Chapter 5
Choice Under
Uncertainty
Chapter 5 Slide 2
Topics to be Discussed
? Describing Risk
? Preferences Toward Risk
? Reducing Risk
? The Demand for Risky Assets
Chapter 5 Slide 3
Introduction
? Choice with certainty is reasonably
straightforward,
? How do we choose when certain
variables such as income and prices are
uncertain (i.e,making choices with
risk)?
Chapter 5 Slide 4
Describing Risk
? To measure risk we must know,
1) All of the possible outcomes,
2) The likelihood that each outcome will
occur (its probability),
Chapter 5 Slide 5
Describing Risk
? Interpreting Probability
? The likelihood that a given outcome will
occur
Chapter 5 Slide 6
Describing Risk
? Interpreting Probability
? Objective Interpretation
?Based on the observed frequency of
past events
Chapter 5 Slide 7
Describing Risk
? Interpreting Probability
? Subjective
?Based on perception or experience with
or without an observed frequency
?Different information or different abilities to
process the same information can influence
the subjective probability
Chapter 5 Slide 8
Describing Risk
? Expected Value
? The weighted average of the payoffs or
values resulting from all possible
outcomes,
?The probabilities of each outcome are
used as weights
?Expected value measures the central
tendency; the payoff or value expected
on average
Chapter 5 Slide 9
Describing Risk
? An Example
? Investment in offshore drilling exploration,
? Two outcomes are possible
?Success -- the stock price increase from
$30 to $40/share
?Failure -- the stock price falls from $30
to $20/share
Chapter 5 Slide 10
Describing Risk
? An Example
? Objective Probability
?100 explorations,25 successes and 75
failures
?Probability (Pr) of success = 1/4 and the
probability of failure = 3/4
Chapter 5 Slide 11
Describing Risk
? An Example,
e)) ( $ 2 0 / s h a rP r ( f a i l u r ee)) ( $ 4 0 / s h a rP r ( s u c c e s s E V ??
)( $ 2 0 / s h a r e43)( $ 4 0 / s h a r e41 E V ??
$ 2 5 /s h a r e EV ?
Expected Value (EV)
Chapter 5 Slide 12
Describing Risk
? Given,
?Two possible outcomes having payoffs X1
and X2
?Probabilities of each outcome is given by
Pr1 & Pr2
Chapter 5 Slide 13
Describing Risk
? Generally,expected value is written as,
nn2211 XPr.,,XPrXPr E ( X) ????
Chapter 5 Slide 14
Describing Risk
? Variability
?The extent to which possible outcomes of
an uncertain even may differ
Chapter 5 Slide 15
Describing Risk
? A Scenario
? Suppose you are choosing between two
part-time sales jobs that have the same
expected income ($1,500)
? The first job is based entirely on
commission,
? The second is a salaried position,
Variability
Chapter 5 Slide 16
Describing Risk
? A Scenario
? There are two equally likely outcomes in
the first job--$2,000 for a good sales job
and $1,000 for a modestly successful one,
? The second pays $1,510 most of the time
(.99 probability),but you will earn $510 if
the company goes out of business (.01
probability),
Variability
Chapter 5 Slide 17
Income from Sales Jobs
Job 1,Commission,5 2000,5 1000 1500
Job 2,Fixed salary,99 1510,01 510 1500
Expected
Probability Income ($) Probability Income ($) Income
Outcome 1 Outcome 2
Describing Risk
Chapter 5 Slide 18
1 5 0 0$, 5 ( $ 1 0 0 0 ),5 ( $ 2 0 0 0 ))E ( X 1 ???
? Job 1 Expected Income
$ 1 5 0 0,0 1 ( $ 5 1 0 ),9 9 ( $ 1 5 1 0 ) )E ( X 2 ???
? Job 2 Expected Income
Income from Sales Jobs
Describing Risk
Chapter 5 Slide 19
? While the expected values are the
same,the variability is not,
? Greater variability from expected values
signals greater risk,
? Deviation
?Difference between expected payoff and
actual payoff
Describing Risk
Chapter 5 Slide 20
Deviations from Expected Income ($)
Job 1 $2,000 $500 $1,000 -$500
Job 2 1,510 10 510 -900
Outcome 1 Deviation Outcome 2 Deviation
Describing Risk
Chapter 5 Slide 21
? Adjusting for negative numbers
? The standard deviation measures the
square root of the average of the
squares of the deviations of the payoffs
associated with each outcome from
their expected value,
Variability
Describing Risk
Chapter 5 Slide 22
Describing Risk
? The standard deviation is written,
? ? ? ?222211 )(Pr)(Pr XEXXEX ?????
Variability
Chapter 5 Slide 23
Calculating Variance ($)
Job 1 $2,000 $250,000 $1,000 $250,000 $250,000 $500.00
Job 2 1,510 100 510 980,100 9,900 99.50
Deviation Deviation Deviation Standard
Outcome 1 Squared Outcome 2 Squared Squared Deviation
Describing Risk
Chapter 5 Slide 24
Describing Risk
? The standard deviations of the two jobs
are,
50.99
9 0 0,9$
0 0 ),0 1 ( $ 9 8 0,1,9 9 ( $ 1 0 0 )
5 0 0
0 0 0,2 5 0$
0,5 ( $ 2 5 0,0 00),5 ( $ 2 5 0,0 0
2
2
2
1
1
1
?
?
??
?
?
??
?
?
?
?
?
?
*Greater Risk
Chapter 5 Slide 25
Describing Risk
? The standard deviation can be used
when there are many outcomes instead
of only two,
Chapter 5 Slide 26
Describing Risk
? Job 1 is a job in which the income
ranges from $1000 to $2000 in
increments of $100 that are all equally
likely,
Example
Chapter 5 Slide 27
Describing Risk
? Job 2 is a job in which the income
ranges from $1300 to $1700 in
increments of $100 that,also,are all
equally likely,
Example
Chapter 5 Slide 28
Outcome Probabilities for Two Jobs
Income
0.1
$1000 $1500 $2000
0.2
Job 1
Job 2
Job 1 has greater
spread,greater
standard deviation
and greater risk
than Job 2,
Probability
Chapter 5 Slide 29
Describing Risk
? Outcome Probabilities of Two Jobs
(unequal probability of outcomes)
?Job 1,greater spread & standard deviation
?Peaked distribution,extreme payoffs are
less likely
Chapter 5 Slide 30
Describing Risk
? Decision Making
?A risk avoider would choose Job 2,same
expected income as Job 1 with less risk,
? Suppose we add $100 to each payoff in
Job 1 which makes the expected payoff =
$1600,
Chapter 5 Slide 31
Unequal Probability Outcomes
Job 1
Job 2
The distribution of payoffs
associated with Job 1 has a
greater spread and standard
deviation than those with Job 2,
Income
0.1
$1000 $1500 $2000
0.2
Probability
Chapter 5 Slide 32
Income from Sales Jobs--Modified ($)
Recall,The standard deviation is the square
root of the deviation squared,
Job 1 $2,100 $250,000 $1,100 $250,000 $1,600 $500
Job 2 1510 100 510 980,100 1,500 99.50
Deviation Deviation Expected Standard
Outcome 1 Squared Outcome 2 Squared Income Deviation
Chapter 5 Slide 33
Describing Risk
? Job 1,expected income $1,600 and a
standard deviation of $500,
? Job 2,expected income of $1,500 and a
standard deviation of $99.50
? Which job?
? Greater value or less risk?
Decision Making
Chapter 5 Slide 34
? Suppose a city wants to deter people
from double parking,
? The alternatives …..,
Describing Risk
Example
Chapter 5 Slide 35
? Assumptions,
1) Double-parking saves a person $5 in
terms of time spent searching for a
parking space,
2) The driver is risk neutral,
3) Cost of apprehension is zero,
Example
Describing Risk
Chapter 5 Slide 36
? A fine of $5.01 would deter the driver
from double parking,
? Benefit of double parking ($5) is less than
the cost ($5.01) equals a net benefit that is
less than 0,
Example
Describing Risk
Chapter 5 Slide 37
? Increasing the fine can reduce
enforcement cost,
? A $50 fine with a,1 probability of being
caught results in an expected penalty of
$5,
? A $500 fine with a,01 probability of being
caught results in an expected penalty of
$5,
Example
Describing Risk
Chapter 5 Slide 38
? The more risk averse drivers are,the
lower the fine needs to be in order to be
effective,
Example
Describing Risk
Chapter 5 Slide 39
Preferences Toward Risk
? Choosing Among Risky Alternatives
? Assume
?Consumption of a single commodity
?The consumer knows all probabilities
?Payoffs measured in terms of utility
?Utility function given
Chapter 5 Slide 40
Preferences Toward Risk
? A person is earning $15,000 and
receiving 13 units of utility from the job,
? She is considering a new,but risky job,
Example
Chapter 5 Slide 41
Preferences Toward Risk
? She has a,50 chance of increasing her
income to $30,000 and a,50 chance of
decreasing her income to $10,000,
? She will evaluate the position by
calculating the expected value (utility) of
the resulting income,
Example
Chapter 5 Slide 42
Preferences Toward Risk
? The expected utility of the new position
is the sum of the utilities associated with
all her possible incomes weighted by
the probability that each income will
occur,
Example
Chapter 5 Slide 43
Preferences Toward Risk
? The expected utility can be written,
? E(u) = (1/2)u($10,000) + (1/2)u($30,000)
= 0.5(10) + 0.5(18)
= 14
? E(u) of new job is 14 which is greater than
the current utility of 13 and therefore
preferred,
Example
Chapter 5 Slide 44
Preferences Toward Risk
? Different Preferences Toward Risk
? People can be risk averse,risk neutral,or
risk loving,
Chapter 5 Slide 45
Preferences Toward Risk
? Different Preferences Toward Risk
? Risk Averse,A person who prefers a
certain given income to a risky income with
the same expected value,
? A person is considered risk averse if they
have a diminishing marginal utility of
income
?The use of insurance demonstrates risk
aversive behavior,
Chapter 5 Slide 46
Preferences Toward Risk
? A Scenario
? A person can have a $20,000 job with
100% probability and receive a utility level
of 16,
? The person could have a job with a,5
chance of earning $30,000 and a,5
chance of earning $10,000,
Risk Averse
Chapter 5 Slide 47
Preferences Toward Risk
? Expected Income = (0.5)($30,000) +
(0.5)($10,000)
= $20,000
Risk Averse
Chapter 5 Slide 48
Preferences Toward Risk
? Expected income from both jobs is the
same -- risk averse may choose current
job
Risk Averse
Chapter 5 Slide 49
Preferences Toward Risk
? The expected utility from the new job is
found,
? E(u) = (1/2)u ($10,000) + (1/2)u($30,000)
? E(u) = (0.5)(10) + (0.5)(18) = 14
?E(u) of Job 1 is 16 which is greater than
the E(u) of Job 2 which is 14,
Risk Averse
Chapter 5 Slide 50
Preferences Toward Risk
? This individual would keep their present
job since it provides them with more
utility than the risky job,
? They are said to be risk averse,
Risk Averse
Chapter 5 Slide 51
Income ($1,000)
Utility The consumer is risk
averse because she
would prefer a certain
income of $20,000 to a
gamble with a,5 probability
of $10,000 and a,5
probability of $30,000,
E
10
10 15 20
13
14
16
18
0 16 30
A
B
C
D
Risk Averse
Preferences Toward Risk
Chapter 5 Slide 52
Preferences Toward Risk
? A person is said to be risk neutral if they
show no preference between a certain
income,and an uncertain one with the
same expected value,
Risk Neutral
Chapter 5 Slide 53
Income ($1,000) 10 20
Utility
0 30
6
A
E
C
12
18
The consumer is risk
neutral and is indifferent
between certain events
and uncertain events
with the same
expected income,
Preferences Toward Risk
Risk Neutral
Chapter 5 Slide 54
Preferences Toward Risk
? A person is said to be risk loving if they
show a preference toward an uncertain
income over a certain income with the
same expected value,
? Examples,Gambling,some criminal
activity
Risk Loving
Chapter 5 Slide 55
Income ($1,000)
Utility
0
3
10 20 30
A
E
C 8
18
The consumer is risk
loving because she
would prefer the gamble
to a certain income,
Preferences Toward Risk
Risk Loving
Chapter 5 Slide 56
Preferences Toward Risk
? The risk premium is the amount of
money that a risk-averse person would
pay to avoid taking a risk,
Risk Premium
Chapter 5 Slide 57
Preferences Toward Risk
? A Scenario
? The person has a,5 probability of earning
$30,000 and a,5 probability of earning
$10,000 (expected income = $20,000),
? The expected utility of these two outcomes
can be found,
?E(u) =,5(18) +,5(10) = 14
Risk Premium
Chapter 5 Slide 58
Preferences Toward Risk
? Question
?How much would the person pay to avoid
risk?
Risk Premium
Chapter 5 Slide 59
Income ($1,000)
Utility
0 10 16
Here,the risk premium
is $4,000 because a
certain income of $16,000
gives the person the same
expected utility as the
uncertain income that
has an expected value
of $20,000,
10
18
30 40
20
14
A
C
E
G
20
F
Risk Premium
Preferences Toward Risk
Risk Premium
Chapter 5 Slide 60
Preferences Toward Risk
? Variability in potential payoffs increase
the risk premium,
? Example,
? A job has a,5 probability of paying $40,000
(utility of 20) and a,5 chance of paying 0
(utility of 0),
Risk Aversion and Income
Chapter 5 Slide 61
Preferences Toward Risk
? Example,
? The expected income is still $20,000,but
the expected utility falls to 10,
? Expected utility =,5u($) +,5u($40,000)
= 0 +,5(20) = 10
Risk Aversion and Income
Chapter 5 Slide 62
Preferences Toward Risk
? Example,
? The certain income of $20,000 has a utility
of 16,
? If the person is required to take the new
position,their utility will fall by 6,
Risk Aversion and Income
Chapter 5 Slide 63
Preferences Toward Risk
? Example,
? The risk premium is $10,000 (i.e,they
would be willing to give up $10,000 of the
$20,000 and have the same E(u) as the
risky job,
Risk Aversion and Income
Chapter 5 Slide 64
Preferences Toward Risk
? Therefore,it can be said that the greater
the variability,the greater the risk
premium,
Risk Aversion and Income
Chapter 5 Slide 65
Preferences Toward Risk
? Combinations of expected income &
standard deviation of income that yield
the same utility
Indifference Curve
Chapter 5 Slide 66
Risk Aversion and
Indifference Curves
Standard Deviation of Income
Expected
Income
Highly Risk Averse:An
increase in standard
deviation requires a
large increase in
income to maintain
satisfaction,
U1
U2
U3
Chapter 5 Slide 67
Risk Aversion and
Indifference Curves
Standard Deviation of Income
Expected
Income
Slightly Risk Averse,
A large increase in standard
deviation requires only a
small increase in income
to maintain satisfaction,
U1
U2
U3
Chapter 5 Slide 68
Business Executives
and the Choice of Risk
? Study of 464 executives found that,
? 20% were risk neutral
? 40% were risk takers
? 20% were risk adverse
? 20% did not respond
Example
Chapter 5 Slide 69
? Those who liked risky situations did so
when losses were involved,
? When risks involved gains the same,
executives opted for less risky
situations,
Example
Business Executives
and the Choice of Risk
Chapter 5 Slide 70
? The executives made substantial efforts
to reduce or eliminate risk by delaying
decisions and collecting more
information,
Example
Business Executives
and the Choice of Risk
Chapter 5 Slide 71
Reducing Risk
? Three ways consumers attempt to
reduce risk are,
1) Diversification
2) Insurance
3) Obtaining more information
Chapter 5 Slide 72
Reducing Risk
? Diversification
? Suppose a firm has a choice of selling air
conditioners,heaters,or both,
? The probability of it being hot or cold is 0.5,
? The firm would probably be better off by
diversification,
Chapter 5 Slide 73
Income from Sales of Appliances
Air conditioner sales $30,000 $12,000
Heater sales 12,000 30,000
* 0.5 probability of hot or cold weather
Hot Weather Cold Weather
Chapter 5 Slide 74
Reducing Risk
? If the firms sells only heaters or air
conditioners their income will be either
$12,000 or $30,000,
? Their expected income would be,
? 1/2($12,000) + 1/2($30,000) = $21,000
Diversification
Chapter 5 Slide 75
Reducing Risk
? If the firm divides their time evenly
between appliances their air
conditioning and heating sales would be
half their original values,
Diversification
Chapter 5 Slide 76
Reducing Risk
? If it were hot,their expected income
would be $15,000 from air conditioners
and $6,000 from heaters,or $21,000,
? If it were cold,their expected income
would be $6,000 from air conditioners
and $15,000 from heaters,or $21,000,
Diversification
Chapter 5 Slide 77
Reducing Risk
? With diversification,expected income is
$21,000 with no risk,
Diversification
Chapter 5 Slide 78
Reducing Risk
? Firms can reduce risk by diversifying
among a variety of activities that are not
closely related,
Diversification
Chapter 5 Slide 79
Reducing Risk
? Discussion Questions
?How can diversification reduce the risk of
investing in the stock market?
?Can diversification eliminate the risk of
investing in the stock market?
The Stock Market
Chapter 5 Slide 80
Reducing Risk
? Risk averse are willing to pay to avoid
risk,
? If the cost of insurance equals the
expected loss,risk averse people will
buy enough insurance to recover fully
from a potential financial loss,
Insurance
Chapter 5 Slide 81
The Decision to Insure
No $40,000 $50,000 $49,000 $9,055
Yes 49,000 49,000 49,000 0
Insurance Burglary No Burglary Expected Standard
(Pr =,1) (Pr =,9) Wealth Deviation
Chapter 5 Slide 82
Reducing Risk
? While the expected wealth is the same,
the expected utility with insurance is
greater because the marginal utility in
the event of the loss is greater than if no
loss occurs,
? Purchases of insurance transfers wealth
and increases expected utility,
Insurance
Chapter 5 Slide 83
Reducing Risk
? Although single events are random and
largely unpredictable,the average
outcome of many similar events can be
predicted,
The Law of Large Numbers
Chapter 5 Slide 84
Reducing Risk
? Examples
? A single coin toss vs,large number of
coins
? Whom will have a car wreck vs,the
number of wrecks for a large group of
drivers
The Law of Large Numbers
Chapter 5 Slide 85
Reducing Risk
? Assume,
?10% chance of a $10,000 loss from a
home burglary
?Expected loss =,10 x $10,000 = $1,000
with a high risk (10% chance of a $10,000
loss)
?100 people face the same risk
Actuarial Fairness
Chapter 5 Slide 86
Reducing Risk
? Then,
?$1,000 premium generates a $100,000
fund to cover losses
?Actual Fairness
? When the insurance premium = expected
payout
Actuarial Fairness
Chapter 5 Slide 87
The Value of Title Insurance
When Buying a House
? A Scenario,
?Price of a house is $200,000
?5% chance that the seller does not own the
house
Example
Chapter 5 Slide 88
The Value of Title Insurance
When Buying a House
? Risk neutral buyer would pay,
Example
000,190]0[05.]000,200[95(,??
Chapter 5 Slide 89
The Value of Title Insurance
When Buying a House
? Risk averse buyer would pay much less
? By reducing risk,title insurance
increases the value of the house by an
amount far greater than the premium,
Example
Chapter 5 Slide 90
Reducing Risk
? Value of Complete Information
?The difference between the expected value
of a choice with complete information and
the expected value when information is
incomplete,
The Value of Information
Chapter 5 Slide 91
Reducing Risk
? Suppose a store manager must
determine how many fall suits to order,
? 100 suits cost $180/suit
? 50 suits cost $200/suit
? The price of the suits is $300
The Value of Information
Chapter 5 Slide 92
Reducing Risk
? Suppose a store manager must
determine how many fall suits to order,
? Unsold suits can be returned for half cost,
? The probability of selling each quantity is
.50,
The Value of Information
Chapter 5 Slide 93
The Decision to Insure
1,Buy 50 suits $5,000 $5,000 $5,000
2,Buy 100 suits 1,500 12,000 6,750
Expected
Sale of 50 Sale of 100 Profit
Chapter 5 Slide 94
Reducing Risk
? With incomplete information,
?Risk Neutral,Buy 100 suits
?Risk Averse,Buy 50 suits
Chapter 5 Slide 95
Reducing Risk
? The expected value with complete
information is $8,500,
? 8,500 =,5(5,000) +,5(12,000)
? The expected value with uncertainty
(buy 100 suits) is $6,750,
The Value of Information
Chapter 5 Slide 96
Reducing Risk
? The value of complete information is
$1,750,or the difference between the
two (the amount the store owner would
be willing to pay for a marketing study),
The Value of Information
Chapter 5 Slide 97
? Per capita milk consumption has fallen
over the years
? The milk producers engaged in market
research to develop new sales
strategies to encourage the
consumption of milk,
Reducing Risk
The Value of Information,Example
Chapter 5 Slide 98
? Findings
? Milk demand is seasonal with the greatest
demand in the spring
? Ep is negative and small
? EI is positive and large
Reducing Risk
The Value of Information,Example
Chapter 5 Slide 99
? Milk advertising increases sales most in the
spring,
? Allocating advertising based on this
information in New York increased sales by
$4,046,557 and profits by 9%,
? The cost of the information was relatively low,
while the value was substantial,
Reducing Risk
The Value of Information,Example
Chapter 5 Slide 100
? Assets
?Something that provides a flow of money
or services to its owner,
?The flow of money or services can be
explicit (dividends) or implicit (capital
gain),
The Demand for Risky Assets
Chapter 5 Slide 101
? Capital Gain
?An increase in the value of an asset,while
a decrease is a capital loss,
The Demand for Risky Assets
Chapter 5 Slide 102
The Demand for Risky Assets
? Risky Asset
?Provides an uncertain flow of money or
services to its owner,
?Examples
?apartment rent,capital gains,corporate
bonds,stock prices
Risky & Riskless Assets
Chapter 5 Slide 103
The Demand for Risky Assets
? Riskless Asset
?Provides a flow of money or services that is
known with certainty,
? Examples
?short-term government bonds,short-
term certificates of deposit
Risky & Riskless Assets
Chapter 5 Slide 104
The Demand for Risky Assets
? Asset Returns
? Return on an Asset
?The total monetary flow of an asset as a
fraction of its price,
? Real Return of an Asset
?The simple (or nominal) return less the
rate of inflation,
Chapter 5 Slide 105
The Demand for Risky Assets
? Asset Returns
P r i c e P u r c h a s e
F l o wM o n e t a r y R e tu r n A s s e t ?
%10??? $1,000$100/y r,P r i c e B ond F l ow R etur n A s s et
Chapter 5 Slide 106
The Demand for Risky Assets
? Expected Return
?Return that an asset should earn on
average
Expected vs,Actual Returns
Chapter 5 Slide 107
The Demand for Risky Assets
? Actual Return
?Return that an asset earns
Expected vs,Actual Returns
Chapter 5 Slide 108
Investments--Risk
and Return (1926-1999)
Common stocks (S&P 500) 9.5 20.2
Long-term corporate bonds 2.7 8.3
U.S,Treasury bills 0.6 3.2
Risk
Real Rate of (standard
Return (%) deviation,%)
Chapter 5 Slide 109
The Demand for Risky Assets
? Higher returns are associated with
greater risk,
? The risk-averse investor must balance
risk relative to return
Expected vs,Actual Returns
Chapter 5 Slide 110
The Demand for Risky Assets
? An investor is choosing between T-Bills
and stocks,
? T-bills (riskless) versus Stocks (risky)
? Rf = the return on risk free T-bills
?Expected return equals actual return
when there is no risk
The Trade-Off Between Risk and Return
Chapter 5 Slide 111
The Demand for Risky Assets
? An investor is choosing between T-Bills
and stocks,
? Rm = the expected return on stocks
? rm = the actual returns on stock
The Trade-Off Between Risk and Return
Chapter 5 Slide 112
The Demand for Risky Assets
? At the time of the investment decision,
we know the set of possible outcomes
and the likelihood of each,but we do
not know what particular outcome will
occur,
The Trade-Off Between Risk and Return
Chapter 5 Slide 113
The Demand for Risky Assets
? The risky asset will have a higher
expected return than the risk free asset
(Rm > Rf),
? Otherwise,risk-averse investors would
buy only T-bills,
The Trade-Off Between Risk and Return
Chapter 5 Slide 114
The Demand for Risky Assets
? How to allocate savings,
b = fraction of savings in the stock
market
1 - b = fraction in T-bills
The Investment Portfolio
Chapter 5 Slide 115
The Demand for Risky Assets
? Expected Return,
Rp,weighted average of the expected
return on the two assets
Rp = bRm + (1-b)Rf
The Investment Portfolio
Chapter 5 Slide 116
The Demand for Risky Assets
? Expected Return,
If Rm = 12%,Rf = 4%,and b = 1/2
Rp = 1/2(.12) + 1/2(.04) = 8%
The Investment Portfolio
Chapter 5 Slide 117
The Demand for Risky Assets
? Question
?How risky is their portfolio?
The Investment Portfolio
Chapter 5 Slide 118
The Demand for Risky Assets
? Risk (standard deviation) of the portfolio
is the fraction of the portfolio invested in
the risky asset times the standard
deviation of that asset,
mp b ?? ?
The Investment Portfolio
Chapter 5 Slide 119
The Demand for Risky Assets
? Determining b,
fmp RbbRR )1( ???
)( fmfp RRbRR ???
The Investor’s Choice Problem
Chapter 5 Slide 120
The Demand for Risky Assets
? Determining b,
p
m
fm
fp
RR
RR ?
?
)( ?
??
mpb ?? /?
The Investor’s Choice Problem
Chapter 5 Slide 121
The Demand for Risky Assets
? Observations
1) The final equation
is a budget line describing the trade-
off between risk and expected
return,
Risk and the Budget Line
p
m
fm
fp
)R(RRR ???
)( p?
)p(R
Chapter 5 Slide 122
The Demand for Risky Assets
? Observations,
2) Is an equation for a straight line,
3)
c o n s t a n t s a r e and,R,R mfm ?
mfm )/R(R S l o p e ???
Risk and the Budget Line
p
m
fm
fp
)R(RRR ???
Chapter 5 Slide 123
The Demand for Risky Assets
? Observations
3) Expected return,RP,increases as
risk increases,
4) The slope is the price of risk or the
risk-return trade-off,
Risk and the Budget Line
Chapter 5 Slide 124
Choosing Between
Risk and Return
0 p R e t u r n,of D e v ia t io n S t a n d a r d ?
Expected
Return,Rp
U2 is the optimal
choice of those
obtainable,since it
gives the highest
return for a
given risk and is
tangent to the
budget line,
Rf
Budget Line
m?
Rm
??
R*
U2
U1
U3
Chapter 5 Slide 125
Rf
Budget line
The Choices of
Two Different Investors
0
Expected
Return,Rp
p R e t u r n,of
D e v ia t io n S t a n d a r d ?
Given the same budget
line,investor A chooses
low return-low risk,
while investor B
chooses high return-
high risk,
UA
RA
A?
UB
RB
m?
Rm
B?
Chapter 5 Slide 126
Rf
Budget line
Buying Stocks on Margin
0
Expected
Return,Rp
UA
RA
A?
UA,High risk aversion
--Stock & T-bill portfolio
p R e t u r n,of
D e v ia t io n S t a n d a r d ?
UB
RB
m?
Rm
B?
UA,Low risk aversion
--The investor would
invest more than
100% of their wealth
by borrowing or
buying on the margin,
Chapter 5 Slide 127
Investing in the Stock Market
? Observations
?Percent of American families who had
directly or indirectly invested in the stock
market
?1989 = 32%
?1995 = 41%
Chapter 5 Slide 128
Investing in the Stock Market
? Observations
?Share of wealth in the stock market
?1989 = 26%
?1995 = 40%
Chapter 5 Slide 129
Investing in the Stock Market
? Observations
?Participation in the stock market by age
?Less than 35
? 1989 = 23%
? 1995 = 29%
?More than 35
? Small increase
Chapter 5 Slide 130
Investing in the Stock Market
? What Do You Think?
?Why are more people investing in the stock
market?
Chapter 5 Slide 131
Summary
? Consumers and managers frequently
make decisions in which there is
uncertainty about the future,
? Consumers and investors are
concerned about the expected value
and the variability of uncertain
outcomes,
Chapter 5 Slide 132
Summary
? Facing uncertain choices,consumers
maximize their expected utility,and
average of the utility associated with
each outcome,with the associated
probabilities serving as weights,
? A person may be risk averse,risk
neutral or risk loving,
Chapter 5 Slide 133
Summary
? The maximum amount of money that a
risk-averse person would pay to avoid
risk is the risk premium,
? Risk can be reduced by diversification,
purchasing insurance,and obtaining
additional information,
Chapter 5 Slide 134
Summary
? The law of large numbers enables
insurance companies to provide
actuarially fair insurance for which the
premium paid equals the expected
value of the loss being insured against,
? Consumer theory can be applied to
decisions to invest in risky assets,
End of Chapter 5
Choice Under
Uncertainty
Choice Under
Uncertainty
Chapter 5 Slide 2
Topics to be Discussed
? Describing Risk
? Preferences Toward Risk
? Reducing Risk
? The Demand for Risky Assets
Chapter 5 Slide 3
Introduction
? Choice with certainty is reasonably
straightforward,
? How do we choose when certain
variables such as income and prices are
uncertain (i.e,making choices with
risk)?
Chapter 5 Slide 4
Describing Risk
? To measure risk we must know,
1) All of the possible outcomes,
2) The likelihood that each outcome will
occur (its probability),
Chapter 5 Slide 5
Describing Risk
? Interpreting Probability
? The likelihood that a given outcome will
occur
Chapter 5 Slide 6
Describing Risk
? Interpreting Probability
? Objective Interpretation
?Based on the observed frequency of
past events
Chapter 5 Slide 7
Describing Risk
? Interpreting Probability
? Subjective
?Based on perception or experience with
or without an observed frequency
?Different information or different abilities to
process the same information can influence
the subjective probability
Chapter 5 Slide 8
Describing Risk
? Expected Value
? The weighted average of the payoffs or
values resulting from all possible
outcomes,
?The probabilities of each outcome are
used as weights
?Expected value measures the central
tendency; the payoff or value expected
on average
Chapter 5 Slide 9
Describing Risk
? An Example
? Investment in offshore drilling exploration,
? Two outcomes are possible
?Success -- the stock price increase from
$30 to $40/share
?Failure -- the stock price falls from $30
to $20/share
Chapter 5 Slide 10
Describing Risk
? An Example
? Objective Probability
?100 explorations,25 successes and 75
failures
?Probability (Pr) of success = 1/4 and the
probability of failure = 3/4
Chapter 5 Slide 11
Describing Risk
? An Example,
e)) ( $ 2 0 / s h a rP r ( f a i l u r ee)) ( $ 4 0 / s h a rP r ( s u c c e s s E V ??
)( $ 2 0 / s h a r e43)( $ 4 0 / s h a r e41 E V ??
$ 2 5 /s h a r e EV ?
Expected Value (EV)
Chapter 5 Slide 12
Describing Risk
? Given,
?Two possible outcomes having payoffs X1
and X2
?Probabilities of each outcome is given by
Pr1 & Pr2
Chapter 5 Slide 13
Describing Risk
? Generally,expected value is written as,
nn2211 XPr.,,XPrXPr E ( X) ????
Chapter 5 Slide 14
Describing Risk
? Variability
?The extent to which possible outcomes of
an uncertain even may differ
Chapter 5 Slide 15
Describing Risk
? A Scenario
? Suppose you are choosing between two
part-time sales jobs that have the same
expected income ($1,500)
? The first job is based entirely on
commission,
? The second is a salaried position,
Variability
Chapter 5 Slide 16
Describing Risk
? A Scenario
? There are two equally likely outcomes in
the first job--$2,000 for a good sales job
and $1,000 for a modestly successful one,
? The second pays $1,510 most of the time
(.99 probability),but you will earn $510 if
the company goes out of business (.01
probability),
Variability
Chapter 5 Slide 17
Income from Sales Jobs
Job 1,Commission,5 2000,5 1000 1500
Job 2,Fixed salary,99 1510,01 510 1500
Expected
Probability Income ($) Probability Income ($) Income
Outcome 1 Outcome 2
Describing Risk
Chapter 5 Slide 18
1 5 0 0$, 5 ( $ 1 0 0 0 ),5 ( $ 2 0 0 0 ))E ( X 1 ???
? Job 1 Expected Income
$ 1 5 0 0,0 1 ( $ 5 1 0 ),9 9 ( $ 1 5 1 0 ) )E ( X 2 ???
? Job 2 Expected Income
Income from Sales Jobs
Describing Risk
Chapter 5 Slide 19
? While the expected values are the
same,the variability is not,
? Greater variability from expected values
signals greater risk,
? Deviation
?Difference between expected payoff and
actual payoff
Describing Risk
Chapter 5 Slide 20
Deviations from Expected Income ($)
Job 1 $2,000 $500 $1,000 -$500
Job 2 1,510 10 510 -900
Outcome 1 Deviation Outcome 2 Deviation
Describing Risk
Chapter 5 Slide 21
? Adjusting for negative numbers
? The standard deviation measures the
square root of the average of the
squares of the deviations of the payoffs
associated with each outcome from
their expected value,
Variability
Describing Risk
Chapter 5 Slide 22
Describing Risk
? The standard deviation is written,
? ? ? ?222211 )(Pr)(Pr XEXXEX ?????
Variability
Chapter 5 Slide 23
Calculating Variance ($)
Job 1 $2,000 $250,000 $1,000 $250,000 $250,000 $500.00
Job 2 1,510 100 510 980,100 9,900 99.50
Deviation Deviation Deviation Standard
Outcome 1 Squared Outcome 2 Squared Squared Deviation
Describing Risk
Chapter 5 Slide 24
Describing Risk
? The standard deviations of the two jobs
are,
50.99
9 0 0,9$
0 0 ),0 1 ( $ 9 8 0,1,9 9 ( $ 1 0 0 )
5 0 0
0 0 0,2 5 0$
0,5 ( $ 2 5 0,0 00),5 ( $ 2 5 0,0 0
2
2
2
1
1
1
?
?
??
?
?
??
?
?
?
?
?
?
*Greater Risk
Chapter 5 Slide 25
Describing Risk
? The standard deviation can be used
when there are many outcomes instead
of only two,
Chapter 5 Slide 26
Describing Risk
? Job 1 is a job in which the income
ranges from $1000 to $2000 in
increments of $100 that are all equally
likely,
Example
Chapter 5 Slide 27
Describing Risk
? Job 2 is a job in which the income
ranges from $1300 to $1700 in
increments of $100 that,also,are all
equally likely,
Example
Chapter 5 Slide 28
Outcome Probabilities for Two Jobs
Income
0.1
$1000 $1500 $2000
0.2
Job 1
Job 2
Job 1 has greater
spread,greater
standard deviation
and greater risk
than Job 2,
Probability
Chapter 5 Slide 29
Describing Risk
? Outcome Probabilities of Two Jobs
(unequal probability of outcomes)
?Job 1,greater spread & standard deviation
?Peaked distribution,extreme payoffs are
less likely
Chapter 5 Slide 30
Describing Risk
? Decision Making
?A risk avoider would choose Job 2,same
expected income as Job 1 with less risk,
? Suppose we add $100 to each payoff in
Job 1 which makes the expected payoff =
$1600,
Chapter 5 Slide 31
Unequal Probability Outcomes
Job 1
Job 2
The distribution of payoffs
associated with Job 1 has a
greater spread and standard
deviation than those with Job 2,
Income
0.1
$1000 $1500 $2000
0.2
Probability
Chapter 5 Slide 32
Income from Sales Jobs--Modified ($)
Recall,The standard deviation is the square
root of the deviation squared,
Job 1 $2,100 $250,000 $1,100 $250,000 $1,600 $500
Job 2 1510 100 510 980,100 1,500 99.50
Deviation Deviation Expected Standard
Outcome 1 Squared Outcome 2 Squared Income Deviation
Chapter 5 Slide 33
Describing Risk
? Job 1,expected income $1,600 and a
standard deviation of $500,
? Job 2,expected income of $1,500 and a
standard deviation of $99.50
? Which job?
? Greater value or less risk?
Decision Making
Chapter 5 Slide 34
? Suppose a city wants to deter people
from double parking,
? The alternatives …..,
Describing Risk
Example
Chapter 5 Slide 35
? Assumptions,
1) Double-parking saves a person $5 in
terms of time spent searching for a
parking space,
2) The driver is risk neutral,
3) Cost of apprehension is zero,
Example
Describing Risk
Chapter 5 Slide 36
? A fine of $5.01 would deter the driver
from double parking,
? Benefit of double parking ($5) is less than
the cost ($5.01) equals a net benefit that is
less than 0,
Example
Describing Risk
Chapter 5 Slide 37
? Increasing the fine can reduce
enforcement cost,
? A $50 fine with a,1 probability of being
caught results in an expected penalty of
$5,
? A $500 fine with a,01 probability of being
caught results in an expected penalty of
$5,
Example
Describing Risk
Chapter 5 Slide 38
? The more risk averse drivers are,the
lower the fine needs to be in order to be
effective,
Example
Describing Risk
Chapter 5 Slide 39
Preferences Toward Risk
? Choosing Among Risky Alternatives
? Assume
?Consumption of a single commodity
?The consumer knows all probabilities
?Payoffs measured in terms of utility
?Utility function given
Chapter 5 Slide 40
Preferences Toward Risk
? A person is earning $15,000 and
receiving 13 units of utility from the job,
? She is considering a new,but risky job,
Example
Chapter 5 Slide 41
Preferences Toward Risk
? She has a,50 chance of increasing her
income to $30,000 and a,50 chance of
decreasing her income to $10,000,
? She will evaluate the position by
calculating the expected value (utility) of
the resulting income,
Example
Chapter 5 Slide 42
Preferences Toward Risk
? The expected utility of the new position
is the sum of the utilities associated with
all her possible incomes weighted by
the probability that each income will
occur,
Example
Chapter 5 Slide 43
Preferences Toward Risk
? The expected utility can be written,
? E(u) = (1/2)u($10,000) + (1/2)u($30,000)
= 0.5(10) + 0.5(18)
= 14
? E(u) of new job is 14 which is greater than
the current utility of 13 and therefore
preferred,
Example
Chapter 5 Slide 44
Preferences Toward Risk
? Different Preferences Toward Risk
? People can be risk averse,risk neutral,or
risk loving,
Chapter 5 Slide 45
Preferences Toward Risk
? Different Preferences Toward Risk
? Risk Averse,A person who prefers a
certain given income to a risky income with
the same expected value,
? A person is considered risk averse if they
have a diminishing marginal utility of
income
?The use of insurance demonstrates risk
aversive behavior,
Chapter 5 Slide 46
Preferences Toward Risk
? A Scenario
? A person can have a $20,000 job with
100% probability and receive a utility level
of 16,
? The person could have a job with a,5
chance of earning $30,000 and a,5
chance of earning $10,000,
Risk Averse
Chapter 5 Slide 47
Preferences Toward Risk
? Expected Income = (0.5)($30,000) +
(0.5)($10,000)
= $20,000
Risk Averse
Chapter 5 Slide 48
Preferences Toward Risk
? Expected income from both jobs is the
same -- risk averse may choose current
job
Risk Averse
Chapter 5 Slide 49
Preferences Toward Risk
? The expected utility from the new job is
found,
? E(u) = (1/2)u ($10,000) + (1/2)u($30,000)
? E(u) = (0.5)(10) + (0.5)(18) = 14
?E(u) of Job 1 is 16 which is greater than
the E(u) of Job 2 which is 14,
Risk Averse
Chapter 5 Slide 50
Preferences Toward Risk
? This individual would keep their present
job since it provides them with more
utility than the risky job,
? They are said to be risk averse,
Risk Averse
Chapter 5 Slide 51
Income ($1,000)
Utility The consumer is risk
averse because she
would prefer a certain
income of $20,000 to a
gamble with a,5 probability
of $10,000 and a,5
probability of $30,000,
E
10
10 15 20
13
14
16
18
0 16 30
A
B
C
D
Risk Averse
Preferences Toward Risk
Chapter 5 Slide 52
Preferences Toward Risk
? A person is said to be risk neutral if they
show no preference between a certain
income,and an uncertain one with the
same expected value,
Risk Neutral
Chapter 5 Slide 53
Income ($1,000) 10 20
Utility
0 30
6
A
E
C
12
18
The consumer is risk
neutral and is indifferent
between certain events
and uncertain events
with the same
expected income,
Preferences Toward Risk
Risk Neutral
Chapter 5 Slide 54
Preferences Toward Risk
? A person is said to be risk loving if they
show a preference toward an uncertain
income over a certain income with the
same expected value,
? Examples,Gambling,some criminal
activity
Risk Loving
Chapter 5 Slide 55
Income ($1,000)
Utility
0
3
10 20 30
A
E
C 8
18
The consumer is risk
loving because she
would prefer the gamble
to a certain income,
Preferences Toward Risk
Risk Loving
Chapter 5 Slide 56
Preferences Toward Risk
? The risk premium is the amount of
money that a risk-averse person would
pay to avoid taking a risk,
Risk Premium
Chapter 5 Slide 57
Preferences Toward Risk
? A Scenario
? The person has a,5 probability of earning
$30,000 and a,5 probability of earning
$10,000 (expected income = $20,000),
? The expected utility of these two outcomes
can be found,
?E(u) =,5(18) +,5(10) = 14
Risk Premium
Chapter 5 Slide 58
Preferences Toward Risk
? Question
?How much would the person pay to avoid
risk?
Risk Premium
Chapter 5 Slide 59
Income ($1,000)
Utility
0 10 16
Here,the risk premium
is $4,000 because a
certain income of $16,000
gives the person the same
expected utility as the
uncertain income that
has an expected value
of $20,000,
10
18
30 40
20
14
A
C
E
G
20
F
Risk Premium
Preferences Toward Risk
Risk Premium
Chapter 5 Slide 60
Preferences Toward Risk
? Variability in potential payoffs increase
the risk premium,
? Example,
? A job has a,5 probability of paying $40,000
(utility of 20) and a,5 chance of paying 0
(utility of 0),
Risk Aversion and Income
Chapter 5 Slide 61
Preferences Toward Risk
? Example,
? The expected income is still $20,000,but
the expected utility falls to 10,
? Expected utility =,5u($) +,5u($40,000)
= 0 +,5(20) = 10
Risk Aversion and Income
Chapter 5 Slide 62
Preferences Toward Risk
? Example,
? The certain income of $20,000 has a utility
of 16,
? If the person is required to take the new
position,their utility will fall by 6,
Risk Aversion and Income
Chapter 5 Slide 63
Preferences Toward Risk
? Example,
? The risk premium is $10,000 (i.e,they
would be willing to give up $10,000 of the
$20,000 and have the same E(u) as the
risky job,
Risk Aversion and Income
Chapter 5 Slide 64
Preferences Toward Risk
? Therefore,it can be said that the greater
the variability,the greater the risk
premium,
Risk Aversion and Income
Chapter 5 Slide 65
Preferences Toward Risk
? Combinations of expected income &
standard deviation of income that yield
the same utility
Indifference Curve
Chapter 5 Slide 66
Risk Aversion and
Indifference Curves
Standard Deviation of Income
Expected
Income
Highly Risk Averse:An
increase in standard
deviation requires a
large increase in
income to maintain
satisfaction,
U1
U2
U3
Chapter 5 Slide 67
Risk Aversion and
Indifference Curves
Standard Deviation of Income
Expected
Income
Slightly Risk Averse,
A large increase in standard
deviation requires only a
small increase in income
to maintain satisfaction,
U1
U2
U3
Chapter 5 Slide 68
Business Executives
and the Choice of Risk
? Study of 464 executives found that,
? 20% were risk neutral
? 40% were risk takers
? 20% were risk adverse
? 20% did not respond
Example
Chapter 5 Slide 69
? Those who liked risky situations did so
when losses were involved,
? When risks involved gains the same,
executives opted for less risky
situations,
Example
Business Executives
and the Choice of Risk
Chapter 5 Slide 70
? The executives made substantial efforts
to reduce or eliminate risk by delaying
decisions and collecting more
information,
Example
Business Executives
and the Choice of Risk
Chapter 5 Slide 71
Reducing Risk
? Three ways consumers attempt to
reduce risk are,
1) Diversification
2) Insurance
3) Obtaining more information
Chapter 5 Slide 72
Reducing Risk
? Diversification
? Suppose a firm has a choice of selling air
conditioners,heaters,or both,
? The probability of it being hot or cold is 0.5,
? The firm would probably be better off by
diversification,
Chapter 5 Slide 73
Income from Sales of Appliances
Air conditioner sales $30,000 $12,000
Heater sales 12,000 30,000
* 0.5 probability of hot or cold weather
Hot Weather Cold Weather
Chapter 5 Slide 74
Reducing Risk
? If the firms sells only heaters or air
conditioners their income will be either
$12,000 or $30,000,
? Their expected income would be,
? 1/2($12,000) + 1/2($30,000) = $21,000
Diversification
Chapter 5 Slide 75
Reducing Risk
? If the firm divides their time evenly
between appliances their air
conditioning and heating sales would be
half their original values,
Diversification
Chapter 5 Slide 76
Reducing Risk
? If it were hot,their expected income
would be $15,000 from air conditioners
and $6,000 from heaters,or $21,000,
? If it were cold,their expected income
would be $6,000 from air conditioners
and $15,000 from heaters,or $21,000,
Diversification
Chapter 5 Slide 77
Reducing Risk
? With diversification,expected income is
$21,000 with no risk,
Diversification
Chapter 5 Slide 78
Reducing Risk
? Firms can reduce risk by diversifying
among a variety of activities that are not
closely related,
Diversification
Chapter 5 Slide 79
Reducing Risk
? Discussion Questions
?How can diversification reduce the risk of
investing in the stock market?
?Can diversification eliminate the risk of
investing in the stock market?
The Stock Market
Chapter 5 Slide 80
Reducing Risk
? Risk averse are willing to pay to avoid
risk,
? If the cost of insurance equals the
expected loss,risk averse people will
buy enough insurance to recover fully
from a potential financial loss,
Insurance
Chapter 5 Slide 81
The Decision to Insure
No $40,000 $50,000 $49,000 $9,055
Yes 49,000 49,000 49,000 0
Insurance Burglary No Burglary Expected Standard
(Pr =,1) (Pr =,9) Wealth Deviation
Chapter 5 Slide 82
Reducing Risk
? While the expected wealth is the same,
the expected utility with insurance is
greater because the marginal utility in
the event of the loss is greater than if no
loss occurs,
? Purchases of insurance transfers wealth
and increases expected utility,
Insurance
Chapter 5 Slide 83
Reducing Risk
? Although single events are random and
largely unpredictable,the average
outcome of many similar events can be
predicted,
The Law of Large Numbers
Chapter 5 Slide 84
Reducing Risk
? Examples
? A single coin toss vs,large number of
coins
? Whom will have a car wreck vs,the
number of wrecks for a large group of
drivers
The Law of Large Numbers
Chapter 5 Slide 85
Reducing Risk
? Assume,
?10% chance of a $10,000 loss from a
home burglary
?Expected loss =,10 x $10,000 = $1,000
with a high risk (10% chance of a $10,000
loss)
?100 people face the same risk
Actuarial Fairness
Chapter 5 Slide 86
Reducing Risk
? Then,
?$1,000 premium generates a $100,000
fund to cover losses
?Actual Fairness
? When the insurance premium = expected
payout
Actuarial Fairness
Chapter 5 Slide 87
The Value of Title Insurance
When Buying a House
? A Scenario,
?Price of a house is $200,000
?5% chance that the seller does not own the
house
Example
Chapter 5 Slide 88
The Value of Title Insurance
When Buying a House
? Risk neutral buyer would pay,
Example
000,190]0[05.]000,200[95(,??
Chapter 5 Slide 89
The Value of Title Insurance
When Buying a House
? Risk averse buyer would pay much less
? By reducing risk,title insurance
increases the value of the house by an
amount far greater than the premium,
Example
Chapter 5 Slide 90
Reducing Risk
? Value of Complete Information
?The difference between the expected value
of a choice with complete information and
the expected value when information is
incomplete,
The Value of Information
Chapter 5 Slide 91
Reducing Risk
? Suppose a store manager must
determine how many fall suits to order,
? 100 suits cost $180/suit
? 50 suits cost $200/suit
? The price of the suits is $300
The Value of Information
Chapter 5 Slide 92
Reducing Risk
? Suppose a store manager must
determine how many fall suits to order,
? Unsold suits can be returned for half cost,
? The probability of selling each quantity is
.50,
The Value of Information
Chapter 5 Slide 93
The Decision to Insure
1,Buy 50 suits $5,000 $5,000 $5,000
2,Buy 100 suits 1,500 12,000 6,750
Expected
Sale of 50 Sale of 100 Profit
Chapter 5 Slide 94
Reducing Risk
? With incomplete information,
?Risk Neutral,Buy 100 suits
?Risk Averse,Buy 50 suits
Chapter 5 Slide 95
Reducing Risk
? The expected value with complete
information is $8,500,
? 8,500 =,5(5,000) +,5(12,000)
? The expected value with uncertainty
(buy 100 suits) is $6,750,
The Value of Information
Chapter 5 Slide 96
Reducing Risk
? The value of complete information is
$1,750,or the difference between the
two (the amount the store owner would
be willing to pay for a marketing study),
The Value of Information
Chapter 5 Slide 97
? Per capita milk consumption has fallen
over the years
? The milk producers engaged in market
research to develop new sales
strategies to encourage the
consumption of milk,
Reducing Risk
The Value of Information,Example
Chapter 5 Slide 98
? Findings
? Milk demand is seasonal with the greatest
demand in the spring
? Ep is negative and small
? EI is positive and large
Reducing Risk
The Value of Information,Example
Chapter 5 Slide 99
? Milk advertising increases sales most in the
spring,
? Allocating advertising based on this
information in New York increased sales by
$4,046,557 and profits by 9%,
? The cost of the information was relatively low,
while the value was substantial,
Reducing Risk
The Value of Information,Example
Chapter 5 Slide 100
? Assets
?Something that provides a flow of money
or services to its owner,
?The flow of money or services can be
explicit (dividends) or implicit (capital
gain),
The Demand for Risky Assets
Chapter 5 Slide 101
? Capital Gain
?An increase in the value of an asset,while
a decrease is a capital loss,
The Demand for Risky Assets
Chapter 5 Slide 102
The Demand for Risky Assets
? Risky Asset
?Provides an uncertain flow of money or
services to its owner,
?Examples
?apartment rent,capital gains,corporate
bonds,stock prices
Risky & Riskless Assets
Chapter 5 Slide 103
The Demand for Risky Assets
? Riskless Asset
?Provides a flow of money or services that is
known with certainty,
? Examples
?short-term government bonds,short-
term certificates of deposit
Risky & Riskless Assets
Chapter 5 Slide 104
The Demand for Risky Assets
? Asset Returns
? Return on an Asset
?The total monetary flow of an asset as a
fraction of its price,
? Real Return of an Asset
?The simple (or nominal) return less the
rate of inflation,
Chapter 5 Slide 105
The Demand for Risky Assets
? Asset Returns
P r i c e P u r c h a s e
F l o wM o n e t a r y R e tu r n A s s e t ?
%10??? $1,000$100/y r,P r i c e B ond F l ow R etur n A s s et
Chapter 5 Slide 106
The Demand for Risky Assets
? Expected Return
?Return that an asset should earn on
average
Expected vs,Actual Returns
Chapter 5 Slide 107
The Demand for Risky Assets
? Actual Return
?Return that an asset earns
Expected vs,Actual Returns
Chapter 5 Slide 108
Investments--Risk
and Return (1926-1999)
Common stocks (S&P 500) 9.5 20.2
Long-term corporate bonds 2.7 8.3
U.S,Treasury bills 0.6 3.2
Risk
Real Rate of (standard
Return (%) deviation,%)
Chapter 5 Slide 109
The Demand for Risky Assets
? Higher returns are associated with
greater risk,
? The risk-averse investor must balance
risk relative to return
Expected vs,Actual Returns
Chapter 5 Slide 110
The Demand for Risky Assets
? An investor is choosing between T-Bills
and stocks,
? T-bills (riskless) versus Stocks (risky)
? Rf = the return on risk free T-bills
?Expected return equals actual return
when there is no risk
The Trade-Off Between Risk and Return
Chapter 5 Slide 111
The Demand for Risky Assets
? An investor is choosing between T-Bills
and stocks,
? Rm = the expected return on stocks
? rm = the actual returns on stock
The Trade-Off Between Risk and Return
Chapter 5 Slide 112
The Demand for Risky Assets
? At the time of the investment decision,
we know the set of possible outcomes
and the likelihood of each,but we do
not know what particular outcome will
occur,
The Trade-Off Between Risk and Return
Chapter 5 Slide 113
The Demand for Risky Assets
? The risky asset will have a higher
expected return than the risk free asset
(Rm > Rf),
? Otherwise,risk-averse investors would
buy only T-bills,
The Trade-Off Between Risk and Return
Chapter 5 Slide 114
The Demand for Risky Assets
? How to allocate savings,
b = fraction of savings in the stock
market
1 - b = fraction in T-bills
The Investment Portfolio
Chapter 5 Slide 115
The Demand for Risky Assets
? Expected Return,
Rp,weighted average of the expected
return on the two assets
Rp = bRm + (1-b)Rf
The Investment Portfolio
Chapter 5 Slide 116
The Demand for Risky Assets
? Expected Return,
If Rm = 12%,Rf = 4%,and b = 1/2
Rp = 1/2(.12) + 1/2(.04) = 8%
The Investment Portfolio
Chapter 5 Slide 117
The Demand for Risky Assets
? Question
?How risky is their portfolio?
The Investment Portfolio
Chapter 5 Slide 118
The Demand for Risky Assets
? Risk (standard deviation) of the portfolio
is the fraction of the portfolio invested in
the risky asset times the standard
deviation of that asset,
mp b ?? ?
The Investment Portfolio
Chapter 5 Slide 119
The Demand for Risky Assets
? Determining b,
fmp RbbRR )1( ???
)( fmfp RRbRR ???
The Investor’s Choice Problem
Chapter 5 Slide 120
The Demand for Risky Assets
? Determining b,
p
m
fm
fp
RR
RR ?
?
)( ?
??
mpb ?? /?
The Investor’s Choice Problem
Chapter 5 Slide 121
The Demand for Risky Assets
? Observations
1) The final equation
is a budget line describing the trade-
off between risk and expected
return,
Risk and the Budget Line
p
m
fm
fp
)R(RRR ???
)( p?
)p(R
Chapter 5 Slide 122
The Demand for Risky Assets
? Observations,
2) Is an equation for a straight line,
3)
c o n s t a n t s a r e and,R,R mfm ?
mfm )/R(R S l o p e ???
Risk and the Budget Line
p
m
fm
fp
)R(RRR ???
Chapter 5 Slide 123
The Demand for Risky Assets
? Observations
3) Expected return,RP,increases as
risk increases,
4) The slope is the price of risk or the
risk-return trade-off,
Risk and the Budget Line
Chapter 5 Slide 124
Choosing Between
Risk and Return
0 p R e t u r n,of D e v ia t io n S t a n d a r d ?
Expected
Return,Rp
U2 is the optimal
choice of those
obtainable,since it
gives the highest
return for a
given risk and is
tangent to the
budget line,
Rf
Budget Line
m?
Rm
??
R*
U2
U1
U3
Chapter 5 Slide 125
Rf
Budget line
The Choices of
Two Different Investors
0
Expected
Return,Rp
p R e t u r n,of
D e v ia t io n S t a n d a r d ?
Given the same budget
line,investor A chooses
low return-low risk,
while investor B
chooses high return-
high risk,
UA
RA
A?
UB
RB
m?
Rm
B?
Chapter 5 Slide 126
Rf
Budget line
Buying Stocks on Margin
0
Expected
Return,Rp
UA
RA
A?
UA,High risk aversion
--Stock & T-bill portfolio
p R e t u r n,of
D e v ia t io n S t a n d a r d ?
UB
RB
m?
Rm
B?
UA,Low risk aversion
--The investor would
invest more than
100% of their wealth
by borrowing or
buying on the margin,
Chapter 5 Slide 127
Investing in the Stock Market
? Observations
?Percent of American families who had
directly or indirectly invested in the stock
market
?1989 = 32%
?1995 = 41%
Chapter 5 Slide 128
Investing in the Stock Market
? Observations
?Share of wealth in the stock market
?1989 = 26%
?1995 = 40%
Chapter 5 Slide 129
Investing in the Stock Market
? Observations
?Participation in the stock market by age
?Less than 35
? 1989 = 23%
? 1995 = 29%
?More than 35
? Small increase
Chapter 5 Slide 130
Investing in the Stock Market
? What Do You Think?
?Why are more people investing in the stock
market?
Chapter 5 Slide 131
Summary
? Consumers and managers frequently
make decisions in which there is
uncertainty about the future,
? Consumers and investors are
concerned about the expected value
and the variability of uncertain
outcomes,
Chapter 5 Slide 132
Summary
? Facing uncertain choices,consumers
maximize their expected utility,and
average of the utility associated with
each outcome,with the associated
probabilities serving as weights,
? A person may be risk averse,risk
neutral or risk loving,
Chapter 5 Slide 133
Summary
? The maximum amount of money that a
risk-averse person would pay to avoid
risk is the risk premium,
? Risk can be reduced by diversification,
purchasing insurance,and obtaining
additional information,
Chapter 5 Slide 134
Summary
? The law of large numbers enables
insurance companies to provide
actuarially fair insurance for which the
premium paid equals the expected
value of the loss being insured against,
? Consumer theory can be applied to
decisions to invest in risky assets,
End of Chapter 5
Choice Under
Uncertainty
