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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 ProbabilityThe likelihood that a given outcome will
occur
Chapter 5 Slide 6
Describing Risk
Interpreting ProbabilityObjective Interpretation
Based on the observed frequency of past events
Chapter 5 Slide 7
Describing Risk
Interpreting ProbabilitySubjective
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 ValueThe weighted average of the payoffs or
values resulting from all possible outcomes.The probabilities of each outcome are
used as weightsExpected value measures the central
tendency; the payoff or value expected on average
Chapter 5 Slide 9
Describing Risk
An ExampleInvestment in offshore drilling exploration:
Two outcomes are possibleSuccess -- the stock price increase from
$30 to $40/shareFailure -- the stock price falls from $30
to $20/share
Chapter 5 Slide 10
Describing Risk
An ExampleObjective 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))($20/sharPr(failuree))($40/sharPr(success EV
)($20/share43)($40/share41 EV
$25/share EV
Expected Value (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 ScenarioSuppose 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.
VariabilityVariability
Chapter 5 Slide 16
Describing Risk
A ScenarioThere 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 $510 if the company goes out of business (.01 probability).
VariabilityVariability
Chapter 5 Slide 17
Income from Sales Jobs
Job 1: Commission .5 2000 .5 1000 1500
Job 2: Fixed salary .99 1510 .01 510 1500
ExpectedProbability Income ($) Probability Income ($) Income
Outcome 1 Outcome 2
Describing Risk
Chapter 5 Slide 18
1500$ .5($1000).5($2000))E(X1
Job 1 Expected Income
$1500.01($510).99($1510) )E(X2
Job 2 Expected Income
Income from Sales Jobs
Describing Risk
Chapter 5 Slide 19
While the expected values are the same, the variability is not.
Greater variability from expected values signals greater risk.
Deviation
Difference between expected payoff and actual payoff
Describing Risk
Chapter 5 Slide 20
Deviations from Expected Income ($)
Job 1 $2,000 $500 $1,000 -$500
Job 2 1,510 10 510 -900
Outcome 1 Deviation Outcome 2 Deviation
Describing Risk
Chapter 5 Slide 21
Adjusting for negative numbers
The standard deviation measures the square root of the average of the squares of the deviations of the payoffs associated with each outcome from their expected value.
VariabilityVariability
Describing Risk
Chapter 5 Slide 22
Describing Risk
The standard deviation is written:
222
211 )(Pr)(Pr XEXXEX
VariabilityVariability
Chapter 5 Slide 23
Calculating Variance ($)
Job 1 $2,000 $250,000 $1,000 $250,000 $250,000 $500.00
Job 2 1,510 100 510 980,100 9,900 99.50
Deviation Deviation Deviation Standard Outcome 1 Squared Outcome 2 Squared Squared Deviation
Describing Risk
Chapter 5 Slide 24
Describing Risk
The standard deviations of the two jobs are:
50.99
900,9$
00).01($980,1.99($100)
500
000,250$
0.5($250,000).5($250,00
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.
ExampleExample
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.
ExampleExample
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 deviationand 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 payoffsassociated 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 MakingDecision Making
Chapter 5 Slide 34
Suppose a city wants to deter people from double parking.
The alternatives …...
Describing Risk
ExampleExample
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.
ExampleExample
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.
ExampleExample
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.
ExampleExample
Describing Risk
Chapter 5 Slide 38
The more risk averse drivers are, the lower the fine needs to be in order to be effective.
ExampleExample
Describing Risk
Chapter 5 Slide 39
Preferences Toward Risk
Choosing Among Risky AlternativesAssume
Consumption of a single commodityThe consumer knows all probabilitiesPayoffs measured in terms of utilityUtility 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.
ExampleExample
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.
ExampleExample
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.
ExampleExample
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.
ExampleExample
Chapter 5 Slide 44
Preferences Toward Risk
Different Preferences Toward RiskPeople can be risk averse, risk neutral, or
risk loving.
Chapter 5 Slide 45
Preferences Toward Risk
Different Preferences Toward RiskRisk 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 ScenarioA 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 AverseRisk Averse
Chapter 5 Slide 47
Preferences Toward Risk
Expected Income = (0.5)($30,000) +
(0.5)
($10,000) =
$20,000
Risk AverseRisk Averse
Chapter 5 Slide 48
Preferences Toward Risk
Expected income from both jobs is the same -- risk averse may choose current job
Risk AverseRisk 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 AverseRisk 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 AverseRisk Averse
Chapter 5 Slide 51
Income ($1,000)
Utility The consumer is riskaverse because she
would prefer a certainincome of $20,000 to a
gamble with a .5 probabilityof $10,000 and a .5
probability of $30,000.
E
10
10 15 20
1314
16
18
0 16 30
AB
C
D
Risk AverseRisk 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 NeutralRisk Neutral
Chapter 5 Slide 53
Income ($1,000)10 20
Utility
0 30
6A
E
C
12
18
The consumer is riskneutral and is indifferentbetween certain eventsand uncertain events
with the same expected income.
Preferences Toward Risk
Risk NeutralRisk 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 LovingRisk Loving
Chapter 5 Slide 55
Income ($1,000)
Utility
0
3
10 20 30
A
E
C8
18The consumer is riskloving because she
would prefer the gamble to a certain income.
Preferences Toward Risk
Risk LovingRisk 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 PremiumRisk Premium
Chapter 5 Slide 57
Preferences Toward Risk
A ScenarioThe 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 PremiumRisk Premium
Chapter 5 Slide 58
Preferences Toward Risk
Question
How much would the person pay to avoid risk?
Risk PremiumRisk Premium
Chapter 5 Slide 59
Income ($1,000)
Utility
0 10 16
Here , the risk premiumis $4,000 because a
certain income of $16,000gives the person the same
expected utility as the uncertain income thathas an expected value
of $20,000.
10
18
30 40
20
14
A
CE
G
20
F
Risk Premium
Preferences Toward Risk
Risk PremiumRisk 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 IncomeRisk 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 IncomeRisk 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 IncomeRisk 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 IncomeRisk 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 IncomeRisk Aversion and Income
Chapter 5 Slide 65
Preferences Toward Risk
Combinations of expected income & standard deviation of income that yield the same utility
Indifference CurveIndifference Curve
Chapter 5 Slide 66
Risk Aversion andIndifference Curves
Standard Deviation of Income
ExpectedIncome
Highly Risk Averse:Anincrease in standarddeviation requires a large increase in income to maintainsatisfaction.
U1
U2
U3
Chapter 5 Slide 67
Risk Aversion andIndifference Curves
Standard Deviation of Income
ExpectedIncome
Slightly Risk Averse:A large increase in standarddeviation requires only a small increase in incometo maintain satisfaction.
U1
U2
U3
Chapter 5 Slide 68
Business Executivesand 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
ExampleExample
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.
ExampleExample
Business Executivesand 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.
ExampleExample
Business Executivesand 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
DiversificationDiversification
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.
DiversificationDiversification
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.
DiversificationDiversification
Chapter 5 Slide 77
Reducing Risk
With diversification, expected income is $21,000 with no risk.
DiversificationDiversification
Chapter 5 Slide 78
Reducing Risk
Firms can reduce risk by diversifying among a variety of activities that are not closely related.
DiversificationDiversification
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 MarketThe 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.
InsuranceInsurance
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.
InsuranceInsurance
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 NumbersThe Law of Large Numbers
Chapter 5 Slide 84
Reducing Risk
ExamplesA 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 NumbersThe 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 FairnessActuarial 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 FairnessActuarial Fairness
Chapter 5 Slide 87
The Value of Title InsuranceWhen Buying a House
A Scenario:
Price of a house is $200,000
5% chance that the seller does not own the house
ExampleExample
Chapter 5 Slide 88
The Value of Title InsuranceWhen Buying a House
Risk neutral buyer would pay:
ExampleExample
000,190]0[05.]000,200[95(.
Chapter 5 Slide 89
The Value of Title InsuranceWhen 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.
ExampleExample
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 InformationThe 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 InformationThe 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 InformationThe 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
ExpectedSale 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 InformationThe 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 InformationThe 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: ExampleThe Value of Information: Example
Chapter 5 Slide 98
FindingsMilk 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: ExampleThe 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: ExampleThe 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.
Examplesapartment rent, capital gains, corporate
bonds, stock prices
Risky & Riskless AssetsRisky & 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.
Examplesshort-term government bonds, short-
term certificates of deposit
Risky & Riskless AssetsRisky & Riskless Assets
Chapter 5 Slide 104
The Demand for Risky Assets
Asset ReturnsReturn on an Asset
The total monetary flow of an asset as a fraction of its price.
Real Return of an AssetThe simple (or nominal) return less the
rate of inflation.
Chapter 5 Slide 105
The Demand for Risky Assets
Asset Returns
Price Purchase
FlowMonetary Return Asset
%10$1,000
$100/yr.
Price Bond
Flow Return Asset
Chapter 5 Slide 106
The Demand for Risky Assets
Expected Return
Return that an asset should earn on average
Expected vs. Actual ReturnsExpected vs. Actual Returns
Chapter 5 Slide 107
The Demand for Risky Assets
Actual Return
Return that an asset earns
Expected vs. Actual ReturnsExpected 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
RiskReal Rate of (standardReturn (%) 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 ReturnsExpected 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 ReturnThe 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 ReturnThe 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 ReturnThe 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 ReturnThe 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 PortfolioThe 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 PortfolioThe 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 PortfolioThe Investment Portfolio
Chapter 5 Slide 117
The Demand for Risky Assets
Question
How risky is their portfolio?
The Investment PortfolioThe 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 PortfolioThe Investment Portfolio
Chapter 5 Slide 119
The Demand for Risky Assets
Determining b:
fmp RbbRR )1(
)( fmfp RRbRR
The Investor’s Choice ProblemThe Investor’s Choice Problem
Chapter 5 Slide 120
The Demand for Risky Assets
Determining b:
pm
fmfp
RRRR
)(
mpb /
The Investor’s Choice ProblemThe 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 LineRisk and the Budget Line
pm
fmfp
)R(RRR
)( p)p(R
Chapter 5 Slide 122
The Demand for Risky Assets
Observations:
2) Is an equation for a straight line:
3)
constants are and ,R ,R mfm
mfm )/R(R Slope
Risk and the Budget LineRisk and the Budget Line
pm
fmfp
)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 LineRisk and the Budget Line
Chapter 5 Slide 124
Choosing BetweenRisk and Return
0 p Return, of
Deviation Standard
ExpectedReturn,Rp
U2 is the optimal choice of those
obtainable, since it gives the highest
return for agiven risk and istangent to the
budget line.
Rf
Budget Line
m
Rm
R*
U2U1
U3
Chapter 5 Slide 125
Rf
Budget line
The Choices ofTwo Different Investors
0
ExpectedReturn,Rp
p Return, of
Deviation Standard
Given the same budgetline, 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
ExpectedReturn,Rp
UA
RA
A
UA: High risk aversion --Stock & T-bill portfolio
p Return, of
Deviation Standard
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
ObservationsPercent 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
ObservationsShare of wealth in the stock market
1989 = 26%1995 = 40%
Chapter 5 Slide 129
Investing in the Stock Market
ObservationsParticipation 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