In economics, insurance and finance, value at risk (VaR) is a widely used measure of the risk of loss on a specific portfolio of financial assets. For a given portfolio, time horizon, and probability α, the 100α% VaR is defined as a threshold loss value, such that the probability that the loss on the portfolio over the given time horizon exceeds this value is α. That is to say, it is a quantile of the distribution of the losses, which has both good analytic properties and easy interpretation as a risk measure. However, its extension to the multivariate framework is not unique because a unique definition of multivariate quantile does not exist. In the current literature, the multivariate quantiles are related to a specific partial order considered in R n , or to a property of the univariate quantile that is desirable to be extended to R n . In this work, we introduce a multivariate value at risk as a vector-valued directional risk measure, based on a directional multivariate quantile, which has recently been introduced in the literature. The directional approach allows the manager to consider external information or risk preferences in her/his analysis. We have derived some properties of the risk measure and we have compared the univariate VaR over the marginals with the components of the directional multivariate VaR. We have also analyzed the relationship between some families of copulas, for which it is possible to obtain closed forms of the multivariate VaR that we propose. Finally, comparisons with other alternative multivariate VaR given in the literature, are provided in terms of robustness.

Several environmental phenomena can be described by different correlated variables that must be considered jointly in order to be more representative of the nature of these phenomena. For such events, identification of extremes is inappropriate if it is based on marginal analysis. Extremes have usually been linked to the notion of quantile, which is an important tool to analyze risk in the univariate setting. We propose to identify multivariate extremes and analyze environmental phenomena in terms of the directional multivariate quantile, which allows us to analyze the data considering all the variables implied in the phenomena, as well as look at the data in interesting directions that can better describe an environmental catastrophe. Since there are many references in the literature that propose extremes detection based on copula models, we also generalize the copula method by introducing the directional approach. Advantages and disadvantages of the non-parametric proposal that we introduce and the copula methods are provided in the paper. We show with simulated and real data sets how by considering the first principal component direction we can improve the visualization of extremes. Finally, two cases of study are analyzed: a synthetic case of flood risk at a dam (a 3-variable case), and a real case study of sea storms (a 5-variable case). Keywords Directional Multivariate Extremes in Environmental PhenomenaRaúl Torres (ratorres@est-econ.uc3m.es) Carlo De Michele (carlo.demichele@polimi.it) Henry Laniado (hlaniado@est-econ.uc3m.es) Rosa E. Lillo (lillo@est-econ.uc3m.es) AbstractSeveral environmental phenomena can be described by different correlated variables that must be considered jointly in order to be more representative of the nature of these phenomena. For such events, identification of extremes is inappropriate if it is based on marginal analysis. Extremes have usually been linked to the notion of quantile, which is an important tool to analyze risk in the univariate setting. We propose to identify multivariate extremes and analyze environmental phenomena in terms of the directional multivariate quantile, which allows us to analyze the data considering all the variables implied in the phenomena, as well as look at the data in interesting directions that can better describe an environmental catastrophe. Since there are many references in the literature that propose extremes detection based on copula models, we also generalize the copula method by introducing the directional approach. Advantages and disadvantages of the non-parametric proposal that we introduce and the copula methods are provided in the paper. We show with simulated and real data sets how by considering the first principal component direction we can improve the visualization of extremes. Finally, two cases of study are analyzed: a synthetic case of flood risk at a dam (a 3−variable case), and a real case study of sea storms (a 5−variable case).

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Air transportation growth is a reality described by different sources (e.g. The World Bank [1], the latest Eurocontrol report [2]). One essential initiative required to improve air traffic capacity while maintaining or increasing safety is to introduce predictive analytics that enable a dynamic adaptation of airline operations in a preemptive manner to an ever changing environment. An important part of this task is to model airport operations and plan accordingly. Particularly runway usage and/or configuration are important aspects of these operations. For example, prior knowledge of runway usage could improve flight plan optimizers. Of course, to create any model or predictor, ground truth data is required. However most of the time, detailed information about runway historical usage/configuration is inaccessible, unreliable or it belongs to national ATC services providers. Then, thinking on a high-scale forecast methodology there is an important drawback given the lack of a feasible target for most of the airports. Thus, the goal of this work is to introduce an accessible, easy to implement algorithm that allows historical reconstruction of runway usage/configuration for any airport based on data transmitted from aircrafts through either Radar or ADS-B technologies, even when the track data is not consistent. We study the quality of the assessment performed by the two parts of the algorithm: 1) Measuring the flight usage accuracy in comparison to the report given by the Spanish ATC service provider (ENAIRE) for each flight landing to or takingoff from two Spanish airports, Madrid-Barajas and Barcelona-El Prat, during October 2016. 2) Comparing the Netherlands-Schiphol runway configuration reported by the Netherlands airspace regulator (LVNL) for three different months, February, April and August 2018. The results provide values above 97% of accuracy for both types of assessment.