The Tail Mean-Variance Model and Extended Efficient Frontier
محورهای موضوعی : Statistical Methods in Financial ManagementEsmat Jamshidi Eini 1 , Hamid Khaloozadeh 2
1 - Department of Systems and Control, K.N. Toosi University of Technology, Tehran, Iran
2 - Department of Systems and Control, K.N. Toosi University of Technology, Tehran, Iran
کلید واژه: Skew-Elliptical Distributions, Tail Mean-Variance criterion, Optimal portfolio selection, Efficient Frontier,
چکیده مقاله :
In portfolio theory, it is well-known that the distributions of stock returns often have non-Gaussian characteristics. Therefore, we need non-symmetric distributions for modeling and accurate analysis of actuarial data. For this purpose and optimal portfolio selection, we use the Tail Mean-Variance (TMV) model, which focuses on the rare risks but high losses and usually happens in the tail of return distribution. The proposed TMV model is based on two risk measures the Tail Condition Expectation (TCE) and Tail Variance (TV) under Generalized Skew-Elliptical (GSE) distribution. We first apply a convex optimization approach and obtain an explicit and easy solution for the TMV optimization problem, and then derive the TMV efficient frontier. Finally, we provide a practical example of implementing a TMV optimal portfolio selection in the Tehran Stock Exchange and show TCE-TV efficient frontier.
[1] Artzner, P., Delbaen, F., Eber, J., and Heath, D., Coherent measures of risk, Mathematical Finance, 1999, 9(3), P. 203-228. Doi: 10.1111/1467-9965.00068.
[2] Azzalini, A., A class of distributions which includes the normal ones, Scandinavian Journal of statistics, 1985, 12, P. 171-178. Doi: 10.2307/4615982.
[3] Azzalini, A., and Capitanio, A., Statistical applications of the multivariate skew-normal distribution, Journal of the Royal Statistical Society, B, 1999, 61, P. 579-602. Doi: 10.1111/1467-9868.00194.
[4] Azzalini, A., and Dalla Valle, A., The multivariate skew-normal distribution, Biometrika, 1996, 83(4), P. 715-726. Doi: 10.1093/biomet/83.4.715.
[5] Bauder, D., Bodnar, T., Parolya, N., and Schmid, W., Bayesian mean–variance analysis: optimal portfolio selection under parameter uncertainty, Quantitative Finance, 13 May, 2020, Advance online publications. Doi: 10.1080/14697688.2020.1748214.
[6] Branco, M., and Dey, D., A general class of multivariate skew-elliptical distributions, Journal of Multivariate Analysis, 2001, 79(1), P. 99-113. Doi: 10.1006/jmva.2000.1960.
[7] Chow, K.V., Li, J., and Sopranzetti, B., The Predictive Power of Tail Risk Premia on Individual Stock Returns, 2018, 30th Annual Conference on Financial Economics and Accounting at NYU Stern.
[8] Darabi, R., Baghban, M., Application of Clayton Copula in Portfolio Optimization and its Comparison with Markowitz Mean-Variance Analysis, Advances in Mathematical Finance and Applications, 2018, 3(1), P. 33-51. Doi: 10.22034/amfa.2018.539133.
[9] Furman, E., and Landsman, Z., Tail variance premium with applications for elliptical portfolio, ASTIN Bulletin, 2006, 36(2), P. 433-462. Doi: 10.2143/AST.36.2.2017929.
[10] Granger, C., Time series concept for conditional distributions, Oxford Bulletin of Economics and Statistics, 2003, 65(s1), P. 689-701. Doi: 10.1046/j.0305-9049.2003.00094.x.
[11] Hu, W., and Kercheval, A.N., A Portfolio optimization for student t and skewed t returns, Quantitative finance journal, 2010, 10(1), P. 91-105. Doi: 10.1080/14697680902814225.
[12] Ignatieva, K., and Landsman, Z., Conditional tail risk measures for the skewed generalized hyperbolic family, Insurance: Mathematics and Economics, 2019, 86(C), P. 98-114. Doi: 10.1016/j.insmatheco.2019.02.008.
[13] Ignatieva, K., and Landsman, Z., Estimating the tails of loss severity via conditional risk measures for the family of symmetric generalised hyperbolic distributions, Insurance: Mathematics and Economics, 2015, 65(C), P. 172-186. Doi: 10.1016/j.insmatheco.2015.09.007.
[14] Ignatieva, K., and Platen, E., Modeling co-movements and tail dependency in the international stock market via copulae, Asia-Pacific Financial Markets, 2010, 17(3), P. 261-302. Doi: 10.1007/s10690-010-9116-2.
[15] Jamshidi Eini, E., Khaloozadeh, H., Tail Variance for Generalized Skew-Elliptical distribution, Communications in Statistics - Theory and Methods, 16 April, 2020, Advance online publications. Doi: 10.1080/03610926.2020.1751853.
[16] Jiang, C., Peng, H., and Yang, Y., Tail variance of portfolio under generalized Laplace distribution, Applied Mathematics and Computation, 2016, 282(C), P. 187-203. Doi: 10.1016/j.amc.2016.02.005.
[17] Kelker, D., Distribution theory of spherical distributions and location-scale parameter generalization, Sankhya: The Indian Journal of Statistics, 1970, 32(4), P. 419-430. Doi: 10.2307/25049690.
[18] Kim, J.H.T., and Kim, S-Y., Tail risk measures and risk allocation for the class of multivariate normal mean–variance mixture distributions, Insurance: Mathematics and Economics, 2019, 86, P. 145–157. Doi: 10.1016/j.insmatheco.2019.02.010.
[19] Landsman, Z., On the Tail Mean-Variance optimal portfolio selection, Insurance: Mathematics and Economics, 2010, 46, P. 547-533. Doi: 10.1016/j.insmatheco.2010.02.001.
[20] Landsman, Z., Makov, U., and Shushi, T., Tail Conditional Expectations for Generalized Skew-Elliptical Distributions, SSRN Electronic Journal, 30 June, 2013. Doi:1 10.2139/ssrn.2298265.
[21] Landsman, Z., Makov, U., and Shushi, T., Extended Generalized Skew-Elliptical Distributions and their Moments, The Indian Journal of Statistics, 2017, 79(1), P. 76-100. Doi: 10.1007/s13171-016-0090-2.
[22] Landsman, Z., Pat, N., and Dhaene, J., Tail Variance premiums for log-elliptical distributions, Insurance: Mathematics and Economics, 2013, 52(3), P. 441-447. Doi: 10.1016/j.insmatheco.2013.02.012.
[23] Landsman, Z., and Valdez, E., Tail Conditional Expectations for Elliptical Distributions, North American Actuarial Journal, 2003, 7(4), P. 55-71. Doi: 10.1080/10920277.2003.10596118.
[24] Luo, J., Chen, L., and Liu, H., Distribution characteristics of stock market liquidity, Physica A Statistical Mechanics and Its Applications, 2013, 392(23), P. 6004–6014. Doi: 10.1016/j.physa.2013.07.046.
[25] Markowitz, H., Portfolio selection, Journal of Finance, 1952, 7(1), P. 77-91. Doi: 10.1111/j.1540-6261.1952.tb01525.x.
[26] Merton, R., An analytic derivation of the efficient portfolio frontier, Journal of Financial and Quantitative Analysis, 1972, 7(4), P. 1851-1872. Doi: 10.2307/2329621.
[27] Miryekemami, S., Sadeh, E., Sabegh, Z., Using Genetic Algorithm in Solving Stochastic Programming for Multi-Objective Portfolio Selection in Tehran Stock Exchange, Advances in Mathematical Finance and Applications, 2017, 2(4), P.107-120. Doi: 10.22034/amfa.2017.536271.
[28] Morgan/Reuters, J., Risk Metrics Technical Document, Fourth Edition New York, 1996.
[29] Nematollahi, N., Farnoosh, R., and Rahnamaei, Z., Location-scale mixture of skew-elliptical distributions: Looking at the robust modeling, Statistical Methodology, 2016, 32(C), P. 131-146. Doi: 10.1016/j.stamet.2016.05.001.
[30] Owadally, I., and Landsman, Z., A characterization of optimal portfolios under the tail mean–variance criterion, Insurance: Mathematics and Economics, 2013, 52(2), P. 213-221. Doi: 10.1016/j.insmatheco.2012.12.004.
[31] Panjer, H., Measurement of risk, solvency requirements and allocation of capital within financial conglomerates, Institute of Insurance and Pension Research, University of Waterloo, Research Report, 2002, P. 1-15.
[32] Panjer, H., et al, Financial Economics, Schaumburg, Illinois: The Actuarial Foundation, 1998.
[33] Rahmani, M., Khalili Eraqi, M., and Nikoomaram, H., Portfolio Optimization by Means of Meta-Heuristic Algorithms, Advances in Mathematical Finance and Applications, 2019, 4(4), P. 83-97. Doi: 10.22034/amfa.2019.579510.1144.
[34] Shushi, T., Generalized skew-elliptical distributions are closed under affine transformations, Statistics and Probability Letters, 2018, 134(C), P. 1-4. Doi: 10.1016/j.spl.2017.10.012.
[35] Vernic, R., On the multivariate Skew-Normal distribution and its scale mixtures, An. St. Univ. Ovidius Constanta, 2005, 13(2), P. 83-96.
[36] Wang, Q., Huang, W., Wu, X., and Zhang, Ch., How Effective is the Tail Mean-Variance Model in the Fund of Fund Selection? An Empirical Study Using Various Risk Measures, Finance Research Letters, 2019, 29, P. 239-244, Doi: 10.1016/j.frl.2018.08.012.
[37] Xu, M., and Mao, T., Optimal capital allocation based on the tail mean-variance model, Insurance: Mathematics and Economics, 2013, 53(3), P. 533-543. Doi: 10.1016/j.insmatheco.2013.08.005.
