A Discussion on Recent Risk Measures with Application to Credit Risk: Calculating Risk Contributions and Identifying Risk Concentrations
Abstract
:1. Introduction and Motivation
2. Credit Risk and Credit Portfolio Modeling
3. Risk Measures beyond VaR: A Comparative Analysis
3.1. Desirable Requirements to Risk Measures
 monotonicity: $\rho \left({L}_{1}\right)\le \rho \left({L}_{2}\right)\forall {L}_{1},{L}_{2}$ with ${L}_{1}\le {L}_{2},$
 cash invariance: $\rho \left(La\right)=\rho \left(L\right)a$ for $a\in \mathbb{R},$
 positive homogeneity: $\rho \left(cL\right)=c\rho \left(L\right)$ for $c\ge 0,$
 subadditivity: $\rho \left({L}_{1}+{L}_{2}\right)\le \rho \left({L}_{1}\right)+\rho \left({L}_{2}\right)$.
 5.
 convex: $\rho \left(c{L}_{1}+(1c){L}_{2}\right)\le c\rho \left({L}_{1}\right)+(1c)\rho \left({L}_{2}\right)$ for $c\ge 0.$
 6.
 comonotonic additivity: $\rho \left({L}_{1}^{c}+{L}_{2}^{c}\right)=\rho \left({L}_{1}^{c}\right)+\rho \left({L}_{2}^{c}\right)$ for comonotonic random losses ${L}_{1}^{c}{=}_{d}{t}_{1}\left(Z\right),\phantom{\rule{0.166667em}{0ex}}{L}_{2}^{c}{=}_{d}{t}_{2}\left(Z\right)$ with nondecreasing functions ${t}_{1},{t}_{2}$ and a positive random variable Z.
 7.
 law invariance: ${L}_{1}{\sim}_{d}{L}_{2}\Rightarrow \rho \left[{L}_{1}\right]=\rho \left[{L}_{2}\right]$.
 8.
 elicitability: $\rho \left[{F}_{L}\right]=\underset{x\in \mathbb{R}}{argmin}\phantom{\rule{0.166667em}{0ex}}\int S(x,l)\phantom{\rule{0.166667em}{0ex}}{F}_{L}\left(dl\right),\phantom{\rule{0.166667em}{0ex}}\forall {F}_{L}\in \mathbb{F}$.
 9.
 robustness: $\rho \left[{F}_{L}\right]\rho \left[{G}_{L}\right]=o\left(1\right)\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}d({F}_{L},{G}_{L})=o\left(1\right),\phantom{\rule{4.pt}{0ex}}\mathrm{mit}\phantom{\rule{4.pt}{0ex}}{F}_{L}{G}_{L}\in \mathbb{F},$
3.2. Classes of Risk Measures
3.3. Risk Measures beyond VaR
4. Risk Contribution and Euler Allocation
5. Application to Credit Risk
5.1. Data Description and Portfolio Structure
5.2. Research Questions and Calibration of the Risk Measures
 How sensitive is the overall portfolio risk w.r.t. changes of the credit quality across the risk measures under consideration?
 How sensitive are the risk contributions w.r.t. sector and name concentrations across the risk measures under consideration?
 Are there differences between the risk measures under consideration w.r.t. capital allocation?
5.3. Empirical Results
6. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
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1  In a broader sense, losses may also arise from rating migrations if the rating of the counterparty changes; see, for instance, Tsaig et al. (2011). 
2  Eckert et al. (2016) discuss a credit portfolio framework that allows for dependence between PD, LGD and EAD; see also Kaposty et al. (2017) and Farinelli and Shkolnikov (2012). 
3  Typically, counterparties are assigned to predefined industry and/or country sectors. 
4  A technical implementation (see, in particular, Algorithm 1 from Jakob and Fischer (2016)) can be found in the R package GCPM, which was used to generate the loss distribution for the hypothetical portfolios in the empirical part. 
5  If the confidence level is high and losses from the tail of the distribution are of special interest, Importance Sampling (IS) might came to application in order to increase the speed of the calculation. With IS, the future economic scenarios are not generated randomly, but the “bad” scenarios have a higher chance of being selected than the “good” scenarios and the bias that is thus introduced is corrected later, see Glassermann (2005), Glassermann and Li (2005) or Chen et al. (2017). 
6  For a multivariate extensions of the VaR, we refer to Cousin and Di Bernardino (2013). 
7  
8  Newey and Powell (1987) and Breckling and Chambers (1988) have already introduced a similar notion of generalized quantiles in a different context. For the choice ${\varphi}_{1}={\varphi}_{2}={x}^{2}$, the generalized quantile is an expectile. 
9  Except for a correction term for a discontinuity in the distribution, the ES equals the conditional Expectation $E\left[L\mid L\ge {\mathrm{VaR}}_{\alpha}\left(L\right)\right]$. It is also well known as AverageVaR, TailVaR and ConditionalVaR. 
10  Cousin and Di Bernardino (2014) discuss multivariate extensions of ES. 
11  Refering to Rockafellar and Uryasev (2002), the $\alpha $tail distribution ${F}_{\alpha ,L}$ is defined as
$$\begin{array}{c}\hfill \left\{\begin{array}{cc}0,\hfill & l<{\mathrm{VaR}}_{\alpha}\left(L\right),\hfill \\ \frac{{F}_{L}\left(L\right)\alpha}{1\alpha},\hfill & l\ge {\mathrm{VaR}}_{\alpha}\left(L\right).\hfill \end{array}\right.\end{array}$$

12  MaumeDeschamps et al. (2017) discuss multivariate extensions of expectiles. 
13  $\overline{\alpha}={sup}_{l\ge 0}\alpha \left(L\right)$ and $\underline{\alpha}={inf}_{l\ge 0}\alpha \left(L\right)$. 
14  Wang (2000) pointed out that this distortion function combines four different approaches: (1) Classical insurance premium calculation; (2) Theory of distortion risk measures; (3) CapitalAssetPricingModel; (4) Option pricing (BlackScholes). Furthermore, Wang (2001) has showed that for a normal or lognormal distributed loss this distortion function results in a normal or lognormal distribution. 
15  The forcing conditions for the parameters such that the two presented inequalities hold are listed in BellesSampera et al. (2014a). 
16  BellesSampera et al. (2014a) calculate some useful analytical closed forms of the GlueVaR for Normal, Lognormal, Exponential, Pareto and TypeII Pareto distributions. Furthermore, they deliver conditions under which the GlueVaR fulfills the special property of tailsubadditivity also introduced in the same paper. 
17  The contribution discusses different approaches for capital allocation methods for the GlueVaR. 
18  Alternative allocation methods: Shapley, AumannShapley, Euler, Activity based, beta method, incremental method, cost gap method, Nucleonlus method 
19  The axiomatic approach follows three key properties of the allocation principle: (1) complete allocation; (2) diversification; (3) continuity with respect to the allocation principle. 
20  We take this portfolio size to guarantee efficient simulations and avoid working memory problems. 
21  The MSCI EMU Index (European Economic and Monetary Union) captures large and mid cap representation across the 10 Developed Markets countries in the EMU. 
22  
23  The simulation results are generated with a simulation number of 100,000. 
RM  Coherence  Convex  Comonotonic Add.  Law Invariant  Elicitability  Robustness 

VaR  x  x  ✓  ✓  ✓  Weak Topology 
ES  ✓  ✓  ✓  ✓  x  Wasserstein 
MS  x  x  ✓  ✓  ✓  Weak Topology 
ExVaR  ✓  ✓  x  ✓  ✓  Wasserstein 
LVaR  x  x  x  ✓  ✓  $\mathbf{C}$robust 
RMBLD  x  x  x  ✓  x  $\mathbf{C}$robust 
RVaR  x  x  ✓  ✓  x  $\mathbf{C}$robust 
Wang  ✓  ✓  ✓  ✓  x  
GlueVaR  x  x  ✓  ✓  x  Wasserstein 
EVaR  ✓  ✓  ✓  ✓  x 
RM  $\mathbf{p}\left[{\mathit{L}}_{\mathit{i}}\right]$  $\mathbf{p}\left[{\mathit{L}}_{\mathit{i}}\mid \mathit{L}\right]$  Source 

VaR  ${q}_{\alpha}^{}\left({L}_{i}\right)$  $\mathbf{E}\left[{L}_{i}\mid L={\mathrm{VaR}}_{\alpha}\left(L\right)\right]$  Tasche (1999) 
ES  $\frac{1}{1\alpha}{\int}_{\alpha}^{1}{q}_{u}^{}\left({L}_{i}\right)du$  $\mathbf{E}\left[{L}_{i}\mid L\ge {\mathrm{VaR}}_{\alpha}\left(L\right)\right]$  Tasche (1999) 
MS  ${q}_{\frac{1\alpha}{2}}^{}\left({L}_{i}\right)$  $\mathbf{E}\left[{L}_{i}\mid L={\mathrm{MS}}_{\alpha}\left(L\right)\right]$  Moser (2016) 
ExVaR  $\mathbf{E}\left[{S}_{\mathrm{ExVaR}}(x,l)\right]$  $\frac{(1\alpha )\mathbf{E}\left[{L}_{i}{1}_{\left\{X>{\mathrm{ExVaR}}_{\alpha}\left(L\right)\right\}}\right]+\alpha \mathbf{E}\left[{L}_{i}{1}_{\left\{L\le {\mathrm{ExVaR}}_{\alpha}\left(L\right)\right\}}\right]}{(1\alpha )P\left[L>{\mathrm{ExVaR}}_{\alpha}\left(L\right)\right]+\alpha P\left[L\le {\mathrm{ExVaR}}_{\alpha}\left(L\right)\right]}$  Emmer et al. (2015) 
RVaR  ${\int}_{0}^{\infty}{g}_{(\alpha ,\beta )}\left(1{F}_{{L}_{i}}\left(l\right)\right)dl$  $\mathbf{E}\left[{L}_{i}\mid {\mathrm{VaR}}_{\beta}\left(L\right)\ge L\ge {\mathrm{VaR}}_{\alpha}\left(L\right)\right]$  Moser (2016) 
Wang  ${\int}_{0}^{\infty}{g}_{\lambda}\left(1{F}_{{L}_{i}}\left(l\right)\right)dl$  $\mathbf{E}\left[{L}_{i}{g}_{\lambda}^{{}^{\prime}}(1{F}_{L}\left(l\right))\right]$  Tsanakas and Barnett (2003) 
GlueVaR  ${\int}_{0}^{\infty}{g}_{\alpha ,\beta}^{{h}_{1},{h}_{2}}\left(1{F}_{{L}_{i}}\left(l\right)\right)dl$  $\mathbf{E}\left[{L}_{i}{g}_{\alpha ,\beta}^{{}^{\prime},{h}_{1},{h}_{2}}(1{F}_{L}\left(l\right))\right]$  Tsanakas and Barnett (2003) 
EVaR  $\frac{E(exp\left({m}_{0}{L}_{p}\right){L}_{p})}{E(exp\left({m}_{p}{L}_{p}\right))}$  $\frac{E(exp\left({m}_{0}{L}_{p}\right){L}_{i})}{E(exp\left({m}_{p}{L}_{p}\right))}$  Zheng and Chen (2015) 
LVaR  x  x  not positive homogeneous 
BLD  x  x  not positive homogeneous 
Sector  BenchmarkPF  PF 1  PF 2  PF 3  PF 4 

Energy  0  0  0  0  0 
Materials  6  2  1  2  1 
Capital goods  12  71  82  71  82 
Commercial services and supplies  34  11  7  11  7 
Transportation  7  2  1  2  1 
Consumer Discretionary  15  5  3  5  3 
Consumer staples  6  2  1  2  1 
Health Care  9  3  2  3  2 
Information technology  3  1  1  1  1 
Telecommunication.services  1  1  1  1  1 
Utilities  7  2  1  2  1 
Number of Counterparties  200  200  200  100  100 
HHI  17.86  52.15  67.92  52.15  67.92 
Risk Measures  BenchmarkPF  PF1  PF2  Risk Measures  BenchmarkPF  PF1  PF2 

VaR (PD 2.0)  9.23  +24  +32  BLD (PD 2.0)  9.23  +24  +32 
VaR (PD 3.5)  12.83  +25  +28  BLD (PD 3.5)  12.83  +25  +28 
VaR (Mixed PD)  6.98  +26  +32  BLD (Mixed PD)  6.98  +26  +32 
ExVaR (PD 2.0)  10.47  +27  +34  ES (PD 2.0)  11.01  +27  +35 
ExVaR (PD 3.5)  14.10  +23  +28  ES (PD 3.5)  14.83  +25  +29 
ExVaR (Mixed PD)  8.02  +25  +33  ES (Mixed PD)  8.49  +25  +32 
EVaR (PD 2.0)  8.72  +31  +36  MS (PD 2.0)  10.80  +27  +33 
EVaR (PD 3.5)  13.74  +22  +27  MS (PD 3.5)  14.85  +21  +29 
EVaR (Mixed PD)  6.98  +23  +30  MS (Mixed PD)  8.10  +25  +33 
Wang (PD 2.0)  9.57  +29  +34  RVaR (PD 2.0)  10.76  +26  +33 
Wang (PD 3.5)  13.11  +22  +27  RVaR (PD 3.5)  14.44  +25  +29 
Wang (Mixed PD)  7.46  +29  +37  RVaR (Mixed PD)  8.20  +24  +33 
LVaR (PD 2.0)  9.45  +26  +31  GlueVaR (PD 2.0)  10.82  +27  +34 
LVaR (PD 3.5)  13.05  +26  +33  GlueVaR (PD 3.5)  14.71  +24  +29 
LVaR ( Mixed PD)  6.75  +33  +47  GlueVaR (Mixed PD)  8.34  +26  +33 
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Fischer, M.; Moser, T.; Pfeuffer, M. A Discussion on Recent Risk Measures with Application to Credit Risk: Calculating Risk Contributions and Identifying Risk Concentrations. Risks 2018, 6, 142. https://doi.org/10.3390/risks6040142
Fischer M, Moser T, Pfeuffer M. A Discussion on Recent Risk Measures with Application to Credit Risk: Calculating Risk Contributions and Identifying Risk Concentrations. Risks. 2018; 6(4):142. https://doi.org/10.3390/risks6040142
Chicago/Turabian StyleFischer, Matthias, Thorsten Moser, and Marius Pfeuffer. 2018. "A Discussion on Recent Risk Measures with Application to Credit Risk: Calculating Risk Contributions and Identifying Risk Concentrations" Risks 6, no. 4: 142. https://doi.org/10.3390/risks6040142