Abstract
Many risk measures have been recently introduced which (for discrete random variables) result in Linear Programs (LP). While some LP computable risk measures may be viewed as approximations to the variance (e.g., the mean absolute deviation or the Gini’s mean absolute difference), shortfall or quantile risk measures are recently gaining more popularity in various financial applications. In this paper we study LP solvable portfolio optimization models based on extensions of the Conditional Value at Risk (CVaR) measure. The models use multiple CVaR measures thus allowing for more detailed risk aversion modeling. We study both the theoretical properties of the models and their performance on real-life data.
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References
Acerbi, C. (2002).“Spectral Measures of Risk: A Coherent Representation of Subjective Risk Aversion.” Journal of Banking & Finance, 26, 1505–1518.
Acerbi, C. and P. Simonetti. (2002). “Portfolio Optimization with Spectral Measures of Risk.” Working Paper (http://gloriamundi.org).
Andersson, F., H. Mausser, D. Rosen, and S. Uryasev. (2001). “Credit Risk Optimization with Conditional Value-at-Risk Criterion.” Mathematical Programming, 89, 273–291.
Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. (1999). “Coherent Measures of Risk.” Mathematical Finance, 9, 203–228.
Mansini, R. and M.G. Speranza. (2005). “An Exact Approach for the Portfolio Selection Problem with Transaction Costs and Rounds.” IIE Transactions, 37, 919–929.
Chiodi, L., R. Mansini, and M.G. Speranza. (2003). “Semi-Absolute Deviation Rule for Mutual Funds Portfolio Selection.” Annals of Operations Research, 124, 245–265.
Embrechts, P., C. Klüppelberg, and T. Mikosch. (1997). Modelling Extremal Events for Insurance and Finance. New York: Springer-Verlag.
Haimes, Y.Y. (1993). “Risk of Extreme Events and the Fallacy of the Expected Value.” Control and Cybernetics, 22, 7–31.
Jorion, P. (2001). Value-at-Risk: The New Benchmark for Managing Financial Risk. NY: McGraw-Hill.
Kellerer, H., R. Mansini, and M.G. Speranza. (2000). “Selecting Portfolios with Fixed Costs and Minimum Transaction Lots.” Annals of Operations Research, 99, 287–304.
Konno, H., and H. Yamazaki. (1991). “Mean-Absolute Deviation Portfolio Optimization Model and Its Application to Tokyo Stock Market.” Management Science, 37, 519–531.
Konno, H., and A. Wijayanayake. (2001). “Portfolio Optimization Problem under Concave Transaction Costs and Minimal Transaction Unit Constraints.” Mathematical Programming, 89, 233–250.
Levy, H., and Y. Kroll. (1978). “Ordering Uncertain Options with Borrowing and Lending.” Journal of Finance, 33, 553–573.
Mansini, R., W. Ogryczak, and M.G. Speranza. (2003a). “On LP Solvable Models for Portfolio Selection.” Informatica, 14, 37–62.
Mansini, R., W. Ogryczak, and M.G. Speranza. (2003b). “LP Solvable Models for Portfolio Optimization: A Classification and Computational Comparison.” IMA J. of Management Mathematics, 14, 187–220.
Mansini, R., W. Ogryczak, and M.G. Speranza. (2003c). “Conditional Value at Risk and Related Linear Programming Models for Portfolio Optimization.” Tech. Report 03–02, Warsaw Univ. of Technology.
Mansini, R., and M.G. Speranza. (1999). “Heuristic Algorithms for the Portfolio Selection Problem with Minimum Transaction Lots.” European J. of Operational Research, 114, 219–233.
Markowitz, H.M. (1952). “Portfolio Selection.” Journal of Finance, 7, 77–91.
Ogryczak, W. (1999). “Stochastic Dominance Relation and Linear Risk Measures.” In A.M.J. Skulimowski (ed.), Financial Modelling—Proceedings of the 23rd Meeting of the EURO Working Group on Financial Modelling. Cracow: Progress & Business Publ., 191–212.
Ogryczak, W. (2000). “Multiple Criteria Linear Programming Model for Portfolio Selection.” Annals of Operations Research, 97, 143–162.
Ogryczak, W. (2002). “Multiple Criteria Optimization and Decisions under Risk.” Control and Cybernetics, 31, 975–1003.
Ogryczak, W. and A. Ruszczyński. (1999). “From Stochastic Dominance to Mean-Risk Models: Semideviations as Risk Measures.” European J. of Operational Research, 116, 33–50.
Ogryczak, W. and A. Ruszczyński. (2001). “On Stochastic Dominance and Mean-Semideviation Models.” Mathematical Programming, 89, 217–232.
Ogryczak, W. and A. Ruszczyński. (2002a). “Dual Stochastic Dominance and Related Mean-Risk Models.” SIAM J. on Optimization, 13, 60–78.
Ogryczak, W. and A. Ruszczyński. (2002b). “Dual Stochastic Dominance and Quantile Risk Measures.” International Transactions in Operational Research, 9, 661–680.
Ogryczak, W. and A. Tamir. (2003). “Minimizing the Sum of the k-Largest Functions in Linear Time.” Information Processing Letters, 85, 117–122.
Pflug, G.Ch. (2000). “Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk.” In S.Uryasev (ed.), Probabilistic Constrained Optimization: Methodology and Applications, Dordrecht: Kluwer A.P.
Rockafellar, R.T. and S. Uryasev. (2000). “Optimization of Conditional Value-at-Risk.” Journal of Risk, 2, 21–41.
Rockafellar, R.T. and S. Uryasev. (2002). “Conditional Value-at-Risk for General Distributions.” Journal of Banking & Finance, 26, 1443–1471.
Rockafellar, R.T., S. Uryasev, and M. Zabarankin. (2002). “Deviation Measures in Generalized Linear Regression.” Research Report 2002-9, Univ. of Florida, ISE.
Rothschild, M. and J.E. Stiglitz. (1969). “Increasing Risk: I. A Definition.” Journal of Economic Theory, 2, 225–243.
Shalit, H. and S. Yitzhaki. (1994). “Marginal Conditional Stochastic Dominance.” Management Science, 40, 670–684.
Sharpe, W.F. (1971a). “A Linear Programming Approximation for the General Portfolio Analysis Problem.” Journal of Financial and Quantitative Analysis, 6, 1263–1275.
Sharpe, W.F. (1971b). “Mean-Absolute Deviation Characteristic Lines for Securities and Portfolios.” Management Science, 8, B1–B13.
Shorrocks, A.F. (1983). “Ranking Income Distributions.” Economica, 50, 3–17.
Simaan, Y. (1997). “Estimation Risk in Portfolio Selection: The Mean Variance Model and the Mean-Absolute Deviation Model.” Management Science, 43, 1437–1446.
Speranza, M.G. (1993). “Linear Programming Models for Portfolio Optimization.” Finance, 14, 107–123.
Topaloglou, N., H. Vladimirou, and S.A. Zenios. (2002). “CVaR Models with Selective Hedging for International Asset Allocation.” Journal of Banking & Finance, 26, 1535–1561.
Whitmore, G.A., and M.C. Findlay (eds.). (1978). Stochastic Dominance: An Approach to Decision–Making Under Risk. Lexington MA: D.C.Heath.
Yaari, M.E. (1987). “The Dual Theory of Choice under Risk.” Econometrica, 55, 95–115.
Yitzhaki, S. (1982). “Stochastic Dominance, Mean Variance, and Gini’s Mean Difference.” American Economic Revue, 72, 178–185.
Young, M.R. (1998). “A Minimax Portfolio Selection Rule with Linear Programming Solution.” Management Science, 44, 673–683.
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Mansini, R., Ogryczak, W. & Speranza, M.G. Conditional value at risk and related linear programming models for portfolio optimization. Ann Oper Res 152, 227–256 (2007). https://doi.org/10.1007/s10479-006-0142-4
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DOI: https://doi.org/10.1007/s10479-006-0142-4