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Multivariate location-scale mixtures of normals and mean-variance-skewness portfolio allocation

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  • Mencía, Javier
  • Sentana, Enrique

Abstract

We show that the distribution of any portfolio whose components jointly follow a location-scale mixture of normals can be characterised solely by its mean, variance and skewness. Under this distributional assumption, we derive the mean-variance-skewness frontier in closed form, and show that it can be spanned by three funds. For practical purposes, we derive a standardised distribution, provide analytical expressions for the log-likelihood score and explain how to evaluate the information matrix. Finally, we present an empirical application in which we obtain the mean-variance-skewness frontier generated by the ten Datastream US sectoral indices, and conduct spanning tests.

Suggested Citation

  • Mencía, Javier & Sentana, Enrique, 2009. "Multivariate location-scale mixtures of normals and mean-variance-skewness portfolio allocation," Journal of Econometrics, Elsevier, vol. 153(2), pages 105-121, December.
    • Handle: RePEc:eee:econom:v:153:y:2009:i:2:p:105-121
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    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. Walter Briec & Kristiaan Kerstens & Octave Jokung, 2007. "Mean-Variance-Skewness Portfolio Performance Gauging: A General Shortage Function and Dual Approach," Management Science, INFORMS, vol. 53(1), pages 135-149, January.
    3. Kon, Stanley J, 1984. "Models of Stock Returns-A Comparison," Journal of Finance, American Finance Association, vol. 39(1), pages 147-165, March.
    4. Ruud, Paul A., 1991. "Extensions of estimation methods using the EM algorithm," Journal of Econometrics, Elsevier, vol. 49(3), pages 305-341, September.
    5. Chen, Ying & Härdle, Wolfgang & Jeong, Seok-Oh, 2008. "Nonparametric Risk Management With Generalized Hyperbolic Distributions," Journal of the American Statistical Association, American Statistical Association, vol. 103(483), pages 910-923.
    6. Dilip B. Madan & Frank Milne, 1991. "Option Pricing With V. G. Martingale Components1," Mathematical Finance, Wiley Blackwell, vol. 1(4), pages 39-55, October.
    7. Fiorentini, Gabriele & Sentana, Enrique & Calzolari, Giorgio, 2003. "Maximum Likelihood Estimation and Inference in Multivariate Conditionally Heteroscedastic Dynamic Regression Models with Student t Innovations," Journal of Business & Economic Statistics, American Statistical Association, vol. 21(4), pages 532-546, October.
    8. Stephen A. Ross, 2005. "Mutual Fund Separation in Financial Theory—The Separating Distributions," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 10, pages 309-356, World Scientific Publishing Co. Pte. Ltd..
    9. Eric Jondeau & Michael Rockinger, 2006. "Optimal Portfolio Allocation under Higher Moments," European Financial Management, European Financial Management Association, vol. 12(1), pages 29-55, January.
    10. Bauwens, Luc & Laurent, Sebastien, 2005. "A New Class of Multivariate Skew Densities, With Application to Generalized Autoregressive Conditional Heteroscedasticity Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 23, pages 346-354, July.
    11. Berk, Jonathan B., 1997. "Necessary Conditions for the CAPM," Journal of Economic Theory, Elsevier, vol. 73(1), pages 245-257, March.
    12. Barone-Adesi, Giovanni, 1985. "Arbitrage Equilibrium with Skewed Asset Returns," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 20(3), pages 299-313, September.
    13. Andrew J. Patton, 2004. "On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation," Journal of Financial Econometrics, Oxford University Press, vol. 2(1), pages 130-168.
    14. Gallant, A. Ronald & Tauchen, George, 1996. "Which Moments to Match?," Econometric Theory, Cambridge University Press, vol. 12(4), pages 657-681, October.
    15. Neil Shephard & Ole E. Barndorff-Nielsen & University of Aarhus, 2001. "Normal Modified Stable Processes," Economics Series Working Papers 72, University of Oxford, Department of Economics.
    16. Dilip B. Madan & Frank Milne, 1991. "Option Pricing With V. G. Martingale Components," Working Paper 1159, Economics Department, Queen's University.
    17. Bertrand Maillet & Emmanuel Jurczenko, 2006. "Multi-moment Asset Allocation and Pricing Models," Post-Print hal-00308990, HAL.
    18. Guidolin, Massimo & Timmermann, Allan, 2007. "Asset allocation under multivariate regime switching," Journal of Economic Dynamics and Control, Elsevier, vol. 31(11), pages 3503-3544, November.
    19. Engle, Robert F & Kozicki, Sharon, 1993. "Testing for Common Features," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(4), pages 369-380, October.
    20. French, Kenneth R & Poterba, James M, 1991. "Investor Diversification and International Equity Markets," American Economic Review, American Economic Association, vol. 81(2), pages 222-226, May.
    21. Bertrand Maillet & Emmanuel Jurczenko & Paul Merlin, 2006. "Hedge Funds Portfolio Selection with Higher-order Moments: A Non-parametric Mean-Variance-Skewness-Kurtosis Efficient Frontier," Post-Print hal-00308993, HAL.
    22. Javier Mencía & Enrique Sentana, 2012. "Distributional Tests in Multivariate Dynamic Models with Normal and Student-t Innovations," The Review of Economics and Statistics, MIT Press, vol. 94(1), pages 133-152, February.
    23. de Athayde, Gustavo M. & Flores, Renato Jr., 2004. "Finding a maximum skewness portfolio--a general solution to three-moments portfolio choice," Journal of Economic Dynamics and Control, Elsevier, vol. 28(7), pages 1335-1352, April.
    24. Engle, Robert F & Kozicki, Sharon, 1993. "Testing for Common Features: Reply," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(4), pages 393-395, October.
    25. Enrique Sentana, 1995. "Quadratic ARCH Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 62(4), pages 639-661.
    26. Sentana, Enrique, 2004. "Factor representing portfolios in large asset markets," Journal of Econometrics, Elsevier, vol. 119(2), pages 257-289, April.
    27. Bertrand Maillet & Emmanuel Jurczenko & Paul Merlin, 2006. "Hedge Funds Portfolio Selection with Higher-order Moments: A Non-parametric Mean-Variance-Skewness-Kurtosis Efficient Frontier," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00308993, HAL.
    28. Chamberlain, Gary, 1983. "A characterization of the distributions that imply mean--Variance utility functions," Journal of Economic Theory, Elsevier, vol. 29(1), pages 185-201, February.
    29. François Longin & Bruno Solnik, 2001. "Extreme Correlation of International Equity Markets," Journal of Finance, American Finance Association, vol. 56(2), pages 649-676, April.
    30. Christian Pedersen & Stephen Satchell, 2002. "On the foundation of performance measures under asymmetric returns," Quantitative Finance, Taylor & Francis Journals, vol. 2(3), pages 217-223.
    31. Flavio Pressacco & Patrizia Stucchi, 2000. "Linearity properties of a three-moments portfolio model," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 23(2), pages 133-150.
    32. Bertrand Maillet & Emmanuel Jurczenko, 2006. "Multi-moment Asset Allocation and Pricing Models," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00308990, HAL.
    33. Davidson, Russell & MacKinnon, James G., 1993. "Estimation and Inference in Econometrics," OUP Catalogue, Oxford University Press, number 9780195060119.
    34. Lim, Kian-Guan, 1989. "A New Test of the Three-Moment Capital Asset Pricing Model," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 24(2), pages 205-216, June.
    35. H. D. Vinod & B. D. McCullough, 1999. "The Numerical Reliability of Econometric Software," Journal of Economic Literature, American Economic Association, vol. 37(2), pages 633-665, June.
    36. Gibbons, Michael R & Ross, Stephen A & Shanken, Jay, 1989. "A Test of the Efficiency of a Given Portfolio," Econometrica, Econometric Society, vol. 57(5), pages 1121-1152, September.
    37. Jondeau, Eric & Rockinger, Michael, 2003. "Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements," Journal of Economic Dynamics and Control, Elsevier, vol. 27(10), pages 1699-1737, August.
    38. Yusif Simaan, 1993. "Portfolio Selection and Asset Pricing---Three-Parameter Framework," Management Science, INFORMS, vol. 39(5), pages 568-577, May.
    39. H. D. Vinod & B. D. McCullough, 1999. "Corrigenda: The Numerical Reliability of Econometric Software," Journal of Economic Literature, American Economic Association, vol. 37(4), pages 1565-1565, December.
    40. Hogan, William W. & Warren, James M., 1974. "Toward the Development of an Equilibrium Capital-Market Model Based on Semivariance," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 9(1), pages 1-11, January.
    41. Owen, Joel & Rabinovitch, Ramon, 1983. "On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice," Journal of Finance, American Finance Association, vol. 38(3), pages 745-752, June.
    42. King, Mervyn & Sentana, Enrique & Wadhwani, Sushil, 1994. "Volatility and Links between National Stock Markets," Econometrica, Econometric Society, vol. 62(4), pages 901-933, July.
    43. Andrea Gamba & Francesco Rossi, 1998. "A three-moment based portfolio selection model," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 21(1), pages 25-48, June.
    44. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
    45. Kraus, Alan & Litzenberger, Robert H, 1976. "Skewness Preference and the Valuation of Risk Assets," Journal of Finance, American Finance Association, vol. 31(4), pages 1085-1100, September.
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    More about this item

    Keywords

    Generalised hyperbolic distribution Maximum likelihood Portfolio frontiers Sortino ratio Spanning tests Tail dependence;

    JEL classification:

    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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