Background

Problem 1. Maximizing  Exponential  utility function

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

Problem 2. Maximizing  Linear-Quadratic utility function

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

Problem 3. Maximizing  Logarithmic utility function

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

 

 

Background

 

Utility functions are quite popular in various financial applications. This case study compares portfolio optimization problems with Exponential, Logarithmic, and Linear-Quadratic utility functions. The rate of return dataset for a portfolio is provided for benchmarking purposes by EpiRisk Research company via Drs. Roger Wets and Michael Tian. The EpiRisk Research relies on letting the manager of a fixed-income portfolio solve a sequence of so-called tacking (optimization) models, described below, to shape the returns' distribution. The shape of the distribution is adjusted by selecting the coefficients of the appraisal (~ utility) function.

 

Problem 1

Maximizing Exponential  utility function.

 

Simplified Problem Statement

 

Maximize Exponential_Utility(Return)

 subject to

Linear ≤ Const

Box constraints

 

where

 

Return = Portfolio rate of return

Logarithmic_Utility(Return) = expected value of exponential function of Return

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 function tbpsg_run (PSG Subroutine Interface):

 

 

Input Files to run CS:

 

 

Problem 2

Maximizing Linear-Quadratic utility function.

 

Simplified Problem Statement

 

Maximize Linear_Quadratic_Utility(Return)

 subject to

Linear ≤ Const

Box constraints

 

where

 

Return = Portfolio rate of return

Linear_Quadratic_Utility(Return) = expected value of piecewise linear-quardatic-linear function of Return

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 function tbpsg_run (PSG Subroutine Interface):

 

 

Input Files to run CS:

 

Problem 3

Maximizing Logarithmic utility function.

 

Simplified Problem Statement

 

Maximize Logarithmic_Quadratic_Utility(Return)

 subject to

Linear ≤ Const

Box constraints

 

where

 

Return = Portfolio rate of return

Logarithmic_Utility(Return) = expected value of piecewise linear-quardatic-linear function of Return

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 function tbpsg_run (PSG Subroutine Interface):

 

 

Input Files to run CS: