Annuity

Annuity
- is a series of uniform payments made at equal intervals of time. When an annuity has
a fixed time span it is known as annuity certain.
Annuities are established for the following purposes:
1. As payment of a debt by a series of equal payment at equal time intervals also known
as amortization.
2. To accumulate a certain amount in the future by depositing equal amounts at equal time
intervals. These amounts are called sinking fund.
3. As a substitute periodic payment for a future lump sum payment.
Elements of Annuity
A = periodic payment
P = present worth of all periodic payments
F or S = future worth or lump sum of all the periodic payments after the last payment is
made
i = interest rate per payment
n = number of payments
Types of Annuity
1. Ordinary Annuity – the payment is made at the end of each period starting from the first
period as in the diagram shown.

P F
A A A A A A
Annuity is based on the principles of compound interest. Hence, computation of
the sum of Annuity may be done using the formulas for geometric progression.
The future worth
F = A { [(1+i)n-1)] / i };
(1+i)n-1) / i is called uniform series compound amount factor denoted by (F/A, i, n)
The present worth
P = {A [(1 – (1+i)-n] / i };
[1-(1+i)-n]/i] is known as uniform series present-worth factor and is denoted by
(P/A, i, n).
The value of A with known F is: A = F{i/[(1+i)n -1]};
i/[(1+i)n -1] is called the sinking fund factor denoted by (A/F, i%, n) read as “A
given F at i% in n interest periods”
The value of A with known P is: A = Pi [1- (1+i)-n ]
2. Deferred Annuity – the first payment is deferred a certain number of periods after the first.
P P1 F
A A A A
F = A [(1+i)n-1) / i ]
3. Annuity Due – the payment is made at the beginning of each period starting from the first
period.
P F
A A A A A A A
F = [A(1+i)n-1)] / i
P = {A[(1- ( 1+i)-n] / i } + A
4. Perpetuity – is an Annuity where the payment periods extend forever or the periodic
payments continue indefinitely.
If the payment is made at the end of each period starting from the first period, the
present worth of a perpetuity of A is
P = A/i
Examples:
1. What are the present worth and the accumulated amount of 10- yr annuity paying P10 000
at the end of each year, with interest at 15% compounded annually. (P = 50 188, F= 203
037).
P F
A A A A A A A A A A
P = A[1-(1+i)n]/i ; 10 000 [1-(1+.15)-10]/.15 ; = 50 187.68
F = A[(1+i)n-1]/i ; 10 000 [(1+.15)10-1]/.15 ; = 203 037.18
2. What is the present worth of P500 deposited at the end of every three months for 6 years if
the interest rate is 12% compounded semiannually. (P = 8 504)

Solution:
Solving for i ; (1+ i)4 -1 = (1+ 0.12/2)2 – 1 ; i = 2.96%
P = A (P/A, 2.96%, 24) ; n= 4 * 6 years =24 periods
= 500 [1-(1+.0296)-24]/.0296 = 8 504.37
3. If 10000 is deposited each year for 9 years, how much annuity can a person get annually
from the bank every year for 8 years starting 1 year after the 9 th deposit is made. Cost of
money is 14%. (A= P34675)
F1= 1000[(1 + i)n-1]/i = 160 853.47
P = A [1- (1+i)-n]/i ;
A = 160 853.47/[1-(1.14)-8]/0.14
= 34 675.19
Problems for Practice:
1. A chemical engineer wishes to set up a special fund by making uniform semiannual
end-of-period deposits for 20 years. The fund is to provide P 100 000 at the end of
each of the last 5 years of the 20-year period. If interest is 8% compounded
semiannually, what is the semiannual deposit to be made? (A= P6193.39)
2. A debt of P40 000, whose interest rate is 15% compounded semiannually, is to be
discharged by a series of 10 semiannual payments, the first payment to be made 6
months after consummation of the loan. The first 6 payments will be P6000 each, while
the remaining 4 payments will be equal and of such amount that the final payment will
liquidate the debt. What is the amount of the last 4 payments? (A= P5454)
3. A house and lot can be acquired by a down payment of P500 000 and a yearly payment
of P100 000 at the end of each year for a period of 10 years starting at the end of 5
years from the date of purchase. If money is worth 14% compounded annually, what
is the cash price of the property? (cash price = 808 835.92)
4. Find the present value in pesos of perpetuity of P15 000 payable semiannually if money
is worth 8% compounded quarterly. (P = 371 287.13)
5. Find the annual payment to extinguish a debt of P10 000 payable for 6 years at 12%
interest annually. (A= 2432.26)

Annuity

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Annuity

a fixed time span it is known as annuity certain.

Annuities are established for the following purposes:

intervals. These amounts are called sinking fund.

3. As a substitute periodic payment for a future lump sum payment.

P = present worth of all periodic payments

i = interest rate per payment

period as in the diagram shown.

Annuity is based on the principles of compound interest. Hence, computation of

The future worth

(1+i)n-1) / i is called uniform series compound amount factor denoted by (F/A, i, n)

The present worth

[1-(1+i)-n]/i] is known as uniform series present-worth factor and is denoted by

The value of A with known F is: A = F{i/[(1+i)n -1]};

given F at i% in n interest periods”

The value of A with known P is: A = Pi [1- (1+i)-n ]

payments continue indefinitely.

present worth of a perpetuity of A is

P = A[1-(1+i)n]/i ; 10 000 [1-(1+.15)-10]/.15 ; = 50 187.68

F = A[(1+i)n-1]/i ; 10 000 [(1+.15)10-1]/.15 ; = 203 037.18

the interest rate is 12% compounded semiannually. (P = 8 504)

Solving for i ; (1+ i)4 -1 = (1+ 0.12/2)2 – 1 ; i = 2.96%

P = A (P/A, 2.96%, 24) ; n= 4 * 6 years =24 periods

= 500 [1-(1+.0296)-24]/.0296 = 8 504.37

money is 14%. (A= P34675)

F1= 1000[(1 + i)n-1]/i = 160 853.47

Problems for Practice:

1. A chemical engineer wishes to set up a special fund by making uniform semiannual

each of the last 5 years of the 20-year period. If interest is 8% compounded

semiannually, what is the semiannual deposit to be made? (A= P6193.39)

2. A debt of P40 000, whose interest rate is 15% compounded semiannually, is to be

discharged by a series of 10 semiannual payments, the first payment to be made 6

is the cash price of the property? (cash price = 808 835.92)

is worth 8% compounded quarterly. (P = 371 287.13)

interest annually. (A= 2432.26)

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