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Hal VarianIntermediate Microeconomics
Chapter Thirteen
Risky Assets
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 av. value of the r.v.;
E[ ] .w ww s ss
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 ss
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. dev[ ] ( ) .w ww w s w ss
S
2 2
1
Mean and Variance
Probability
Random Variable Values
Two distributions with the samevariance and different means.
Mean and Variance
Probability
Random Variable Values
Two distributions with the samemean 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?
dUUd
Ud
Ud
Ud
dd
UU
0
//
.
Preferences over Risky Assets
Mean Return,
St. Dev. of Return,
Preferred Higher mean return is a good.Higher risk is a bad.
dd
UU
//
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 ss
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 stockto 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 stockto 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 ss
Sxm x r r2 2
11
( ( ) ) .
Budget Constraints for Risky Assets
Portfolio’s rate-of-return variance is
x s f x ss
Sxm x r r2 2
11
( ( ) ) .
r xr x rx m f ( ) .1
Budget Constraints for Risky Assets
Portfolio’s rate-of-return variance is
x s f x ss
Sxm x r r2 2
11
( ( ) ) .
r xr x rx m f ( ) .1
x s f m f ss
Sxm x r xr x r2 2
11 1
( ( ) ( ) )
Budget Constraints for Risky Assets
Portfolio’s rate-of-return variance is
x s f x ss
Sxm x r r2 2
11
( ( ) ) .
r xr x rx 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
( ( ) ( ) )
( )
Budget Constraints for Risky Assets
Portfolio’s rate-of-return variance is
x s f x ss
Sxm x r r2 2
11
( ( ) ) .
r xr x rx 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
( ( ) ( ) )
( ) ( )
Budget Constraints for Risky Assets
Portfolio’s rate-of-return variance is
x s f x ss
Sxm x r r2 2
11
( ( ) ) .
r xr x rx 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
( ( ) ( ) )
( ) ( ) .
Budget Constraints for Risky Assets
x mx2 2 2Variance
x mx .so st. deviation
Budget Constraints for Risky Assets
x mx2 2 2
x = 0 and x = 1 x 0 x m .
Variance
x mx .so st. deviation
Budget Constraints for Risky Assets
x mx2 2 2
x = 0 and x = 1 x 0 x 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 x 0 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 x 0 0,
m0
rmx r rx m x m 1 ,
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 x 0 0,
m0
rmx r rx m x m 1 ,
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 x 0 0,
m0
rmx r rx m x m 1 ,
Budget line, slope =
rf
r rm f
m
Mean Return,
St. Dev. of Return,
Choosing a Portfolio
m0
rmBudget line, slope =
rf
r rm f
m
Mean Return,
St. Dev. of Return,
is the price of risk relative tomean return.
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,
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,
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,
Choosing a Portfolio
m0
rmBudget line, slope =
rf
r rMRSm f
m
Where is the most preferredreturn/risk combination?
rx
x
Mean Return,
St. Dev. of Return,
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,
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
rmBudget 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 andhigher 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.
Measuring Risk
Asset A’s risk relative to risk in the whole stock market is measured by
Arisk of asset A
risk of whole market .
Measuring Risk
Asset A’s risk relative to risk in the whole stock market is measured by
Arisk of asset A
risk of whole market .
AAcovariance(
variance(
r rr
m
m
, ))
where is the market’s rate-of-returnand 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 1A .
A 1
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 Am.
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 Am. Price of risk is
So cost of asset A’s risk is pAm.
pr rm f
m
.
Equilibrium in Risky Asset Markets
Risk adjustment for asset A is
Risk adjusted rate-of-return for asset A is
pr r
r rmm f
mm m f
A 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 ( )