Ordinary Annuity

Module 7 Ordinary Annuity

Objectives: • Identify the different types of annuity • Solve problems involving ordinary 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 1. Annuity Certain • Annuity payable for a definite duration • Payments begin and end at fixed times

Types of Annuity 1. Annuity Certain • Annuity payable for a definite duration • Payments begin and end at fixed times

2. 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

Kinds of Annuity Certain Simple Annuity »The interest conversion period ( 𝒎 ) is equal to the payment interval (𝒑𝒊) »𝒎 = 𝒑𝒊

Kinds of Annuity Certain 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 (𝑨) is made at the end of each payment interval.

Annuity Due (𝑨 − 𝒅𝒖𝒆) Annuity in which periodic payment (𝑨) is made at the beginning of each payment interval

Deferred Annuity (𝑨 − 𝒅𝒆𝒇.) Annuity in which periodic payment (𝑨) is neither at the beginning nor end of each payment interval but some later date.

Classification of Simple Annuity Ordinary Annuity Annuity in which periodic payment (𝑨) is made at the end of each payment interval.

Annuity Due (𝑨 − 𝒅𝒖𝒆) Annuity in which periodic payment (𝑨) is made at the beginning of each payment interval

Deferred Annuity (𝑨 − 𝒅𝒆𝒇.) Annuity in which periodic payment (𝑨) is neither at the beginning nor end of each payment interval but some later date.

Classification of Simple Annuity Ordinary Annuity Annuity in which periodic payment (𝑨) is made at the end of each payment interval.

Annuity Due (𝑨 − 𝒅𝒖𝒆) Annuity in which periodic payment (𝑨) is made at the beginning of each payment interval

Deferred Annuity (𝑨 − 𝒅𝒆𝒇.) Annuity in which periodic payment (𝑨) is neither at the beginning nor end of each payment interval but some later date.

ORDINARY ANNUITY FORMULAS • Finding F when given A

 (1  i ) n  1 F  A  i   • Finding P when given A

 (1  i )  1 P  A n   i (1  i )  n

Finding A when given F

  i A  F  n  (1  i )  1 Finding A when given P

 i (1  i )  A  P  n  (1  i )  1 n

FORMULAS • Finding F when given A

 (1  i ) n  1 F  A  i   • Finding P when given A

 (1  i )  1 P  A n   i (1  i )  n

Finding A when given F

  i Uniform Series A  F  n Compound ( 1  i )  1   amount factor Finding A when given P

 i (1  i )  A  P  n  (1  i )  1 n

FORMULAS • Finding F when given A

 (1  i ) n  1 F  A  i   • Finding P when given A

 (1  i )  1 P  A n   i (1  i )  n

Finding A when given F

  i A fund  F Sinking  n factor  (1  i )  1 Finding A when given P

 i (1  i )  A  P  n  (1  i )  1 n

Ordinary Annuity - FORMULAS • Finding F when given A

 (1  i ) n  1 F  A  i   • Finding P when given A

 (1  i )  1 P  A n   i (1  i )  n

Finding A when given F

  i A  F  n  (1  i )  1 Finding A when given P

 Uniform  i (1  i )  P seriesApresent  n worth factor (1  i )  1 n

FORMULAS • Finding F when given A

 (1  i ) n  1 F  A  i   • Finding P when given A

Finding A when given F

  i A  F  n  (1  i )  1 Finding A when given P

 i (1  i )   (1  i )  1 Capital A  P P  A   n n  recovery ( 1  i )  1 i ( 1  i )     factor n

n

A man deposits P 12,200 every end of 6 months in an account paying 5.5% compounded semiannually. What amount is in the account at the end of 9 years and 6 months?

A man deposits P 12,200 every end of 6 months in an account paying 5.5% compounded semiannually. What amount is in the account at the end of 9 years and 6 months? (1 + 𝑖)𝑛 −1 𝐹=𝐴 𝑖

A man deposits P 12,200 every end of 6 months in an account paying 5.5% compounded semiannually. What amount is in the account at the end of 9 years and 6 months? (1 + 𝑖)𝑛 −1 𝐹=𝐴 𝑖 𝐴 = 𝑃12,200

𝑖=

5.5% = 2.75% 2

6 𝑛= 9 12

2 = 19

A man deposits P 12,200 every end of 6 months in an account paying 5.5% compounded semiannually. What amount is in the account at the end of 9 years and 6 months? (1 + 𝑖)𝑛 −1 (1 + 0.0275)19 −1 𝐹=𝐴 = 𝑃12,200 𝑖 0.0275 𝐴 = 𝑃12,200

𝑖=

5.5% = 2.75% 2

6 𝑛= 9 12

2 = 19

A man deposits P 12,200 every end of 6 months in an account paying 5.5% compounded semiannually. What amount is in the account at the end of 9 years and 6 months? (1 + 𝑖)𝑛 −1 (1 + 0.0275)19 −1 𝐹=𝐴 = 𝑃12,200 𝑖 0.0275 𝐴 = 𝑃12,200

𝑖=

5.5% = 2.75% 2

6 𝑛= 9 12

2 = 19

𝑭 = 𝑷𝟐𝟗𝟗, 𝟏𝟖𝟎. 𝟕𝟖

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.

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. 𝑫𝑷 = 𝑷𝟐𝟎𝟎, 𝟎𝟎𝟎

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. 𝑫𝑷 = 𝑷𝟐𝟎𝟎, 𝟎𝟎𝟎 𝑨 = 𝑷𝟏𝟖, 𝟎𝟎𝟎

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. 𝑫𝑷 = 𝑷𝟐𝟎𝟎, 𝟎𝟎𝟎 𝑨 = 𝑷𝟏𝟖, 𝟎𝟎𝟎 𝒏 = 𝟑 𝟏𝟐 = 𝟑𝟔 𝟏𝟐% 𝒊= = 𝟏% 𝟏𝟐

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. 𝑫𝑷 = 𝑷𝟐𝟎𝟎, 𝟎𝟎𝟎 𝑨 = 𝑷𝟏𝟖, 𝟎𝟎𝟎 𝒏 = 𝟑 𝟏𝟐 = 𝟑𝟔 𝟏𝟐% 𝒊= = 𝟏% 𝟏𝟐 𝑪𝑽 =?

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. 𝑫𝑷 = 𝑷𝟐𝟎𝟎, 𝟎𝟎𝟎 𝑨 = 𝑷𝟏𝟖, 𝟎𝟎𝟎 𝒏 = 𝟑 𝟏𝟐 = 𝟑𝟔 𝟏𝟐% 𝒊= = 𝟏% 𝟏𝟐 𝑪𝑽 =?

𝐶𝑉 = 𝐷𝑃 + 𝑃

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. 𝑫𝑷 = 𝑷𝟐𝟎𝟎, 𝟎𝟎𝟎 𝑨 = 𝑷𝟏𝟖, 𝟎𝟎𝟎 𝒏 = 𝟑 𝟏𝟐 = 𝟑𝟔 𝟏𝟐% 𝒊= = 𝟏% 𝟏𝟐 𝑪𝑽 =?

𝐶𝑉 = 𝐷𝑃 + 𝑃 (1 + 𝑖)𝑛 −1 𝐶𝑉 = 𝐷𝑃 + 𝐴 𝑖(1 + 𝑖)𝑛

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. 𝑫𝑷 = 𝑷𝟐𝟎𝟎, 𝟎𝟎𝟎 𝑨 = 𝑷𝟏𝟖, 𝟎𝟎𝟎 𝒏 = 𝟑 𝟏𝟐 = 𝟑𝟔 𝟏𝟐% 𝒊= = 𝟏% 𝟏𝟐 𝑪𝑽 =?

𝐶𝑉 = 𝐷𝑃 + 𝑃 (1 + 𝑖)𝑛 −1 𝐶𝑉 = 𝐷𝑃 + 𝐴 𝑖(1 + 𝑖)𝑛 (1 + 0.01)36 −1 𝐶𝑉 = 𝑃200,000 + 𝑃18,000 0.01(1 + 0.01)36

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. 𝑫𝑷 = 𝑷𝟐𝟎𝟎, 𝟎𝟎𝟎 𝑨 = 𝑷𝟏𝟖, 𝟎𝟎𝟎 𝒏 = 𝟑 𝟏𝟐 = 𝟑𝟔 𝟏𝟐% 𝒊= = 𝟏% 𝟏𝟐 𝑪𝑽 =?

𝐶𝑉 = 𝐷𝑃 + 𝑃 (1 + 𝑖)𝑛 −1 𝐶𝑉 = 𝐷𝑃 + 𝐴 𝑖(1 + 𝑖)𝑛 (1 + 0.01)36 −1 𝐶𝑉 = 𝑃200,000 + 𝑃18,000 0.01(1 + 0.01)36 𝑪𝑽 = 𝑷𝟕𝟒𝟏, 𝟗𝟑𝟓. 𝟎𝟗

An LED 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. 𝑪𝑽 = 𝑷𝟏𝟐𝟓, 𝟑𝟕𝟔. 𝟕𝟕

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, a.) how much did Mrs. Alvarez pay in total? b.) what is the cash value of the house and lot? 𝒂. 𝑷𝟏, 𝟑𝟒𝟗, 𝟔𝟎𝟖 𝒃. 𝑪𝑽 = 𝑷𝟏, 𝟎𝟔𝟗, 𝟒𝟑𝟔. 𝟒𝟎

How much monthly deposit must be made for 5 years and 5 months in order to accumulate P 120,000 at 15% compounded monthly?

𝑨 = 𝑷𝟏, 𝟐𝟎𝟕. 𝟓𝟐

What sum 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? 𝑨 = 𝑷𝟐, 𝟔𝟔𝟒. 𝟖𝟐

Dino wants to buy a car worth P740,000. 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? 𝑨 = 𝑷𝟏𝟎, 𝟓𝟔𝟐. 𝟕𝟑

Pam wants to have P 75,000 at the end of 5 years for her graduation expenses. She plans to deposit a certain sum, to achieve this at the end of each month. If her bank pays 15% compounded monthly, what should be the amount of her monthly deposit? 𝑨 = 𝑷𝟖𝟒𝟔. 𝟕𝟒

How much must be paid for 48 months to settle an obligation of P 123,400, if money is worth 12% compounded monthly?

𝑨 = 𝑷𝟑, 𝟐𝟒𝟗. 𝟔𝟎

Assignment

1.

2.

A man deposits P500 every end of the month in an account paying 5 ½% compounded monthly. What amount is in the account at the end of 9 years and 6 months? 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?

3. On April 30, 2020, Connie borrowed P 185,000 at 10% compounded monthly. The loan is to be paid out in 90 equal monthly payments with the first payment on May 31, 2020. What is the size of each monthly payment? 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?