Annuity Due
Annuity Due – a type of annuity were
equal payments are made at the
beginning of each period.
𝑃 =𝐴+
𝐹=
𝐴 (1 + 𝑖)𝑛−1 − 1
[
]
(1 + 𝑖)𝑛−1
𝑖
𝐴
[(1 + 𝑖)𝑛+1 − 1] − 𝐴
𝑖
Where:
P = value or sum of money at present
F = value or sum of money at some future
time
A = series of periodic equal amount of
payments
i = interest rate per interest period
n = number of interest periods/number
of equal payments
SAMPLE PROBLEMS
Example 1: If money is worth 4%
compounded semiannually, find the
present amount of an annuity due paying
P 5,000 semiannually for a term of 3.5
years.
Ans. P 33,007.15
Example 2: A man agrees to make equal
payments at the beginning of each 6
months for 10 years to discharge a debt
of P 50,000 due now. If money is worth
8% compounded semiannually, find the
semiannual payment.
Ans. P 3,537.58
Example 3: To accumulate a fund of P
80,000 at the end of 10 years, a man will
make equal annual deposits in the fund
at the beginning of each year. How much
should he deposit if the fund is invested
at 5% compounded annually?
Ans. P 6,057.49
Example 4: Determine the present worth
and the accumulated amount of an
annuity consisting of 6 payments of
P120, 000 each, the payment are made at
the beginning of each year. Money is
worth 15% compounded annually.
Ans. P = P 522,259; F = P 1,208,016
Example 5: A farmer bought a tractor
costing P 25,000 payable in 10 semiannual 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 4,077.20
Example 6: 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 P 200,000 each year for 5 years, each
payment being made at the beginning of
each year. The second bid offered to pay
P 120,000 the first year, P 180,000 the
second year, and P 270,000 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
the plant accept?
Ans. second bid, Present worth = P
859,727.18
Deferred Annuity
Deferred Annuity – a type of annuity were
the first payment is made several periods
after the beginning of annuity.
𝐴 (1 + 𝑖)𝑛 − 1
𝑃= [
] (1 + 𝑖)−𝑚
𝑛
(1
𝑖
+ 𝑖)
Where:
P = value or sum of money at present
F = value or sum of money at some future
time
A = series of periodic equal amount of
payments
i = interest rate per interest period
n = number of interest periods/number
of equal payments
m = number of interest periods when
there is no payment made
SAMPLE PROBLEMS
Example 1: The present value of an
annuity of R pesos payable annually for 8
years, with the 1st payment at the end of
10 years is P 187,481.25. Find the value
of R if money if money is worth 5%.
Ans. P 45,000
Example 2: A parent on the day that child
is born wishes to determine what lump
sum would have to be paid into an
account bearing interest of 5%
compounded annually, in order to
withdraw P 20,000 each on the child’s
18th, 19th , 20th and 21th birthdays?
Ans. P 30,941.73
Example 3: An asphalt road requires no
upkeep until the end of 2 years when
P60, 000 will be needed for repairs. After
this P90, 000 will be needed for repairs
at the end of each year 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
was the equivalent uniform annual cost
for the 12-year period?
Ans. P 79,245.82
Example 4: 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. P 2,285.25
Example 5: A lathe for a machine shop
costs P 60,000, if paid in cash. On the
installment plan, a purchaser should pay
P 20,000 downpayment and 10 quarterly
installments, the first due at the end of
the first year after purchase. If money is
worth 15% compounded quarterly,
determine the quarterly installment.
Ans. P 5,439.18
Example 6: A man invests P 10,000 now
for the college education of his 2 year old
son. If the fund earns 14% effective
interest rate, how much will his son get
each year starting from his 18th to the
22nd birthday?
Ans. P 20,791.64