Time Value of Money

1. Time Value of Money
R Singh

2. The Time Value of Money
❑Simple Interest and Compound Interest
Present Value and Future Value
❑Present Value Annuity and Future Value
Annuity
❑Mixed Cash Flows
❑Compounding More Than Once per Year
❑Effective Rate of Interest
❑Perpetuity
❑Sinking Fund
❑Amortization Schedule

3. The Interest Rate
Which would you prefer -- Rs.1000
today or Rs.1000 after one year?

4. The Interest Rate
Which would you prefer -- Rs.1000
today or Rs.1000 after one year?
Obviously, Rs.1000 today.
You already recognize that there is
TIME VALUE TO MONEY!!

5. TIME allows you the opportunity to
postpone consumption and earn
INTEREST.
Why TIME?
Why is TIME such an important
element in your decision?

6. Types of Interest
Simple Interest
Interest paid (earned) on only the original
amount, or principal, borrowed (lent).
Compound Interest
Interest paid (earned) on any previous
interest earned, as well as on the
principal borrowed (lent).

7. Simple Interest Formula
Formula SI = P0(r)(n)
SI: Simple Interest
P0: Deposit today (t=0)
r: Interest Rate per Period
n: Number of Time Periods

8. Simple Interest Example
Assume that you deposit Rs.1,000 in an
account earning 7% simple interest for
2 years. What is the accumulated
interest at the end of the 2nd year?

9. SI = P0(r)(n)
= Rs.1,000(.07)(2)
= Rs.140
Simple Interest Example
Assume that you deposit Rs.1,000 in an
account earning 7% simple interest for
2 years. What is the accumulated
interest at the end of the 2nd year?

10. Simple Interest (FV)
What is the Future Value (FV) of the
deposit?

11. Simple Interest (FV)
What is the Future Value (FV) of the
deposit?
FV = P0 + SI
= Rs.1,000 + Rs.140
= Rs.1,140
Future Value is the value at some future
time of a present amount of money, or a
series of payments, evaluated at a given
interest rate.

12. Simple Interest (PV)
What is the Present Value (PV) of the
previous problem?

13. Simple Interest (PV)
What is the Present Value (PV) of the
previous problem?
The Present Value is simply the
Rs.1,000 you originally deposited.
That is the value today!
Present Value is the current value of a
future amount of money, or a series of
payments, evaluated at a given interest
rate.

14. Why Compound Interest?
Future
Value
(Indian
Rupees)

15. Assume that you deposit Rs.1,000
at a compound interest rate of 7%
for 2 years.
Future Value
Single Deposit (Graphic)
0 1 2
Rs.1,000
FV2
7%

16. FV1 = P0 (1+r)1 = Rs.1,000 (1.07)
= Rs.1,070
Compound Interest
You earned Rs.70 interest on your
Rs.1,000 deposit over the first year.
This is the same amount of interest you
would earn under simple interest.
Future Value
Single Deposit (Formula)

17. FV1 = P0 (1+r)1 = Rs.1,000 (1.07)
= Rs.1,070
FV2 = FV1 (1+r)1
= P0 (1+r)(1+r) = Rs.1,000(1.07)(1.07)
= P0 (1+r)2 = Rs.1,000(1.07)2
= Rs.1,144.90
You earned an EXTRA Rs.4.90 in Year 2 with
compound over simple interest.
Future Value
Single Deposit (Formula)

18. General Future
Value Formula
FV1 = P0(1+r)1
FV2 = P0(1+r)2
etc.
General Future Value Formula:
FVn = P0 (1+r)n
or FVn = P0 (CVIFr,n) -- See Table I

19. CVIF/FVIFr,n is found on Table I
Valuation Using Table I
Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469

20. FV2 = Rs.1,000 (FVIF7%,2)
= Rs.1,000 (1.145)
= Rs.1,145 [Due to Rounding]
Using Future Value Tables
Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469

21. Problem
Mr. Prakash wants to know how large her
deposit of Rs.10,000 today will become at a
compound annual interest rate of 10% for 5
years.

22. Problem
Mr. Prakash wants to know how large her deposit
of Rs.10,000 today will become at a compound
annual interest rate of 10% for 5 years.
0 1 2 3 4 5
10%
Rs.10,000
FV5

23. = Rs.16,110 [Due to Rounding]
Solution
Calculation based on general formula:
FVn = P0 (1+r)n
FV5 = Rs.10,000 (1+ 0.10)5
= Rs.16,105.10
Calculation based on Table I:
FV5 = Rs.10,000 (FVIF10%, 5)
= Rs.10,000 (1.611)

24. = Rs.16,110 [Due to Rounding]
Solution
Calculation based on general formula:
FVn = P0 (1+r)n
FV5 = Rs.10,000 (1+ 0.10)5
= Rs.16,105.10
Calculation based on Table I:
FV5 = Rs.10,000 (FVIF10%, 5)
= Rs.10,000 (1.611)

25. Assume that you need Rs.1,000 in 2 years.
Let’s examine the process to determine
how much you need to deposit today at a
discount rate of 7% compounded annually.
Present Value
Single Deposit (Graphic)

26. Assume that you need Rs.1,000 in 2 years.
Let’s examine the process to determine
how much you need to deposit today at a
discount rate of 7% compounded annually.
0 1 2
Rs.1,000
7%
PV1
PV0
Present Value
Single Deposit (Graphic)

27. PV0 = FV2 / (1+r)2 = Rs.1,000 / (1.07)2
= FV2 / (1+r)2 = Rs.873.44
Present Value
Single Deposit (Formula)
0 1 2
Rs.1,000
7%
PV0

28. General Present
Value Formula
PV0 = FV1 / (1+r)1
PV0 = FV2 / (1+r)2
etc.
General Present Value Formula:
PV0 = FVn / (1+r)n
or PV0 = FVn (PVIFi,n) -- See Table II

29. PVIFr,n is found on Table II
at the end of the book.
Valuation Using Table II
Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681

30. PV2 = Rs.1,000 (PVIF7%,2)
= Rs.1,000 (.873)
= Rs.873 [Due to Rounding]
Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681
Present Value Table

31. PVIFr,n is found on Table II
Valuation Using Table II
Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681

32. Problem
Ms Radha wants to know how large of a
deposit to make so that the money will
grow to Rs.10,000 in 5 years at a
discount rate of 10%.

33. Problem
Ms Radha wants to know how large of a
deposit to make so that the money will
grow to Rs.10,000 in 5 years at a
discount rate of 10%.
0 1 2 3 4 5
10%
Rs.10,000
PV0

34. Calculation based on general formula:
PV0 = FVn / (1+r)n
PV0 = Rs.10,000 / (1+ 0.10)5
Rs.10,000 (.621)
= Rs.6,209.21
Calculation based on Table I: PV0
= Rs.10,000 (PVIF10%, 5)
= Rs.10,000 (.621)
= Rs.6,210.00 [Due to Rounding]
Solution :-

35. Double Your Money!!!
Quick! How long does it take to
double Rs.5,000 at a compound
rate of 12% per year (approx.)?
We will use the “Rule-of-72”.

36. The “Rule-of-72”
Quick! How long does it take to
double Rs.5,000 at a compound
rate of 12% per year (approx.)?
Approx. Years to Double = 72 / r%
72 / 12% = 6 Years
[Actual Time is 6.12 Years]

37. 1. Read problem thoroughly
2. Create a time line
3. Put cash flows and arrows on time line
4. Determine if it is a PV or FV problem
5. Determine if solution involves a single
CF, annuity stream(s), or mixed flow
6. Solve the problem
Steps to Solve Time Value
of Money Problems

38. Mixed Flows Example
Julie Miller will receive the set of cash
flows below. What is the Present
Value at a discount rate of 10%.
0 1 2 3 4 5
10%
Rs.600 Rs.600 Rs.400 Rs.400 Rs.100
PV0

39. “Piece-At-A-Time”
4 5
Rs.400 Rs.100
0 1 2 3
10%
Rs.600 Rs.600 Rs.400
Rs.545.45
Rs.495.87
Rs.300.53
Rs.273.21
Rs. 62.09
Rs.1677.15 = PV0 of the Mixed Flow

40. What is the minimum amount which a person
should be ready to accept today from a debtor
who otherwise has to pay a sum of Rs. 5000
Today Rs. 6000, Rs. 8000 Rs.9000 and Rs.
10000 at the end of year 1,2,3,4 respectively from
Today. The rate of Interest may be taken at 14 %.
Illustration 2.7

41. “Piece-At-A-Time”
0 1 2 3 4
14%
5000 6000 8000 9000 10000
5000
5262
6152
6075
5920
Rs.28409 = PV0 of the Mixed Flow

42. An Annuity represents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.
Ordinary Annuity: Payments or receipts
occur at the end of each period.
Annuity Due: Payments or receipts
occur at the beginning of each period.
Annuity

43. Examples of Annuities
Student Loan Payments
Car Loan Payments
Insurance Premiums
Mortgage Payments
Retirement Savings

44. Parts of an Annuity
2 3
(Ordinary Annuity)
End of Period 1
0 1
End of
Period 2
Today
Rs.100 Rs.100 Rs.100
Equal Cash Flows
Each 1 Period Apart
End of
Period 3

45. Parts of an Annuity
0 1 2 3
(Annuity Due)
Beginning of
Period 1
Beginning of
Period 2
Rs.100
Today
Rs.100 Rs.100
Equal Cash Flows
Each 1 Period Apart
Beginning of
Period 3

46. FVAn = A(1+r)n-1 + A(1+r)n-2 +
... + A(1+r)1 + A(1+r)0
Overview of an
Ordinary Annuity -- FVA
A A A
0 1 2 n n+1
FVAn
A = Annuity
Periodic
Cash Flow
Cash flows occur at the end of the period
r% . . .

47. FVA3 = Rs.1,000(1.07)2 +
Rs.1,000(1.07)1 + Rs.1,000(1.07)0
= Rs.1,145 + Rs.1,070 + Rs.1,000
= Rs.3,215
Example of an
Ordinary Annuity -- FVA
Rs.1,000 Rs.1,000
0
Rs.3,215 =
FVA3
7%
Rs.1,000
Rs.1,070
Rs.1,145
Cash flows occur at the end of the period
1 2 3

48. FVAn
FVA3
= A (FVIFAr%,n)
= Rs.1,000 (FVIFA7%,3)
= Rs.1,000 (3.215) = Rs.3,215
Valuation Using Table III
Period 6% 7% 8%
1 1.000 1.000 1.000
2 2.060 2.070 2.080
3 3.184 3.215 3.246
4 4.375 4.440 4.506
5 5.637 5.751 5.867

49. Valuation Using Formula

50. FVADn = A(1+r)n + A(1+r)n-1 +
... + A(1+r)2 + A(1+r)1
= FVAn (1+r)
Overview View of an
Annuity Due -- FVAD
A A
0 1 2 3
A A A
n-1 n
FVADn
r% . . .
Cash flows occur at the beginning of the period

51. FVAD3 = Rs.1,000(1.07)3 +
Rs.1,000(1.07)2 + Rs.1,000(1.07)1
= Rs.1,225 + Rs.1,145 + Rs.1,070
= Rs.3,440
Example of an
Annuity Due -- FVAD
Rs.1,000
0 4
Rs.3,440 =
FVAD 3
7%
Rs.1,000 Rs.1,000
Rs.1,070
Rs.1,145
Rs.1,225
Cash flows occur at the beginning of the period
1 2 3

52. FVADn
FVAD3
= A (FVIFAr%,n)(1+r)
= Rs.1,000 (FVIFA7%,3)(1.07)
= Rs.1,000 (3.215)(1.07) = Rs.3,440
Valuation Using Table III
Period 6% 7% 8%
1 1.000 1.000 1.000
2 2.060 2.070 2.080
3 3.184 3.215 3.246
4 4.375 4.440 4.506
5 5.637 5.751 5.867

53. + ... + A/(1+r)n
Overview of an
Ordinary Annuity -- PVA
A A A
0 1 2 n n+1
PVAn
PVAn = A/(1+r)1 + A/(1+r)2
A = Periodic
Cash Flow
r% . . .
Cash flows occur at the end of the period

54. Calculate Present value
Annuity -- PVA
Rs.1,000 Rs.1,000 Rs.1,000
0 4
7%
Cash flows occur at the end of the period
1 2 3

55. PVA3 = Rs.1,000/(1.07)1 +
Rs.1,000/(1.07)2 +
Rs.1,000/(1.07)3
= Rs.934.58 + Rs.873.44 +Rs.816.30
= Rs.2,624.32
Example of an
Ordinary Annuity -- PVA
Rs.1,000 Rs.1,000 Rs.1,000
0 4
7%
Rs.934.58
Rs.873.44
Rs.816.30
Rs.2,624.32 =
PVA3
Cash flows occur at the end of the period
1 2 3

56. PVA3 = Rs.1,000/(1.07)1 +
Rs.1,000/(1.07)2 +
Rs.1,000/(1.07)3
= Rs.934.58 + Rs.873.44 + Rs.816.30
= Rs.2,624.32
Example of an
Ordinary Annuity -- PVA
Rs.1,000 Rs.1,000 Rs.1,000
0 4
7%
Rs.934.58
Rs.873.44
Rs.816.30
Rs.2,624.32 =
PVA3
Cash flows occur at the end of the period
1 2 3

57. PVAn = A (PVIFAr%,n)
PVA3 = Rs.1,000 (PVIFA7%,3)
= Rs.1,000 (2.624) = Rs.2,624
Valuation Using Table IV
Period 6% 7% 8%
1 0.943 0.935 0.926
2 1.833 1.808 1.783
3 2.673 2.624 2.577
4 3.465 3.387 3.312
5 4.212 4.100 3.993

58. Valuation Using Table IV

59. PVADn = A/(1+r)0 + A/(1+r)1 + ,. + A/(1+r)n-1
= PVAn (1+r)
Overview of an
Annuity Due -- PVAD
A A A A
0 1 2 n-1 n
PVADn
A: Periodic
Cash Flow
r% . . .
Cash flows occur at the beginning of the period

60. Calculate Present Value
Annuity Due -- PVAD
Rs.1,000.00 Rs.1,000 Rs.1,000
0 4
7%
Cash flows occur at the beginning of the period
1 2 3

61. Example of an
Annuity Due -- PVAD
Rs.1,000.00 Rs.1,000 Rs.1,000
0 4
7%
Rs. 934.58
Rs. 873.44
Rs.2,808.02 = PVADn
PVADn = Rs.1,000/(1.07)0 + Rs.1,000/(1.07)1 +
Rs.1,000/(1.07)2 = Rs.2,808.02
Cash flows occur at the beginning of the period
1 2 3

62. PVADn = A (PVIFAr%,n)(1+r)
PVAD3 = Rs.1,000 (PVIFA7%,3)(1.07)
= Rs.1,000 (2.624)(1.07) = 2,808
Valuation Using Table IV
Period 6% 7% 8%
1 0.943 0.935 0.926
2 1.833 1.808 1.783
3 2.673 2.624 2.577
4 3.465 3.387 3.312
5 4.212 4.100 3.993

63. Valuation Using Table IV

64. General Formula:
FVn = PV0(1 + [r/m])mn
n :- Number of Years
m:- Compounding Periods per Year
r:- Annual Interest Rate today
FVn,m :- FV at the end of Year n
PV0 :- PV of the Cash Flow
Frequency of
Compounding

65. Mr. John has Rs.1,000 to invest for 2
Years at an annual interest rate of
12%.
Calculate Future Value if compounded
a) Annually,
b) Semi Annually,
c) Quarterly,
d) Monthly and
e) daily
Impact of Frequency

66. Mr. John has Rs.1,000 to invest for 2
Years at an annual interest rate of
12%.
Annual FV2
Semi FV2
= 1,000(1+ [.12/1])(1)(2)
= 1,254.40
= 1,000(1+ [.12/2])(2)(2)
= 1,262.48
Impact of Frequency

67. Qrtly
Monthly
Daily
FV2 = 1,000(1+ [.12/4])(4)(2)
= 1,266.77
FV2 = 1,000(1+ [.12/12])(12)(2)
= 1,269.73
FV2 = 1,000(1+[.12/365])(365)(2)
= 1,271.20
Impact of Frequency

68. Effective Annual Interest Rate
The actual rate of interest earned (paid)
after adjusting the nominal rate for
factors such as the number of
compounding periods per year.
(1 + [ r / m ] )m - 1
Effective Annual
Interest Rate

69. Basket Wonders (BW) has a Rs.1,000 CD at
the bank. The interest rate is 6%
compounded quarterly for 1 year. What is
the Effective Annual Interest Rate (EAR)?
EAR = ( 1 + 6% / 4 )4 - 1
= 1.0614 - 1 = .0614 or 6.14%!
BWs Effective
Annual Interest Rate

70. Perpetuity mean indefinite / forever.
Perpetuity may be defined as an infinite
series of equal cash flows occurring at
regular intervals
PVp = Annual Cash Flow / r
= A
r
Other Cash Flows:
Perpetuity

71. It is a kind of reserve by which a provision
is made to reduce future liability. For
Example Redemption of debenture or
repayment of liability
Specific some of money is kept aside from
the profit every year to accumulate the total
amount to be paid at the time of maturity.
SINKING FUND

72. SINKING FUND
An amount of Rs 100000 is required at the
end of 5 years from now to pay a debenture
Liability. What amount should be accumulated
every year at 10% rate of interest so that it
ultimately becomes Rs 100000 after 5 years

73. Mr. John is borrowing $10 million at a
compound annual interest rate of 12%.
Amortize the loan if annual payments are
made for 5 years.Calculate annuity.
Amortizing a Loan Example

74. Mr. John is borrowing $10 million at a
compound annual interest rate of 12%. Amortize
the loan if annual payments are made for 5
years.
Step 1: Payment
PV0 = A (PVIFA r%,n)
$10 = A (PVIFA 12%,5)
$10 = A (3.605)
A = $10 / 3.605 = $ 2.774 million
Amortizing a Loan Example

75. A loan of Rs 50000 is to be repaid in equal
annual instalment of Rs 14000. The loan
carries a 6 % interest rate. How many
payment are required to repay this loan?
Prepare Amortization Schedule
Practical Problem

76. Practical Problem
Amortisation Schedule
Year
Principal
Owed
Interest
@6% Total AnnuityInterest
Principal
Repayment
1 50000 3000 53000 14000 3000 11000
2 39000 2340 41340 14000 2340 11660
3 27340 1640 28980 14000 1640 12360
4 14980 899 15879 14000 899 13101
5 1879 113 1992 1992 113 1879
Total 57992 7992 50000

77. Solved All unsolved Problem Page no 47
P2.1 to P2.5,P2.7, P2.8 and P2.9
Practical Problem