Optimal Risky Portfolio Optimal Risky Portfolios Portfolio of Two Risky Assets Suppose you hold a proportion w in asset A and 1 w in asset B The portfolio expected return and risk is given by E RP wE RA 1 w E RB p 2 w2 A 2 1 w 2 B 2 2w 1 w A B AB An Example Suppose you hold two assets in your portfolio GE and IBM Let the portfolio weight of GE be w and then the portfolio weight of IBM be 1 w If w 1 you hold only GE If w 0 you hold only IBM If w 0 5 you have an equally weighted or naively diversified portfolio When choosing the optimal allocation between a risk free asset and a risky portfolio we have assumed that we have already selected the optimal risky portfolio In this section we learn how to determine the optimal risky portfolio We start from two risky assets Most of the intuition carries to the case of more than two risky assets Return and Risk of A Portfolio Given the expected returns of A and B variances of A and B and the covariance correlation between A and B we compute the expected return and variance of the portfolio for a series of portfolio weights Then we plot the expected returns against variances An Example Based on data for 1982 2001 we find that The average monthly return is 1 68 for GE and 1 22 for IBM The standard deviation is 6 49 for GE and 8 10 for IBM The correlation between GE and IBM is 0 377 1 Equally weighted portfolio Equally weighted portfolio p 2 w2 A 2 1 w 2 B 2 2w 1 w A B AB The expected return of the equally weighted portfolio is 0 5 1 68 0 5 1 22 1 45 The standard deviation of this portfolio is more complicated 0 52 6 49 2 0 52 8 10 2 2 0 5 0 5 6 49 8 10 0 377 0 00368 p 6 07 Another Portfolio Diversification Benefit Calculate the expected return and standard deviation of the portfolio consisting of 80 GE and 20 IBM 0 018 Variance 0 017 1 000 GE 0 IBM 0 016 Standard Deviation The portfolio risk is lower than either of the individual stocks This is called the benefit of diversification I repeat the above steps for other portfolio weights w 0 0 1 0 2 0 9 1 0 I plot the expected return against standard deviation Diversification and Risk Portfolios of GE and IBM 800 GE 200 IBM 0 015 NonSystematic 600 GE 400 IBM 400 GE 600 IBM 0 014 200 GE 800 IBM 0 013 0 012 This portfolio is less risky than either of GE and IBM 0 GE 1000 IBM Investment Opportunity Set Systematic 0 011 49 47 45 43 41 39 37 35 33 31 29 23 27 25 17 0 095 21 0 09 15 0 085 19 0 08 11 0 075 13 0 07 9 0 065 7 0 06 5 0 055 3 1 0 01 0 05 Number of Assets Expected Return 2 Feasible Portfolios With N Risky Assets Efficient Frontier Investment Opportunity Set or Feasible Set Expected Return E R You can construct the efficient frontier using one of two equivalent approaches B C A Efficient frontier Std dev You can do this using the Solver in Excel What if a risk free asset is available Utility Maximization Higher Utility Indifference Curves C Expected Return E R For given expected return find the minimum variance For given variance find the maximum expected return A B We have covered the capital allocation problem between a risk free asset and a risky asset Recall that the capital allocation line is the straight line through the risk free asset and the risky asset A is the Utility maximizing risky asset portfolio Std dev The Optimal Risky Portfolio has the Highest Sharpe Ratio Capital Market Line Expected Return E R Optimal Risky Portfolio D C E Riskless Asset CALB B What if a risk free asset is available CALA A The feasible set of portfolios becomes more attractive We can identify an optimal risky portfolio which dominates all other risky portfolios irrespective of risk preferences The optimal tangency portfolio has the highest Sharpe ratio among all feasible portfolios Std dev 3 Optimal Risky Portfolio is the Market Portfolio Utility Maximization with a Risk free Asset Expected Return E Ri Optimal Risky Portfolio Riskless Asset E Capital Market Line D Efficient frontier Everybody holds a combination of risk free asset and the optimal risky portfolio This optimal risky portfolio is the same for everybody regardless of how risk averse you are It must be the market portfolio If not then there must be some assets that no one holds which cannot be true Std dev i Passive Strategy is Efficient The optimal risky portfolio is the same for every investor and is the market portfolio No need for stock selection Investors need only to adjust the mix of riskfree asset and the market portfolio based on risk aversion The Optimal Risky Portfolio Key Question How do we find the optimal risky portfolio By choosing asset weights wi that maximize the Sharpe Ratio Max S p wi The Optimal Risky Portfolio with 2 risky assets For two risky assets we know that the portfolio return and standard deviation are given by E R p R f p The Optimal Risky Portfolio with 2 risky assets Therefore we need to maximize the ratio Sp E RP wE RA 1 w E RB wE RA 1 w E RB R f w A 2w 1 w AB 1 w 2 B 2 2 2 p w2 A 2 2w 1 w AB 1 w 2 B2 by choosing w appropriately This can be done using Solver in Microsoft Excel 4 The Optimal Risky Portfolio with 2 risky assets The Optimal Risky Portfolio with N risky assets Or by the following formula which gives the weights for the optimal portfolio comprised of only two assets WA E R R E R R E R R E R R E R R A E R R A f 2 B 2 B f B WB f B 2 A f A AB f B f 1 WA The Optimal Risky Portfolio with N risky assets AB Steps to solve for the optimal risky portfolio weights using Solver in Microsoft Excel Identify all of the risky assets to be included in the investment universe Compute return series from prices for each risky asset and the risk free asset and determine the average return for each Solver Compute the covariance matrix of the risky assets Enter the formula for the Sharpe ratio into a cell Set up a column of cells for the portfolio weights Use Solver to maximize the Sharpe ratio by changing the weights subject to the constraint that the weights sum to one Solver Solver 5 Solver Solver Solver Solver Solver Solver 6 Solver Solver Solver Solver 7