risky assets
TRANSCRIPT
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Hal Varian
Intermediate Microeconomics
Chapter Thirteen
Risky Assets
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Mean of a Distribution
x A random variable (r.v.) wtakesvalues w
1,,w
Swith probabilities
1,..., S ( 1 + + S = 1).x The mean (expected value) of the
distribution is the av. value of the
r.v.; E[ ] .w ww s ss
S= == 1
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Variance of a Distribution
x The distributions variance is the r.v.s av.squared deviation from the mean;
x Variance measures the r.v.s variation.var[ ] ( ) .w ww s w s
s
S= =
= 2 2
1
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Standard Deviation of a
Distributionx The distributions standard deviation
is the square root of its variance;
x St. deviation also measures the r.v.svariability.
st. dev[ ] ( ) .w ww w s w ss
S
= = = =
2 2
1
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Mean and Variance
Probability
Random Variable Values
Two distributions with the samemean and different variances.
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Preferences over Risky Assets
x Higher mean return is preferred.
x Less variation in return is preferred
(less risk).x Preferences are represented by a
utility function U( , ).x U as mean return .x U as risk .
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Preferences over Risky Assets
Preferred Higher mean return is a good.
Higher risk is a bad.
Mean Return,
St. Dev. of Return,
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Preferences over Risky Assets
Preferred Higher mean return is a good.
Higher risk is a bad.
Mean Return,
St. Dev. of Return,
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Preferences over Risky Assets
x How is the MRS computed?
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Preferences over Risky Assets
x How is the MRS computed?
dUU
dU
d
Ud
Ud
d
d
U
U
= + =
=
=
0
/
/.
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Preferences over Risky Assets
Mean Return,
St. Dev. of Return,
Preferred Higher mean return is a good.
Higher risk is a bad.
d
d
U
U
=
/
/
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Budget Constraints for Risky
Assets
x Two assets.
x Risk-free assets rate-or-return is rf.
x Risky stocks rate-or-return is ms ifstate s occurs, with prob. s.
x Risky stocks mean rate-of-return is
r mm s ss
S== .1
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Budget Constraints for Risky
Assets
x A bundle containing some of therisky stock and some of the risk-freeasset is a portfolio.
xxis the fraction of wealth used tobuy the risky stock.
x
Givenx
, the portfolios av. rate-of-return is r xr x r x m f = + ( ) .1
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Budget Constraints for Risky
Assets
r xr x r x m f = + ( ) .1
x= 0 r rx f= andx= 1 r rx m= .
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Budget Constraints for Risky
Assets
r xr x r x m f = + ( ) .1
x= 0 r rx f= andx= 1 r rx m= .Since stock is risky and risk is a bad, for stockto be purchased must have r rm f> .
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Budget Constraints for Risky
Assets
r xr x r x m f = + ( ) .1
x= 0 r rx f= andx= 1 r rx m= .Since stock is risky and risk is a bad, for stockto be purchased must have r rm f> .
So portfolios expected rate-of-return rises withx
(more stock in the portfolio).
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Budget Constraints for Risky
Assets
x Portfolios rate-of-return variance is
x s f x s
s
S xm x r r 2 2
1
1= +
=
( ( ) ) .
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Budget Constraints for Risky
Assets
x Portfolios rate-of-return variance is
x s f x s
s
S xm x r r 2 2
1
1= +
=
( ( ) ) .
r xr x r x m f = + ( ) .1
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Budget Constraints for Risky
Assets
x Portfolios rate-of-return variance is
x s f x s
s
S xm x r r 2 2
1
1= +
=
( ( ) ) .
r xr x r x m f = + ( ) .1
x s f m f ss
S xm x r xr x r 2 2
1
1 1= + =
( ( ) ( ) )
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Budget Constraints for Risky
Assets
x Portfolios rate-of-return variance is
x s f x s
s
S xm x r r 2 2
1
1= +
=
( ( ) ) .
r xr x r x m f = + ( ) .1
x s f m f ss
S
s m ss
S
xm x r xr x r
xm xr
2 2
1
2
1
1 1= +
=
=
=
( ( ) ( ) )
( )
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Budget Constraints for Risky
Assets
x Portfolios rate-of-return variance is
x s f x s
s
S xm x r r 2 2
1
1= +
=
( ( ) ) .
r xr x r x m f = + ( ) .1
x s f m f ss
S
s m ss
Ss m s
s
S
xm x r xr x r
xm xr x m r
2 2
1
2
1
2 2
1
1 1= +
= =
=
=
=
( ( ) ( ) )
( ) ( )
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Budget Constraints for Risky
Assets
x Portfolios rate-of-return variance is
x s f x s
s
S xm x r r 2 2
1
1= +
=
( ( ) ) .
r xr x r x m f = + ( ) .1
x s f m f ss
S
s m ss
Ss m s
s
Sm
xm x r xr x r
xm xr x m r x
2 2
1
2
1
2 2
1
2 2
1 1= +
= = =
=
=
=
( ( ) ( ) )
( ) ( ) .
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Budget Constraints for Risky
Assets
x mx2 2 2
=Variance
x mx= .so st. deviation
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Budget Constraints for Risky
Assets
x mx2 2 2
=
x= 0 andx= 1 x= 0 x m= .
Variance
x mx= .so st. deviation
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Budget Constraints for Risky
Assets
x mx2 2 2
=
x= 0 andx= 1 x= 0 x m= .
Variance
x mx= .so st. deviation
So risk rises withx(more stock in the portfolio).
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Budget Constraints for Risky
AssetsMean Return,
St. Dev. of Return,
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Budget Constraints for Risky
Assets
r xr x r x m f = + ( ) .1 x mx= .
x r r x f x = = =0 0,
0
rf
Mean Return,
St. Dev. of Return,
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Budget Constraints for Risky
Assets
r xr x r x m f = + ( ) .1 x mx= .
x r r x f x = = =0 0,
m0
rm
x r r x m x m= = =1 ,
rf
Mean Return,
St. Dev. of Return,
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Budget Constraints for Risky
Assets
r xr x r x m f = + ( ) .1 x mx= .
x r r x f x = = =0 0,
m0
rm
x r r x m x m= = =1 ,
Budget line
rf
Mean Return,
St. Dev. of Return,
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Budget Constraints for Risky
Assets
r xr x r x m f = + ( ) .1 x mx= .
x r r x f x = = =0 0,
m0
rm
x r r x m x m= = =1 ,
Budget line, slope =
rf
r rm f
m
Mean Return,
St. Dev. of Return,
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Choosing a Portfolio
m0
rmBudget line, slope =
rf
r rm f
m
Mean Return,
St. Dev. of Return,
is the price of riskrelative to
mean return.
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Choosing a Portfolio
m0
rmBudget line, slope =
rf
r rm f
m
Where is the most preferredreturn/risk combination?
Mean Return,
St. Dev. of Return,
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Choosing a Portfolio
m0
rmBudget line, slope =
rf
r rm f
m
Where is the most preferredreturn/risk combination?
Mean Return,
St. Dev. of Return,
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Choosing a Portfolio
m0
rmBudget line, slope =
rf
r rm f
m
Where is the most preferredreturn/risk combination?
rx
x
Mean Return,
St. Dev. of Return,
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Choosing a Portfolio
m0
rmBudget line, slope =
rf
r r MRSm f
m
=
Where is the most preferredreturn/risk combination?
rx
x
Mean Return,
St. Dev. of Return,
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Choosing a Portfolio
m0
rmBudget line, slope =
rf
r r UU
m f
m
=
/
/
Where is the most preferredreturn/risk combination?
rx
x
Mean Return,
St. Dev. of Return,
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Choosing a Portfolio
x Suppose a new risky asset appears,with a mean rate-of-return ry > rmand a
st. dev. y > m.x Which asset is preferred?
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Choosing a Portfolio
x Suppose a new risky asset appears,with a mean rate-of-return ry > rmand a
st. dev. y > m.x Which asset is preferred?
x Supposer r r r y f
y
m f
m
>
.
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Choosing a Portfolio
m0
rm
rf
rx
x
Budget line, slope =r rm f
m
Mean Return,
St. Dev. of Return,
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Choosing a Portfolio
m0
rmBudget line, slope =
rf
r rm f
m
rx
x
ry
y
Mean Return,
St. Dev. of Return,
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Choosing a Portfolio
m0
rm
rf
Budget line, slope =r rm f
m
rx
x
ry
y
Budget line, slope =r ry f
y
Mean Return,
St. Dev. of Return,
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Choosing a Portfolio
m0
rm
rf
Budget line, slope = r rm f
m
rx
x
ry
y
Budget line, slope =r ry f
y
Higher mean rate-of-return and
higher risk chosen in this case.
Mean Return,
St. Dev. of Return,
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Measuring Risk
x Quantitatively, how risky is an asset?
x Depends upon how the assets value
depends upon other assets values.x E.g. Asset As value is $60 with
chance 1/4 and $20 with chance 3/4.
x
Pay at most $30 for asset A.
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Measuring Risk
x Asset As value is $60 with chance 1/4and $20 with chance 3/4.
x Asset Bs value is $20 when asset Asvalue is $60 and is $60 when asset Asvalue is $20 (perfect negativecorrelation of values).
x Pay up to $40 > $30 for a 50-50 mix ofassets A and B.
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Measuring Risk
x Asset As risk relative to risk in thewhole stock market is measured by
Arisk of asset A
risk of whole market= .
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Measuring Risk
x Asset As risk relative to risk in thewhole stock market is measured by
A
risk of asset A
risk of whole market= .
AAcovariance(
variance(=
r r
rm
m
, )
)
where is the markets rate-of-returnand is asset As rate-of-return.rA
rm
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Measuring Risk
x asset As return is not
perfectly correlated with the wholemarkets return and so it can be usedto build a lower risk portfolio.
+1 1A .
A < + 1
E ilib i i Ri k A t
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Equilibrium in Risky Asset
Markets
x At equilibrium, all assets risk-adjusted rates-of-return must beequal.
x How do we adjust for riskiness?
E ilib i i Ri k A t
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Equilibrium in Risky Asset
Markets
x Riskiness of asset A relative to totalmarket risk is
A.
x
Total market risk is m.x So total riskiness of asset A is
A m.
E ilib i i Ri k A t
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Equilibrium in Risky Asset
Markets
x Riskiness of asset A relative to totalmarket risk is A.
x
Total market risk is m.x So total riskiness of asset A is A m.x Price of risk is
x So cost of asset As risk is p A m.p
r rm f
m
=
.
E ilib i i Ri k A t
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Equilibrium in Risky Asset
Markets
x Risk adjustment for asset A is
x Risk adjusted rate-of-return for assetA is
pr r
r rmm f
mm m f
A A A=
= ( ).
r r rm fA A ( ).
E ilib i i Ri k A t
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Equilibrium in Risky Asset
Markets
x At equilibrium, all risk adjusted rates-of-return for all assets are equal.
x The risk-free assets = 0 so itsadjusted rate-of-return is just
x Hence,
for every risky asset A.
r r r r
r r r r
f m f
f m f
=
= +
A A
A Ai.e.
( )
( )
rf.
E ilib i i Ri k A t
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Equilibrium in Risky Asset
Markets
x Thatat equilibrium in asset markets is themain result of the Capital AssetPricing Model (CAPM), a model usedextensively to study financialmarkets.
r r r r f m fA A= + ( )