Case Study: Portfolio Optimization, CVaR vs. ST_DEV

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Case study background and problem formulations

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

PROBLEM 1: problem_min_cvar_dev_2p9 Minimize Cvar_dev (minimizing portfolio Cvar deviation)nsubject to
Linear = 1 (budget constraint) Linear ≥ Const (constraint on the portfolio rate of return) Box constraints (lower bounds on weights) ——————————————————————– Cvar_dev = CVaR Deviation for Loss
Box constraints = constraints on individual decision variables

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Data and solution in Run-File Environmentn

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 2.66GHz (sec)
Dataset1	Problem Statement	Data	Solution	10	1,000	0.03631774	<0.01

Data and solution in MATLAB Environmentn

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.50GHz (sec)
Dataset1	Matlab code	Data	Solution	10	1,000	0.0363177	<0.01

Data and solution in R Environmentn

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.50GHz (sec)
Dataset1	R code	Data		10	1,000	0.0363177	<0.01

PROBLEM 2: problem_st_dev_covariances_2p9 Minimize Sqrt_quadratic (minimizing risk measured by standard deviation calculated with the covariance matrix))nsubject to
Linear = 1 (budget constraint) Linear ≥ Const (constraint on the portfolio rate of return) Box constraints (lower bounds on weights) ——————————————————————– Sqrt_quadratic = PSG function which implements Standard Deviation calculated with covariance matrix
Box constraints = constraints on individual decision variables

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Data and solution in Run-File Environmentn

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 2.66GHz (sec)
Dataset1	Problem Statement	Data	Solution	10	10 * 10	0.00874964	<0.01

Data and solution in MATLAB Environmentn

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.50GHz (sec)
Dataset1	Matlab code	Data	Solution	10	10 * 10	0.00874964	<0.01

Data and solution in R Environmentn

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.50GHz (sec)
Dataset1	R code	Data		10	10 * 10	0.00874964	<0.01

PROBLEM 3: problem_st_dev_scenarios_2p9 Minimize St_dev (minimizing risk measured by standard deviation calculated with the matrix of scenarios)nsubject to
Linear = 1 (budget constraint) Linear ≥ Const (constraint on the portfolio rate of return) Box constraints (lower bounds on weights) ——————————————————————– St_dev = Standard Deviatio
Box constraints = constraints on individual decision variables

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Data and solution in Run-File Environmentn

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 2.66GHz (sec)
Dataset1	Problem Statement	Data	Solution	10	1,000	0.008749650	<0.01

Data and solution in MATLAB Environmentn

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.50GHz (sec)
Dataset1	Matlab code	Data	Solution	10	1,000	0.008749650	0.02

Data and solution in R Environmentn

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.50GHz (sec)
Dataset1	R code	Data		10	1,000	0.008749650	0.02

CASE STUDY SUMMARYn This case study compares three setups of a single-period portfolio optimization problem when risk is measured by CVaR Deviation, Standard Deviation calculated with the matrix of scenarios, and Standard Deviation calculated with the covariance matrix. In the third setup we use sqrt_quadratic PSG function. The second and the third setups are equivalent representations of the Markowitz (1952) problem trading-off mean and variance of portfolio return. The original Markowitz problem finds a minimum-variance portfolio under restriction on mean return. Here we keep a similar setup but with the CVaR deviation as a replacement to the Standard deviation.