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Robust Omega ratio optimization using regular vines

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Abstract

We study the robust portfolio optimization model for the Omega ratio when the joint ambiguity in the returns distributions is modeled utilizing copulas. We propose the copula formulation of the Omega ratio and use it to formulate the worst-case robust optimization model. Furthermore, we propose a Markov chain predicated filtering strategy to filter a set of fewer assets from a large pool of available assets in the market to amend the performance of the conventional Omega ratio model. We propose an R-vine copula Autoregressive Moving Average Generalized Autoregressive Conditional Heteroskedasticity (ARMA-GARCH) model for the joint distribution of assets returns. We obtain the standardized residuals for each return series of the filtered assets using the ARMA-GARCH model and, subsequently, exploit the regular vine copulas to model the joint dependence among the transformed residuals. The tree structure in the regular vines is accomplished using Kendall’s tau. The dependence structure so obtained is used to simulate scenarios for the returns of assets. The simulated data is used to obtain optimal portfolios in the worst-case Copula Omega ratio model. Empirical evidence shows the aptness of the proposed filtering strategy and analyzes the performance of the worst-case copula Omega portfolios on several datasets using a rolling window approach. The optimal portfolios from the worst-case copula Omega model are found to outperform the portfolios from the Gaussian copula Omega ratio model by having a higher information ratio, Value-at-risk ratio, and Rachev ratio.

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Notes

  1. The choice of copulas considered for the formation of mixture copulas is inspired from Rustem ([27]) and the earlier works ([21, 22]), seen along with the research in ([19, 20]). Each chosen copula better describes different dependency. Clayton, Joe and Gumbel are non-symmetric to explain stronger dependence below and above the 50-th percentile, respectively; Frank copula is symmetric but it has different properties to the Gaussian copula. Hence, by using them in a mixture structure we can cover a large spectrum of possible dependencies.

  2. A Naive portfolio of n assets has 1/n weight allocated to each asset.

  3. The performance measures used are mean return, Sortino ratio (SR), standard deviation (Std Dev), downside deviation (DD), Information ratio, Omega Sharpe ratio, Sharpe ratio with standard deviation, VaR and CVaR as risk measures, VaR, CVaR, Rachev ratios, and VaR ratios, at 97 and 95% level of confidence

  4. We first select a copula family among these five. Then determine the best pair-copula at each node amongst the various rotations of the chosen copula family which are available in the R package VineCopula. We follow this procedure with each of the five copulas.

References

  1. Developments in modelling risk aggregation, Basel Committee on Banking Supervision. Bank for International Settlements (2010)

  2. Bacon, C.R.: Practical Portfolio Performance Measurement and Attribution, vol. 568. Wiley, Hoboken (2011)

    Google Scholar 

  3. Bedford, T., Cooke, R.M.: Probability density decomposition for conditionally dependent random variables modeled by vines. Ann. Math. Artif. Intell. 32(1–4), 245–268 (2001)

    Article  MathSciNet  Google Scholar 

  4. Bedford, T., Cooke, R.M.: Vines: a new graphical model for dependent random variables. Ann. Stat. 30(4), 1031–1068 (2002)

    Article  MathSciNet  Google Scholar 

  5. Charnes, A., Cooper, W.W.: Programming with linear fractional functionals. Naval Res. Logist. 9(3–4), 181–186 (1962)

    Article  MathSciNet  Google Scholar 

  6. Cherubini, U., Luciano, E., Vecchiato, W.: Copula Methods in Finance. Wiley, Chichester (2004)

    Book  Google Scholar 

  7. Chollete, L., Heinen, A., Valdesogo, A.: Modeling international financial returns with a multivariate regime-switching copula. J. Financ. Econom. 7(4), 437–480 (2009)

    Article  Google Scholar 

  8. Dennis, P., Mayhew, S., Stivers, C.: Stock returns, implied volatility innovations, and the asymmetric volatility phenomenon. J. Financ. Quant. Anal. 41(2), 381–406 (2006)

    Article  Google Scholar 

  9. van Dijk, D.: Brooks-introductory econometrics for finance. De Economist (2003)

  10. Dissmann, J., Brechmann, E.C., Czado, C., Kurowicka, D.: Selecting and estimating regular vine copulae and application to financial returns. Comput. Stat. Data Anal. 59, 52–69 (2013)

    Article  MathSciNet  Google Scholar 

  11. Engle, R.F.: Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econom. J. Econom. Soc. 50, 987–1007 (1982)

    MathSciNet  MATH  Google Scholar 

  12. Engle, R.F., Bollerslev, T.: Modelling the persistence of conditional variances. Econom. Rev. 5(1), 1–50 (1986)

    Article  MathSciNet  Google Scholar 

  13. Fabozzi, F.J., Huang, D., Zhou, G.: Robust portfolios: contributions from operations research and finance. Ann. Oper. Res. 176(1), 191–220 (2010)

    Article  MathSciNet  Google Scholar 

  14. Fishburn, P.C.: Mean-risk analysis with risk associated with below-target returns. Am. Econ. Rev. 67(2), 116–126 (1977)

    Google Scholar 

  15. Goel, A., Sharma, A., Mehra, A.: Index tracking and enhanced indexing using mixed conditional value-at-risk. J. Comput. Appl. Math. 335, 361–380 (2018). https://doi.org/10.1016/j.cam.2017.12.015. http://www.sciencedirect.com/science/article/pii/S0377042717306350

  16. Goel, A., Sharma, A., Mehra, A.: Robust optimization of mixed cvar starr ratio using copulas. J. Comput. Appl. Math. (2018). https://doi.org/10.1016/j.cam.2018.08.001

    Article  MATH  Google Scholar 

  17. Gouriéroux, C.: ARCH Models and Financial Applications. Springer, New York (2012)

    MATH  Google Scholar 

  18. Gregoriou, G.N., Gueyie, J.P.: Risk-adjusted performance of funds of hedge funds using a modified Sharpe ratio. J. Wealth Manag. 6(3), 77–83 (2003)

    Article  Google Scholar 

  19. Hasselblad, V.: Estimation of parameters for a mixture of normal distributions. Technometrics 8(3), 431–444 (1966)

    Article  MathSciNet  Google Scholar 

  20. Hasselblad, V.: Estimation of finite mixtures of distributions from the exponential family. J. Am. Stat. Assoc. 64(328), 1459–1471 (1969)

    Article  Google Scholar 

  21. Hu, L.: Dependence patterns across financial markets: methods and evidence. Preprint, Department of Economic, Yale University (2002)

  22. Hu, L.: Dependence patterns across financial markets: a mixed copula approach. Appl. Financ. Econ. 16(10), 717–729 (2006)

    Article  Google Scholar 

  23. Hyndman, R.J., Khandakar, Y., et al.: Automatic time series for forecasting: the forecast package for R. 6/07. Department of Econometrics and Business Statistics, Monash University (2007)

  24. Linsmeier, T.J., Pearson, N.D.: Value at risk. Financ. Anal. J. 56(2), 47–67 (2000)

    Article  Google Scholar 

  25. Joe, H.: Families of \(m\)-variate distributions with given margins and \(m(m-1)/2\) bivariate dependence parameters. Lecture Notes-Monograph Series, pp. 120–141 (1996)

  26. Joe, H., Li, H., Nikoloulopoulos, A.K.: Tail dependence functions and vine copulas. J. Multivar. Anal. 101(1), 252–270 (2010)

    Article  MathSciNet  Google Scholar 

  27. Kakouris, I., Rustem, B.: Robust portfolio optimization with copulas. Eur. J. Oper. Res. 235(1), 28–37 (2014)

    Article  MathSciNet  Google Scholar 

  28. Kapsos, M., Christofides, N., Rustem, B.: Worst-case robust omega ratio. Eur. J. Oper. Res. 234(2), 499–507 (2014)

    Article  MathSciNet  Google Scholar 

  29. Kapsos, M., Zymler, S., Christofides, N., Rustem, B.: Optimizing the omega ratio using linear programming. J. Comput. Finance 17(4), 49–57 (2014)

    Article  Google Scholar 

  30. Keating, C., Shadwick, W.F.: A universal performance measure. J. Perform. Meas. 6(3), 59–84 (2002)

    Google Scholar 

  31. Konno, H., Yamazaki, H.: Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Manag. Sci. 37(5), 519–531 (1991)

    Article  Google Scholar 

  32. Markowitz, H.: Portfolio selection. J. Finance 7(1), 77–91 (1952)

    Google Scholar 

  33. Mohammadi, H., Su, L.: International evidence on crude oil price dynamics: applications of ARIMA-GARCH models. Energy Econ. 32(5), 1001–1008 (2010)

    Article  Google Scholar 

  34. Nelsen, R.B.: An Introduction to Copulas. Springer, New York (2006)

    MATH  Google Scholar 

  35. Peel, D., McLachlan. G.J.: Robust mixture modelling using the t distribution. Stat. Comput. 10(4), 339–348 (2000)

  36. Prim, R.C.: Shortest connection networks and some generalizations. Bell Syst. Tech. J. 36(6), 1389–1401 (1957)

    Article  Google Scholar 

  37. Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–42 (2000)

    Article  Google Scholar 

  38. Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26(7), 1443–1471 (2002)

    Article  Google Scholar 

  39. Sharma, A., Mehra, A.: Extended omega ratio optimization for risk-averse investors. Int. Trans. Oper. Res. 24(3), 485–506 (2017)

    Article  MathSciNet  Google Scholar 

  40. Sharma, A., Utz, S., Mehra, A.: Omega-CVaR portfolio optimization and its worst case analysis. OR Spectr. 39(2), 505–539 (2017)

    Article  MathSciNet  Google Scholar 

  41. Sharpe, W.F.: The sharpe ratio. J. Portf. Manag. 21(1), 49–58 (1994)

    Article  Google Scholar 

  42. Sklar, M.: Fonctions de répartition à n dimensions et leurs marges. Université Paris 8 (1959)

  43. Sortino, F.A., Price, L.N.: Performance measurement in a downside risk framework. J. Invest. 3(3), 59–64 (1994)

    Article  Google Scholar 

  44. Treynor, J.L.: How to rate management of investment funds. Harv. Bus. Rev. 43(1), 63–75 (1965)

    Google Scholar 

  45. Tsay, R.S.: Analysis of Financial Time Series. Wiley, Hoboken (2005)

    Book  Google Scholar 

  46. Zhu, S., Fukushima, M.: Worst-case conditional value-at-risk with application to robust portfolio management. Oper. Res. 57(5), 1155–1168 (2009)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first author is thankful to the Council of Scientific and Industrial Research, India, for providing the financial grant in initiating the present research and IIT Delhi for extending the financial support to conclude the same. The authors are grateful to the editor and referees for their valuable support and constructive suggestions.

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  1. Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India

    Anubha Goel & Aparna Mehra

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  1. Anubha Goel
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  2. Aparna Mehra
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Appendix I: Empirical Analysis of the Proposed Filtering Strategy and Optimization Models

Appendix I: Empirical Analysis of the Proposed Filtering Strategy and Optimization Models

See Figs. 4, 5, 6, 7, 8 and Tables 2, 3, 4, 5, 6.

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Cumulative returns BSE (1260, 524)

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Cumulative returns BSE (1260, 252)

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Cumulative returns BSE (504, 252)

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Fig. 7
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Cumulative returns BSE (252,252)

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Fig. 8
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Cumulative returns BSE (252, 126)

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Table 2 Out-of-sample performance analysis of the classical Omega ratio optimization model on the subsets of stocks filtered by the proposed Markov chain strategy (portfolio FOmegaP) and on the entire sets of assets in the datasets (portfolio OmegaP
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Table 3 The window-wise out-of-sample performance of optimal portfolios from the worst-case (WCCOR) model and Gaussian (GCOR) model for \((d_1,d_2)=(1260, 504)\) and (1260,252) for the SPC Index
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Table 4 The out-of-sample performance of optimal portfolios from the worst-case (WCCOR) model, Gaussian (GCOR) model, Naive Strategy, Minimum Variance Markowitz (Min) model for each of the in-sample and out-of-sample periods
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Table 5 The out-of-sample performance of optimal portfolios from the worst-case (WCCOR) model, Gaussian (GCOR) model, Naive Strategy, Minimum Variance Markowitz (Min) model for each of the in-sample and out-of-sample periods
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Table 6 The out-of-sample performance of optimal portfolios from the worst-case (WCCOR) model, Gaussian (GCOR) model, Naive Strategy, Minimum Variance Markowitz (Min) model for each of the in-sample and out-of-sample periods
Full size table

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Goel, A., Mehra, A. Robust Omega ratio optimization using regular vines. Optim Lett 15, 2067–2108 (2021). https://doi.org/10.1007/s11590-020-01629-5

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