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Chapter 2
The Two Key Concepts in Finance
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It’s what we learn after we think we know it
all that counts.
- Kin Hubbard
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Outline
 Introduction
 Time
value of money
 Safe dollars and risky dollars
 Relationship between risk and return
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Introduction
 The
occasional reading of basic material in
your chosen field is an excellent
philosophical exercise
• Do not be tempted to include that you “know it
all”
– E.g., what is the present value of a growing
perpetuity that begins payments in five years
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Time Value of Money
 Introduction
 Present
and future values
 Present and future value factors
 Compounding
 Growing income streams
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Introduction
 Time
has a value
• If we owe, we would prefer to pay money later
• If we are owed, we would prefer to receive
money sooner
• The longer the term of a single-payment loan,
the higher the amount the borrower must repay
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Present and Future Values
 Basic
time value of money relationships:
PV  FV  DF
FV  PV  CF
where PV = present value;
FV = future value;
DF = discount factor = 1/(1  R )t
CF = compounding factor = (1  R )t
R = interest rate per period; and
t = time in periods
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Present and Future Values
(cont’d)
 A present
value is the discounted value of
one or more future cash flows
 A future value is the compounded value of
a present value
 The discount factor is the present value of a
dollar invested in the future
 The compounding factor is the future value
of a dollar invested today
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Present and Future Values
(cont’d)
 Why
is a dollar today worth more than a
dollar tomorrow?
• The discount factor:
– Decreases as time increases
• The farther away a cash flow is, the more we discount it
– Decreases as interest rates increase
• When interest rates are high, a dollar today is worth much
more than that same dollar will be in the future
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Present and Future Values
(cont’d)
 Situations:
• Know the future value and the discount factor
– Like solving for the theoretical price of a bond
• Know the future value and present value
– Like finding the yield to maturity on a bond
• Know the present value and the discount rate
– Like solving for an account balance in the future
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Present and Future Value
Factors
 Single
sum factors
 How we get present and future value tables
 Ordinary annuities and annuities due
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Single Sum Factors
 Present
value interest factor and future
value interest factor:
PV  FV  PVIF
FV  PV  FVIF
where
1
PVIF 
(1  R)t
FVIF  (1  R)t
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Single Sum Factors (cont’d)
Example
You just invested $2,000 in a three-year bank certificate of
deposit (CD) with a 9 percent interest rate.
How much will you receive at maturity?
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Single Sum Factors (cont’d)
Example (cont’d)
Solution: Solve for the future value:
FV  $2, 000 1.093
 $2, 000 1.2950
 $2,590
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How We Get Present and
Future Value Tables
 Standard
time value of money tables present
factors for:
•
•
•
•
Present value of a single sum
Present value of an annuity
Future value of a single sum
Future value of an annuity
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How We Get Present and
Future Value Tables (cont’d)
 Relationships:
• You can use the present value of a single sum to
obtain:
– The present value of an annuity factor (a running
total of the single sum factors)
– The future value of a single sum factor (the inverse
of the present value of a single sum factor)
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Ordinary Annuities
and Annuities Due
 An
annuity is a series of payments at equal
time intervals
 An
ordinary annuity assumes the first
payment occurs at the end of the first year
 An
annuity due assumes the first payment
occurs at the beginning of the first year
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Ordinary Annuities
and Annuities Due (cont’d)
Example
You have just won the lottery! You will receive $1 million
in ten installments of $100,000 each. You think you can
invest the $1 million at an 8 percent interest rate.
What is the present value of the $1 million if the first
$100,000 payment occurs one year from today? What is
the present value if the first payment occurs today?
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Ordinary Annuities
and Annuities Due (cont’d)
Example (cont’d)
Solution: These questions treat the cash flows as an
ordinary annuity and an annuity due, respectively:
PV of ordinary annuity  $100, 000  6.7100  $671, 000
PV of annuity due  $100, 000  ($100, 000  6.2468)  $724, 680
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Compounding
 Definition
 Discrete
versus continuous intervals
 Nominal versus effective yields
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Definition
 Compounding
refers to the frequency with
which interest is computed and added to the
principal balance
• The more frequent the compounding, the higher
the interest earned
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Discrete Versus
Continuous Intervals

Discrete compounding means we can count the
number of compounding periods per year
• E.g., once a year, twice a year, quarterly, monthly, or
daily

Continuous compounding results when there is
an infinite number of compounding periods
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Discrete Versus
Continuous Intervals (cont’d)
 Mathematical
adjustment for discrete
compounding:
FV  PV (1  R / m) mt
R  annual interest rate
m  number of compounding periods per year
t  time in years
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Discrete Versus
Continuous Intervals (cont’d)
 Mathematical
equation for continuous
compounding:
FV  PVe Rt
e  2.71828
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Discrete Versus
Continuous Intervals (cont’d)
Example
Your bank pays you 3 percent per year on your savings
account. You just deposited $100.00 in your savings
account.
What is the future value of the $100.00 in one year if
interest is compounded quarterly? If interest is
compounded continuously?
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Discrete Versus
Continuous Intervals (cont’d)
Example (cont’d)
Solution: For quarterly compounding:
FV  PV (1  R / m) mt
 $100.00(1  0.03 / 4)4
 $103.03
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Discrete Versus
Continuous Intervals (cont’d)
Example (cont’d)
Solution (cont’d): For continuous compounding:
FV  PVe
Rt
 $100.00  e0.03
 $103.05
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Nominal Versus
Effective Yields
 The
stated rate of interest is the simple rate
or nominal rate
• 3.00% in the example
 The
interest rate that relates present and
future values is the effective rate
• $3.03/$100 = 3.03% for quarterly compounding
• $3.05/$100 = 3.05% for continuous
compounding
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Growing Income Streams
 Definition
 Growing
annuity
 Growing perpetuity
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Definition
 A growing
stream is one in which each
successive cash flow is larger than the
previous one
• A common problem is one in which the cash
flows grow by some fixed percentage
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Growing Annuity
 A growing
annuity is an annuity in which
the cash flows grow at a constant rate g:
C
C (1  g ) C (1  g ) 2
C (1  g ) n
PV 


 ... 
2
3
(1  R) (1  R)
(1  R)
(1  R) n 1
N

C1
 1 g  

1  
 
R  g   1  R  
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Growing Perpetuity
 A growing
perpetuity is an annuity where
the cash flows continue indefinitely:
C
C (1  g ) C (1  g ) 2
C (1  g ) 
PV 


 ... 
2
3
(1  R) (1  R)
(1  R)
(1  R) 
Ct (1  g )t 1
C1


t
(1  R)
Rg
t 1

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Safe Dollars and Risky Dollars
 Introduction
 Choosing
among risky alternatives
 Defining risk
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Introduction
 A safe
dollar is worth more than a risky
dollar
• Investing in the stock market is exchanging
bird-in-the-hand safe dollars for a chance at a
higher number of dollars in the future
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Introduction (cont’d)
 Most
investors are risk averse
• People will take a risk only if they expect to be
adequately rewarded for taking it
 People
have different degrees of risk
aversion
• Some people are more willing to take a chance
than others
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Choosing Among
Risky Alternatives
Example
You have won the right to spin a lottery wheel one time.
The wheel contains numbers 1 through 100, and a pointer
selects one number when the wheel stops. The payoff
alternatives are on the next slide.
Which alternative would you choose?
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Choosing Among
Risky Alternatives (cont’d)
A
[1-50]
[51-100]
Avg.
payoff
B
$110 [1-50]
$90 [51-100]
$100
C
$200 [1-90]
$0 [91-100]
$100
D
$50 [1-99]
$500 [100]
$100
$1,000
-$89,000
$100
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Choosing Among
Risky Alternatives (cont’d)
Example (cont’d)
Solution:
 Most people would think Choice A is “safe.”
 Choice B has an opportunity cost of $90 relative
to Choice A.
 People who get utility from playing a game pick
Choice C.
 People who cannot tolerate the chance of any
loss would avoid Choice D.
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Choosing Among
Risky Alternatives (cont’d)
Example (cont’d)
Solution (cont’d):
 Choice A is like buying shares of a utility stock.
 Choice B is like purchasing a stock option.
 Choice C is like a convertible bond.
 Choice D is like writing out-of-the-money call
options.
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Defining Risk
 Risk
versus uncertainty
 Dispersion and chance of loss
 Types of risk
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Risk Versus Uncertainty
 Uncertainty
involves a doubtful outcome
• What you will get for your birthday
• If a particular horse will win at the track
 Risk
involves the chance of loss
• If a particular horse will win at the track if you
made a bet
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Dispersion and Chance of Loss
 There
are two material factors we use in
judging risk:
• The average outcome
• The scattering of the other possibilities around
the average
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Dispersion and Chance of Loss
(cont’d)
Investment value
Investment A
Investment B
Time
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Dispersion and Chance of Loss
(cont’d)
 Investments A and
B have the same
arithmetic mean
 Investment
B is riskier than Investment A
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Types of Risk
 Total
risk refers to the overall variability of
the returns of financial assets
 Undiversifiable
risk is risk that must be
borne by virtue of being in the market
• Arises from systematic factors that affect all
securities of a particular type
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Types of Risk (cont’d)
 Diversifiable
risk can be removed by proper
portfolio diversification
• The ups and down of individual securities due
to company-specific events will cancel each
other out
• The only return variability that remains will be
due to economic events affecting all stocks
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Relationship Between Risk and
Return
 Direct
relationship
 Concept of utility
 Diminishing marginal utility of money
 St. Petersburg paradox
 Fair bets
 The consumption decision
 Other considerations
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Direct Relationship
 The
more risk someone bears, the higher the
expected return
 The appropriate discount rate depends on
the risk level of the investment
 The risk-less rate of interest can be earned
without bearing any risk
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Direct Relationship (cont’d)
Expected return
Rf
0
Risk
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Direct Relationship (cont’d)
 The
expected return is the weighted
average of all possible returns
• The weights reflect the relative likelihood of
each possible return
 The
risk is undiversifiable risk
• A person is not rewarded for bearing risk that
could have been diversified away
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Concept of Utility
 Utility
measures the satisfaction people get
out of something
• Different individuals get different amounts of
utility from the same source
– Casino gambling
– Pizza parties
– CDs
– Etc.
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Diminishing Marginal
Utility of Money
 Rational
people prefer more money to less
• Money provides utility
• Diminishing marginal utility of money
– The relationship between more money and added
utility is not linear
– “I hate to lose more than I like to win”
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Diminishing Marginal
Utility of Money (cont’d)
Utility
$
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St. Petersburg Paradox
 Assume
the following game:
• A coin is flipped until a head appears
• The payoff is based on the number of tails
observed (n) before the first head
• The payoff is calculated as $2n
 What
is the expected payoff?
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St. Petersburg Paradox
(cont’d)
Number of Tails
Before First
Head
0
1
Probability
(1/2)1 = 1/2
(1/2)2 = 1/4
Payoff
$1
$2
Probability
x Payoff
$0.50
$0.50
2
3
4
(1/2)3 = 1/8
(1/2)4 = 1/16
(1/2)5 = 1/32
$4
$8
$16
$0.50
$0.50
$0.50
n
(1/2)n + 1
1.00
$2n
$0.50
Total

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St. Petersburg Paradox
(cont’d)
 In
the limit, the expected payoff is infinite
 How
much would you be willing to play the
game?
• Most people would only pay a couple of dollars
• The marginal utility for each additional $0.50
declines
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Fair Bets
 A fair
bet is a lottery in which the expected
payoff is equal to the cost of playing
• E.g., matching quarters
• E.g., matching serial numbers on $100 bills
 Most
people will not take a fair bet unless
the dollar amount involved is small
• Utility lost is greater than utility gained
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The Consumption Decision
 The
consumption decision is the choice to
save or to borrow
• If interest rates are high, we are inclined to save
– E.g., open a new savings account
• If interest rates are low, borrowing looks
attractive
– E.g., a higher home mortgage
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The Consumption
Decision (cont’d)
 The
equilibrium interest rate causes savers
to deposit a sufficient amount of money to
satisfy the borrowing needs of the economy
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Other Considerations
 Psychic
return
 Price risk versus convenience risk
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Psychic Return
 Psychic
return comes from an individual
disposition about something
• People get utility from more expensive things,
even if the quality is not higher than cheaper
alternatives
– E.g., Rolex watches, designer jeans
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Price Risk Versus
Convenience Risk

Price risk refers to the possibility of adverse
changes in the value of an investment due to:
• A change in market conditions
• A change in the financial situation
• A change in public attitude

E.g., rising interest rates and stock prices, a
change in the price of gold and the value of the
dollar
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Price Risk Versus
Convenience Risk (cont’d)
 Convenience
risk refers to a loss of
managerial time rather than a loss of dollars
• E.g., a bond’s call provision
– Allows the issuer to call in the debt early, meaning
the investor has to look for other investments
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