Ph. D. thesis summary. by Aleksandra Rutkowska, M. Sc. Stock portfolio optimization in the light of Liu s credibility theory

Transcription

1 Ph. D. thesis summary by Aleksandra Rutkowska, M. Sc. Stock portfolio optimization in the light of Liu s credibility theory The thesis concentrates on the concept of stock portfolio optimization in the light of Liu s credibility theory. Axiomatic credibility theory, which is an expansion of the possibility theory 1, was published in The theory is based on the concept of credibility measure, proposed in as a measure of a particular event happening. The goal of this work is to synthesize elements of the fuzzy variable theory for the needs of portfolio analysis. The hitherto publications regarding portfolio analysis commonly leave out elements that are important from the science and investor s points of view. This causes incoherence in the theory and difficulties in empirical applications. The following auxiliary goals are important for meeting the work s main goal: 1. Classify uncertainty in a consensual approach. 2. Determine methodology for plotting membership and credibility functions of a fuzzy variable. 3. Propose a uni-criterion, intuitive optimization model. 4. Compare and analyze portfolio optimization fuzzy tasks. The first goal was realised through literature studies. They were a basis for formulating an answer to the following question: can a coherent typology for the concepts of uncertainty and risk be achieved on the basis of available literature? In order to meet the second goal, the following auxiliary research questions were asked: 1. How to set a membership function of a fuzzy return rate? 2. Does the shape of membership functions influence results of portfolio optimization tasks? 3. Does the approximation of different shapes with a linear shape significantly influence the change of choosing stocks for a given portfolio? Reaching the third goal will be possible after answering the following questions: 1. How do investors choose stocks for their portfolio? 1 Proposed by Zadeh in 1978 ( Zadeh, L., 1978, Fuzzy Sets as the Basis for a Theory of Possibility, Fuzzy Sets and Systems 1:3 28, republished in Fuzzy Sets and Systems 100 (Supplement): 9 34, Liu, B., 2004, Uncertainty Theory: An Introduction to Its Axiomatic Foundations, Studies in Fuzziness and Soft Computing, Springer, Berlin. 3 Liu, B., Liu, Y.-K., 2002, Expected value of fuzzy variable and fuzzy expected value models, Fuzzy Systems, IEEE Transactions on, vol. 10, nr 4, s

2 2. Can a uni-criterion optimization task be constructed? 3. How to reflect investor s preferences in an optimization task? The fourth goal will be realised through answering the following auxiliary questions: 1. Which of the optimization tasks are equivalent and under what assumptions? 2. How to evaluate the effectiveness of optimization tasks and which criteria to use? 3. Can it be determined which of the proposed tasks is more effective? 4. Are the optimization tasks that take investor preferences under account as effective as those based on uncertainty and return measures? The first part presents theoretical frames of a fuzzy return rate defined by credibility and elaborates characteristics of optimization tasks. The starting points were the ways of modelling uncertainty and different approaches towards portfolio optimization problems connected with it. Topics regarding the rate of return and investment uncertainties were presented. The uncertainty typology was discussed and an attempt to synthesize classifications presented in literature made. Then, the concept of a fuzzy return rate in light of Liu s credibility theory was characterized. The theoretical basics of a fuzzy variable were presented and a concept of a fuzzy rate of return including uncertainty measures was introduced. A fuzzy variable is defined as follows. Let Θ be a non-empty set, P(Θ) the power of the set Θ and Cr a credibility measure. Then (Θ, P(Θ), Cr) is called a credibility space and ξ fuzzy variable a function from the credibility space into a set of real numbers. Through a fuzzy rate of return we understand a function plotted on a credibility space returning the value of a future rate of return. A membership function of a fuzzy variable is plotted on a set of future rate values, allowing each of the values to happen. Cr credibility function of a fuzzy variable is set from a subset of elementary events into a set of real numbers and defines the credibility of an event, that the variable adopts a value from a given section. This part also proved that fuzzy rates of return are independent and discussed the problem of determining a fuzzy variable that comes to the concept of determining a membership function of a fuzzy set. The work concentrates on and compares three types of transformations based on statistical data: 1. a method proposed by Dubois and Prade 4 called bijective transformation method, assuming that the level of necessity of event A i is the probability value connected to elementary events from set A i in comparison to the probability value attributed to the most common event not in set A i. 4 D. Dubois, H. Prade. 1983, Unfair coins and necessity measures: Towards a possibilistic interpretation of histograms. Fuzzy Sets and Systems, 10(1 3):15 20,.

3 2. Klir s concept 5, called conservation of uncertainty method, assuming that when transforming measures from one theory to another, the uncertainty value should be kept and each particular values in on theory must be transformed to their equivalents in the second theory using some scale. 3. author s method based on the principle of maximum weighted entropy, assuming that all available information need to be considered while additional information should be omitted. The next part covers optimization tasks that are based on the fuzzy variable concept in the light of credibility theory. Together with the credibility theory developed numerous financial applications surfaced, especially dynamic models of portfolio optimization. In 2005 first models based on the credibility measure were proposed: mean-variance model, optimistic value model and maximum credibility model 6. In 2006 Huang 7 developed the concept of maximizing the credibility measure. In acceptability curve of investment loss and then fuzzy variable entropy 9 were introduced, in order to limit the risk. The mean-variance model evolved into the minmax mean-variance model 10 and mean-variance-skewness model saw the presentation of the regret minimization model 12. The tasks were divided into those using uncertainty measures and those using the investor s preferences and ratio measures. Examples for particular tasks were also presented. The second group presents the author s optimization model based on the satisfaction measure. In this case, by satisfaction we understand the similarity of a return result to expectations, while ex ante, it will be the similarity of the investor s expectations, plotted with a fuzzy set, to the credibility of particular return rates from different portfolios happening. Using fuzzy sets will allow us to 5 G.J. Klir. 1990, A principle of uncertainty and information invariance. International Journal of General Systems, 17: J. Peng, H. M. K. Mok, W.-M. Tse, 2005, Credibility programming approach to fuzzy portfolio selection problems, Proceedings of 2005 International Conference on Machine Learning and Cybernetics, X. Huang, 2006, Credibility Based Fuzzy Portfolio Selection. IEEE International Conference on Fuzzy Systems, X. Huang, 2006, Fuzzy chance-constrained portfolio selection. Applied Mathematics and Computation, 177(2): X. Huang, 2008, Risk curve and fuzzy portfolio selection. Computers & Mathematics with Applications, 55(6): , 9 X. Huang, 2008, Mean-Entropy Models for Fuzzy Portfolio Selection. IEEE Transactions on Fuzzy Systems, 16: , 10 X. Huang, 2010, Minimax mean-variance models for fuzzy portfolio selection. Soft Computing, 15(2): , X. Li, Z. Qin, S. Kar, 2010, Mean-variance-skewness model for portfolio selection with fuzzy returns. European Journal of Operational Research, 202(1): , 12 X. Li, B. Shou, Z. Qin. An expected regret minimization portfolio selection model. European Journal of Operational Research, 218(2): , 2012.

4 determine preferences and using the similarity index proposed by Tverski 13 the introduction of a risk aversion and regret parameters determining the importance of an investment s safety or concentrating on high returns. So, for an investor who concentrates only on profits, with no risk aversion, the similarity function will only check the level of realizing expectations, considering the common part and unmet expectations. In an opposite case, for an investor with a maximum risk aversion, the expectations for a given portfolio will be considered together with the risk of a result below expectations. The research part of the thesis consists of 4 studies. Two of them a study of sensitivity of optimization results to a change in shape of the membership function and a study of the influence of the used membership function plotting method on a determined rate of return are a preliminary research. The aim of the first study is to check whether a triangular shape of a fuzzy variable membership function is an adequate approximation of fuzzy rates of return. The study covers 4 different membership function shapes: triangular, parabolic, normal and the SZ shape. The shapes will be studied in two cases: with an assumption of fixed function parameters and a fixed field under the plot. The second research aimed at checking whether the parameters of the generated fuzzy variable membership function differ significantly depending on the used method of plotting. Preliminary results allowed for the following conclusions. Approximation of a membership function with a triangular shape is justifiable in case of the SZ shape. In case of other shapes, approximation mistakes depend on the considered task type and assumptions regarding field area or task parameters. Results from the study s second part show that values of discrete membership functions plotted with methods based on historical data are similar and that linear approximation of continuous functions decreases differences. On the basis of preliminary results a main empirical research was carried out; it consisted of determining and testing portfolios chosen with different tasks over 22 test periods of different length: monthly (preceded with a quarterly market observation period), quarterly (preceded with a half-yearly observation period), half-yearly (preceded with a yearly observation period), yearly (preceded with a half-yearly period of data observation). The research includes the following tasks: - mean-variance task: the task of minimizing variance with an assumed expected value level and the task of maximizing expected value with an assumed variance level (EV), 13 A. Tversky, 1977, Features of similarity, Psychological Review, 84:

5 - mean-semivariance task: the task of minimizing semivariance with an assumed expected value level and the task of maximizing expected value with an assumed semivariance level, - mean-entropy task: the task of minimizing entropy with an assumed expected value level and the task of maximizing expected value with an assumed entropy level, - mean-var task: the task of minimizing value at risk with an assumed expected value level and the task of maximizing expected value with an assumed value at risk level, - mean-cvar task: the task of minimizing conditional value at risk with an assumed expected value level and the task of maximizing expected value with an assumed conditional value at risk level, - maximizing satisfaction (S) for three different levels of risk aversion and regret parameters. Mean return rates of portfolios set using the optimization tasks did not differ significantly from the benchmark level. Moreover, it can be noted that return rates of portfolios set with optimization tasks showed, on average, better results than those of portfolios set with the uncertainty minimization method. On average, return rates of portfolios chosen with optimization tasks over shorter test periods were higher than the return rates of portfolios chosen and testes over longer periods. The fourth part is independent and covers a survey research among individual investors, concentrating on portfolio selection methods, ways of determining criteria and preferences. The survey was carried out among over 300 individual investors through the Association of Individual Investors. As the surveys show, the investors, when estimating a portfolio s risk mainly use intuition and experts opinions, while every 7 th person does not evaluate risk at all. The least popular analysis, even among people with higher education, is the portfolio analysis almost a half does not use it at all or rarely. It can suggest a low knowledge of the method or its insignificant usefulness. The survey showed a low percentage of people estimating or checking risk measures with low evaluations of their interpretation, particularly the ones regarding semi-variance and value at risk measures. However, over a half of the investors tries to (always or often) determine the expected value of the investment. The investment results are most commonly compared with the assumed goals the investors do not consider the best or worst market results of a particular period. The results they expect are determined within a given section and depend on the market conditions at the moment of making an investment decision.

6 The last part of the thesis presents a synthetic evaluation of optimization task, based on the author s measure, considering the following criteria: - intuitive and usefulness of optimization criteria, - task s sensitivity to membership functions, - calculation complexity, - results from real data. The tasks of maximizing expected value while limiting value at risk, semivariance and conditional value at risk got the highest evaluation. The tasks using value at risk had the best evaluation due to the informational usefulness of uncertainty measures and low calculation complexity; minor deviations below the WIG20 index during loss periods also had good evaluation. The task of maximizing expected value while limiting uncertainty with semivariance measure had high evaluation thanks to both, low sensitivity to the change of shape of the membership function as well as due to the informational usefulness; but most of all it had the best results in empirical effectiveness on the basis of return rates from during test periods. The proposed task of maximizing satisfaction was highly evaluated due to the compatibility of the optimization criterion and intuition, however it is depreciated due to a lack of information on portfolio uncertainty, calculation complexity and did not achieve satisfying empirical results during loss periods. The task of minimizing entropy, despite low sensitivity to the change in membership function s shape and calculation simplicity, was given the lowest evaluation. The ending of the thesis presents its most important results, conclusions and possible future research destinations.