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Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
2
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. Month days
1 January 31
2 February 28/29
3 March 31
4 April 30
5 May 31
6 June 30
7 July 31
8 August 31
9 September 30
10 October 31
11 November 30
12 December 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.
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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Solution : January 1
February
March
April 1
Time 3 months
I = P. r. t
0.612
308.0300 I $
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 t= 1/12 in the
next formula
I = P. r. t
0.50012
112.0500000 I $
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.
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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Solution :
Substituting p= 500, I= 12.50, and t= 2/12 in the
next formula
I = P. r. t
12
25005.12 r $
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 t= 6/12 in the
next formula
I = P. r. t
12
606.075.63 P
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
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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t 06.050000.50
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
0.20101.02000 I
The total payment for the first month is $120. and the new
unpaid balance is $ 1900.
For the second month the interest is
0.19101.01900 I
After 19 payments the debt is down to $100, and the interest
payment is
0.1101.0100 I
The interest payments are $20, $19, $18, …, $1.0
Total interest = 20 + 19 + 18 + … + 1
According to arithmetic progression
Sum = 0.210)10.20(2
20)(
21 naa
n$
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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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
25.2612/615.0350 I
S == P + I = 350.0 + 26.25 = $376.25
Another solution:
S = P ( 1 + r . t ) = 350 ( 1 + 0.15 * 6/12) = $376.25
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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Example 1 page 21: The current annual dividend rate )معدل الرباد ( of a savings and
loan association is 5.5%. dividend are credited to a persons
account on June 30 and December 31. Money put in the 10th
of
the month earns dividends for the entire month. If money is put
in after the 10th
. it starts earning dividends the following
month. A person opens an account on January 7 with 450$. 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
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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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 50.7360/6015.0300 I
Exact inertest 40.7365/6015.0300 I
Note that the ordinary interest is greater than exact intertest.
Exact and approximate time:
Exact time includes all days except the first
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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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 5- March 31- 5 = 26
4 April 30
5 May 31
6 June 30
7 July 31
8 August 31
9 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)الممارسات التجارية(
There are four ways to compute simple interest:
1- Ordinary interest and exact time ( BANKERS Rule's)
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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2- Exact interest and exact time
3- Ordinary interest and approximate
4- 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 30-15=15
December 31
January 1994 31
20- February 20
TOTAL 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
21.20360
9715.0500 I
2. Exact interest and exact time
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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93.19365
9715.0500 I
3.Ordinary interest and approximate
79.19360
9515.0500 I
4.Exact interest and approximate time
79.19365
9515.0500 I
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 :
Time Month days
1
yea r 1- March 1944 31
April1994 30
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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May1994 31
9 m
on
ths
1-June 1944 30
July1994 31
August1994 31
September1994 30
5 m
on
ths
1- October 1994 31
Novembe1994r 30
3 m
on
ths 1- December 1994 31
January 1955 30
February 1955 28
1- March 1995 30
Interest of each part of the loan is
45000115.0300000 I
2250012
915.0200000 I
12500012
515.0200000 I
375012
315.0100000 I
Total interest = $ 83750
Amount of loan = 800000 + 83750 = $883750
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 days
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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5 4-May 31-4=27
6 June 30
7 July 31
8 August 31
9 September 30
10 October 31
11 November 30
12 15-December 15
Total 225
Time at second period 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.
trSP
1
Example :
If money is worth 5 % , what is the present value of %105
due in 1 year?
Solution:
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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Substituting r = 0.05, S= 105 , P= ? , and t= 1 year in
the next formula
0.100105.01
1051
trSP
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 :
73.5046107.01
54001
trSP
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
Present value
of $5400 due
in 1 year
Better plan
7% $5046.73 Save $ 46.73 now by paying cash
8% $5000.00 Planes are equivalent
9% $4954.13 Save $45.87 now by paying $5400
Exercise 1c: 1-10 page 38
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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Present value of interest –bearing debt
Example : if we want to find the current value of an interest-
bearing 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 :
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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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%
48.1990)12/5(12.01
20901
trSP
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 :
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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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 :
73.437$24.25449.183
118.01300
12
618.01
200
x
Example 2:
Solve the preceding problem using 12 months hence as the
focal date.
We set the equation of value :
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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98.438$18.1
518
51830021818.1
300)12
618.01(200)118.01(
x
x
x
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 :
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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71.532$14.2
1140
14.21140$
035.1105.11140$
)12
314.01()
12
914.01()114.01(1000
x
x
xx
xx
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 :
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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24.1111$
200079979.1
2000869565.0930233.0
200015.1075.1
2000115.01
12
615.01
x
x
x
xx
xx
Example 5:
Work example 4 with the focal date on June 1, 1997.
Solution :
The equation of value is
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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43.1108$075.2
2300
2300075.2
2300075.1
)115.01(2000)2
115.01(
x
x
xx
xx
Example 6: A person borrowed $ 6000 on September 15, 1992
agreeing to pay $2000 on January 15, 1993, and $ 2000
on may 15, 1993. if the interest rate was 18% , how
much paid on September 15, 1993 to settle the debt ( put
the focal date September 15, 1993.
Solution :
The equation of value is
2720$212022407080
212022407080
)3
118.01(2000)
3
218.01(2000)118.01(6000
x
x
x
Exercise 1 d: 1-10
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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Partial payments
There are two common ways to allow interest credit
on short-term transactions: Merchant Rule and
United state rule.
Example 1:
A debt of $1000 is due in 1 year with interest at 15%
the debtor pays $300 in 4 months and $200 in 10
months. Find the balance due in1 year using
Merchant Rule and United state rule.
Solution:
a) By Merchant Rule
Put the focal date at the final settlement
1000(1+0.15*1)= 300(1+0.15*8/12)
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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+ 200(1+0.15*2/12) + x
X= 615$
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; $1000 on January
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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12, 1996. What is the balance due on march 10 ,1996,
by united state rule?
Solution:
I1 = 5000*0.16*25/360=55.56<2000
S1 = 5000+55.56-2000= 3055.56
I2 = 3055.56*0.16*133/360 = 180.62 >50
I2* = 3055.56*0.16*186/360= 252.56<1000
S2 = 3055.56+252.56-(1000+50)=2258.15$
I3 = 2258.15*0.16*58/360= 58.21
S3 = 2258.15+58.21=2316.36$
Mathematics of FinanceIslamic University of Gaza Dr. Nafez M. Barakat
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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)