ANNUITIES

Notes for Annuity
Present Value
𝑃=𝑅
(1)
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
P2500 and interest is at 9% compounded monthly?
Answer: P=28, 587.28
(2)
Suppose Mr. Gran wants to purchase a
cellular phone. He decided to pay monthly for 1 year
starting at the beginning of the month. How much is
the cost of the cellular phone if his monthly payment
is P2500 and interest is at 9% compounded monthly?
Answer: P20, 788.56
(3)
Suppose Mr. Gran wants to purchase a
cellular phone. He decided to pay monthly for 1 year
starting at the 4th month to the end of the 15th month.
How much is the cost of the cellular phone if his
monthly payment is P2500 and interest is at 9%
compounded monthly?
Answer: P=27, 953.60
Formula for Ordinary Annuity
 Future Value
𝐹=𝑅
(1+𝑗)𝑛 −1
𝑗
1−(1+𝑗)−𝑛
𝑗
Where R is the regular payment
j is the interest rate per period
n is the number of payments
(1)
In order to save for her high school
graduation, Maria decided to save P200 at the end of
each month. If the bank pays 0.25% compounded
monthly, how much will her money be at the end of
6 years?
ANSWER: P14, 507.02
(2)
Suppose Mrs. Remoto would like to know the
present value of her monthly deposit of P3, 000 when
interest 9% compounded monthly. How much is the
present value of her savings at the end of 6 months?
ANSWER: P17, 536.79
ANSWER: P38, 803.35
Formula for R in Ordinary Annuity
 Given that we have Future Value,
𝐹𝑗
𝑅=
(1 + 𝑗)𝑛 − 1

Given that we have Present Value
𝑃𝑗
𝑅=
1 − (1 + 𝑗)−𝑛
(5)
A high student would like to save P50, 000 for
his graduation. How much should he deposit in
savings account every month for 5.5 years if interest
is at 0.25% compounded monthly?
P752.46
(3)
Mr. Ribaya paid P 200,000 as down payment
of for a car. The remaining amount is to be settled by
paying P 16, 200 at the end of each month for 5 years.
If interest is 10.5% compounded monthly, what is the
cash price of his car.
ANSWER: P953, 702.20
Formula for Annuity Due
Present Value
𝑃̅ = 𝑅 (1 +
1 − (1 + 𝑗)−(𝑛−1)
)
𝑗
Future Value
DEFINITION
The cash value or cash price is equal to the down
payment plus the value of the installment payments.
1|P age
(4)
Paolo borrowed P100, 000. He agrees to pay
the principal plus interest by paying an equal amount
of money each year for 3 years. What should be his
annual payment if the interest is 8% compounded
annually?
𝐹̅ = 𝑅 (
(1 + 𝑗)𝑛+1 − 1
− 1)
𝑗
ANNUITIES
(1)
For a house and lot, a man made cash down
payment for 2 years ago, and agreed to pay 3, 500
monthly for 10 years starting now. Find the present
value of his monthly payments now if money is
worth 9% compounded monthly.
Answer: 𝑃̅ = 278,368.14
(2)
If 3000 is deposited at every beginning of
each 6 months for 10 years at 7% compounded semiannually, how much is in the fund (a) at the end of 6
½ years after the payment; (b) at the end of 10 years?
Answer: (a) 𝐹̅ = 50, 030.96(b) 𝐹̅ = 87,808.41
Formula for R in Annuity Due

Given that we have Future Value,
𝑅=

(1 + 𝑗)𝑛+1 − 1
(
− 1)
𝑗
𝑃̅
1 − (1 + 𝑗)−(𝑛−1)
(1 +
)
𝑗
(3)
To discharge a debt amounting to 80, 000
pesos, Mr. S.A. Mal agreed to make equal monthly
deposit at the beginning of the month for 5 years.
How much should he deposit monthly if the money is
worth 4%compounded monthly?
Answer: 𝑅 = 1, 468.43
2|P age
Answer: 𝑅 = 1, 115.05
Formula for Present Value of Deferred Annuity
𝑃∗ = 𝑅
1 − (1 + 𝑗)−𝑛
𝑗(1 + 𝑗)𝑘
(1)
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?
Answer: P=107, 088.20
𝐹̅
Given that we have Present Value,
𝑅=
(4)
Mrs. Cha Knock deposits at the beginning of
each quarter to accumulate 100, 000 at the end of 15
years. How much is his quarterly at 5% compounded
quarterly?
Formula for R in Deferred Annuity
𝑅=
𝑃∗ 𝑗(1 + 𝑗)𝑘
1 − (1 + 𝑗)−𝑛
(2)
A house costs P 1,300,000 cash. A buyer
bought it by paying P 300, 000 down payment
and would pay 48 monthly installments, the
first of which is due at the end of one year. If
the rate of interest is 20.4% compounded
monthly, what is the monthly installments?
Answer: P 36,887.08
SEATWORK!!
Determine the kind of annuity used in each problem
then solve.
1.
Deku is looking for an apartment near his
school. An apartment with a monthly rent of P18, 000
is payable at the beginning of each month. If money
is worth 15% compounded monthly. What is the cash
equivalent of 3 years rent?
2.
Peter started to deposit P 5, 000 quarterly at
the end of each term in a fund that pays 1%
compounded quarterly. How much will be the fund
after 6 years?
3.
To create a fund worth 500, 000 at the end of
5 years, 15 members of a cooperative association
contributed an equal amount every beginning of the
month. How much will each member contribute if
money is worth 9%?
4.
Mr. and Mrs. Mercado decided to sell their
house 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 will be in college. How is the fund
deposited if the interest rate is 3% converted
annually?
5.
The buyer of a lot pays P 50, 000 cash and
P10, 000 every month for 10 years. If money is 8%
compounded monthly, how much is the cash value of
the lot?