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

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  is the optimal solution

Applications StockExp. ReturnVariance IBM0.400% MSFT0.513% AAPL4.085% DGX1.006% BAC1.236%

IBMMSFTAAPLDGXBAC IBM MSFT AAPL DGX BAC

Example  When equals to 50, the optimal portfolio strategy shows that the investor should invest  % in IBM  30.37% in MSFT  3.19% in AAPL  22.60% in DGX  59.78% in BAC

 If cases involving of short selling are excluded in this example, the investor should invest  19.77% in MSFT  2.05% in AAPL  16.96% in DGX  61.22% in BAC

The risk aversion parameter 



Proof:

The End Thanks!