Background

Problem 1. Maximizing estimated return with probability constraint

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

Problem 2. Maximizing estimated return with VaR constraint

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

 

 

Background

This case study demonstrates the equivalence between chance constraints and VaR constraints, as explained in Sarykalin et al. (2008). Several engineering applications deal with probabilistic constraints such as the reliability of a system or a delivery system likelihood to meet a demand. In portfolio management, often it is required that portfolio loss with high reliability should not exceed some value. In these cases an optimization model can be set up so that constraints are required to be satisfied with some probability level rather than almost surely. Chance constraints and VaR (percentile) constraints are closely related. We will illustrate numerically the equivalence of the constraints:

 

is equivalent to

 

i.e., the constraint assuring that the probability that loss exceeding is less or equal than is equivalent to the constraint that VaR (percentile) with confidence level is less or equal than . The PSG function “Probability Exceeding Penalty for Loss” implements the function  and PSG function “VaR Risk for Loss” implements .

 

We solved two portfolio optimization problems. In both cases we maximized the estimated return of the portfolio. In the first problem, we imposed a constraint on probability; in the second problem, we imposed an equivalent constraint on VaR. For two problems we obtained at optimality the same objective function values and similar optimal portfolios.

 

Problem 1

Maximizing estimated return with probability constraint.

 

Simplified Problem Statement

 

Maximize Linear (maximizing estimated return)

 subject to:

Pr_pen ≤ Const1 (probability constraint)

Linear = 1 (budget constraint)

Box constraints (upper bounds on positions)

 

where

 

Pr_pen = Probability Exceeding Penalty 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 riskconstrprog PSG subroutine (General (Text) Format of PSG in MATLAB):

 

Input Files to run CS:

 

 

Problem 2

Maximizing estimated return with VaR constraint.

 

Simplified Problem Statement

 

Maximize Linear (maximizing estimated return)

 subject to

Var_risk ≤ Const2 (VaR constraint)

Linear = 1 (budget constraint)

Box constraints (upper bounds on positions)

 

where

 

Var_risk = VaR Risk for Los

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 riskconstrprog PSG subroutine (General (Text) Format of PSG in MATLAB):

 

Input Files to run CS: