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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 * w11,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.