Module-7-Annuities(1)

```SENIOR HIGH SCHOOL
General Mathematics
Quarter 2 – Module 7
Annuities
Department of Education
Republic of the Philippines
Alternative Delivery Mode
Quarter 2– Module 7: Annuities
Republic Act 8293, section 176 states that: “No copyright shall subsist in any work of the
Government of the Philippines. However, prior approval of the government agency or office
wherein the work is created shall be necessary for exploitation of such work for profit. Such agency
or office may, among other things, impose as a condition, payment of royalties.
Borrowed materials included in this module are owned by the respective copyright holders.
Effort has been exerted to locate and seek permission to use these materials from the respective
copyright owners. The publisher and author do not represent nor claim ownership over them.
Secretary:
Undersecretary:
Assistant Secretary:
Development Team of the Module
Authors: Angelie D. Enterina
Editor:
Reviewers:
Illustrator:
Layout Artist:
Management Team:
Printed in the Philippines by _____________________________
Department of Education – Bureau of Learning Resources (DepEd – BLR)
______________________________________
Telefax:
______________________________________
______________________________________
SENIOR HIGH SCHOOL
General Mathematics
Quarter 2 – Module 7:
Annuities
This instructional material was collaboratively developed and reviewed
by educators from public and private schools, colleges, and or/universities. We
encourage teachers and other education stakeholders to email their feedback,
comments, and recommendations to the Department of Education at
action@deped.gov.ph.
We value your feedback and recommendations.
Department of Education
Republic of the Philippines
Lesson
1
Simple Annuity
What I need to know…
At the end
✓
✓
✓
of the lesson, the learner will be able to:
Illustrate simple and general annuities
Distinguish between simple and general annuities
Computes the future value, present value and periodic
payment of 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
__________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
__________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
f.) 10
g.) 11
h.) 13
periods
periods
periods
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
f.) 6
g.) 5
h.) 7
semi-annual
semi-annual
semi-annual
semi-annual
intervals
intervals
intervals
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
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
An ANNUITY is a sequence of equal payments (or deposits) made at a
regular interval of time.
ANNUITY
According to
payment interval
and interest period
Simple Annuity – an
annuity where the
payment interval is the
same as the interest
period
General Annuity – an
annuity where the
payment interval is not the
same as the interest
period.
According to time of
payment
Ordinary Annuity (Annuity Immediate) – a type of
annuity in which the payments are made at the end
of each payment interval
According to
duration
Annuity Certain – an annuity in which payments
begin and end at definite times.
❖ Term of an Annuity (t)
The time between the first payment interval and the last payment interval.
❖ Regular or Periodic Payment (R)
The amount of each payment.
❖ Amount (Future Value) of an annuity (F)
The sum of future value of all the payments to be made during the entire
term of the annuity.
❖ Present Value of an annuity (P)
The sum of present value of all the payments to be made during the entire
term of the annuity.
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........ ......... ..R
1
2
3
4
5
n
Time Diagram for an n-Payment ordinary annuity
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?
Given:
Periodic payment (R) = P3,000
Term (t) = 6 months
Interest rate per annum (annually) (i) = 0.09/9%
Number of conversion per year (m) = 12
𝑖
0.09
Interest rate per period 𝑗 =
=
= 0.0075
𝑚
12
(1) Illustrate the cash flow in time diagram and Find the future value of
all the payments at the end of term (t=6).
Time
0
(in months)
Payment/
Deposit
1
2
3,000
3,000
3
3,000
4
5
6
3,000
3,000
3,000
3,000
3,000 (1 + 0.0075)
3,000 (1 + 0.0075) 2
3,000 (1 + 0.0075) 3
3,000 (1 + 0.0075) 4
3,000 (1 + 0.0075) 5
(2) Add all the future values obtained from the cash flow.
3,000
3,000
3,000
3,000
3,000
3,000
(1
(1
(1
(1
(1
+
+
+
+
+
=
=
=
=
=
=
0.0075)
0.0075) 2
0.0075) 3
0.0075) 4
0.0075) 5
Thus, the amount of this annuity is
3,000
3,022.50
3,045.17
3,068.01
3,091.02
3,114.20
P18,340.89
FORMULA 1: FUTURE VALUE
a. The future value of an ordinary annuity with regular payments
R at a nominal interest rate I compounded m times a year after
t years is
𝐹 = 𝑅[
Note: j =
𝑖
𝑚
n = mt
(1+
𝑖 𝑚𝑡
) −1
𝑚
𝑖
𝑚
]
𝐹 = 𝑅[
(1+ 𝑗)𝑛 −1
𝑗
]
(3) Solution using formula 1
Given:
A(t) = ?
R = 3,000
𝑖 𝑚𝑡
𝐹 = 𝑅[
(1+ 𝑚)
−1
𝑖
𝑚
= 3,000 [
= 3,000 [
= 3,000 [
m = 12
t (annually) = 6/12
]
0.09 12(0.5)
)
−1
12
0.09
12
(1+ 0.0075)6 −1
= 3,000 [
= 3,000 [
i = 0.09
(1+
]
0.0075
(1.0075)6 −1
]
0.0075
1.045852235−1
0.0075
0.458522351
0.0075
]
]
]
= 3,000 ( 6.113631347)
F = 18, 340.89
Therefore, the amount of future value of Mrs. Manda’s savings after 6 months
is P18,340.89.
Thus, using different kinds of processes in finding the future value of
an ordinary annuity comes up with the same answer.
EXAMPLE 2:
To start a business, Jake wants to save a certain amount of money at
the end of every month to put in an account providing 2% interest compounded
monthly. His estimated start-up capital is P150,000. If he wants to start a
business in 1.5 years, how much monthly deposit must he put into the account?
SOLUTION:
Since the deposits are made at the end of every month, then this Is an
example of an ordinary annuity. Use FORMULA 1 with:
GIVEN:
i = 0.02, m = 12,
𝐹 = 𝑅[
(1+
t = 1.5, and A = P150,000.
𝑖 𝑚𝑡
) −1
𝑚
𝑖
𝑚
0.02 12(1.5)
)
−1
12
0.02
12
(1+ 0.001666)18 −1
150,000 = 𝑅 [
150,000 = 𝑅 [
(1+
]
]
0.001666
150,000 = 𝑅 [
150,000 = 𝑅 [
150,000 = 𝑅 [
]
(1.001666)18 −1
0.001666
1.030428801−1
0.001666
0.0304288015
0.001666
]
]
]
150,000
= R ( 18.2572809)
18.2572809
18.2572809
8,215.90 = R
Thus, Jake must deposit P8,215.90 at the end of each month.
EXAMPLE 3:
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?
Given:
Periodic payment (R) = P3,000
Term (t) = 6 months
Interest rate per annum (annually) (i) = 0.09/9%
Number of conversion per year (m) = 12
𝑖
0.09
Interest rate per period 𝑗 =
=
= 0.0075
𝑚
12
(1) Illustrate the cash flow in time diagram and Find the Present value
of all the payments at the end of term (t=6).
Time
0
(in months)
Payment/
Deposit
1
2
3,000
3,000
3
3,000
4
5
6
3,000
3,000
3,000
3,000 (1 + 0.0075) -1
3,000 (1 + 0.0075) -2
3,000 (1 + 0.0075) -3
3,000 (1 + 0.0075) -4
3,000 (1 + 0.0075) -5
3,000 (1 + 0.0075) -6
(2) Add all the present values obtained from the cash flow.
3,000
3,000
3,000
3,000
3,000
3,000
(1
(1
(1
(1
(1
(1
+
+
+
+
+
+
0.0075) -1
0.0075) -2
0.0075) -3
0.0075) -4
0.0075) -5
0.0075) -6
=
=
=
=
=
=
Thus, the amount of this annuity is
2,977.667
2,955.501
2,933.50
2,911.663
2,889.988
2,868.474
P17,536.79
FORMULA 2: PRESENT VALUE
b. The present value P of an ordinary annuity with regular
payments R at a nominal interest rate I compounded m times
a year after t years is
𝑃 = 𝑅[
Note: j =
𝑖
𝑚
n = mt
𝑖 −𝑚𝑡
)
𝑚
𝑖
𝑚
1−(1+
]
𝑃 = 𝑅[
1−(1+ 𝑗)−𝑛
𝑗
]
(3) Solution using formula 2
Given:
P = ? R = 3,000 i = 0.09
𝑃 = 𝑅[
𝑖 −𝑚𝑡
)
𝑚
𝑖
𝑚
1−(1+
= 3,000 [
= 3,000 [
= 3,000 [
1−(1+
1−(1.0075)−6
]
0.0075
1−0.9561580178
0.0075
0.04384198223
0.0075
]
]
0.0075
= 3,000 [
t (annually) = 6/12
]
0.09 −(12(0.5))
)
12
0.09
12
1−(1+ 0.0075)−6
= 3,000 [
m = 12
]
]
= 3,000 ( 5.84559763)
P = 17,536.79
Therefore, the amount of Present value of Mrs. Manda’s savings after 6
months is P17,536.79.
Thus, using different kinds of processes in finding the Present value of
an ordinary annuity comes up with the same answer.
EXAMPLE 4:
A certain fund currently has P100,000 and is invested at 3% interest
compounded annually. How much withdrawal can be made at the end of each
year so that the fund will have zero balance at the end of 12 years?
SOLUTION:
Since withdrawals are made every end of the year, then this ordinary annuity.
Given:
Periodic payment (R) = P100,000
Term (t) = 12 years
Interest rate per annum (annually) (i) = 0.03/3%
Number of conversion per year (m) = 1
𝑖
0.03
Interest rate per period 𝑗 =
=
= 0.03
𝑚
1
𝑃 = 𝑅[
𝑖 −𝑚𝑡
)
𝑚
𝑖
𝑚
1−(1+
0.03 −(12(1))
)
1
0.03
1
1−(1+ 0.03)−12
1−(1+
100,000 = 𝑅 [
100,000 = 𝑅 [
1−(1.03)−12
100,000 = 𝑅 [
]
]
0.03
100,000 = 𝑅 [
100,000 = 𝑅 [
]
]
0.03
1−0.7013798802
0.03
0.2986201198
0.03
100,000
9.954003994
]
]
= R ( 9.954003994)
9.954003994
10,046.21 = R
Hence, the amount of yearly withdrawal is P10,046.21.
PERIODIC PAYMENT R OF AN ANNUITY:
Periodic payment R can also be solved using the formula for amount
Future value F or Present Value P of an annuity.
𝑖 𝑚𝑡
𝐹 = 𝑅[
𝑃 = 𝑅[
Note: j =
(1+ 𝑚)
−1
𝑖
𝑚
𝑖 −𝑚𝑡
)
𝑚
𝑖
𝑚
1−(1+
]
𝐹
𝑅=
൬1+
൦
]
𝑅
𝑖
𝑚
n = mt
where R is the regular payment
P is the present value of an annuity
F is the future value of an annuity
j is the interest rate per period
n is the number of payments
𝑖 𝑚𝑡
൰
−1
𝑚
൪
𝑖
𝑚
‫ۍ‬
‫ې‬
‫ۑ‬
𝑃
= ‫ێێ‬
𝑖 −𝑚𝑡 ‫ۑ‬
‫ ێ‬1−൬1+ 𝑚൰
‫ۑ‬
𝑖
‫ۏ‬
‫ے‬
𝑚
What is it…
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..
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.
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.
What I can do…
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?
1. In a certain account providing an interest rate of r compounded quarterly,
P2,500 is deposited every end of the quarter. What value of r will make the
future value of the account P5,200 in six months?
Lesson
2
General Annuity
What I need to know…
At the end of the lesson, the learner will be able to:
✓ Illustrate general annuities
✓ Find the future and present values of general annuities
and compute the periodic payment of a general annuity
✓ Calculate the fair market value of a cash flow stream
that includes an 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
A GENERAL ANNUITY is an annuity where the length of the payment
interval is not the same as the length of the interest compounding period.
❖ GENERAL ORDINARY ANNUITY
A general annuity in which the periodic payment is made at the end of the
payment interval.
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.
Future and Present Value of a General Ordinary Annuity
The Future value F and present value P of a general ordinary annuity
is given by:
𝐹 = 𝑅[
(1+
𝑖 𝑚𝑡
) −1
𝑚
𝑖
𝑚
and
]
𝑃 = 𝑅[
1− (1+
𝑖 −𝑚𝑡
)
𝑚
𝑖
𝑚
]
Note: j =
𝑖
𝑚
Where:
R = is the regular payment
j = is the equivalent interest rate per payment interval
converted from the interest rate per period
n = the number of payments
, n = mt
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.
Cash Flow
F
0
1,000
1,000
1
2
1,000 . . . . . . . . . . . 1,000
3 . . . . . . . . . . . . . 179
1,000
180
(1) Convert 6% compounded quarterly to its equivalent interest rate for
monthly payment interval.
F1 = F2
𝑖12
P (1 +
(1 +
(1 +
(1 +
(1 +
(1 +
𝑖 12
12
𝑖 12
12
12
12𝑡
=
)
𝑖12
12𝑡
𝑖12
12
P (1 +
=
)
12
𝑖4
4
4𝑡
)
4𝑡
𝑖4
(1 + 4 )
0.06 4
=
(1 +
)
12
=
(1.015)4
𝑖12
)
=
((1.015)4 )12
)
12
=
((1.015)3
)
12
𝑖12
12
12
𝑖12
4
)
1
1
1
= (1.015)3 − 1
= 0.00497521 = j
Thus, the interest rate per monthly payment interval is 0.00497521%.
(2) Apply the formula in finding the future value of an ordinary annuity
using the computed equivalent rate.
𝐹 = 𝑅[
(1+ 𝑗)𝑛 −1
𝑗
𝐹 = 1,000 [
]
(1+ 0.00497521)180 −1
0.00497521
]
𝑭 = 𝟐𝟗𝟎, 𝟎𝟖𝟐. 𝟓𝟏
Thus, Cris will have P290,082.51 in the fund after 20 years.
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.
Cash Flow
P=?
R = 38,973.76
0
1
R = 38,973.76
2
R = 38,973.76
3
(1) Convert 8% compounded quarterly to its equivalent interest rate for each
payment interval
F 1 = F2
P (1 +
𝑖1
𝑖1
1𝑡
𝑖1
1
(1 + 1 )
(1 + 1 )
𝑖1
1
(1 + 1 )
𝑖1
(1 + 1 )
𝑖1
(1 + 1 )
𝑖1
1
𝑖1
1
1𝑡
)
1
=
=
4𝑡
𝑖4
P (1 +
)
4
𝑖4
4𝑡
(1 + 4 )
0.08 4
=
(1 +
=
(1.02)4
=
((1.02)4 )
=
((1.02)4
4
)
1
= (1.02)4 − 1
= 0.082432 = j = 8.24%
Thus, the interest rate per payment interval is 0.082432 or 8.24%.
(2) Apply the formula in finding the present value of an ordinary annuity
using the computed equivalent rate j = 0.082432.
𝑃 = 𝑅[
1−(1+ 𝑗)−𝑛
𝑗
𝑃 = 38,973.76 [
𝑃 = 38,973.76 [
𝑃 = 38,973.76 [
𝑃 = 38,973.76 [
]
1−(1+0.082432)−3
0.082432
1−(1+0.082432)−3
0.082432
1−0.7284462444
0.082432
0.2715537556
0.082432
]
]
]
]
𝑃 = 38,973.76[2.565829711]
P = 100,000
Hence, Ken borrowed P100,000 from Kat
A cash flow is a term that refers to payments received (cash inflows)
or payments or deposits made (cash outflows). Cash inflows can be
represented by positive numbers and cash outflows can be
represented by negative numbers.
The fair market value or economic value of a cash flow (payment
stream) on a particular date refers to a single amount that is
equivalent to the value of the payment stream at that date. This
particular date is called focal date.
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
Find the market value of each offer.
SOLUTION:
We illustrate the cash flows of the two offer using time diagram
Mr. Ocampo’s Offer
50,000
1 million
0
1
2
3
4
5
Mr. Cruz’s Offer
50,000
40,000
0
1
40,000
2
40,000
. . . . . . . .. . 40,000
3 . . . . . . . . . . . . . . . . 20
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 = P833,526.20
Mr. Cruz’s Offer: We first compute for the present value of a general annuity
with quarterly payments but with annual compounding period at 5%.
Solve the equivalent rate, compounded quarterly of 5% compounded annually.
F 1 = F2
P (1 +
𝑖4
4(5)
)
4
𝑖4
20
𝑖4
20
(1 + 4 )
(1 + 4 )
20
𝑖4
(1 + 4 )
𝑖4
(1 + 4 )
𝑖4
(1 + 4 )
=
𝑖1
P (1 +
𝑖4
1(5)
)
1
5
=
(1 + 1 )
=
(1 +
=
(1.05)5 -1
=
(1.05)5(20) -1
=
0.012272234
0.05 5
1
)
1
The present value of an annuity is given by
𝑃 = 𝑅[
1−(1+ 𝑗)−𝑛
𝑗
𝑃 = 40,000 [
]
1−(1+0.012272)−20
0.01227222
]
P = 705,572.70
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…
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 value of 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..
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
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.
What I can do…
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 semiannually?
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
3
Deferred Annuity
What I need to know…
At the end
✓
✓
✓
of the lesson, the learner will be able to:
Illustrate a Deferred Annuity
Find the present value of a deferred annuity
Calculate the period of deferral of a 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.
❖ DEFERRED ANNUITY
Annuity (t)
A DEFERRED ANNUITY is a kind of annuity whose payments (or deposits)
start in more than one period from the present.
❖ PERIOD OF DEFERRAL
The time between the purchase of an annuity and the start of the payments
for the deferred annuity.
ILLUSTRATION
0
R*
R . . . . . . . . . . . . . . . R*
R
R. . .
R
1
2
k+1
k+2
k+n
k
Time Diagram for a Deferred Annuity
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
The present value of a k-year deferred annuity at interest rate i
compounded m times ayear with regular payments R for t years is given
by:
𝑃 = (1 +
𝑖
𝑚
−𝑘𝑚
)
𝑅[
𝑖 −𝑚𝑡
)
𝑚
𝑖
𝑚
1−(1+
]
Where:
P = Present value of the deferred annuity
R = regular payment
m = compounding periods
i = interest rate
k = period of deferral
t = time
The only difference of the formula above to the formula of the
present value of an ordinary annuity is the factor (1 +
𝑖
−𝑘𝑚
)
𝑚
.
EXAMPLE 1:
A certain fund is to be established today in order to pay for the P5,000
worth of monthly rent for a commercial space. If the payments for rent will start
next year and the fund must be sufficient to pay for the monthly rental for 2
years, how much must be deposited at 2.5% interest compounded monthly?
SOLUTION:
Consider a 3-year timeline for the illustration. Since the payment will
start next year, then the first year ( 12 compounding periods) is known as the
period of deferral.
The payment will start at the end of the 12th month and end at the end
of the 36th month.
Time
(in months)
0
1
2
. . . . . .. . .
12 13 14 15 16 . . . .
Period of Deferral
35 36
Payment Period
GIVEN: R = 3,000, i = 0.025, m = 12, t = 2 ( since the payment period is 2
years)
𝑃 = (1 +
𝑃 = (1 +
−𝑘𝑚
𝑖
𝑚
)
𝑅[
0.025 −1(12)
12
)
𝑖 −𝑚𝑡
)
𝑚
𝑖
𝑚
1−(1+
5,000 [
]
1−(1+
𝑃 = (1.002083333)−12 (5,000) [
0.025 −(12(2))
)
12
0.025
12
]
1−(1.002083333)−24
0.002083333
]
𝑃 = (0.9753352758) (5,000)[23.38612786]
𝑃 = (0.9753352758) (𝟏𝟏𝟔, 𝟗𝟑𝟎. 𝟔𝟑𝟗𝟑)
𝑷 = 𝟏𝟏𝟒, 𝟎𝟒𝟔. 𝟓𝟖
Thus, the amount of deposit needed today is P114,046.58.
Payment Period
Time
0
(in months)
1
2
. . . . . .. . .
12 13 14 15 16 . . . .
35 36
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…
sheet of paper.
1. Differentiate Deferred Annuity and Period of Deferrral.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2. What is a Deferred Annuity?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
3. What is a period of deferral?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
4. 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..
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.
What I can do…
separate sheet of paper.
3. 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?
4. 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?
5. 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
__________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?
a.) 12
b.) 10
c.) 11
d.) 13
periods
periods
periods
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
b.) 6
c.) 5
d.) 7
semi-annual
semi-annual
semi-annual
semi-annual
intervals
intervals
intervals
intervals
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?
ITEM NO.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
a
c
c
d
b
a
c
c
d
a
b
a
b
c
a
References:
General Mathematics Book pg. 106-112
C &amp; E Publishing, Inc.
By: Lynie Dimasuay, jeric Alcala, Jane Palacio and Alleli Ester Domingo
General Mathematics pg. 168-205
Department of Education Teachers Materials
```