REPUBLIC OF THE PHILIPPINES
DEPARTMENT OF EDUCATION
REGION X
DIVISION OF VALENCIA CITY
VP-GREENVALE ACADEMY,INC.
P17C,HAGKOL VALENCIA CITY,BUKIDNON
SCHOOL ID NO:405069
GENERAL MATHEMATICS
Quarter 2 - Module 6:
Simple and Compound Interests
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Lesson
1
Simple Interest
In many aspects of modern life, Mathematics plays an important role. In the field of business,
mathematics is essential in analyzing markets, predicting stock market prices, business decision
making, forecasting production, financial analysis, and in business operation in general. This module
will introduce the students to the basic concepts of business mathematics such as the simple and
compound interests.
What I Know (Pretest)
Directions: Read each statement carefully. Choose the letter of the correct answer and write it on a 1
whole sheet of paper.
1.) This refers to the accumulated amount obtained by adding the principal and the compound
interest.
A. Compound amount
C. Present value
B. Compound interest
D. Simple interest
2.) Date on which the money borrowed or loaned is to be completely repaid.
A. Conversion period
C. Maturity date
B. Loan date
D. Origin date
3.) What is the formula in computing the simple interest on a given financial transaction?
A.
C.
B.
D.
4.) This refers to the interest rate per conversion period.
A. Compound interest
C. Rate of interest
B. Periodic rate
D. Simple interest
5.) This refers to the amount paid or earned for the use of money.
A. Conversion period
C. Principal
B. Interest
D. Rate
6.) 30 months is equivalent to
A. 2.5 years
C. 3 years
B. 2.75 years
D. 3.25 years
7.) How much is the simple interest on this financial transaction, P = ₱5,000.00, r = 6%, and t = 2
years?
A. ₱120.00
C. ₱1,200.00
B. ₱600.00
D. ₱6,000.00
8.) What is the total number of conversion periods when a certain amount is borrowed at 10%
compounded monthly for 5 years?
A. 12
C. 24
B. 50
D. 60
9.) How much was the interest if Sophia borrowed ₱45,000.00 and paid a total of ₱55,500.00 at the
end of the term?
A. ₱10,500.00
C. ₱11,500.00
B. ₱45,000.00
D. ₱100,500.00
10.) What is the interest rate per conversion period if ₱25,900.00 was invested at 3.5%
compounded annually for 4 years and 6 months?
A. 0.035
C. 0.140
B. 0.350
D. 0.460
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11.) John borrowed ₱45,400.00 at 10% simple interest rate. How much should he repay after 3
years?
A. ₱13,620.00
C. ₱59,020.00
B. ₱46,762.00
D. ₱104,420.00
12.) An interest of ₱760 was earned on an investment for 9 months at 3% interest rate. How much
was invested?
A. ₱2,052.00
C. ₱20,520.00
B. ₱2,814.81
D. ₱33,777.78
13.) At what simple interest rate was ₱18,350.00 invested if it earned ₱1,025.00 interest for 1.5
years?
A. 0.0372%
C. 3.72%
B. 0.1193%
D. 11.93%
14.) How much will be the compound interest if ₱30,220.00 is invested at 7% compounded
quarterly for 2 years and 9 months?
A. ₱4,499.21
C. ₱36,574.05
B. ₱6,354.05
D. ₱147,899.31
15.) When is ₱78,800.00 due if its present value of ₱61,500.00 is invested at
% compounded
monthly?
A. 2. 32 years
B. 6.95 years
C. 10.75 years
D. 27.79 years
What’s In
On the previous modules, the basic concepts on functions were introduced. Functions were used
as mathematical models. These are abstract models that use mathematical language to describe
relationships. With the notion of mathematical modeling, mathematics is concerned not only with the
measures of the physical world but it has also expanded its applicability to sciences, both social and
biological, business, and finance. So with this, lessons relating to business and finance will then be
introduced specifically on simple interest.
What’s New
“When you saved money in the bank, you will gained an interest paid by the bank. On the other
hand, when you borrow money, you are charged an interest on the amount you borrowed. How does
gained and charged interests computed?”
A debtor pay the bank an amount which is more than the amount they borrowed. An investor
may withdraw from the bank more than the amount deposited. This additional sum is called
INTEREST.
Definition of terms:
Lender or creditor – person (or institution) who invests the money or makes the funds available.
Borrower or debtor – person (or institution) who owes the money or avails of the funds from the
lender.
Origin or loan date – date on which money is received by the borrower.
Repayment date or maturity date – date on which the money borrowed or loaned is to be completely
repaid.
Time or term (t) – amount of time in years the money is borrowed or invested; length of time between
the origin and maturity dates.
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Principal or present value (P) – amount of money borrowed or invested on the origin date.
Rate of interest or simply rate (r) – annual rate, usually in percent, charged by the lender, or rate of
increase of the investment.
Interest (I) – amount paid or earned for the use of money.
Maturity Value or Future Value (F) – amount after t years that the lender receives from the borrower
on the maturity date; equal to the sum of principal and the interest earned.
What is It
Simple Interest (Is)
For every financial transaction, whether you borrowed or invested a certain amount P, a
corresponding percentage of the principal called interest is being paid. Simple Interest (Is) is the
interest charged on the principal alone for the entire duration or period t of the loan or investment, at a
particular rate r. After the term of the loan or investment, the maturity value or future value F is
computed by getting the sum of the principal and the interest due.
Formulas:



or


***where
simple interest
principal
rate of interest or simply rate
time (in year)
future value (or maturity value)
Note: If the given time is in months, it can be converted to year(s) by using the formula
Example.
Directions: Complete the table below by solving the unknown quantities in each row.
Principal
(P)
Rate
(r)
Time
(t)
1.) ₱500,000.00
12.5%
10 years
2.)
2.5%
4 years
₱1,500.00
1 year and
6 months
₱4,860.00
3.) ₱36,000.00
4.) ₱250,000.00
5.) ₱10,000.00
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0.5%
4%
Simple Interest
(Is)
₱1,400.00
5 months
Future Value
(F)
Solution:
1.) Given:
P = ₱500,000.00 ;
r = 12.5% or 0.125 ; t = 10 years
2.) Given:
r = 2.5% or 0.025 ;
t = 4 years ;
3.) Given:
P = ₱36,000.00 ;
t=1
4.) Given:
or 9%
P = ₱250,000.00 ;
r = 0.5% or 0.005 ;
5.) Given:
years
P = ₱10,000.00 ;
r = 4% or 0.04 ;
= ₱1,500.00
years or 1.5 years ;
= ₱4,860.00
= ₱1,400.00
t=
( )
What’s More
I.
Complete the table below by solving the unknown quantities in each row. Write your
complete solutions and answers on a 1 whole sheet of paper.
Principal
Rate
Time
Simple Interest
Future Value
(P)
(r)
(t)
(Is)
(F)
1.) ₱40,000.00
2%
3 years
2.)
10%
5 years
₱2,500.00
1.5 years
₱3,600.00
₱15,400.00
3.) ₱100,000.00
4.) ₱250,000.00
5.) ₱12,345.00
II.
4.5%
8.25%
9 months
Solve the future value (refer on test I) using the alternative formulas:
What I Have Learned
Problems Involving Simple Interest
1. A bank offers 1.5% annual simple interest rate for a particular deposit. How much interest will
be earned if 1 million pesos is deposited in this savings account for 1 year?
Solution:
Given:
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r = 1.5% or 0.015 ;
P = ₱1,000,000.00 ; t = 1 year
.00
Therefore, an interest amounting to
.00 will be earned if 1 million pesos is deposited in
a savings account for 1 year with 1.5% annual simple interest rate.
2. When invested at an annual interest rate of 7%, the amount earned ₱11,200.00 of simple
interest in 2.5 years. How much money was originally invested?
Solution:
Given:
r = 7% or 0.07 ;
= ₱11,200.00 ;
t = 2.5 years
= ₱64,000.00
Therefore, the amount of money originally invested was ₱64,000.00.
3. Ricky borrowed ₱25,000.00 and paid ₱1,250.00 interest for 6 months. What was the rate of
interest?
Solution:
Given:
P = ₱25,000.00 ;
= ₱1,250.00 ;
t=
year or 0.5 year
0.1 or 10%
Therefore, the rate of interest was 0.1 or 10%.
4. How long in years will it take for ₱17,300.00 to amount to ₱20,000.00 at
interest?
simple
Solution:
Given:
P = ₱17,300.00 ;
F = ₱20,000.00 ;
r = 11.25% or 0.1125
years
Therefore, it will take 1.39 years for ₱17,300.00 to amount to ₱20,000.00.
What I Can Do
Answer the following problems involving simple interest. Write your complete solutions and
answers on a 1 whole sheet of paper.
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1. Find the simple interest on a loan of ₱65,000.00 if the loan is given at a rate of 2% and is due in
5 years and 3 months?
2. How much money will you have after 4 years if you deposited ₱10,000.00 in a bank that pays
6% simple interest?
Additional Activities
Formulate own three problems that involve simple interest. Write the problems and its complete
solutions and answers on 1 whole sheet of paper.
Answer Key
What I Know (Pretest)
1.
A
2.
C
3.
A
4.
B
5.
B
6.
A
7.
B
8.
D
9.
A
10. A
11. C
12. D
13. C
14. B
15. A
Lesson
2
What’s More
I.
1.
Is = ₱2,400.00
F = ₱42,400.00
2.
P = ₱5,000.00
F = ₱7,500.00
3.
r = 0.024 or 2.4% F = ₱103,600.00
4.
t = 1.37 years
F = ₱265,400.00
5.
Is = ₱763.85
F = ₱13,108.85
II.
*The same answers on test I (values of F).
What I Can Do
1.
Is = ₱6,825.00
2.
F = ₱12,400.00
Additional Activities
*Answers may vary
Compound Interest
What’s In
Problems involving simple interest were discussed on the previous lesson. Simple interest is the
interest charged on the principal alone for the entire length of the loan or investment. Several formulas
were introduced to solve problems involving simple interest. The second type of interest that will be
discussed on this lesson is the compound interest. For many long-term financial transactions,
compound interest is used instead of simple interest.
What’s New
“Suppose you won
and you plan to invest if for 5 years. A cooperative group offers
simple interest rate per year. A bank offers
compounded annually. Which will you choose and
why?”
Definition of terms:
Compound amount (F) – also called maturity value, it is an accumulated amount obtained by adding
the principal and the compound interest.
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Conversion period (m) – the number of times in a year the interest will be compounded.
The following are the common conversion periods in a year:
annually
:
m=1
semi-annually :
m=2
quarterly
:
m=4
monthly
:
m = 12
Number of conversion periods (n) – the total number of times interest is calculated for the entire term
of the investment or loan.
Annual interest rate or nominal rate (r) – the stated rate of interest per year.
Periodic rate (i) – the interest rate per conversion period.
Present value of F (P) – this is the principal P, that will accumulate to F if there is an interest at
periodic rate i for n conversion periods.
What is It
Compound Interest (Ic)
Compound interest (Ic) is usually used by banks in calculating interest for long-term
investments and loans such as savings account and time deposits. In this type of interest, the interest
due at stipulated interval is added to the principal and earns interest thereafter. It implies that the
principal increases over a period of time, resulting to an increase in interest earned at every
compounding period. Thus, compound interest is an interest resulting from the periodic addition of
simple interest to the principal amount or simply the difference between the compound amount and the
original principal.
The problem below is an example of compound interest.
Example:
₱50,000.00 was loaned for a period of 3 years with 5% interest compounded annually. What
amount of money will be needed to repay the loan?
Principal at
the start of
the year
Interest
Amount at the end of the
year
First Year
₱50,000.00
₱50,000 × 0.05 × 1 = ₱2,500.00
₱50,000 + 2 500 =
₱52,500.00
Second Year
₱52,500.00
₱52,500 × 0.05 × 1 = ₱2,625.00
₱52,500 + 2625 =
₱55,125.00
Third Year
₱55,125.00
₱55,125 × 0.05 × 1 = ₱2,756.25
₱55,125 + 2 756.25 =
₱57,881.25
The required answer to the problem is ₱57,881.25.
As shown in the table, the amount at the end of the year is equal to the sum of the principal and
the interest for that year.
Thus,
Amount for First Year :
A = 50000 + (50000 × 0.05)
= 50000 (1 + 0.05)
Amount for Second Year:
A = 50000 (1 + 0.05) + (50000 (1 + 0.05)(0.05))
= 50000 (1 + 0.05) (1 + 0.05)
= 50000 ( 1 + 0.05)2
Amount for Third Year:
A = 50000 (1 + 0.05)2 + (50000 ( 1 + 0.05)2(0.05))
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= 50000 (1 + 0.05)2 (1 + 0.05)
= 50000 (1 + 0.05)3
Generally, when interest is compounded annually for n years, the amount A = P( 1 + i) n.
Computation of the compound amount by the method shown above is tedious and timeconsuming. The formulas below will greatly ease computations.
Formulas:

(
)

(
)
or

[( )
]
( )

***where
compound interest
present value of F
annual interest rate
time (per year)
compound amount or maturity value
conversion period
annually
:
m=1
semi-annually :
m=2
quarterly
:
m=4
monthly
:
m = 12
total number of conversion periods
periodic rate (
)
Examples:
1.) Find the compound amount and interest earned on ₱15,000.00 for 1 year at
(a) 7% compounded semi-annually and
(b) 7% compounded quarterly.
Solution:
(a) Given:
P = ₱15,000.00
r = 7% or 0.07
t = 1 year
m=2
Therefore, the compound amount and the interest are
Alternative solution for solving the compound interest:
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and
, respectively.
(b) Given:
P = ₱15,000.00
t = 1 year
m=4
r = 7% or 0.07
Therefore, the compound amount and the interest are
Alternative solution for solving the compound interest:
2.) Find the present value of
monthly.
Solution:
(a) Given:
and
due in 3 years if the interest rate is 6% compounded
F=
t = 3 years
m = 12
r = 6% or 0.06
Therefore, the present value is
.
3.) At what rate of interest compounded semi-annually will
in 2 years and 6 months?
accumulate to
Solution:
Given:
P=
F=
[( )
[(
m=2
t=
years or 2.5 years
]
)
]
or
%
Therefore, the rate of interest will be 0.0704 or 7.04%.
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, respectively.
4.) How many years will it take for
annually?
to become
at 12.5% compounded
Solution:
Given:
P=
F=
m=1
r = 12.5% or 0.125
( )
(
)
3.66 years
Therefore,
will become
in 3.66 years.
What’s More
Solve what is asked in each item. Write your complete solutions and answers on a 1 whole
sheet of paper.
1.) Find the final or compound amount of ₱15,900.00 at 5.5% interest compounded annually for 18
months.
2.) Find the interest on ₱25,750.00 for 3 years at 8% compounded quarterly.
3.) Find the present value of ₱150,000.00 at 15% interest compounded monthly for 6 years.
4.) At what rate of interest compounded semi-annually will
accumulate to
in 2 years?
5.) How long will it take for
accumulate to
at 4.5% compounded quarterly?
What I Have Learned
Problems Involving Compound Interest
1. Joseph borrows
and promise to pay the principal and interest at 12% compounded
monthly. How much must he repay after 6 years?
Solution:
Given:
P = ₱50,000.00
r = 12% or 0.12
Therefore, Joseph must repay
2. A loan
m = 12
t = 6 years
after 6 years.
at 8% compounded quarterly was paid back with an amount of
at the end of the period. For how long was the money borrowed?
Solution:
Given:
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P=
F=
r = 8% or 0.08
m=4
( )
(
)
4.32 years
Therefore, the money was borrowed for 4.32 years.
3. How much must be invested today in a savings account in order to have
and 9 months if money earns 5.4% compounded semi-annually?
in 6 years
Solution:
Given:
F=
r = 5.4% or 0.054
Therefore, an amount of
t=
years or 6.75 years
m=2
must be invested today.
What I Can Do
Answer the following problems involving compound interest. Write your complete solutions
and answers on a 1 whole sheet of paper.
1. In a certain bank, Justine invested
in a time deposit that pays 0.5% compounded
annually. How much will be his money after 5 years? How much interest will he gain?
2. Recca borrows
and agrees to pay
after 2 years. At what rate,
compounded monthly, is the interest computed?
Additional Activities
Directions: Answer the following activities and write answers on a 1 whole sheet of paper.
I.
Arrange the jumbled letters to form a word/s related to business mathematics.
1.) T M T U R I Y A A E D T
2.) N E E T T R I S E R T A
3.) I I P P C N L R A
4.) O O U D C M N P N U T M A O
5.) I P S E M L T E N T S R E I
6.) E V M N T T N E I S
7.) T E U Y A Q R L R
8.) C M C U E T U L A A
9.) R W O R B O R E
10.) M O C U D O N P T N T R E E S I
II.
Suppose you are working in a bank that encourages customers to save money for their
future. Your bank manager requested you to form a team and formulate a slogan to be used
for this savings campaign. Write a short phrase or slogan to persuade people to open a
savings account in the bank where you work.
III.
Complete the Simple Interest Vs Compound Interest Comparison Chart below.
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Simple Interest Vs Compound Interest
Basis for Comparison
Simple Interest
Compound Interest
1. Meaning
2. Formula
3. Principal
4. Amount of Interests
Assessment (Post-test)
Directions: Read each statement carefully. Choose the letter of the correct answer and write it on a
1 whole sheet of paper.
1.) Date on which money is received by the borrower.
A. Conversion period
C. Maturity date
B. Loan date
D. Repayment date
2.)
% is equivalent to
A. 0.0032
C. 0.32
B. 0.032
D. 3.2
3.) This refers to the interest charged on the principal alone for the entire duration or period of the
loan or investment.
A. Compound interest
C. Interest rate
B. Future value
D. Simple interest
4.) This refers to the number of years for which the money is borrowed or invested.
A. Conversion period
C. Principal
B. Interest rate
D. Time
5.) An interest resulting from the periodic addition of simple interest to the principal amount.
A. Compound amount
C. Interest rate
B. Compound interest
D. Simple interest
6.) What is the formula in computing the present value of F in a financial transaction involving
compound interest?
A.
C.
B.
D.
7.) How much was the interest if Althea invested ₱30,400.00 and received a total of ₱40,300.00 at
the end of the term?
A. ₱9,900.00
C. ₱40,300.00
B. ₱30,400.00
D. ₱70,700.00
8.) How much is the future value on this financial transaction, P = ₱10,000.00, r = 5%, and t = 3
years?
A. ₱1,500.00
C. ₱21,500.00
B. ₱11,500.00
D. ₱25,000.00
9.) What is the total number of conversion periods when a certain amount is borrowed at 5.5%
compounded quarterly for 4 years?
A. 4
C. 16
B. 12
D. 22
10.) What is the interest rate per conversion period if ₱29,500.00 was invested at 2.5%
compounded semi-annually for 5 years and 4 months?
A. 0.0025
C. 0.025
B. 0.0125
D. 2.5
11.) Edgardo invested ₱15,600.00 at 10.25% interest rate. How long will take for his investment to
earn an interest of ₱5,055.00?
A. 0.32 years
C. 6.29 years
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B. 3.16 years
D. 30.11 years
12.) Find the simple interest on a loan of ₱65,000.00 if the loan is given at a rate of 8% and is due
in 6 years and 3 months.
A. ₱3,250.00
C. ₱32,500.00
B. ₱31,200.00
D. ₱46,800.00
13.) Jamaico made a loan of ₱20,450.00 from a bank that charges 3% simple interest. How much
must he pay the bank after 2 years?
A. ₱1,227.00
C. ₱32,720.00
B. ₱21,677.00
D. ₱42,127.00
14.) At what interest rate compounded semi-annually will
accummulate to
in 10 years?
A. 2.05%
C. 4.05%
B. 2.59%
D. 5.17%
15.) ABC University anticipates additional expenses of
for a new equipment needed
for offering a new course 5 years from now. How much should be invested in an account that
earns 12% compounded monthly?
A. ₱62,427.83
C. ₱202,455.37
B. ₱165,344.63
D. ₱668,181.05
Answer Key
What’s More
1.
F=
2.
3.
P=
4.
r = 0.1130 or 11.30%
5.
t = 13.93 years
What I Can Do
1.
F=
2.
r = 0.1479 or 14.79%
=
Additional Activities
I.
1.
MATURITY DATE
2.
INTEREST RATE
3.
PRINCIPAL
4.
COMPOUND AMOUNT
5.
SIMPLE INTEREST
6.
INVESTMENT
7.
QUARTERLY
8.
ACCUMULATE
9.
BORROWER
10. COMPOUND INTEREST
II.
*Answers may vary
III.
1. Definitions of Simple Interest and Compound Interest
2. Formulas of Simple Interest and Compound Interest
3. Simple Interest – Constant;
Compound Interest – Changing during the entire term of loan or
investment
4. Simple Interest – uniform; Compound Interest – Increasing rapidly
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Assessment (Post-test)
1.
B
2.
B
3.
D
4.
D
5.
B
6.
A
7.
A
8.
B
9.
C
10. B
11. B
12. C
13. B
14. D
15. C
SENIOR HIGH SCHOOL
General Mathematics
Quarter 2 – Module 7
Annuities
Lesson
Simple Annuity
What I know…
PRE-TEST
Direction: Choose the letter of the correct answer and write on the separate
sheet of paper.
1. It is an annuity where the payment interval is the same as the
interest period.
a.) Simple Annuity
b.) General Annuity
c.) Annuity Certain
d.) Contingent annuity
2. It is a sequence of payments made at equal (fixed) intervals or
periods of time.
a.) Future Value of an annuity
b.) Present Value of an annuity
c.) Annuity
d.) Periodic Payment
3. The sum of future values of all the payments to be made during
the entire term of annuity
a.) Annuity
b.) Present Value of an annuity
c.) Future Value of an annuity
d.) Periodic Payment
4. The sum of all present values of all the payments to be made
during the entire term of the annuity.
a.) Periodic Payment
b.) Time of an Annuity
c.) Future Value of an annuity
d.) Present Value of an annuity
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5. Find the future value of an ordinary annuity with a regular
payment of P1,000 AT 5% interest rate compounded quarterly for
3 years.
a.) P12,806.63
b.) P12,860.36
c.) P12,860.63
d.) P12,806.36
6. Find the present value of an ordinary annuity with regular
quarterly payments worth P1,000 at 3% annual interest rate
compounded quarterly at the end of 4 years.
a.) P15,024.31
b.) P15,204.31
c.) P15,402.31
d.) P15,420.31
7. It is a term that refers to payments received (cash inflow).
a.) General Annuity
b.) General Ordinary Annuity
c.) Cash Flow
d.) Annuity Certain
8. It is refers to a single amount that is equivalent to the value of
the payment stream that shall date.
a.) Future Value of a general annuity
b.) Present Value of a general annuity
c.) Fair market value
d.) Periodic Payment
9. What is the other term for fair market value?
a.) Cash flow
b.) Present Value of a general annuity
c.) Future Value of a general annuity
d.) Economic Value
10. A teacher saves P5,000 every 6 months in the bank that pays
0.25% compounded monthly. How much will be her savings after
10 years?
a.) P101,197.06
b.) P101,179.06
c.) P101,971.06
d.) P101,791.06
11. It is an annuity that does not begin until a given time interval
has passed.
a.) Period of Deferral
b.) Deferred Annuity
c.) Present value of a deferred annuity
d.) Contingent annuity
12. It is a time between the purchase of an annuity and the start of
the payments for the deferred annuity.
a.) Period of deferral
b.) General Ordinary Annuity
c.) Deferred annuity
d.) Present value of a deferred annuity
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13. Melvin availed of a loan from a bank that gave him an option to
pay P20,000 monthly for 2 years . The first payment is due after 4
months. How much is the present value of the loan if the interest
rate is 10% converted monthly?
e.) P422,795.78
f.) P422,759.78
g.) P422,579.78
h.) P422,597.78
14. Annual payments of P2,500 for 24 years that will start 12 years
from now. What is the period of deferral in the deferred annuity?
e.) 12 periods
f.) 10 periods
g.) 11 periods
h.) 13 periods
15. Semi-annual payments of P6,000 for 13 years that will start 4
years from now. What is the period of deferral in the deferred
annuity?
e.) 8 semi-annual intervals
f.) 6 semi-annual intervals
g.) 5 semi-annual intervals
h.) 7 semi-annual intervals
What’s in…
REVIEW
You use money in everyday life. In order to buy what you need, you do
transactions involving money.
In the previous lessons, you learned the methods of solving the value of
money under compound and simple interest environment. You have learned to
illustrate and distinguish between simple and compound. You also learned how
to compute for the interest, present value and future value in a simple and
compound interest environment. As well as solve problems involving real life
situations of simple and compound interest.
What’s new…
Ma’am Angel wants to start a business with an initial capital of P100,000.
She decided to put up a fund with deposits made at the end of each month. If
she wants to gain the initial capital after 4 years, how much monthly deposit
must be made?
In most cases where house or cars are purchased, a series of payments is
needed at certain points in time. Such Transaction is called ANNUITY.
 ANNUITY
ANNUITY
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Simple Annuity – an
annuity where the
payment interval is the
same as the interest
period
According to
payment interval
and interest period
General Annuity – an
annuity where the
payment interval is not the
same as the interest
period.
Ordinary Annuity (Annuity Immediate) – a type of
annuity in which the payments are made at the end
of each payment interval
Annuity Certain – an annuity in which payments
begin and end at definite times.
According to time of
payment
According to
duration




Annuities may be illustrated using a time diagram. The time
diagram for an ordinary annuity (i.e., payments are made at the end of the
year) is given below.
ILLUSTRATION
0
R
R
R
R
R.................
1
2
3
4
5
..R
n
EXAMPLE 1:
Suppose Mrs. Manda would like to deposit P3,000 every month in a
fund that gives 9%, compounded monthly. How much is the amount of future
value of her savings after 6 months?
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Given:
Periodic payment (R) = P3,000
Term (t) = 6 months
Interest rate per annum (annually) (i) = 0.09/9%
0.0075
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What is it…
Activity 1: Question and Answer
Directions: Answer the questions briefly. Write your answers in a separate
sheet of paper.
1. Differentiate Simple Annuity and General Annuity?
2. What is an Ordinary Annuity?
3. What is the formula in finding the future value of an ordinary annuity?
Identify each variable represents.
4. What is the formula in finding the present value of an ordinary annuity?
Identify each variable represents.
5. What is the periodic payment formula of an annuity?
What’s more..
Answer as indicated. Write your answers in a separate sheet of paper.
1. Find the future value of an ordinary annuity with a regular payment of
P1,000 at 5% compounded quarterly for 3 years.
2. Find the present value of an ordinary annuity with regular quarterly
opayments worth P1,000 at 3% annual interest rate compounded
quarterly at the end of 4 years.
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What have I have learned..
Complete the sentence below. Write your answers on a separate sheet of paper.
1.
is a sequence of payments made at
equal (fixed) intervals or periods of time.
2.
is the sum of present value of all
the payments to be made during the entire term of the annuity.
3.
is an annuity where the payment
interval is the same as the interest period.
4.
is a type of annuity in which the
payments are made at the end of each payment interval.
5.
is the sum of future values of all
payments to be made during the entire term of the annuity.
Solve for the following problems. Answer as indicated. Write your answers in a
separate sheet of paper.
1. Mr. Ribaya paid P200,000 as downpayment for a car. The remaining amount is
to be settled by paying P16,200 at the end of each month for 5 years. If interest
is 10.5% compounded monthly, what is the cash price of his car?
2. In order to save for her high school graduation, Marie decided to save P200
at the end of each month. If the bank pays 0.250% compounded monthly,
how much will her money be at the end of 6 years?
3. Paolo borrowed P100,000. He agrees to pay the principal plus interest by
paying an equal amount of money each year for 3 years. What should be his
annual payment if interest is 8% compounded annually?
Lesson
General Annuity
What’s in…
REVIEW
In the previous lessons, you learned to illustrate a Simple Annuity and you
solve the present and future values of simple Annuity. You also compute for the
periodic payment of simple annuity. As well as solve problems involving real life
situations on simple Annuities.
What’s new…
 GENERAL ANNUITY
 GENERAL ORDINARY ANNUITY
Examples of General annuity:
1. Monthly installment payment of a car, lo or house with an interest rate that is
compounded annually.
2. Paying a debt semi-annually when the interest is compounded monthly.
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Future and Present Value of a General Ordinary Annuity
EXAMPLE 1:
Cris started to deposit P1,000 monthly in a fund that pays 6% compounded
quarterly. How much will be in the fund after 15 years?
GIVEN: R = 1,000, n = 12(15) = 180 payments, i (4) = 0.06m = 4
Find F
SOLUTION:
The Cash Flow for this problem is shown in the diagram below.
F
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EXAMPLE 2:
Ken borrowed an amount of money from Kat. He agrees to pay the
principal plus interest by paying P38, 973.76 each year for 3 years. How much
money did he borrow if the interest is 8% compounded quarterly?
GIVEN: R = 38,973.76, i(4) = 0.08, m = 4, n = 3 payments
Find P, Present Value
SOLUTION
The Cash Flow for this problem is shown in the diagram below.
P=?
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EXAMPLE 3:
Mr. Ribaya received two offers on a lot that he wants to sell. Mr.
Ocampo has offered P50,000 and a P1million lump sum payment 5 years from
now. Mr. Cruz has offered P50,000 plus P40,000 every quarter for five years.
Compare the fair market value of the two offers if money can earn 5%
compounded annually. Which offer has a higher market value?
Mr. ocampo’s Offer
P50,000 down payment
P1,000,000 after 5 years
Mr. Cruz’s Offer
P50,000 down payment
P40,000 every quarter for 5 years
SOLUTION:
50,000
0
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1 million
1
2
3
4
5
Choose a focal date and determine the values of the two offers at that focal date.
For example the focal date can be the date at the start of the term.
Since the focal date is at t = 0, compute for the present value of each offer.
Mr. Ocampo’s Offer: Since P50,000 is offered today, then its present value is still
P50,000. The present value of P1,000,000 offred 5 years from now is
P = F (1 + j)-n
P = 1,000,000 (1 + 0.05) -5 P =
P783, 526.20
Fair Market value (FMV) = DOWNPAYMENT + PRESENT VALUE
FMV = 50,000 + 783, 526.20
FMV = P833,526.2
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FAIR MARKET VALUE (FMV) = DOWNPAYMENT + PRESENT VALUE
FMV = 50,000 + 705,572.70
FMV = 755,572.70
Hence, Mr. ocampo’s Offer has a higher market value. The difference
between the market values of the two offers at the start of the term is
833,526.20 – 756,572.70 = P77,953.50
What is it…
Activity 1: Question and Answer
Directions: Answer the questions briefly. Write your answers in a separate
sheet of paper.
1. Differentiate General Annuity and General Ordinary Annuity?
2. What is a General Ordinary Annuity?
3. Express the process in finding the Present and future valueof General
ordinary annuity.
4. What is the formula in finding the Fair Market Value?
5. Express the process in finding the Fair Market Value.
What’s more..
Answer as indicated. Write your answers in a separate sheet of paper.
1. Which Offer has a better Fair Market Value?
Company A offers P150,000 at the end of 3 years plus P300,000 at the end
of 5 years. Company B offers P25,000 at the end of each quarter for the
next 5 years. Assume that money is worth 8% compounded annually.
COMPANY A
P150,000 at the end of 3 years
P300,000 at the end of 5 years
COMPANY B
P25,000 at the end of each
quarter for 5 years
2. ABC bank pays interest at the rate of 2% compounded quarterly. How much
will Ken have in the bank at the end of 5 years if he deposits P3,000 every
month?
What have I have learned..
Complete the sentence below. Write your answers on a separate sheet of paper.
1.
is an annuity where length of the
payment interval is not the same as the length of the interest compounding
period.
2.
is general annuity in which the
periodic payment is made at the end of the payment interval.
3.
is a term that refers to payments
received or payments or deposits made.
4.
of a cash flow on a particular date
refers to a single amount that is equivalent to the value of the payment stream
at that date.
5.
installments payment of a car, lot or
house with an interest rate that is compounded annually.
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What I can do…
Solve for the following problems. Answer as indicated. Write your answers in a
separate sheet of paper.
1. Mrs. Remoto would like to buy a television (TV) set payable for 6 months
starting at the end of the month. How much is the cost of the TV set if her
monthly payment is P3,000 and interest is 9% compounded semi- annually?
2. Kat received two offers for investments. The first one is P150,000 every year
for 5 years at 9% compounded annually. The other investment scheme is
P12,000 per month for 5 years with the same interest rate. Which fair market
value between these offers is preferable?
Lesson
Deferred Annuity
What’s in…
REVIEW
In the previous lessons, you learned the methods of solving the value of
money under General annuities. You were able to find the future and present
value of general annuities and compute the periodic payment of a general
annuity. And you also solve for the fair market value of a cash flow stream that
includes an annuity. As well as solve problems involving real life situations of
General annuities.
What’s new…
In this section, you will explore annuities whose payments do not
necessarily start at the beginning or at the end of the next compounding period.
For instance, for certain employee who will retire in 20 years, his pension will
only start after 20 years.


ILLUSTRATION
R*
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R....................................... R*
In the time diagram the period of deferral is k because the regular payments of R start
at the time k+1.
The rotation R* represent k”artificial payments”, each equal to R but are not
actually paid during the period of deferral.
PRESENT VALUE OF A DEFERRED ANNUITY
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Payment Period
Time
0
1
2
. . . . . .. . .
12 13 14 15 16 . . . .
35 36
(in months)
P116,930.64
P114,046.58
Notice that there are two stages in finding the present value of a
deferred annuity: (1) find the value of the payment at the start of the payment
period by using the formula for the present value of an annuity, and then (2) fin
the value of the amount to be obtained at the start (or time 0) by using the
formula for the present value of a single amount given in the formula of the
resent value of a deferred annuity.
If the period is k-years, you call the annuity a k-year deferred annuity
What is it…
Activity 1: Question and Answer
Directions: Answer the questions briefly. Write your answers in a separate
sheet of paper.
1.
2.
3.
4.
Differentiate Deferred Annuity and Period of Deferrral.
What is a Deferred Annuity?
What is a period of deferral?
What is the formula in finding the present value of a deferred annuity?
Identify each variable represents
5. Draw the time diagram for a deferred annuity.
What’s more..
Answer as indicated. Write your answers in a separate sheet of paper.
1. Find the present value of a 2-year deferred annuity at 4% interest compounded
quarterly with payments of P1,000 made every quarter for 3 years.
2. Find the present value of a 3-year deferred annuity with regular payments of
P10,000 compounded annually at an interest rate of 3%.
What have I have learned..
Complete the sentence below. Write your answers on a separate sheet of paper.
1.
is a kind of annuity whose
payments (or deposits) start in more than one period from the present.
2.
is the time between the purchase of
an annuity and the start of the payments for the deferred annuity.
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What I can do…
Solve for the following problems. Answer as indicated. Write your answers in a
separate sheet of paper.
1. Mariel purchased a smart television set through the credit cooperative of their
company. The cooperative provides an option for a deferred payment. Mariel
decided to pay after 2 months of purchase. Her monthly payment is computed
as P3,800 payable in 12 months. How much is the cash value of the television
set of the interest rate is 12% convertible monthly?
2. Melvin availed of a loan from a bank that gave him an option to pay
P20,000 monthly for 2 years . The first payment is due after 4 months.
How much is the present value of the loan if the interest rate is 10%
converted monthly?
3. Quarterly payments of 300 for 9 years that will start 1 year from now,
What is the period of deferral in the deferred annuity?
Assessment…
POST-TEST
Direction: Choose the letter of the correct answer and write on the separate
sheet of paper.
1. It is an annuity where the payment interval is the same as the
interest period.
a.) Simple Annuity
b.) General Annuity
c.) Annuity Certain
d.) Contingent annuity
2. It is a sequence of payments made at equal (fixed) intervals or
periods of time.
a.) Future Value of an annuity
b.) Present Value of an annuity
c.) Annuity
d.) Periodic Payment
3. The sum of future values of all the payments to be made during
the entire term of annuity
a.) Annuity
b.) Present Value of an annuity
c.) Future Value of an annuity
d.) Periodic Payment
4. The sum of all present values of all the payments to be made
during the entire term of the annuity.
a.) Periodic Payment
b.) Time of an Annuity
c.) Future Value of an annuity
d.) Present Value of an annuity
5. Find the future value of an ordinary annuity with a regular
payment of P1,000 AT 5% interest rate compounded quarterly for
3
years.
a.) P12,806.63
b.) P12,860.36
c.) P12,860.63
d.) P12,806.36
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6. Find the present value of an ordinary annuity with
regular quarterly payments worth P1,000 at 3% annual interest rate
compounded quarterly at the end of 4 years.
a.) P15,024.31
b.) P15,204.31
c.) P15,402.31
d.) P15,420.31
7. It is a term that refers to payments received (cash inflow).
a.) General Annuity
b.) General Ordinary
Annuity c.) Cash Flow
d.) Annuity Certain
8. It is refers to a single amount that is equivalent to the
value of the payment stream that shall date.
a.) Future Value of a general annuity
b.) Present Value of a general annuity
c.) Fair market value
d.) Periodic Payment
9. What is the other term for fair market value?
a.) Cash flow
b.) Present Value of a general annuity
c.) Future Value of a general annuity
d.) Economic Value
10. A teacher saves P5,000 every 6 months in the bank that
pays 0.25% compounded monthly. How much will be her
savings after 10 ears?
a.) P101,197.06
b.) P101,179.06
c.) P101,971.06
d.) P101,791.06
11. It is an annuity that does not begin until a given time
interval has passed.
a.) Period of Deferral
b.) Deferred Annuity
c.) Present value of a deferred annuity
d.) Contingent annuity
12. It is a time between the purchase of an annuity and the
start of the payments for the deferred annuity.
a.) Period of deferral
b.) General Ordinary Annuity
c.) Deferred annuity
d.) Present value of a deferred annuity
13. Melvin availed of a loan from a bank that gave him an
option to pay P20,000 monthly for 2 years . The first
payment is due after 4 months. How much is the present
value of the loan if the interest rate is 10% converted
monthly?
a.) P422,795.78 \
b.) P422,759.78
c.) P422,579.78
d.) P422,597.78
14. Annual payments of P2,500 for 24 years that will start 12
years from now. What is the period of deferral in the
deferred annuity?
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a.) 12 periods
b.) 10 periods
c.) 11 periods
d.) 13 periods
15. Semi-annual payments of P6,000 for 13 years that will
start 4 years from now. What is the period of deferral in
the deferred annuity?
a.) 8 semi-annual intervals
b.) 6 semi-annual intervals
c.) 5 semi-annual intervals
d.) 7 semi-annual intervals
Additional Activities…
Answer as indicated. Write your answers in a separate sheet of paper.
1. Mr. Quijano decided to sell their farm and to deposit the fund in a
bank. After computing the interest, they learned that they may
withdraw P480,000 yearly for 8 years starting at the end of 6 years
when it is time for him to retire. How much is the fund deposited if
the interest rate is 5% converted annually?
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