Ordinary annuity and annuity due

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2. BY:ASSOCIATE PROFESSOR NADEEM UDDIN
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.5000as advance and rest of the value payable in 24
installment of Rs.1000 per month, it is an example of annuity certain. When
the term of an annuity depends upon an uncertain event, the annuity is called
contingent annuity.
Ordinary annuity
When the payments are made at the end of each payment interval, the annuity is
called an ordinary annuity.
Amount of annuity (Sum of annuity)
The value of all payments at the end of the term of the annuity. For computation
following formula is used:
( )
n
1 r 1
A R
r
+ −
=
 
 
  
where,
R= value of an annuity(Payment Per Period)
r=
i
m
=interest rate per compounding period
n=m×t= number of annuity payments.

3. Example -16
Find the amount of an annuity (sum of an annuity) of Rs.1000 payable at the end of
each year for 10 years, if the interest rate is 6% compounded semi-annually.
Solution:
( )
( )
( )
n
20
20
R 1000
6
i 6% 0.06
100
t 10
m 2
we know that
i 0.06
r= 0.03
m 2
n m t 2 10 20
1 r 1
A Sum R
r
1 0.03 1
A 1000
0.03
1.03 1
A 1000
0.03
=
= = =
=
=
= =
=  =  =
 + −
= =  
  
 + −
=  
  
 −
=  
  
1.806111 1 0.806111
A 1000 1000
0.03 0.03
A Rs.26870.37
−   
= =      
=

4. Example -17
If you want to accumulate Rs.26870 in 10 years, how much money should you
deposit in Bank every month that earns interest 6% compounded semi-annualy.
Solution:
( )
( ) ( )
n
n 20
A 26870
6
i 6% 0.06
100
t 10
m 2
we know that
i 0.06
r= 0.03
m 2
n m t 2 10 20
1 r 1
A R
r
A 26870
R
1 r 1 1 0.03 1
r 0.03
26870
R 999.99 Rs.1000
26.870367
=
= = =
=
=
= =
=  =  =
 + −
=  
  
= =
   + − + −
   
      
= = =
PRESENT VALUE OF 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;

5. ( )
( )
( )
n
n
n
1 r 1
p R
r 1 r
OR
1 1 r
p R
r
where
R=Payment Per Period
i
r= int erest rate per compounding period.
m
n=m t=number of annuity payments(number of compounding period
−
+ −
=
+
− +
=
=

 
 
  
 
 
  
s)
p=Present value of the annuity
Example-18
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
m 2
we know that
i 0.06
r= 0.03
m 2
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
+ − + −
=
+ +
=
   
   
      
 
=  

6. Example 19
Determine the quarterly payment necessary to repay the Rs.10,000 loan, if the
interest compounded quarterly is 10%, for the period of 05 years. Also find
interest.
Solution:
( )
( )
( )
( )
n
n
n
n
p 10000
10
i 10% 0.10
100
t 5years
m 4
we know that
i 0.10
r= 0.025
m 4
n m t 4 5 20
where
1 r 1
p=R
r 1 r
p
R
1 r 1
r 1 r
=
= = =
=
=
= =
=  =  =
+ −
+
=
+ −
+
 
 
  
 
 
  
( )
( )
( )
( )
20
20
20
20
1000
R
1 0.025 1
0.025 1 0.025
1000
R Rs.641.50
15.589156
There will be 20 payments=20 641.50=Rs12830
Interest = 12830-10000 = Rs.2830
1 0.025 1
from table 15.589156
0.025 1 0.025
=
+ −
+
= =

 
 
  
 + −
= 
+  

7. Annuity due (Amount of annuity) (Sum of annuity)
When the payments are made at the beginning of each payment interval, the
annuity is called an annuity due.
For computation following formula is used:
A = R [
( 1+𝑟 ) 𝑛+1 −1
𝑟
] − 𝑅
OR
A = R [
( 1+𝑟 ) 𝑛 −1
𝑟
] × (1 + 𝑟)
Where,
R= value of an annuity(Payment Per Period)
r=
i
m
=interest rate per compounding period
n=m×t= number of annuity payments.
Example -20
Find the amount of an annuity (sum of an annuity) of Rs.1000 payable at the
beginning of each year for 10 years, if the interest rate is 6% compounded semi-
annually.
Solution:
R = 1000
i = 6% =0.06
t = 10
m = 2
We know that
r =
0.06
2
= 0.03
n = m × t = 2 ×10 = 20
A = R [
( 1+𝑟 ) 𝑛+1 −1
𝑟
] − 𝑅

8. A = 1000 [
( 1+0.03 )20+1 −1
0.03
] − 1000
A = 1000 [
( 1.03 )21 −1
0.03
] − 1000
A = 1000 [
1.860295 −1
0.03
] − 1000
A = 1000 [
0.860295
0.03
] − 1000
A = 28676.5 – 1000
A = Rs.27676.5
PRESENT VALUE OF ANNUITY DUE
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;
P = R [
1− ( 1+𝑟 )− ( 𝑛− 1)
𝑟
] + 𝑅
OR
P = R [
1− ( 1+𝑟 )− 𝑛
𝑟
] × (1 + 𝑟)
Example-21
If Rs.1000 payable at the beginning of each year for 10 years, if the interest rate is
6% compounded semi-annually, find its present value.
Solution:
R = 1000
i = 6% =0.06
t = 10
m = 2
We know that
r =
0.06
2
= 0.03

9. n = m × t = 2 ×10 = 20
P = R [
1− ( 1+𝑟 )− ( 𝑛− 1)
𝑟
] + 𝑅
P = 1000 [
1− ( 1+0.03 )− ( 20− 1)
0.03
] + 1000
P = 1000 [
1− ( 1.03 )− ( 19)
0.03
] + 1000
P = 1000 [
1−0.570286
0.03
] + 1000
P = 1000 [
0.4297139
0.03
] + 1000
P = 14323.80 + 1000
P = Rs.15323.80