https://rb.gy/n89u77 Discuss the role of time value in finance, the use of computational tools, and the basic patterns of cash flow. Understand the concepts of future value and present value, their calculation for single amounts, and the relationship between them.

1. 3-1
Chapter 5
Time Value
of Money
Qasim Raza Khan
Lecturer in Department of Management Sciences
COMSATS-Lahore.
You Tube Link for lecture:
https://rb.gy/n89u77

2. 3-2
The Time Value of Money
 The Interest Rate
 Simple Interest
 Compound Interest
 Amortizing a Loan

3. 3-3
Obviously, $10,000 today.
You already recognize that there is
TIME VALUE TO MONEY!!
The Interest Rate
Which would you prefer -- $10,000
today or $10,000 in 5 years?

4. 3-4
TIME allows you the opportunity to
postpone consumption and earn
INTEREST.
Why TIME?
Why is TIME such an important
element in your decision?

5. 3-5
Types of Interest
Simple Interest
Interest paid (earned) on only the original
amount, or principal borrowed (lent).

6. 3-6
Types of Interest
Compound Interest
Interest paid (earned) on any previous
interest earned, as well as on the
principal borrowed (lent).

7. 3-7
0
5000
10000
15000
20000
1st Year 10th
Year
20th
Year
30th
Year
Future Value of a Single $1,000 Deposit
10% Simple
Interest
7% Compound
Interest
10% Compound
Interest
Why Compound Interest?
FutureValue(U.S.Dollars)

8. 3-8
We will Learn
How we can calculate FV with
simple Interest and compound
interest?

9. 3-9
Simple Interest Formula
Formula SI = P0(i)(n)
SI: Simple Interest
P0: Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods

10. 3-10
SI = P0(i)(n)
= $1,000(.07)(2)
= $140
Simple Interest Example
Assume that you deposit $1,000 in an
account earning 7% simple interest for
2 years. What is the accumulated
interest at the end of the 2nd year?

11. 3-11
FV = P0 + SI
= $1,000 + $140
= $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.
Simple Interest (FV)
What is the Future Value (FV) of the
deposit?

12. 3-12
We will Learn
How we can calculate Present
Value (PV) with discount rate?
PV=?

13. 3-13
The Present Value is simply the
$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.
Simple Interest (PV)
What is the Present Value (PV) of the
previous problem?

14. 3-14
Assume that you deposit $1,000 at
a compound interest rate of 7% for
2 years.
Future Value
Single Deposit (Graphic)
0 1 2
$1,000
FV2
7%

15. 3-15
General Future Value Formula:
FVn = P0 (1+i)n
or
FVn = P0 (FVIFi,n) -- See Table I
General Future
Value Formula
.

16. 3-16
FV2 = P0 (1+i)2 = $1,000(1.07)2
= $1,144.90
You earned an EXTRA $4.90 in Year 2 with
compound over simple interest.
Future Value
Single Deposit (Formula)

17. 3-17
FVIFi,n is found on Table I at the end
of the book or on the card insert.
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

18. 3-18
FV2 = $1,000 (FVIF7%,2)
= $1,000 (1.145)
= $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

19. 3-19
Julie Miller wants to know how large her deposit
of $10,000 today will become at a compound
annual interest rate of 10% for 5 years.
Story Problem Example
(Practice)
0 1 2 3 4 5
$10,000
FV5
10%

20. 3-20
Calculation based on Table I:
FV5 = $10,000 (FVIF10%, 5)
= $10,000 (1.611)
= $16,110 [Due to Rounding]
Story Problem Solution
 Calculation based on general formula:
FVn = P0 (1+i)n
FV5 = $10,000 (1+ 0.10)5
= $16,105.10

21. 3-21
Assume that you need $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
$1,000
7%
PV1PV0
Present Value
Single Deposit (Graphic)

22. 3-22
General Present Value Formula:
PV0 = FVn / (1+i)n
or
PV0 = FVn (PVIFi,n) -- See Table II
General Present
Value Formula

23. 3-23
PV0 = FV2 / (1+i)2 = $1,000 / (1.07)2
= $873.44
Present Value
Single Deposit (Formula)
0 1 2
$1,000
7%
PV0

24. 3-24
PVIFi,n is found on Table II at the end
of the book or on the card insert.
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

25. 3-25
PV2 = $1,000 (PVIF7%,2)
= $1,000 (.873)
= $873 [Due to Rounding]
Using Present Value Tables
Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681

26. 3-26
Julie Miller wants to know how large of a
deposit to make so that the money will
grow to $10,000 in 5 years at a discount
rate of 10%.
Story Problem Example
(Practice)
0 1 2 3 4 5
$10,000
PV0
10%

27. 3-27
 Calculation based on general formula:
PV0 = FVn / (1+i)n
PV0 = $10,000 / (1+ 0.10)5
= $6,209.21
 Calculation based on Table I:
PV0 = $10,000 (PVIF10%, 5)
= $10,000 (.621)
= $6,210.00 [Due to Rounding]
Story Problem Solution

28. 3-28

29. 3-29
What we will learn…
 1. Annuity
 2. Mixed Cash Stream
 3. Frequency of compounding
 4. Loan Amortization

30. 3-30
Annuity
 A stream of equal periodic cash flows
over a specified time period. These cash
flows can be inflows of returns earned
on investments or outflows of funds
invested to earn future returns.
 An Annuity represents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.

31. 3-31
Types of Annuities
 Ordinary Annuity: Payments or receipts
occur at the end of each period.
 Annuity Due: Payments or receipts
occur at the beginning of each period.

32. 3-32
Examples of Annuities
 Student Loan Payments
 Car Loan Payments
 Insurance Premiums
 Mortgage Payments
 Retirement Savings

33. 3-33
Parts of an Annuity
0 1 2 3
$100 $100 $100
(Ordinary Annuity)
End of
Period 1
End of
Period 2
Today Equal Cash Flows
Each 1 Period Apart
End of
Period 3

34. 3-34
Parts of an Annuity
0 1 2 3
$100 $100 $100
(Annuity Due)
Beginning of
Period 1
Beginning of
Period 2
Today Equal Cash Flows
Each 1 Period Apart
Beginning of
Period 3

35. 3-35
Section-1
Future Value (FV):
1. Ordinary Annuity
2. Annuity Due

36. 3-36
FUTURE VALUE OF AN
ORDINARY ANNUITY….Case 1
Or FVAn = R (FVIFAi%,n)
General Formula
Using Tables

37. 3-37
FUTURE VALUE OF AN ORDINARY ANNUITY
Fran Abrams wishes to determine how much
money she will have at the end of 5 years if he
chooses the ordinary annuity. It represents
deposits of $1,000 annually, at the end of each of
the next 5 years, into a savings account paying
7% annual interest.

38. 3-38
FUTURE VALUE OF AN
ORDINARY ANNUITY
General Formula
Using Tables
FVA5 = $1,000 (FVIFA7%,5)
= $1,000 (5.7507) = $5750

39. 3-39

40. 3-40
FUTURE VALUE OF AN
ANNUITY Due ….Case 2
Or FVADn= R (FVIFAi%,n)(1+i)
General Formula
Using Tables

41. 3-41
FUTURE VALUE OF AN
ANNUITY DUE
Fran Abrams wishes to determine how much
money she will have at the end of 5 years if he
chooses annuity due. It represents deposits of
$1,000 annually, at the start of each of the next 5
years, into a savings account paying 7% annual
interest.

42. 3-42
FUTURE VALUE OF AN
ANNUITY DUE
General Formula
FVAD5 = 6153
Using Tables
FVAD5 = $1,000 (FVIFA7%,5) (1+i)
= $1,000 (5.7507)(1.07) = $6152.5

43. 3-43
Comparison of an Annuity Due
with an Ordinary Annuity Future Value

44. 3-44
Section-2
Present Value (PV):
1. Ordinary Annuity
2. Annuity Due

45. 3-45
Present Value- Ordinary
Annuity ….. Case 3
OR PVAn = R (PVIFAi%,n)
Using Tables
General Formula

46. 3-46
Present Value- Ordinary
Annuity
Braden Company, wants to determine the most it
should pay to purchase a particular ordinary
annuity. The annuity consists of cash flows of $700 at
the end of each year for 5 years. The firm requires the
annuity to provide a minimum return of 8%.

47. 3-47

48. 3-48

49. 3-49
Present Value Annuity Due –
PVAD …. Case 4
Using Tables
OR PVADn = R (PVIFAi%,n)(1+i)
General Formula

50. 3-50
Present Value- Annuity DUE
Braden Company, a small producer of plastic toys,
wants to determine the most it should pay to
purchase a particular annuity due. The annuity consists
of cash flows of $700 at the start of each year for 5
years. The firm requires the annuity to provide a
minimum return of 8%.

51. 3-51
Comparison of an Annuity Due
with an Ordinary Annuity Present
Value

52. 3-52
1. Read problem thoroughly
2. Determine if it is a PV or FV problem
3. Create a time line
4. Put cash flows and arrows on time line
5. Determine if solution involves a single
CF, annuity stream(s), or mixed flow
6. Solve the problem
7. Check with financial calculator (optional)
Steps to Solve Time Value
of Money Problems

53. 3-53
Section-3
Mixed Streams

54. 3-54
Julie Miller will receive the set of cash
flows below.
Mixed Flows Example
0 1 2 3 4 5
$600 $600 $400 $400 $100
10%

55. 3-55
Future Value of a Mixed
Stream

56. 3-56
Mixed Flows Example

57. 3-57
Mixed Flows Example

58. 3-58
Present Value of a mixed
stream

59. 3-59
Mixed Flows Example

60. 3-60
Mixed Flows Example

61. 3-61
Section-4
Frequency of Compounding

62. 3-62
Interest is often compounded more
frequently than once a year. Savings
institutions compound interest
semiannually, quarterly, monthly, weekly,
daily, or even continuously.
Frequency of
Compounding

63. 3-63
General Formula:
FVn = PV0(1 + [i/m])mn
n: Number of Years
m: Compounding Periods per Year
i: Annual Interest Rate
FVn,m: FV at the end of Year n
PV0: PV of the Cash Flow today
Frequency of
Compounding

64. 3-64
Julie Miller has $1,000 to invest for 2
years at an annual interest rate of
12%.
Annual FV2 = 1,000(1+ [.12/1])(1)(2)
= 1,254.40
Semi FV2 = 1,000(1+ [.12/2])(2)(2)
= 1,262.48
Impact of Frequency

65. 3-65
Qrtly FV2 = 1,000(1+ [.12/4])(4)(2)
= 1,266.77
Monthly FV2 = 1,000(1+ [.12/12])(12)(2)
= 1,269.73
Daily FV2 = 1,000(1+[.12/365])(365)(2)
= 1,271.20
Impact of Frequency

66. 3-66
Nominal rate of interest
The nominal, or stated, annual
rate is the contractual annual rate
of interest charged by a lender or
promised by a borrower

67. 3-67
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.
EAR=(1 + [ i / m ] )m - 1
Effective Annual
Interest Rate

68. 3-68
Basket Wonders (BW) has a $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 + 0.06/ 4 )4 - 1
= 1.0614 - 1 = .0614 or 6.14%!
BW’s Effective
Annual Interest Rate

69. 3-69

70. 3-70
Section-5
Loan Amortization

71. 3-71
 The determination of the equal periodic loan
payments necessary to provide a lender with
a specified interest return and to repay the
loan principal over a specified period.
 Loan amortization schedule
A schedule of equal payments to repay a loan.
It shows the allocation of each loan payment to
interest and principal.
LOAN AMORTIZATION

72. 3-72
1. Calculate the payment per period.
2. Determine the interest in Period t.
(Loan balance at t-1) x (i% / m)
3. Compute principal payment in Period t.
(Payment - interest from Step 2)
4. Determine ending balance in Period t.
(Balance - principal payment from Step 3)
5. Start again at Step 2 and repeat.
Steps to Amortizing a Loan

73. 3-73
Julie Miller is borrowing $10,000 at a
compound annual interest rate of 12%.
Amortize the loan if annual payments are
made for 5 years.
Step 1: Payment
PV0 = R (PVIFA i%,n)
$10,000 = R (PVIFA 12%,5)
$10,000 = R (3.605)
R = $10,000 / 3.605 = $2,774
Amortizing a Loan Example

74. 3-74
Amortizing a Loan Example
End of
Year
Payment Interest Principal Ending
Balance
0 --- --- --- $10,000
1 $2,774 $1,200 $1,574 8,426
2 2,774 1,011 1,763 6,663
3 2,774 800 1,974 4,689
4 2,774 563 2,211 2,478
5 2,775 297 2,478 0
$13,871 $3,871 $10,000
[Last Payment Slightly Higher Due to Rounding]