Multivariate Location-Scale Mixtures of Normals and Mean-Variance-Skewness Portfolio Allocation

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.

[1]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[2]  S. Satchell,et al.  On the foundation of performance measures under asymmetric returns , 2002 .

[3]  Gustavo Athayde and Renato Flores.  Finding a maximum skewness portfolio , 2001 .

[4]  A. Gallant,et al.  Which Moments to Match? , 1995, Econometric Theory.

[5]  K. French,et al.  Investor Diversification and International Equity Markets , 1991 .

[6]  J. MacKinnon,et al.  Estimation and inference in econometrics , 1994 .

[7]  Enrique Sentana,et al.  Maximum Likelihood Estimation and Inference in Multivariate Conditionally Heteroscedastic Dynamic Regression Models With Student t Innovations , 2003 .

[8]  Eric Jondeau,et al.  Conditional volatility, skewness, and kurtosis: existence, persistence, and comovements , 2003 .

[9]  O. Barndorff-Nielsen.  Exponentially decreasing distributions for the logarithm of particle size , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[10]  A. Timmermann,et al.  Asset Allocation Under Multivariate Regime Switching , 2005 .

[11]  Andrew J. Patton.  (IAM Series No 001) On the Out-Of-Sample Importance of Skewness and Asymetric Dependence for Asset Allocation , 2002 .

[12]  D. Madan,et al.  1option Pricing with V. G. Martingale Components , 1991 .

[13]  Enrique Sentana.  Quadratic Arch Models , 1995 .

[14]  Stephen A. Ross,et al.  Mutual fund separation in financial theory—The separating distributions , 1978 .

[15]  Enrique Sentana,et al.  Factor Representing Portfolios in Large Asset Markets , 2000 .

[16]  M. Tanner.  Tools for statistical inference: methods for the exploration of posterior distributions and likeliho , 1994 .

[17]  Robert F. Engle,et al.  Testing for Common Features: Reply , 1993 .

[18]  Giovanni Barone-Adesi,et al.  Arbitrage Equilibrium with Skewed Asset Returns , 1985, Journal of Financial and Quantitative Analysis.

[19]  P. Blæsild.  The two-dimensional hyperbolic distribution and related distributions, with an application to Johannsen's bean data , 1981 .

[20]  R. Engle,et al.  Testing for Common Features , 1990 .

[21]  Stephen A. Ross,et al.  A Test of the Efficiency of a Given Portfolio , 1989 .

[22]  K. Lim.  A New Test of the Three-Moment Capital Asset Pricing Model , 1989, Journal of Financial and Quantitative Analysis.

[23]  G. Chamberlain.  A characterization of the distributions that imply mean—Variance utility functions☆ , 1983 .

[24]  Andrew J. Patton.  On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation , 2002 .

[25]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[26]  Enrique Sentana,et al.  Volatiltiy and Links between National Stock Markets , 1990 .

[27]  Joel Owen,et al.  On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice , 1983 .

[28]  Bertrand B. Maillet,et al.  Hedge Funds Portfolio Selection with Higher-order Moments: A Non-parametric Mean-Variance-Skewness-Kurtosis Efficient Frontier , 2006 .

[29]  F. Longin,et al.  Extreme Correlation of International Equity Markets , 2000 .

[30]  William W. Hogan,et al.  Toward the Development of an Equilibrium Capital-Market Model Based on Semivariance , 1974, Journal of Financial and Quantitative Analysis.

[31]  Kjersti Aas,et al.  Risk Estimation using the Multivariate Normal Inverse Gaussian Distribution , 2006 .

[32]  Jonathan B. Berk,et al.  Necessary Conditions for the CAPM , 1997 .

[33]  Luc Bauwens,et al.  A New Class of Multivariate Skew Densities, With Application to Generalized Autoregressive Conditional Heteroscedasticity Models , 2005 .

[34]  P. Carr,et al.  The Variance Gamma Process and Option Pricing , 1998 .

[35]  Decisions in economics and finance , 2000 .

[36]  Bruce D. McCullough,et al.  The Numerical Reliability of Econometric Software , 1999 .

[37]  B. Jørgensen.  Statistical Properties of the Generalized Inverse Gaussian Distribution , 1981 .

[38]  R. Flôres,et al.  Finding a maximum skewness portfolio--a general solution to three-moments portfolio choice , 2004 .

[39]  John C. Liechty,et al.  Portfolio selection with higher moments , 2004 .

[40]  J. Wooldridge,et al.  Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances , 1992 .

[41]  Distributional Tests in Multivariate Dynamic Models with Normal and Student t Innovations , 2009 .

[42]  Stanley J. Kon.  Models of Stock Returns—A Comparison , 1984 .

[43]  M. Rockinger,et al.  Optimal Portfolio Allocation Under Higher Moments , 2004 .

[44]  Paul A. Ruud,et al.  Extensions of estimation methods using the EM algorithm , 1991 .

[45]  N. Shephard,et al.  Normal Modified Stable Processes , 2001 .

[46]  Phhilippe Jorion.  Value at Risk: The New Benchmark for Managing Financial Risk , 2000 .

[47]  Robert F. Engle,et al.  Testing for Common Features , 1993 .

[48]  R. Litzenberger,et al.  SKEWNESS PREFERENCE AND THE VALUATION OF RISK ASSETS , 1976 .

[49]  Yusif Simaan,et al.  Portfolio Selection and Asset Pricing-Three-Parameter Framework , 1993 .

[50]  Flavio Pressacco,et al.  Linearity properties of a three-moments portfolio model , 2000 .

[51]  W. Härdle,et al.  Nonparametric Risk Management With Generalized Hyperbolic Distributions , 2005 .

[52]  M. Crowder.  Maximum Likelihood Estimation for Dependent Observations , 1976 .

[53]  Andrea Gamba,et al.  A three-moment based portfolio selection model , 1996 .

[54]  Bruce D. McCullough,et al.  Corrigenda: The Numerical Reliability of Econometric Software , 1999 .

[55]  Kristiaan Kerstens,et al.  Mean-Variance-Skewness Portfolio Performance Gauging: A General Shortage Function and Dual Approach , 2007, Manag. Sci..