Bi- Objective Optimization Background In 1952, Markowitz proposed a portfolio optimization model that aimed at maximizing returns while simultaneously minimizing risk (defined as portfolio variance). This is a bi-objective optimization problem. Indices Index i of Investment Options with i E I Decision Variable Proportion Invested in Each Investment: w; E W Data Random Variables for Return of Each Investment Option: r; ER Expected Value for Each Investment Option: E(r;) = M; Vector of Expected Values of Investment Options: E(R) = m = (M1,M2, . . Mi)T Covariance of Investment Options: cov(R) = > Calculated Variables Rate of Return: R = rTw, Expected Value: E(R) = mow Variance: E(R) = WEw Parameters Minimum Baseline Rate of Return (Required Returns): MBFormulation What we seek to do then is minimize the portfolio variability subject to the constraint that we achieve at least the minimum baseline rate of return and that our investment proportions sum to l. Minimize Z = inEw. (This is I/:. of the portfolio variance. You need not have the 1/2. in the OF.) Subject to: C1: mTw 2 pig. (The sum of expected returns X proportions invested must be greater than baseline.) C2: eTw = 1. (Here, 3 E as a vector of 1's. This is equivalent to Ziwi = 1.) Homework Data and Tasks We have ve investment options with the following return rates (actual monthly returns for various lnds), and we seek Markowitz models for required returns (baseline rates) of .005, .01 , .015, .02, .025, and .03. A B C D E 0.12% -0.32% 4.48% 0.58% 5.10% 0.13% 0.30% -0.69% -5.03% -4.66% 0.13% -0.04% 7.00% 5.43% 2.46% 0.11% -0.86% -4.65% -4.00% -2.81% 0.11% -0.18% 3.03% 2.00% 1.76% 0.13% 1.15% 2.37% -1.24% 0.72% 0.12% 0.74% 2.33% 3.46% -1.44% 0.13% 0.34% 0.69% -0.66% 3.61% 0.13% 0.82% 5.33% 4.23% 3.09% 0.11% -1.23% 4.38% -0.39% 2.35% 0.08% -1.45% 2.76% 5.21% 2.26% 0.07% -0.71% -1.01% 2.85% -1.09% 0.97% 7.50% 18.31% 31.85% 8.17% 0.07% 0.14% 3.84% 7.24% 4.64% 0.07% 0.99% 10.95% 18.26% 15.54% 0.06% -0.42% -2.66% 0.50% -3.97% 0.06% -0.03% -3.80% -3.04% -2.60% 0.05% -0.81% 7.19% 7.20% 5.12% 0.06% 1.49% 5.64% 5.71% 2.33% 0.06% 0.63% 1.99% 4.00% 3.44% 0.06% 0.46% 4.76% 8.79% 4.50% 0.07% 1.78% 12.81% 15.81% 6.42% 0.11% -0.64% -12.40% -21.40% -13.87% 0.13% 1.82% -8.24% -8.01% -7.74% 0.17% 1.91% -0.04% -0.62% -2.73% Formulate and solve the biobjective Markowitz problems for the values of the required (baseline) returns. Then plot the required rates of return (baseline rates) versus the variance obtained (objective lnction) to generate a plot called the efficient frontier