Annuity
Content
Annuity – a series of equal payments occurring at equal period of time.
A. APPLICATION OF ANNUITY
1. Installment purchase
2. Amortization of loan (amortization – payment of debt by installment usually by equal amounts
and at equal intervals of time)
3. Depreciation
4. Payment of insurance premiums
B. TYPES OF ANNUITY
1. Ordinary annuity - Is one where the payments are made at the end of each period.
P= A [
( 1+i )n−1
]
n
i ( 1+ i )
F=A [
( 1+i )n−1
]
i
2.
Annuity Due - Payments occur at the beginning of each period.
P= A [
( 1+i )n−1 −1
+1]
i ( 1+ i )n −1
F=A [
( 1+i )n+1 −1
−1]
i
3. Deferred annuity - Is one where the first payment is made several periods after the
beginning of annuity.
P= A [
( 1+i )n−1
] (1+ i)−m
n
i ( 1+ i )
F=A [
( 1+i )n−1
]
i
4.
P=
A
i
F=P(1+i)n
Perpetuity – an annuity in which the periodic payments continues indefinitely. (n → ∞)
Problems:
Ordinary Annuity:
1. How much must you invest today in order to withdraw Php 2,000 annually for 10 years if the
interest rate is 9%?
2. Money borrowed today is to be paid in 6 equal payments at the end of 6 quarters. If the interest is
12% compounded quarterly, how much was initially borrowed if quarterly payment is Php
2,000.00?
3. What is the accumulated amount of the 5-year annuity paying Php 6,000 at the end of each year
with interest at 15% compounded annually?
4. A merchant loaned Php 500,000 payable in ten years at an interest rate of 12% compounded
annually. What is the monthly amortization for ten years?
5. What is the present worth of Php 500 deposited at the end of every three months for 6 years if the
interest rate is 12% compounded semiannually?
6. A businessman needs Php 50,000 for his operations. One financial institution is willing to lend him
the money for one year at 12.5% interest per annum (discounted). Another lender is charging
14%, with principal and interest payable at the end of one year. A third financier is willing to lend
him Php 50,000 payable in 12 equal monthly installments of Php 4,600. Which offer is best for
him?
Annuity Due
1. Engr. Dizon borrows P100,000.00 at 10% effective annual interest. He must pay back the loan
over 30 years with uniform monthly payments due on the first day of each month. What does
Engr. Dizon pay each month?
2. Ralph wishes to have P35,000 when he retires 15 years from now. If he can expect to receive 4%
annual interest, how much he set aside in each of 15 equal annual beginning of year deposits?
3. On retirement, a workman finds that his company pension calls for payment of P300 to him or his
estate if he dies at the beginning of each month for 20 years. Find the present value of this
pension at 5% compounded monthly.
4. Under a workman deposits P25 at the beginning of each month for 4 years, and the management
guarantees accumulations at 6% compounded monthly. How much stands to the work man’s
credit at the end of 4 years.
5. A certain property is being sold and the owner received two bids. The first bidder offered to pay
Php 400,000 each year for 5 years, each payment is to be made at the beginning of each year.
The second bidder offered to pay Php 240,000 first year, Php 360,000 second year and Php
540,000 each year for the next 3 years, all payment will be made at the beginning of each year.
If money is worth 20% compounded annually, which bid should the owner of the property accept?
Deferred Annuity
1. On the day his grandson was born, a man deposited to a trust company a sufficient amount of
money so that the boy could receive five annual payments of Php 10,000 each for his college
tuition fees, starting with his 18 th birthday. Interest at the rate of 12% per annum was to be paid on
all amounts on deposit. There was also a provision that the grandson could elect to withdraw no
annual payments and receive a single lump amount on his 25 th birthday. The grandson chose this
option.
a. How much did the boy receive as the single payment?
b. How much did the grandfather deposit?
2. If Php 10,000 is deposited each year for 9 years, how much annuity can a person get annually
from a bank every year for 8 years starting 1 year after the 9 th deposit is made. Cost of money is
14%.
3. A debt of Php 40,000 whose interest rate is 15% compounded semiannually, is to be discharged
by a series of 10 semiannual payments, the first payment to be made 6 months after
consummation of the loan. The first 6 payments will be Php 6,000 each, while the remaining 4
payments will be equal and of such amount that the final payment will liquidate the debt. What is
the amount of the last 4 payments?
Perpetuity:
1. What amount of money is invested today at 15% interest can provide the following scholarships:
Php 30,000 at the end of each year for 6 years; Php 40,000 for the next 6 years and Php 50,000
thereafter?
A. APPLICATION OF ANNUITY
1. Installment purchase
2. Amortization of loan (amortization – payment of debt by installment usually by equal amounts
and at equal intervals of time)
3. Depreciation
4. Payment of insurance premiums
B. TYPES OF ANNUITY
1. Ordinary annuity - Is one where the payments are made at the end of each period.
P= A [
( 1+i )n−1
]
n
i ( 1+ i )
F=A [
( 1+i )n−1
]
i
2.
Annuity Due - Payments occur at the beginning of each period.
P= A [
( 1+i )n−1 −1
+1]
i ( 1+ i )n −1
F=A [
( 1+i )n+1 −1
−1]
i
3. Deferred annuity - Is one where the first payment is made several periods after the
beginning of annuity.
P= A [
( 1+i )n−1
] (1+ i)−m
n
i ( 1+ i )
F=A [
( 1+i )n−1
]
i
4.
P=
A
i
F=P(1+i)n
Perpetuity – an annuity in which the periodic payments continues indefinitely. (n → ∞)
Problems:
Ordinary Annuity:
1. How much must you invest today in order to withdraw Php 2,000 annually for 10 years if the
interest rate is 9%?
2. Money borrowed today is to be paid in 6 equal payments at the end of 6 quarters. If the interest is
12% compounded quarterly, how much was initially borrowed if quarterly payment is Php
2,000.00?
3. What is the accumulated amount of the 5-year annuity paying Php 6,000 at the end of each year
with interest at 15% compounded annually?
4. A merchant loaned Php 500,000 payable in ten years at an interest rate of 12% compounded
annually. What is the monthly amortization for ten years?
5. What is the present worth of Php 500 deposited at the end of every three months for 6 years if the
interest rate is 12% compounded semiannually?
6. A businessman needs Php 50,000 for his operations. One financial institution is willing to lend him
the money for one year at 12.5% interest per annum (discounted). Another lender is charging
14%, with principal and interest payable at the end of one year. A third financier is willing to lend
him Php 50,000 payable in 12 equal monthly installments of Php 4,600. Which offer is best for
him?
Annuity Due
1. Engr. Dizon borrows P100,000.00 at 10% effective annual interest. He must pay back the loan
over 30 years with uniform monthly payments due on the first day of each month. What does
Engr. Dizon pay each month?
2. Ralph wishes to have P35,000 when he retires 15 years from now. If he can expect to receive 4%
annual interest, how much he set aside in each of 15 equal annual beginning of year deposits?
3. On retirement, a workman finds that his company pension calls for payment of P300 to him or his
estate if he dies at the beginning of each month for 20 years. Find the present value of this
pension at 5% compounded monthly.
4. Under a workman deposits P25 at the beginning of each month for 4 years, and the management
guarantees accumulations at 6% compounded monthly. How much stands to the work man’s
credit at the end of 4 years.
5. A certain property is being sold and the owner received two bids. The first bidder offered to pay
Php 400,000 each year for 5 years, each payment is to be made at the beginning of each year.
The second bidder offered to pay Php 240,000 first year, Php 360,000 second year and Php
540,000 each year for the next 3 years, all payment will be made at the beginning of each year.
If money is worth 20% compounded annually, which bid should the owner of the property accept?
Deferred Annuity
1. On the day his grandson was born, a man deposited to a trust company a sufficient amount of
money so that the boy could receive five annual payments of Php 10,000 each for his college
tuition fees, starting with his 18 th birthday. Interest at the rate of 12% per annum was to be paid on
all amounts on deposit. There was also a provision that the grandson could elect to withdraw no
annual payments and receive a single lump amount on his 25 th birthday. The grandson chose this
option.
a. How much did the boy receive as the single payment?
b. How much did the grandfather deposit?
2. If Php 10,000 is deposited each year for 9 years, how much annuity can a person get annually
from a bank every year for 8 years starting 1 year after the 9 th deposit is made. Cost of money is
14%.
3. A debt of Php 40,000 whose interest rate is 15% compounded semiannually, is to be discharged
by a series of 10 semiannual payments, the first payment to be made 6 months after
consummation of the loan. The first 6 payments will be Php 6,000 each, while the remaining 4
payments will be equal and of such amount that the final payment will liquidate the debt. What is
the amount of the last 4 payments?
Perpetuity:
1. What amount of money is invested today at 15% interest can provide the following scholarships:
Php 30,000 at the end of each year for 6 years; Php 40,000 for the next 6 years and Php 50,000
thereafter?
