PortfolioOptimization with MultiObjective Evolutionary Algorithms in the case

Portfolio-Optimization with Multi-Objective Evolutionary Algorithms in the case of Complex Constraints 26. 08. 2005 Benedikt Scheckenbach

Outline 1. 2. 3. 4. Basics of Portfolio Optimization Examples of Complex Constraints Existing MOEAs Idea of thesis 2

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 3

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%. 4

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. 5

Diversification Effect in case of two shares • Mean of Portfolio: • Variance of Portfolio: • Standard-Deviation: – 2 special cases: 6

Mean-Variance Portfolio Optimization • „Classic“ optimization problem: Pareto Front V(x) • Without further constraints there exists an analytical solution. Minimum-Variance Portfolio E(x) • In reality, further constraints have to be considered: – Additional requirements regarding the portfolio‘s weights. – Cardinality constraints. 7

Additional Requirements regarding the weigths • 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%. 8

Cardinality Constraints or • 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. 9

Extended Optimization Problem • Very large search space because of the combinatorial constraints. Application of MOEAs. 10

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. 11

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. Basic Problem Specification of basic Problem • A and b specify linear constraints that fulfill basic constraints 12

Critical Line Algorithm (2) • Example: Possible Matrix and RHS that fulfill German Investment Law: 13

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. 14

Idea of the Diploma thesis • Using Critical Line Algorithm as decoding function. • New geno- and phenotypes. 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 New non-dominated sorting, crossover, mutation. 15

„Modified“ Non-dominated Sorting • Build „aggregated“ Fronts (Set of Pareto-Fronts), that are not dominated by remaining Pareto Fronts. 1. agg. Front V(x) 2. agg. Front 3. agg. Front E(x) • Diversity sorting Criteria: Contribution of Pareto-Front to aggregated Front in form of length. 16

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. V(x) E(x) 17

Variance and Return within a Segment V(x) E(x) 18

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 Dominated Area V(x) E(x) 19

Calculation of intersection-points (1) • First Idea: – Intersection with other segments is only possible, if intervals on return -axis overlap. V(x) E(x) 20

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. V(x) E(x) 21

Update of Population • Similar to NSGA-2 old pop Modified non-domiated sorting offspring agg. Front 1 agg. Front 2 agg. Front 3 Diversity sorting Form new offsprings agg Front 3 … agg. Front k 22

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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“ 24