arXiv 1502.009082015-02-03q-fin.RMstat.AP

一个方向性多变量风险价值

A Directional Multivariate Value at Risk

Raúl Torres, Rosa E. Lillo, Henry Laniado

备注：: 30 pages, 9 figures
发表信息：: Insurance: Mathematics and Economics, Volume 65, November 2015, Pages 111-123
DOI：: 10.1016/j.insmatheco.2015.09.002

摘要：

在经济学、保险学和金融学中，风险价值（VaR）是一种广泛使用的衡量特定金融资产组合损失风险的方法。对于给定的投资组合、时间范围和概率$α$，$100α\%$ VaR被定义为阈值损失值，使得投资组合在给定时间范围内的损失超过该值的概率为$α$。也就是说，它是损失分布的分位数，既具有良好的分析性质，又易于解释为风险度量。然而，它的多元框架的扩展是不是唯一的，因为一个唯一的定义多元分位数不存在。在目前的文献中，多变量分位数与$\mathbb{R}^{n}$中考虑的特定偏序相关，或者与期望扩展到$\mathbb{R}^{n}$的单变量分位数的属性相关。在这项工作中，我们引入了一个多变量风险价值作为向量值的方向性风险度量，基于一个方向性多变量分位数，这是最近在文献中介绍。定向方法允许经理在分析中考虑外部信息或风险偏好。我们推导了风险度量的一些性质，并将边际上的单变量\textit{VaR}与方向性多变量VaR的分量进行了比较。我们还分析了一些Copula族之间的关系，对于这些Copula族，我们可以得到我们提出的多元VaR的封闭形式。最后，与文献中给出的其他替代多元风险值的比较，提供的鲁棒性。

英文摘要：

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 $\mathbb{R}^{n}$, 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.

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