Investment Management EffiCient DiversifiCation Efficient-Diversification 2

The Investment Opportunity Set with the Stock and Bond Funds (Assuming Correlation ρBS = 0.2) Efficient Diversification Fall 2018 13 / 36 The Investment Opportunity Sets for Bonds and Stocks with Various Correlation Coefficients Efficient Diversification Fall 2018 14 / 36 Minimum Variance Portfolio (MVP) Minimum Variance Portfolio (MVP) Gives lowest risk portfolio Formal optimization problem: Choose w1 such that portfolio variance is lowest. Solution: Set derivative (with respect to w1 ) to 0 w1 = w2 = σ22 − Cov (r1 , r2 ) σ12 + σ22 − 2Cov (r1 , r2 ) 1 − w1 Diversification: For ρ < +1, minimum variance portfolio has less risk than either asset Efficient Diversification Fall 2018 15 / 36 Efficient and Dominated Portfolios Portfolios that lie below the minimum-variance portfolio in the figure (on the downward-sloping portion of the curve) are inefficient. Any such portfolio is dominated by the portfolio that lies directly above it on the upward-sloping portion of the curve since that portfolio has higher expected return and equal standard deviation Efficient Set: Portfolios that are not dominated The best choice among the portfolios on the upward-sloping portion of the curve is not as obvious, because in this region higher expected return is accompanied by greater risk Efficient Diversification Fall 2018 16 / 36 Extending Concepts to All Securities (Many Risky Assets) Consider all possible combinations of securities, with all possible different weightings and keep track of combinations that provide more return for less risk or the least risk for a given level of return and graph the result The set of portfolios that provide the optimal trade-offs are described as the efficient frontier The efficient frontier portfolios are dominant or the best diversified possible combinations All investors should want a portfolio on the efficient frontier (... Until we add the riskless asset) Efficient Diversification Fall 2018 17 / 36 Asset Allocation with Many Risky Assets Find portfolios that minimize risk for given expected return (use computer) Minimum-variance frontier: Looks like previous opportunity set with 2 assets Efficient frontier: Minimum variance portfolios that are not dominated (looks like previous efficient set) Efficient Diversification Fall 2018 18 / 36 The Efficient Frontier of Risky Assets Efficient Diversification Fall 2018 19 / 36 Asset Allocation with Many Risky Assets and a Risk-free Asset Combinations of riskless asset + any risky portfolio = Straight line (CAL) Each risky portfolio −→ Different CAL Optimal risky portfolio −→ Point of tangency between CAL & efficient frontier Gives highest feasible reward-to-variability ratio (slope of CAL) Note: Optimal risky portfolio is independent of risk aversion!!! Portfolio manager offers same risky portfolio to all = separation property Efficient Diversification Fall 2018 20 / 36 The Opportunity set of Stocks, Bonds and a Risk-free Asset with two Capital Allocation Lines Efficient Diversification Fall 2018 21 / 36 The Optimal Capital Allocation Line with Stocks, Bonds and a Risk-free Asset Efficient Diversification Fall 2018 22 / 36 The Optimal Capital Allocation Line with Stocks, Bonds and a Risk-free Asset Formally: Find weights that result in highest slope of CAL max wB E [ rp − rf ] s.t. ∑ wi = 1 σp i Solution: wB is given by the expression (E [rB ] − rf ) σS2 − (E [rS ] − rf )σB σS ρBS (E [rB ] − rf ) σS2 + (E [rS ] − rf )σB2 − (E [rB ] − rf + E [rS ] − rf )σB σS ρBS wS = 1 − wB Efficient Diversification Fall 2018 23 / 36 Combining the Optimal Risky Asset with the Risk-free Asset One possible choice for the preferred complete portfolio, C where the investor places 55% of wealth in portfolio O and 45% in Treasury bills Efficient Diversification Fall 2018 24 / 36