deposit $44, 870.10 in the account to accumulate
$80, 000 at the end of 8 years.
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De…nition
(Present Value) The present value of an amount A which is to be paid
or accumulated after n periods of time at the interest rate of i per period
is the principal amount, P, which, when invested at the interest rate of i
per period for n periods, grows to A.
Theorem
Let P be the present value of an amount A which is to be paid or
accumulated after n periods of time at the interest rate of i per period.
Then
A
.
P=
(1 + i )n
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E¤ective Rate (Annual Percent Yield)
De…nition
Suppose that the interest is r percent per year compounded m times
a year.
Let Ir denote the interest earned at the end of the year with this
interest rate.
Suppose that the interest rate is re percent per year compounded
annually and let Ie be the interest earned at the end of the year with
this interest rate.
Then re is called the e¤ective rate or annual percent yield
corresponding to the interest rate r if
Ir = Ie .
Also, r is called the nominal rate.
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Theorem
( E¤ective Rate or Annual Percent Yield)
(i) The e¤ective rate or the annual percent yield, re , corresponding to a
given interest rate, r , compounded m periods per year is given by
re = 1 +
r
m
m
1.
(ii) Suppose that P dollars increases to an amount of A dollars in t years.
Then the e¤ective rate or annual percent yield re is
re =
A
P
1
t
1.
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Example
We …nd the e¤ective rate for an account in which the interest is
compounded monthly at the rate of 7.5% per year.
Here r = 0.075 and m = 12. Thus,
r
re = 1 +
m
m
1=
0.075
1+
12
12
1 = 0.07763.
Hence, the e¤ective rate is 7.763%.
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Exercise: What is the e¤ective rate for an account in which interest is
compounded quarterly at the rate of 8% per year?
Solution: Here r = 0.08 and m = 4. Thus,
re = 1 +
r
m
m
1=
1+
0.08
4
4
1
0.0824.
Hence, the e¤ective rate is 8.24%.
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