(5) Simple Annuities.pdf

SIMPLE ANNUITIES

Learning Objectives • Identify the types of annuities • Find the amount of an ordinary annuity • Find the present value and amount of an annuity payment • Calculate the known rate, time and periodic payment • Find the equivalent cash price • Find the amount of annuity due and deferred annuity

Annuity Annuity is a series/sequence of payments (usually equal) made at equal intervals of time.  Installment payments  Monthly payments/ rentals  Insurance premiums  Monthly retirement benefits  Weekly/ monthly wages

Types of Annuity  Annuity Certain  Annuity payable for a definite duration  Payments begin and end at fixed times  Contingent Annuity or Annuity Uncertain  Annuity payable for an indefinite duration (beginning or termination is dependent on some certain event)  Payments are not certain to be made

Types of Annuity  Simple Annuity  The interest conversion period (𝑚) is equal to the payment interval (𝑝𝑖)  𝑚 = 𝑝𝑖  General Annuity  The interest conversion period (𝑚) is not equal to the payment interval (𝑝𝑖)  𝑚 ≠ 𝑝𝑖

Classification of Simple Annuity  Ordinary Annuity  Annuity in which periodic payment (𝑅) is made at the end of each payment interval.  Annuity Due (A-due)  Annuity in which periodic payment (𝑅) is made at the beginning of each payment interval  Deferred Annuity (A-def)  Annuity in which periodic payment (𝑅) is neither at the beginning nor end of each payment interval but some later date.

Definition of Terms  Payment Interval (𝒑𝒊) or Payment Period  the period of time between successive payments of annuity  monthly (𝑚 = 12)  quarterly (𝑚 = 4)  semi-annually (𝑚 = 2)  annually (𝑚 = 1)  Term of Annuity  the time from the beginning of the first payment interval to the end of the last payment interval  Periodic Payment  the size of each annuity payment

Notations 𝑆 𝐴 𝑅𝑠 𝑅𝑎 𝑡 𝑟 𝑛 𝑚 𝑖

sum or amount of annuity present value of annuity periodic payment of the sum periodic payment of the present value term of annuity rate of an annuity conversion periods of the whole term (𝑡 × 𝑚) number of conversion periods per year interest per conversion period

𝑟

𝑚

Notations 𝑝𝑖 payment interval 𝐴𝑜 ordinary annuity 𝐴(𝑑𝑢𝑒) annuity due 𝐴(𝑑𝑒𝑓) deferred annuity

Ordinary Annuity R

0

1

R

R

R

R

R

2

3

𝒏−𝟐

𝒏−𝟏

𝒏

periods

Term of Ordinary Annuity

Time Value of Money (TVM) The idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This core principle of finance holds that, provided money can earn interest, any amount of money is worth more the sooner it is received

Getting the Amount of Annuity (S) Find the amount of a P5000 ordinary annuity payable annually for four years if money is worth 5% effective.

To get the amount means to accumulate 𝐹 = 𝑃(1 + 𝑖)𝑛

Getting the Amount of Annuity (S)

𝑛

(1 + 𝑖) −1 𝑆 = 𝑅𝑠 𝑖

Getting the Present Value (A) Find the present value of a P5000 ordinary annuity payable annually for 4 years if money is worth 5% effective.

To get the present value means to discount 𝑃 = 𝐹(1 + 𝑖)−𝑛

Getting the Present Value (A)

𝐴 = 𝑅𝑎

1 − (1 + 𝑖) 𝑖

−𝑛

Ordinary Annuity Example 1 Find the amount and present value of P1,500 payable every three months for 6 years and 6 months if money is worth 6%.

Ordinary Annuity Example 2 A car was bought with a down payment of P200,000 and P18,000 at the end of every month for 3 years to discharge all principal and interest at the rate of 12% compounded monthly. Find the cash value of the car.

Ordinary Annuity Example 3

A man deposits P12,200 every end of 6 months in an account paying 5 ½% compounded semi-annually. What amount is in the account at the end of 9 years and 6 months?

Ordinary Annuity Example 4 A home video entertainment set is offered for sale for P18,000 down payment and P1800 every 3 months for the balance, for 18 months. If interest is to be computed at 10% converted quarterly, what is the cash price equivalent of the set?

Ordinary Annuity Example 5 Mrs. Alvarez pays P250,000 cash and the balance in 24 quarterly payments of P45,817 for a house and lot. If money is worth 10% converted quarterly, what is the cash value of the house and lot?

Ordinary Annuity Example 6 At the end of each 6 months for 5 years, a father will deposit P10,000 in a trust fund to provide for his daughter’s education at the end of 5 years. If the money accumulates at 5.24% compounded semiannually, how much will be in the fund a.) at the end of 2 years? b.) after the 7th deposit? c.) after the last deposit?

Ordinary Annuity Example 7 An LCD television set is purchased with a down payment of P30,000 and P4,624.50 at the end of each month for 2 years to discharge all principal and interest at 15% compounded monthly. Find the cash value of the television set.

Periodic Payment (R) of an Ordinary Annuity

𝑆×𝑖 𝑅𝑠 = 𝑛 (1 + 𝑖) −1 𝐴×𝑖 𝑅𝑎 = −𝑛 1 − (1 + 𝑖)

Periodic Payment (R) of an Ordinary Annuity Example 1 How much monthly deposit must be made

for 5 years and 5 months in order to accumulate P 120,000 at 15% compounded monthly?

Periodic Payment (R) of an Ordinary Annuity Example 2 What amount of money will be paid at the end of each quarter for 6 years and 6 months, if the present value is P 50,500 and interest is paid at 10% compounded quarterly?

Periodic Payment (R) of an Ordinary Annuity Example 3 Dino wants to buy a car worth P 740,000. He can pay 40% of the price as down payment and the balance payable every end of the month for 60 months, how much must he pay monthly at 15% compounded monthly?

Periodic Payment (R) of an Ordinary Annuity Example 4 Pam wants to have P 750,000 at the end of 5 years for her graduation expenses. To achieve this, she plans to deposit a certain sum at the end of each month. If her bank pays 15% compounded monthly, what should be the amount of her monthly deposit?

Periodic Payment (R) of an Ordinary Annuity Example 5 On May 31, 2007, Connie invested P 185,000 at 10% compounded monthly. The investment is to be paid out in 90 equal monthly payments with the first payment on June 30,2007. What is the size of each monthly payment?

Periodic Payment (R) of an Ordinary Annuity Example 6 How much must be paid for 48 months to

settle an obligation of P 123,400, if money is worth 12% compounded monthly?

SEATWORK: 1. A fund is to be created by investing P2,800 at the end of every month for 5 years and 10 months. If money is worth 10% compounded monthly, how much is in the fund at the end of the term? 2. Find the cash value of a sala set that can be bought for P15,000 down payment and P1,500 a month for 48 months, if money is worth 14% compounded monthly. 3. An alumnus of a certain school wants to provide a P 180,000 research fellowship at the end of each year for the next 10 years. If the school can invest money at 10% m = 1, how much should the man give now to set up a fund for the scholarship?

SEATWORK: 4. Cocoy wants to accumulate P 230,000 in 9.5 years. Equal deposits are made at the end of each quarter in an account that pays 15% compounded quarterly. What is the size of each deposit? 5. A P50,000 loan is payable in 3 years. To repay the loan, the debtor must pay an amount every 6 months with an interest rate of 6% compounded semi-annually. How much should he pay every six months?

Annuity Due (𝑨 − 𝒅𝒖𝒆)  Annuity in which periodic payment (𝑅) is made at the beginning of each payment period.  The term of an annuity due starts at the time of the first payment and ends one payment period after the date of the last payment.

Annuity Due (𝑨 − 𝒅𝒖𝒆) R

R

R

R

R

0

1

2

3

𝒏−𝟐

A

R

𝒏−𝟏

𝒏

S

Annuity Due (𝑨 − 𝒅𝒖𝒆) ഥ ) of 𝐴 − 𝑑𝑢𝑒 is the value on  The present value (𝐀 the day of the first payment. ത is the sum of the accumulated  The amount (𝐒) value of the payments at the end of the term

Sum and Present Value of 𝑨 − 𝒅𝒖𝒆 𝑆ҧ = 𝑅𝑠 𝐴ҧ = 𝑅𝑎

(1 + 𝑖)

(𝑛+1)

𝑖 1 − (1 + 𝑖

−1

−1

−(𝑛−1) 𝑖)

+1

Annuity Due (𝑨 − 𝒅𝒖𝒆) NOTE: Problems that involve expenses and cash are

A problems, and problems that involve income or

revenue are S problems.

Annuity Due (𝑨 − 𝒅𝒖𝒆) Example 1 Heart wants to buy a computer set within a year. She decides to make regular deposits of P3,000 at the start of every month, her money earning 5% compounded monthly. How much will she have in her savings a year after? S Problem

Annuity Due (𝑨 − 𝒅𝒖𝒆) Example 2 Find the cash equivalent of an item that was purchased for P18,000 down payment and P2,500 at the beginning of each six months for 3½ years, if interest is 5½ % compounded semi-annually.

A Problem

Annuity Due (𝑨 − 𝒅𝒖𝒆) Example 3 Nette bought a brand new car. What is the cash price of the car if she has to make 36 quarterly payments of P22,000 at the beginning of each quarterly period at 10.5% compounded quarterly?

A Problem

Deferred Annuity (𝑨 − 𝒅𝒆𝒇)  Annuity in which periodic payment (𝑅) is neither at the beginning nor end of each payment interval but some later date.  Deferment period is the length of time for which there are no payments.  First payment occur after the deferment period.

Deferred Annuity (𝑨 − 𝒅𝒆𝒇) R

0

1

2

3

𝒌

R

𝒌+𝟏

𝒌+𝟐

A

R

𝒌+𝒏

S Period of deferment

Ordinary Annuity of 𝑛 payments

Present Value of 𝑨 − 𝒅𝒆𝒇  The value of the annuity at point 0  Compute the present value (𝑨) of the ordinary annuity of the 𝒏 payments of 𝑹 at point 𝒅  Discount 𝑨 for 𝒅 periods 𝒏 = number of payments made 𝒅 = number of payments missed

1− 1+𝑖 𝐴𝑑 = 𝑅 𝑖

−𝑛

1+𝑖

−𝑑

Sum or Amount of 𝑨 − 𝒅𝒆𝒇

𝑆𝑑 = 𝐴𝑑 1 + 𝑖

𝑑+𝑛

Deferred Annuity (𝑨 − 𝒅𝒆𝒇) Example 1 Find the present value of a deferred annuity of P900 every three months for 5 years that is deferred for 3 years, if money is worth 10% compounded quarterly.

Deferred Annuity (𝑨 − 𝒅𝒆𝒇) Example 2 Find the present value of a deferred annuity of P4,800 every six months for 7 years, if the first payment is made after 4 years, and money is worth 11% compounded semi-annually.

Deferred Annuity (𝑨 − 𝒅𝒆𝒇) Example 3 In a series of quarterly payments of P5,700 each, the first payment is due at the end of 5 years and the last at the end of 10 years and 9 months. If money is worth 6% compounded quarterly, find the present value of the deferred annuity.

Deferred Annuity (𝑨 − 𝒅𝒆𝒇) Example 4 Find the present value of an annuity of P33,000 payable at the end of each year if the first payment is made at the end of 3 years and the last payment is made at the end of 9 years. Assume money is worth 10% effective.

Deferred Annuity (𝑨 − 𝒅𝒆𝒇) Example 5 Find the present value of 10 semi-annual payments of P3,000 each if the first payment is due at the end of 3 1/2 years and money is worth 12% compounded semi-annually.

Deferred Annuity (𝑨 − 𝒅𝒆𝒇) Example 6 Find the present value of a P4,500 annuity payable annually for 7 years and is deferred for 2 years if money is worth 8% effective.

Deferred Annuity (𝑨 − 𝒅𝒆𝒇) Example 7 A house costs P1.3 million cash. A buyer bought it by paying P300,000 down payment and would pay 48 monthly installments, the first of which is due at the end of 1 year. If the rate of interest is 20.4% compounded monthly, what is the monthly installment?

Deferred Annuity (𝑨 − 𝒅𝒆𝒇) Example 8 Find the quarterly payment for 21 quarters to discharge an obligation of P120,000 if money is worth 4 1/2% compounded quarterly and the first payment is due at the end of 3 years and 9 months.

SEATWORK: 1. An investment of P5,500 is made at the beginning of each month for 5 years and 5 months. If interest is 15% compounded monthly, how much will the investment be worth at the end of the term? 2. Polly purchased a car. He paid P 150,000 as a down payment, and P 5,500 payable at the beginning of each month for 48 months. If money is worth 12% compounded monthly, what is the equivalent cash price of the car? 3. Pompei will pay off a debt of P 120,000 by equal payments every beginning of each quarter for 10 years. If interest is charged at 11% compounded quarterly, what will be the size of each payment?

SEATWORK: 4. Find the present value of a series of quarterly payments of P 950 each, the first payment is due at the end of 2 years and 3 months, and the last at the end of 5 years and 6 months, if money is worth 15% compounded quarterly. 5. Find the monthly payment for 36 periods to discharge an obligation of P88,000, if money is worth 12%, m=12 and the first payment is due at the end of 1 year and 3 months.