General Mathematics Quarter 2 – Module 7

Annuities

Department of Education Republic of the Philippines

SENIOR HIGH SCHOOL

General Mathematics – Grade 11 Alternative Delivery Mode Quarter 2– Module 7: Annuities

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General Mathematics Quarter 2 – Module 7:

Annuities

Department of Education Republic of the Philippines

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Simple Annuity Lesson

1

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 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

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.

❖ ANNUITY

An ANNUITY is a sequence of equal payments (or deposits) made at a

regular interval of time.

❖ Term of an Annuity (t)

The time between the first payment interval and the last payment interval.

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.

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

Interest rate per period 𝑗 =𝑖

𝑚 =

0.09

12 = 0.0075

❖ 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.

ILLUSTRATION

R R R R R . . . . . . . . . . . . . . . . . . . R

0 1 2 3 4 5 n

Time Diagram for an n-Payment ordinary annuity

EXAMPLE 1:

(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 1 2 3 4 5 6 (in months) Payment/ Deposit 3,000 3,000 3,000 3,000 3,000 3,000

(2) Add all the future values obtained from the cash flow.

3,000 = 3,000 3,000 (1 + 0.0075) = 3,022.50

3,000 (1 + 0.0075) 2 = 3,045.17

3,000 (1 + 0.0075) 3 = 3,068.01 3,000 (1 + 0.0075) 4 = 3,091.02

3,000 (1 + 0.0075) 5 = 3,114.20

Thus, the amount of this annuity is P18,340.89

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

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 [(1+

𝑖

𝑚)

𝑚𝑡−1

𝑖

𝑚

] 𝐹 = 𝑅 [(1+ 𝑗)𝑛−1

𝑗]

Note: j = 𝑖

𝑚

n = mt

(3) Solution using formula 1

Given:

A(t) = ? R = 3,000 i = 0.09 m = 12 t (annually) = 6/12

𝐹 = 𝑅 [(1+

𝑖

𝑚)

𝑚𝑡−1

𝑖

𝑚

]

= 3,000 [(1+

0.09

12)

12(0.5)−1

0.09

12

]

= 3,000 [(1+ 0.0075)6−1

0.0075]

= 3,000 [(1.0075)6−1

0.0075]

= 3,000 [1.045852235−1

0.0075]

= 3,000 [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.

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:

EXAMPLE 2:

GIVEN:

i = 0.02, m = 12, t = 1.5, and A = P150,000.

𝐹 = 𝑅 [(1+

𝑖

𝑚)

𝑚𝑡−1

𝑖

𝑚

]

150,000 = 𝑅 [(1+

0.02

12)

12(1.5)−1

0.02

12

]

150,000 = 𝑅 [(1+ 0.001666)18−1

0.001666]

150,000 = 𝑅 [(1.001666)18−1

0.001666]

150,000 = 𝑅 [1.030428801−1

0.001666]

150,000 = 𝑅 [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.

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

Interest rate per period 𝑗 =𝑖

𝑚 =

0.09

12 = 0.0075

EXAMPLE 3:

(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 1 2 3 4 5 6 (in months) Payment/ Deposit 3,000 3,000 3,000 3,000 3,000 3,000

(2) Add all the present values obtained from the cash flow.

3,000 (1 + 0.0075) -1 = 2,977.667 3,000 (1 + 0.0075) -2 = 2,955.501 3,000 (1 + 0.0075) -3 = 2,933.50

3,000 (1 + 0.0075) -4 = 2,911.663 3,000 (1 + 0.0075) -5 = 2,889.988

3,000 (1 + 0.0075) -6 = 2,868.474

Thus, the amount of this annuity is 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 [1−(1+

𝑖

𝑚)

−𝑚𝑡

𝑖

𝑚

] 𝑃 = 𝑅 [1−(1+ 𝑗)−𝑛

𝑗]

Note: j = 𝑖

𝑚

n = mt

3,000 (1 + 0.0075) -1

3,000 (1 + 0.0075) -4

3,000 (1 + 0.0075) -5

3,000 (1 + 0.0075) -3

3,000 (1 + 0.0075) -2

3,000 (1 + 0.0075) -6

(3) Solution using formula 2

Given:

P = ? R = 3,000 i = 0.09 m = 12 t (annually) = 6/12

𝑃 = 𝑅 [1−(1+

𝑖

𝑚)

−𝑚𝑡

𝑖

𝑚

]

= 3,000 [1−(1+

0.09

12)

−(12(0.5))

0.09

12

]

= 3,000 [1−(1+ 0.0075)−6

0.0075]

= 3,000 [1−(1.0075)−6

0.0075]

= 3,000 [1−0.9561580178

0.0075]

= 3,000 [0.04384198223

0.0075]

= 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.

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

Interest rate per period 𝑗 =𝑖

𝑚 =

0.03

1 = 0.03

EXAMPLE 4:

𝑃 = 𝑅 [1−(1+

𝑖

𝑚)

−𝑚𝑡

𝑖

𝑚

]

100,000 = 𝑅 [1−(1+

0.03

1)

−(12(1))

0.03

1

]

100,000 = 𝑅 [1−(1+ 0.03)−12

0.03]

100,000 = 𝑅 [1−(1.03)−12

0.03]

100,000 = 𝑅 [1−0.7013798802

0.03]

100,000 = 𝑅 [0.2986201198

0.03]

100,000 = R ( 9.954003994)

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.

𝐹 = 𝑅 [(1+

𝑖

𝑚)

𝑚𝑡−1

𝑖

𝑚

] 𝑅 = 𝐹

൦൬1+

𝑖𝑚

൰𝑚𝑡

−1

𝑖𝑚

൪

𝑃 = 𝑅 [1−(1+

𝑖

𝑚)

−𝑚𝑡

𝑖

𝑚

] 𝑅 =

ۏێێێۍ

𝑃

1−൬1+ 𝑖

𝑚൰−𝑚𝑡

𝑖𝑚 ے

ۑۑۑې

Note: j = 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

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.

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…

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?

Additional Activities…

Answer as indicated. Write your answers in a separate sheet of paper.

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?

General Annuity Lesson

2

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…

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.

❖ 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.

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.

1,000 1,000 1,000 . . . . . . . . . . . 1,000 1,000

0 1 2 3 . . . . . . . . . . . . . 179 180

Cash Flow

EXAMPLE 1:

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

𝑚)

𝑚𝑡−1

𝑖

𝑚

] and 𝑃 = 𝑅 [1− (1+

𝑖

𝑚)

−𝑚𝑡

𝑖

𝑚

]

Note: j = 𝑖

𝑚 , n = mt

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

F

(1) Convert 6% compounded quarterly to its equivalent interest rate for

monthly payment interval.

F1 = F2

P (1 + 𝑖12

12)

12𝑡

= P (1 + 𝑖4

4)

4𝑡

(1 + 𝑖12

12)

12𝑡

= (1 + 𝑖4

4)

4𝑡

(1 + 𝑖12

12)

12

= (1 + 0.06

4)

4

(1 + 𝑖12

12)

12

= (1.015)4

(1 + 𝑖12

12) = ((1.015)4)

112

(1 + 𝑖12

12) = ((1.015)

13

𝑖12

12 = (1.015)

1

3 − 1

𝑖12

12 = 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.

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.

(1) Convert 8% compounded quarterly to its equivalent interest rate for each

payment interval

F1 = F2

P (1 + 𝑖1

1)

1𝑡

= P (1 + 𝑖4

4)

4𝑡

(1 + 𝑖1

1)

1𝑡

= (1 + 𝑖4

4)

4𝑡

(1 + 𝑖1

1)

1

= (1 + 0.08

4)

4

(1 + 𝑖1

1)

1

= (1.02)4

(1 + 𝑖1

1) = ((1.02)4)

1

(1 + 𝑖1

1) = ((1.02)4

𝑖1

1 = (1.02)4 − 1

𝑖1

1 = 0.082432 = j = 8.24%

Thus, the interest rate per payment interval is 0.082432 or 8.24%.

EXAMPLE 2:

P = ? R = 38,973.76 R = 38,973.76 R = 38,973.76

0 1 2 3

Cash Flow

(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 [1−(1+0.082432)−3

0.082432]

𝑃 = 38,973.76 [1−(1+0.082432)−3

0.082432]

𝑃 = 38,973.76 [1−0.7284462444

0.082432]

𝑃 = 38,973.76 [0.2715537556

0.082432]

𝑃 = 38,973.76[2.565829711]

P = 100,000

Hence, Ken borrowed P100,000 from Kat

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 Mr. Cruz’s Offer

P50,000 down payment P1,000,000 after 5 years

P50,000 down payment P40,000 every quarter for 5 years

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:

Find the market value of each offer.

SOLUTION:

We illustrate the cash flows of the two offer using time diagram

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

50,000 1 million

0 1 2 3 4 5

Mr. Ocampo’s Offer

50,000 40,000 40,000 40,000 . . . . . . . .. . 40,000

0 1 2 3 . . . . . . . . . . . . . . . . 20

Mr. Cruz’s Offer

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.

F1 = F2

P (1 + 𝑖4

4)

4(5)

= P (1 + 𝑖1

1)

1(5)

(1 + 𝑖4

4)

20

= (1 + 𝑖4

1)

5

(1 + 𝑖4

4)

20

= (1 + 0.05

1)

5

(1 + 𝑖4

4)

20

= (1.05)5-1

(1 + 𝑖4

4) = (1.05)5(

1

20)-1

(1 + 𝑖4

4) = 0.012272234

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…

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 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..

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 COMPANY B

P150,000 at the end of 3 years

P300,000 at the end of 5 years

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.

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?

Deferred Annuity Lesson

3

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.

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.

ILLUSTRATION

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)

−𝑘𝑚𝑅 [

1−(1+ 𝑖

𝑚)

−𝑚𝑡

𝑖

𝑚

]

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

𝑚)

−𝑘𝑚.

R* R . . . . . . . . . . . . . . . R* R R. . . R

0 1 2 k k+1 k+2 k+n

Time Diagram for a Deferred Annuity

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 0 1 2 . . . . . . . . . 12 13 14 15 16 . . . . 35 36 (in months)

Period of Deferral Payment Period

GIVEN: R = 3,000, i = 0.025, m = 12, t = 2 ( since the payment period is 2

years)

𝑃 = (1 + 𝑖

𝑚)

−𝑘𝑚𝑅 [

1−(1+ 𝑖

𝑚)

−𝑚𝑡

𝑖

𝑚

]

𝑃 = (1 + 0.025

12)

−1(12)5,000 [

1−(1+ 0.025

12)

−(12(2))

0.025

12

]

𝑃 = (1.002083333)−12(5,000) [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.

EXAMPLE 1:

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. 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..

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.

What I can do…

Solve for the following problems. Answer as indicated. Write your answers in a

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 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?

Key Answers…

ITEM NO. ANSWER

1 a

2 c

3 c

4 d

5 b

6 a

7 c

8 c

9 d

10 a

11 b

12 a

13 b

14 c

15 a

References:

General Mathematics Book pg. 106-112 C & E Publishing, Inc. By: Lynie Dimasuay, jeric Alcala, Jane Palacio and Alleli Ester Domingo

General Mathematics pg. 168-205

Department of Education Teachers Materials