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”.