GlueVaR risk measures

Jaume Belles-Sampera, Montserrat Guillén & Miguel Santolino

GlueVaR risk measures combine Value-at-Risk and Tail Value-at-Risk at different tolerance levels and have analytical closed-form expressions for the most frequently used distribution functions in many applications, i.e. Normal, Log-normal, Student-t and Generalized Pareto distributions. A subfamily of GlueVaR risk measures fulfils the property of tail-subadditivity. Here, we present these basic risk measures and implement them based on the empirical dstribution function.

A methodological overview can be found in:


In this example, we use the daily prices of CAC-40, DAX and IBEX-35 from January of 2005 to May of 2014. "fImport" R-packaged is used to import the data.

Name Content description
2386 observations from the index of the Spanish Continuous Market.
CAC.csv 2395 observations from the most widely-used indicator of the Paris market.
DAX.csv 2392 observations from German blue chip stocks traded on the Frankfurt Stock Exchange.

VaR (Value at Risk)

Given a risk X and a probability level α ∈ (0,1), the corresponding VaR, denoted by VaRα(X), is defined as

\begin{equation} \mathrm{VaR}_{\alpha}(X) = F^{-1}_{X}(\alpha) \end{equation}

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TVaR (Tail Value at Risk)

Given a risk X and a probability level α, the corresponding TVaR, denoted by TVaRα(X), is defined as

\begin{equation} \mathrm{TVaR}_{\alpha}(X) = \frac{1}{1-\alpha} \int\limits_{\alpha}^1 \mathrm{VaR}_{\xi}(X)d\xi, \hspace{1cm} 0 < \alpha < 1 \end{equation}

We thus see that TVaRα(X) can be viewed as the 'arithmetic average' of the VaRs of X, from α on, if X is a continuous random variable.

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GlueVaR (Generalized Value at Risk mesures defined by Belles-Sampera et al., 2013)

We define a new family of risk measures, named GlueVaR.

\begin{equation} \mathrm{GlueVaR}_{\beta,\alpha}^{h_1,h_2}\left(X\right)=\displaystyle \int_{-\infty}^{0}\left[\kappa _{\beta ,\alpha }^{h_{1},h_{2}}\left(S_{X}\left(x\right)\right)-1\right]dx + \int_{0}^{+\infty}\kappa _{\beta ,\alpha }^{h_{1},h_{2}}\left(S_{X}\left(x\right)\right)dx \end{equation}
Any GlueVar risk measure can be described by means of its distortion function. Given a confidence level α the distortion function for GlueVaR is:

\begin{equation} \kappa _{\beta ,\alpha }^{h_{1},h_{2}}\left( u\right) = \left\{ \begin{array}{l} \displaystyle \frac{h_{1}}{1-\beta} \cdot u , \quad \mbox{if} \quad 0\leq u<1-\beta \\ \\ h_1+ \displaystyle \frac{h_2-h_1}{ \beta - \alpha} \cdot \left[u - \left(1- \beta \right)\right] ,\\ \quad \quad \quad \mbox{if} \quad 1-\beta \leq u < 1-\alpha \\ \\ 1, \quad \mbox{if} \quad 1-\alpha \leq u \leq 1 \\ \end{array} \right. \end{equation}

where α,β ∈ [0,1] so that α ≤ β, h1 ∈ [0,1], and h2 ∈ [h1,1]. Parameter β is the additional confidence level besides α.

Some examples of distortion functions of GlueVaR risk measures are shown below:

If the following notation is used,

\begin{equation} \left\{ \begin{array}{ll} \omega _{1}= & h_{1}-\displaystyle\frac{\left( h_{2}-h_{1}\right) \cdot \left( 1-\beta \right) }{\beta -\alpha }\\ \omega _{2}= & \displaystyle\frac{h_{2}-h_{1}}{\beta -\alpha }\cdot \left( 1-\alpha \right) \\ \omega _{3}= & 1-\omega _{1}-\omega _{2}\quad = 1-h_{2}, \end{array} \right. \label{Weights} \end{equation}

then GlueVaR is a risk measure that can be expressed as a linear combination of three risk measures: TVaR at confidence levels β and α and VaR at confidence level α, \begin{equation} \begin{array}{c} \mathrm{GlueVaR}_{\beta ,\alpha }^{h_{1},h_{2}}\left( X\right) = \omega _{1}\cdot \mathrm{TVaR}_{\beta }\left( X\right) + \omega _{2}\cdot \mathrm{TVaR}_{\alpha }\left( X\right) +\omega _{3}\cdot \mathrm{VaR}_{\alpha }\left( X\right) . \label{gluevar_wa_risa} \end{array} \end{equation}

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  • Universitat de Barcelona - Last Updated: 06-03-2014