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

1 nx x x

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:

2 T

p x V x

*p

*1 1T Tx and p x p

////////////////////////// //

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

1

minn

i ii

F x a f x

Multi-objective Formulation Minimize w.r.t. Subject to:

Assign two weighting coefficients

Minimize: Subject to:

,T Tp x x V x////////////////////////////////////////////////////////

1 1Tx

1 1Tx

T Tp x x V x ////////////////////////////////////////////////////////

x

1 21 0and

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:

T

px p x////////////////////////////

2 T

p x V x

1 1Tx

1 1Tx

0When

When

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

*p

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

1 1T T T

L x p x x V x x ////////////////////////////////////////////////////////////////////////////// //////

1* 1 2

1 1

1 11 2

1 2( )1

2 2

1 1 1T T

aVx V p

a a

where a V and a V p

////////////////////////// //

///////////// /

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 *x

ApplicationsStock 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

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

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

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

1 2 3 4 5 6 7 8 9 10-1

-0.5

0

0.5

1

1.5

IBM

MSFT

AAPLDGX

BAC

The risk aversion parameter

Proof:

The End

Thanks!