Portfolio Selection Based on Modified CoVaR in Gaussian Framework
Abstract
1. Introduction
“The higher the value of X, the better”.
2. Notation
2.1. CoVaR in Copula Setting
2.2. Portfolio Selection
3. The Mean-CoVaR Model
- The symmetric matrix is positively defined, i.e., is a matrix of a scalar product on ;
- The vectors and are linearly independent (i.e., not parallel);
- E is any real number.
4. Proofs and Auxiliary Results
4.1. Gaussian Copulas
- 1.
- ;
- 2.
- ;
- 3.
- .
4.2. The Optimization Problems
4.2.1. The Basic Problem
4.2.2. The Markowitz Problem
4.2.3. Critical Plane
4.2.4. Auxiliary Optimization Problem
4.2.5. Proof of Theorem 2
4.2.6. Proof of Theorem 3
5. Examples
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Alexander, G.J.; Francis, J.C. Portfolio Analysis, 3rd ed.; Prentice-Hall: New Jersey, NJ, USA, 1986. [Google Scholar]
- Black, F. Capital market equilibrium with restricted borrowing. J. Bus. 1972, 45, 444–455. [Google Scholar] [CrossRef]
- Tobin, J. Liquidity preference as behavior towards risk. Rev. Econ. Stud. 1958, 25, 65–86. [Google Scholar] [CrossRef]
- Frost, P.A.; Savarino, J.E. An Empirical Bayes Approach to Efficient Portfolio Selection. J. Financ. Quant. Anal. 1986, 3, 293–305. [Google Scholar] [CrossRef]
- Frost, P.A.; Savarino, J.E. For better performance: Constrain portfolio weights. J. Portf. Manag. 1988, 15, 29. [Google Scholar] [CrossRef]
- Petukhina, A.; Klochkov, Y.; Härdle, W.K.; Zhivotovskiy, N. Robustifying Markowitz. J. Econom. 2024, 239, 105387. [Google Scholar] [CrossRef]
- Black, F.; Litterman, R. Global Portfolio Optimization. Financ. Anal. J. 1922, 48, 28–43. [Google Scholar] [CrossRef]
- Idzorek, T. A step-by-step guide to the Black-Litterman model: Incorporating user-specified confidence levels. In Forecasting Expected Returns in the Financial Markets; Elsevier: Amsterdam, The Netherlands, 2007; pp. 17–38. [Google Scholar]
- Palczewski, A.; Palczewski, J. Black–Litterman model for continuous distributions. Eur. J. Oper. Res. 2019, 273, 708–720. [Google Scholar] [CrossRef]
- Jagannathan, R.; Tongshu, M. Risk reduction in large portfolios: Why imposing the wrong constraints helps. J. Financ. 2003, 58, 1651–1683. [Google Scholar] [CrossRef]
- Adrian, T.; Brunnermeier, M.K. CoVaR (Working Paper 17454); National Bureau of Economic Research Working Paper Series; National Bureau of Economic Research: Cambridge, MA, USA, 2011. [Google Scholar]
- Adrian, T.; Brunnermeier, M.K. CoVaR. Am. Econ. Rev. 2016, 106, 1705–1741. [Google Scholar] [CrossRef]
- Bernardi, M.; Durante, F.; Jaworski, P. Covar of families of copulas. Stat. Probab. Lett. 2017, 120, 8–17. [Google Scholar] [CrossRef]
- Föllmer, H.; Schied, A. Stochastic Finance. An Introduction in Discrete Time, 2nd ed.; de Gruyter: Berlin, Germany, 2004. [Google Scholar]
- Jaworski, P. On the Conditional Value at Risk (CoVaR) for tail-dependent copulas. Depend. Model. 2017, 5, 1–15. [Google Scholar] [CrossRef]
- Jaworski, P. On the Conditional Value-at-Risk (CoVaR) in copula setting. In Copulas and Dependence Models with Applications; Úbeda Flores, M., de Amo Artero, E., Durante, F., Fernández-Sánchez, J., Eds.; Springer: Cham, Switzerland, 2017; pp. 95–117. [Google Scholar]
- Zalewska, A. On peculiarities of covar-based portfolio selection. Appl. Math. 2018, 45, 181–197. [Google Scholar]
- Bernardi, M.; Durante, F.; Jaworski, P.; Petrella, L.; Salvadori, G. Conditional Risk based on multivariate Hazard Scenarios. Stoch. Environ. Res. Risk Assess. 2018, 32, 203–211. [Google Scholar] [CrossRef]
- Hakwa, B.; Jäger-Ambrozewicz, M.; Rüdiger, B. Analysing systemic risk contribution using a closed formula for conditional Value at Risk through copula. Commun. Stoch. Anal. 2015, 9, 131–158. [Google Scholar] [CrossRef]
- Mainik, G.; Schaanning, E. On dependence consistency of CoVaR and some other systemic risk measures. Stat. Risk Model. 2014, 31, 49–77. [Google Scholar] [CrossRef]
- Girardi, G.; Ergün, T.A. Systemic risk measurement: Multivariate GARCH estimation of CoVar. J. Bank. Financ. 2013, 37, 3169–3180. [Google Scholar] [CrossRef]
- Nelsen, R.B. An Introduction to Copulas, 2nd ed.; Springer Series in Statistics; Springer: New York, NY, USA, 2006. [Google Scholar]
- Cherubini, U.; Luciano, E.; Vecchiato, W. Copula Methods in Finance; John Wiley & Sons Ltd.: Hoboken, NJ, USA, 2004. [Google Scholar]
- Durante, F.; Sempi, C. Principles of Copula Theory; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Embrechts, P. Copulas: A personal view. J. Risk Insur. 2009, 76, 639–650. [Google Scholar] [CrossRef]
- Joe, H. Dependence Modeling with Copulas; Chapman & Hall/CRC: London, UK, 2014. [Google Scholar]
- McNeil, A.J.; Frey, R.; Embrechts, P. Quantitative Risk Management. Concepts, Techniques and Tools; Princeton Series in Finance; Princeton University Press: Princeton, NJ, USA, 2005. [Google Scholar]
- Mai, J.; Scherer, M. Simulating Copulas: Stochastic Models, Sampling Algorithms, and Applications; Series in Quantitative Finance; Imperial College Press: London, UK, 2012. [Google Scholar]
- Meyer, C. The bivariate normal copula. Commun. Stat. Theory Methods 2013, 42, 2402–2422. [Google Scholar] [CrossRef]
- Sheppard, W.F. On the calculation of the double integral expressing normal correlation. Trans. Camb. Phil. Soc. 1900, 19, 23–68. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Jaworski, P.; Zalewska, A. Portfolio Selection Based on Modified CoVaR in Gaussian Framework. Mathematics 2024, 12, 3766. https://doi.org/10.3390/math12233766
Jaworski P, Zalewska A. Portfolio Selection Based on Modified CoVaR in Gaussian Framework. Mathematics. 2024; 12(23):3766. https://doi.org/10.3390/math12233766
Chicago/Turabian StyleJaworski, Piotr, and Anna Zalewska. 2024. "Portfolio Selection Based on Modified CoVaR in Gaussian Framework" Mathematics 12, no. 23: 3766. https://doi.org/10.3390/math12233766
APA StyleJaworski, P., & Zalewska, A. (2024). Portfolio Selection Based on Modified CoVaR in Gaussian Framework. Mathematics, 12(23), 3766. https://doi.org/10.3390/math12233766