For portfolios with a large number of assets, the single index model allows for expressing the large number of covariances between individual asset returns through a significantly smaller number of parameters. This avoids the constraint of having very large samples to estimate the mean and the covariance matrix of the asset returns, which practically would be unrealistic given the dynamic of market conditions. The traditional way to estimate the regression parameters in the single index model is the maximum likelihood method. Although the maximum likelihood estimators have desirable theoretical properties when the model is exactly satisfied, they may give completely erroneous results when outliers are present in the data set. In this paper, we define minimum pseudodistance estimators for the parameters of the single index model and using them we construct new robust optimal portfolios. We prove theoretical properties of the estimators, such as consistency, asymptotic normality, equivariance, robustness, and illustrate the benefits of the new portfolio optimization method for real financial data.

Mean-risk models have been widely used in solving portfolio selection problems in the last years. Since the mean-variance theory of Markowitz, an enormous amount of papers have been published extending or modifying the basic model in several directions like the simplification of the type and amount of input data, the introduction of alternative measures of risk, the incorporation of additional criteria or constraints. Recently, risk measures concerned with the left tails of distributions that evaluate the extremely unfavourable outcomes are used. The most used risk measure for such purposes is Value-at-Risk (VaR). In this paper we concentrate on the second direction of incorporating of a new risk measure in portfolio modeling. We define a new risk measure, which take into consideration the values exceeding a certain threshold in the extreme tail of the loss distribution, called Limited Value-at-Risk (LVaR). We study the properties of this risk measure. We build a new model for portfolio selection, named mean-LVaR model, in which risk is evaluated using LVaR risk measure. We study the properties of the new mean-risk model and compare it with the classical mean-VaR model. We derive the analytical form of LVaR risk measure in the case of normal distribution. We provide computational results and analyze the implications of using the mean-LVaR risk model in portfolio optimization problem.