Chapter Thirteen
Risky Assets
Main Issue
Mean-Variance Utility
Budget Constraints for Risky Assets
Measuring Risk
Capital Asset Pricing Model
Mean of a Distribution
A random variable (r.v.) w takes
values w1,…,wS with probabilities
1,...,?S (?1 + · · · +?S = 1).
The mean (expected value) of the
distribution is the average value of
the r.v.; E [ ],w w
w s s
s
S
1
Variance of a Distribution
The distribution’s variance is the
r.v.’s av,squared deviation from the
mean;
Variance measures the r.v.’s
variation,
var [ ] ( ),w ww s w s
s
S
2 2
1
Standard Deviation of a
Distribution
The distribution’s standard deviation
is the square root of its variance;
St,deviation also measures the r.v.’s
variability,
st,de v [ ] ( ),w ww w s w s
s
S
2 2
1
Mean and Variance
Probability
Random Variable Values
Two distributions with the same
variance and different means.
Mean and Variance
Probability
Random Variable Values
Two distributions with the same
mean and different variances.
Preferences over Risky Assets
Higher mean return is preferred.
Less variation in return is preferred
(less risk).
Preferences over Risky Assets
Higher mean return is preferred.
Less variation in return is preferred
(less risk).
Preferences are represented by a
utility function U(?,?).
U? as mean return.
U? as risk.
Preferences over Risky Assets
Preferred Higher mean return is a good.
Higher risk is a bad.
Mean Return,?
St,Dev,of Return,?
Preferences over Risky Assets
Preferred Higher mean return is a good.
Higher risk is a bad.
Mean Return,?
St,Dev,of Return,?
Preferences over Risky Assets
How is the MRS computed?
Preferences over Risky Assets
How is the MRS computed?
dU
U
d
U
d
U
d
U
d
d
d
U
U
0
/
/
.
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
/
/
Budget Constraints for Risky
Assets
Two assets.
Risk-free asset’s rate-or-return is rf,
Risky stock’s rate-or-return is ms if
state s occurs,with prob,?s,
Risky stock’s mean rate-of-return is
r mm s s
s
S
,
1
Budget Constraints for Risky
Assets
A bundle containing some of the
risky stock and some of the risk-free
asset is a portfolio.
x is the fraction of wealth used to
buy the risky stock.
Given x,the portfolio’s av,rate-of-
return is r xr x rx m f( ),1
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x = 0? r rx f? and x = 1? r rx m?,
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x = 0? r rx f? and x = 1? r rx m?,
Since stock is risky and risk is a bad,for stock
to be purchased must have r rm f?,
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x = 0? r rx f? and x = 1? r rx m?,
Since stock is risky and risk is a bad,for stock
to be purchased must have r rm f?,
So portfolio’s expected rate-of-return rises with x
(more stock in the portfolio).
Budget Constraints for Risky
Assets
Portfolio’s rate-of-return variance is
x s f x s
s
S
xm x r r2 2
1
1
( ( ) ),
Budget Constraints for Risky
Assets
Portfolio’s rate-of-return variance is
x s f x s
s
S
xm x r r2 2
1
1
( ( ) ),
r xr x rx m f( ),1
Budget Constraints for Risky
Assets
Portfolio’s rate-of-return variance is
x s f x s
s
S
xm x r r2 2
1
1
( ( ) ),
r xr x rx m f( ),1
x s f m f s
s
S
xm x r xr x r2 2
1
1 1
( ( ) ( ) )
Budget Constraints for Risky
Assets
Portfolio’s rate-of-return variance is
x s f x s
s
S
xm x r r2 2
1
1
( ( ) ),
r xr x rx m f( ),1
x s f m f s
s
S
s m s
s
S
xm x r xr x r
xm xr
2 2
1
2
1
1 1
( ( ) ( ) )
( )
Budget Constraints for Risky
Assets
Portfolio’s rate-of-return variance is
x s f x s
s
S
xm x r r2 2
1
1
( ( ) ),
r xr x rx m f( ),1
x s f m f s
s
S
s m s
s
S
s 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
( ( ) ( ) )
( ) ( )
Budget Constraints for Risky
Assets
Portfolio’s rate-of-return variance is
x s f x s
s
S
xm x r r2 2
1
1
( ( ) ),
r xr x rx m f( ),1
x s f m f s
s
S
s m s
s
S
s m s
s
S
m
xm x r xr x r
xm xr x m r x
2 2
1
2
1
2 2
1
2 2
1 1
( ( ) ( ) )
( ) ( ),
Budget Constraints for Risky
Assets
x mx2 2 2?Variance
x mx?,so st,deviation
Budget Constraints for Risky
Assets
x mx2 2 2?
x = 0? and x = 1 x? 0x m?,
Variance
x mx?,so st,deviation
Budget Constraints for Risky
Assets
x mx2 2 2?
x = 0? and x = 1 x? 0x m?,
Variance
x mx?,so st,deviation
So risk rises with x (more stock in the portfolio),
Budget Constraints for Risky
Assets
Mean Return,?
St,Dev,of Return,?
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x mx?,
x r rx f x0 0,?
0
rf
Mean Return,?
St,Dev,of Return,?
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x mx?,
x r rx f x0 0,?
m0
rm x r rx m x m1,
rf
Mean Return,?
St,Dev,of Return,?
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x mx?,
x r rx f x0 0,?
m0
rm x r rx m x m1,
Budget line
rf
Mean Return,?
St,Dev,of Return,?
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x mx?,
x r rx f x0 0,?
m0
rm x r rx m x m1,
Budget line,slope =
rf
r rm f
m
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r rm f
m
Mean Return,?
St,Dev,of Return,?
is the price of risk relative to
mean return.
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r rm f
m
Where is the most preferred
return/risk combination?
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r rm f
m
Where is the most preferred
return/risk combination?
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r rm f
m
Where is the most preferred
return/risk combination?
rx
x
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r r M R Sm f
m
Where is the most preferred
return/risk combination?
rx
x
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r r U
U
m f
m
/
/
Where is the most preferred
return/risk combination?
rx
x
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
Suppose a new risky asset appears,
with a mean rate-of-return ry > rm and
a st,dev,?y >?m.
Which asset is preferred?
Choosing a Portfolio
Suppose a new risky asset appears,
with a mean rate-of-return ry > rm and
a st,dev,?y >?m.
Which asset is preferred?
Suppose
r r r ry f
y
m f
m
.
Choosing a Portfolio
m0
rm
rf
rx
x
Budget line,slope =
r rm f
m
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r rm f
m
rx
x
ry
y
Mean Return,?
St,Dev,of Return,?
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,?
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,?
Measuring Risk
Quantitatively,how risky is an asset?
Depends upon how the asset’s value
depends upon other assets’ values.
E.g,Asset A’s value is $60 with
chance 1/4 and $20 with chance 3/4.
Pay at most $30 for asset A.
Measuring Risk
Asset A’s value is $60 with chance
1/4 and $20 with chance 3/4.
Asset B’s value is $20 when asset
A’s value is $60 and is $60 when
asset A’s value is $20 (perfect
negative correlation of values).
Pay up to $40 > $30 for a 50-50 mix of
assets A and B.
Budget Constraints for Risky
Assets
Asset A’s risk relative to risk in the
whole stock market is measured by
A r isk o f a s se t Ar isk o f wh ole m ar ke t?,
Measuring Risk
Asset A’s risk relative to risk in the
whole stock market is measured by
A r isk o f a s se t Ar isk o f wh ole m ar ke t?,
A Ac o v a r ia nc e (v a r ia nc e (? r rr m
m
,)
)
where is the market’s rate-of-return
and is asset A’s rate-of-return.rA
rm
Measuring Risk
asset A’s return is not
perfectly correlated with the whole
market’s return and so it can be used
to build a lower risk portfolio,
1 1? A,
A1
Equilibrium in Risky Asset
Markets
At equilibrium,all assets’ risk-
adjusted rates-of-return must be
equal.
How do we adjust for riskiness?
Equilibrium in Risky Asset
Markets
Riskiness of asset A relative to total
market risk is?A.
Total market risk is?m.
So total riskiness of asset A is?A?m.
Equilibrium in Risky Asset
Markets
Riskiness of asset A relative to total
market risk is?A.
Total market risk is?m.
So total riskiness of asset A is?A?m.
Price of risk is
So cost of asset A’s risk is p?A?m.
p r rm f
m
,
Equilibrium in Risky Asset
Markets
Risk adjustment for asset A is
Risk adjusted rate-of-return for asset
A is
p r r r rm m f
m
m m fA A A?
( ).
r r rm fA A ( ).
Equilibrium in Risky Asset
Markets
At equilibrium,all risk adjusted rates-
of-return for all assets are equal.
The risk-free asset’s? = 0 so its
adjusted rate-of-return is just
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.
Equilibrium in Risky Asset
Markets
That
at equilibrium in asset markets is the
main result of the Capital Asset
Pricing Model (CAPM),a model used
extensively to study financial
markets.
r r r rf m fA A ( )
Risky Assets
Main Issue
Mean-Variance Utility
Budget Constraints for Risky Assets
Measuring Risk
Capital Asset Pricing Model
Mean of a Distribution
A random variable (r.v.) w takes
values w1,…,wS with probabilities
1,...,?S (?1 + · · · +?S = 1).
The mean (expected value) of the
distribution is the average value of
the r.v.; E [ ],w w
w s s
s
S
1
Variance of a Distribution
The distribution’s variance is the
r.v.’s av,squared deviation from the
mean;
Variance measures the r.v.’s
variation,
var [ ] ( ),w ww s w s
s
S
2 2
1
Standard Deviation of a
Distribution
The distribution’s standard deviation
is the square root of its variance;
St,deviation also measures the r.v.’s
variability,
st,de v [ ] ( ),w ww w s w s
s
S
2 2
1
Mean and Variance
Probability
Random Variable Values
Two distributions with the same
variance and different means.
Mean and Variance
Probability
Random Variable Values
Two distributions with the same
mean and different variances.
Preferences over Risky Assets
Higher mean return is preferred.
Less variation in return is preferred
(less risk).
Preferences over Risky Assets
Higher mean return is preferred.
Less variation in return is preferred
(less risk).
Preferences are represented by a
utility function U(?,?).
U? as mean return.
U? as risk.
Preferences over Risky Assets
Preferred Higher mean return is a good.
Higher risk is a bad.
Mean Return,?
St,Dev,of Return,?
Preferences over Risky Assets
Preferred Higher mean return is a good.
Higher risk is a bad.
Mean Return,?
St,Dev,of Return,?
Preferences over Risky Assets
How is the MRS computed?
Preferences over Risky Assets
How is the MRS computed?
dU
U
d
U
d
U
d
U
d
d
d
U
U
0
/
/
.
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
/
/
Budget Constraints for Risky
Assets
Two assets.
Risk-free asset’s rate-or-return is rf,
Risky stock’s rate-or-return is ms if
state s occurs,with prob,?s,
Risky stock’s mean rate-of-return is
r mm s s
s
S
,
1
Budget Constraints for Risky
Assets
A bundle containing some of the
risky stock and some of the risk-free
asset is a portfolio.
x is the fraction of wealth used to
buy the risky stock.
Given x,the portfolio’s av,rate-of-
return is r xr x rx m f( ),1
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x = 0? r rx f? and x = 1? r rx m?,
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x = 0? r rx f? and x = 1? r rx m?,
Since stock is risky and risk is a bad,for stock
to be purchased must have r rm f?,
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x = 0? r rx f? and x = 1? r rx m?,
Since stock is risky and risk is a bad,for stock
to be purchased must have r rm f?,
So portfolio’s expected rate-of-return rises with x
(more stock in the portfolio).
Budget Constraints for Risky
Assets
Portfolio’s rate-of-return variance is
x s f x s
s
S
xm x r r2 2
1
1
( ( ) ),
Budget Constraints for Risky
Assets
Portfolio’s rate-of-return variance is
x s f x s
s
S
xm x r r2 2
1
1
( ( ) ),
r xr x rx m f( ),1
Budget Constraints for Risky
Assets
Portfolio’s rate-of-return variance is
x s f x s
s
S
xm x r r2 2
1
1
( ( ) ),
r xr x rx m f( ),1
x s f m f s
s
S
xm x r xr x r2 2
1
1 1
( ( ) ( ) )
Budget Constraints for Risky
Assets
Portfolio’s rate-of-return variance is
x s f x s
s
S
xm x r r2 2
1
1
( ( ) ),
r xr x rx m f( ),1
x s f m f s
s
S
s m s
s
S
xm x r xr x r
xm xr
2 2
1
2
1
1 1
( ( ) ( ) )
( )
Budget Constraints for Risky
Assets
Portfolio’s rate-of-return variance is
x s f x s
s
S
xm x r r2 2
1
1
( ( ) ),
r xr x rx m f( ),1
x s f m f s
s
S
s m s
s
S
s 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
( ( ) ( ) )
( ) ( )
Budget Constraints for Risky
Assets
Portfolio’s rate-of-return variance is
x s f x s
s
S
xm x r r2 2
1
1
( ( ) ),
r xr x rx m f( ),1
x s f m f s
s
S
s m s
s
S
s m s
s
S
m
xm x r xr x r
xm xr x m r x
2 2
1
2
1
2 2
1
2 2
1 1
( ( ) ( ) )
( ) ( ),
Budget Constraints for Risky
Assets
x mx2 2 2?Variance
x mx?,so st,deviation
Budget Constraints for Risky
Assets
x mx2 2 2?
x = 0? and x = 1 x? 0x m?,
Variance
x mx?,so st,deviation
Budget Constraints for Risky
Assets
x mx2 2 2?
x = 0? and x = 1 x? 0x m?,
Variance
x mx?,so st,deviation
So risk rises with x (more stock in the portfolio),
Budget Constraints for Risky
Assets
Mean Return,?
St,Dev,of Return,?
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x mx?,
x r rx f x0 0,?
0
rf
Mean Return,?
St,Dev,of Return,?
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x mx?,
x r rx f x0 0,?
m0
rm x r rx m x m1,
rf
Mean Return,?
St,Dev,of Return,?
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x mx?,
x r rx f x0 0,?
m0
rm x r rx m x m1,
Budget line
rf
Mean Return,?
St,Dev,of Return,?
Budget Constraints for Risky
Assets
r xr x rx m f( ),1
x mx?,
x r rx f x0 0,?
m0
rm x r rx m x m1,
Budget line,slope =
rf
r rm f
m
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r rm f
m
Mean Return,?
St,Dev,of Return,?
is the price of risk relative to
mean return.
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r rm f
m
Where is the most preferred
return/risk combination?
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r rm f
m
Where is the most preferred
return/risk combination?
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r rm f
m
Where is the most preferred
return/risk combination?
rx
x
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r r M R Sm f
m
Where is the most preferred
return/risk combination?
rx
x
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r r U
U
m f
m
/
/
Where is the most preferred
return/risk combination?
rx
x
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
Suppose a new risky asset appears,
with a mean rate-of-return ry > rm and
a st,dev,?y >?m.
Which asset is preferred?
Choosing a Portfolio
Suppose a new risky asset appears,
with a mean rate-of-return ry > rm and
a st,dev,?y >?m.
Which asset is preferred?
Suppose
r r r ry f
y
m f
m
.
Choosing a Portfolio
m0
rm
rf
rx
x
Budget line,slope =
r rm f
m
Mean Return,?
St,Dev,of Return,?
Choosing a Portfolio
m0
rm
Budget line,slope =
rf
r rm f
m
rx
x
ry
y
Mean Return,?
St,Dev,of Return,?
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,?
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,?
Measuring Risk
Quantitatively,how risky is an asset?
Depends upon how the asset’s value
depends upon other assets’ values.
E.g,Asset A’s value is $60 with
chance 1/4 and $20 with chance 3/4.
Pay at most $30 for asset A.
Measuring Risk
Asset A’s value is $60 with chance
1/4 and $20 with chance 3/4.
Asset B’s value is $20 when asset
A’s value is $60 and is $60 when
asset A’s value is $20 (perfect
negative correlation of values).
Pay up to $40 > $30 for a 50-50 mix of
assets A and B.
Budget Constraints for Risky
Assets
Asset A’s risk relative to risk in the
whole stock market is measured by
A r isk o f a s se t Ar isk o f wh ole m ar ke t?,
Measuring Risk
Asset A’s risk relative to risk in the
whole stock market is measured by
A r isk o f a s se t Ar isk o f wh ole m ar ke t?,
A Ac o v a r ia nc e (v a r ia nc e (? r rr m
m
,)
)
where is the market’s rate-of-return
and is asset A’s rate-of-return.rA
rm
Measuring Risk
asset A’s return is not
perfectly correlated with the whole
market’s return and so it can be used
to build a lower risk portfolio,
1 1? A,
A1
Equilibrium in Risky Asset
Markets
At equilibrium,all assets’ risk-
adjusted rates-of-return must be
equal.
How do we adjust for riskiness?
Equilibrium in Risky Asset
Markets
Riskiness of asset A relative to total
market risk is?A.
Total market risk is?m.
So total riskiness of asset A is?A?m.
Equilibrium in Risky Asset
Markets
Riskiness of asset A relative to total
market risk is?A.
Total market risk is?m.
So total riskiness of asset A is?A?m.
Price of risk is
So cost of asset A’s risk is p?A?m.
p r rm f
m
,
Equilibrium in Risky Asset
Markets
Risk adjustment for asset A is
Risk adjusted rate-of-return for asset
A is
p r r r rm m f
m
m m fA A A?
( ).
r r rm fA A ( ).
Equilibrium in Risky Asset
Markets
At equilibrium,all risk adjusted rates-
of-return for all assets are equal.
The risk-free asset’s? = 0 so its
adjusted rate-of-return is just
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.
Equilibrium in Risky Asset
Markets
That
at equilibrium in asset markets is the
main result of the Capital Asset
Pricing Model (CAPM),a model used
extensively to study financial
markets.
r r r rf m fA A ( )
