Portfolio-Optimization with Multi-Objective Evolutionary Algorithms in the case of

Complex Constraints

26.08.2005

Benedikt Scheckenbach

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Outline

1. Basics of Portfolio Optimization2. Examples of Complex Constraints3. Existing MOEAs4. Idea of the thesis

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Price of a share

• Price of a share can be regarded as a stochastic process.

• We define the return at a future date as

• Central Assumption: The returns are normally distributed

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Definition of a Portfolio

• A portfolio is a bundle of shares.• Self-similarity property of Normal Distribution:

Returns of shares a normal distributed Return of portfolio is normally distributed.

• The money invested in each share is a portion (weight) between 0 and 100% of the portfolio price.

• The sum of all weights has to be 100%.

10%

20%

40%

5%

25%

Share 1

Share 2

Share 3

Share 4

Share 5

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Why Portfolio Optimization?

• Diversification Portfolio might have lower variance than every single share.

• Individuality Each investor can adjust variance and mean to his needs.

• Simple Example: 2 Shares – Bivariate normal distribution of single returns.– Portfolio return is a convolution of single returns.

Correlation between two shares is important.

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Diversification Effect in case of two shares

• Mean of Portfolio:

• Variance of Portfolio:

• Standard-Deviation:– 2 special cases:

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Mean-Variance Portfolio Optimization

• „Classic“ optimization problem:

• Without further constraints there exists an analytical solution.

• In reality, further constraints have to be considered:– Additional requirements regarding the portfolio‘s weights.– Cardinality constraints.

E(x)

V(x)

Pareto Front

Minimum-Variance Portfolio

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• In-house requirements:– Parts of the portfolio shall be invested in specific countries, sectors or

branches.– Each share is required to have a minimum weight to reduce transaction

costs (Buy-in threshold).

• Legal requirements:– German Investment Law §60 (1):

• The weight of each share has to be below 10%.• The sum of all weights above 5% may not exceed 40%.

Additional Requirements regarding the weigths

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Cardinality Constraints

• Index-Tracking:– Financial products often have a share index as underlying.– Sometomes not all shares have an sufficient turnover volume.– To price the product one has to rebuild the index with only a few

shares. We need to find a portfolio that matches expected return and variance

of the index as close as possible with a maximum given number of shares.

or

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Extended Optimization Problem

• Very large search space because of the combinatorial constraints. Application of MOEAs.

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Existing MOEAs

• Focus on Cardinality Constraints, only buy-in thresholds as additional requirements regarding the weights.

• Phenotype: One Point in space

• Genotype: Mostly real-valued representation of weights.

• Non-dominated sorting according to NSGA-II.

• Critic:– Slow Convergence.– Algorithms don‘t incorporate special features of portfolio optimization.

• Critical Line Algorithm: Calculates the Pareto-Front for a given set of linear constraints.

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Critical Line Algorithm (1)

• In the following: no cardinality constraints.• Input for Critical Line Algorithm: Concrete specification of basic

constraints as a system of linear inequalities.

• A and b specify linear constraints that fulfill basic constraints

Basic Problem Specification of basic Problem

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Critical Line Algorithm (2)

• Example: Possible Matrix and RHS that fulfill German Investment Law:

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Critical Line Algorithm (3)

• Output of Critical Line Algorithm: Weights of specific „Corner Portfolios“ that lie on Pareto Front for given constraints.

• All other portfolios of the Pareto-Front can be constructed as linear combinations of neighbored Corner-Portfolios.

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Idea of the Diploma thesis

• Using Critical Line Algorithm as decoding function.• New geno- and phenotypes.

New non-dominated sorting, crossover, mutation.

Diploma thesis Present Algorithms

Genotype Set of linear constraints that fulfill basic constraints

Weights that fulfill basis constraints

Phenotype Complete Pareto-Front, i.e. best portfolios for given constraints

Single, suboptimal portfolio

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„Modified“ Non-dominated Sorting

• Build „aggregated“ Fronts (Set of Pareto-Fronts), that are not dominated by remaining Pareto Fronts.

• Diversity sorting Criteria: Contribution of Pareto-Front to aggregated Front in form of length.

1. agg. Front2. agg. Front3. agg. Front

V(x)

E(x)

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Calculation of intersection- and jump-points

• Basic Idea– Each Pareto Front is a set of segments– Segment := Part of Pareto-Front, which starts and ends at two

neighboured Corner-Portfolios.– Start with segment that contains Corner-Portfolio with highest

expected return.– Run through all segments until segment with lowest return has been

reached– Check at each segment if there is an intersection or a jump to another

segment

• Every segment defines intervals on return and variance axis.

E(x)

V(x)

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Variance and Return within a Segment

E(x)

V(x)

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Dominated Area

Calculation of jump-points

• Two cases where jumps are possible:1. Another Pareto-Front starts within the return-interval defined by the

current segment.2. The current segment is the most left one: jump to next best Pareto-

Front. Further Pareto-Fronts can only be counted to aggregated Front if there is no domination by variance of best-known portfolio

E(x)

V(x)

E(x)

V(x)

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• First Idea:– Intersection with other segments is only possible, if intervals on return-

axis overlap.

Calculation of intersection-points (1)

V(x)

E(x)

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Calculation of intersection-points(2)

• We need to check if return and variance of two segments are equal:

• Subistitute . Possible intersection is solution of quadratic equation depending on .

• depends on the position of the two segments.• Better alternative: construct artificial segments, that have equal

return-intervals.

E(x)

V(x)

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agg Front 3

Update of Population

• Similar to NSGA-2

oldpop

off-spring

Modifiednon-domiated

sorting

agg. Front 1

agg. Front 2

agg.Front 3

…

agg. Front k

Diversity sorting

agg. Front 1

agg. Front 2

Form new offsprings

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Literature

• Streichert, Ulmer und Zell: „Evalutating a Hybrid Encoding and Three Crossover Operators on the Constrained Portfolio Selection Problem“

• Streichert, Ulmer und Zell: „Comparing Discrete and Continuos Genotypes on the Constrained Portfolio Selection Problem“

• Streichert, Ulmer und Zell: „Evolutionary Algorithms and the Cardinality Constrained Portfolio Optimization Problem“

• Derigs und Nickel: „Meta-heuristic based decision support for portfolio optimization with a case study on tracking error minimization in passive portfolio management“