Deferred Annuity, Summaries of Mathematics

Download Deferred Annuity and more Summaries Mathematics in PDF only on Docsity!

Deferred Annuity

by CHED on January 12, 2017

lesson duration of 5 minutes

under General Mathematics

generated on January 12, 2017 at 06:51 am

Tags: Annuities

K-12 Teacher's Resource Community

Generated:Generated: Jan 12,2017 02:51 PMJan 12,2017 02:51 PM

Deferred Annuity Deferred Annuity ( 5 mins )( 5 mins )

Written By: Written By: (^) CHED on May 27, 2016CHED on May 27, 2016

Subjects: Subjects: General MathematicsGeneral Mathematics

Tags: Tags: (^) AnnuitiesAnnuities

ResourcesResources

n/a n/a

Content Standard Content Standard

The learner demonstrates understanding of key concepts of simple and compound interests, and simple and general The learner demonstrates understanding of key concepts of simple and compound interests, and simple and general annuities.annuities.

Performance Standard Performance Standard

The learner is able to investigate, analyze and solve problems involving simple and compound interests and simple The learner is able to investigate, analyze and solve problems involving simple and compound interests and simple and general annuities using appropriate business and financial instruments.and general annuities using appropriate business and financial instruments.

Learning Competencies Learning Competencies

The learner calculates the present value and period of deferral of a deferred annuity. The learner calculates the present value and period of deferral of a deferred annuity.

Introduction/Motivation Introduction/Motivation 1 mins1 mins

Ask the students what they would like to do on their 18th birthday. Then describe to them the following situation. Ask the students what they would like to do on their 18th birthday. Then describe to them the following situation.

Iya, who is celebrating her 17th birthday today, does not want an extravagant party for her 18th birth- day. Instead, sheIya, who is celebrating her 17th birthday today, does not want an extravagant party for her 18th birth- day. Instead, she asks her parents if she could receive P500 per month until her 21st birthday. Iya's mom decided to save today so thatasks her parents if she could receive P500 per month until her 21st birthday. Iya's mom decided to save today so that she can provide extra allowance every month after Iya's 18th birth- day. But how much should the mother save todayshe can provide extra allowance every month after Iya's 18th birth- day. But how much should the mother save today so that she will have P200 every month for 3 years which is due exactly one year from now?so that she will have P200 every month for 3 years which is due exactly one year from now?

Tell the students that this is an example of a deferred annuity. Tell them that deferred annuities are series ofTell the students that this is an example of a deferred annuity. Tell them that deferred annuities are series of payments, as what they have already learned in the past lessons on annuities, but will start on a later date.payments, as what they have already learned in the past lessons on annuities, but will start on a later date.

Discuss some examples of this type of annuity in real life.Discuss some examples of this type of annuity in real life.

A credit card company offering its clients to purchase today but to start paying monthly with their choice ofA credit card company offering its clients to purchase today but to start paying monthly with their choice of term after 3 months.term after 3 months. A real estate agent is urging a condominium unit buyer to purchase now and start paying after 3 years whenA real estate agent is urging a condominium unit buyer to purchase now and start paying after 3 years when the condominium is ready for occupancy.the condominium is ready for occupancy.

K-12 Teacher's Resource Community

(d) Define deferred annuity and illustrate. Give the formula for finding the present value.(d) Define deferred annuity and illustrate. Give the formula for finding the present value.

a. Discuss the following problem as an introduction to deferred annuity. Emphasize that for deferred annuities, the starta. Discuss the following problem as an introduction to deferred annuity. Emphasize that for deferred annuities, the start of the payments is on a later date.of the payments is on a later date.

Recall the previous example. What if Mr. Gran is considering another cellular phone that has a different paymentRecall the previous example. What if Mr. Gran is considering another cellular phone that has a different payment scheme? In this scheme, he has to pay P2,500 for 1 year starting at the end of the fourth month. If the interest rate isscheme? In this scheme, he has to pay P2,500 for 1 year starting at the end of the fourth month. If the interest rate is also 9% converted monthly, how much is the cash value of the cellular phone?also 9% converted monthly, how much is the cash value of the cellular phone?

Note that the two payment schemes have the same number of payments n and the same interest rate per period j.Note that the two payment schemes have the same number of payments n and the same interest rate per period j. Their main difference is the start of the payments. The first scheme started at the end of the first interval which makesTheir main difference is the start of the payments. The first scheme started at the end of the first interval which makes it anit an ordinary annuityordinary annuity. The second scheme started on a later date. This annuity is called. The second scheme started on a later date. This annuity is called deferred annuitydeferred annuity ..

In this example, Mr. Gran pays starting atIn this example, Mr. Gran pays starting at (^) the end of the 4th month to the end of the 15th month.the end of the 4th month to the end of the 15th month. (^) The timeThe time diagram for this option is given by:diagram for this option is given by:

Thus, the present value of the cellular phone is P27,953.60. Thus, the present value of the cellular phone is P27,953.60.

Comparing the present values of the two schemes, the present value in the second scheme is lower than the presentComparing the present values of the two schemes, the present value in the second scheme is lower than the present value in the first because the payments in the second scheme will be received on a later date.value in the first because the payments in the second scheme will be received on a later date.

Derive the formula for calculating the present value of a deferred annuity by generalizing the procedure from theDerive the formula for calculating the present value of a deferred annuity by generalizing the procedure from the previous example.previous example.

K-12 Teacher's Resource Community

Consider the following time diagram where k artificial payments of R? are placed in the period of deferral.Consider the following time diagram where k artificial payments of R? are placed in the period of deferral.

EXAMPLE 1. EXAMPLE 1. On his 40th birthday, Mr. Ramos decided to buy a pension plan for himself. This plan will allow him toOn his 40th birthday, Mr. Ramos decided to buy a pension plan for himself. This plan will allow him to claim P10,000 quarterly for 5 years starting 3 months after his 60th birthday. What one-time payment should he makeclaim P10,000 quarterly for 5 years starting 3 months after his 60th birthday. What one-time payment should he make on his 40th birthday to pay off this pension plan, if the interest rate is 8% compounded quarterly?on his 40th birthday to pay off this pension plan, if the interest rate is 8% compounded quarterly?

The annuity is deferred for 20 years and it will go on for 5 years. The first payment is due three months (one quarter) The annuity is deferred for 20 years and it will go on for 5 years. The first payment is due three months (one quarter) after his 60th birthday, or at the end of the 81st conversion period. Thus, there are 80 artificial payments.after his 60th birthday, or at the end of the 81st conversion period. Thus, there are 80 artificial payments.

K-12 Teacher's Resource Community

SeatworkSeatwork 1 mins1 mins

Seatwork 1. Find the period of deferral in each of the following deferred annuity problem (one way to find the period of Seatwork 1. Find the period of deferral in each of the following deferred annuity problem (one way to find the period of deferral is to count the number of artificial payments)deferral is to count the number of artificial payments)

1.1. (a) Monthly payments of P2,000 for 5 years that will start 7 months from now Solution. The first payment is at(a) Monthly payments of P2,000 for 5 years that will start 7 months from now Solution. The first payment is at time 7. The period of deferral will be from time 0 to 6, which is equivalent to 6 periods or 6 months.time 7. The period of deferral will be from time 0 to 6, which is equivalent to 6 periods or 6 months. 2.2. (b) Annual payments of P8,000 for 12 years that will start 5 years from now Solution. Five years from now is(b) Annual payments of P8,000 for 12 years that will start 5 years from now Solution. Five years from now is at time 5. The period of deferral will be from time 0 to time 4. Thus, the period of deferral is 4 periods or 4at time 5. The period of deferral will be from time 0 to time 4. Thus, the period of deferral is 4 periods or 4 years.years. 3.3. (c) Quarterly payments of P 5,000 for 8 years that will start two years from now. Solution. Two years from(c) Quarterly payments of P 5,000 for 8 years that will start two years from now. Solution. Two years from now will be at time 8 if one quarter is considered as one period. Thus, the period of deferral is from time 0 tonow will be at time 8 if one quarter is considered as one period. Thus, the period of deferral is from time 0 to time 7, which is equivalent to 7 quarters or 7 periods.time 7, which is equivalent to 7 quarters or 7 periods. 4.4. (d) Semi-annual payments of P60,000 for 3 years that will start 5 years from now Solution. The first payment(d) Semi-annual payments of P60,000 for 3 years that will start 5 years from now Solution. The first payment is due five years from now which is equivalent to time 10 if payments are made semi-annually. The period ofis due five years from now which is equivalent to time 10 if payments are made semi-annually. The period of deferral will be from time 0 to time 9, which is equivalent to 9 semi-annual intervals or 9 periods.deferral will be from time 0 to time 9, which is equivalent to 9 semi-annual intervals or 9 periods. 5.5. (e) Payments of P3,000 every 2 years for 10 years starting at the end of 6 years Solution. The first payment(e) Payments of P3,000 every 2 years for 10 years starting at the end of 6 years Solution. The first payment is due at the end of 6 years which is at time 3 if payments are made every 2 years. The period of deferral isis due at the end of 6 years which is at time 3 if payments are made every 2 years. The period of deferral is from time 0 to time 2, which is equivalent to 2 periods or 2 two-year intervals.from time 0 to time 2, which is equivalent to 2 periods or 2 two-year intervals.

Seatwork 2. Answer the following problems completely.Seatwork 2. Answer the following problems completely.

1.1. (a) Emma availed of a cash loan that gave her an option to pay P10,000 monthly for 1 year. The first(a) Emma availed of a cash loan that gave her an option to pay P10,000 monthly for 1 year. The first payment is due after 6 months. How much is the present value of the loan if the interest rate is 12%payment is due after 6 months. How much is the present value of the loan if the interest rate is 12% converted monthly? Answer: P107,088.20converted monthly? Answer: P107,088. 2.2. (b) Adrian purchased a laptop through the credit cooperative of their company. The cooperative provides an(b) Adrian purchased a laptop through the credit cooperative of their company. The cooperative provides an option for a deferred payment. Adrian decided to pay after 4 months of purchase. His monthly payment isoption for a deferred payment. Adrian decided to pay after 4 months of purchase. His monthly payment is computed as P3,500 payable in 12 months. How much is the cash value of the laptop if the interest rate iscomputed as P3,500 payable in 12 months. How much is the cash value of the laptop if the interest rate is 8% convertible monthly? Answer: P39,441.148% convertible monthly? Answer: P39,441. 3.3. (c) Mr. and Mrs. Mercado decided to sell their house and to deposit the fund in a bank. After computing the(c) Mr. and Mrs. Mercado decided to sell their house and to deposit the fund in a bank. After computing the interest, they found out that they may withdraw P350,000 yearly for 4 years starting at the end of 7 yearsinterest, they found out that they may withdraw P350,000 yearly for 4 years starting at the end of 7 years when their child will be in college. How much is the fund deposited if the interest rate is 3% convertedwhen their child will be in college. How much is the fund deposited if the interest rate is 3% converted annually? Answer: P1,089,533.99annually? Answer: P1,089,533. 4.4. A group of employees decided to invest a portion of their bonus. After 3 months from today, they want toA group of employees decided to invest a portion of their bonus. After 3 months from today, they want to withdraw from this fund P5,000 monthly for 12 months to fund their gathering that they decide to do everywithdraw from this fund P5,000 monthly for 12 months to fund their gathering that they decide to do every month. How much is the total deposit now if the interest rate is 5% converted monthly? Answer: P57,922.41month. How much is the total deposit now if the interest rate is 5% converted monthly? Answer: P57,922. 5.5. Anna converted her loan to light payments which gives her an option to pay P1,500 every month for 2 years.Anna converted her loan to light payments which gives her an option to pay P1,500 every month for 2 years. The first payment is due 3 months from now. How much is the amount of the loan if the interest rate is 9%The first payment is due 3 months from now. How much is the amount of the loan if the interest rate is 9% converted monthly? Answer: P32,346.70converted monthly? Answer: P32,346.

Evaluation Evaluation 1 mins1 mins

(a) Read the following annuity problem. Fill in the blanks in the statements that follow. (a) Read the following annuity problem. Fill in the blanks in the statements that follow.

A loan of P30,000 is to be repaid monthly for 5 years that will start at the end of 4 years. If interest rate is 12%A loan of P30,000 is to be repaid monthly for 5 years that will start at the end of 4 years. If interest rate is 12% converted monthly, how much is the monthly payment?converted monthly, how much is the monthly payment?

1.1. The type of annuity illustrated in the problem is a _____The type of annuity illustrated in the problem is a _____ 1.1.^ Answer:Answer: deferred annuitydeferred annuity

K-12 Teacher's Resource Community

2.2. The total number of payments is ________.The total number of payments is ________. 1.1. Answer:Answer: 6060 3.3. The number of conversion periods in the period of deferral is ________The number of conversion periods in the period of deferral is ________ 1.1. Answer:Answer: (^) 4747 4.4. The interest rate per period is _________.The interest rate per period is _________. 1.1.^ Answer:Answer:^ 0.010. 5.5. The present value of the loan is ________.The present value of the loan is ________. 1.1.^ Answer:Answer: P30,000P30,

(b) Find the period of deferral in the following problems.(b) Find the period of deferral in the following problems.

1.1. Monthly payments of P10,000 for 8 years that will start 6 months from nowMonthly payments of P10,000 for 8 years that will start 6 months from now 1.1. Answer:Answer: 5 months or 5 periods5 months or 5 periods 2.2. Semi-annual payments of P15,000 for 10 years that will start 5 years from nowSemi-annual payments of P15,000 for 10 years that will start 5 years from now 1.1. Answer:Answer: 9 periods or 9 half-year intervals9 periods or 9 half-year intervals 3.3. Payments of P5,000 every 4 months for 10 years that will start ????ve years from now.Payments of P5,000 every 4 months for 10 years that will start ????ve years from now. 1.1. Answer:Answer: 14 periods or 14 4-month intervals14 periods or 14 4-month intervals 4.4. Annual payments of P600 for 20 years that will start 10 years from nowAnnual payments of P600 for 20 years that will start 10 years from now

1.1. Answer:Answer: 9 years or 9 periods9 years or 9 periods 5.5. Payments of P3,000 every 3 years for 12 years starting at the end of 9 yearsPayments of P3,000 every 3 years for 12 years starting at the end of 9 years

1.1. Answer:Answer: (^) 2 periods or 2 three-year intervals2 periods or 2 three-year intervals

(c) Solve the following problems completely.(c) Solve the following problems completely.

1.1. A loan is to be repaid quarterly for 5 years that will start at the end of 2 years. If interest rate is 6% convertedA loan is to be repaid quarterly for 5 years that will start at the end of 2 years. If interest rate is 6% converted quarterly, how much is the loan if the quarterly payment is P10,000?quarterly, how much is the loan if the quarterly payment is P10,000? 1.1.^ Answer:Answer:^ P154,694.03P154,694. 2.2. A car is to be purchased in monthly payments of P17,000 for 4 years starting at the end of 4 months. HowA car is to be purchased in monthly payments of P17,000 for 4 years starting at the end of 4 months. How much is the cash value of the car if the interest rate used is 12% converted monthly?much is the cash value of the car if the interest rate used is 12% converted monthly? 1.1.^ Answer:Answer: P626,571.56P626,571. 3.3. A mother's savings may allow her to withdraw P50,000 semi-annually for 5 years starting at the end of 5A mother's savings may allow her to withdraw P50,000 semi-annually for 5 years starting at the end of 5 years. How much is the mother's savings if the interest rate is 8% converted semi-annually?years. How much is the mother's savings if the interest rate is 8% converted semi-annually? 1.1. Answer:Answer: P284,930.39P284,930.

Evaluation Evaluation 1 mins1 mins

(a) Read the following annuity problem. Fill in the blanks in the statements that follow. (a) Read the following annuity problem. Fill in the blanks in the statements that follow.

A loan of P30,000 is to be repaid monthly for 5 years that will start at the end of 4 years. If interest rate is 12%A loan of P30,000 is to be repaid monthly for 5 years that will start at the end of 4 years. If interest rate is 12% converted monthly, how much is the monthly payment?converted monthly, how much is the monthly payment?

1.1. The type of annuity illustrated in the problem is a _____The type of annuity illustrated in the problem is a _____ 1.1. Answer:Answer: deferred annuitydeferred annuity 2.2. The total number of payments is ________.The total number of payments is ________. 1.1. Answer:Answer: 6060 3.3. The number of conversion periods in the period of deferral is ________The number of conversion periods in the period of deferral is ________ 1.1. Answer:Answer: 4747 4.4. The interest rate per period is _________.The interest rate per period is _________. 1.1. Answer:Answer: 0.010. 5.5. The present value of the loan is ________.The present value of the loan is ________. 1.1. Answer:Answer: (^) P30,000P30,

(b) Find the period of deferral in the following problems.(b) Find the period of deferral in the following problems.