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 one gives more weights to small cap stocks? Which one is more sensitive to “market impact”? Which one does better in crisis? Which one does better in bull market? In bear market? Which one is more sensitive to size effect? Which one is more sensitive to value effect? To momentum effect? Which one is more sensitive to liquidity premium?

predictive mean predictive variances and covariances “target” volatility ⇒ Mean- variance optimiation portfolio weights

Problem of finding maximum-mean portfolio subject to and

Problem of finding minimum- variance portfolio subject to and minimize

Mean-variance efficient frontier standard deviation mean global minimum variance portfolio

standard deviation mean zero beta portfolio

standard deviation mean risk free asset maximum Sharpe ratio portfolio

Short-sale constraints subject to minimize

Beta constraints subject to minimize

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” predictive variances and covariances “target” volatility Mean- variance optimiation portfolio weights market portfolio weights

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.