  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 $\alpha$, the $100\alpha\%$ 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 $\alpha$.
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 $\mathbb{R}^{n}$, or to a property of the univariate quantile
that is desirable to be extended to $\mathbb{R}^{n}$. 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
\textit{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.
