Bi-Criteria Portfolio Optimization Models with Percentile and Symmetric Risk Measures by Mathematical Programming

• Autorzy: Sawik B.

Warianty tytułu

PL

Zaimplementowane metodami programowania matematycznego dwukryterialne modele optymalizacji portfelowej z percentylowymi oraz symetrycznymi miarami ryzyka

• Języki publikacji: EN

Abstrakty

EN

The purpose of this paper is to compare three different bi-criteria portfolio optimization models. The first model is constructed with the use of percentile risk measure Value-at-Risk and solved by mixed integer programming. The second one is constructed with the use of percentile risk measure Conditional Value-at-Risk and solved by linear programming. The third one is constructed with the use of a symmetric measure of risk - variance of return - as in the Markowitz portfolio and solved by quadratic programming. Computational experiments are conducted for bi-criteria portfolio stock exchange investments. The results obtained prove, that the bi-objective portfolio optimization models with Value-at-Risk and Conditional Value-at-Risk could be used to shape the distribution of portfolio returns. The decision maker can assess the value of portfolio return and the risk level, and can decide how to invest in a real life situation comparing with ideal (optimal) portfolio solutions. The proposed scenario-based portfolio optimization problems under uncertainty, formulated as a bi-objective linear, mixed integer or quadratic program are solved using commercially available software (AMPL/CPLEX) for mathematical programming.

PL

W pracy zaprezentowano problem portfelowy, dla którego zaproponowane zostały trzy dwukryterialne modele optymalizacji. W kazdym z zaimplementowanych modeli funkcja celu jest wazona suma dwóch kryteriów, które podlegają optymalizacji dla konkretnego portfela. W pierwszym modelu z warunkową wartoscią zagrozoną zwrotu (Conditional Value-at-Risk) kryteria decyzyjne to maksymalizacja CVaR i przewidywanego zwrotu portfela. W drugim modelu z wartoscią zagrozoną zwrotu (Value-at-Risk) minimalizowane jest prawdopodobienstwo ryzyka straty oraz maksymalizacja oczekiwanego zwrotu z portfela. Trzeci model to omawiany szeroko w literaturze model portfela Markowitza zmodyfikowany tak, by funkcja kryterialna była wazona sumą kryteriów i minimalizowała ryzyko portfela zdefiniowanego jako macierz kowariancji historycznych zwrotów oraz maksymalizowała przewidywany zwrot portfela. Modele te zostały zaimplementowane: w pierwszym - uzywając metody programowania mieszanego całkowitoliczbowego, a w drugi i trzeci z uzyciem programowania liniowego i kwadratowego. Efektywnosc zaproponowanych modeli została zweryfikowana eksperymentalnie, zwracając szczególną uwagę na czas obliczen oraz przewidywany zwrot portfela. Zamieszczono wyniki eksperymentów obliczeniowych przeprowadzonych z zastosowaniem optymizatora CPLEX i jezyka modelowania algebraicznego AMPL.

Słowa kluczowe

EN

multi-criteria decision making; portfolio optimization; percentile and symmetric risk measures; weighting approach; mathematical programming

PL

wielokryterialne podejmowanie decyzji; optymalizacja portfelowa; percentylowe i symetryczne miary ryzyka; waźona funkcja celu; programowanie matematyczne

• Wydawca: Wydawnictwo SIGMA-NOT
• Czasopismo: Przegląd Elektrotechniczny
• Rocznik: 2012
• Tom: R. 88, nr 10b
• Strony: 176--180
• Opis fizyczny: Bibliogr. 26 poz., rys.

Twórcy

autor

Sawik B.

• Department of Applied Computer Science Faculty of Management AGH University of Science & Technology Al. Mickiewicza 30, 30-059 Kraków,

Bibliografia

• [1] Markowitz H.M.: Portfolio selection, Journal of Finance, 7, pp. 77-91, 1952.
• [2] Alexander G.J., Baptista A.M.: A Comparison of VaR and CVaR Constraints on Portfolio Selection with the Mean-Variance Model, Management Science, 50(9), pp. 1261–1273, 2004.
• [3] Benati S., Rizzi R.: A mixed integer linear programming formulation of the optimal mean/Value-at-Risk portfolio problem, European Journal of Operational Research, 176, pp. 423–434, 2007.
• [4] Mansini R., Ogryczak W., Speranza M.G.: Conditional value at risk and related linear programming models for portfolio optimization, Annals of Operations Research, 152, pp. 227–256, 2007.
• [5] Ogryczak W.: Multiple criteria linear programming model for portfolio selection, Annals of Operations Research, 97, pp. 143–162, 2000.
• [6] Sawik B.: Downside Risk Approach for Multi-Objective Portfolio Optimization, In: Klatte D., Lüthi H.-J., Schmedders K. (Eds.), Operations Research Proceedings 2011, Operations Research Proceedings, Springer-Verlag, Berlin, Heidelberg, Germany, pp. 191–196, 2012.
• [7] Sawik B.: Conditional Value-at-Risk vs. Value-at-Risk to Multi-Objective Portfolio Optimization, In: Lawrence K.D., Kleinman G. (Eds.), Applications of Management Science, Volume 15, Emerald Group Publishing Limited, Bingley, UK, Vol. 15, Ch. 13, pp. 277–305, 2012.
• [8] Sawik B.: A Review of Multi-Criteria Portfolio Optimization by Mathematical Programming, In: Dash G.H., Thomaidis N. (Eds.), Recent Advances in Computational Finance, NovaScience Publishers, New York, USA, (forthcomming 2012).
• [9] Sawik B.: Conditional value-at-risk and value-at-risk for portfolio optimization model with weighting approach, Automatyka, AGH, 15(2), pp. 429–434, 2011.
• [10] Sawik B.: A Bi-Objective Portfolio Optimization with Conditional Value-at-Risk, Decision Making in Manufacturing and Services, 4(1-2), pp. 47–69, 2010.
• [11] Sawik B.: Selected Multi-Objective Methods for Multi-Period Portfolio Optimization by Mixed Integer Programming, In: Lawrence K.D., Kleinman G. (Eds.), Applications of Management Science, Volume 14, Applications in Multi-Criteria Decision Making, Data Envelopment Analysis and Finance, Emerald Group Publishing Limited, Bingley, UK, Vol. 14, Ch. 1, pp. 3–34, 2010.
• [12] Sawik B.: A Reference Point Approach to Bi-Objective Dynamic Portfolio Optimization, Decision Making in Manufacturing and Services, 3(1-2), pp. 73–85, 2009.
• [13] Sawik B.: Lexicographic and Weighting Approach to Multi-Criteria Portfolio Optimization by Mixed Integer Programming, In: Lawrence K.D., Kleinman G. (Eds.), Applications of Management Science, Volume 13, Financial Modeling Applications and Data Envelopment Applications, Emerald Group Publishing Limited, Bingley, UK, Vol. 13, Ch. 1, pp. 3–18, 2009.
• [14] Sawik B.: A weighted-sum mixed integer program for biobjective dynamic portfolio optimization, Automatyka, AGH, 13(2), pp. 563–571, 2009.
• [15] Sawik B.: A Three Stage Lexicographic Approach for Multi-Criteria Portfolio Optimization by Mixed Integer Programming, Przegla˛d Elektrotechniczny (Electrical Review), 84(9), pp. 108-112, 2008.
• [16] Speranza M.G.: Linear programming models for portfolio optimization, Finance, 14, pp. 107-123, 1993.
• [17] Rockafellar R.T., Uryasev S.: Conditional value-at-risk for general loss distributions, The Journal of Banking and Finance, 26, pp. 1443-1471, 2002.
• [18] Sarykalin S., Serraino G., Uryasev S.: Value-at- Risk vs. Conditional Value-at-Risk in Risk Management and Optimization, In: Chen Z-L., Raghavan S., Gray P. (Eds.), Tutorials in Operations Research, INFORMS Annual Meeting, Washington D.C., USA, October 12–15, 2008.
• [19] Uryasev S.: Conditional value-at-risk: optimization algorithms and applications, Financial Engineering News, Issue 14, February 2000.
• [20] Rockafellar R.T. , Uryasev S.: Optimization of conditional value-at-risk, The Journal of Risk, 2(3), pp. 21-41, 2000.
• [21] Steuer R.E.: Multiple Criteria Optimization: Theory, Computation and Application, A John Wiley & Sons, New York, USA, 1986.
• [22] Alves M.J., Climaco J.: A review of interactive methods for multiobjective integer and mixed-integer programming, European Journal of Operational Research, 180, pp. 99-115, 2007.
• [23] Sawik B.: Weighting vs. Lexicographic Approach to Multi-Objective Portfolio Optimization by Mixed Integer Programming, In: Kochan E. (Ed.), Problems of Mechanical Engineering and Robotics, Monographic of Faculty of Mechanical Engineering and Robotics, AGH, 36, pp. 201–208, 2007.
• [24] Crescenzi P., Kann V. (Eds. ): A compendium of NP optimization problems, [web page] http://www.nada.kth.se/~viggo/wwwcompendium/, 2005.
• [25] Merris R.: Combinatorics, Wiley-Interscience, A John Wiley & Sons, Inc., Publication, Hoboken, New Jersey, USA, 2003.
• [26] Nemhauser G.L., Wolsey L.A.: Integer and Combinatorial Optimization, A John Wiley & Sons, Toronto, Canada, 1999.