Future value of an annuity & present value of an annuity

1. Basics of personal financial
planning (concept of time
value of money)
Lecture 5
By: Swasti Suvasweta Satpathy
Assistant professor,LPU,Jalandhar

2. • Future value of single cash flow
• Future value of Annuity
• Present value of Annuity

3. Compound interest (future value)
• FVn = P0 (1+i)n (if compounded once annually)
• FVn = P0 (1+i/m)nm (if compounded more than
once in a year)
• FVn = P0 (1+i/2)n*2 (if compounded half yearly)
• FVn = P0 (1+i/4)n*4 (if compounded quarterly)

4. Questions to solve
• If you deposit $4000 into an
account paying 6% annual interest
compounded quarterly, how much money will
be in the account after 5 years?
• If you deposit $6500 into an
account paying 8% annual interest
compounded monthly, how much money will
be in the account after 7 years?

5. Answers
• FV =4000(1+0.06/4)4*5
=5387.42
• FV=6500(1+0.08/12)12*6
=11358.24

6. • How much money would you need to
deposit today at 9% annual interest
compounded monthly to have $12000 in
the account after 6 years?

7. • 1200=P(1+0.09/12)12*6
P= 7007.08

8. Annuity
• An annuity is a series of even cash
flows(inflow/outflow).
• Rate of interest remains same during the annuity
period
• Open a RD account with bank, where you have to
deposit Rs100 per month for a period of 5 years & the
bank offers 12% per annum for the investment period.
• If your cash flow occur at end of each period we call it
regular annuity. If the cash flow occurs at beginning of
each period we call it as annuity due.

9. Future value of an annuity
• Suppose 1$ is to be invested every year for the
next 3 years. The investment is earning return
@ 8% per annum. Assuming further that each
investment is made on last day of the year,
How much money would you receive at the
end of 3rd year?
Answer:
{1*(1+0.08)^2}+{1*(1+0.08)^1}+1= 3.2464

10. • As each investment is being made at the end of
each period, it is a regular annuity.
• The future value of a regular annuity earning
interest @ ‘r’ % for ‘n’ years is
FVRA = A [ (1+r)^n – 1 ]
r
Where A=Contribution in each period
r= Rate of interest per period
n=no of periods in annuity

11. “ [ (1+r)^n – 1 ] “
r
is called as future value interest factor of
annuity(FVIA).
• If it is the future value of an annuity due :
FVAD = FVRA (1+r)

12. Example
• Ramesh opens a 5 years RD account with SBI
which offering interest 8 % p.a. Ramesh
decides to deposit a sum of 10,000/- p.a.
How much money can Ramesh expect to get
at the end of 5 years assuming
a)Each deposit is made at the end of each year ?
b)At the beginning of each year ?

13. • a) Regular annuity :
FVRA = A [ (1+r)^n – 1 ]
r
= 10,000 * [(1+0.08)^5 – 1]
0.08
= 58,666/-
“This can also be solved using FVIFA table as
FV = A* FVIFA(8%,5)”

14. • b)Annuity due :
FVAD = FVRA (1+r)
= 58,666(1+0.08)
=63,359.29

15. Present value of annuity
• Regular Annuity :
The present value of an annuity earning interest
at the rate of ‘r’ % at the end of ‘n’ years is :
PVRA = A* [ (1+r)^n -1 ]
[ r *(1+r)^n]
“ [ (1+r)^n -1 ] “
[ r *(1+r)^n] is called the Present value
interest factor of annuity.(PVIFA)

16. • Annuity due :
PVAD = PVRA (1+r)

17. • Rohan is planning to buy a pension plan which
would provide him an annual pension of
20,000/- for the next 30 years. How much
should he be willing to pay for this pension
plan if he wants a return of 9% on his
investments?
• Assume that the pension is received
a)at the end of each year ?
b)in the beginning of each year ?

18. • a) Regular Annuity :
PVRA = A* [ (1+r)^n -1 ]
[ r *(1+r)^n]
=20,000 * [ (1+0.09)^30 -1 ]
[ 0.09 *(1+0.09)^30]
=20,000 * 10.2736
=205,472/-

19. • This can also be solved using PVIFA table.
• The PVIFA (9%, 30 Years) is 10.2737.
Therefore PV = A* PVIFA (9%, 30)
=20,000*10.2737
=2,05,474/-

20. • b) Annuity Due :
PVAD = PVRA (1+r)
PVAD = 205,474* (1+0.09)
= 223,966 /-
Thus Rohan should be willing to pay 2,05,474/-
or 2,23,966 /- for buying the pension plan.