Heuristic optimization of complex Heuristic optimization of complex constrained portfolio sets with short salesconstrained portfolio sets with short sales

G A G A Vijayalakshmi PaiPai

Dept of Math. & Computer Applns.Dept of Math. & Computer Applns.PSG College of TechnologyPSG College of TechnologyCoimbatore, INDIACoimbatore, INDIA

paigav@mca.psgtech.ac.inpaigav@mca.psgtech.ac.in

Thierry Michel

Tactical Asset Allocation and OverlayLombard Odier Darrier Hentsch

Paris, FRANCE

Thierry.Michel@lodh.com

Machine Intelligence Research day, India Workshop, Nagpur, Jan 24, 2009

OutlineOutlinePortfolio OptimizationPortfolio Optimization

Computational Intelligence based solution methodsComputational Intelligence based solution methods

KK--means cluster analysis for portfolio optimizationmeans cluster analysis for portfolio optimization

Evolution Strategy with Hall of Fame Evolution Strategy with Hall of Fame –– design and design and implementationimplementation

Differential Evolution (rand/1/bin)Differential Evolution (rand/1/bin)-- design and design and implementationimplementation

Experimental Studies and ResultsExperimental Studies and Results

ConclusionsConclusions22

Portfolio OptimizationPortfolio Optimization

A A financial portfolio financial portfolio is a basket of tradable assets such asis a basket of tradable assets such asbonds, stocks, securities etcbonds, stocks, securities etc

The problem of The problem of portfolio optimizationportfolio optimization incorporates the twin incorporates the twin

objectives of objectives of maximizing returnmaximizing return and/or and/or minimizing riskminimizing risk

Obtain the Obtain the optimal weights optimal weights

Trace the Trace the efficient frontierefficient frontier which is a riskwhich is a risk--return tradeoff return tradeoff curvecurve 33

Efficient frontier

0 20 40 60 80 100 120 14020

40

60

80

100

120

140

160

180

Annualized risk(%)

Exp

ecte

d po

rtfo

lio a

nnua

l ret

urn(

%)

10 20 30 40 5010

20

30

40

50

60

70

Annualized risk (%)

Exp

ecte

d po

rtfo

lio a

nnua

l ret

urn(

%)

Bombay Stock Exchange BSE200 dataset

(July 2001-July 2006)

Tokyo Stock Exchange Nikkei225 dataset

(March 2002-March 2007)

44

MathematicalMathematical formulationformulation

iii

N

i

N

jijji WWW )1(

1 1

minimize

]1,0[,...2,1,10

11

NiW

W

i

N

iiSubject to

(basic constraints)

Markowitz Mean-Variance model

55

Markowitz (1952) framework assumed a perfect marketMarkowitz (1952) framework assumed a perfect market

For such markets, the mathematical formulation reduced to a For such markets, the mathematical formulation reduced to a quadratic optimization problemquadratic optimization problem and traditional methods of and traditional methods of solution could be directly appliedsolution could be directly applied

In reality, portfolio selection problems are difficult to solve especially when market frictions, investor preferences are considered

The mathematical model calls for augmenting the objective function with several constraints, rendering the optimization problem complex enough for direct solving by analytical methods.

66

Inclusion of cardinality constraintInclusion of cardinality constraint

Empirical findings have revealed Empirical findings have revealed that investors prefer to that investors prefer to invest in rather a limited number of assetsinvest in rather a limited number of assets, considering , considering the fact that transaction costs, portfolio management the fact that transaction costs, portfolio management fees and information gathering costs are dependent on fees and information gathering costs are dependent on the number of assets that are included for investment. the number of assets that are included for investment.

To achieve this the To achieve this the cardinality constraintcardinality constraint calls for the calls for the inclusion of only K assets out of a universe of N assets, inclusion of only K assets out of a universe of N assets, for a prefor a pre--specified value of K chosen by the investor.specified value of K chosen by the investor.

77

Mathematical formulation Mathematical formulation –– turns complex!turns complex!

iii

N

i

N

jijji WWW )1(

1 1min

]1,0[,...2,1,10

11

NiW

W

i

N

iiSubject to

otherwise

WifZwhereKZ i

iN

ii ,0

01

1

(basic constraints)

(cardinality constraints)88

Computational Intelligence based solution methodsComputational Intelligence based solution methods

A A brute force approachbrute force approach calling for a choice from calling for a choice from

different alternate solutions to the problem different alternate solutions to the problem

combinatorial explosion for large values of N combinatorial explosion for large values of N

[[Farrell (1997)] Farrell (1997)] grouping available assets into asset classesgrouping available assets into asset classes

based based on the industry, size, geographical aspects etc and on the industry, size, geographical aspects etc and

making a selection from each of these classesmaking a selection from each of these classes

inferior solutions largely due to ignoriinferior solutions largely due to ignoring ng the correlations between the assets the correlations between the assets

KCN

99

MultiMulti--agent methodsagent methods such as Ant systems, Genetic algorithms such as Ant systems, Genetic algorithms

and Evolutionary and Evolutionary algorithms or algorithms or hybrid search methods hybrid search methods

(Maringer, 2005) (Maringer, 2005)

near optimal or mixed results near optimal or mixed results

Single agent local search algorithms such as Simulated Annealing

and Threshold Accepting (TA) (Chang et al., 2000; Winker 2001)

perils of getting stuck in local optimaperils of getting stuck in local optima

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[Fernandez and Gomez (2007)] [Fernandez and Gomez (2007)] Hopfield neural Hopfield neural networks for the solution of the problem networks for the solution of the problem

no one heuristic approach outperformed others inall kinds of investment policies

Pai and Michel (2007, 2009) Pai and Michel (2007, 2009) kk--means cluster means cluster analysis analysis to tackle cardinality constraintto tackle cardinality constraint

1111

kk--means cluster analysis for portfolio means cluster analysis for portfolio optimization optimization -- a brief reviewa brief review

Cluster analysisCluster analysis could be viewed as an exploratory data could be viewed as an exploratory data

analysis tool which can classify objects into clusters analysis tool which can classify objects into clusters

depending on the maximum degree of association shared depending on the maximum degree of association shared

by objects belonging to the same cluster.by objects belonging to the same cluster.

kk--means clusteringmeans clustering unlike other clustering techniques unlike other clustering techniques

addresses the need when exactly K different clusters addresses the need when exactly K different clusters with the greatest possible distinction are required for a with the greatest possible distinction are required for a prepre--specified value of K. specified value of K.

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Observations Observations

k-means clustering though primarily adopted to eliminate the

cardinality constraint yields a simplified mathematical model which

is amenable for direct solving by traditional optimization methods

such as Quadratic Programming, besides heuristic strategies

The reliability [ Tola, 2005] of k-means clustered portfolio sets turn

out to be better than those obtained by the Markowitz and RMT

filtered counterparts

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For large K, the efficient frontiers traced by the k-means clustered

portfolio sets were found to be in close proximity to the exact or

ideal efficient frontier.

Heuristic approaches are sensitive to the number of design

variables in the problem. k-means clustering promotes

dimensionality reduction which could be put to its best use by the

approaches concerned, thereby leading to faster convergence.

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An investable universe

defined as a set of K assets each of which is

randomly or preferentially chosen from cluster

is extracted

},1|{ ij

ij

ij CaKiaI

jia

  KiCi 1,

1515

Mathematical formulation Mathematical formulation –– turns complex!turns complex!

iii

N

i

N

jijji WWW )1(

1 1min

]1,0[

11

N

iiWSubject to

otherwise

WifZwhereKZ i

iN

ii ,0

01

1

(basic constraints)

(cardinality constraints)1616

Inclusion of other constraintsInclusion of other constraints

 

Kb

Ka

NibWa i31;3

,...2,1,

(Short sales)

j

10

jj

jji

ij iW

For each class of assets

,

(class constraints)

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Min( risk) and Max(return)

Subject to

Basic constraints Cardinality constraint Short sales Class constraints

K-means cluster analysis

Min( risk) and Max(return)Subject to Basic constraints Short sales Class constraints

Heuristic Optimization techniques

Weight standardization algorithms

Optimal weights

Heuristic Portfolio Optimization processHeuristic Portfolio Optimization process

0 20 40 60 80 100 120 14020

40

60

80

100

120

140

160

180

Annualized risk(%)

Exp

ecte

d po

rtfo

lio a

nnua

l ret

urn(

%)

Efficient frontier1818

Heuristic Portfolio Optimization strategiesHeuristic Portfolio Optimization strategies

Evolution Strategy with Hall of Fame

Differential Evolution (rand/1/bin)

Quadratic Programming

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Evolution Strategy with Hall of Fame: Evolution Strategy with Hall of Fame: Design and implementationDesign and implementation

Parent Population

Offspring Population

Arithmetic variable point cross over

Real number uniform mutation (Andrzej Osyczka, 2002)

New population

s : u Hall of Fame

2020

Each chromosome in the population comprises of K genes representative of the portfolio weights that need to be optimized.

During each generation the chromosomes undergo a weight standardization

The weight standardization ensures that each chromosome in each population represents a feasible solution, before they compete among themselves and the one in the Hall of Fame to determine the best solution.

The objective function of the portfolio optimization problem yields the fitness value that is used to rank chromosomes

That chromosome reporting the minimum fitness value is declared as the best fit chromosome

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Differential Evolution (rand/1/bin): Differential Evolution (rand/1/bin): Design and implementationDesign and implementation

Parent Population

Offspring Population

Mutation operator

New population

Trial vectorPopulation

Binary Cross over operator

Deterministic selection

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Experimental Studies and ResultsExperimental Studies and ResultsPortfolio optimization problem definition

over BSE200 data setcardinality constraint K=20

short selling bounds (-0.15, +1.15)

N=200 assets in the BE200 data set

Class constraints:

Banks:

Technology:

Oil and Gas:

3.001.0 t

tw

4.001.0 s

sw

25.001.0 r

rw

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Number of Assets in the chosen asset class and their respective class constraints

Banks Technology Oil and Gas

Investable

universe 13 3 2

Investable

universe 22 4 1

Investable

universe 35 3 1

Table 1: Composition of assets in the Investable universes (K =20) of BSE200 data set, satisfying the class constraints

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Parameter Values set

Chromosome length (number of genes) K = 20

Population size 100

Generations 2000

Parent, child chromosomes ratio in a generation (1:2)

Choice of values for the Risk aversion

parameter λ

to graph the efficient frontier

18 points

Table 2 Choice of parameters defining the Evolution Strategy with Hall of Fame during the experimental studies

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Table 3 Choice of parameters defining Differential Evolution (rand/1/bin) during the experimental

studies

Parameter Values set

Chromosome length (number of genes) K = 20

Population size 200

Generations 1000

Scaling factor β 0.5

Probability of recombination 0.87

Choice of values for the Risk aversion parameter

λ

to graph the efficient frontier

18 points

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Experiment 1

Efficient frontiers for Short selling and Class constrained Investable universes of BSE200 data set (K=20) using Quadratic Programming

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Experiment 2

Efficient frontiers traced by Evolution Strategy with Hall of Fame for various runs on an Investable universe of the BSE200 (July 2001-July 2006) data set

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Experiment 3

Efficient frontiers traced by Differential Evolution (rand/1/bin) for various runs on an Investable universe of the BSE200 (July 2001-July 2006) data set

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Efficient frontiers traced by the Evolution Strategy with Hall of Fame, Differential Evolution (rand/1/bin) and Quadratic Programming for a specific investable universe of BSE200 (July 2001-July 2006) data set

Experiment 4

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Data Envelopment Analysis (DEA) is a non-parametric, deterministic

methodology to asses the relative efficiencies and performances of a

collection of comparable entities called Decision Making Units (DMUs)

which transform inputs to outputs.

DEA determines the efficiency scores of each DMU relative to the

others and makes use of linear programming to compute the

efficiency scores.

To obtain the efficiency scores of the optimal portfolios each risk

return couple is chosen to represent a DMU. The efficient frontiers

graphed in Experiment 4 were the source inputs to the DEA.

Data Envelopment Analysis (DEA) of optimal portfolios

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Portfolio optimization method

Number of DMUs (n) considered by the DEA

Average efficiency score (µ)

Standard deviation of the efficiency scores (σ)

Coefficient of Variation (%)

Evolution Strategy with Hall of Fame

18 0.8322 0.0788 9.4702

Differential Evolution (rand/1/bin)

18 0.8502 0.0703 8.2640

Summary of the DEA of constrained optimal portfolios obtained by ES HoF and DE(rand/1/bin) for the BSE200 data set

100

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ConclusionsConclusions

the simplified problem model obtained after the simplified problem model obtained after kk--means clustering is means clustering is

amenable for solution by both traditional and heuristic methodsamenable for solution by both traditional and heuristic methods

kk--means clustering means clustering promotes dimensionality reductionpromotes dimensionality reduction which which

could be exploited by the heuristic approaches in could be exploited by the heuristic approaches in reducing the reducing the

design variables and consequently leading to faster design variables and consequently leading to faster

convergenceconvergence

Both the heuristic methods yielded Both the heuristic methods yielded consistent results consistent results

irrespective of the investable universesirrespective of the investable universes3333

Conclusions (Contd..)Conclusions (Contd..)

A Data Envelopment Analyses of the optimal portfolios obtained bA Data Envelopment Analyses of the optimal portfolios obtained by the y the

two competing heuristic approaches revealed that the two competing heuristic approaches revealed that the optimal portfolios optimal portfolios

obtained by DE(rand/1/bin) approach were robust in comparison toobtained by DE(rand/1/bin) approach were robust in comparison to

those obtained by ES those obtained by ES HoFHoF

Statistical inferences drawn with regard to the technical efficiStatistical inferences drawn with regard to the technical efficiencies of encies of

the optimal portfolios, obtained by ES the optimal portfolios, obtained by ES HoFHoF and DE(rand/1/bin) and DE(rand/1/bin)

concluded that there was a concluded that there was a significant difference in the means of the significant difference in the means of the

efficiency scores of the two methods and hence these are distincefficiency scores of the two methods and hence these are distinctively tively

different in behavior.different in behavior.

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ReferencesReferences

Chang T J, N Meade, J B Beasley and Y M Chang T J, N Meade, J B Beasley and Y M SharaihaSharaiha, Heuristics for , Heuristics for cardinality constrained portfolio optimization, cardinality constrained portfolio optimization, Computers and Operations Computers and Operations ResearchResearch, 27: 1271, 27: 1271--1302, 2000.1302, 2000.

MaringerMaringer DietmarDietmar, , Portfolio management with heuristic optimizationPortfolio management with heuristic optimization, , Springer, 2005.Springer, 2005.

PaiPai VijayalakshmiVijayalakshmi G A and Thierry Michel, On measuring the reliability G A and Thierry Michel, On measuring the reliability of kof k--means clustered financial portfolio sets for cardinality constrameans clustered financial portfolio sets for cardinality constrained ined portfolio optimization, in portfolio optimization, in Mathematical and Computational ModelsMathematical and Computational Models, , (eds.) R (eds.) R NadarajanNadarajan, R , R AnithaAnitha and C and C PorkodiPorkodi, pp. 313, pp. 313--323, 323, NarosaNarosa Publishing House, 2007.Publishing House, 2007.

PaiPai VijayalakshmiVijayalakshmi G A and Thierry Michel, Evolutionary optimization of G A and Thierry Michel, Evolutionary optimization of constrained constrained kk--means clustered assets for diversification in small means clustered assets for diversification in small portfoliosportfolios, , IEEE Trans. On Evolution ComputationIEEE Trans. On Evolution Computation, 2009 (to appear), 2009 (to appear)

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