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Multi-Objective Portfolio Optimization

Jeremy Eckhause

AMSC 698SProfessor S. Gabriel6 December 2004

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Portfolio Optimization: Overview

Objective:

“I want to maximize a quantity that measures historical portfolio performance subject to constraints such as:1. Not too much concentration in any onesector, industry, or individual stock. 2. Reasonable capital usage3. Low correlation with market indices,Etc.

The quantities that change will be the weights of the portfolio constituents.”

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Portfolio Optimization: Overview

Constraints:

• Not too much concentration in any one sector, industry, or individual stock

• Reasonable capital usage • Low correlation with market indices

(Change to covariance to get convex constraints. This constraint will be added, but not in time for the results of this presentation. These constraints are not difficult to add, despite being nonlinear, as they are convex.)

(Linear Constraints)(Linear Constraints)

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Portfolio Optimization: Overview

Multi-Objective Optimization:

“I want to maximize a quantity that measures historical portfolio performance subject to constraints.”

“Quantity” is the Sharpe Ratio.

Sharpe Ratio = mean / standard deviation

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Portfolio Optimization: Overview

Multi-Objective Optimization:

“I want to maximize a quantity that measures historical portfolio performance subject to constraints.”

“Quantity: is the Sharpe Ratio.

Sharpe Ratio = mean / standard deviation

Maximize return while minimizing risk. Using Markowitz’s method, we can generate a Pareto curve!

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Portfolio Optimization: Definitions

Return:

The monthly historic returns over the past five years for each possible investment.

In reality, the optimization uses historic data to show the best portfolio over the past five years. The hope is that these trends will continue in the future.

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Portfolio Optimization: Notation

Data:

s

N

dbdb

t

H

M

u

R

k

ik

u

i

ij

maxmin

return of investment i in month j

“capital usage” of investment i (inverse of standard deviation over five year returns)

maximum capital usage allowed

if investment i is in in sector k

threshold level for sector k

“directional bias” within some range

no. of months

vector of 1 (long) and –1 (short) for each investment

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Portfolio Optimization: Formulation

0

)(

110

maxmin

1

2

1

x

dbsxudb

ktHxMxu

xx

NyyyRx

y

Nyy

T

kuT

ii

N

jj

N

jj

s.t.

min

max Make second objectivefunction a constraint!

(1)

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0

)(

110

maxmin

1

2

1

x

dbsxudb

ktHxMxu

xx

NyyyRx

ly

Nyy

T

kuT

ii

N

jj

N

jj

s.t.

min

mean must be above some return

(2)

Portfolio Optimization: Approach

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We know that from the constraint method that:

If is a binding constraint, then the optimal solution to (2) is along the efficient Pareto frontier for (1).

What are those values for where the constraint is binding?

ly

ly l

Objective function changes for 3.23 < l < 6.702

Portfolio Optimization: Approach

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Portfolio Optimization: Model

Optimization model built in LINGO

• Change value for lower bound return

• Quickly calculates the global optimum for even large-scale problems

• Output to Excel to generate graphs of Pareto optimal values

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Portfolio Optimization: Model

LINGO Code!Objective function;

MIN = SHARPEVAR; !Min variance;

SHARPEVAR = (@SUM(PD(J): ((Y(J) - MEANY)^2))/N);

!Constraints;

@FOR(PD(I): @SUM(ST(J): M5(I,J)*X(J) ) = Y(I)); !Assigns y_i to the sum of each row;

MEANY = (@SUM(PD(I): Y(I))/N); !Mean of y is the average return;

MEANY >= LB; !Return above some lower bound;

@SUM(ST(I): CU(I)*XX(I)) <= MAXCU; !Below some maximum capital usage;

CU(1)*X(1) + CU(4)*X(4) <= THR(1); !Threshold limits for each sector;

CU(2)*X(2) <= THR(2);

CU(3)*X(3) <= THR(3);

@SUM(ST(I): CU(I)*X(I)) <= DBMAX;

@SUM(ST(I): CU(I)*X(I)) >= DBMIN; ! Within some directional bias range;

Etc.

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Portfolio Optimization: Data

S1 S2 S3 S4 S5 (Index 1) S6 (Index 2) PortfolioStance 1 1 1 -1 -1 -1

Sector A B C ANominal Weight 0.21806 0.3018 0.34254 0.092326 0.045275 0Actual Weight 0.21806 0.3018 0.34254 -0.092326 -0.045275 0

0.0325 0.0516 0.0510 -0.0081 -0.0067 0.0000Return 0.1126 0.1788 0.1768 -0.0281 -0.0234 0.0000 0.4167

Variance 0.6262 0.5484 0.4367 0.6982 0.3643 0.4877 0.14798Capital Usage 1.2637 1.3504 1.5133 1.1968 1.6568 1.4320

9/30/2004 0.670 1.283 0.894 -0.011 0.908 0.477 0.79938/31/2004 -0.492 1.258 0.932 -0.027 1.633 0.820 0.52037/31/2004 0.634 -0.514 0.063 0.646 0.250 0.549 -0.06606/30/2004 0.552 -0.110 0.257 0.533 -0.191 0.873 0.13465/31/2004 1.101 -1.022 0.447 0.229 0.629 1.222 0.03504/30/2004 -0.048 2.367 0.557 2.252 1.022 0.684 0.64063/31/2004 -0.898 0.726 -1.209 0.332 0.427 0.441 -0.44072/29/2004 1.567 0.141 0.655 0.061 0.108 1.020 0.59811/31/2004 0.027 0.306 1.008 -0.106 0.473 0.905 0.432112/31/2003 2.722 2.028 0.619 0.468 -0.153 0.802 1.381511/30/2003 1.327 -0.756 0.093 0.093 0.839 0.529 0.046410/31/2003 0.688 0.810 0.227 0.358 1.068 1.139 0.3908

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Portfolio Optimization: Results

Minimized Variance for Given Return

0

0.05

0.1

0.15

0.2

0.25

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

Annualized Return

Mo

nth

ly V

aria

nce

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Portfolio Optimization: Conclusions

Multi-objective optimization tool for maximizing return while minimizing variance works well.

Relatively easy to minimize variance and find the “knee” of the Pareto frontier when changing the minimum return.

Allows the decision maker on Wall Street to pick his or her risk tolerance vs. expected payoff.

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Portfolio Optimization: Future Work

Add additional constraints, such as considering some minimum on recent return. Some may require using integer variables, although this is expected to be a small number.

Perform analysis on actual data (in progress). will be in the 100 variable range. Matrix should be about 100x100. Additional constraints may add another 100-500 variables. All easily handled by a LINGO or similar solver.

Enhance model to generate Pareto frontier efficiently, automatically providing each strategy (i.e., ) during the output phase.

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