The main contribution of the paper is to employ the financial market network as a useful tool to improve the portfolio selection process, where nodes indicate securities and edges capture the dependence structure of the system. Three different methods are proposed in order to extract the dependence structure between assets in a network context. Starting from this modified structure, we formulate and then we solve the asset allocation problem. We find that the portfolios obtained through a network-based approach are composed mainly of peripheral assets, which are poorly connected with the others. These portfolios, in the majority of cases, are characterized by an higher trade-off between performance and risk with respect to the traditional Global Minimum Variance (GMV) portfolio. Additionally, this methodology benefits of a graphical visualization of the selected portfolio directly over the graphic layout of the network, which helps in improving our understanding of the optimal strategy. IntroductionModern portfolio theory, which originated with Harry Markowitz's seminal paper in 1952 (see [23]) has stood the test of time and continues to be the intellectual foundation for real-world portfolio management. In this static framework, investors optimally allocate their wealth across a set of assets considering only the first and second moment of the returns' distribution. Despite the profound changes derived from this publication, the out-of-sample performance of Markowitz's prescriptions is often not as promising as expected. The poor performance of Markowitz's rule stems from the large estimation errors on the vector of expected returns (see [26]) and on the covariance matrix (see [17]) leading to the well-documented error-maximizing property discussed in [27]. The magnitude of this problem is evident when we acknowledge the modest improvements achieved by those models specifically designed to tackle the estimation risk (see [10]). Moreover, the evidence indicates that the simple yet effective equally-weighted portfolio rule has not been consistently out-performed by more sophisticated alternatives, as reported in [2,10]. The literature on portfolio selection has been extended on several directions. On one hand, some extensions modify the optimal problem, considering higher moments of returns' distribution (see [24] and the reference therein), exploiting alternative risk measures (see [6,19]) and as well as utility functions (see [37,13]), or proposing dynamic approaches (see among others [39,4]). On the other hand, there is now a vast literature on how to deal with the problem, that focuses on improved estimation procedures, considering that the optimal allocation is very sensitive to the estimation of moments and co-moments 1 .1 This sensitivity has generally been attributed to the tendency of the optimization to magnify the effects of estimation error. Michaud in [27] referred to "portfolio optimization "as "error maximization ". Efforts to improve parameters estimation procedure include among others the arXiv:18...

This paper presents the first methodological proposal of estimation of the ΛVaR. Our approach is dynamic and calibrated to market extreme scenarios, incorporating the need of regulators and financial institutions in more sensitive risk measures. We also propose a simple backtesting methodology by extending the VaR hypothesis-testing framework. Hence, we test our ΛVaR proposals under extreme downward scenarios of the financial crisis and different assumptions on the profit and loss distribution. The findings show that our ΛVaR estimations are able to capture the tail risk and react to market fluctuations significantly faster than the VaR and expected shortfall. The backtesting exercise displays a higher level of accuracy for our ΛVaR estimations.

In the equity context different Smart Beta strategies (such as the equally weighted, global minimum variance, equal risk contribution and maximum diversified ratio) have been proposed as alternatives to the capweighted index. These new approaches have attracted the attention of equity managers as different empirical analyses demonstrate the superiority of these strategies with respect to cap-weighted and to strategies that consider only mean and variance. In this paper we focus our attention to hedge fund index portfolios and analyze if the results reported in the equity framework are still valid. We consider hedge fund index and equity portfolios, the approaches used for portfolio selection are the four 'Smart Beta' strategies, mean-variance and meanvariance-skewness. In the two latter approaches the Taylor approximation of a CARA expected utility function and the Polynomial Goal Programing (PGP) have been used. The obtained portfolios are analyzed in the in-sample as well as in the out-of-sample perspectives.

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