A directional multivariate value at risk

Abstract

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.

Directional multivariate extremes in environmental phenomena

“…Note that an oriented orthant is a translation and a rotation of the nonnegative Euclidean orthant toward a new vertex in the point x and a new direction u . As is explained in Torres et al (), R u is not unique for n ⩾ 3. Then, in order to guarantee uniqueness in the orthogonal transformation, the QR‐oriented orthant was defined in (Torres et al, ) as follows, let u be a unit vector with non‐null components and let M u and M e be matrices defined as, where u i , i = 1,…, n is the i ‐th component of u , s g n (·) is the scalar sign function and e i is the vector with all its components equal to zero except the i ‐th component equal to one.…”

“…Hence, we get the directional level sets by applying the inverse of the rotation R u to the elements belonging to the sets defined in Equations where the copula modeling of R u X has been used. All this thanks to Property 3.8 in Torres et al () and relationship , (the result also holds through when the alternative definition based on joint distributions is used).…”

“…Such duality can be also seen in the approaches AND and OR defined in De Michele et al (), or the UPPER and LOWER differentiation given in Embrechts and Puccetti (); Cousin and Di Bernardino (). But, without loss of generality, we have decided to implement the extreme detection analysis in terms of the survival analogy, because a key relationship can be established between the extremes given by and those associated to the arguments (1 − α , − u ) reversing the inequalities (see Corollary 4.3 in Torres et al, ); that is, Then, in terms of risks, relation allows us to consider risk aversion; that is, we would expect more extreme events that corresponds to a conservative position. Now, we describe a nonparametric procedure to estimate the extreme thresholds, that is, the directional multivariate quantiles, as well as, the lower and upper level sets for a dataset.…”

“…There exist infinite directions to look at the data from a reference point that could help with the accuracy of the analysis and the interpretation of the results. Attempts have been made considering alternative directions, for instance, Laniado et al (2012) and Torres et al (2015) developed a financial application where the risk of losses is analyzed considering the direction of the investment weight composition in a portfolio; Cascos and Molchanov (2007); Hallin et al (2010) and Kong and Mizera (2012) have applied a directional setting to define quantile trimmings. Hence, a second contribution of this paper is to outline a general approach to detect directional multivariate extremes, which can be useful in other statistical areas apart from environmental sciences.…”

Estimation of extreme quantiles conditioning on multivariate critical layers

“…In all other cases, adaptations should be made in order to operate in the correct area of the copula (such as the directional multivariate return levels proposed by Torres et al. ()).…”