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:


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 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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REFERENCES

[1] Artzner, P., Delbaen, F., Eber, J-M., Heath, D. (1999) Coherent measures of risk. Mathematical Finance, 9(3):203-228.

[2] Balbás, A., Garrido, J., Mayoral, S. (2009) Properties of distorsion risk measures. Methodology and Computing in Applied Probability, 11(3):385-399.

[3] 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.

[4] Belles-Sampera, J., Guillén, M., Santolino, M. (2013) The use of flexible quantile-based measures in risk assessment. Institut de Recerca en Economia Aplicada, Regional i Pública IREA-UB Working papers, 2013/23.

[5] Belles-Sampera, J., Guillén, M. and Santolino, M. (2013) Generalizing some usual risk measures in financial and insurance applications, in M. Fernández-Izquierdo, M. Muñoz-Torres and R. León, eds, Modeling and Simulation in Engineering, Economics and Management. Proceedings of the MS 2013 International Conference, Vol. 145 of Lecture Notes in Business Information Processing, Springer-Verlag, pp. 75-­­82.

[6] Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2005). Actuarial Theory for Dependent RisksMeasures, Orders and Models. Chichester: John Wiley & Sons Ltd.

[7] Szëgo, G. (2002) Measures of risk. Journal of Banking and Finance, 26(7):1253-1272.

[8] Wang S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26(1):71-92.



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