##### Document Text Contents

Page 2

Lecture Notes in Computer Science 6025

Commenced Publication in 1973

Founding and Former Series Editors:

Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board

David Hutchison

Lancaster University, UK

Takeo Kanade

Carnegie Mellon University, Pittsburgh, PA, USA

Josef Kittler

University of Surrey, Guildford, UK

Jon M. Kleinberg

Cornell University, Ithaca, NY, USA

Alfred Kobsa

University of California, Irvine, CA, USA

Friedemann Mattern

ETH Zurich, Switzerland

John C. Mitchell

Stanford University, CA, USA

Moni Naor

Weizmann Institute of Science, Rehovot, Israel

Oscar Nierstrasz

University of Bern, Switzerland

C. Pandu Rangan

Indian Institute of Technology, Madras, India

Bernhard Steffen

TU Dortmund University, Germany

Madhu Sudan

Microsoft Research, Cambridge, MA, USA

Demetri Terzopoulos

University of California, Los Angeles, CA, USA

Doug Tygar

University of California, Berkeley, CA, USA

Gerhard Weikum

Max-Planck Institute of Computer Science, Saarbruecken, Germany

Page 252

Active Portfolio Management 223

Nowadays, there exist a plethora of methodologies for structuring index track-

ing portfolios with cardinality constraints (some references are given in following

sections; see also [1] for a comprehensive survey). Despite the numerous applica-

tions of passive portfolio selection, active tracking formulations - especially those

incorporating cardinality constraints - have not yet attracted full attention. In

[8] we consider three alternative formulations of active portfolio management,

based on the traditional mean-variance view of portfolio selection. In this paper,

we expand upon this issue by making an attempt to incorporate non-standard

objectives related to portfolio performance. These focus, for example, on the

probability that the index portfolio delivers positive return relatively to the

benchmark or on restricting the total risk of the investment strategy. Such tar-

gets/constraints on portfolio performance can be effectively formulated within

the framework of fuzzy mathematical multi-objective programming.

The rest of the article is structured as follows: enhanced index tracking with

cardinality constraints is studied in section 2, while section 3 discusses the

nature-inspired optimisation heuristics employed in our study to solve active

portfolio formulations. Section 4 details an alternative, “fuzzy”, conceptualisa-

tion of active portfolio management structured around approximate investment

targets and portfolio constraints. In section 5 we evaluate the performance of

fuzzy portfolios in terms of actively reproducing the American DJIA index. Sec-

tion 6 summarises the main findings and proposes future research directions.

2 Index Tracking with Cardinality Constraints

Consider the following investment problem whereby a fund manager has to de-

cide the best subset of K stocks (K < N) that can actively reproduce returns

on a benchmark index I as well as the appropriate percentage of capital wi that

should be invested in each stock i. The active portfolio optimisation problem can

be stated in its general form as maximise

w∈RN , s∈{0,1}N

f(w, s) subject to

∑N

i=1 wi = 1,

wl ≤ wi ≤ wu, i = 1, ..., N and

∑N

i=1 si ≤ K ≤ N , where f(w, s) is the port-

folio’s active objective, which is a function of weights w = (w1, w2, ..., wN )′, a

vector of binary variables s = (s1, s2, ..., sN )′ and sample data. Several choices

for f(., .) are later discussed in section 4. Each si is an indicator function that

takes the value 1 if asset i is included in the portfolio and 0 otherwise. All asset

weights sum to one, implying that the initial capital is fully invested, and the

maximum number of assets allowed in the portfolio should not exceed K ≤ N ,

which is the cardinality constraint. Depending on the cardinality of the portfolio

P , some of the wi’s may be zero and if an asset i is included then the fund man-

ager may impose a lower or upper limit on its weight, wl and wu, respectively,

the so-called floor and ceiling constraint.

The introduction of cardinality constraints significantly increases the com-

putational effort associated with deriving optimal portfolio allocations. In fact,

one ends up with a mixed nonlinear-integer programming problem, which even

Page 253

224 N.S. Thomaidis

for small values of N becomes a challenge for gradient-based optimisation tech-

niques. In this paper, we adopt a solution strategy that overcomes the difficulties

of handling integer constraints by introducing a transformation of the decision

variables (see [8,6] for details). The underlying idea is to solve an unconstrained

optimisation programme with decision variables (x1, ...., xN ) ∈ RN and use a

deterministic function to map the values of (x1, ...., xN ) onto a feasible portfolio

allocation (w1, w2, ..., wN ), where K out of N weights are zero.

3 Nature-Inspired Optimisation Algorithms

Many real-life optimisation problems in finance are considered “hard” due to

combinatorial explosion or the ruggedness of the optimisation landscape. Tradi-

tionally, practitioners approach these problems adopting techniques that make

use of relaxation and decomposition principles, such as branch & bound, dy-

namic programming and quadratic line-search. In the last decades, a number of

intelligent computational algorithms, such as simulated annealing, tabu search,

genetic algorithms, ant colonies and particle swarms, have been developed to

solve a wide range of practical optimisation problems.

In this study, we experiment with three popular nature-inspired computa-

tional schemes in the task of solving active portfolio optimisation problems. We

first present a trajectory-based strategy, namely simulated annealing, and then

introduce two evolutionary computational heuristics, genetic algorithms and par-

ticle swarm optimisation.

Simulated annealing (SA) took its name and inspiration from the annealing,

a technique used in metallurgy involving heating and controlled cooling of a

solid ([5]). The central idea is to start with some arbitrary solution and have

it modified by randomly generating a number (or else population) of new so-

lutions in its vicinity. To overcome the possibility of premature convergence to

local optima, the algorithm also accepts solutions that come with impairment,

yet with decreasing probability. Genetic algorithms (GAs) employ an alterna-

tive solution-search strategy by forcing a parallel exploration of the solution

space by means of several agents that interact with each other. Inspired by the

process of natural selection that drives biological evolution, a genetic algorithm

repeatedly modifies a population of solutions until it“evolves” towards a “fit”

generation ([7]). Among the numerous forms in which genetic algorithms ap-

pear in the literature, in this paper we adopt a simple version of the algorithm

that encodes solutions into vectors of real numbers and uses three genetic opera-

tors to move towards fitter populations: elitist selection, crossover and mutation.

Particle swarm optimisation (PSO) is another computational heuristic inspired

by the behaviour of biological flocks or swarms [3,4]. Instead of applying ge-

netic operators, PSO flows “particles” in the search space with a velocity and

direction that are dynamically adjusted according to each particle’s learning ex-

perience (the best solution detected by the particle in its own exploration) and

the swarm’s collective memory (the global best solution found by any particle

in the swarm).

Page 503

Author Index 475

Oh, Jae C. II-261

Olague, Gustavo I-344

Oliveto, Pietro Simone I-61

Öncan, Temel II-431

O’Neill, Michael I-161, II-192,

II-251, II-341

Oranchak, David I-181

Pacut, Andrzej II-71, II-91

Pantano, Pietro I-211

Pasquet, Olivier II-391

Pasquier, Philippe I-131

Payne, Joshua L. I-41

Peña, José-Maŕıa I-422

Pizzo, Christian II-41

Piazzolla, Alessandro I-320

Pizzuti, Stefano II-151

Pospichal, Petr I-442

Protopapas, Mattheos K. I-191

Qin, Peiyu II-111

Quattrone, Aldo I-211

Ramirez, Adriana II-462

Ramtohul, Tikesh II-212

Rees, Jonathan I-312

Reynoso-Meza, Gilberto I-532

Richter, Hendrik I-552

Rimmel, Arpad I-201

Rocchisani, Jean-Marie I-292

Rolet, Philippe I-592

Romero, Juan II-271

Runarsson, Thomas Philip I-121

Ryan, Conor II-202

Şahin, Cem Şafak II-41

Sánchez, Ernesto I-11, I-412

Sánchez, Pablo Garćıa I-171

Sánchez-Pérez, Juan Manuel II-61

Sanchis, Javier I-532

Sarasola, Briseida I-572

Sbalzarini, Ivo F. I-432

Schettini, Raimondo I-282

Schumann, Enrico II-242

Schwarz, Josef I-442

Scionti, Alberto I-412

Scott, Cathy II-141, II-421

Seo, Kisung I-352, I-381

Serquera, Jaime II-381

Shao, Jianhua II-341

Sharabati, Anas I-361

Shukla, Pradyumn Kumar I-21

Silva, Sara I-272, II-131

Şima Uyar, A. II-1, II-81, II-121

Smit, S.K. I-542

Sorenson, Nathan I-131

Sossa, Humberto I-344

Squillero, Giovanni I-11, I-412

Staino, Andrea I-211

Stanley, Kenneth O. I-141, II-331

Stramandinoli, Francesca I-211

Süral, Haldun II-431

Swafford, John Mark I-161

Szeto, K.Y. I-151

Tamimi, Hashem I-361

Tenne, Yoel I-582

Tettamanzi, Andrea G.B. II-161

Teytaud, Fabien I-201, I-452

Teytaud, Olivier I-592

Thomaidis, Nikos S. II-222

Tirronen, Ville I-471

Togelius, Julian I-141

Tonda, Alberto I-11, I-412

Torre, Antonio II-351

Trucco, Emanuele I-241

Tsviliuk, Olena II-232

Urquhart, Neil II-141, II-421

Urrea, Elkin II-41

Uyar, M. Ümit II-41

Vanneschi, Leonardo I-282

Vasconcelos, Maria J. I-272

Vega-Rodŕıguez, Miguel Angel II-61

Vidal, Franck Patrick I-292

Villegas-Cortez, Juan I-344

Voß, Stefan II-462

Vrahatis, Michael N. II-411

Waldock, Antony I-461

Wang, Xingwei II-111

Watson, Richard A. I-1

Weber, Matthieu I-471

Wong, Ka-Chun I-481

Wong, Man-Hon I-481

Wu, Chun-Ho I-302

Wu, Degang I-151

Page 504

476 Author Index

Yang, Shengxiang I-491, I-562

Yannakakis, Georgios N. I-141

Yao, Xin I-61

Yayımlı, Ayşegül II-81

Yung, Kei-Leung I-302

Zaccagnino, Rocco II-351

Zajec, Edward II-261

Zhang, Di II-232

Zhang, Mengjie II-51

Zincir-Heywood, Nur II-101

Lecture Notes in Computer Science 6025

Commenced Publication in 1973

Founding and Former Series Editors:

Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board

David Hutchison

Lancaster University, UK

Takeo Kanade

Carnegie Mellon University, Pittsburgh, PA, USA

Josef Kittler

University of Surrey, Guildford, UK

Jon M. Kleinberg

Cornell University, Ithaca, NY, USA

Alfred Kobsa

University of California, Irvine, CA, USA

Friedemann Mattern

ETH Zurich, Switzerland

John C. Mitchell

Stanford University, CA, USA

Moni Naor

Weizmann Institute of Science, Rehovot, Israel

Oscar Nierstrasz

University of Bern, Switzerland

C. Pandu Rangan

Indian Institute of Technology, Madras, India

Bernhard Steffen

TU Dortmund University, Germany

Madhu Sudan

Microsoft Research, Cambridge, MA, USA

Demetri Terzopoulos

University of California, Los Angeles, CA, USA

Doug Tygar

University of California, Berkeley, CA, USA

Gerhard Weikum

Max-Planck Institute of Computer Science, Saarbruecken, Germany

Page 252

Active Portfolio Management 223

Nowadays, there exist a plethora of methodologies for structuring index track-

ing portfolios with cardinality constraints (some references are given in following

sections; see also [1] for a comprehensive survey). Despite the numerous applica-

tions of passive portfolio selection, active tracking formulations - especially those

incorporating cardinality constraints - have not yet attracted full attention. In

[8] we consider three alternative formulations of active portfolio management,

based on the traditional mean-variance view of portfolio selection. In this paper,

we expand upon this issue by making an attempt to incorporate non-standard

objectives related to portfolio performance. These focus, for example, on the

probability that the index portfolio delivers positive return relatively to the

benchmark or on restricting the total risk of the investment strategy. Such tar-

gets/constraints on portfolio performance can be effectively formulated within

the framework of fuzzy mathematical multi-objective programming.

The rest of the article is structured as follows: enhanced index tracking with

cardinality constraints is studied in section 2, while section 3 discusses the

nature-inspired optimisation heuristics employed in our study to solve active

portfolio formulations. Section 4 details an alternative, “fuzzy”, conceptualisa-

tion of active portfolio management structured around approximate investment

targets and portfolio constraints. In section 5 we evaluate the performance of

fuzzy portfolios in terms of actively reproducing the American DJIA index. Sec-

tion 6 summarises the main findings and proposes future research directions.

2 Index Tracking with Cardinality Constraints

Consider the following investment problem whereby a fund manager has to de-

cide the best subset of K stocks (K < N) that can actively reproduce returns

on a benchmark index I as well as the appropriate percentage of capital wi that

should be invested in each stock i. The active portfolio optimisation problem can

be stated in its general form as maximise

w∈RN , s∈{0,1}N

f(w, s) subject to

∑N

i=1 wi = 1,

wl ≤ wi ≤ wu, i = 1, ..., N and

∑N

i=1 si ≤ K ≤ N , where f(w, s) is the port-

folio’s active objective, which is a function of weights w = (w1, w2, ..., wN )′, a

vector of binary variables s = (s1, s2, ..., sN )′ and sample data. Several choices

for f(., .) are later discussed in section 4. Each si is an indicator function that

takes the value 1 if asset i is included in the portfolio and 0 otherwise. All asset

weights sum to one, implying that the initial capital is fully invested, and the

maximum number of assets allowed in the portfolio should not exceed K ≤ N ,

which is the cardinality constraint. Depending on the cardinality of the portfolio

P , some of the wi’s may be zero and if an asset i is included then the fund man-

ager may impose a lower or upper limit on its weight, wl and wu, respectively,

the so-called floor and ceiling constraint.

The introduction of cardinality constraints significantly increases the com-

putational effort associated with deriving optimal portfolio allocations. In fact,

one ends up with a mixed nonlinear-integer programming problem, which even

Page 253

224 N.S. Thomaidis

for small values of N becomes a challenge for gradient-based optimisation tech-

niques. In this paper, we adopt a solution strategy that overcomes the difficulties

of handling integer constraints by introducing a transformation of the decision

variables (see [8,6] for details). The underlying idea is to solve an unconstrained

optimisation programme with decision variables (x1, ...., xN ) ∈ RN and use a

deterministic function to map the values of (x1, ...., xN ) onto a feasible portfolio

allocation (w1, w2, ..., wN ), where K out of N weights are zero.

3 Nature-Inspired Optimisation Algorithms

Many real-life optimisation problems in finance are considered “hard” due to

combinatorial explosion or the ruggedness of the optimisation landscape. Tradi-

tionally, practitioners approach these problems adopting techniques that make

use of relaxation and decomposition principles, such as branch & bound, dy-

namic programming and quadratic line-search. In the last decades, a number of

intelligent computational algorithms, such as simulated annealing, tabu search,

genetic algorithms, ant colonies and particle swarms, have been developed to

solve a wide range of practical optimisation problems.

In this study, we experiment with three popular nature-inspired computa-

tional schemes in the task of solving active portfolio optimisation problems. We

first present a trajectory-based strategy, namely simulated annealing, and then

introduce two evolutionary computational heuristics, genetic algorithms and par-

ticle swarm optimisation.

Simulated annealing (SA) took its name and inspiration from the annealing,

a technique used in metallurgy involving heating and controlled cooling of a

solid ([5]). The central idea is to start with some arbitrary solution and have

it modified by randomly generating a number (or else population) of new so-

lutions in its vicinity. To overcome the possibility of premature convergence to

local optima, the algorithm also accepts solutions that come with impairment,

yet with decreasing probability. Genetic algorithms (GAs) employ an alterna-

tive solution-search strategy by forcing a parallel exploration of the solution

space by means of several agents that interact with each other. Inspired by the

process of natural selection that drives biological evolution, a genetic algorithm

repeatedly modifies a population of solutions until it“evolves” towards a “fit”

generation ([7]). Among the numerous forms in which genetic algorithms ap-

pear in the literature, in this paper we adopt a simple version of the algorithm

that encodes solutions into vectors of real numbers and uses three genetic opera-

tors to move towards fitter populations: elitist selection, crossover and mutation.

Particle swarm optimisation (PSO) is another computational heuristic inspired

by the behaviour of biological flocks or swarms [3,4]. Instead of applying ge-

netic operators, PSO flows “particles” in the search space with a velocity and

direction that are dynamically adjusted according to each particle’s learning ex-

perience (the best solution detected by the particle in its own exploration) and

the swarm’s collective memory (the global best solution found by any particle

in the swarm).

Page 503

Author Index 475

Oh, Jae C. II-261

Olague, Gustavo I-344

Oliveto, Pietro Simone I-61

Öncan, Temel II-431

O’Neill, Michael I-161, II-192,

II-251, II-341

Oranchak, David I-181

Pacut, Andrzej II-71, II-91

Pantano, Pietro I-211

Pasquet, Olivier II-391

Pasquier, Philippe I-131

Payne, Joshua L. I-41

Peña, José-Maŕıa I-422

Pizzo, Christian II-41

Piazzolla, Alessandro I-320

Pizzuti, Stefano II-151

Pospichal, Petr I-442

Protopapas, Mattheos K. I-191

Qin, Peiyu II-111

Quattrone, Aldo I-211

Ramirez, Adriana II-462

Ramtohul, Tikesh II-212

Rees, Jonathan I-312

Reynoso-Meza, Gilberto I-532

Richter, Hendrik I-552

Rimmel, Arpad I-201

Rocchisani, Jean-Marie I-292

Rolet, Philippe I-592

Romero, Juan II-271

Runarsson, Thomas Philip I-121

Ryan, Conor II-202

Şahin, Cem Şafak II-41

Sánchez, Ernesto I-11, I-412

Sánchez, Pablo Garćıa I-171

Sánchez-Pérez, Juan Manuel II-61

Sanchis, Javier I-532

Sarasola, Briseida I-572

Sbalzarini, Ivo F. I-432

Schettini, Raimondo I-282

Schumann, Enrico II-242

Schwarz, Josef I-442

Scionti, Alberto I-412

Scott, Cathy II-141, II-421

Seo, Kisung I-352, I-381

Serquera, Jaime II-381

Shao, Jianhua II-341

Sharabati, Anas I-361

Shukla, Pradyumn Kumar I-21

Silva, Sara I-272, II-131

Şima Uyar, A. II-1, II-81, II-121

Smit, S.K. I-542

Sorenson, Nathan I-131

Sossa, Humberto I-344

Squillero, Giovanni I-11, I-412

Staino, Andrea I-211

Stanley, Kenneth O. I-141, II-331

Stramandinoli, Francesca I-211

Süral, Haldun II-431

Swafford, John Mark I-161

Szeto, K.Y. I-151

Tamimi, Hashem I-361

Tenne, Yoel I-582

Tettamanzi, Andrea G.B. II-161

Teytaud, Fabien I-201, I-452

Teytaud, Olivier I-592

Thomaidis, Nikos S. II-222

Tirronen, Ville I-471

Togelius, Julian I-141

Tonda, Alberto I-11, I-412

Torre, Antonio II-351

Trucco, Emanuele I-241

Tsviliuk, Olena II-232

Urquhart, Neil II-141, II-421

Urrea, Elkin II-41

Uyar, M. Ümit II-41

Vanneschi, Leonardo I-282

Vasconcelos, Maria J. I-272

Vega-Rodŕıguez, Miguel Angel II-61

Vidal, Franck Patrick I-292

Villegas-Cortez, Juan I-344

Voß, Stefan II-462

Vrahatis, Michael N. II-411

Waldock, Antony I-461

Wang, Xingwei II-111

Watson, Richard A. I-1

Weber, Matthieu I-471

Wong, Ka-Chun I-481

Wong, Man-Hon I-481

Wu, Chun-Ho I-302

Wu, Degang I-151

Page 504

476 Author Index

Yang, Shengxiang I-491, I-562

Yannakakis, Georgios N. I-141

Yao, Xin I-61

Yayımlı, Ayşegül II-81

Yung, Kei-Leung I-302

Zaccagnino, Rocco II-351

Zajec, Edward II-261

Zhang, Di II-232

Zhang, Mengjie II-51

Zincir-Heywood, Nur II-101