# Case Study: Portfolio Credit-Risk Optimization Modeled by Scenarios and Mixtures of Normal Distributions

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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_var_LLN

Minimize Var_risk (minimizing Value-at-Risk)
subject to
Linear = 1 (budget constraint)
Linear ≥ Const (portfolio return constraint)
Box constraints (bounds on decision variables)
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Var = VaR Risk for Loss
Box constraints = constraints on individual decision variables
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Dataset 3,000 10,000 2.808627866 18.57 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data R R Code Data
PROBLEM 2: problem_cvar_LLN
Minimize Cvar_risk (minimizing Conditional Value-at-Risk)
subject to
Linear = 1 (budget constraint)
Linear ≥ Const (portfolio return constraint)
Box constraints (bounds on decision variables)
——————————————————————–
Cvar_risk = CVaR Risk for Loss
Box constraints = constraints on individual decision variables

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Dataset 3,000 10,000 3.41392772129 1.16 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data R R Code Data
PROBLEM 3: problem_avg_var_CLT
Minimize Avg_var_risk_ni (minimizing Average Value-at-Risk for Normal Independent Distribution)
subject to
Linear = 1 (budget constraint)
Linear ≥ Const (portfolio return constraint)
Box constraints (bounds on decision variables)
——————————————————————–
Avg_var_risk_ni = Average Value-at-Risk for Multivariate Normal Independent Distribution
Box constraints = constraints on individual decision variables

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Dataset 3,000 10,000 3.13761873657 371.68 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data R R Code Data
PROBLEM 4: problem_avg_cvar_CLT
Minimize Avg_cvar (minimizing Average Conditional Value-at-Risk for Normal Independent Distribution)
subject to
Linear = 1 (budget constraint)
Linear ≥ Const (portfolio return constraint)
Box constraints (bounds on decision variables)
——————————————————————–
Avg_cvar_risk_ni = Average Conditional Value-at-Risk for Multivariate Normal Independent Distribution
Box constraints = constraints on individual decision variables

——————————————————————–

Dataset 3,000 10,000 3.648889845209 241.56 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data R R Code Data
PROBLEM 5: problem_avg_var_CLT (alternative formulation)
Minimize Xvar (minimizing Average Value-at-Risk for Normal Independent Distribution using alternative formulation of Problem 3)
subject to
Linear = 1 (budget constraint)
Avg_pr_pen_ni ≤ 1-α (probability constraint)
Linear ≥ Const (portfolio return constraint)
Box constraints (bounds on decision variables)
——————————————————————–
Xvar = additional variable which is equal to VaR at optimality
Avg_pr_pen_ni = Average Probability Exceeding Penalty for Loss Normal Independent
Box constraints = constraints on individual decision variables
Box constraints = constraints on individual decision variables

——————————————————————–

Dataset 3,001 10,000 3.13761873657 371.68 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data R R Code Data
PROBLEM 6: problem_avg_cvar_CLT (alternative formulation)
Minimize Xvar + 1/(1-α )*Avg_pm_pen_ni (minimizing Average Conditional Value-at-Risk for Normal Independent Distribution using alternative formulation of Problem 4)
subject to
Linear = 1 (budget constraint)
Linear ≥ Const (portfolio return constraint)
Box constraints (bounds on decision variables)
——————————————————————–
Xvar = additional variable which is equal to VaR at optimality
Avg_pm_pen_ni = Average Partial Moment Penalty for Loss Normal Independent
Box constraints = constraints on individual decision variables

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