Islamic University of Gaza
Dr. Nafez M. Barakat
Chapter 1
Simple Interest
1
Mathematics of Finance
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
Simple interest is defined as the product of principal, rate, and time.
This definition leads to the simple interest formula.
I = P. r. t
I : simple interest in (dollars) or other monetary unit)
P: principal in dollars
r: interest rate
t: time in units that correspond to the rate
table 1
No.
1
2
3
4
5
6
7
8
9
10
11
12
Month
days
January
February
March
April
May
June
July
August
September
October
November
December
31
28/29
31
30
31
30
31
31
30
31
30
31
Example (1)
A bank pays 8% per annum on saving accounts. A
person opens an account with a deposit of $300 on
January 1. how much interest will the person receive
on April 1.
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Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
Solution :
January
February
March
April
Time
1
1
3 months
I = P. r. t
I  300  0.08 
3
 6.0 $
12
Example (2) a couple buys a home and gets a loan for
$ 50000. the annual interest rate is 12%. The term of
the loan is 30 years, and the monthly payments is
$514.31. find the interest for the first month and the
amount of the house purchased with the first
payment.
Solution :
Substituting p= 50000, r= 0.12, and
next formula
I = P. r. t
I  500000  0.12 
t= 1/12 in the
1
 500.0 $
12
So the 514.31 payment buys only 514.31-500.0 = $14.31
worth of house
Example (3) the interest paid on a loan of $500.0 for 2
months was $ 12.5. what was the interest rate.
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Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
Solution :
Substituting p= 500, I= 12.50, and
next formula
I = P. r. t
12.5  500  r 
2
12
t= 2/12 in the
$
r = 15.0%
Example (4)
A person gets $ 63.75 every 6 months from an
investment that pays 6% interest. How much money is
invested?
Solution:
Substituting r= 0.06, I= 63.75, and
next formula
t= 6/12 in the
I = P. r. t
63.75  P  0.06 
6
12
P= $2125.0
Example (5)
How long will it take $ 5000 to earn $ 50 interest at 6%.
Solution:
Substituting r= 0.06, I= $50, P= $5000 , and t= ? in
the next formula
I = P. r. t
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Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
50.0  5000  0.06  t
t= 1/6 years or 2 months
example (6)
a woman borrows $ 2000 from a credit union. Each month she is
to pay $ 100 on the principal. She also pays interest at rate of
1% a month on the unpaid balance at the beginning of the
month. Find the total interest.
Solution : note that the rate is monthly rate.
The first month's interest is
I  2000  0.01  1  20.0
The total payment for the first month is $120. and the new
unpaid balance is $ 1900.
For the second month the interest is
I  1900  0.01  1  19.0
After 19 payments the debt is down to $100, and the interest
payment is
I  100  0.01  1 1.0
The interest payments are $20, $19, $18, …, $1.0
Total interest = 20 + 19 + 18 + … + 1
According to arithmetic progression
Sum =
n
20
( a1  a n ) 
(20.0  1)  210.0 $
2
2
5
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
Amount :
The sum of the principal and the interest is called the amount,
designated by the symbols S.
S= P + I
= P + P. r . t = P ( 1 + r . t )
Example
A man borrows $ 350 for 6 months at 15%. What a
mount must he repay?
Solution :
Substituting r= 0.15, I= ?, P= $350 , and t= 6/12 in
the next formula
I = P. r. t
I  350  0.15  6 / 12  26.25
S == P + I = 350.0 + 26.25 = $376.25
Another solution:
S = P ( 1 + r . t ) = 350 ( 1 + 0.15 * 6/12) = $376.25
6
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
Example 1 page 21:
The current annual dividend rate )‫ ( معدل الرباد‬of a savings and
loan association is 5.5%. dividend are credited to a persons
account on June 30 and December 31. Money put in the 10 th of
the month earns dividends for the entire month. If money is put
in after the 10th . it starts earning dividends the following
month. A person opens an account on January 7 with 450$. On
February 25, $300 is added, and on June 10, $ 240 is placed in
the association. What is the amount in the account on June 30?
Solution:
S = 400 (1+0.055*6/12) + 300 (1+0.055*4/12)
+ 240 ( 1+.005*1/*12) = $957
Exercise 1 a: page(25)
1, 5,7, 8, 9, 10, 11,12, 13, 14, 15,16, 17, 18, 19 , 20,21, 22
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Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
Exact and ordinary interest
When the time is in days and the rate is an annual rate , it is
necessary to convert the dayes to a fractional part of a year when
substituting in the simple interest formulas
 Interest computed using a divisor 360 is called ordinary
interest.
 Interest computed using a divisor 365 or 366 is called
exact interest.
Example: figure the ordinary and exact interest on a 60 days
loan of $ 300 if the rate is 15%.
Solution:
Substituting r= 0.15, I= ?, P= $300 , and t= 60.0 days
in the next formula
I = P. r. t
Ordinary inertest I  300 0.15  60 / 360  7.50
Exact inertest I  300  0.15  60 / 365  7.40
Note that the ordinary interest is greater than exact intertest.
Exact and approximate time:
Exact time includes all days except the first
8
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
Approximate time based on the assumption that all the full
months contain 30 days
Example:
Find the exact and approximate time between 5 March and 28
September .
Exact time :
3
4
5
6
7
8
9
5- March
31- 5 = 26
April
30
May
31
June
30
July
31
August
31
28 -September
28
Total
207 days
Approximate time : we count the number of months from 5
March to 5 September which is equal 6 months, and equal 6 *
30 = 180 days, and we add the 23 days from September 5 to
September 28, so the total approximate time equal 203 days.
Commercial Practice)‫التجارية‬
There are four ways to compute simple interest:
1-
Ordinary interest and exact time ( BANKERS Rule's)
9
‫سات‬
Islamic University of Gaza
234-
Dr. Nafez M. Barakat
Mathematics of Finance
Exact interest and exact time
Ordinary interest and approximate
Exact interest and approximate time
Example:
On November 15 , 1993, a woman borrowed $500 at 15 %. The
debt is repaid on February20, 1994. find the simple interest
using the four methods.
Solution :
First we get the exact and the approximate time:
Exact time
Month
days
15 -November
December
January 1994
20- February
TOTAL
30-15=15
31
31
20
97 DAYS
Approximate time :
From 15 November to 15 February
there is three months which is equal
90 days.
And from 15 to 20 February there is
5 days
Total time ………………………….. 95 days
1. Ordinary interest and exact time ( BANKERS Rule's)
I = P. r. t
97
 20.21
360
Exact interest and exact time
I  500  0.15 
2.
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Islamic University of Gaza
I  500  0.15 
Dr. Nafez M. Barakat
Mathematics of Finance
97
19.93
365
3.Ordinary interest and approximate
I  500  0.15 
95
19.79
360
4.Exact interest and approximate time
95
19.79
365
I  500  0.15 
Example :
The builder of an apartment building obtained an $800000
construction loan at an annual rate of 15%. The money was
advanced as follows:
March 1, 1994
$ 300000
June 1, 1994
$ 200000
October 1, 1994 $ 200000
December 1, 1994 $ 100000
The building was completed in February of 1995 and the loan
repaid on march 1, 1995. find the amount using ordinary interest
and approximate time.
Solution :
1
yea
r
Time
Month
days
1- March 1944
April1994
31
30
11
9 months
5 months
3 months
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
May1994
1-June 1944
July1994
August1994
September1994
1- October 1994
Novembe1994r
1- December 1994
January 1955
February 1955
1- March 1995
31
30
31
31
30
31
30
31
30
28
30
Interest of each part of the loan is
I  300000 0.15  1  45000
9
 22500
12
5
I  200000  0.15 
125000
12
3
I  100000  0.15 
 3750
12
Total interest
= $ 83750
Amount of loan = 800000 + 83750 = $883750
I  200000  0.15 
Exercise 1b (q12. page 34)
On may 4,1991, a person borrows $1850 and promises to repay
the debits in 120 day’s with interest at 12%. If the loan is not
paid on time the contract requires the borrower to pay 10% on
the unpaid amount for the time after the due date. Determine
how much this person must pay to settle the debt on December
15, 1991.
Solution :
No.
Month
12
days
Islamic University of Gaza
5
6
7
8
9
10
11
12
Dr. Nafez M. Barakat
4-May
June
July
August
September
October
November
15-December
Total
Time at second period
Mathematics of Finance
31-4=27
30
31
31
30
31
30
15
225
225-120=105
days
S1 = P(1 + r . t ) = 1850 ( 1 + 0.12 * 120/360) = $1924
S2 = P(1 + r . t ) = 1924 ( 1 + 0.10 * 105/360) = $1980
H.W: 1-14 page 34
Present value at simple interest
If we know the amount and we want to obtain the principal,
we solve the formula for P.
P 
S
1 r t
Example :
If money is worth 5 % , what is the present value of %105
due in 1 year?
Solution:
13
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
Substituting r = 0.05, S= 105 , P= ? , and t= 1 year in
the next formula
P 
S
1 r t

105
100.0
1  0.05 1
Example:
A person can buy a piece of property for $5000
cash or $54000 in a year. the prospective buyer has
cash and invest it in at 7%. Which method of
payment is better and by how much now?
Solution :
P 
S
1 r t
5400

 5046.73
1  0.07  1
This mean that the buyer would have to invest $5046.73 now
at 7% to have $5400 in a year.
So by paying cash the buyer save $46.73 now.
If another rate of return on the money was available, the
decision might be different. For example:
Rate of
return
7%
8%
9%
Present value
of $5400 due
Better plan
in 1 year
$5046.73
Save $ 46.73 now by paying cash
$5000.00
Planes are equivalent
$4954.13
Save $45.87 now by paying $5400
Exercise 1c: 1-10 page 38
14
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
Present value of interest –bearing debt
Example :
if we want to find the current value of an interestbearing debt in the future, we must find the maturity
value of the debt, using the stated interest rate for the
term of the ;loan. Then we compute the present value
of this maturity value for the time between the day it
is discounted and the due date.
Example :
A debtor signs a note for $2000 due in 6 months with
interest at 9%. One month after the debt is
contracted, the holder of the note sells it to a thirty
party, who determine the present value at 12%. How
much is received for the note?
Solution :
15
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
Step 1: the maturity value is obtained at 9%
S = P ( 1 + r . t ) = 2000 ( 1 + 0.09 * 6/12) = $2090
Step 2: the maturity value is discounted for 5 months at 12%
P 
S
1 r t

2090
1990.48
1  0.12  ( 5 / 12)
Equations of value
There are two ways to move the money backward and
forward, lock at any time diagram. If a sum is to be moved
forward use an amount formula, and if backward use a
present value formula.
Note : always bring obligations to the same point using the
specified rate before combining them. This common point is
called a focal date.
Example :
16
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
A person owes $200 due in 6 months and $300 due in 1 year.
The creditor will accept a cash settlement of both debts
using a simple interest rate of 18% and putting the focal
date now. Determine the size of the cash settlement.
Solution :
We set the equation of value :
x 
200
1  0.18 
6
12
 1  0300
.18  1
 183.49  254.24  $437.73
Example 2:
Solve the preceding problem using 12 months hence as the
focal date.
We set the equation of value :
17
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
6
)  300
12
1.18 x  218  300  518
x (1  0.18  1)  200 (1  0.18 
x
518
 $438.98
1.18
Example 3:
A person owes $ 1000 due in 1 year with interest at 14% .
two equal payments in 3 and 9 months, respectively, will be
used to discharge this obligation. What will be the size of
these payments if the person and the creditor agree to use an
interest rate of 14% and a focal date 1 year hence.
Solution :
We set the equation of value :
18
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
Example 4:
A person borrows $ 2000 at 15% interest on June 1,
1996. the debt will be repaid with two equal payments,
one on December 1 , 1996 and the other on June 1, 1997.
put the focal date on June 1, 1996 and find the size of the
payments
We set the equation of value :
19
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
+ 200(1+0.15*2/12) + x
X= 615$
b) by United state rule
I1 = 1000*0.05*4/12 = 50 < 300
S1 = 1000+50-300 = 750$
I2 = 750*0.05*6/12 = 56.25 < 200
S2 = 750+56.25-200 = 606.25
I3 = 606.25*0.15* 2/12 = 15.18
S3 = 606.25+15.18 = 621.41$
Example2 :
On June 15, 1995 a Pearson borrows $500 at 165.
Payments are made as follows : $2000 on July 10,
1995, $50 on November 201995; $1000 on January
23
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
12, 1996. What is the balance due on march 10 ,1996,
by united state rule?
Solution:
I1 = 5000*0.16*25/360=55.56<2000
S1 = 5000+55.56-2000= 3055.56
I2 = 3055.56*0.16*133/360 = 180.62 >50
I2* = 3055.56*0.16*186/360= 252.56<1000
S2 = 3055.56+252.56-(1000+50)=2258.15$
I3 = 2258.15*0.16*58/360= 58.21
S3 = 2258.15+58.21=2316.36$
24
Islamic University of Gaza
Dr. Nafez M. Barakat
Mathematics of Finance
Example3:
A couple gets an $80000, 30-year, 12$ loan. The
monthly payments is $822.90. how much of the first
two payments goes to interest and how much to
principle?
Solution:
I= 80000*0.12*1/12 = $800.0
Payments to principal = $822.90 -$800.0 = $22.90
Balance at the end of the first month = 80000-22.9 =
$79977.10
I2 = 79977.1*0.12*1/12 = $799.7
Payments to principal = 822.9-799.77 = $23.13
Total interest at the first two months = 800 + 799.7 =
$1599.7 and (22.9+23.13 = 46.03) goes to principal.
WH : Page 60-61
: Problems (1-10)
25