26.08.2005 Benedikt Scheckenbach

1. Portfolio-Optimization with Multi-Objective Evolutionary Algorithms in the case of Complex Constraints 26.08.2005 Benedikt Scheckenbach

2. Outline • Basics of Portfolio Optimization • Examples of Complex Constraints • Existing MOEAs • 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. Pareto Front Minimum-Variance Portfolio 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. V(x) E(x)

8. 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%.

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 Problem Specification 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.

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

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

19. Dominated Area Calculation of jump-points • Two cases where jumps are possible: • Another Pareto-Front starts within the return-interval defined by the current segment. • 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 V(x) V(x) E(x) E(x)

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

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

23. Thank you

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“