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```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.;
S
E[ w]   w   ws s .
s 1
Variance of a Distribution
 The
distribution’s variance is the
r.v.’s av. squared deviation from the
mean;
S
2
2
var[ w]   w   ( ws   w )  s .
s 1
 Variance
measures the r.v.’s
variation.
Standard Deviation of a
Distribution
 The
distribution’s standard deviation
is the square root of its variance;
2
st. dev[ w]   w   w 
 St.
S
2
 ( ws   w )  s .
s 1
deviation also measures the r.v.’s
variability.
Mean and Variance
Probability
Two distributions with the same
variance and different means.
Random Variable Values
Mean and Variance
Probability
Two distributions with the same
mean and different variances.
Random Variable Values
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
Mean Return, 
Preferred
Higher mean return is a good.
St. Dev. of Return, 
Preferences over Risky Assets
Mean Return, 
Preferred
Higher mean return is a good.
St. Dev. of Return, 
Preferences over Risky Assets
 How
is the MRS computed?
Preferences over Risky Assets
 How
is the MRS computed?
U
U
d  0
d 
dU 


U
U
d
d  



U / 
d
.


U / 
d
Preferences over Risky Assets
Mean Return, 
Preferred
Higher mean return is a good.
d
U / 

d
U / 
St. Dev. of Return, 
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
S
rm   ms 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
 Given x, the portfolio’s av. rate-ofreturn is
r  xr  (1  x )r .
x
m
f
Budget Constraints for Risky
Assets
x=0
rx  xrm  (1  x )r f .
rx  r f and x = 1  rx  rm .
Budget Constraints for Risky
Assets
x=0
rx  xrm  (1  x )r f .
rx  r f and x = 1  rx  rm .
Since stock is risky and risk is a bad, for stock
to be purchased must have rm  r f .
Budget Constraints for Risky
Assets
x=0
rx  xrm  (1  x )r f .
rx  r f and x = 1  rx  rm .
Since stock is risky and risk is a bad, for stock
to be purchased must have rm  r 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
S
 2x   ( xms  (1  x )r f  rx ) 2  s .
s 1
Budget Constraints for Risky
Assets
 Portfolio’s
rate-of-return variance is
S
 2x   ( xms  (1  x )r f  rx ) 2  s .
s 1
rx  xrm  (1  x )r f .
Budget Constraints for Risky
Assets
 Portfolio’s
rate-of-return variance is
S
 2x   ( xms  (1  x )r f  rx ) 2  s .
s 1
rx  xrm  (1  x )r f .
S
2
2
 x   ( xms  (1  x )r f  xrm  (1  x )r f )  s
s 1
Budget Constraints for Risky
Assets
 Portfolio’s
rate-of-return variance is
S
 2x   ( xms  (1  x )r f  rx ) 2  s .
s 1
rx  xrm  (1  x )r f .
S
2
2
 x   ( xms  (1  x )r f  xrm  (1  x )r f )  s
s 1
S
2
  ( xms  xrm )  s
s 1
Budget Constraints for Risky
Assets
 Portfolio’s
rate-of-return variance is
S
 2x   ( xms  (1  x )r f  rx ) 2  s .
s 1
rx  xrm  (1  x )r f .
S
2
2
 x   ( xms  (1  x )r f  xrm  (1  x )r f )  s
s 1
S
2
2 S
2
  ( xms  xrm )  s  x  (ms  rm )  s
s 1
s 1
Budget Constraints for Risky
Assets
 Portfolio’s
rate-of-return variance is
S
 2x   ( xms  (1  x )r f  rx ) 2  s .
s 1
rx  xrm  (1  x )r f .
S
2
2
 x   ( xms  (1  x )r f  xrm  (1  x )r f )  s
s 1
S
2
2 S
2
2 2
  ( xms  xrm )  s  x  (ms  rm )  s  x  m .
s 1
s 1
Budget Constraints for Risky
Assets
Variance
so st. deviation
2
2 2
 x  x m
 x  x m.
Budget Constraints for Risky
Assets
Variance
so st. deviation
x=0
2
2 2
 x  x m
 x  x m.
x 0
and x = 1 
 x   m.
Budget Constraints for Risky
Assets
Variance
so st. deviation
x=0
2
2 2
 x  x m
 x  x m.
x 0
and x = 1 
 x   m.
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
Mean Return, 
rf
rx  xrm  (1  x )r f .
 x  x m.
x  0  rx  r f , x  0
0
St. Dev. of Return, 
Budget Constraints for Risky
Assets
rx  xrm  (1  x )r f .
 x  x m.
Mean Return, 
x  1  rx  rm , x   m
rm
rf
x  0  rx  r f , x  0
0
m
St. Dev. of Return, 
Budget Constraints for Risky
Assets
rx  xrm  (1  x )r f .
 x  x m.
Mean Return, 
x  1  rx  rm , x   m
rm
Budget line
rf
x  0  rx  r f , x  0
0
m
St. Dev. of Return, 
Budget Constraints for Risky
Assets
rx  xrm  (1  x )r f .
 x  x m.
Mean Return, 
x  1  rx  rm , x   m
rm
Budget line, slope =
rf
rm  r f
m
x  0  rx  r f , x  0
0
m
St. Dev. of Return, 
Choosing a Portfolio
Mean Return, 
rm
Budget line, slope =
rm  r f
m
is the price of risk relative to
mean return.
rf
0
m
St. Dev. of Return, 
Choosing a Portfolio
Mean Return, 
Where is the most preferred
return/risk combination?
rm
Budget line, slope =
rm  r f
m
rf
0
m
St. Dev. of Return, 
Choosing a Portfolio
Mean Return, 
Where is the most preferred
return/risk combination?
rm
Budget line, slope =
rm  r f
m
rf
0
m
St. Dev. of Return, 
Choosing a Portfolio
Mean Return, 
Where is the most preferred
return/risk combination?
rm
Budget line, slope =
rx
rm  r f
m
rf
0
x
m
St. Dev. of Return, 
Choosing a Portfolio
Mean Return, 
Where is the most preferred
return/risk combination?
rm
Budget line, slope =
rx
rm  r f
m
rf
0
x
m
St. Dev. of Return, 
 MRS
Choosing a Portfolio
Mean Return, 
Where is the most preferred
return/risk combination?
rm
rm  r f
U / 

Budget line, slope =
m
U / 
rx
rf
0
x
m
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?
rm  r f
 Suppose ry  r f

.
y
m
Choosing a Portfolio
Mean Return, 
rm
Budget line, slope =
rx
rm  r f
rf
0
x
m
St. Dev. of Return, 
m
Choosing a Portfolio
Mean Return, 
ry
rm
Budget line, slope =
rx
rm  r f
rf
0
x
 m y
St. Dev. of Return, 
m
Choosing a Portfolio
Mean Return, 
ry
Budget line, slope =
rm
Budget line, slope =
rx
ry  r f
y
rm  r f
rf
0
x
 m y
St. Dev. of Return, 
m
Choosing a Portfolio
Mean Return, 
ry
Budget line, slope =
rm
ry  r f
y
rm  r f
rx
Budget line, slope =
rf
Higher mean rate-of-return and
higher risk chosen in this case.
0
x
 m y
St. Dev. of Return, 
m
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
risk of asset A
A 
.
risk of whole market
Measuring Risk
 Asset
A’s risk relative to risk in the
whole stock market is measured by
risk of asset A
A 
.
risk of whole market
covariance( rA , rm )
A 
variance( rm )
where rm is the market’s rate-of-return
and rA is asset A’s rate-of-return.
Measuring Risk
1   A  1.
 A
 1  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.
Equilibrium in Risky Asset
Markets
equilibrium, all assets’ riskadjusted rates-of-return must be
equal.
 How do we adjust for riskiness?
 At
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.
rm  r f
 Price of risk is
p
.
m
 So
cost of asset A’s risk is pAm.
Equilibrium in Risky Asset
Markets
 Risk
p A m 
 Risk
A is
rm  r f
m
 A m   A (rm  r f ).
rA   A (rm  r f ).
Equilibrium in Risky Asset
Markets
 At
equilibrium, all risk adjusted ratesof-return for all assets are equal.
 The risk-free asset’s  = 0 so its
adjusted rate-of-return is just r f .
 Hence, r  r   ( r  r )
f
A
A m
f
i.e. rA  r f   A (rm  r f )
for every risky asset A.
Equilibrium in Risky Asset
Markets
 That
rA  r f   A (rm  r f )
at equilibrium in asset markets is the
main result of the Capital Asset
Pricing Model (CAPM), a model used
extensively to study financial
markets.
```
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