ANNUITIES
-a series of equal payments occurring at equal periods of time
Kinds of Annuities:
a. Ordinary Annuity
b. Deferred Annuity
c. Annuity Due
d. Perpetuity
ORDINARY ANNUITY
-payments are made at the end of each period
P
1
2
3
4
n-1
n
______________________________......._________
0
A
A
A
A
A
A
Where:
P = present value of money
F = future value of money
A = periodic payment
n = number of interest periods
i = interest rate per period
Finding P when A is given:
Write the equation of value...
P = A(1+i)-1 +A(1+i)-2 + A(1+i)-3 +…+A(1+i)-(n-1) + A(1+i)-n
(eq. 1)
Multiply both sides by (1+i)…
P(1+i) = A + A(1+i)-1 +A(1+i)-2 + … + A(1+i)-n+2 + A(1+i)-n+1
(eq. 2)
Subtract (eq. 1) from (eq. 2):
P + Pi = A + A(1+i)-1 +A(1+i)-2 + … + A(1+i)-n+2 + A(1+i)-n+1
(eq. 2)
P = A(1+i)-1 +A(1+i)-2 + A(1+i)-3 +…+A(1+i)-(n-1) + A(1+i)-n
_____________________________________________________
(eq. 1)
Pi = A – A (1+i)-n
Pi = A[1 - (1+i)-n]
P = A {[1- (1+i)-n] / i }
Or if the factor (1+i) / (1+i) is multiplied to the formula we have the result:
P = A{[(1+i)n – 1] / [i(1+i)n]}
The factor {[(1+i)n – 1] / [i(1+i)n]} is called the “uniform series present worth factor” and
is sometimes represented as (P/A,i%, n) and is read as “P given A at i percent in n interest
periods.
Finding A when P is given:
A = P{i / [1 – (1+i)-n]}
The factor {i / [1 – (1+i)-n]} is called the “capital recovery factor” and is sometimes
represented as (A/P, i%, n) and is read as “A given P at i percent in n interest periods.”
Finding F when A is given:
1
2
3
4
n-1
F
______________________________......._________ n
0
A
A
A
A
A
A
F = A(1+i)n-1 +A(1+i)n-2 + A(1+i)n-3 +…+A(1+i)2 + A(1+i) + A
(eq. 1)
Multiply both sides by (1+i)…
F + Fi = A(1+i)n +A(1+i)n-1 + A(1+i)n-2 +…+A(1+i)2 + A(1+i)
(eq. 2)
Subtract (eq. 1) from (eq. 2):
F + Fi = A(1+i)n +A(1+i)n-1 + A(1+i)n-2 +…+A(1+i)2 + A(1+i)
(eq. 2)
F = A(1+i)n-1 +A(1+i)n-2 + A(1+i)n-3 +…+A(1+i)2 + A(1+i) + A
(eq. 1)
Fi = A(1+i)n – A
Or
F = A{[(1+i)n – 1] / i}
The factor [(1+i)n – 1] / i is called the “uniform series compound amount factor” and is
sometimes represented as (F/A, i%, n) and is read as “F given A, at i per cent in n interest
periods”.
Finding A when F is given:
A = F{i / [(1+i)n – 1]}
The factor {i / [(1+i)-n - 1]} is called the “sinking fund factor” and is sometimes represented
as (A/F, i%, n) and is read as “A given P, at i percent in n interest periods”.
Relation between Capital Recovery Factor & Sinking Fund Factor:
{ i / [(1+i)n – 1] } + i = [ i + i (1 + i)n - i ] / [(1+ i)n – 1]
= [ i (1 + i)n ] / [(1+ i)n – 1] X [(1 + i)-n / (1 + i)-n ]
{ i / [(1+i)n – 1] } + i = i / [ 1 – (1 + i)-n ]
Sinking fund factor + i = capital recovery factor
Examples:
1. A factory operator bought a generator set for P10,000 and agreed to pay the dealer a
uniform sum at the end of each year for 5 years at 8% interest compounded annually so that
the final payment will cancel the debt for principal and interest. What is the annual payment?
Solution:
P 10,000
1
2
3
4
5
_____________________________________
0
A
A
i = 8% compounded annually
A
A
A
A = P {i / [1 – (1+i)-n]} = 10,000 { 0.08 / [1 - (1+0.08)-5}
A = P 2, 504.57 (answer)
2. A man bought a laptop for P21,000 on installment basis at the rate of 12% per year on the
unpaid balance. If he paid a down payment of P6,000 in cash and proposes to pay the balance
in 20 monthly payments, what should these monthly payments be?
Solution:
r = 12% per year
i = r/m = 12% / 12 = 1%
n = 20
P 21,000
1
2
3
4
20
________________________________ ……….._____
P 6,000
A
A
A
A
P = 21,000 – 6,000 = 15,000
A = P {i / [1 – (1+i)-n]} = 15,000 { 0.01 / [1 - (1+0.01)-20}
A = P831.23 (Answer)
A
Exercise:
Juan dela Cruz borrowed P2,400 at 1% per month payable in 24 equal payments. How much of
the loan remains unpaid immediately after he has paid the 12 th payment?
Answer: P1,271.60