Case Study: VaR vs Difference of CVaRs Minimization

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)nsubject 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 ——————————————————————– Download other datasets in Run-File Environment. 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	-0.002140167	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)nsubject 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 ——————————————————————– Download other datasets in Run-File Environment. 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	-0.002115116	0.07

CASE STUDY SUMMARYn 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.