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

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

3 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

4 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%.

5 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.

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

7 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

8 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

9 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

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

11 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.

12 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 ProblemSpecification of basic Problem

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

14 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.

15 Idea of the Diploma thesis Using Critical Line Algorithm as decoding function. New geno- and phenotypes.  New non-dominated sorting, crossover, mutation. Diploma thesisPresent Algorithms GenotypeSet of linear constraints that fulfill basic constraints Weights that fulfill basis constraints PhenotypeComplete Pareto- Front, i.e. best portfolios for given constraints Single, suboptimal portfolio

16 „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. Front 2. agg. Front 3. agg. Front V(x) E(x)

17 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)

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

19 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)

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

21 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)

22 agg Front 3 Update of Population Similar to NSGA-2 old pop off- spring Modified non-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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24 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“