1. Multi-objective Approach to Portfolio Optimization 童培俊 张帆

2. CONTENTS  Introduction  Motivation  Methodology  Application  Risk Aversion Index

3. Key Concept  Reward and risk are measured by expected return and variance of a portfolio  Decision variable of this problem is asset weight vector

4. 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:

5.  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

6. 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

7. Multi-objective Formulation  Minimize w.r.t.  Subject to:  Assign two weighting coefficients  Minimize:  Subject to:

8. 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

9.  Model 2: Maximize expected return (disregard risk)  Maximize:  Subject to:  Model 3: Minimize risk (disregard expected return)  Minimize:  Subject to:

10. 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

12. 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

13. Solving Multi-objective Optimization  Using Lagrangian multiplier  The optimized solution for the portfolio weight vector is

14. 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

15. Applications StockExp. ReturnVariance IBM0.400%0.006461 MSFT0.513%0.0039 AAPL4.085%0.012678 DGX1.006%0.005598361 BAC1.236%0.001622897

16. IBMMSFTAAPLDGXBAC IBM0.00646 1 0.00298 3 0.00235 487 0.00096 889 MSFT0.00298 3 0.00390.00095 937 - 0.00019 87 0.00063 459 AAPL0.00235 5 0.00095 9 0.01267 778 0.00135 712 0.00134 481 DGX0.00235 5 -0.00020.00135 712 0.00559 836 0.00041 942 BAC0.00096 9 0.00063 5 0.00134 481 0.00041 942 0.00162 29

17. Example  When equals to 50, the optimal portfolio strategy shows that the investor should invest  -15.94% in IBM  30.37% in MSFT  3.19% in AAPL  22.60% in DGX  59.78% in BAC

18.  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

21. The risk aversion parameter 

22. 

23. Proof:

24. The End Thanks!