Present value of ordinary annuity

Present value of ordinary annuity

2. Annuity

Most commonly the practice of making a series of payments to
an individual from a capital investment is known as annuity. It is
also considered as a form of insurance.
➢ The time between successive payments of an annuity is
called the payment interval.

➢ The time from the beginning of the first payment interval to
the end of the last payment interval is called term of an
annuity, when the term of an annuity is fixed, the annuity is
called an annuity certain.


For example, if Abu Tariq gets a mobile set from a shop
by making a payment of Rs.5000, as advance and rest of
the value payable in 24 installment of Rs.1000 per month, it
is an example of annuity certain.


Ordinary annuity

When the payments are made at the end of each payment
interval, the annuity is called an ordinary annuity.

3. PRESENT VALUE OF ORDINARY ANNUITY

The Present value of an annuity is an amount of money today
which is equivalent to a series of equal payments in future. For
computation following formula is used;

??????=??????{
(1+�)
??????
−1
�(1+�)
??????
}

Or

??????=??????{
1−(1+�)
−??????
�
}

Where,

R = Value of an annuity (Payment Per Period)

i = rate of interest

t = number of years

m = numbers of times in a year when the interest is compounded.

r = i
m = interest rate per compounding period

n = m×t = number of annuity payments.

4. Example -1
A depositor pays Rs.1000 at the end of each year for a period
of 5 years. The rate of interest is 15% compounded yearly, find
the Present value of an annuity.

Solution:

R = 1000

i = 15% = 15/100 = 0.15

t = 5

m = 1

r = i
m =
0.15
1
= 0.15

n = m×t = 1×5 = 5

??????=??????{
1−(1+�)
−??????
�
}


??????=1000{
1−(1+0.15)
−5
0.15
}

??????=1000{
1−(1.15)
−5
0.15
}

5. ??????=1000{
1−0.4971674
0.15
}

??????=1000{
0.5028326
0.15
}

??????=1000{3.352217}

??????=??????�.3352.22

6. Example -2.
If Rs.1000 payable at the end of each year for 10 years, if the
interest rate is 6% compounded semi-annually, find its present
value.

Solution:
R 1000
6
i 6% 0.06
100
t 10years
m2
we know that
i 0.06
r= 0.03
m2
n m t 2 10 20
=
= = =
=
=
==
=  =  =



()
()
( )
( )
n 20
n 20
where
1 r 1 1 0.03 1
p=R 1000
r 1 r 0.03 1 0.03
0.80611
p 1000
0.05418
Rs.14877.47
+ − + −
=
++
=
   
   
      

=



7. Example -3.
Find the Present value of an annuity of Rs.8500 invested at the
end of each quarter for 04 years at 6% compounded quarterly.

Solution:

R = 8500

i = 6% = 6/100 = 0.06

t = 4

m = 4

r = i
m =
0.06
4
=0.015 =

n = m×t = 4×4 = 16
??????=??????{
1−(1+�)
−??????
�
}


??????=8500{
1−(1+0.015)
−16
0.015
}

??????=8500{
1−(1.015)
−16
0.015
}

??????=8500{
1−0.78803104
0.015
}

8. ??????=8500{
0.21196896
0.015
}

??????=8500{14.131264}

??????=??????�.120115.74


Example -4.
If Rs.1000 is paid at the end of each month for next five years
in the account earning 8% interest compounded monthly. What
will be the Present value of annuity.

Solution:

R = 1000

i = 8% = 8/100 = 0.08

t = 5

m = 12

r = i
m =
0.08
12
=0.0067

n = m×t = 12×5 = 60
??????=??????{
1−(1+�)
−??????
�
}

9. ??????=1000{
1−(1+0.0067)
−60
0.0067
}

??????=1000{
1−(1.0067)
−60
0.0067
}

??????=1000{
1−0.66987825
0.0067
}

??????=1000{
0.33012174
0.0067
}

??????=1000{49.27190155}

??????=??????�.49271.90