Chapter Twelve
Uncertainty
Structure
State contingent consumption (依情形而定的消费）
State-contingent budget constraint
Preferences under uncertainty
Choice under uncertainty
Risk aversion
Diversification and risk spreading
Uncertainty is Pervasive
What is uncertain in economic
systems?
–tomorrow’s prices
–future wealth
–future availability of commodities
–present and future actions of other
people.
Uncertainty is Pervasive
What are rational responses to
uncertainty?
–buying insurance (health,life,auto)
–a portfolio of contingent
consumption goods.
States of Nature
Possible states of Nature:
–“car accident” (a)
–“no car accident” (na).
Accident occurs with probability?a,
does not with probability?na ;
a +?na = 1,
Accident causes a loss of $L.
Contingencies
A contract implemented only when a
particular state of Nature occurs is
state-contingent.
E.g,the insurer pays only if there is
an accident.
Contingencies
A state-contingent consumption plan
is implemented only when a
particular state of Nature occurs.
E.g,take a vacation only if there is no
accident.
State-Contingent Budget
Constraints
Each $1 of accident insurance costs?.
Consumer has $m of wealth.
Cna is consumption value in the no-
accident state.
Ca is consumption value in the
accident state.
State-Contingent Budget
Constraints
Cna
Ca
State-Contingent Budget
Constraints
Cna
Ca
20
17
A state-contingent consumption
with $17 consumption value in the
accident state and $20 consumption
value in the no-accident state.
State-Contingent Budget
Constraints
Without insurance,
Ca = m - L
Cna = m.
State-Contingent Budget
Constraints
Cna
Ca
m The endowment bundle.
m L?
State-Contingent Budget
Constraints
Buy $K of accident insurance.
Cna = m -?K.
Ca = m - L -?K + K = m - L + (1-?)K.
State-Contingent Budget
Constraints
Buy $K of accident insurance.
Cna = m -?K.
Ca = m - L -?K + K = m - L + (1-?)K.
So K = (Ca - m + L)/(1-?)
State-Contingent Budget
Constraints
Buy $K of accident insurance.
Cna = m -?K.
Ca = m - L -?K + K = m - L + (1-?)K.
So K = (Ca - m + L)/(1-?)
And Cna = m -? (Ca - m + L)/(1-?)
State-Contingent Budget
Constraints
Buy $K of accident insurance.
Cna = m -?K,（?K 为 premium(保费 )）
Ca = m - L -?K + K = m - L + (1-?)K.
So K = (Ca - m + L)/(1-?)
And Cna = m -? (Ca - m + L)/(1-?)
I.e,C
m L C
na a?
1 1
State-Contingent Budget
Constraints
Cna
Ca
m The endowment bundle.
m L
C m L Cna a1 1
m L?
State-Contingent Budget
Constraints
Cna
Ca
m The endowment bundle.
sl o pe1
C m L Cna a1 1
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m The endowment bundle.
Where is the
most preferred
state-contingent
consumption plan?
C m L Cna a1 1
sl o pe1
m L
m L?
Preferences Under Uncertainty
2 states of nature:
–At probability?a,consumption is
ca
–At probability?na,consumption is
cna
–?a +?na = 1.
Utility is U(ca,cna,?a,?na).
Preferences Under Uncertainty
a n aE U U ( c ) U ( c )a n a
Expected utility function
Von-Neuman-Morgenstern
utility function
Preferences Under Uncertainty
State-contingent consumption plans
that give equal expected utility are
equally preferred.
Preferences Under Uncertainty
Cna
Ca
EU1
EU2
EU3
Indifference curves
EU1 < EU2 < EU3
Preferences Under Uncertainty
What is the MRS of an indifference
curve?
For constant EU,dEU = 0.
Preferences Under Uncertainty
a n aE U U ( c ) U ( c )a n a
Preferences Under Uncertainty
a a n a n ad E U M U ( c ) d c M U ( c ) d ca n a
a n aE U U ( c ) U ( c )a n a
Preferences Under Uncertainty
a a n a n ad E U 0 M U ( c ) d c M U ( c ) d c 0a n a
a n aE U U ( c ) U ( c )a n a
a a n a n ad E U M U ( c ) d c M U ( c ) d ca n a
Preferences Under Uncertainty
a a n a n aM U ( c ) d c M U ( c ) d ca n a
a n aE U U ( c ) U ( c )a n a
a a n a n ad E U M U ( c ) d c M U ( c ) d ca n a
a a n a n ad E U 0 M U ( c ) d c M U ( c ) d c 0a n a
Preferences Under Uncertainty
n a a
a n a
d c M U ( c ),
d c M U ( c )
a
na
a n aE U U ( c ) U ( c )a n a
a a n a n ad E U M U ( c ) d c M U ( c ) d ca n a
a a n a n ad E U 0 M U ( c ) d c M U ( c ) d c 0a n a
a a n a n aM U ( c ) d c M U ( c ) d ca n a
Preferences Under Uncertainty
Cna
Ca
EU1
EU2
EU3
Indifference curves
EU1 < EU2 < EU3dc
dc
M U ( c )
M U ( c )
na
a
a
na
a
na
Choice Under Uncertainty
Q,How is a rational choice made
under uncertainty?
A,Choose the most preferred
affordable state-contingent
consumption plan.
State-Contingent Budget
Constraints
Cna
Ca
m The endowment bundle.
C m L Cna a1 1
Where is the
most preferred
state-contingent
consumption plan?
sl o pe1
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m The endowment bundle.
Where is the
most preferred
state-contingent
consumption plan?Affordableplans
C m L Cna a1 1
sl o pe1
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m
Where is the
most preferred
state-contingent
consumption plan?
More preferred
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m
Most preferred affordable plan
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m
Most preferred affordable plan
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m
Most preferred affordable plan
MRS = slope of budget
constraint
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m
Most preferred affordable plan
MRS = slope of budget
constraint; i.e.
m L
m L?
1
a
na
M U ( c )
M U ( c )
a
na
Risk Aversion
Think of a lottery.
Win $90 with probability 1/2 and win
$0 with probability 1/2,
U($90) = 12,U($0) = 2.
Expected utility is
EU U ( $90) U ( $0)
1
2
1
2
1
2
12
1
2
2 7,
Preferences Under Uncertainty
Think of a lottery.
Win $90 with probability 1/2 and win
$0 with probability 1/2,
Expected money value of the lottery
is EM $90 $01
2
1
2 45$,
Preferences Under Uncertainty
EU = 7 and EM = $45.
U($45) > 7? $45 for sure is preferred to
the lottery? risk-aversion (规避风险）,
U($45) < 7? the lottery is preferred to $45
for sure? risk-loving （ 喜爱风险）,
U($45) = 7? the lottery is preferred
equally to $45 for sure? risk-neutrality （
风险中性）,
Preferences Under Uncertainty
Wealth$0 $90
2
12
$45
EU=7
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
U($45)
U($45) > EU? risk-aversion.
2
EU=7
$45
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
U($45)
U($45) > EU? risk-aversion.
2
EU=7
$45
MU declines as wealth
rises.
U”<0.
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
2
EU=7
$45
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
U($45) < EU? risk-loving.
2
EU=7
$45
U($45)
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
U($45) < EU? risk-loving.
2
EU=7
$45
MU rises as wealth
rises.
U”>0.
U($45)
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
2
EU=7
$45
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
U($45) = EU? risk-neutrality.
2
U($45)=
EU=7
$45
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
U($45) = EU? risk-neutrality.
2
$45
MU constant as wealth
rises.
U”=0.U($45)=
EU=7
Utility
Example,Competitive Insurance
Suppose entry to the insurance
industry is free.
Expected economic profit = 0.
I.e,?K -?aK - (1 -?a)0 = (? -?a)K = 0.
I.e,free entry =?a.
If price of $1 insurance = accident
probability,then insurance is fair.
Competitive Insurance
When insurance is fair,rational
insurance choices satisfy?
1 1
a
a
a
na
M U ( c )
M U ( c )
a
na
Competitive Insurance
When insurance is fair,rational
insurance choices satisfy
I.e,M U ( c ) M U ( c )a na?
1 1
a
a
a
na
M U ( c )
M U ( c )
a
na
Competitive Insurance
When insurance is fair,rational
insurance choices satisfy
I.e.
Marginal utility of income must be
the same in both states.
1 1
a
a
a
na
M U ( c )
M U ( c )
a
na
M U ( c ) M U ( c )a na?
Competitive Insurance
How much fair insurance does a risk-
averse consumer buy?
M U ( c ) M U ( c )a na?
Competitive Insurance
How much fair insurance does a risk-
averse consumer buy?
Risk-aversion? MU(c)? as c?.
M U ( c ) M U ( c )a na?
Competitive Insurance
How much fair insurance does a risk-
averse consumer buy?
Risk-aversion? MU(c)? as c?.
Hence
M U ( c ) M U ( c )a na?
c c,a na?
Competitive Insurance
How much fair insurance does a risk-
averse consumer buy?
Risk-aversion? MU(c)? as c?.
Hence
I.e,full-insurance.
M U ( c ) M U ( c )a na?
c c,a na?
,Unfair” Insurance
Suppose insurers make positive
expected economic profit.
I.e,?K -?aK - (1 -?a)0 = (? -?a)K > 0.
,Unfair” Insurance
Suppose insurers make positive
expected economic profit.
I.e,?K -?aK - (1 -?a)0 = (? -?a)K > 0.
Then >?a?
1 1
a
a
.
,Unfair” Insurance
Rational choice requires?
1
a
na
M U ( c )
M U ( c )
a
na
,Unfair” Insurance
Rational choice requires
Since
1
a
na
M U ( c )
M U ( c )
a
na?
1 1
a
a
,M U ( c ) > M U ( c )a na
,Unfair” Insurance
Rational choice requires
Since
Hence for a risk-averter.
1
a
na
M U ( c )
M U ( c )
a
na?
1 1
a
a
,M U ( c ) > M U ( c )a na
c < ca na
,Unfair” Insurance
Rational choice requires
Since
Hence for a risk-averter.
I.e,a risk-averter buys less than full
“unfair” insurance.
1
a
na
M U ( c )
M U ( c )
a
na?
1 1
a
a
,M U ( c ) > M U ( c )a na
c < ca na
Uncertainty is Pervasive
What are rational responses to
uncertainty?
–buying insurance (health,life,auto)
–a portfolio of contingent
consumption goods.
Uncertainty is Pervasive
What are rational responses to
uncertainty?
–buying insurance (health,life,auto)
–a portfolio of contingent
consumption goods.
Uncertainty is Pervasive
What are rational responses to
uncertainty?
–buying insurance (health,life,auto)
–a portfolio of contingent
consumption goods.
Diversification (多样化）
Two firms,A and B,Shares cost $10.
With prob,1/2 A’s profit is $100 and
B’s profit is $20.
With prob,1/2 A’s profit is $20 and
B’s profit is $100.
You have $100 to invest,How?
Diversification
Buy only firm A’s stock?
$100/10 = 10 shares.
You earn $1000 with prob,1/2 and
$200 with prob,1/2.
Expected earning,$500 + $100 = $600
Diversification
Buy only firm B’s stock?
$100/10 = 10 shares.
You earn $1000 with prob,1/2 and
$200 with prob,1/2.
Expected earning,$500 + $100 = $600
Diversification
Buy 5 shares in each firm?
You earn $600 for sure.
Diversification has maintained
expected earning and lowered risk.
Diversification
Buy 5 shares in each firm?
You earn $600 for sure.
Diversification has maintained
expected earning and lowered risk.
Typically,diversification lowers
expected earnings in exchange for
lowered risk.
Risk Spreading/Mutual Insurance
100 persons each independently risk
a $10,000 loss.
Loss probability = 0.01.
Initial wealth is $40,000.
No insurance,expected wealth is0 99 40 000 0 01 40 000 10 000
39 900
$,($,$,)
$,.
Risk Spreading/Mutual Insurance
Mutual insurance,Expected loss is
Each of the 100 persons pays $100
into a mutual insurance fund.
Mutual insurance,each person has a
certain wealth of
Risk-spreading benefits everyone.
0 01 10 000 100$,$,
$ 4 0,0 0 0 $ 1 0 0 $ 3 9,9 0 0,
Uncertainty
Structure
State contingent consumption (依情形而定的消费）
State-contingent budget constraint
Preferences under uncertainty
Choice under uncertainty
Risk aversion
Diversification and risk spreading
Uncertainty is Pervasive
What is uncertain in economic
systems?
–tomorrow’s prices
–future wealth
–future availability of commodities
–present and future actions of other
people.
Uncertainty is Pervasive
What are rational responses to
uncertainty?
–buying insurance (health,life,auto)
–a portfolio of contingent
consumption goods.
States of Nature
Possible states of Nature:
–“car accident” (a)
–“no car accident” (na).
Accident occurs with probability?a,
does not with probability?na ;
a +?na = 1,
Accident causes a loss of $L.
Contingencies
A contract implemented only when a
particular state of Nature occurs is
state-contingent.
E.g,the insurer pays only if there is
an accident.
Contingencies
A state-contingent consumption plan
is implemented only when a
particular state of Nature occurs.
E.g,take a vacation only if there is no
accident.
State-Contingent Budget
Constraints
Each $1 of accident insurance costs?.
Consumer has $m of wealth.
Cna is consumption value in the no-
accident state.
Ca is consumption value in the
accident state.
State-Contingent Budget
Constraints
Cna
Ca
State-Contingent Budget
Constraints
Cna
Ca
20
17
A state-contingent consumption
with $17 consumption value in the
accident state and $20 consumption
value in the no-accident state.
State-Contingent Budget
Constraints
Without insurance,
Ca = m - L
Cna = m.
State-Contingent Budget
Constraints
Cna
Ca
m The endowment bundle.
m L?
State-Contingent Budget
Constraints
Buy $K of accident insurance.
Cna = m -?K.
Ca = m - L -?K + K = m - L + (1-?)K.
State-Contingent Budget
Constraints
Buy $K of accident insurance.
Cna = m -?K.
Ca = m - L -?K + K = m - L + (1-?)K.
So K = (Ca - m + L)/(1-?)
State-Contingent Budget
Constraints
Buy $K of accident insurance.
Cna = m -?K.
Ca = m - L -?K + K = m - L + (1-?)K.
So K = (Ca - m + L)/(1-?)
And Cna = m -? (Ca - m + L)/(1-?)
State-Contingent Budget
Constraints
Buy $K of accident insurance.
Cna = m -?K,（?K 为 premium(保费 )）
Ca = m - L -?K + K = m - L + (1-?)K.
So K = (Ca - m + L)/(1-?)
And Cna = m -? (Ca - m + L)/(1-?)
I.e,C
m L C
na a?
1 1
State-Contingent Budget
Constraints
Cna
Ca
m The endowment bundle.
m L
C m L Cna a1 1
m L?
State-Contingent Budget
Constraints
Cna
Ca
m The endowment bundle.
sl o pe1
C m L Cna a1 1
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m The endowment bundle.
Where is the
most preferred
state-contingent
consumption plan?
C m L Cna a1 1
sl o pe1
m L
m L?
Preferences Under Uncertainty
2 states of nature:
–At probability?a,consumption is
ca
–At probability?na,consumption is
cna
–?a +?na = 1.
Utility is U(ca,cna,?a,?na).
Preferences Under Uncertainty
a n aE U U ( c ) U ( c )a n a
Expected utility function
Von-Neuman-Morgenstern
utility function
Preferences Under Uncertainty
State-contingent consumption plans
that give equal expected utility are
equally preferred.
Preferences Under Uncertainty
Cna
Ca
EU1
EU2
EU3
Indifference curves
EU1 < EU2 < EU3
Preferences Under Uncertainty
What is the MRS of an indifference
curve?
For constant EU,dEU = 0.
Preferences Under Uncertainty
a n aE U U ( c ) U ( c )a n a
Preferences Under Uncertainty
a a n a n ad E U M U ( c ) d c M U ( c ) d ca n a
a n aE U U ( c ) U ( c )a n a
Preferences Under Uncertainty
a a n a n ad E U 0 M U ( c ) d c M U ( c ) d c 0a n a
a n aE U U ( c ) U ( c )a n a
a a n a n ad E U M U ( c ) d c M U ( c ) d ca n a
Preferences Under Uncertainty
a a n a n aM U ( c ) d c M U ( c ) d ca n a
a n aE U U ( c ) U ( c )a n a
a a n a n ad E U M U ( c ) d c M U ( c ) d ca n a
a a n a n ad E U 0 M U ( c ) d c M U ( c ) d c 0a n a
Preferences Under Uncertainty
n a a
a n a
d c M U ( c ),
d c M U ( c )
a
na
a n aE U U ( c ) U ( c )a n a
a a n a n ad E U M U ( c ) d c M U ( c ) d ca n a
a a n a n ad E U 0 M U ( c ) d c M U ( c ) d c 0a n a
a a n a n aM U ( c ) d c M U ( c ) d ca n a
Preferences Under Uncertainty
Cna
Ca
EU1
EU2
EU3
Indifference curves
EU1 < EU2 < EU3dc
dc
M U ( c )
M U ( c )
na
a
a
na
a
na
Choice Under Uncertainty
Q,How is a rational choice made
under uncertainty?
A,Choose the most preferred
affordable state-contingent
consumption plan.
State-Contingent Budget
Constraints
Cna
Ca
m The endowment bundle.
C m L Cna a1 1
Where is the
most preferred
state-contingent
consumption plan?
sl o pe1
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m The endowment bundle.
Where is the
most preferred
state-contingent
consumption plan?Affordableplans
C m L Cna a1 1
sl o pe1
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m
Where is the
most preferred
state-contingent
consumption plan?
More preferred
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m
Most preferred affordable plan
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m
Most preferred affordable plan
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m
Most preferred affordable plan
MRS = slope of budget
constraint
m L
m L?
State-Contingent Budget
Constraints
Cna
Ca
m
Most preferred affordable plan
MRS = slope of budget
constraint; i.e.
m L
m L?
1
a
na
M U ( c )
M U ( c )
a
na
Risk Aversion
Think of a lottery.
Win $90 with probability 1/2 and win
$0 with probability 1/2,
U($90) = 12,U($0) = 2.
Expected utility is
EU U ( $90) U ( $0)
1
2
1
2
1
2
12
1
2
2 7,
Preferences Under Uncertainty
Think of a lottery.
Win $90 with probability 1/2 and win
$0 with probability 1/2,
Expected money value of the lottery
is EM $90 $01
2
1
2 45$,
Preferences Under Uncertainty
EU = 7 and EM = $45.
U($45) > 7? $45 for sure is preferred to
the lottery? risk-aversion (规避风险）,
U($45) < 7? the lottery is preferred to $45
for sure? risk-loving （ 喜爱风险）,
U($45) = 7? the lottery is preferred
equally to $45 for sure? risk-neutrality （
风险中性）,
Preferences Under Uncertainty
Wealth$0 $90
2
12
$45
EU=7
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
U($45)
U($45) > EU? risk-aversion.
2
EU=7
$45
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
U($45)
U($45) > EU? risk-aversion.
2
EU=7
$45
MU declines as wealth
rises.
U”<0.
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
2
EU=7
$45
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
U($45) < EU? risk-loving.
2
EU=7
$45
U($45)
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
U($45) < EU? risk-loving.
2
EU=7
$45
MU rises as wealth
rises.
U”>0.
U($45)
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
2
EU=7
$45
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
U($45) = EU? risk-neutrality.
2
U($45)=
EU=7
$45
Utility
Preferences Under Uncertainty
Wealth$0 $90
12
U($45) = EU? risk-neutrality.
2
$45
MU constant as wealth
rises.
U”=0.U($45)=
EU=7
Utility
Example,Competitive Insurance
Suppose entry to the insurance
industry is free.
Expected economic profit = 0.
I.e,?K -?aK - (1 -?a)0 = (? -?a)K = 0.
I.e,free entry =?a.
If price of $1 insurance = accident
probability,then insurance is fair.
Competitive Insurance
When insurance is fair,rational
insurance choices satisfy?
1 1
a
a
a
na
M U ( c )
M U ( c )
a
na
Competitive Insurance
When insurance is fair,rational
insurance choices satisfy
I.e,M U ( c ) M U ( c )a na?
1 1
a
a
a
na
M U ( c )
M U ( c )
a
na
Competitive Insurance
When insurance is fair,rational
insurance choices satisfy
I.e.
Marginal utility of income must be
the same in both states.
1 1
a
a
a
na
M U ( c )
M U ( c )
a
na
M U ( c ) M U ( c )a na?
Competitive Insurance
How much fair insurance does a risk-
averse consumer buy?
M U ( c ) M U ( c )a na?
Competitive Insurance
How much fair insurance does a risk-
averse consumer buy?
Risk-aversion? MU(c)? as c?.
M U ( c ) M U ( c )a na?
Competitive Insurance
How much fair insurance does a risk-
averse consumer buy?
Risk-aversion? MU(c)? as c?.
Hence
M U ( c ) M U ( c )a na?
c c,a na?
Competitive Insurance
How much fair insurance does a risk-
averse consumer buy?
Risk-aversion? MU(c)? as c?.
Hence
I.e,full-insurance.
M U ( c ) M U ( c )a na?
c c,a na?
,Unfair” Insurance
Suppose insurers make positive
expected economic profit.
I.e,?K -?aK - (1 -?a)0 = (? -?a)K > 0.
,Unfair” Insurance
Suppose insurers make positive
expected economic profit.
I.e,?K -?aK - (1 -?a)0 = (? -?a)K > 0.
Then >?a?
1 1
a
a
.
,Unfair” Insurance
Rational choice requires?
1
a
na
M U ( c )
M U ( c )
a
na
,Unfair” Insurance
Rational choice requires
Since
1
a
na
M U ( c )
M U ( c )
a
na?
1 1
a
a
,M U ( c ) > M U ( c )a na
,Unfair” Insurance
Rational choice requires
Since
Hence for a risk-averter.
1
a
na
M U ( c )
M U ( c )
a
na?
1 1
a
a
,M U ( c ) > M U ( c )a na
c < ca na
,Unfair” Insurance
Rational choice requires
Since
Hence for a risk-averter.
I.e,a risk-averter buys less than full
“unfair” insurance.
1
a
na
M U ( c )
M U ( c )
a
na?
1 1
a
a
,M U ( c ) > M U ( c )a na
c < ca na
Uncertainty is Pervasive
What are rational responses to
uncertainty?
–buying insurance (health,life,auto)
–a portfolio of contingent
consumption goods.
Uncertainty is Pervasive
What are rational responses to
uncertainty?
–buying insurance (health,life,auto)
–a portfolio of contingent
consumption goods.
Uncertainty is Pervasive
What are rational responses to
uncertainty?
–buying insurance (health,life,auto)
–a portfolio of contingent
consumption goods.
Diversification (多样化）
Two firms,A and B,Shares cost $10.
With prob,1/2 A’s profit is $100 and
B’s profit is $20.
With prob,1/2 A’s profit is $20 and
B’s profit is $100.
You have $100 to invest,How?
Diversification
Buy only firm A’s stock?
$100/10 = 10 shares.
You earn $1000 with prob,1/2 and
$200 with prob,1/2.
Expected earning,$500 + $100 = $600
Diversification
Buy only firm B’s stock?
$100/10 = 10 shares.
You earn $1000 with prob,1/2 and
$200 with prob,1/2.
Expected earning,$500 + $100 = $600
Diversification
Buy 5 shares in each firm?
You earn $600 for sure.
Diversification has maintained
expected earning and lowered risk.
Diversification
Buy 5 shares in each firm?
You earn $600 for sure.
Diversification has maintained
expected earning and lowered risk.
Typically,diversification lowers
expected earnings in exchange for
lowered risk.
Risk Spreading/Mutual Insurance
100 persons each independently risk
a $10,000 loss.
Loss probability = 0.01.
Initial wealth is $40,000.
No insurance,expected wealth is0 99 40 000 0 01 40 000 10 000
39 900
$,($,$,)
$,.
Risk Spreading/Mutual Insurance
Mutual insurance,Expected loss is
Each of the 100 persons pays $100
into a mutual insurance fund.
Mutual insurance,each person has a
certain wealth of
Risk-spreading benefits everyone.
0 01 10 000 100$,$,
$ 4 0,0 0 0 $ 1 0 0 $ 3 9,9 0 0,
