Case Study: Portfolio Optimization with Exponential, Logarithmic, and Linear-Quadratic Utilities

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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_1 Maximize Exp_eut (maximizing Exponential Utility)nsubject to
Linear = Const (budget constraint) Box constraints (bounds on positions) ——————————————————————– Exp_eut = xponential Utility
Box constraints = constraints on individual decision variables ——————————————————————– nData and solution in Run-File Environment

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 2.66GHz (sec)
Dataset1	Problem Statement	Data	Solution	12	199,554	-15.03056674	1.56

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	12	199,554	-15.031	0.98

Data and solution in R Environmentn

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.50GHz (sec)
Dataset1	R code	Data		12	199,554	-15.031	0.98

PROBLEM 2: problem_2 Maximize Linear-Quadratic Utilitynsubject to
Linear = Const (budget constraint) Box constraints (bounds on positions) ——————————————————————– Linear_Quadratic_Utility(Return) = expected value of piecewise linear-quardatic-linear function of Retur
Box constraints = constraints on individual decision variables ——————————————————————– nData 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	12	199,554	20.03335814	0.17

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	12	199,554	20.03335814	0.28

Data and solution in R Environmentn

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.50GHz (sec)
Dataset1	R code	Data		12	199,554	20.03335814	0.28

PROBLEM 3: problem_3 Maximize Log_eut (maximizing Logarithmic Utility)nsubject to
Linear = Const (budget constraint) Box constraints (bounds on positions) ——————————————————————– Log_eut = Logarithmic Utility
Box constraints = constraints on individual decision variables ——————————————————————– nData 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	12	199,554	0.22436597	0.18

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	12	199,554	0.22436597	0.14

Data and solution in R Environmentn

Problem Datasets				# of Variables	# of Scenarios	Objective Value	Solving Time, PC 3.50GHz (sec)
Dataset1	R code	Data		12	199,554	0.22436597	0.14

CASE STUDY SUMMARYn 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.