Portfolio Selection with Common Correlation Mixture Models

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].