Developing Portfolio Using the Dow Jones Industrial Average as benchmark, we will implement a portfolio Optimization M。des二m steps and to construct realistic, optimal BY BOB TAYLOR Estimating Asset Return moments Portfolio optimization was first developed in the 1950s, but The mean and covariance of asset returns are a number of practical and theoretical problems have limited Estimating these moments involves three its use by investment managers. For example, it is often tasks: acquiring data, dealing with missing data, and setting up a suitable benchmark. difficult to obtain sufficient his lity historical data for Acquiring Data orough analysis. In addition, the efficient frontier where We use MATLAB and Datafeed Toolbox to obtain return data for stocks and mar- optimal portfolios lie tends to shift over time, quickly making ket indexes. In our example we acquire these portfolios suboptimal onthly total return data on 44 blue-chip stocks and the dow Jones Industrial Aver- Modern data analysis tools, such as MATLABe and age (D)JIA) from Yahoo! Finance Financial Toolbox, can overcome these challeng Dealing with Missing Data Fortunately, historical financial data is often messy and incomplete. We use the Financial Toolbox function ecmmmle to deal with data Mean-variarce Eficient Frontier and Random Portfolios sets that have missing values(represented as NaNs in MATLAB). This function uses all available data to obtain best estimates for asset return moments in the presence of NaNs-a ce alternative to the usual ad hoc approaches. We use a market index as our benchmark, 3013 since market return is the main driver of asset returns in capital asset pricing. By removing market returns from the data an focus on non-market returns and risks n our example we subtract the return of the DJIA from individual asset returns Using Classic Mean-Variance Analysis Risk (Standard Deviation) ariance analysis, expected return is ns)for rufolo. We igure 1. The efficient frontier. ate random combinations of portfolio weights produce a scatterplot of the expectedreturm igure 1).Developing Portfolio Optimization Models Portfolio optimization was first developed in the 1950s, but a number of practical and theoretical problems have limited its use by investment managers. For example, it is often difficult to obtain sufficient high-quality historical data for thorough analysis. In addition, the efficient frontier where optimal portfolios lie tends to shift over time, quickly making these portfolios suboptimal. Modern data analysis tools, such as MATLAB® and Financial Toolbox, can overcome these challenges. Figure 1. The efficient frontier. By Bob Taylor Using the Dow Jones Industrial Average as a benchmark, we will implement a portfolio optimization methodology based on capital asset pricing and mean-variance analysis. Our goals are to use consistent, repeatable steps and to construct realistic, optimal portfolios that are stable over time. Estimating Asset Return Moments The mean and covariance of asset returns are primary inputs for portfolio optimization. Estimating these moments involves three tasks: acquiring data, dealing with missing data, and setting up a suitable benchmark. Acquiring Data We use MATLAB and Datafeed Toolbox to obtain return data for stocks and mar￾ket indexes. In our example we acquire monthly total return data on 44 blue-chip stocks and the Dow Jones Industrial Aver￾age (DJIA) from Yahoo! Finance. Dealing with Missing Data Unfortunately, historical financial data is often messy and incomplete. We use the Financial Toolbox function ecmnmle to deal with data sets that have missing values (represented as NaNs in MATLAB). This function uses all available data to obtain best estimates for asset return moments in the presence of NaNs—a nice alternative to the usual ad hoc approaches. Setting up a Benchmark We use a market index as our benchmark, since market return is the main driver of asset returns in capital asset pricing. By removing market returns from the data we can focus on non-market returns and risks. In our example we subtract the return of the DJIA from individual asset returns. Using Classic Mean-Variance Analysis In mean-variance analysis, expected return is plotted against risk (the standard deviation of asset returns) for a given portfolio. We gener￾ate random combinations of portfolio weights to produce a scatter plot of the expected return and risk for each portfolio (Figure 1)