** Case study background and problem formulations**

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, PSG MATLAB Subroutines and PSG R.

**PROBLEM 1: problem_cvar2_err**

Minimize cvar2_err (Minimizing CVaR (Superquantile) error)

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cvar2_err = CVaR (Superquantile) error)

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Run-File and Matlab Toolbox files contain problem formulation and data for one minimization problem (estimation of CVaR with linear regression by minimizing CVaR2 error). Matlab Subroutines: 1) Link “Matlab Code” contains zip file with m-file subroutine for minimizing CVaR2 error “minimize_CVaR2_err.m” and m-file “cvar_2err_programmatically.m” which runs “minimize_CVaR2_err” in cycle (for 10 banks); 2) Link “Data” contains zipped data file “data_CVaR2_err.mat”.

# of Variables |
# of Scenarios |
Objective Value |
Pseudo R2 |
Solving Time, PC 3.14GHz (sec) |
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Dataset | 4 | 1,264 | 4.67908485068 | 0.85652496 | 0.03 | |||
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Environments |
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Run-File | Problem Statement | Data | Solution | |||||

Matlab Toolbox | Data | |||||||

Matlab Subroutines | Matlab Code | Data | ||||||

R | R Code | Data |

**PROBLEM 2: problem_KB_err**

Minimize KB_err

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KB_err = Koenker and Basset error function

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Run-File and Matlab Toolbox files contain problem formulation and data for one minimization problem (estimation of VaR with linear regression by minimizing Koenker and Basset error, KB_err). Matlab Subroutines: 1) Link “Matlab Code” contains zip file with m-file subroutine for minimizing KB_err error “minimize_KB_err.m” and m-file “quantile_regression_programmatically.m” which runs “minimize_KB_err” in cycle (for 10 banks); 2) Link “Data” contains zipped data file “data_Quantile_regression.mat”.

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
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Dataset | 4 | 1,264 | 0.012484193442 | 0.02 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab Subroutines | Matlab Code | Data | |||||

R | R Code | Data |

**CASE STUDY SUMMARY**

This Case Study considers the new systemic risk measure, Conditional Value-at-Risk of the financial system conditional on institution being under distress, which is called CoCVaR. The CoCVaR is estimated with CVaR linear regression (Problem 1. Minimization of CVaR (Superquantile) error). Institution is considered to be in distress if it is at VaR level. VaR is estimated with quantile regression by minimizing Koenker and Basset error (Problem 2). CoCVaR was calculated for 10 largest publicly traded banks in the United States.

**References**

• Huang, W., Pavlikov, K. and S. Uryasev (2016): Systemic Risk Contribution Measurement: CoCVaR Approach. Working paper.

• Rockafellar,R. T. , Royset, J. O., and S. I. Miranda (2014): Superquantile Regression with Applications to Buffered Reliability, Uncertainty Quantification and Conditional Value-at- Risk. European J. Operations Research 234 (2014), 140-154.

• Adrian, T., and Brunnermeier, M. (2008). CoVaR. Federal Reserve Bank of New York Staff Report, 348.

• Borri, N., Caccavaio, M., Giorgio, G. D., and Sorrentino, A. M. (2014). Systemic risk in the Italian banking industry. Economic Notes, 1, 21-38.

• Girardi, G., and Ergun, A. T. (2013). Systemic risk measurement: multivariate GARCH estimation of CoVaR. Journal of Banking and Finance, 8, 3169-3180.

• Reboredo, J. C., and Ugolini, A. (2015). Systemic risk in European sovereign debt markets: A CoVaR-copula approach. Journal of International Money and Finance, 51, 214-244.