This document provides an overview of key concepts related to time value of money including: - Defined benefit and defined contribution pension plans and how benefits are calculated under each. - Common retirement account types like 401(k) plans which allow tax-deferred contributions and earnings. - The core concept that money has a time value because it can be invested and earn returns over time. - Key time value of money calculations like present value, future value, and determining interest rates or time periods for investments to double in value. - The differences between simple and compound interest and how compound interest leads to exponential growth. - How annuities represent a stream of regular cash flows and the calculations

1. 1
Chapter 4
Time Value of Money
Prepared by the student
Mahmoud y. Al- Saftawi

6. 6
DB
 A defined benefit pension plan is a major type of
pension plan in which an employer/sponsor promises
a specified monthly benefit on retirement that is
predetermined by a formula based on the employee's
earnings history, tenure of service and age, rather
than depending directly on individual investment
returns. It is 'defined' in the sense that the benefit
formula is defined and known in advance.
Conversely, for a "defined contribution pension plan",
the formula for computing the employer's and
employee's contributions is defined and known in
advance, but the benefit to be paid out is not known
in advance .

7. DB
 The most common type of formula used is
based on the employee’s terminal earnings
(final salary). Under this formula, benefits are
based on a percentage of average earnings
during a specified number of years at the end
of a worker’s career .

8. .
 In the private sector, defined benefit plans are often funded
exclusively by employer contributions. For very small
companies with one owner and a handful of younger
employees, the business owner generally receives a high
percentage of the benefits. In the public sector, defined
benefit plans usually require employee contributions.
Over time, these plans may face deficits or surpluses between
the money currently in their plans and the total amount of
their pension obligations. Contributions may be made by the
employee, the employer, or both. In many defined benefit
plans the employer bears the investment risk and can benefit
from surpluses.

9. DC
 a defined contribution plan is a type of employer's annual
contribution is specified. Individual accounts are set up for
participants and benefits are based on the amounts credited to
these accounts (through employer contributions and, if
applicable, employee contributions) plus any investment
earnings on the money in the account. Only employer
contributions to the account are guaranteed, not the future
benefits. In defined contribution plans, future benefits fluctuate
on the basis of investment earnings. The most common type of
defined contribution plan is a savings and thrift plan. Under this
type of plan, the employee contributes a predetermined portion
of his or her earnings (usually pretax) to an individual account

10. A 401(k)
 A 401(k) is a type of retirement savings account in the
U.S., which takes its name from subsection 401(k) of
the Internal Revenue Code (Title 26 of the United
States Code). 401(k) are "defined contribution plans"
with annual contributions limited, currently, to
$17,500. Contributions are tax-deferred ,deducted
from paychecks before taxes and then taxed when a
withdrawal is made from the 401(k) account.
Depending on the employer's program a portion of the
employee's contribution may be matched by the
employer.

11. What is Time Value?
 We say that money has a time value
because that money can be invested
with the expectation of earning a
positive rate of return
 In other words, “a dollar received today
is worth more than a dollar to be
received tomorrow”

12. TVM
 Time value of money quantifies the value of a dollar
through time
 Which would you prefer -- $10,000 today or
 $10,000 in 5 years?
Obviously, $10,000 today.
You already recognize that there is
TIME VALUE TO MONEY!!

13. Why TIME?
Why is TIME such an important element in your
decision?
 TIME allows you the opportunity to postpone
consumption and earn INTEREST.
 For present need.
 For re-investment purpose.
 Future uncertainties.

15. Time Value of Money
 The time value of money is the value of
money figuring in a given amount of interest
earned or inflation accrued over a given amount
of time. The ultimate principle suggests that a
certain amount of money today has different
buying power than the same amount of money in
the future. This notion exists both because there
is an opportunity to earn interest on the money
and because inflation will drive prices up, thus
changing the "value" of the money. The time
value of money is the central concept in finance
theory

16. TVM

17. Uses of Time Value of Money
 Time Value of Money, or TVM, is a concept that is
used in all aspects of finance including:
 Bond valuation.
 Stock valuation.
 Accept/reject decisions for project management.
 retirement planning.
 loan payment schedules.
 decisions to invest (or not) in new equipment.
 And many others

18. Time Value Topics
 Future value
 Present value
 Rates of return
 Amortization

19. The Terminology of Time
Value
 Present Value - An amount of money today,
or the current value of a future cash flow
 Future Value - An amount of money at
some future time period
 Period - A length of time (often a year, but
can be a month, week, day, hour, etc.)
 Interest Rate - The compensation paid to a
lender (or saver) for the use of funds
expressed as a percentage for a period
(normally expressed as an annual rate)

20. Methods of time value of
money
 Compounding techniques.
 Discounting techniques

21. Types of Interest
Simple Interest
Interest paid (earned) on only the original amount,
or principal.
Compound Interest
Interest paid (earned) on any previous interest
earned, as well as on the principal borrowed .
when interest is earned on the interest earned in prior
periods, we call it compound interest. If interest is
earned only on the principal, we call it simple
interest.

22. Simple Interest Formula
 Formula SI = P0(i)(N)
 SI: Simple Interest
 P0: principal (t=0)
 i: Interest Rate per Period
 n: Number of Time Periods

23. 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?
 SI = P0(i)(n)
= $1,000(.07)(2)
= $140

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

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

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

27. Types of TVM Calculations
 There are many types of TVM calculations
 The basic types will be covered in this review
module and include:
 Present value of a lump sum
 Future value of a lump sum
 Present and future value of cash flow streams
 Present and future value of annuities

28. Types of TVM Calculations
 Present value The current worth of a future sum of
money or stream of cash flows given a specified rate
of return. Future cash flows are discounted at the
discount rate, and the higher the discount rate, the
lower the present value of the future cash flows.
Determining the appropriate discount rate is the key
to properly valuing future cash flows, whether they
be earnings or obligations.
 Present value of an annuity An annuity is a series
of equal payments or receipts that occur at evenly
spaced intervals. Leases and rental payments are
examples. The payments or receipts occur at the end
of each period for an ordinary annuity while they
occur at the beginning of each period for an annuity
due.

29. Types of TVM Calculations
 Present value of a perpetuity is an infinite
and constant stream of identical cash flows.
 Future value is the value of an asset or cash
at a specified date in the future that is
equivalent in value to a specified sum today.
 Future value of an annuity (FVA) is the
future value of a stream of payments
(annuity), assuming the payments are
invested at a given rate of interest.

30. Timelines
v A timeline is a diagram used to clarify the
timing of the cash flows for an investment
v Each tick represents one time period
PV FV
0 1 2 3 4 5
Today

31. Time lines show timing of cash
flows.
TIME LINES
0 1 2 3
i%
CF0 CF1 CF3 CF2
Tick marks at ends of periods, so Time 0
is today; Time 1 is the end of Period 1; or
the beginning of Period 2.

32. TIME LINES
The intervals from 0 to 1, 1 to 2, and 2
to 3 are time periods such as years
or months. Time 0 is today, and it is
the beginning of Period 1; Time 1 is one
period from today, and it is both the
end of Period 1 and the beginning of
Period 2; and so on. they could also
be quarters or months or even days.

33. .
We can use four different procedures to
solve time value problems.
 Step-by-Step Approach. TIME LINES
 Formula Approach.
 Financial Calculators.
 Spreadsheets

34. Future Value
A dollar in hand today is worth more than a
dollar to be received in the future—if
you had the dollar now you could invest it,
earn interest, and end up with more
than one dollar in the future. The process of
going forward, from present values
(PVs) to future values (FVs), is called
compounding.

35. Future Value of a Lump Sum
 You can think of future value as the
opposite of present value
 Future value determines the amount
that a sum of money invested today will
grow to in a given period of time
 The process of finding a future value is
called “compounding” (hint: it gets
larger)

36. Example of FV of a Lump Sum
How much money will you have in 5
years if you invest $100 today at a
10% rate of return?
1. Draw a timeline
$100
i = 10%
?
0 1 2 3 4 5

37. .
2. Write out the formula using symbols:
FV = PV * (1+i)N
3. Substitute the numbers into the formula:
FV = $100 * (1+0.1)5
4. Solve for the future value:
FV = $161.05

38. FV of an initial $100 after
3 years (i = 10%)
 .
0 1 2 3
FV = ?
10%
100
Finding FVs (moving to the right
on a time line) is called compounding.

39. After 1 year
 FV1= PV(1 + i)
 = $100(1.10)
 = $110.00

40. After 2 years
 FV2=
 = PV(1+i)
 = $100(1.10)
 = $121.00

41. After 3 years
 FV3=
 = PV(1+i)
 = $100(1.10)
 = $133.10
 In general,
 FVN = PV(1 +i)N

42. Growth of $100 at Various
Interest Rates and Time Periods

43. Present Value
 Finding present values is called
discounting, and as previously noted, it
is the reverse
 of compounding: If you know the PV,
you can compound to find the FV; or if
you know
 the FV, you can discount to find the PV.

44. Present Value of a Lump Sum
 Present value calculations determine what the
value of a cash flow received in the future
would be worth today (time 0)
 The process of finding a present value is
called “discounting” (hint: it gets smaller)
 The interest rate used to discount cash flows
is generally called the discount rate

45. Example of PV of a Lump Sum
 How much would $100 received five years from now
be worth today if the current interest rate is 10%?
1. Draw a timeline
The arrow represents the flow of money and the
numbers under the timeline represent the time
period.
Note that time period zero is today
i = 10%
? $100
0 1 2 3 4 5

46. .
2. Write out the formula using symbols:
PV = FV / (1+i)N
3. Insert the appropriate numbers:
PV = 100 / (1 +0 .1)5
4. Solve the formula:
PV = $62.09

47. What’s the PV of $100 due in
3 years if I/YR = 10%?
 .
Finding PVs is discounting, and it’s the
reverse of compounding.
0 1 2 3
10%
100
PV = ?

48. 48
Present Value of $1 at Various
Interest Rates and Time Periods

49. 49
Finding the interest rate (i)
0 1 2 3
?%
2
-1
FV = PV(1 + i)N
$2 = $1(1 + i)3
(2)(1/3) = (1 + i)
1.2599 = (1 + i)
i = 0.2599 = 25.99%

50. 50
Finding the Time to Double
Finding the number of years (N)
0 1 2 ?
20%
2
-1
FV = PV(1 + i)N
Continued on next slide

51. 51
Finding the Time to Double
Finding the number of years (N)
We can working with natural logs
$2 = $1(1 + 0.20)N
2 = ( 1.20 )
( 2 ) = ( 1.20 )
1 log 2 = N log 1.20
بالقسمة على log للطرفين 1.20
N =1 log 2 /log 1.20
=3.8
N
N

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

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

54. annuities
 An annuity is a cash flow stream in which the
cash flows are all equal and occur at regular
intervals.
such as bonds provide a series of cash inflows
over time, and obligations such as auto
loans, student loans, and mortgages call for a
series of payments. If the payments
are equal and are made at fixed intervals,
then we have an annuity

55. annuities
If payments occur at the end of each period, then we
have an ordinary (or deferred) annuity.
Payments on mortgages, car loans, and student loans
are generally made at the ends of the periods and
thus are ordinary annuities. If the payments are
made at the beginning of each period, then we have
an annuity due. Rental lease payments, life insurance
premiums, and lottery payoffs (if you are lucky
enough to win one!) are examples of annuities due.
Ordinary annuities are more common in finance, so
when we use the term “annuity” in this book, you
may assume that the payments occur at the ends of
the periods unless we state otherwise.

56. Types of Annuities
u 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.

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

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

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

60. What’s the FV of a 3-year
ordinary annuity of $100 at 10%?
60
0 1 2 3
100 100 100
10%
110
121
FV = 331

61. 61
FV ordinary (deferred) annuity
Formula
 The future value of an annuity with N
periods and an interest rate of r can be
found with the following formula:
= PMT
(1+i)N -1
i
= $100
(1+0.10)3 -1
0.10
= $331

62. FUTURE VALUE OF AN ANNUITY DUE
Because each payment occurs one period
earlier with an annuity due, the
payments
will all earn interest for one additional
period. Therefore, the FV of an annuity
due
will be greater than that of a similar
ordinary annuity.

63. FUTURE VALUE OF ANANNUITY DUE
FORMULA
 FVAD =
PMT * [ (1 + i ) – 1 ] * (1 + i )
ــــــــــــــــــــــــــــ
i
N

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

65. 65
What’s the PV of this ordinary
annuity?
0 1 2 3
100 100 100
10%
90.91
82.64
75.13
248.69 = PV

66. 66
PV ordinary annuity? Formula
 The present value of an annuity with N
periods and an interest rate of I can be
found with the following formula:
PMT * [ 1 - (1/1+i) ]
ـــــــــــــــــــــــــــــ
i
N

67. Present Value of Annuities
Due
 Because each payment for an annuity
due occurs one period earlier, the
payments
 will all be discounted for one less
period. Therefore, the PV of an annuity
due must
 be greater than that of a similar
ordinary annuity.

68. PRESENT VALUE OF ANANNUITY
DUE FORMULA
PVAD =
PMT * [ 1 - (1/1+i) ] * (1+i)
ـــــــــــــــــــــــــــــ
i
N

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

70. PERPETUITIES
some securities promise to make payments
forever. For example, in the mid-1700s the British
government issued some bonds that never
matured and whose proceeds were used to pay off
other British bonds. Since this action consolidated
the government’s debt, the new bonds were called
“consols. ”The term stuck, and now any bond that
promises to pay interest perpetually is called a
consol ,or a perpetuity. The interest rate on the
consols was 2.5%, so a consol with a face value of
$1,000 would pay $25per year in perpetuity
.

71. perpetuity
 A consol, or perpetuity, is simply an annuity whose promised
payments extend out forever. Since the payments go on
forever, you can’t apply the step-by-step approach. However,
it’s easy to find the PV of a perpetuity with the following
formula:
 PV of a perpetuity =
PMT
ـــــــــــــــــــــــ
I

72. UNEVEN, OR IRREGULAR,CASHFLOWS

73. What is the PV of this
uneven cash flow stream?
.
0
1
100
2
300
3
300
10%
4
-50
90.91
247.93
225.39
-34.15
530.08 = PV

74. PRESENT VALUE OF AN UNEVEN CASH
FLOW STREAM FORMULA

75. 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%
$600 $600 $400 $400 $100
PV0

76. How to Solve?
1. Solve a “piece-at-a-time” by
discounting each piece back to t=0.
2. Solve a “group-at-a-time” by first
breaking problem into groups of
annuity streams and any single
cash flow groups. Then discount
each group back to t=0.

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

78. “Group-At-A-Time” (#1)
.
0 1 2 3 4 5
10%
$600 $600 $400 $400 $100
$1,041.60
$ 573.57
$ 62.10
$1,677.27 = PV0 of Mixed Flow [Using Tables]
$600(PVIFA10%,2) = $600(1.736) = $1,041.60
$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57
$100 (PVIF10%,5) = $100 (0.621) = $62.10

79. “Group-At-A-Time” (#2)
.
0 1 2 3 4
$400 $400 $400 $400
PV0 equals
$1677.30.
0 1 2
$200 $200
0 1 2 3 4 5
$100
$1,268.00
Plus
$347.20
Plus
$62.10

80. FUTURE VALUE OF AN UNEVEN CASH
FLOW STREAM
 The future value of an uneven cash
flow stream (sometimes called the
terminal, or horizon, value) is found by
compounding each payment to the end
of the stream and then summing the
future values

81. FUTURE VALUE OF AN UNEVEN CASH
FLOW STREAM FORMULA

82. Steps to Solve Time Value
of Money Problems
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

83. SEMIANNUAL AND OTHER
COMPOUNDING PERIODS
 In most of our examples thus far, we
assumed that interest is compounded
once a year, or annually. This is annual
compounding. Suppose, however, that
you put $1,000 into a bank that pays a
6% annual interest rate but credits
interest each 6 months. This is
semiannual compounding

84. .
 quarterly compounding.
 monthly compounding.
 weekly compounding.
 daily compounding.

85. .
 bonds pay interest semiannually; most stocks
pay dividends quarterly; most mortgages,
student loans, and auto loans
involve monthly payments; and most money
fund accounts pay interest daily. Therefore,
it is essential that you understand how to
deal with non annual compounding.

86. .
N * (M) COMPOUNDING PERIOD PER YEAR
I / (M) COMPOUNDING PERIOD PER YEAR

87. Types of Interest Rates
When we move beyond annual compounding,
we must deal with the following four
types of interest rates:
• Nominal annual rates, given the symbol INOM
• Annual percentage rates, termed APR rates
• Periodic rates, denoted as IPER
• Effective annual rates, given the symbol EAR
or EFF% (or Equivalent) Annual Rate

88. 88
Nominal rate (INOM)
 Stated in contracts, and quoted by
banks and brokers.
 Not used in calculations or shown on
time lines
 Periods per year (M) must be given.
 Examples:
 8%; Quarterly
 8%, Daily interest (365 days)

89. NOTE
 Note that the nominal rate is never
shown on a time line, and it is never
used as an input in a financial calculator
(except when compounding occurs only
once a year). If more frequent
compounding occurs, you must use
periodic rates

90. 90
Periodic rate (IPER )
 IPER = INOM /M, where M is number of compounding
periods per year. M = 4 for quarterly, 12 for monthly,
and 360 or 365 for daily compounding.
 Used in calculations, shown on time lines.
 Examples:
 8% quarterly: IPER = 8%/4 = 2%.
 8% daily (365): IPER = 8%/365 = 0.021918%.

91. 91
The Impact of Compounding
 Will the FV of a lump sum be larger or
smaller if we compound more often,
holding the stated I% constant?
 Why?

92. 92
The Impact of Compounding
(Answer)
 LARGER!
 If compounding is more frequent than
once a year--for example, semiannually,
quarterly, or daily--interest is earned on
interest more often.

93. 93
FV Formula with Different
Compounding Periods
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

94. $100 at a 12% nominal rate with
semiannual compounding for 5 years
94
I FV NOM N = PV 1 +
M
M N
0.12
FV5S = $100 1 +
2
2x5
= $100(1.06)10 = $179.08

95. FV of $100 at a 12% nominal rate for
5 years with different compounding
FV(Ann.) = $100(1.12)5 = $176.23
FV(Semi.) = $100(1.06)10 = $179.08
FV(Quar.) = $100(1.03)20 = $180.61
FV(Mon.) = $100(1.01)60 = $181.67
FV(Daily) = $100(1+(0.12/365))(5x365) = $182.19
95

96. FV Formula with Different
Compounding Periods
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

97. .
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

98. 98
Effective Annual Rate (EAR =
EFF%)
 The EAR is the annual rate that causes
PV to grow to the same FV as under
multi-period compounding.

99. Effective Annual Rate Example
99
 Example: Invest $1 for one year at 12%,
semiannual:
FV = PV(1 + INOM/M)M
FV = $1 (1.06)2 = $1.1236.
 EFF% = 12.36%, because $1 invested for
one year at 12% semiannual compounding
would grow to the same value as $1 invested
for one year at 12.36% annual compounding.

100. 100
Comparing Rates
 An investment with monthly payments
is different from one with quarterly
payments. Must put on EFF% basis to
compare rates of return. Use EFF%
only for comparisons.
 Banks say “interest paid daily.” Same
as compounded daily.

101. EFF% for a nominal rate of 12%,
compounded semiannually
101
INOM
M
M
EFF% = 1 + − 1
0.12
2
2
= 1 + − 1
= (1.06)2 - 1.0
= 0.1236 = 12.36%.

102. Can the effective rate ever be
equal to the nominal rate?
 Yes, but only if annual compounding is
used, i.e., if M = 1.
102
 If M > 1, EFF% will always be
greater than the nominal rate.

103. 103
When is each rate used?
INOM: Written into contracts, quoted
by banks and brokers. Not used
in calculations or shown
on time lines.

104. 104
When is each rate used?
(Continued)
IPER: Used in calculations, shown on
time lines.
If INOM has annual compounding,
then IPER = INOM/1 = INOM.

105. 105
When is each rate used?
(Continued)
 EAR (or EFF%): Used to compare
returns on investments with different
payments per year.
 Used for calculations if and only if
dealing with annuities where payments
don’t match interest compounding
periods.

106. FRACTIONAL TIME PERIODS
 For example, suppose you deposited $100in a bank
that pays a nominal rate of 10%, compounded daily,
based on a 365-day year. How much would you have
after 9months? The answer of $107.79 is found as
follows:

107. FRACTIONAL TIME PERIODS
 Now suppose that instead you borrow $100 at a
nominal rate of 10% per year,
simple interest, which means that interest is not earned
on interest. If the loan is out-standing for 274 days
(or 9 months), how much interest would you have to
pay? The interest owed is equal to the principal
multiplied by the interest rate times the number of
periods. In this case, the number of periods is equal
to a fraction of a year:
N = 274/365 = 0.7506849.
Interest owed = $100(10%)(0.7506849) = $7.51

108. Amortization
Steps to Amortizing a Loan
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.

109. 109
Amortization
For example,
suppose a company borrows $100,000, with the loan
to be repaid in 5 equal payments at the end of each
of the next 5 years. The lender charges 6% on the
balance at the beginning of each year. Here’s a
picture of the situation:

110. It is possible to solve the
annuity formula
PV= PMT * [ 1 - (1/1+i) ]
ـــــــــــــــــــــــــــــ
i
100000 = PMT * [ 1 - (1/1+0.06) ] = $23,739.64
ـــــــــــــــــــــــــــــ
0.06
N

112. 112
 Amortization tables are widely
used--for home mortgages, auto
loans, business loans, retirement
plans, and more. They are very
important!
 Financial calculators (and
spreadsheets) are great for setting
up amortization tables.

113. 113
Non-matching rates and periods
 What’s the value at the end of Year 3 of
the following CF stream if the quoted
interest rate is 10%, compounded
semiannually?

114. 114
Time line for non-matching
rates and periods
0 1
2 3
100
5%
4 5 6 6-mos.
periods
100 100

115. 115
Non-matching rates and periods
 Payments occur annually, but
compounding occurs each 6 months.
 So we can’t use normal annuity
valuation techniques.

116. 116
1st Method: Compound Each
CF
0 1
2 3
100
5%
4 5 6
100 100.00
110.25
121.55
331.80
FVA3 = $100(1.05)4 + $100(1.05)2 + $100
= $331.80

117. 117
2nd Method: Treat as an
annuity, use financial calculator
Find the EFF% (EAR) for the quoted rate:
0.10
2
2
EFF% = 1 + − 1 = 10.25%

118. .
 PV
= PMT
(1+i)N -1
i
= $100
(1+0.1025)3 -1
0.1025
= $331.8

119. 119
What’s the PV of this stream?
0
1
100
5%
2 3
100 100
90.70
82.27
74.62
247.59

120. .
2
 100 / 1.05 = 90.7
4
 100 / 1.05 = 82.27
6
 100 / 1.05 = 74.62

121. 121
Comparing Investments
 You are offered a note that pays
$1,000 in 15 months (or 456 days) for
$850. You have $850 in a bank that
pays a 6.76649% nominal rate, with
365 daily compounding, which is a daily
rate of 0.018538% and an EAR of
7.0%. You plan to leave the money in
the bank if you don’t buy the note.
The note is riskless.
 Should you buy it?

122. Daily time line
0 365 456 days
122
IPER = 0.018538% per day.
1,000
-850
… …

123. 123
Three solution methods
 1. Greatest future wealth: FV
 2. Greatest wealth today: PV
 3. Highest rate of return: EFF%

124. 124
1. Greatest Future Wealth
Find FV of $850 left in bank for
15 months and compare with
note’s FV = $1,000.
FVBank = $850(1.00018538)456
= $924.97 in bank.
Buy the note: $1,000 > $924.97.

125. 125
2. Greatest Present Wealth
Find PV of note, and compare
with its $850 cost:
PV = $1,000/(1.00018538)456
= $918.95
Buy the note: $918.95 > $850

126. 126
3. Rate of Return
Find the EFF% on note and compare
with 7.0% bank pays, which is your
opportunity cost of capital:
FVN = PV(1 + I)N
$1,000 = $850(1 + I)456
Now we must solve for I.

127. .
بالقسمة على 850 لطرفي المعادلة
1.176 = (1 + I )
456
( 1.176 ) = 1 + I
1.000355 = 1 + I
I = 0.0355 * 365
= 13.01 %
456

128. 128
Using interest conversion
P/YR =365
NOM% =0.035646(365) = 13.01%
EFF% = [ 1 + (0.1301 /365 ) – 1
= 13.89 %
365
Since 13.89% > 7.0% opportunity cost,
buy the note.