General Markowitz Optimization Problems

Affiliation(s)

Department of Mathematical Sciences, University of New Brunswick, Saint John, Canada.

Department of Mathematical Sciences, University of New Brunswick, Saint John, Canada.

ABSTRACT

We solve two Markowitz optimization problems for the one-step financial model with a finite number of assets. In our results, the classical (inefficient) constraints are replaced by coherent measures of risk that are continuous from below. The methodology of proof requires optimization techniques based on functional analysis methods. We solve explicitly both problems in the important case of Tail Value at Risk.

Cite this paper

G. Stoica, "General Markowitz Optimization Problems,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 2038-2040. doi: 10.4236/am.2012.312A281.

G. Stoica, "General Markowitz Optimization Problems,"

References

[1] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, “Coherent Measures of Risk,” Mathematical Finance, Vol. 9, No. 3, 1999, pp. 203-228. doi:10.1111/1467-9965.00068

[2] F. Delbaen, “Coherent Risk Measures on General Probability Spaces,” In: K. Sandmann, et al., Eds., Advances in Finance and Stochastics, Springer-Verlag, Berlin, 2002, pp. 1-37.

[3] G. Stoica, “Relevant Coherent Measures of Risk,” Journal of Mathematical Economics, Vol. 42, No. 6, 2006, pp. 794-806. doi:10.1016/j.jmateco.2006.03.006

[4] L. A. Konovalov, “Coherent Risk Measures and a Limit Pass,” Theory of Probability and its Applications, Vol. 54, No. 3, 2010, pp. 403-423. doi:10.1137/S0040585X97984309

[5] C. Acerbi, “Coherent Representations of Subjective Risk Aversion,” In: G. Szeg?, Ed., Risk Measures for the 21st Century, Wiley, New York, 2004, pp. 147-207.

[6] R. T. Rockafellar and S. Uryasev, “Optimization of Conditional Value-at-Risk,” Journal of Risk, Vol. 2, 2000, pp. 21-41.

[7] R. T. Rockafellar, S. Uryasev and M. Zabarankin, “Master Funds in Portfolio Analysis with General Deviation Measures,” Journal of Banking and Finance, Vol. 30, No. 2, 2006, pp. 743-778. doi:10.1016/j.jbankfin.2005.04.004

[8] A. S. Cherny, “Equilibrium with Coherent Risk,” Preprint, 2006.

[9] A. Grothendieck, “Topological Vector Spaces,” Gordon and Breach, Philadelphia, 1992.

[10] S. Jaschke and U. Küchler, “Coherent Risk Measures and Good-Deal Bounds,” Finance and Stochstics, Vol. 5, No. 2, 2001, pp. 181-200. doi:10.1007/PL00013530

[11] S. Kusuoka, “On Law Invariant Coherent Risk Measures,” Advances in Mathematical Economics, Vol. 3, 2001, pp. 83-95.

[12] A. S. Cherny, “Pricing with Coherent Risk,” Theory of Probability and Its Applications, Vol. 52, No. 3, 2008, pp. 389-415. doi:10.1137/S0040585X97983158

[1] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, “Coherent Measures of Risk,” Mathematical Finance, Vol. 9, No. 3, 1999, pp. 203-228. doi:10.1111/1467-9965.00068

[2] F. Delbaen, “Coherent Risk Measures on General Probability Spaces,” In: K. Sandmann, et al., Eds., Advances in Finance and Stochastics, Springer-Verlag, Berlin, 2002, pp. 1-37.

[3] G. Stoica, “Relevant Coherent Measures of Risk,” Journal of Mathematical Economics, Vol. 42, No. 6, 2006, pp. 794-806. doi:10.1016/j.jmateco.2006.03.006

[4] L. A. Konovalov, “Coherent Risk Measures and a Limit Pass,” Theory of Probability and its Applications, Vol. 54, No. 3, 2010, pp. 403-423. doi:10.1137/S0040585X97984309

[5] C. Acerbi, “Coherent Representations of Subjective Risk Aversion,” In: G. Szeg?, Ed., Risk Measures for the 21st Century, Wiley, New York, 2004, pp. 147-207.

[6] R. T. Rockafellar and S. Uryasev, “Optimization of Conditional Value-at-Risk,” Journal of Risk, Vol. 2, 2000, pp. 21-41.

[7] R. T. Rockafellar, S. Uryasev and M. Zabarankin, “Master Funds in Portfolio Analysis with General Deviation Measures,” Journal of Banking and Finance, Vol. 30, No. 2, 2006, pp. 743-778. doi:10.1016/j.jbankfin.2005.04.004

[8] A. S. Cherny, “Equilibrium with Coherent Risk,” Preprint, 2006.

[9] A. Grothendieck, “Topological Vector Spaces,” Gordon and Breach, Philadelphia, 1992.

[10] S. Jaschke and U. Küchler, “Coherent Risk Measures and Good-Deal Bounds,” Finance and Stochstics, Vol. 5, No. 2, 2001, pp. 181-200. doi:10.1007/PL00013530

[11] S. Kusuoka, “On Law Invariant Coherent Risk Measures,” Advances in Mathematical Economics, Vol. 3, 2001, pp. 83-95.

[12] A. S. Cherny, “Pricing with Coherent Risk,” Theory of Probability and Its Applications, Vol. 52, No. 3, 2008, pp. 389-415. doi:10.1137/S0040585X97983158