Background

Problem 1. Minimizing CVaR deviation

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

Problem 2. Minimizing standard deviation calculated with the covariance matrix

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

Problem 3. Minimizing standard deviation calculated with the matrix of scenarios

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

 

 

Background

 

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. 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.

 

 

Problem 1

Single-period portfolio optimization problem when risk is measured by CVaR deviation.

 

Simplified Problem Statement

 

Minimize Cvar_dev (minimizing Cvar deviation)

 subject to

Linear = 1 (budget constraint)

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

Box constraints (lower bounds on weights)

 

where

 

Cvar_dev = CVaR Deviation for Loss

Box constraints = constraints on individual decision variables

 

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with PSG MATLAB subroutine riskparam (General (Text) Format of PSG in MATLAB):
 

 

Input Files to run CS:

 

 

Problem 2

Single-period portfolio optimization problem when risk is measured by standard deviation calculated by using the covariance matrix.

 

Simplified Problem Statement

 

Minimize Sqrt_quadratic (minimizing standard deviation calculated with the covariance matrix)

 subject to

Linear = 1 (budget constraint)

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

Box constraints (lower bounds on weights)

 

where

 

Sqrt_quadratic = PSG function which implements Standard Deviation calculated with covariance matrix

Box constraints = constraints on individual decision variables

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with PSG MATLAB subroutine riskparam (General (Text) Format of PSG in MATLAB):
 

 

Input Files to run CS:

 

 

Problem 3

Single-period portfolio optimization problem when risk is measured by standard deviation calculated with the matrix of scenarios.

 

Simplified Problem Statement

 

Minimize St_dev (minimizing standard deviation calculated with the matrix of scenarios)

 subject to

Linear = 1 (budget constraint)

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

Box constraints (lower bounds on weights)

 

where

 

St_dev = Standard Deviation

Box constraints = constraints on individual decision variables

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with PSG MATLAB subroutine riskparam (General (Text) Format of PSG in MATLAB):
 

 

Input Files to run CS: