LESSON 6 & 7
COMPOUND INTEREST
ANNUITIES, SINKING FUNDS
ANDMATHEMATICS OF BUYING
Learning Outcomes

By the end of this lesson, should be able
to:
◦ Understand the definition of compound
interest.
◦ Identify interest rate per compounding period
and number of compounding periods.
◦ Calculate the future value and the compound
interest using the Compound Interest Table
and formula.
Learning Outcomes

By the end of this lesson, students should be able to:
◦ Calculate the present value of an ordinary annuity
Calculate sinking funds
Know how an invoice looks like.
Calculate single trade discounts and net cost
Calculate series of trade discounts and net cost
Express a series discount as an equivalent single
discount
◦ Find the list price
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Learning Outcome (contd.)
◦ Understand cash discount
◦ Determine the cash discount due dates
◦ Calculate cash discount using ordinary dating
method
◦ Calculate cash discount using post-dated (AS
OF) method
◦ Calculate cash discounts using end-of-month
(EOM) dating method
◦ Calculate cash discount using receipt-of-goods
(ROG) dating method
Learning Objectives ()
◦ Calculate the present value of a compound
interest using Compound Interest Table and
formula.
◦ Understand the underlying hukum for
compound interest in Islam
◦ Calculate the future value of an ordinary
annuity
◦ Calculate the future value of an annuity due
Compound interest
Compound interest is interest calculated on
any interest previously credited to an account in
addition to the original principal.
 It is the amount of interest earned on principal
and on previously earned interest that has been
added to the principal.
 It is used in many long-term investments, such as
savings accounts or certificates of deposit at a
bank. The amount an investment will be worth at
the end of some future period of time when
interest is compounded is called the future
value, the compounded amount or the
maturity value.

Identifying interest rate per
compounding period and number of
compounding periods
For compound interest, interest is usually
compounded daily, monthly, quarterly,
semi-annually, or yearly. The number of
interest payment period in each year for
each method is shown in the next slide in
table form.
 The formulas involved in calculating the
interest rate per compounding period and
number of compounding period per year
include: i =R/k and n=kT

Contd…
i – Interest per compounding period
 R – Annual interest rate
 k – No. of compounding period per year
 T – Time in years
 n – Total number of compounding periods

Interest Compounded
Compound at the
End of Every
Number of Compounding Periods in 1 Year,
k
Annually
12 months
Once a year
Semi-annually
6 months
2 times a year
Quarterly
3 months
4 times a year
Monthly
1 month
12 times a year
Daily
1 day
365 times a year
Future value of a compound
interest
Maturity value is the future value of the
investment, or the amount an investment
will be worth at the end of future period
of time when interest is compounded.
 There are 2 ways of calculating future
value or maturity value and compound
interest (1) by using the formula (2) by
using the Compound Interest Table.

Maturity value
Interest
M  P1  i 
n
I M P
M – Maturity value
P – Principal
i – Interest rate per compounding period.
n – Total number of compounding periods
I – Interest
M – Maturity
P – Principal
Finding Future Value and Compound Interest using the
Compound Interest Table

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To calculate the compound amount with a future value table,
the following steps are involved:
Find the period interest rate, i =R/k
Find the number of compounding period, n=kT
Find the factor of the future value of RM 1 at the
intersection of the period interest rate and the number of
periods (i,n) in the Compound Interest table.
Multiply the factor by the principal amount. This will give the
compound amount, or the future value, or the maturity
value.
Maturity value = Principal x factor from the Compound
Interest table.
To find the amount of interest earned: l =M-P
Present value of a compound
interest
Present value
It is the amount of money to be deposited
today to produce the needed amount in the
future at a specified date.
 For example, a manufacturing company
would want to know the amount of money
to keep aside or invest today in order to
have enough money to buy new equipments
in 3 years.
 2 ways are involve in calculating the present
value (1) by using the formula (2)the Present
Value of a RM 1 Table


Finding Present Value using the Formula
,
PV 
M
1  i n
where
PV – Present Value
M – Maturity value / Future value
i – Interest rate per compounding
period
n – Total number of compounding
periods
i
R
, n  kT
k
where
R – Interest rate
k – Number of compounding
period per year
T – Time in years
I M P
where
I – Interest,
M – Maturity value
P - Principal
Finding Present Value using the Present Value of a RM 1
Table
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To calculate the present value with a present value
table, the following steps are involved:
Find the period interest rate, i =R/k
Find the number of compounding period, n= kT
Look at the intersection of the two columns (i, n) in
the present value table to get the factor for the
compound amount of RM 1.
Multiply the factor by the future value amount. This
will give the present value.
PV = M x number from Present Value of a dollar Table
To find the interest earned, I = Future Value – Present
Value.
Compound interest in Islam
During the pre-Islamic era, when interest is
considered normal, a person who fails to pay back
the principal and interest charged on him, the
lender extends the loan on the condition that the
interest will also become part of the loan
(essentially Compound Interest).
 Quran verses were revealed in order to stop the
people from such practices:


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"O believers, take not doubled and redoubled
interest, and fear God so that you may
prosper." (Quran 3:130-1)
Annuity payment
An annuity is a series of equal payments made
or received at regular time intervals. Examples
of annuities are monthly payments of housing
loans, car loans, and insurance.
 Each payment earns compound interest until
the end of the term of the annuity. The total
amount in the account at the end of the term is
called the amount of the annuity or future
value of the annuity.

Contd…
The time between payments is called payment
period and the time from the beginning of the
first payment period through the end of the last
payment period is the term of the annuity.
 There are 2 types of annuity:
 Ordinary annuity: Investments or payments
that are made at the end of each period. For
example: salary.
 Annuity due: Investments or payments that
are made at the beginning of each period. For
example: life insurance, house rent.

Ordinary annuity
(Using formula Approach)
Table Approach
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To calculate the future value of an ordinary annuity
using Amount of an Annuity Table, use the following
steps:
Find the period interest rate.
Find the number of compounding period.
Look at the intersection of the two columns in the
Amount of an Annuity Table to get the factor for future
value of an annuity.
Multiply the factor by the amount of annuity payment.
This will give the amount of an annuity, or the future
value of the annuity.
To get the interest earned:
interest = Amount – (n x payment)
Main differences

The main difference between an ordinary
annuity and annuity due is that an
ordinary annuity will not earn interest in
the first period because the investment is
made at the end of the period. An annuity
due earns interest in the first period, and
that interest will be compounded over
the entire length of the investment
period.
Present value of an ordinary annuity
The present value of an annuity finds the value of a
lump sum amount that must be deposited today for it
to grow to a desired value at a specific date in the
future.
 Suppose a firm needs RM 250,000 at a specific date in
the future. The firm can achieve that goal either by
making periodic payments into an account for several
years, or depositing lump sum into an account and
letting the funds grows. The lump sum that can be
deposited today is the present value of the annuity
involving the periodic payments.
 There are 2 methods to calculate the present value of
an ordinary annuity

Using formula
where;
1 1  i  
PVA  Payment  

i


n
– Present value of an annuity
Payment – the amount paid
i – Interest rate per compounding period
n – number of compounding period
Using present value of an annuity table

The steps involved to find the present value
from an annuity table are as follows:
 Find the period interest rate, i =R/k and the number of
compounding periods, n =kT
 Find the intersection of the interest rate per
compounding period and the number of compounding
periods per year.
 Calculate the Present Value of an Annuity using the
following formula.

PVA = Payment x number from
Present Value of an Annuity table.
Find the interest earned : I = Amount - (n x payment)
Sinking funds
A sinking fund is a fund set up to receive
periodic investments to achieve a specific
future value at a specific future date.
 A sinking fund table is used to determine
the periodic amount to be invested on a
regular basis to accumulate a known, lump
sum future value.
 There are 2 ways of calculating sinking
fund

Using Formula
1. Formula to calculate sinking funds.
where;
2. Formula to calculate Interest.
SF – Sinking Funds
M – Maturity value
SF  M 
i
1  i n
1
I  M  n  payment 
i – interest per compounding period
n – number of compounding periods
Using Table
The steps you need to follow are:
 Find the period interest rate and number
of compounding periods per year.
 Find the intersection of the periodic
payment and the number of periods per
year.
 Find the periodic amount to be invested.

SF = M x Number of sinking fund table

Calculate interest earned:
I = M - (n x payment)
Invoice
An invoice is a printed document for
record the transaction between seller and
buyer. For seller, it is a sales invoice, and
for buyers, it is a purchase invoice.
 The invoice identifies the seller and buyer,
describes the items purchased, quantity
purchased, unit list price, discount terms,
and shipping and insurance charges.

Trade discounts and net
cost

Trade discount is a business term used for
slashing off the original selling price (list price)
of an item and not related to early payment,
resulting in net cost or net price.

Formulas to find the trade discounts and net
cost are as follows:
 Trade discount = List price x Trade discount
rate
 Net Cost/Price = List Price – Trade Discount

Cash discounts
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Business sellers often grant time to business customers
(example 30 days) to pay for the merchandise
purchased.
Sellers encourage the buyers to pay promptly as they
require the cash for their operations. Therefore, cash
discount is offered by sellers to encourage prompt
payment of bills by customers.
Cash discount is shown in the invoice as part of the
terms of sales. Thus, it is important to understand the
method to calculate cash discounts.
Methods of computing cash
discounts
Ordinary dating method
This method is the most commonly used method for determining
cash discounts. The date and net payment date are counted from
the date of the invoice.
 Cash discounts are generally shown in an invoice such as:
 3/10, n/40 OR 3/10, net 40
 Read as “three ten, net forty”
 First digit is the rate of discount (3%)
 Second digit is number of days allowed to take the discount (10
days)
 n/40, net 40 is the total number of days given to pay the invoice in
full.

Contd….
• Post-dated (AS OF) method
Sometimes an invoice is post-dated to
give the buyer more time to take the cash
discount. This method calculates the
discount period and the net payment date
starting from the given AS OF date. For
example, an invoice dated July 25 AS OF
August 1 will have both cash discount
date and net payment date counted from
August 1.
Contd…
• End-of-month (EOM) dating method
End-of month dating is similar to
proximo dating. It is written as such
5/10 EOM, and 5/10 prox. which means
that the buyer will get 5% cash discount if
payment is made by the 10th of the month
that follows sale.
 If the invoice is dated on 20 March with
terms of 3/10 EOM, then the cash
discount period is up until 10 April.

Receipt-of-goods (ROG)
dating method
The discount period and the net payment period begin
when delivery is made.
 Abbreviated as ROG, this dating method gives the
buyer time to receive and inspect the goods to ensure
the merchandise received was in good condition.
 The method also allowed the buyer to benefit from a
cash discount.
 An invoice with terms of 6/15 ROG, means 6% cash
discount is given if the invoice is paid within 15 days
from the receipt of goods.
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Lessons Summary
The lesson teaches students how to
calculate compound interest through two
approaches- by using the formula and the
Compound Interest table.
 Islam prohibits the act of charging interest
to the borrower. Both borrower and
lender are answerable to God for their
deeds.

Lesson Summary
The topic explains the basis for annuity
payments and its calculation.
 There are a few cases where may be
involved in annuity payments in life.
 Students should be familiar with way of
computing annuity in order to
accumulates money for some personal
reasons in the future.

Contd..
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Similarly, students have learnt the
knowledge involving trade and cash
discounts that usually accompanies buying
and selling of goods between business to
business buyers. Students are reminded to
use the number of each day of the year
table to facilitate calculation for discounts
when offered.