LESSON 2.3.3. ANNUITY


Annuity is defined as a series of equal payments occurring at equal interval of
time.
When an annuity has a fixed time span, it is known as annuity certain.
 Types of annuity:
1. Ordinary Annuity
is a type of annuity where the payments are made at the end of each
period beginning from the first period.
2. Annuity Due
is a type of annuity where the payments are made at the beginning of
each period starting from the first period.
3. Deferred Annuity
is a type of annuity where the first payment does not begin until some
later date in the cash flow.
4. Perpetuity
When an annuity does not have a fixed time span but continues
indefinitely, then it is referred to as a perpetuity. The sum of perpetuity is an
infinite value
Unit 2: Money-Time Relationships and Equivalence
37
 Formulating the Equation (Ordinary Annuity)
 Ordinary Annuity Formulas
Unit 2: Money-Time Relationships and Equivalence
38
Example Problem #1
What is the accumulated amount of the five year annuity paying 6,000 at the
end of each school year, with interest at 15% compounded annually?
Solution:
Example Problem #2
A 2001 model car can be purchased with a down payment of 109,000 and
equal monthly installments of 12,000 for 20 months. If money is worth 11%
compounded monthly, what is the equivalent cash price of the car?
Solution:
Example Problem #3
A mother wish to earn 50,000 from an investment after 6 years so that she will
have enough money to celebrate her daughter’s 7 th birthday. What equal amounts
should the mother invest every year for 6 years if interest on the investment is 9%
compounded quarterly?
Solution:
Unit 2: Money-Time Relationships and Equivalence
39
 Annuity Due Formulas
Example Problem #1
A 12,000.00 loan is payable at the beginning of each month for 1 year. If the
interest is 10% compounded monthly, how much is the monthly payment?
Solution:
Example Problem #2
Under a company savings plan, each employee deposits 3,000 at the
beginning of each quarter for 5 years. How much will each employee receive if the
interest is 15% compounded quarterly at the end of 5 years?
Solution:
Example Problem #3
A warehouse can be leased at 50,000.00 a month for 5 years. However, with
a 2M loan payable at the beginning of each month for 5 years, the company could
purchase a new warehouse. Should the company lease or buy, considering that their
expenses for the warehouse should not exceed 50,000.00 and the interest for a loan
is 6% compounded monthly?
Unit 2: Money-Time Relationships and Equivalence
40
Solution:
Example Problem #4
A man owes 10,000.00 with interest at 6% payable semi-annually. What equal
payments at the beginning of each 6 months for 8 years will discharge his debt?
Solution:
Example Problem #5
How much equal deposit should you make at the beginning of each year for
10 years to be able to withdraw 24,000.00 yearly for 8 years?
The first withdrawal is a year after your last deposit. Interest is 13% compounded
annually.
Unit 2: Money-Time Relationships and Equivalence
41
Solution:
Unit 2: Money-Time Relationships and Equivalence
42
POST-TEST 3
Ordinary and Annuity Due
1. A manufacturing firm wishes to give each 80 employees a holiday bonus. How
much is needed to invest monthly for a year at 12% nominal interest rate,
compounded monthly, so that each employee will receive 2,000 bonus?
ANS. P 12,615.81
2. A man wishes to provide a fund for his retirement such that from his 60th to 70th
birthdays he will be able to withdraw equal sums of P18, 000 for his yearly
expenses. He invests equal amount for his 41st to 59th birthdays in a fund
earning 10% compounded annually. How much should each of these amounts
be?
ANS. 2,285.25
3. A civil engineer plans to own a 300 m 2 lot after 5 years for an estimated cost of
570,000. To accumulate this amount, he will make equal year-end deposits in a
fund earning 12% compounded annually. However, at the end of the 2 nd year, he
married his girlfriend and decided to build a 250,000 worth house on the lot he is
planning to buy. What should be his annual deposits for the last 3 years?
ANS. P 163,810.80
4. How much could BTU Oil & Gas Fracking afford to spend on new equipment
every start of the year for 3 years, if it expects a profit of P50 million at the
beginning of the 3rd year, assumed rate of return is 20% per year.
ANS. 13,736,263.74
5. A certain manufacturing plant is being sold and was submitted for bidding. Two
bids were submitted by interested buyers. The first bid offered to pay 200,000.00
each year for 5 years, each payment being made at the beginning of each year.
The second bidder offered to pay 120,000.00 the first year, 180,000.00 the
second year and 270,000.00 each year for the next 3 years, all payments being
made at the beginning of each year. If money is worth 12 % compounded
annually, which bid should the owner of plant accept?
ANS. BID 2 MUST BE ACCEPTED; bid 1 = 807,469.87; bid 2 = 859,727.18
6. A farmer bought a tractor costing 25,000.00 payable in 10 semi-annually
payments, each installment payable at the beginning of each period. If the rate of
interest is 26% compounded semi-annually, determine the amount of each
installment.
ANS. P 4077.20
Unit 2: Money-Time Relationships and Equivalence
43
 Deferred Annuity Formulas
Example Problem #1
Engr. Garcia deposited 100,000 now so that his 2 years old daughter will
receive 5 equal amounts of money yearly starting on her 17 th birthday. If money
earns 11% compounded annually, how much will the girl receive yearly?
Solution:
Example Problem #2
A 100 m2 lot in a certain subdivision costs 120,000 in cash. On the installment
basis, the seller wants a 40,000 down payment and 12 monthly installments, the first
due at the end of the first year after purchase. Determine the amount of the monthly
installments if money is worth 3% compounded monthly.
Solution:
Unit 2: Money-Time Relationships and Equivalence
44
Example Problem #3
Refer to problem #2. If in case the seller agrees to the buyer’s proposal to pay
the supposedly 40,000 down payment on 12 monthly instalments, what would be the
amount of the monthly payments using the same rate of interest? The first payment
is to be made at the end of the first month after the purchase.
Solution:
Example Problem #4
A ten-wheeler, second-hand truck is offered for sale. The owner offers two
methods of payment. The first method requires 200,000 down payment and 15,000
monthly instalments for two years. The second method requires 100,000 down
payment and monthly payments of 30,000 for the first year and 15,000 for the
second and third year. If the rate of interest for both methods of payment is 5%
compounded monthly, which method of payment is better for the buyer and how
much?
Unit 2: Money-Time Relationships and Equivalence
45
Solutions:
Option 1:
Option 2:
Example Problem #5
A member of the cooperative loaned an amount of 80,000 payable in 12 equal
quarterly instalments. The first payment was made a year after the money was
borrowed. How much is each of the quarterly instalments if the rate is 15%
compounded bi-monthly?
Solution:
Unit 2: Money-Time Relationships and Equivalence
46
 Perpetuity Formulas
Example Problem #1
The maintenance of newly acquired equipment is estimated to cost P
5,000.00 at the end of each month. If the money is worth 8% compounded monthly.
What amount of money should be set aside now to take care of all the future
maintenance cost?
Solution:
Example Problem #2
What is the present worth of a pension of P 10,000.00 monthly if money is
worth 7% compounded monthly?
Solution:
Example Problem #3
Find the present value in pesos of a perpetuity of P 15,000.00 payable semiannually if money is worth 8% compounded quarterly.
Solution:
Unit 2: Money-Time Relationships and Equivalence
47
Example Problem #4
What amount of money invested today at 15% interest compounded annually
can provide the following scholarships: P30,000 at the end of each year for 6 years;
P40,000 for the next 6 years; and P50,000 thereafter?
Unit 2: Money-Time Relationships and Equivalence
48
POST-TEST 4
Deferred and Perpetuity
1. Mr. Reyes borrows P600, 000 at 12% compounded annually, agreeing to repay
the loan in 15 equal annual payments. How much of the original principal is still
unpaid after he has made the 8th payment?
ANS. 162,378.06
2. If money is worth 5% compounded semi-annually, find the present value of a
sequence of 12 semi-annually payments of 500 each, the first of which is due to
the end of 4 ½ years.
ANS. P 4209.51
3. An asphalt road requires no repairs until the end of 2 years. At the end of 3 rd
year, P90, 000 will be needed for repairs for the next 5 years, then P120, 000 at
the end of each year for the next 5 years. If money is worth 14% compounded
annually, what is the present value of the repair cost and its equivalent uniform
annual cost for the 12-year period?
ANS. P = 402,386.46, A = 71,089.35
4. P 45,000.00 is deposited in a savings account that pays 5% interest compounded
semi-annually. Equal annual withdrawals are to be made from the account,
beginning one year from now and continuing forever. Compute the maximum
amount of the equal annual withdrawal.
ANS. P 2278.13
5. If money is worth 4%, find the present value of perpetuity of 100.00 payable at
the beginning of each year.
ANS. P 2600
6. What perpetual amount will you receive annually starting next year if you were
able to deposit 130,000.00 five years ago at 10.5% interest compounded
annually?
ANS. P 22,487.65
Unit 2: Money-Time Relationships and Equivalence
49