Simple Deferred Annuity (for Demo).pptx

1. TIME’S UP! Find the cash price of Television Set. monthly payments of P2,000 for 1 year with interest rate of 8% compounded monthly

2. TIME’S UP! Find the present value of a loan: Semi-annual payments of Php8,000 for 10 years with interest rate of 6% compounded semi-annually

Deferred Payment! 3 months

₱2,325.36/

month

1 year

Deferred Payment! 3 months

₱2,325.36/month 1 year

Deferred Annuity - an annuity that does not begin until a given time interval has passed.

Problem. Suppose Mr. Gran wants to purchase a cellular phone. He decided to pay monthly for 1 year starting at the end of the month. How much is the cost of the cellular phone if his monthly payment is P2,500 and interest is at 9% compounded monthly?

Given: R = P2,500 t = 1 year r = 9% or 0.09 m = 12 n = 12

Find: P

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 is also 9% converted monthly, how much is the cash value of the cellular phone?

In this example, Mr. Gran pays starting at the end of the 4th month to the end of the 15th month.

Now, how do we get the present value of this annuity?

Step 1: there are no skipped payments.

In this example, Mr. Gran pays starting at the end of the 4th month to the end of the 15th month. The time diagram for this option is given by: Artificial Payments

Given: R = P2,500 r = 9% or 0.09 m = 12 n = 15 Find: P

Step 1: there are no skipped payments. Step 2: Calculate the present value of the payments made during the period of deferral.

Artificial Payments

Given: R = P2,500

r = 9% or 0.09 m = 12 n=3 Find: P

Step 3: Since the payments in the period of deferral are artificial payments, we subtract the present value of these payments. We obtain:

35,342.49 7,388.89

Assumed no skipped payments Artificial payments

27,953.60 *Thus, the present value of the cellular phone is P27,953.60

Formula to find the present value of deferred simple annuity: 𝑃=

𝑅 1 + 𝑟/𝑚

−𝑘

− 1 + 𝑟/𝑚 𝑟 𝑚

− 𝑘+𝑛

Where: 𝑛 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑜. 𝑎𝑐𝑡𝑢𝑎𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑘 = 𝑛𝑜. 𝑜𝑓 𝑎𝑟𝑡𝑖𝑓𝑖𝑐𝑖𝑎𝑙 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠

Using the formula we can easily compute the deferred annuity

Given: R = P2,500 r = 9% or 0.09 m = 12 n = 12 k= 3 Find: P

𝑃=

𝑃=

𝑅 1 + 𝑟/𝑚

−𝑘

2,500 1 + 0.09/12

− 1 + 𝑟/𝑚 𝑟 𝑚

−3

− 𝑘+𝑛

− 1 + 0.09/12 0.09 12

− 3+12

Period of Deferral - time between the purchase of an annuity and the start of the payments for the deferred annuity.

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 (𝑘)

Monthly payments of P10,000 for 8 years that will start 6 months from now. P 0

1

2

3

4

5

6

7

8

5 months or 5 periods

9 10

Semi-annual payments of P15,000 for 10 years that will start 5 years from now P 0 0.5 1 1.5 2 2.5 . . . 4.5 5 5.5 6

9 periods or 9 half-year intervals

Payments of P5,000 every 4 months for 3 years that will start five years from now. P 0 4m

8m

1y 1y,4m . . . 4y,8m 5y

14 periods or 14 4-month intervals

Payments of P2,000 every 3 months for 8 years that will start after 6 years. P 0 3m 6m 9m 1y

1y,3m

. . . 6y

24 periods or 24 3-month intervals

Payments of P1,000 every other month for 2 years that will start after 3 years . P 0 1m 3m 5m 7m

9m

. . . 3y

18 periods or 18 2-month intervals

Payments of P500 every month for 1 year that will start at the end of the third month . P 0 1m 2m 3m 4m

5m

. . . 13 14

2 periods or 2 months

Payments of P200 every 5 months for 3 years that will start at the end of 5 years. P 0 5m 10m 15m 20m 25m . . . 55m 5y

11 periods or 11 5-month intervals

Open MS Excel

Find the present value (P): R = Php15,000 t = 2 years r = 2.8% m = 12 k=9 n = 24

Let’s try to apply it!

Problem # 1

Given: R = P2,250 r = 8% m = 12 n = 12 Answer: k= 4 Find: P

Problem # 2

Given: R = P750 r = 10% m = 12 n = 24 Answer: k= 3 Find: P

Let’s do some recap!

Question: How is deferred simple annuity differ from an ordinary simple annuity?

Quiz: In a short bond paper, solve the following problems. Show your solution.

1. A savings account may allow the owner to withdraw P30,000 semi-annually for 3 years starting at the end of 3 years. How much is the savings if the interest rate is 4% converted semi-annually?

2. Ruben bought a laptop that is payable by monthly installment of P1,800 for 12 months starting 3 months from now. How much is the cash value of the laptop if interest is at 10% convertible monthly?

ANSWERS

Given: R = P30,000 t = 3 years r = 4% or 0.04 m =2 n =6 k= 5 Find: P

𝑃=

𝑃=

𝑅 1 + 𝑟/𝑚

−𝑘

30,000 1 + 0.04/2

− 1 + 𝑟/𝑚 𝑟 𝑚 −5

− 𝑘+𝑛

− 1 + 0.04/2 0.04 2

− 5+6

Given: R = P1,800 t = 12 months r = 10% or 0.10 m = 12 n = 12 k= 2 Find: P

𝑃=

𝑃=

𝑅 1 + 𝑟/𝑚

−𝑘

1,800 1 + 0.10/12

− 1 + 𝑟/𝑚 𝑟 𝑚

−2

− 𝑘+𝑛

− 1 + 0.10/12 0.10 12

− 2+12

8. Mr. Canlapan deposited his money from selling his old vehicle. The fund would allow him to withdraw P45,000 semi-annually for 5 years starting at the end of 1 year. How much is the amount deposited if the interest rate is 2% converted semi-annually?

9. A cellular phone may be purchased at P1,500 payable monthly for 18 months. The first payment is due after 3 months. How much is the cellular phone if the interest rate is 12% convertible monthly?

Tomorrow bring the following for your PORTFOLIO:

• 2 LONG PAPER FOLDER • 4 BINDER CLIPS • 4 LONG BOND PAPERS *This will be recorded as output (50%).

Time diagram:

Given: R = P10,000 r = 8% or 0.08 m= 4 n = 4 x 5 = 20 k = 4 x 20 = 80 t = 5 years Find: P

𝑃=

𝑃=

𝑅 1 + 𝑟/𝑚

10,000

−𝑘

0.08 1+ 4

− 1 + 𝑟/𝑚 𝑟 𝑚

−80

0.08 − 1+ 4 0.08 4

− 𝑘+𝑛

− 80+20

2. A credit card company offers a deferred payment option for the purchase of any appliance. Rose plans to buy a smart television set with monthly payments of P4,000 for 2 years. The payments will start at the end of 3 months. How much is the cash price of the TV set if the interest rate is 10% compounded quarterly?

Time diagram:

Given: R = P4,000 r = 10% or 0.10 m= 4 𝑚𝑗 = 12 𝑡 = 2 years n = 12 x 2 = 24 k= 2 Find: P

𝑃=

𝑅 1+𝑗

−𝑘

− 1+𝑗 𝑗

− 𝑘+𝑛

4,000 1 + 0.008265 −2 − 1 + 0.008265 𝑃= 0.008265

− 2+24

3. 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% converted monthly?

Time diagram:

Given: R = P10,000 r = 12% or 0.12 m = 12 n = 12 k= 6 Find: P

𝑃=

𝑃=

𝑅 1 + 𝑟/𝑚

10,000

−𝑘

0.12 1+ 12

− 1 + 𝑟/𝑚 𝑟 𝑚 −6

0.12 − 1+ 12 0.12 12

− 𝑘+𝑛

− 6+12

4. 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 is computed as P3,500 payable in 12 months. How much is the cash value of the laptop if the interest rate is 8% convertible monthly?

Time diagram:

Given: R = P3,500 t = 12 months r = 8% or 0.08 m = 12 n = 12 k= 4 Find: P

𝑃=

𝑃=

𝑅 1 + 𝑟/𝑚

3,500

−𝑘

0.08 1+ 12

− 1 + 𝑟/𝑚 𝑟 𝑚

−4

0.08 − 1+ 12 0.08 12

− 𝑘+𝑛

− 4+12

5. 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 years when their child will be in college. How much is the fund deposited if the interest rate is 3% converted semi-annually?

Time diagram:

Given: R = P350,000 r = 3% or 0.03 m= 2 𝑚𝑗 = 1 𝑡 = 4 years n=4 k=6 Find: P

𝑃=

𝑅 1+𝑗

−𝑘

− 1+𝑗 𝑗

− 𝑘+𝑛

4,000 1 + 0.030225 −2 − 1 + 0.030225 𝑃= 0.030225

− 2+24