1. FIN 2732 - Fundamental of
Financial Management
Time Value of Money
2. Coverage
6.1 Time lines and notation
6.2 Future Value of a Single Amount
6.3 Present Value of a Single Amount
6.4 Future Value of an Annuity
6.5 Present Value of an Annuity
Reference: Chapter 6 (Financial Management Prasanna Chandra 8e)
4. TIME LINE
Part A
0 1 2 3 4 5
12% 12% 12% 12% 12%
10,000
Part B
0 1 2 3 4 5
12% 12% 12% 12% 12%
10,000 10,000 10,000 10,000 10,000
5. NOTATION
PV : Present value
FVn : Future value n years hence
Ct : Cash flow occurring at the end of year t
A : A stream of periodic cash flow over a given time
r : Interest rate or discount rate
n : Number of periods over which the cash flows occur.
6. 6.2 FUTURE VALUE OF A SINGLE AMOUNT
Rs.
First year: Principal at the beginning 1,000
Interest for the year
(Rs.1,000 x 0.10) 100
Principal at the end 1,100
Second year: Principal at the beginning 1,100
Interest for the year
(Rs.1,100 x 0.10) 110
Principal at the end 1,210
Third year: Principal at the beginning 1,210
Interest for the year
(Rs.1,210 x 0.10) 121
Principal at the end 1,331
FORMULA FUTURE VALUE = PRESENT VALUE (1+r)n
FV = PV (1+r)n = 1000 (1+0.1) 3 = 1000(1.1 ) 3 = 1000 (1.331)
7. Example: Let us find the value of Rs 1,000 at the end of 3
years given that the rate of interest earned by it is 4%.
7
8. 8
Solution: Future value = Present value (1+r)n
Future value = 1000 (1+0.04)3 = Rs 1,124.86.
FV(1000)
0 1 2 3
Rs 1000
9. DOUBLING PERIOD
Thumb Rule : Rule of 72
72
Interest rate
Interest rate : 15%
72
15
A more accurate thumb rule : Rule of 69
69
Interest rate
Interest rate : 15 percent
69
15
Doubling period =
= 4.8 years
Doubling period =
Doubling period = 0.35 +
Doubling period = 0.35 + = 4.95 years
10. 10
6.3 Present Value of a Single amount
The present value of an amount expected at some time in future is
calculated as:
PV=
A
n
0
PV(A)
n
r)
(1
FV
11. Example: Suppose a particular investment opportunity provides us Rs.2,000 at
the end of three years. What is the present value of this cash inflow, if the
interest rate is 5%?
11
12. 12
Solution: Present value = FV x
= 2000 x = Rs 1,727.68.
n
r)
(1
1
3
)
05
.
0
1
(
1
2000
3
0
PV(2000)
13. 13
Future Value of Multiple Cash Flows
The future value of multiple flows can be computed as
FVn = A1 (1+r)n + A2 (1+r)n-1 +A3(1+r)n-2
where A1, A2 & A3 are the investments at the beginning of the year 1, 2 & 3.
FVn : Future value of the investment at the end of n years
0 1 2 n
A1 A2 A3
FV(A3)+
FV(A2)+
FV(A1)
14. Example: Ram invests Rs 1500 at the beginning of the first year; Rs. 2,000 at the
beginning of the second year and Rs 5,000 at the beginning of third year at a
rate of interest 5% per annum. What will be the accumulated value of all these
cash outflows at the end of the third year?
14
15. 15
Solution:
The accumulated value which Ram will get at the end of
three years will be:
= 1,500 (1+.05)3 + 2,000 (1+0.05)2 + 5,000 (1+0.05)1
= 1,500 (1.158) + 2,000 (1.1025) + 5,000 (1.05)
= 1737+ 2205 + 5250 = Rs 9,192.
0 1 2 3
1500 2000 5000
FV(5000)+
FV(2000)+
FV(1500)
16. 16
6.4 Future Value of an Annuity
Annuity is a pattern of cash flows that are equal in each year.
Future value of an annuity:
FVAn= A (1+r)n + A (1+r)n-1 +…....+A =
r
1
r)
(1 n
A
0 1 2 n
FV(A)
+
FV(A) +
FV(A)
A A A
17. Example: Ram is investing Rs.1500 at the beginning of all
three years. What will be the accumulated amount at the
end of the third year assuming same rate i.e. 5%?
17
18. 18
The accumulated value which Ram will get at the end of
three years = 1500 FVIFA (5%, 3)
= 1500 ((1+.05)3-1)/.05 = 1500 x 3.1525 = Rs 4,728.75.
FV(1500) +
1500 1500 1500
FV(1500) +
FV(1500)
0 1 2 n
19. Suppose you have decided to deposit Rs.30,000 per year in your Public Provident Fund
Account for 30 years. What will be the accumulated amount in your Public Provident
Fund Account at the end of 30 years if the interest rate is 11% ?
20. The accumulated sum will be :
Rs.30,000 (FVIFA11%,30yrs)
= Rs.30,000 (1.11)30 - 1
.11
= Rs.30,000 [199.02]
= Rs.5,970,600
21. HOW MUCH SHOULD YOU SAVE ANNUALLY
You want to buy a house after 5 years when it is expected to cost Rs.2 million. How much
should you save annually if your savings earn a compound return of 12%?
22. The future value interest factor for a 5 year annuity, given an interest rate of 12%, is
:
(1+0.12)5 - 1
2,000,000 = * amount
0.12
The annual savings should be :
Rs.2,000,000 = Rs.314,812
6.353
23. Futura Limited has an obligation to redeem Rs.500 million bonds 6 years hence. How
much should the company deposit annually in a sinking fund account wherein it earns
14% interest to cumulate Rs.500 million in 6 years time?
24. The future value interest factor for a 6 year annuity, given an interest rate of
14% is :
FVIFAn=6, r=14% = (1+0.14)6 – 1 = 8.536
0.14
The annual sinking fund deposit should be :
Rs.500 million = Rs.58.575 million
8.536
25. 25
Present Value of Multiple Cash Flows
If A1, A2, An are the cash flows occurring at the end of the time period
1,2 and n respectively then their present value can be computed as:
PV = A1/(1+r) + A2/(1+r)2 +........+An/(1+r)n
PV(A1) +
PV(A2)+
PV(A3)
0 1 2 n
A1 A2 An
26. Example: A person invested certain amount of money in a project. The project
generates an inflow of Rs.1,500 at the end of first year, Rs.2,000 at the end of
second year & Rs.4,000 at the end of third year. What is the present value of
these future cash inflows given that the rate of interest is 5%?
26
28. 28
6.5 Present Value of An Annuity
The present value of an annuity can be computed as:
PV= A/(1+r) + A/(1+r)2 +……+ A/(1+r)n
PV = A x
1 2 n
A A A
0
PV(A)+
PV(A)+
PV(A)
n
n
r)
r(1
1
r)
(1
29. Example: A person invested certain amount of money in a project. The project
generates an inflow of Rs.2,000 at the end of first, second & third year. What is
the present value of this annuity of Rs.2,000 given that the rate of interest is
5%?
29
31. 31
Example: Mr. A borrowed a loan of Rs 14,000 at a rate of 9% for a
period of three years. Calculate the annual installment, if he has to
liquidate the loan.
32. Solution: The annual installment for a loan of Rs. 14,000 at a rate of
9% can be computed using the capital recovery factor.
14000 = Annual installment x
Annual installment = 14,000 x
= 14,000 x 0.3951 = Rs. 5,531
32
1
)
09
.
1
(
)
09
.
1
(
09
.
3
3
3
3
)
09
.
1
(
09
.
1
)
09
.
1
(
33. Practice questions (page 163, 8e)
6.1 If you invest Rs.5000 today at a compounding interest of 9%, what will be the
its future value after 75 years?
6.2 If the interest rate is 12% what is the doubling period as per the rule of 72
and rule of 69?
6.4 Fifteen annual payments of Rs.5000 are made into a deposit account that
pays 14% interest per year. What is the future value of this annuity?
6.6 What is the present value of Rs.1,000,000 receivable 60 years from now, if
the discount rate is 10%?
6.8 What is the present value of the following cash stream if the discount rate is
14%?
6.9 Mahesh deposits Rs.200,000 in a bank account which pays 10% interest. How
much can he withdraw annually for a period of 15 years?
Year 0 1 2 3 4
Cash flow 5,000 6,000 8,000 6,000 8,000
34. 6.9 Solution: The annual money if he deposits Rs. 200,000 at a rate
of 10% for 15 years.
200,000 = Annual amount x
Annual amount = 200,000 x
= 200,000 x 0.131474 = Rs. 26294
34
1
)
1
.
1
(
)
1
.
1
(
1
.
15
15
15
15
)
1
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1
(
1
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1
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1
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1
(
35. • You can save Rs.2,000 a year for 5 years, and Rs.3,000 a year for 10
years thereafter. What will these savings cumulate to at the end of 15
years, if the interest rate is 10%?
• Mr. Ram plans to send his son for higher studies abroad after 10
years. He expects the cost of the studies to be Rs.1000,000. How
much should he save annually to have a sum of Rs.1000,000 at the
end of 10 years, if the interest rate is 12%?
36. Practice questions
• Theory questions (page 163): 1, 3, 5, 15
• Numerical (page 166): problems based on present and future value of
single cash flow and annuity
