FUZZY PORTFOLIO SELECTION VIA RANKING MODELS IN DEA AND MULTI-CRITERIA DECISION MAKING

Abstract

A portfolio selection problem implemented through an optimization technique is called portfolio optimization. The mathematical model for portfolio optimization al- locates total capital among various assets following investors’ preferences about return/risk. Generally, investors seek to reduce risk while enhancing returns, yet attaining higher expected returns inevitably involves accepting greater levels of risk. Therefore, an investor faces a trade-off between risk and expected return. Hence, portfolio optimization is a technique used to construct an optimal basket of assets, where an optimal is understood in the context of an investor’s objec- tives and desires. In many real-world situations, the return from an asset cannot be anticipated accurately based on historical data. The presence of vagueness and fuzziness in the input and output data can not be resolved by using proba- bility theory. The unpredictable dynamic nature of the financial market motivates researchers to use the concept of fuzzy set theory in the field of portfolio selec- tion. The possibility theory is an uncertainty theory devoted to the handling of incomplete information. Besides an accurate determination of a risk measure of a return distribution, investors also wish to evaluate the performance of their portfolios concerning a benchmark index or to rank different portfolio strategies. Generally, the role of an asset’s performance in optimal portfolio construction has not been considered so far. When selecting assets for a portfolio, an investor considers several fac- tors. Data Envelopment Analysis (DEA) simultaneously accommodates multiple inputs and outputs, providing a composite efficiency score. As DEA measures the relative efficiency of several similar processing units, it also helps in asset selection before portfolio construction. However, the DEA allows each financial asset to evaluate its efficiency relative to other homogeneous financial assets by assigning favorable weights. This often results in unrealistic weight schemes. To address this issue, the DEA cross-efficiency framework is employed, which elim- v inates such unrealistic weight allocations. In financial markets, assets compete for higher efficiency scores, often leading to multiple optimal weights in standard cross-efficiency. DEA game cross-efficiency introduces a noncooperative frame- work where competing assets jointly determine balanced weights, reducing non- uniqueness and producing more stable and fair rankings for portfolio selection. In certain instances, DEA models may yield an efficiency score of one for sev- eral decision-making units (DMUs), making it challenging to rank these DMUs. Further, in DEA, every approach uses a distinct theory and framework to rank the DMUs, so each DMU has a different ranking order. The decision-maker’s reliability of the results is a critical consideration when choosing a ranking sys- tem. Multi-Criteria Decision Making (MCDM) approaches, which differ from DEA models, can be used to solve the problem of ranking efficient DMUs. The challenge of aggregating self- and peer-evaluated cross-efficiencies into a final score has been widely discussed. Notably, using a simple arithmetic average inherently assumes that all DMUs’ evaluations are equally valid or dependable, which is not always true. To address this issue, we propose a novel use of the Ordered Visibility Graph Averaging (OVGA) operator for more meaningful aggre- gation. Furthermore, in the same work, we introduce a portfolio selection model for constructing the most efficient optimal portfolio. In DEA game cross-efficiency, the final score is obtained through an iterative algorithm, where each iteration aggregates the game cross-efficiency scores of all DMUs using the arithmetic averaging method. This thesis presents the OVGA aggregated DEA game cross-efficiency method, which considers the competition among DMUs in portfolio selection. These Game cross-efficiency scores serve as a tool for efficient portfolio selection. Further, a multi-objective portfolio selection model is proposed, where the Maverick index and variance of cross-efficiency are treated as risk metrics, and the OVGA game cross-efficiency scores are used as return characteristics. The Semi-oriented radial measure (SORM) model of DEA effectively handles the negative input-output data. However, it has a limitation of producing negative cross-efficiencies. We propose a modified SORM model to deal with this issue. Also, a novel multi-objective portfolio selection model is introduced, using the maverick index to represent risk and the diversity index to represent return. The maverick index is calculated using the column average of the cross-efficiency vi matrix, while the diversity index is determined using the row average. In another research study in this thesis, an innovative approach is introduced to portfolio selection derived from the RDM cross-efficiency matrix. In practical appli- cations, the column average of the cross-efficiency matrix is commonly employed for decision-making, as it helps identify efficient and consistent performers. How- ever, the row average also provides valuable insight into how fairly or aggressively each DMU evaluates its peers. We provide a method for categorization of the as- sets, which utilizes both row and column averages of the RDM cross-efficiency matrix. An essential aspect of investment management is the unique rating of portfolios, which enables investors to identify and assess the most effective portfolios based on criteria such as risk, return, etc. We present a hybrid approach for ranking investment portfolios by combining the Modified Slack-Based Measure (MSBM) of DEA with a multi-criteria decision-making method. Techniques like the MSBM and TOPSIS incorporate traditional performance metrics while adding flexibility to address fuzzy environments and handle imprecise data. aims to evaluate fuzzy portfolios using the MSBM model, with trapezoidal fuzzy numbers for returns and possibilistic measures for risk and mean return. Efficient portfolios are further ranked using the TOPSIS technique. This thesis entitled “Fuzzy Portfolio Selection via Ranking Models in DEA and Multi-criteria Decision Making” aims to highlight the advantages of DEA as an innovative tool for portfolio optimization, contributing to the development of more robust and efficient investment strategies. The methodologies introduced and developed in this thesis are rigorously tested on real-world case studies, demon- strating their practical applicability and effectiveness in enhancing portfolio selec- tion processes.