# Meta-heuristic solution approaches to the portfolio optimisation problem

Thesis (MCom)--Stellenbosch University, 2018.

Thesis

ENGLISH SUMMARY : The portfolio optimisation problem is a well documented and researched combinatorial problem in the financial and operations research fields. The problem definition is defined as having to decide on which financial assets to invest in so as to minimise the associated risk while still maintaining a desired level of return on the investment. To accomplish this, various models have been formulated to help find accurate methods of quantifying and then minimising this risk. One such model is the Markowitz Mean-Variance model, introduced in 1952 as the initial method of quantifying risk and beginning the renaissance of investing in a diversified portfolio. This paper attempts to solve three problems associated with investing in a diversified portfolio using the Markowitz model. These are; that the time taken to solve the models with traditional mathematical methods are unusable for real world dataset sizes; that the initial investment needed to purchase a fully diversified portfolio is large enough that the common investor may struggle to invest early enough in his or her lifespan; and that the Markowitz model is based on the assumption that fnancial assets expected returns are normally distributed. These problems are solved in three parts. The first part is explaining what unit trusts are and how they can be used a tool for the average investor to use as an aid in efficiently investing in a diversified portfolio. The second is to overcome the estimations errors associated with Markowitz's assumption of normality issue by using a distribution free estimate of the variancecovariance matrix gained through shrinkage theory. An added effect of the shrinkage theory estimate is to attempt to correct the estimation errors that come with financial data due to the high dimensionality property it possesses. The third is to apply and compare a selection of metaheuristics using the adjusted model on the portfolio optimisation problem to see if they provide a usable alternate technique for real world datasets. The meta-heuristics used in this paper are Simulated Annealing (SA), the Artificial Bee Colony (ABC), and the Pareto Envelop-based Selection Algorithm (PESA). A collection of unit trusts were collected and evaluated, before using the listed meta-heuristics to find good solutions to the unit trusts selection problem. This would allow ordinary investors to have access to a diversified portfolio, whose risk may be lowered even further by diversifying between unit trusts. The shrinkage theory estimate was successfully applied to overcome the second problem and preliminary results indicate that there may be some benefit to using the new estimate as it may provide more accurate portfolio covariances and lead to more assured returns in the future. The solutions to two of the meta-heuristics, namely the ABC and the PESA, were found to be within an acceptable range of the true efficient set of portfolio for the data set, while the SA results were not successful. The solutions were all found within a relatively usable time period, namely 2 to 4 hours, and can be concluded to be of use for solving the portfolio optimisation problem for larger data sets in a usable time period.

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