Portfolio Selection with Common Correlation Mixture Models

Conference paper
Part of the Contributions to Economics book series (CE)

The estimation of the covariance matrix of returns on financial assets is a considerable problem in applications of the traditional mean-variance approach to portfolio selection. If the number of assets is large, as is often the case in reality, the estimation error in the (sample) covariance matrix, the number of elements of which increases at a quadratic rate with the number of assets, can seriously distort “optimal” portfolio decisions [8, 31, 32]. In order to mitigate this problem, several alternative approaches have been proposed to filter out the systematic information from historic correlations, e.g., use of factor structures [12], shrinkage techniques [36, 37], and others (see [8] for an overview).

While it is generally found that these methods help to predict return correlations more precisely, empirical research on the distributional characteristics of asset returns comes up with a further challenge for the classical portfolio theory developed by Markowitz [43]. In particular, it has long been known that the distribution of stock returns sampled at a daily, weekly or even monthly frequency is not well described by a (stationary) normal distribution [45]. The empirical return distributions tend to be leptokurtic, that is, they are more peaked and fatter tailed than the normal distribution, properties that are of great importance for risk management. In addition, recent evidence suggests that there are two types of asymmetries in the (joint) distribution of stock returns. The first is skewness in the marginal distribution of the returns of individual stocks [29, 46, 33]. The second relates to the joint distribution of stock returns and is an asymmetry in the dependence between assets. Namely, stock returns appear to be more highly correlated during high-volatility periods, which are often associated with market downturns, i.e., bear markets. Evidence for the asymmetric dependence phenomenon has been reported, among others, in [17, 34, 49, 38, 5, 6, 10, 47, 20].


Risk Aversion Stock Return Portfolio Selection Asset Return Asset Allocation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akaike H (1973) Information Theory and an Extension of the Maximum Likelihood Principle. In: Petrov BN, Csaki F (eds) 2nd International Symposium on Information Theory. Akademiai Kiado, BudapestGoogle Scholar
  2. 2.
    Alexander C (2001) Market Models. A Guide to Financial Data Analysis. Wiley, ChichesterGoogle Scholar
  3. 3.
    Alexander C, Lazar E (2006) Normal Mixture GARCH(1,1): Applications to Exchange Rate Modelling. Journal of Applied Econometrics 21: 307–336CrossRefGoogle Scholar
  4. 4.
    Aneja YP, Chandra R, Gunay E (1989) A Portfolio Approach to Estimating the Average Correlation Coefficient for the Constant Correlation Model. Journal of Finance 44:1435–1438CrossRefGoogle Scholar
  5. 5.
    Ang A, Bekaert G (2002) International Asset Allocation with Regime Shifts. Review of Financial Studies 15:1137–1187CrossRefGoogle Scholar
  6. 6.
    Ang A, Chen J (2002) Asymmetric Correlations of Equity Portfolios. Journal of Financial Economics 63:443–494CrossRefGoogle Scholar
  7. 7.
    Bauwens L, Preminger A, Rombouts J (2006) Regime-Switching GARCH Models. CORE Discussion Paper 2006/11Google Scholar
  8. 8.
    Brandt MW (2005) Portfolio Choice Problems. In: Aït-Sahalia Hansen LP (eds) Handbook of Financial Econometrics. North-Holland, AmsterdamGoogle Scholar
  9. 9.
    Brannolte C (2002) Nichtlineare Regimewechselmodelle. Theoretische und emprirische Evidenz am deutschen Kapitalmarkt. Pro Business, BerlinGoogle Scholar
  10. 10.
    Butler KC, Joaquin DC (2002) Are the Gains from International Portfolio Diversification Exaggerated? The Influence of Downside Risk in Bear Markets. Journal of International Money and Finance 21:981–1011CrossRefGoogle Scholar
  11. 11.
    Campbell R, Koedijk K, Kofman P (2002) Increased Correlation in Bear Markets. Financial Analysts Journal 58:87–94CrossRefGoogle Scholar
  12. 12.
    Chan L, Karceski KCJ, Lakonishok J (1999) On Portfolio Optimization: Forecasting Covariances and Choosing the Risk Model. Review of Financial Studies 12:937–974CrossRefGoogle Scholar
  13. 13.
    Chesnay F, Jondeau E (2001) Does Correlation between Stock Returns Really Increase during Turbulent Periods? Economic Notes 30:53–80CrossRefGoogle Scholar
  14. 14.
    Dempster AP, Laird NM, Rubin DB (1977) Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society B 39:1–38Google Scholar
  15. 15.
    Elton EJ, Gruber MJ (1973) Estimating the Dependence Structure of Share Prices — Implications for Portfolio Selection. Journal of Finance 28:1203–1232CrossRefGoogle Scholar
  16. 16.
    Elton EJ, Gruber MJ, Urich TJ (1978) Are Betas Best? Journal of Finance 33:1375–1384CrossRefGoogle Scholar
  17. 17.
    Erb CB, Harvey CR, Viskanta TE (1994) Forecasting International Equity Correlations. Financial Analysts Journal 50:32–45CrossRefGoogle Scholar
  18. 18.
    Eun CS, Resnick BG (1984) Estimating the Correlation Structure of International Share Prices. Journal of Finance 39:1311–1324CrossRefGoogle Scholar
  19. 19.
    Fraley C, Raftery AE (1998) How Many Clusters? Which Clustering Method? Answers via Model-based Cluster Analysis. Computer Journal 41:578–588CrossRefGoogle Scholar
  20. 20.
    Guidolin M, Timmermann A (2005) Economic Implications of Bull and Bear Regimes in UK Stock and Bond Returns. Economic Journal 115:111–143CrossRefGoogle Scholar
  21. 21.
    Haas M, Mittnik S, Paolella MS (2004) Mixed Normal Conditional Heteroskedasticity. Journal of Financial Econometrics 2:211–250CrossRefGoogle Scholar
  22. 22.
    Haas M, Mittnik S, Paolella MS (2004) A New Approach to Markov-switching GARCH Models. Journal of Financial Econometrics 2:493–530CrossRefGoogle Scholar
  23. 23.
    Haas M, Mittnik S, Paolella MS (2006) Multivariate Normal Mixture GARCH. Center for Financial Studies Working Paper 2006/09Google Scholar
  24. 24.
    Hamilton JD (1989) A New Approach to the Economic Analysis of Non-stationary Time Series and the Business Cycle. Econometrica 57:357–384CrossRefGoogle Scholar
  25. 25.
    Hamilton JD (1990) Analysis of Time Series Subject to Changes in Regime. Journal of Econometrics 45:39–70CrossRefGoogle Scholar
  26. 26.
    Hamilton JD (1994) Time Series Analysis. Princeton University Press, PrincetonGoogle Scholar
  27. 27.
    Hamilton JD (1996) Specification Testing in Markov-switching Time Series Models. Journal of Econometrics 70:127–157CrossRefGoogle Scholar
  28. 28.
    Hamilton JD, Susmel R (1994) Autoregressive Conditional Heteroskedasticity and Changes in Regime. Journal of Econometrics 64:307–333CrossRefGoogle Scholar
  29. 29.
    Harvey CR, Siddique A (1999) Autoregressive Conditional Skewness. Journal of Financial and Quantitative Analysis 34:465–487CrossRefGoogle Scholar
  30. 30.
    Horn RA and Johnson CR (1991) Topics in Matrix Analysis. Cambridge University Press, CambridgeGoogle Scholar
  31. 31.
    Jobson JD, Korkie B (1980) Estimation of Markowitz Efficient Portfolios. Journal of the American Statistical Association 75:544–554CrossRefGoogle Scholar
  32. 32.
    Jobson JD, Korkie B (1981) Putting Markowitz Theory to Work. Journal of Portfolio Management 7:70–74Google Scholar
  33. 33.
    Jondeau E, Rockinger M (2003) Conditional Volatility, Skewness, and Kurtosis: Existence, Persistence, and Comovements. Journal of Economic Dynamics and Control 27:1699–1737CrossRefGoogle Scholar
  34. 34.
    Karolyi GA, Stulz RM (1996) Why do Markets Move Together? An Investigation of U.S.-Japan Stock Return Comovements. Journal of Finance 51: 951–986CrossRefGoogle Scholar
  35. 35.
    Kim CJ (1994) Dynamic Linear Models with Markov-switching. Journal of Econometrics 60:1–22CrossRefGoogle Scholar
  36. 36.
    Ledoit O, Wolf M (2003) Improved Estimation of the Covariance Matrix of Stock Returns with an Application to Portfolio Selection. Journal of Empirical Finance 10:603–621CrossRefGoogle Scholar
  37. 37.
    Ledoit O, Wolf M (2004) Honey, I Shrunk the Sample Covariance Matrix. Journal of Portfolio Management 31:110–119CrossRefGoogle Scholar
  38. 38.
    Longin F, Solnik B (2001) Extreme Correlation of International Equity Markets. Journal of Finance 56:649–676CrossRefGoogle Scholar
  39. 39.
    Mandelbrot B (1963) The Variation of certain Speculative Prices. Journal of Business 36:394–419CrossRefGoogle Scholar
  40. 40.
    McLachlan GJ, Krishnan T (1997) The EM Algorithm and Extensions. Wiley, New YorkGoogle Scholar
  41. 41.
    McLachlan GJ, Peel D (2000) Finite Mixture Models. Wiley, New YorkGoogle Scholar
  42. 42.
    McLachlan GJ, Peel D, Bean RW (2003) Modelling High-Dimensional Data by Mixtures of Factor Analyzers. Computational Statistics and Data Analysis 41:379–388CrossRefGoogle Scholar
  43. 43.
    Markowitz HM (1952) Portfolio Selection. Journal of Finance 7:77–91CrossRefGoogle Scholar
  44. 44.
    Meng XL, Rubin DB (1993) Maximum Likelihood Estimation via the ECM Algorithm: A General Framework. Biometrika 80:267–278CrossRefGoogle Scholar
  45. 45.
    Pagan A (1996) The Econometrics of Financial Markets. Journal of Emprirical Finance 3:15–102CrossRefGoogle Scholar
  46. 46.
    Paolella MS (1999) Tail Estimation and Conditional Modeling of Heteroscedastic Time Series. Pro Business, BerlinGoogle Scholar
  47. 47.
    Patton AJ (2004) On the Out-of-Sample Importance of Skewness and Asymmetric Dependence for Asset Allocation. Journal of Financial Econometrics 2:130–168CrossRefGoogle Scholar
  48. 48.
    Pelletier D (2006) Regime Switching for Dynamic Correlations. Journal of Econometrics 131:445–473CrossRefGoogle Scholar
  49. 49.
    Ramchand L, Susmel R (1998) Volatility and Cross Correlation Across Major Stock Markets. Journal of Empirical Finance 5:397–416CrossRefGoogle Scholar
  50. 50.
    Redner RA, Walker HF (1984) Mixture Densities, Maximum Likelihood and the EM Algorithm. SIAM Review 26: 195–239CrossRefGoogle Scholar
  51. 51.
    Samuelson PA (1967) General Proof that Diversification Pays. Journal of Financial and Quantitative Analysis 2:1–13CrossRefGoogle Scholar
  52. 52.
    Schwarz G (1978) Estimating the Dimension of a Model. Annals of Statistics 6:461–464CrossRefGoogle Scholar
  53. 53.
    Searle SR (1982) Matrix Algebra Useful for Statistics. Wiley, New YorkGoogle Scholar
  54. 54.
    Timmermann A (2000) Moments of Markov Switching Models. Journal of Econometrics 96:75–111CrossRefGoogle Scholar
  55. 55.
    Tipping ME, Bishop CM (1999) Mixtures of Probabilistic Principal Component Analyzers. Neural Computation 11:443–482CrossRefGoogle Scholar
  56. 56.
    Turner CM, Startz R, Nelson CR (1989) A Markov Model of Heteroskedasticity, Risk, and Learning in the Stock Market. Journal of Financial Economics 25:3–22CrossRefGoogle Scholar
  57. 57.
    Whitelaw RF (2000) Stock Market Risk and Return. Review of Financial Studies 13:521–547CrossRefGoogle Scholar
  58. 58.
    Zhang Z, Li WK, Yuen KC (2006) On a Mixture GARCH Time Series Model. Journal of Time Series Analysis 27:577–597CrossRefGoogle Scholar

Copyright information

© Physica-Verlag Heidelberg 2009

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of MunichGermany

Personalised recommendations