This document defines and explains different types of annuities, including ordinary annuity, annuity due, and deferred annuity. It provides formulas to calculate the present value and regular payments of each type of annuity. Key details include that ordinary annuity payments are made at the end of each interval, annuity due payments are made at the beginning of each interval, and deferred annuity payments begin at a later time.

1. Annuity

2. Annuity
Is a sequence of equal
payments made at equal
period.

3. Types of Annuity

4. Annuity Certain
Whose payments begin
and end at fixed time.

5. Contingent Annuity
Whose payment made upon
an event that cannot be
foretold accurately.

6. Simple Annuity
Is an annuity in which the
payment Interval is the same
as the interest period.

7. General Annuity
Is an annuity where the
length of the payments
interval is not the same as
the length of interest
compounding period.

8. Three kinds of
Annuity

9. Ordinary Annuity Annuity Due Deferred Annuity
Payments made at the end
of each interval.
Payments made at the beginning of
each interval.
Is an annuity in which
the first payment is
made at some later
time.
The amount of an ordinary
annuity, S:
-Is the value at the end of
the term.
-Is the value on the last
payment date.
-is the sum of the
accumulated payments at
the end of the term.
𝑺 = 𝑹[
𝟏 + 𝒊 𝒏 − 𝟏
𝒕
]
𝑺 = 𝑹[
𝟏 + 𝒊 𝒏+𝟏
− 𝟏
𝒕
− 𝟏]

10. Ordinary Annuity Annuity Due Deferred Annuity
The present value
of annuity denoted
by A:
- Is the value
one period
before the first
payment.
- -is a sum of
discounted
payments at
the beginning
of the terms
𝑨 = 𝑹[
𝟏−(𝟏+𝒊)−𝒏
𝒊
]
𝑨 = 𝑹[𝟏 +
𝟏−(𝟏+𝒊)−(𝒏−𝟏)
𝒊
]
𝑨 = 𝑹[
𝟏−(𝟏+𝒊)−(𝒏+𝒅)
𝒊
-
𝟏−(𝟏+𝒊)−(𝒅)
𝒊
]

11. Ordinary Annuity Annuity Due Deferred Annuity
Regular Payment
𝑅 =
𝑠
(
1 + 𝑖 𝑛 − 1
𝑖
)
𝑅 =
𝐴
1 − (1 + 𝑖)−𝑛
𝑖
𝑅 =
𝑆
(
1 + 𝑖 𝑛+1 − 1
𝑖
− 1)
𝑅 =
Ā
1 +
1 − (1 + 𝑖)−(𝑛−1)
𝑖
𝑅
=
𝐴𝑑
(
1 − 1 + 𝑖 − 𝑛+𝑑
𝑖
−
1 − (1 + 𝑖)−𝑑
𝑖

12. Symbols/ Terminologies
 The payment period is the time between
successive payments of an annuity.
 The term(n) of an annuity is the time between
the first payment interval and the last payment
interval.
 The periodic payment(R) is the size /amount
of each payment
 j: nominal rate, m=number of conversion
 𝒊 =
𝒋
𝒎
𝒏 = 𝒎𝒕

13. Examples

14. 1. A car bought a down payment
of 300 000 pesos and 20, 000 at
the end of every month for three
years to discharge all principal
and interest at the rate of 12%
compounded monthly. Find the
cash price of a car.

15. 2. The monthly rent of an
apartment is 18, 000 payable at
the beginning of each month. If
money is worth 15%
compounded monthly, What is
the cash equivalent for 3 years
rent?

16. 3. A house cost 1 300 000 cash.
A buyer bought paying 300 000
pesos down payment and would
pay 48 monthly instalments, the
first of which is due at the end of
one year. If the rate interest is
20.4% compounded monthly,
What is the monthly instament?