Multi-objective Approach to Portfolio Optimization

1
Multi-objective Approach to Portfolio Optimization

• ???
• ??

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
• Varying

11
(No Transcript)
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
Stock Exp. Return Variance
IBM 0.400 0.006461
MSFT 0.513 0.0039
AAPL 4.085 0.012678
DGX 1.006 0.005598361
BAC 1.236 0.001622897
16
IBM MSFT AAPL DGX BAC
IBM 0.006461 0.002983 0.00235487 0.00235487 0.00096889
MSFT 0.002983 0.0039 0.00095937 -0.0001987 0.00063459
AAPL 0.002355 0.000959 0.01267778 0.00135712 0.00134481
DGX 0.002355 -0.0002 0.00135712 0.00559836 0.00041942
BAC 0.000969 0.000635 0.00134481 0.00041942 0.0016229
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

19
(No Transcript)
20
(No Transcript)
21
The risk aversion parameter

•  

22

•  

23
Proof
 
24
The End
Thanks!