1. Time Value of Money
Financial Management
Dr K. Krishna Kumar
Professor of Finance & Associate Dean
School of Management
Presidency University, Bengaluru
2. Time Value of Money
• It is a fundamental concept in Finance.
• It is widely used in both Corporate and Personal
Finance.
• Finance is all about effective and efficient
utilization of money which is a scarce resource
with alternate uses.
• Time Value helps in understanding the
implications of utilization of money.
3. Concept of Time Value of Money
• A rupee today is worth more than a rupee
tomorrow.
• It is so because:
1) Money can be invested and returns earned
out of it.
(can have more than a rupee by tomorrow, by investing and earning returns )
2) Money loses Purchasing Power over time.
(you cannot buy the same quantity of goods tomorrw due to increase in prices )
3) Uncertainty ( Risk ) with regard to Future
(today is more certain and tomorrow is always uncertain )
4. Simple & Compound Interest
• Simple Interest Calculation does not consider ‘Time Value of Money’ as your money is
not put to full use.
• On the other hand Compound Interest Calculation considers ‘ Time Value of Money ’.
• All calculations in Finance are based on Compound Interest.
• Calculations that use compound interest are called Time Value Calculations.
Investment based on Simple Interest
Year
Op
Balance
Investment
Today
Returns @ 10%
pa Cl Balance
1 0 10000 1000 11000
2 11000 0 1000 12000
Investment based on Compound Interest
Year
Op
Balance Investment Returns @ 10% Cl Balance
1 0 10000 1000 11000
2 11000 0 1100 12100
5. Time Lines in Compound Interest Calculations
Today
Time 0
Present Value
( PV )
End of 5 Years
Time 5
Future Value
( FV )
1 2 3 4
Process of finding Future Value from the Present Value is called Compounding
Process of finding Present Value from a given Future Value is called Discounting
Lump Sum
Amount
Lump Sum
Amount
6. Computing Future Value of a Lump Sum
• FV = Future Value
• PV = Present Value
• i = Rate of Return per Year ( in Decimals )
• N = Number of Years
0 1 2 3 4 5
X - - - - ?
7. Co. X invests Rs 50000 in a Bank Fixed Deposit which pays an
interest rate of 8% pa compounded annually. What would be the
value of the Deposit at the end of 5 Years ?
Investment based on Annual Compounding
Year Op. Balance Investment Returns @ 10% pa Cl. Balance
1 0 10000 1000 11000
2 11000 0 1100 12100
Future Value of Fixed Deposit = 50000 x ( 1+.08)5 = Rs 73466
• PV = 50000
• I = .08
• N = 5
0 1 2 3 4 5
50000 - - - - ?
8. A Firm deposits Rs 55000 in a Bank Time Deposit @ 10% pa
compounded annually. The deposit was for 10 years. The Bank
allows early withdrawal with a penalty of .5% pa. What would be
the value of the Deposit in case it is withdrawn at the end of 7
Years ?
FV of FD = 55000 x ( 1+.095)7 = Rs 103815
• PV = 55000
• I = (.10-.005) = 0.095 = 9.5%
• N = 7
9. Investment based on Half Yearly Compounding
Compnay X invests Rs 50000 in a Bank Fixed Deposit which pays an
interest rate of 8% pa compounded half yearly. What would be the
value of the Deposit at the end of 5 Years ?
Interest is
computed
twice a
year
When the number Compounding's in a year is not annual, the FV formula needs to be
modified :
m
m m = Number of
Compounding’s in a Year
FV of FD = 50000 x ( 1+ ( .08/2))(5x2)
Rs 74012
Year Op. Balance Investment Returns @ 10% pa Cl. Balance
1st Half Year 0 10000 500 10500
2nd Half Year 10500 0 525 11025
3rd Half Year 11025 0 551 11576
4th Half Year 11576 0 579 12155
• PV = 50000
• I = (.08/2) = .04
• N = (5 x 2 )=10
10. Effect of Frequency of Compounding
Find FV of Rs 50000 invested @ 8% pa for 5 Years if :
With the increase in the Number of Compounding’s in a year, the FV increases but
at a decreasing rate.
Change
0.74%
0.39%
0.26%
0.13%
An extreme form of Compounding is Continuous Compounding
• e = 2.71828 ( base of Natural Logarithm)
• N = Number of Years = 5
• APR = Annual Percentage Rate = .08
FV if Compounded Continuously Rs 74591
FV = 50000 x (2.71828( 5 x .08))
FV = 74591
Compounded Annually 73466
Compounded Half Yearly 74012
Compounded Quarterly
Compounded Monthly
Compounded Daily
74297
74492
74588
11. Effective Annual Return
• Annual Percentage Rate is the rate before considering the effect of Compounding.
• Effective rate is the net return earned by the investor, after considering the effect of
compounding.
• A Bank Fixed Deposit pays 8.5% pa, compounded quarterly. What is the rate of return
earned by the investor ?
• Effective Rate = ( 1+(.085/4)4 -1
• Effective Rate = 8.775% pa
Quarter Op Bal Interest @ 2.125% Cl Bal
1 1000.00 21.25 1021.25
2 1021.25 21.70 1042.95
3 1042.95 22.16 1065.11
4 1065.11 22.63 1087.75
12. A Company is considering borrowing funds. It has two options :
1) C Bank offers the loan @ 21% pa, compounded quarterly
2) X Bank offers the loan @ 22.5% pa, compounded annually
Which option would you recommend ?
Rate of C Bank is not in ‘effective’ terms, so we need to covert the rate into Effective
Effective Rate = ( 1+ (.21/4))4 - 1
Effective Rate on C Bank Loan = 22.71%
As X Banks rate is cheaper, the company should borrow at 22.5% pa from X Bank
13. FV with Multiple Rates
AB Ltd. had a Profit of Rs. 25000 for the Year 2012. The Profits for the next 3 years are
expected to grow @ 10%, 15% and 5% pa respectively . What would be the Profit of the
Company for the ended 2015 ?
FV of Profit = 25000 x ( 1.10) x ( 1.15) x ( 1.05 ) 33206
Profit for the Year ended 2015 is expected to be Rs 33206
14. Finding the Rate of Growth
The Sales of GK Ltd. was 60000 for the Year 2018 and the same is expected to be
100000 for the Year 2021. What is the annual rate of growth in sales achieved by the
Company over the last 4 years ?
Rate of Growth in Sales = ( 100000 / 60000 )(1/4) -1 13.62% pa
15. Solving Future Value using FVIF Tables
Future
Value
Interest
Factor
FVIF for 8%, 5 Years ( 1.08)5 1.469328
FVIF for 8%, 10 Years ( 1.08)10 2.158925
16. Find the Future Value of Rs 25000 invested for 5 years at 10% pa compounded half
yearly.
As the Compounding is half yearly, convert the Time and the Rate accordingly
Rate per Compounding Period
= 10 % / 2 = 5%
Number of Compounding
Periods = 5 x 2 = 10
FVIF = 1.6289
FV = 25000 x 1.6289 = 40722
17. A Company wants to invest its surplus cash of Rs 30000 for 2 years @ 9.5 % pa. What
would be the maturity value of this investment ?
Take the average of FVIF of 9% and 10%
FVIF = 1.1990
FV =30000 x 1.1990 = 35970
18. P Ltd. Sold one of it Manufacturing units for Rs. 1.5 Crs.
The company wants to invest this money in an expansion project to be undertaken
after 5 years which would cost then Rs. 5 Crs.
The Co. intends to invest the sale consideration for 5 years at 11.5% pa.
What will the extra finance that needs to be arranged at the end of 5 years for the
expansion project ?
FV of Investment = 1.5 x ( 1.115)5
FV = 2.5850 Crs
Finance required = 5 Crs ; Available = 2.5850 Crs; Balance = 2.4150 Crs
19. Computing Present Value of a Lump Sum
A Company has a liability of Rs 500000 to be paid at the end of 5 years from now. The
company intends to set aside ( invest ) a lump sum amount today towards this repayment.
If the Company can earn a return of 10% pa, what is the amount that it has to set aside
( invest ) today ?
PV = 500000 / ( 1.10)5
310461 Year
Opening
Balance Investment
Returns @
10% pa
Closing
Balance
1 0 310461 31046 341507
2 341507 0 34151 375658
3 375658 0 37566 413224
4 413224 0 41322 454546
5 454546 0 45455 500001
PVIF
PV = 500000 x .620921
1/ (1.10)5
0 1 2 3 4 5
? - - - - X
20. Find the PV of a sum of Rs 60000 to be received at the end of 6 years if the rate of interest
is 8% pa compounded half yearly
PV = 60000 / ( 1.04)12
37476
PVIF 1/ (1.04)12 .624597
PV = 60000 x .624597 = 37476
21. You want to Buy a Machine. The Supplier offers two Payment Options :
1. Full Down Payment of Rs 500000 today
2. Down Payment of Rs 2.5 Lacs today and Rs 3 Lacs at the end of 4 years
If the Opportunity Cost of Money is 10% pa, which Option would you choose ?
Comparison can be made by finding the PV of the Future Cash Flow
Option : 1 PV0 = 500000
Option : 2 PV0 of CF 1 =250000
PV0 of CF 2=
300000 / ( 1.10)4
In PV terms Option 2 costs lesser, so choose Option 2
204904
Total PV0 = 250000 + 204904 = 454904
Find FV of Differential CF FV4 of 250000 = 366025
At the end of 4 Years Pay Rs 3
Lacs and Pocket Rs 66025
Alternative Solution Through FV
22. A Company is planning an Expansion which will cost Rs 12 Crs today. The investment is
likely to fetch the following incremental cash inflows for the next 10 years:
End of Year Cash Flow ( Crs )
1 2.10
2 3.05
3 4.25
4 4.02
5 3.25
6 3.05
7 2.45
8 1.55
9 1.35
10 1.25
If the required return on investment is 14% pa, should the company go ahead with
the expansion or not ?
Find the PV0 of each Cash Inflow separately and add them up. Compare the Total of
PV0 of Inflows with PV0 of Outflows.
PV @ 14%
1.842
2.347
2.867
2.380
1.688
1.390
0.979
0.543
0.415
0.337
Sum of PV
= 14.79
PV0 of Cash In Flows @ 14% = 14.79 Crs and PV0 of Out Flow is only 12 Crs
As PV0 of Future Cash In Flows is more than Outflows, it is beneficial for the
Company to Expand its Operations.
23. • A Company wants to Replace some of its Machinery 10 Years from now. The Cost of
Replacement if done today is 10 Crs. This cost is expected to go up @ 5% pa. To meet
this objective of Replacement, the Company wants to set aside a fund today, which
will be invested at 10% pa till replacement is undertaken.
• What is the Fund that needs to be set aside today ?
Step 1 : Find the FV of the Replacement Project
Step 2 : Discount the FV @ 10% for 10 Years
16.29 Crs
= 10 ( 1.05)10
=
16.29/(1.10^10)
6.28 Crs
Fund to be set aside today is Rs 6.28 Crs
24. Annuities
Today
Time 0
End of 5 Years
1 2 3 4 5
Cash Flows / Annuities / Payments
Cash Flows can occur at the Beginning of the Year or at the End of the Year
Beginning of the Year Annuities are called ‘ Annuity Due’
If Cash Flows occur starting from Time 0, then it is said to be Beginning of the Year CFs.
If Cash Flows occur starting from Time 1 onwards , then it is said to be End of the Year CFs.
End of the Year Annuities are simply called ‘ Annuities ’
25. Computing Future Value of an Annuity
A Company invests Rs 50000 at the end of
every year for 10 years. If the Investment
earns 10% pa, what is the Future Value ?
FV = 50000 x (( (1.10)10 – 1) /.10 ) = 50000 x 15.9374
FV = 50000 x 15.9374 = 796871
FV = A x FVIFA
0 1 2 3 4 5
- 50000 50000 50000 50000 50000
?
26. Computing Future Value of an Annuity
A Company invests Rs 5000 end of every quarter for 10 years. If the Investment earns 10%
pa, what is the Future Value ?
FVIFA = ( (1.025)40 – 1 ) /.025 = 67.40255
FV = 5000 x 67.40255 = 337013
FV = A x PVIFA
As the Investment is Quarterly, convert the formula accordingly*
*The Assumption made here is that keeping in line the pattern of CF’s, the Compounding
of returns is also done Quarterly.
If this assumption does not hold good, then :
Convert the Annual Compounding Rate to Quarterly rate before Compounding so that the
Compounding formula can be used.
(1.1^(1/4))-1
2.41% per Quarter before
Compounding
FVIFA = (( 1.0241)^40 -1) /.0241
27. Computing Future Value of an Annuity Due
A Company invests Rs 20000 at the Beginning of every year for 5 years. If the Investment
earns 10% pa, what is the Future Value ?
FVIFA = ( (1.10)5 – 1)/.10 = 6.10
FV = 20000 x 6.71 = Rs 134200
• Cash Flows from or into an Investment can occur at the End of the Period or at the
Beginning of the Period.
• Cash Flows occurring at the end of the period are called Annuities.
• Cash Flows occurring at the Beginning of the period are called Annuity Due.
FVIFA Due = 6.1 x 1.1 = 6.71
0 1 2 3 4 5
20000 20000 20000 20000 20000 -
?
28. FV of Annuity Due @ 10% pa for 10 years (Begin)
Year Op Bal Investment Returns Cl Balance
1 0 50000 5000 55000
2 55000 50000 10500 115500
3 115500 50000 16550 182050
4 182050 50000 23205 255255
5 255255 50000 30526 335781
6 335781 50000 38578 424359
7 424359 50000 47436 521794
8 521794 50000 57179 628974
9 628974 50000 67897 746871
10 746871 50000 79687 876558
FV of Annuity @ 10% pa for 10 years (End)
Year Op Bal Investment Returns Cl Balance
1 0 50000 0 50000
2 50000 50000 5000 105000
3 105000 50000 10500 165500
4 165500 50000 16550 232050
5 232050 50000 23205 305255
6 305255 50000 30526 385781
7 385781 50000 38578 474359
8 474359 50000 47436 571794
9 571794 50000 57179 678974
10 678974 50000 67897 796871
Difference in Values =
876558-796871 = 79687
The difference is one
periods interest on the
end value of Ordinary
Annuity
=796871 x 10%
29. Calculating Annuity from FV (using Sinking Fund Factor)
A Company wants have a Fund of Rs 70 Lacs by the end of 8 years to repay a liability. It
intends to create this fund by setting aside a fixed amount every year from its profits. If
the fund can earn a return of 12% pa, what is the fixed amount that needs to be set
aside every year.
Annuity = 70 x ( .12 / ((1.128) -1 ) )
Investment every Year = 70 x 0.081303 = Rs 5.69 lacs
Sinking Fund Annuity Factor = ( 1/FVIFA )
Sinking Fund Factor = 1/12.30
Sinking Fund = 70 x 0.081301 = 5.691 Lacs
Alternately Formula =
0 1 2 3 4 5
- ? ? ? ? ?
X
30. Computing Present Value of an Annuity
You are promised an Annual end of the year Amounts of Rs 50000 for 20 years. If the
interest rates in the economy are 9% pa, how much would be willing to pay for this
Annuity today ?
PVIFA = (1- (1/ (1.09)20 ) / .09 = ( .821569/.09 ) = 9.1285
PV of Annuity = 50000 x 9.1285 = 456427
0 1 2 3 4 5
? X X X X X
31. Computing Present Value of an Annuity Due
You are promised an annual Beginning of the year Payment of Rs 50000 for 20 years. If
the interest rates in the economy are 9% pa, how much would be willing to pay for this
today ?
PVIFA = (1- (1/ (1.09)20 ) / .09 = ( .821569/.09 ) = 9.1285
PV of Annuity = (50000 x 9.1285 ) = 456427
PV of Annuity Due = 456427 x ( 1.09) = Rs 497506
0 1 2 3 4 5
X X X X X X
?
32. Computing Present Value of an Annuity
You are promised an Annual end of the year Payments of Rs 50000 for 10 years and Rs
75000 for the next 10 years. If the interest rates in the economy are 9% pa, how much
would be willing to pay for this Annuity today ?
PV0 of Annuity from Year 1 to 10 = 50000 x 6.4177 = 320885
PV10 of Annuity from Year 11 to 20 = 75000 x 6.4177 = 481328
PV0 of Annuity from Year 11 to 20 = 481328 /1.09^10 =203318
Total of PV0 / Price to be paid Today = 320885 + 203318 = 5.24
33. A Company intends to take a loan of Rs 50 Lacs at an interest rate of 9% pa. This loan
is repayable in 10 equal annual installments inclusive of interest. What would be the
annual installment of this loan ?
Installment = 50 (( .09(1.0910 ))/ (1.0910 -1 ))
Installment = 50 ( .213063 / 1.36736)
Installment = 50 x .15582 = Rs 7.79 Lacs
Capital Recovery Factor = ( 1/ PVIFA )
PVIFA = 6.4177
Capital Recovery Factor = 0.15582
Alternately Formula =
Calculating Annuity from PV (using Capital Recovery Factor)
0 1 2 3 4 5
X ? ? ? ? ?
34. Amortized Loans
• Amortized loans are those in which the principal amount of the loan is repaid over
the life of the loan.
• The loan installments consist of both principal and interest.
• In such loans, initially, the interest component in the installment is high.
• Interest component keeps reducing over time as the principal is being repaid.
Amortization Table for a 50 Lacs Loan of 9% pa repayable in 10 annual
installments
Year Op Bal Interest Payable Instalment Paid Principal Component Closing Balance
1 50 4.50 7.79 3.29 46.71
2 46.71 4.20 7.79 3.59 43.12
3 43.12 3.88 7.79 3.91 39.21
4 39.21 3.53 7.79 4.26 34.95
5 34.95 3.15 7.79 4.65 30.30
6 30.30 2.73 7.79 5.06 25.24
7 25.24 2.27 7.79 5.52 19.72
8 19.72 1.77 7.79 6.02 13.71
9 13.71 1.23 7.79 6.56 7.15
10 7.15 0.64 7.79 7.15 0.00
35. Ms Kanchan is offered a Car Loan of Rs 8 Lacs with an interest rate of 12% pa for a
tenure of 4 Years and 2 months. What will be the EMI payable by her
CRF = 1/39.19612 = 0.025513 Equated Monthly Installment =
0.025513 x 800000 = Rs 20410
36. Time Value Calculations
Present Value
Future Value
Lump sum
Annuity
Lump Sum
Annuity
Annuity
For a given
Present Value
For a given
Future Value
FV
FVA
PVA
PV
Capital
Recovery
Factor
1/ PVA
Sinking
Fund Factor
1/ FVA
37. Future Value of a Growing Annuity
FV of Growing Annuity = A x ( 1+ i)n – ( 1+g)n
( i –g)
A Company wants to invest 10% of its Profits at the end of every year in an
investment earning 10% pa for 10 years. Profit expected is Rs. 5 Crs this year,
and will grow @5% pa every year. How much can the Company accumulate at
the end 10 years ?
= FV of G A Factor = (( 1.10)10 – ( 1.05)10) / ( .10-.05)
FV = 19.2969 x .5 Crs = 9.648 Crs
Year Op Bal Inv Return Cl Bal
1 0 50000 0 50000
2 50000 52500 5000 107500
3 107500 55125 10750 173375
4 173375 57881 17338 248594
5 248594 60775 24859 334228
6 334228 63814 33423 431465
7 431465 67005 43147 541617
8 541617 70355 54162 666133
9 666133 73873 66613 806619
10 806619 77566 80662 964848
Year Op Bal Inv Return Cl Bal
1 0 50000 5000 55000
2 55000 52500 10750 118250
3 118250 55125 17338 190713
4 190713 57881 24859 273453
5 273453 60775 33423 367651
6 367651 63814 43147 474612
7 474612 67005 54162 595778
8 595778 70355 66613 732747
9 732747 73873 80662 887281
10 887281 77566 96485 1061333
Beginning of the Year
End of the Year
9.648 x 1.10 = 10.61 Crs
38. Computing Present Value of a Growing Annuity
An investment promises a payout of Rs 10000 at the end of the year, increasing at 5%
every year for 10 years. If interest rates are 10% pa, what should be the Present Value of
this investment ?
• PV = ( 10000/ ( .10-.05) ) x ( 1- ( 1.0510 ) / 1.1010 )
• PV = 200000 x .371991
• PV = 74398
39. Present Value of a Perpetuity
Perpetuities are cash flows occurring perpetually or for infinite number of years
An investment offers a Cash Flow of Rs 10000 to the investor perpetually at the end of
every year. If the rate of interest is 8% pa, what should be the purchase price of this
perpetuity ?
PV = 10000 / .08
PV = 125000
What is the PV if it is a Perpetuity Due ?
PV of Perpetuity Due = PV of Ordinary Perpetuity + One Perpetuity Amount
PV of Perpetuity Due = 125000 + 10000 = 135000
40. Present Value of a Growing Perpetuity
An investment offers a Cash Flow of Rs 10000 growing at 5% pa to the investor
perpetually at the end of every year. If the rate of interest is 8% pa, what should be the
purchase price of this perpetuity ?
PV = 10000 / ( .08 -.05)
PV =333333
41. Rule of 72 & Rule of 69
Rule of 72 is used to find the approximate time taken for the Investment to double /
grow 100%
Rate Doubling Period as per Rule of 72
8% 9.00 Years
9% 8.00 Years
10% 7.20 Years
Number of Years for Doubling
% As Per Rule 72 ( Approx. ) As Per Rule of 69 ( Approx. ) Actual Time
5 14.40 14.15 14.21
6 12.00 11.85 11.90
7 10.29 10.21 10.24
8 9.00 8.98 9.01
9 8.00 8.02 8.04
10 7.20 7.25 7.27
11 6.55 6.62 6.64
12 6.00 6.10 6.12
13 5.54 5.66 5.67
14 5.14 5.28 5.29
15 4.80 4.95 4.96
42. A Company wants to invest at the end of every year Rs 5 Crs in an investment which
provides 10% pa for 5 Years. At the end of 5 years, by utilizing this value, it wants to
carry out up gradation of its Machinery. What will be value of up gradation that it can
undertake ?
Hint : We need to solve using FV of Annuity
Practice Questions
43. A Company wants to take a Loan of Rs 5 Crs at 12% pa repayable in equal installments
over the next 5 years. What will be annual installment amount ?
Hint : We can find the Installment using the Capital Recovery Factor
44. X Ltd. wants to start a project with a budget of Rs 60 Crs at the end of 3 years from now.
What is the fund that has to be set aside today if the amount can earn 15% pa to have
the required funds ?
Hint : Find the Present Value of a Lump sum
45. X Ltd. has an obligation to pay an amount of Rs. 5 Crs at the end of every year for the
next 5 years. How much should it set aside today to honor this commitment if money
can earn 9% pa ?
Hint : We can solve by using the Present Value of Annuity
46. AB Ltd. recently took over a Company. As per the terms it has to pay its promoters, a
lump sum amount of 22 Crs at the end of 6 years from now. It wants to set aside from
its profits a fixed amount at the end every year for 5 years only. What is the amount
required to be invested every year to achieve the objective if funds can earn 10% pa till
the payment is made ?
Step 1 – Find the PV of Rs 22 Crs at the end of Year 5
Step 2 – Using the above as the FV, compute the Annuity using Sinking Fund Factor
Use the Time Line
0 1 2 3 4 5 6
? ? ? ? ?
? 22 Cr