Multi-objective Approach to Portfolio Optimization：方法的多目标组合优化.ppt

Multi-objective Approach to Portfolio Optimization 童培俊 张帆 CONTENTS Introduction Motivation Methodology Application Risk Aversion Index Key Concept Reward and risk are measured by expected return and variance of a portfolio Decision variable of this problem is asset weight vector Introduction to Portfolio Optimization The Mean Variance Optimization Proposed by Nobel Prize Winner Markowitz in 1990 Model 1: Minimize risk for a given level of expected return Minimize: Subject to: Not be the best model for those who are extremely risk seeking Does not allow to simultaneously minimize risk and maximize expected return Multi-objective Optimization Introduction to Multi-objective Optimization Developed by French-Italian economist Pareto Combine multiple objectives into one objective function by assigning a weighting coefficient to each objective Multi-objective Formulation Minimize w.r.t. Subject to: Assign two weighting coefficients Minimize: Subject to: Risk Aversion Index We can consider as a risk aversion index that measures the risk tolerance of an investor Smaller , more risk seeking Larger , more risk averse Model 2: Maximize expected return (disregard risk) Maximize: Subject to: Model 3: Minimize risk (disregard expected return) Minimize: Subject to: Comparison with Mean Variance Optimization Since the Lagrangian multipliers of both methods are same, their efficient frontiers are also same Different in their approach to producing their efficient frontiers Varying Varying Two comparative advantages For investors who are extremely risk seeking When investors do not want to place any constraints on their investment Provide the entire picture of optimal risk-return trade off Solving Multi-objective Optimization Using Lagrangian multiplier The optimized solution for the portfolio weight vector is Convex Vector Optimization The second derivative of the objective function is positive definite The equality constraint can be expressed in linear form