SIMPLE
ANNUITIES
Eng Econ
SIMPLE ANNUITIES
ANNUITY
Is a series of periodic payments (usually equal) made at regular
intervals of time.
Ex: installment payments, monthly rentals, and life insurance
premiums
Payment Interval, pi
 The period of time between consecutive payments.
Term of an Annuity
The time from the beginning of the first payment interval
to the end of the last payment interval.
Interval maybe of any convenient length of time:
Monthly (m = 12)
Quarterly (m = 4)
Semi-annually (m = 2)
Annually (m = 1)
TYPES OF ANNUITIES:
1. Annuity Certain
An annuity for a definite duration not dependent on some outside
contingency. It means that this annuity begins and ends on a definite
or fixed date.
Ex: Thus, monthly payment on a car form an annuity certain because
the payment starts on a fixed date and continue until the required
number of payments has been made.
2. Annuity Uncertain or Contingent Annuity
An annuity payable for an indefinite duration in which the
beginning or the termination is dependent on some certain
event. It means that this annuity’s first or last payment, or
both, depends upon some events.
Ex: thus, pensions and life insurance policies are examples of
annuity uncertain.
Kinds of Annuity Certain
1. Simple Annuity
An annuity whose interest conversion period (m) is equal or the
same as the payment interval (pi).
2. General Annuity
An annuity whose interest conversion period (m) is unequal or
not the same as the payment interval (pi).
Classification of Simple Annuity
1. Ordinary Annuity (Ao)
 Is an annuity in which the periodic payment (R) is made at the end of each payment
interval.
2. Annuity Due (Adue)
 Is an annuity in which the periodic payment (R) is made at the beginning of each
payment interval.
3. Deferred Annuity (Adef)
 Is an annuity in which the periodic payment (R) is not made at the beginning nor at the
end of each payment interval, but some later date.
Notations used:
S = sum or amount of annuity
Rs = periodic payment of the sum
t = term of an annuity
n = no. of conversion period for the
whole term ( t x m)
pi = payment interval
Ao = ordinary annuity
Adue = annuity due
A = present value of an annuity
Ra = periodic payment of the
present value
r = rate of an annuity
m = no. of conversion period per
year
i = interest per conversion period
(r/m)
Adef = deferred annuity
Amount and Present Value of Ordinary Annuity
Amount of an Annuity, S
Is the total of all the periodic payments of the term.
(𝟏+𝒊)𝒏 −𝟏
S = 𝑹𝒔[
]
𝒊
Present Value, A
Is the total of the present value of all the payments of the annuity.
𝟏−(𝟏+𝒊)−𝒏
A = 𝑹𝒂[
]
𝒊
Sample Problems:
1. Find the amount and present value of P 1,500 payable every three months
for 6 years and 6 months. If money is worth 6%, m = 4. (ans. S = P 47,270.95
and A = 32, 097.95)
2. A man deposits P 12,200 every end of 6 months in an account paying 5 ½
% compounded semi-annually. What amount is in the account at the end of
nine years and 6 months? (ans. S = P299,180.78)
3. A home video entertainment set is offered for whole sale for P 18,000
down payment and P 1,800 every three months for the balance, for 18
months. If interest is to be computed at 10%, what is the cash price
equivalent of the set? (ans. CE = D/P + A = P 27,914.63)
1. Find the amount and present value of P 1,500 payable every three
months for 6 years and 6 months. If money is worth 6%, m = 4.
G: R = 1, 500 r = 6% m = 4 t = 6.5 yrs
R: S, A
S: n = 4(6.5) = 26 i = 0.06/4 = 0.015
S = 𝑹𝒔
(𝟏+𝒊)𝒏 −𝟏
𝒊
A = 𝑹𝒂
𝟏−(𝟏+𝒊)−𝒏
𝒊
= 𝟏, 𝟓𝟎𝟎
= 𝟏, 𝟓𝟎𝟎
𝟏+𝟎.𝟎𝟏𝟓 𝟐𝟔 −𝟏
𝟎.𝟎𝟏𝟓
= 𝑷 𝟒𝟕, 𝟐𝟕𝟎. 𝟗𝟓
𝟏− 𝟏+𝟎.𝟎𝟏𝟓 −𝟐𝟔
𝟎.𝟎𝟏𝟓
= 𝑷 𝟑𝟐, 𝟎𝟗𝟕. 𝟗𝟓
2. A man deposits P 12,200 every end of 6 months in an account
paying 5 ½ % compounded semi-annually. What amount is in the
account at the end of nine years and 6 months?
G: R , m, r, t
S: S = 𝑹𝒔
R: S
(𝟏+𝒊)𝒏 −𝟏
𝒊
= 𝟏𝟐, 𝟐𝟎𝟎
𝑺 = 𝑷 𝟐𝟗𝟗, 𝟏𝟖𝟎. 𝟕𝟖
𝟏+𝟎.𝟎𝟐𝟕𝟓 𝟏𝟗 −𝟏
𝟎.𝟎𝟐𝟕𝟓
3. A home video entertainment set is offered for whole sale for P
18,000 down payment and P 1,800 every three months for the
balance, for 18 months. If interest is to be computed at 10%,
what is the cash price equivalent of the set?
G: D/P, R, m, t, r
R: CE = cash equivalent
S: Solve for A = present value (balance)
A = 𝑹𝒂
𝟏−(𝟏+𝒊)−𝒏
𝒊
= 1, 800
1− 1+0.025 −6
0.025
CE = 18,000+9914.63 = P 27,914.63 ans.
= 9914.63
Note:
What Formula to use?
1. If the payment follow the single sum, it is an A problem, while if
payment precede the single sum, it is an S problem.
2. Problems that involve expenses and cash are A problems, and
problems that involve income or revenue are S problems.