# American Institute of Mathematical Sciences

2012, 8(2): 343-362. doi: 10.3934/jimo.2012.8.343

## Robust portfolio selection with a combined WCVaR and factor model

 1 Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

Received  March 2011 Revised  August 2011 Published  April 2012

In this paper, a portfolio selection model with a combined Worst-Case Conditional Value-at-Risk (WCVaR) and Multi-Factor Model is proposed. It is shown that the probability distributions in the definition of WCVaR can be determined by specifying the mean vectors under the assumption of multivariate normal distribution with a fixed variance-covariance matrix. The WCVaR minimization problem is then reformulated as a linear programming problem. In our numerical experiments, to compare the proposed model with the traditional mean variance model, we solve the two models using the real market data and present the efficient frontiers to illustrate the difference. The comparison reveals that the WCVaR minimization model is more robust than the traditional one in a market recession period and it can be used in a long-term investment.
Citation: Ke Ruan, Masao Fukushima. Robust portfolio selection with a combined WCVaR and factor model. Journal of Industrial & Management Optimization, 2012, 8 (2) : 343-362. doi: 10.3934/jimo.2012.8.343
##### References:
 [1] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk,, Mathematical Finance, 9 (1999), 203. doi: 10.1111/1467-9965.00068. [2] T. S. Beder, VAR: Seductive but dangerous,, Financial Analysts Journal, 51 (1995), 12. doi: 10.2469/faj.v51.n5.1932. [3] A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs,, Operations Research Letter, 25 (1999), 1. doi: 10.1016/S0167-6377(99)00016-4. [4] L. El Ghaoui, M. Oks and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach,, Operations Research, 51 (2003), 543. doi: 10.1287/opre.51.4.543.16101. [5] F. J. Fabozzi, D. Huang and G. Zhou, Robust portfolios: Contributions from operations research and finance,, Annals of Operations Research, 176 (2010), 191. doi: 10.1007/s10479-009-0515-6. [6] E. F. Fama, Efficient capital markets: A review of theory and empirical work, in "Frontiers of Quantitative Economics" (Invited Papers, Econometric Soc. Winter Meetings, New York, 1969),, Contributions to Economic Analysis, (1971), 309. [7] E. F. Fama, Efficient capital markets: II,, Journal of Finance, 46 (1991), 1575. doi: 10.2307/2328565. [8] E. F. Fama and K. R. French, Common risk factors in the returns on stocks and bonds,, Journal of Financial Economics, 33 (1993), 3. doi: 10.1016/0304-405X(93)90023-5. [9] D. Goldfarb and G. Iyengar, Robust portfolio selection problems,, Mathematics of Operations Research, 28 (2003), 1. doi: 10.1287/moor.28.1.1.14260. [10] A. Kreinin, L. Merkoulovitch, D. Rosen and Z. Michael, Measuring portfolio risk using quasi Monte Carlo methods,, Algo Research Quarterly, 1 (1998), 17. [11] J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets,, The Review of Economics and Statistics, 47 (1956), 13. doi: 10.2307/1924119. [12] H. M. Markowitz, Portfolio selection,, Journal of Finance, 7 (1952), 77. doi: 10.2307/2975974. [13] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk,, Journal of Risk, 2 (2000), 21. [14] R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distribution,, Journal of Banking and Finance, 26 (2002), 1443. doi: 10.1016/S0378-4266(02)00271-6. [15] S. A. Ross, The arbitrage theory of capital asset pricing,, Journal of Economic Theory, 13 (1976), 341. doi: 10.1016/0022-0531(76)90046-6. [16] W. F. Sharp, Capital asset prices: A theory of market equilibrium under conditions of risk,, Journal of Finance, 19 (1964), 425. doi: 10.2307/2977928. [17] S. Zhu and M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management,, Operations Research, 57 (2009), 1155. doi: 10.1287/opre.1080.0684.

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##### References:
 [1] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk,, Mathematical Finance, 9 (1999), 203. doi: 10.1111/1467-9965.00068. [2] T. S. Beder, VAR: Seductive but dangerous,, Financial Analysts Journal, 51 (1995), 12. doi: 10.2469/faj.v51.n5.1932. [3] A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs,, Operations Research Letter, 25 (1999), 1. doi: 10.1016/S0167-6377(99)00016-4. [4] L. El Ghaoui, M. Oks and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach,, Operations Research, 51 (2003), 543. doi: 10.1287/opre.51.4.543.16101. [5] F. J. Fabozzi, D. Huang and G. Zhou, Robust portfolios: Contributions from operations research and finance,, Annals of Operations Research, 176 (2010), 191. doi: 10.1007/s10479-009-0515-6. [6] E. F. Fama, Efficient capital markets: A review of theory and empirical work, in "Frontiers of Quantitative Economics" (Invited Papers, Econometric Soc. Winter Meetings, New York, 1969),, Contributions to Economic Analysis, (1971), 309. [7] E. F. Fama, Efficient capital markets: II,, Journal of Finance, 46 (1991), 1575. doi: 10.2307/2328565. [8] E. F. Fama and K. R. French, Common risk factors in the returns on stocks and bonds,, Journal of Financial Economics, 33 (1993), 3. doi: 10.1016/0304-405X(93)90023-5. [9] D. Goldfarb and G. Iyengar, Robust portfolio selection problems,, Mathematics of Operations Research, 28 (2003), 1. doi: 10.1287/moor.28.1.1.14260. [10] A. Kreinin, L. Merkoulovitch, D. Rosen and Z. Michael, Measuring portfolio risk using quasi Monte Carlo methods,, Algo Research Quarterly, 1 (1998), 17. [11] J. Lintner, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets,, The Review of Economics and Statistics, 47 (1956), 13. doi: 10.2307/1924119. [12] H. M. Markowitz, Portfolio selection,, Journal of Finance, 7 (1952), 77. doi: 10.2307/2975974. [13] R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk,, Journal of Risk, 2 (2000), 21. [14] R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distribution,, Journal of Banking and Finance, 26 (2002), 1443. doi: 10.1016/S0378-4266(02)00271-6. [15] S. A. Ross, The arbitrage theory of capital asset pricing,, Journal of Economic Theory, 13 (1976), 341. doi: 10.1016/0022-0531(76)90046-6. [16] W. F. Sharp, Capital asset prices: A theory of market equilibrium under conditions of risk,, Journal of Finance, 19 (1964), 425. doi: 10.2307/2977928. [17] S. Zhu and M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management,, Operations Research, 57 (2009), 1155. doi: 10.1287/opre.1080.0684.
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