# GlueVaR risk measures

## Jaume Belles-Sampera, Montserrat Guillén & Miguel Santolino

A methodological overview can be found in:

- Belles-Sampera, J., Guillén, M. and Santolino, M. (2014) "Beyond value-at-risk in finance and insurance: GlueVaR distortion risk measures" Risk Analysis, 34(1), 121-134.

**DATA DESCRIPTION**

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 |

IBEX.csv |
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 VaR

_{α}(X), is defined as

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

**- Script**

- Results

**TVaR (Tail Value at Risk)**

Given a risk * X* and a probability level

*α*, the corresponding TVaR, denoted by TVaR

_{α}(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 TVaR_{α}(X) can
be viewed as the 'arithmetic average' of the VaRs of * X*,
from

*α*on, if

*is a continuous random variable.*

**X****- Script**

- Results

**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, *h _{1}*
∈ [0,1], and

*h*∈ [

_{2}*h*,1]. Parameter β is the additional confidence level besides α.

_{1}Some examples of distortion functions of GlueVaR risk measures are shown below: