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

Robust portfolio selection with a combined WCVaR and factor model

Ke Ruan 1, and  Masao Fukushima 1,

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.
Keywords: linear programming., multi-factor model, Portfolio selection, worst-case conditional value-at-risk.
Mathematics Subject Classification: Primary: 90B50, 90C05; Secondary: 90C90, 91G1.
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), 203-228. doi: 10.1111/1467-9965.00068.  Google Scholar

[2]

T. S. Beder, VAR: Seductive but dangerous, Financial Analysts Journal, 51 (1995), 12-24. doi: 10.2469/faj.v51.n5.1932.  Google Scholar

[3]

A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Operations Research Letter, 25 (1999), 1-13. doi: 10.1016/S0167-6377(99)00016-4.  Google Scholar

[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), 543-556. doi: 10.1287/opre.51.4.543.16101.  Google Scholar

[5]

F. J. Fabozzi, D. Huang and G. Zhou, Robust portfolios: Contributions from operations research and finance, Annals of Operations Research, 176 (2010), 191-220. doi: 10.1007/s10479-009-0515-6.  Google Scholar

[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, Vol. 71, North-Holland, Amsterdam, (1971), 309-361.  Google Scholar

[7]

E. F. Fama, Efficient capital markets: II, Journal of Finance, 46 (1991), 1575-1617. doi: 10.2307/2328565.  Google Scholar

[8]

E. F. Fama and K. R. French, Common risk factors in the returns on stocks and bonds, Journal of Financial Economics, 33 (1993), 3-56. doi: 10.1016/0304-405X(93)90023-5.  Google Scholar

[9]

D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Mathematics of Operations Research, 28 (2003), 1-38. doi: 10.1287/moor.28.1.1.14260.  Google Scholar

[10]

A. Kreinin, L. Merkoulovitch, D. Rosen and Z. Michael, Measuring portfolio risk using quasi Monte Carlo methods, Algo Research Quarterly, 1 (1998), 17-26. Google Scholar

[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), 13-37. doi: 10.2307/1924119.  Google Scholar

[12]

H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. doi: 10.2307/2975974.  Google Scholar

[13]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41. Google Scholar

[14]

R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distribution, Journal of Banking and Finance, 26 (2002), 1443-1471. doi: 10.1016/S0378-4266(02)00271-6.  Google Scholar

[15]

S. A. Ross, The arbitrage theory of capital asset pricing, Journal of Economic Theory, 13 (1976), 341-360. doi: 10.1016/0022-0531(76)90046-6.  Google Scholar

[16]

W. F. Sharp, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442. doi: 10.2307/2977928.  Google Scholar

[17]

S. Zhu and M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management, Operations Research, 57 (2009), 1155-1168. doi: 10.1287/opre.1080.0684.  Google Scholar

show all references

References:
[1]

P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk, Mathematical Finance, 9 (1999), 203-228. doi: 10.1111/1467-9965.00068.  Google Scholar

[2]

T. S. Beder, VAR: Seductive but dangerous, Financial Analysts Journal, 51 (1995), 12-24. doi: 10.2469/faj.v51.n5.1932.  Google Scholar

[3]

A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Operations Research Letter, 25 (1999), 1-13. doi: 10.1016/S0167-6377(99)00016-4.  Google Scholar

[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), 543-556. doi: 10.1287/opre.51.4.543.16101.  Google Scholar

[5]

F. J. Fabozzi, D. Huang and G. Zhou, Robust portfolios: Contributions from operations research and finance, Annals of Operations Research, 176 (2010), 191-220. doi: 10.1007/s10479-009-0515-6.  Google Scholar

[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, Vol. 71, North-Holland, Amsterdam, (1971), 309-361.  Google Scholar

[7]

E. F. Fama, Efficient capital markets: II, Journal of Finance, 46 (1991), 1575-1617. doi: 10.2307/2328565.  Google Scholar

[8]

E. F. Fama and K. R. French, Common risk factors in the returns on stocks and bonds, Journal of Financial Economics, 33 (1993), 3-56. doi: 10.1016/0304-405X(93)90023-5.  Google Scholar

[9]

D. Goldfarb and G. Iyengar, Robust portfolio selection problems, Mathematics of Operations Research, 28 (2003), 1-38. doi: 10.1287/moor.28.1.1.14260.  Google Scholar

[10]

A. Kreinin, L. Merkoulovitch, D. Rosen and Z. Michael, Measuring portfolio risk using quasi Monte Carlo methods, Algo Research Quarterly, 1 (1998), 17-26. Google Scholar

[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), 13-37. doi: 10.2307/1924119.  Google Scholar

[12]

H. M. Markowitz, Portfolio selection, Journal of Finance, 7 (1952), 77-91. doi: 10.2307/2975974.  Google Scholar

[13]

R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-41. Google Scholar

[14]

R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distribution, Journal of Banking and Finance, 26 (2002), 1443-1471. doi: 10.1016/S0378-4266(02)00271-6.  Google Scholar

[15]

S. A. Ross, The arbitrage theory of capital asset pricing, Journal of Economic Theory, 13 (1976), 341-360. doi: 10.1016/0022-0531(76)90046-6.  Google Scholar

[16]

W. F. Sharp, Capital asset prices: A theory of market equilibrium under conditions of risk, Journal of Finance, 19 (1964), 425-442. doi: 10.2307/2977928.  Google Scholar

[17]

S. Zhu and M. Fukushima, Worst-case conditional value-at-risk with application to robust portfolio management, Operations Research, 57 (2009), 1155-1168. doi: 10.1287/opre.1080.0684.  Google Scholar

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