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Definition of Annuity Due
•An annuity due is a series of
equal periodic payments which
are due at the beginning of the
period.
Definition of Annuity Due
•Hence, in an annuity due, the first
payment is made at once and the
last payment is made one interval
before the end of the term.
Final Value of Annuity Due
•The amount of an annuity due
is the value of the annuity at
the end of its term.
Final Value of Annuity Due
•The first payment is due now; that is, at the
beginning of the term. The nth or last
payment is due at the beginning of the nth
period, that is, one period before the end of
the term.
Final Value of Annuity Due
•To denote the final value of
an annuity due we will use
majuscule letter “𝑆𝑑𝑢𝑒”.
Something to think about…
•What do ordinary annuity
and annuity due have in
common?
Something to think about…
•The number of periodic payments for
an annuity due and an ordinary annuity
are the same if they have similar length
of terms or period of coverage.
Something to think about…
•How do ordinary
annuity and annuity due
differ?
Something to think about…•The last payment for the annuity due is
made at the beginning of the last rent
period. While, the payment for an
ordinary annuity are made at the end of
each rent period.
Formula:
•𝑺𝒅𝒖𝒆 = 𝑹(𝟏+𝒊)𝒏+𝟏−𝟏
𝒊− 𝟏
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟓. 𝟏
Example 5.1
•An annuity contract provides for the payment
of P1,200.00 at the beginning of 6 months for
8 years. If money is worth 7%, compounded
semiannually, what is the amount of annuity
at the end of 8 years?
Final Answer:
•The amount of annuity
at the end of 8 years will
be P26,046.02.
The Present Value of the Annuity Due
•To denote the present value of a
simple annuity due we will use
the majuscule letter “𝐴𝑑𝑢𝑒".
Formula:
•𝑨𝒅𝒖𝒆 = 𝑹𝟏−(𝟏+𝒊)𝟏−𝒏
𝒊+ 𝟏
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟓. 𝟐
Example 5.2:•A Punongbayan & Araullo accounting firm acquired
computers under a capital lease agreement. They
pays the lessor P300,000.00 per year at the beginning
of each year for six years. If they can obtain six-year
financing at 10% compounded annually, what is the
long-term lease liability will they report in their
financial statements?
Example 5.2:
•The initial lease
liability is
P1,437,236.03.
Periodic Payment of Annuity Due
•The periodic payment of a simple
annuity due is the size of the annuity
payment made at the beginning of
the equal interval/period.
Periodic Payment of Annuity Due
•To denote the periodic payment
of a simple annuity due we will
use the majuscule letter “𝑅𝑑𝑢𝑒".
Two Methods:
•The Periodic Payment based from the Final
Value (𝑆𝑑𝑢𝑒) of a Simple Annuity Due; and
•The Periodic Payment based from the
Present Value (𝐴𝑑𝑢𝑒) of a Simple Annuity Due.
Formula for Method 1:
•𝑹𝒅𝒖𝒆 =𝑺𝒅𝒖𝒆𝒊
(𝟏+𝒊)𝒏+𝟏−(𝟏+𝒊)
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟓. 𝟑
Example 5.3:
•If you would like to accumulate P3,500,000.00 in
15 years, what amount must you contribute each
year if the investment earns 13% compounded
annually and the contributions are made at the
beginning of each year?
Final Answer:
•You need to make an
annual contribution of
P76,633.83.
Formula for Method 2:
•𝑹𝒅𝒖𝒆 =𝑨𝒅𝒖𝒆𝒊
𝟏+𝒊 −(𝟏+𝒊)𝟏−𝒏
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟓. 𝟒
Formula for Method 2:•A lease that was 4 years to run is recorded on a
company’s book as a liability of P1,000,000.00. If
the company’s cost of borrowing was 15%
compounded monthly when the lease was
signed, what is the amount of the lease payment
made at the beginning of each month?
Final Answer:
•The monthly lease
payment is P27, 487.16.
Let’s Practice 5.1: Solve the following problems:1. The present value of a series of payments at the beginning of each 3 months
for 7 ½ years is P20,000.00. If money is worth 8% compounded quarterly, what is
the amount of each quarterly payment?
2. How much will Madam Kaycee accumulate in her insurance policy by age of
60 if the first semiannual contribution of P10,000.00 is made on her 28th
birthday and the last is made six months before her birthday? Assume that her
insurance policy earns 11% compounded semiannually.
3. A LCD projector is bought for P2,000.00 a month for 24 months. If the interest
charged at 24% compounded monthly, what is the cash price or present value of
the unit?
Assignment 5.1: Solve the following problems:
•Xian purchased a sports car. He paid P450,000.00 down and
P25,000.00 payable at the beginning of each month for 3 years. If
money is worth 15% compounded monthly, what is the equivalent
cash price of the car?
•Ten years from today, the SGV Company will need P1,000,000.00 to
replace worn out equipment. Beginning today, what bimonthly
deposits must be made in fund paying 18% compounded bimonthly
for 10 years to accumulate this sum?
Definition of Annuity Due
•When an annuity in which the first
payment interval is delayed, or
deferred, for a period of time it is
called deferred annuity.
Interval of Deferment (d)
•It is the time interval to
the beginning of the first
payment interval.
Interval of Deferment (d)•Moreover, it is the length of time
between now and the beginning of the
term of the deferred annuity, which ends
one period before the first payment is
due.
Interval of Deferment (d)
•It is also called as “period of
deferral” and we will denote
this using miniscule letter.
Note:•The first payment here in
deferred annuity is due d+1
period, hence, and the nth
payment d+n periods hence.
Final Value of Deferred Annuity
•The final value of an annuity
for n periods, deferred m
periods, is the value of the
annuity at the end of its term.
Final Value of Deferred Annuity
•This amount is that of an ordinary
annuity for n periods, since no
payments are made during the
interval of the deferment.
Final Value of Deferred Annuity
•We will denote the final
value of the deferred
annuity as 𝑆𝑑𝑒𝑓 .
Note:
•The focal date for the final value of a
deferred annuity is at the time of the last
payment. At this focal date, the deferred
annuity is the same with from an n-payment
ordinary simple annuity ending on this date.
Therefore,
•𝑆𝑑𝑒𝑓 will equal to the final
value of 𝑆𝑛 (final value of
ordinary annuity).
Formula:
•𝑺𝒅𝒆𝒇 = 𝑹𝒅𝒆𝒇𝟏+𝒊 𝒏−𝟏
𝒊
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟓. 𝟓
Present Value of a Deferred Annuity•The present value of an annuity for
n periods, deferred d periods is the
value of the annuity at the
beginning of the interval of
deferment.
Present Value of a Deferred Annuity
•To denote the present value
of an annuity due we will use
majuscule letter “𝐴𝑑𝑒𝑓”.
Present Value of a Deferred Annuity
•This is not the same as the value of an ordinary
annuity for n periods at the beginning of its term.
However, the deferred annuity may be thought of
as an annuity for d+n periods in which the first d
payments are withheld.
Hence,
•Hence, the present value of the
deferred annuity is equal to the
present value of the ordinary
annuity for d periods.
Derivation of Formula:
𝑨𝒅𝒆𝒇 =𝑨𝒏
(𝟏+𝒊)𝒅
Formula:
•𝑨𝒅𝒆𝒇 = 𝑹𝒅𝒆𝒇(𝟏+𝒊)−𝒅−(𝟏+𝒊)−(𝒏+𝒅)
𝒊
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟓. 𝟔
Hence,•Joselito is setting up a fund to help finance his son’s
college education. He wants his son to be able to
withdraw P40,000.00 at the end of every three months
for 2 years starting in the end of 6 years. If the fund
can earn 8% compounded quarterly, what single
amount contributed today will provide for the
payments?
Derivation of Formula:
•𝑨𝒅𝒆𝒇 = 𝑹𝒅𝒆𝒇(𝟏+𝒊)−𝒅−(𝟏+𝒊)−(𝒏+𝒅)
𝒊
Derivation of Formula:
•𝑹𝒅𝒆𝒇 =𝑨𝒅𝒆𝒇𝒊
(𝟏+𝒊)−𝒅−(𝟏+𝒊)−(𝒏+𝒅)
𝑭𝒐𝒓𝒎𝒖𝒍𝒂 𝟓. 𝟕
Example 5.4:•Matibay Appliance Center is planning a promotion on a
washing machine with a price of P7,000.00. Buyers will pay
“no money down and no payments for 6 months.” The first of
12 equal monthly payments are required six months from the
purchased date. What should the monthly payment be if the
store is to earn 18% compounded monthly on its account
receivable during both the deferral period and the repayment
period?
Let’s Practice 5.4: Solve the following problems.
•Find the present value of a man’s pension of P13,000.00 payable
monthly, the first due at the end of 1 year, and the last at the end of 5
years, if money is worth 6% compounded monthly.
•A man borrowed P24,000.00 from SSS calamity loan with interest at
12% compounded monthly. At the end of 3 months he made the first
payment of 24 monthly payments which fully discharged his debt.
Find the monthly payment.
Assignment 5.2: Solve the following problems.
•What price will a finance company pay to a merchant for a
conditional sale contract that requires 18 monthly payments of
P5,600.00 beginning seven months, if the finance company requires a
rate of return of 12% compounded monthly?
•Micah borrowed P38,000.00 from the farmers’ cooperative that
charges interest at 11% compounded quarterly. She promised to pay
of the loan in 9 quarterly payments. The first payment is to be made at
the end of 4 years. Find the quarterly payment.
Definition of Perpetuity
•It is a type of annuity whose
payments begin at a fixed date
and continue forever.
Examples of Perpetuity
•Some examples are: endowments of
charitable institutions, interest
payments on perpetual bonds, and
dividends on preferred stocks.
Furthermore,
•It is sometimes called
as “perpetual annuity”.
Furthermore,
•It is sometimes called
as “perpetual annuity”.