A Survey on Portfolio Optimisation with Metaheuristics

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A Survey on Portfolio Optimisation with
Metaheuristics
Prisadarng Skolpadungket Keshav Dahal School of
Informatics University of Bradford, UK
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Outline

• Introduction.
• Portfolio Optimisation Problems.
• Realistic Constraints
• Metaheuristic Methods and Results
• Conclusion and Future works

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Introduction

• Portfolio optimisation objective to find
  minimum risk a given expect return .
• A portfolio manager task to select K assets
  from a universe of N assets (K lt N) (e.g. all
  listed companies stocks) to form an equity
  portfolio.
• 2 relevant characteristics of any assets
• 1) Expected return
• 2) Risk profile.
• Risk is a relatively vague concept thus can be
  quantitatively represented by many definitions.

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Introduction (Cont.)

• The most popular representations of risk
• Variance of return (or Standard Deviation)
• Value at Risk (VaR).
• Variance and VaR of a portfolio not equal to
  plain summation of the assets returns
• Assets returns may be correlated thus
  portfolios return less than the plain summation.
• The more un-correlated, the less risk the
  portfolio.
• The choices of assets in combination affect
• weight average expected return
• non-linear and inter-related risk

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Portfolio Optimisation Problems
The general model of portfolio optimisation
(modified Markowitz model by Black) is

• xi are selection variables, short-sale allowed.
• xi can be negative (short position of asset i).
  
• If xi are all continuous and differentiable
  then the model has
• a closed form solution (solved by standard
  calculus).
• a Portfolio Frontier represents the set of
  portfolio optima .

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Portfolio Frontier
rp (Return)
r (Return)
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Portfolio Optimisation Problems (Cont.)

• The original Markowitz model has no short-sell
• xi cannot be negative.
• Non-negative constraint is added into the
  general model

• This model cannot be solved by calculus.
• But by some specialised techniques
• e.g. interior point algorithms, branch and cut
  approaches, other numerical techniques using
  structures of the co-variance matrix.

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Realistic Constraints

• Realistic constraints arise
• Regulations
• Market institution
• investment policies
• economic (cost) reasons etc.
• Integer constraints every asset includes in
  the portfolio must be rounded normal trading
  lot).

• Cardinality constraints the maximum number
  and minimum number of assets to include in the
  portfolio

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Realistic Constraints (Cont.)

• Floor and ceiling constraints lower and upper
  limits on the proportion of each asset held.

• Turnover constraints upper bound for
  variations of the asset holding from one period
  to the next.

• Trading constraints limits on buying and
  selling tiny quantities of assets

• Transaction costs associate with purchases
  and sales of assets incorporated in the realistic
  models.

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Metaheuristic Methods and Results

• With realistic constraints
• Portfolio optimisation became Non-Polynomial
  Hard problems (NP-Hard).
• Integer and cardinality constraints
• The problems become combinatorial optimisation
  problems with factorial time complexity (O (C (N,
  k)) ).
• Two ways to cope with the NP hard problems.
• Approximate Models
• Approximate Algorithms
• Heuristics are Approximate Algorithms.
• Simple heuristics tend to end their searches in
  local minima.
• Metaheuristics
• Heuristics with mechanisms to escape local
  minima.
• Allowing temporary moves to inferior points.
• Two categories of Metaheuristics
• Local search metaheuristics (LSMs)
• Evolutionary algorithms (EAs).

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Simulated Annealing

• The oldest among the metaheuristics.
• Allowing moves toward worse solutions to escape
  from local optima.
• The probability of worse moves diminishs like
  molecules slowing down as the temperature cooling
  down.
• Crama and Schyns (Crama 2003)
• With floor, ceiling, turnover, trading and
  cardinality constraints.
• Approximated the optimal portfolio frontier
  (medium size problem with 151 assets) within
  acceptable time.
• Could handle more classes of constraints than
  classical approaches.
• Versatile to apply to different measures of
  risk.
• Needs to customize and fine tune parameters for
  different classes of constraints.

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Tabu Search

• Keeps lists of previous searches (tabu list)
• Forbids moves toward tabu list.
• Used to avoid local optima
• To implement an explorative search strategy
  (Blum 2003.)
• Busetti (Busetti 2000)
• Used Tabu Search/Scatter Search
• Compared with Genetic Algorithms (GA)
• Found that tabu/scatter search is unsuitable for
  cardinality constraint problem (40 assets)
• Concluded that GA is better than tabu/scatter
  search

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Ant Colony Search

• Imitation ants findind shortest path between
  food sources and their nest.
• Ants deposit pheromone on the ground.
• Decide the direction based on the concentration
  of pheromone.
• Maringer (Maringer 2005) applied ant colony
  algorithms for small portfolios (with
  cardinality constraints).
• Similar to Knapsack problems with some
  modifications.
• The value of asset depends on the overall
  structure of portfolio.
• Has to decide jointly
• to include an asset or not
• the amount of the asset.
• Found that AC is more efficient for smaller K
  (3) and with pheromone evaporation

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Hybrid Local Search

• Maringer (Maringer 2003, 2005)
• Applied hybrid local search.
• Combine population based with local search.
• A crystal-like structure represents a portfolio
  of assets.
• All of the crystals represent the population.
• The iterations consist of three stages
• Modification of crystal (portfolio) structure
• Evaluation and ranking of the modified structure
• Replacement of the poorest crystal in the
  population.
• Test the algorithm against SA and SA with a group
  of isolated crystals (GSA).
• Results
• HLS (best) gtgt GSA gtgt SA (worst)
• Conclusion
• Evolutionary strategies improve metaheuristic
  algorithms for portfolio optimisation

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Genetic Algorithms

• GA are population based heuristic algorithms
• Imitating the natural selection of survival of
  the fittest.
• Represented as chromosomes
• breed by crossover
• modified by mutation
• Busetti (Busetti 2000) compared GA with tabu
  search
• Found that GA performs better.
• Streichart et al. (Streichart 2004a) applied the
  Multi-Objective Evolutionary Algorithm (MOEA)
• Use 2 binary bit-string based genotypes
• Gray-code encoding
• Real-valued genotype (32 bits)
• Compare
• GA with and without Lamarckism (can be modified
  not only being removed from the population)
• Knapsack GA (KGA) with and without Larmarckism.
• With constraints cardinality and integer
  (discrete) constraints

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Genetic Algorithms (Cont.)

• The reults
• with constraints, KGA produced better results
  and converged faster than ordinary GA.
• without constraints, both GA and KGA perform
  almost the same.
• GAs without Larmarckism tend to be premature
  convergence and trapped in local minima.
• The real value coding performed worst
• Bit gray-coding and Larmarckism was the best
• Intrepretations
• The mutation and crossover operators are more
  effective in the gray code representation.
• Larmarckism adds performance due to its ability
  to remove neutrality in the search space.

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Conclusion Future Works

• A portfolio optimisation with realistic
  constraints is a NP hard problem.
• Pure search metaheuristics tends to be trapped in
  local optima.
• Population based metaheuristics tends to be time
  inefficient.
• Uses of hybrid algorithms can improve the
  situations.
• A clear trend is heading toward hybrid models.
• Our future work will be toward improving hybrid
  and novel metaheuristic implementations.
• We also plan to extend the techniques to
  implement on other portfolio selection models
  with
• Different definition of risk and return,
• Estimations of the volatility and of the
  returns.

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The End
Thank You !