1Optimal Risky PortfoliosOptimal Risky 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.() () () )(12)(1+)(1ABBA2B22A22pρσσσσσ⋅⋅−+−=−+=wwwwREwRwEREBAPPortfolio 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 byReturn 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 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.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 0.3772Equally weighted portfolio 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 complicatedEqually weighted portfolio This portfolio is less risky than either of GE and IBM!!!%07.600368.0377.0%10.8%49.65.05.02%10.85.0%49.6.50 )(12)(1+2222ABBA2B22A22p=⇒=×××××+×+×=⋅⋅−+−=pwwwwσρσσσσσAnother Portfolio Calculate the expected return and standard deviation of the portfolio consisting of 80% GE and 20% IBMDiversification Benefit 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 deviationPortfolios of GE and IBM$0 GE, $1000 IBM$200 GE, $800 IBM$400 GE, $600 IBM$600 GE, $400 IBM$800 GE, $200 IBM$1,000 GE, $0 IBM0.010.0110.0120.0130.0140.0150.0160.0170.0180.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095Expected ReturnStandard DeviationInvestment Opportunity Set135791113151719212325272931333537394143454749VarianceNumber of AssetsDiversification and RiskSystematicNon-Systematic3Feasible PortfoliosWith N Risky Assets ExpectedReturn E(R)Std dev σEfficientfrontierInvestment Opportunity Set or Feasible SetABCEfficient Frontier You can construct the efficient frontier using one of two equivalent approaches. For given expected return, find the minimum variance For given variance, find the maximum expected return You can do this using the Solver in Excel.Utility MaximizationExpectedReturn E(R)Std dev σA.A is the Utility maximizingrisky-asset portfolioIndifference CurvesBCHigher UtilityWhat if a risk-free asset is available? 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.The Optimal Risky Portfolio has the Highest Sharpe RatioExpectedReturn E(R)Std dev σRisklessAsset••••ABCALACALBCDE••Capital Market LineOptimal Risky PortfolioWhat if a risk-free asset is available? The feasible set of portfolios becomes more attractive  We can identify an optimal risky portfoliowhich dominates all other risky portfolios (irrespective of risk preferences) The optimal (tangency) portfolio has the highest Sharpe ratio among all feasible portfolios4Utility Maximization witha Risk-free AssetExpectedReturn E(Ri)Std dev σiEfficientfrontier..Optimal RiskyPortfolioRisklessAsset..DECapital Market LineOptimal Risky Portfolio is the Market Portfolio 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 truePassive 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 risk-free asset and the market portfolio based on risk aversionThe Optimal Risky Portfolio Key Question: How do we find the optimal risky portfolio? By choosing asset weights withat maximize the Sharpe Ratio:()pfppwRRESiσ−= MaxThe Optimal Risky Portfolio(with 2 risky assets) For two risky assets, we know that the portfolio return and standard deviation are given by() () ()2B2AB2A2p)(1+ )(12)(1σσσσwwwwREwRwEREBAP−−+=−+=The Optimal Risky Portfolio(with 2 risky assets) Therefore, we need to maximize the ratio by choosing w appropriately. This can be done using Solver in Microsoft Excel.()()2B2AB2A2)(1+ )(12)(1σσσwwwwRREwRwESfBAp−−+−−+=5The Optimal Risky Portfolio(with 2 risky assets)♦Or, by the following formula which gives the weights for the optimal portfolio comprised of only two assets:()()()()()[]()[]() ()[]()ABABfBfAAfBBfAABfBBfAAWWRRERRERRERRERRERREW−=−+−−−+−−−−=1222σσσσσThe Optimal Risky Portfolio(with N risky assets) 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 eachThe Optimal Risky Portfolio(with N risky assets) 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 oneSolverSolver Solver6Solver SolverSolverSolverSolverSolver7Solver SolverSolver