# VaR vs Difference of CVaRs Minimization

** Case study background and problem formulations**

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

**PROBLEM 1: VaR minimization**

Miminize Var_risk (minimizing VaR with confidence level alpha)

subject to

Linear ≥ Const (linear constraint)

Box constraints (upper/lower bounds on positions)

——————————————————————–

Var_risk = VaR Risk for Loss

Box constraints = constraints on individual decision variables

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset1 | 23 | 10,000 | -0.001977571995 | 0.11 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

Instructions for importing problems from Run-File to PSG MATLAB.

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset2 | Problem Statement | Data | Solution | 23 | 100,000 | -0.002140167562 | 7.49 |

**PROBLEM 2: Minimization of Difference of CVaRs**

Miminize ((1-alpha_1)/(alpha_2-alpha_1))*CVaR_risk1 – ((1-alpha_2)/(alpha_2-alpha_1))*CVaR_risk2 (minimizing difference of two CVaR functions with confidence levels alpha_1 and alpha_2 satisfying inequalities alpha_1 < alpha < alpha_2)

subject to

Linear ≥ Const (constraint on the portfolio rate of return)

Box constraints (upper/lower bounds on exposures)

——————————————————————–

CVaR_risk = CVaR Risk for Loss

Box constraints = constraints on individual decision variables

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset1 | 23 | 10,000 | -0.001972530367 | 0.01 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

Instructions for importing problems from Run-File to PSG MATLAB.

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset2 | Problem Statement | Data | Solution | 23 | 100,000 | -0.002115116297 | 0.07 |

**CASE STUDY SUMMARY**

Problems 1 minimized VaR using standard PSG risk function VaR_Risk. Let us denote the confidence level of VaR by alpha in Problem 1. Problem 2 minimized the difference of two CVaR functions with confidence levels alpha_1 and alpha_2 satisfying inequalities alpha_1 < alpha < alpha_2. Confidence levels alpha_1 and alpha_2 are close to alpha.