Download Lecture 16

Lecture 16
Portfolio Weights
determine
market capitalization
value-weighting
equal-weighting
mean-variance optimization
capital asset pricing model
market impact
crisis
bull market
bear market
size effect
value effect
momentum effect
liquidity premium
portfolio weights
predictive mean
predictive variance
predictive covariance
volatility
target
maximum
minimum
efficient frontier
risk-free asset
global minimum variance portfolio
zero-beta portfolio
standard deviation
Sharpe ratio
consumer staples sector
short-sale constraint
market beta
limitation
unstable
extreme
garbage
personal
input
output
combine
risky
shift
Bayes rule
Fisher Black
Robert Litterman
Given the list of stocks that we want to buy, how
do we determine portfolio weights?
•
•
•
•
Equal weighting
Market cap weighting (value weighting)
Mean-variance optimization
Other approaches
Market-cap weighting
and capital asset pricing model
Equal-weighting vs. market-cap weighting
•
•
•
•
•
•
•
Which
Which
Which
Which
Which
Which
Which
one
one
one
one
one
one
one
gives more weights to small cap stocks?
is more sensitive to “market impact”?
does better in crisis?
does better in bull market? In bear market?
is more sensitive to size effect?
is more sensitive to value effect? To momentum effect?
is more sensitive to liquidity premium?
predictive mean
E ( r1,T 1 )

E (rN ,T 1 )
predictive variances
and covariances
var( r1,T 1 ) cov( r1,T 1 , r2,T 1 )  cov( r1,T 1 , rN ,T 1 )
var( r2,T 1 )
 cov( r2,T 1 , rN ,T 1 )

var( rN ,T 1 )
“target” volatility
P *
Meanvariance
optimiation
⇒
portfolio
weights
w1 *

wN *
Problem of finding maximummean portfolio
Choose w1 , , w N that
maximize
w1 E (r1,T 1 )    wN E (rN ,T 1 )
subject to
w1    wN  1
and
w1 var( r1,T 1 )    wN var( rN ,T 1 )
2
2
 2w1w2 cov( r1,T 1 , r2,T 1 )    2w1wN cov( r1,T 1 , rN ,T 1 )

 2wN 1wN cov( rN 1,T 1 , rN ,T 1 )
  p *2
Problem of finding minimumvariance portfolio
Choose w1 , , w N that
w1 var( r1,T 1 )    wN var( rN ,T 1 )
2
minimize
2
 2w1w2 cov( r1,T 1 , r2,T 1 )    2w1wN cov( r1,T 1 , rN ,T 1 )

 2wN 1wN cov( rN 1,T 1 , rN ,T 1 )
subject to
w1    wN  1
and
w1 E (r1,T 1 )    wN E (rN ,T 1 )   p *
Mean-variance efficient frontier
mean
global minimum variance portfolio
standard deviation
mean
zero beta portfolio
standard deviation
mean
maximum Sharpe ratio portfolio
risk free asset
standard deviation
Short-sale constraints
Choose w1 , , w N that
w1 var( r1,T 1 )    wN var( rN ,T 1 )
2
minimize
2
 2w1w2 cov( r1,T 1 , r2,T 1 )    2w1wN cov( r1,T 1 , rN ,T 1 )

 2wN 1wN cov( rN 1,T 1 , rN ,T 1 )
subject to
w1  0  wN  0
w1 E (r1,T 1 )    wN E (rN ,T 1 )   p *
Beta constraints
Choose w1 , , w N that
w1 var( r1,T 1 )    wN var( rN ,T 1 )
2
minimize
2
 2w1w2 cov( r1,T 1 , r2,T 1 )    2w1wN cov( r1,T 1 , rN ,T 1 )

 2wN 1wN cov( rN 1,T 1 , rN ,T 1 )
subject to
w1    wN  1
w1 E (r1,T 1 )    wN E (rN ,T 1 )   p *
w11,k   wN  N ,k   p *
Limitations of Mean-Variance
Optimization Approach
• Unstable?
• Extreme weights?
• “Garbage in, garbage out”
Ideas of Black & Litterman
Move the weights toward the value weights
“Black-Litterman alpha”
market
portfolio
weights
E ( r1,T 1 )

E (rN ,T 1 )
predictive variances
and covariances
var( r1,T 1 ) cov( r1,T 1 , r2,T 1 )  cov( r1,T 1 , rN ,T 1 )
var( r2,T 1 )
 cov( r2,T 1 , rN ,T 1 )

var( rN ,T 1 )
“target” volatility
P *
Meanvariance
optimiation
portfolio
weights
w1 *

wN *
Two more steps
• Combine the expected returns implied by the market
capitalization with the investor’s personal view.
• Use the new expected returns as the input to the optimization
process.
What do we get in the end?
True or false?
•
•
•
•
•
•
•
•
•
•
Market cap weighting can be justified by capital asset pricing model.
If the capital asset pricing model is true, then everyone’s portfolio must
be market cap weighted.
Small cap stocks tend to have bigger weights in optimized portfolios
than in equally weighted portfolios.
In crisis, equal weighted portfolios tend to have higher returns than value
weighted portfolios.
When value and momentum perform poorly, equal weighted portfolios
tend to do poorly as well.
A portfolio weight vector is an input to the mean-variance optimization.
Expected returns of each asset are outputs of the mean-variance
optimization.
One of the constraints in the standard mean-variance optimization is that
the sum of stock weights is zero.
Portfolios of risky assets cannot have variance smaller than that of the
global minimum variance portfolio.
The zero-beta portfolio has zero correlation with every efficient portfolio.
True or false?
•
•
•
•
•
•
•
•
•
The covariance between the maximum Sharpe ratio portfolio and any
portfolio whose expected return equals the risk-free asset is zero.
The efficient frontier moves to the left if we impose no short-sale
constraint.
There is analytic solution to the quadratic programming problem with
lower and upper bounds.
As we add beta constraints to the mean-variance optimization, the
efficient frontier shifts to the left.
A common complaints about the mean-variance optimization is that the
resulting portfolio includes too many stocks.
The idea of Black and Litterman is based on the Bayes’ rule.
If you follow the Black Litterman approach, you are more likely to end up
with a smaller number of stocks in the portfolio.
If you follow the Black Litterman approach, the tracking error relative the
market portfolio is likely to go up.
Black and Litterman believed that investors’ personal views should not
influence the portfolio.