# 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 Toolbox, PSG MATLAB Subroutines and PSG R.

**PROBLEM 1: problem_1**

Maximize Exp_eut (maximizing Exponential Utility)

subject to

Linear = Const (budget constraint)

Box constraints (bounds on positions)

——————————————————————–

Exp_eut = xponential Utility

Box constraints = constraints on individual decision variables

——————————————————————– Data 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 | -15.030566742946 | 1.56 |

Data and solution in MATLAB Environment

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 Environment

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 Utility

subject to

Linear = Const (budget constraint)

Box constraints (bounds on positions)

——————————————————————–

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

Box constraints = constraints on individual decision variables

——————————————————————– Data 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 | 20.033358148852 | 0.17 |

Data and solution in MATLAB Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Matlab code | Data | Solution | 12 | 199,554 | 20.033358148852 | 0.28 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | R code | Data | 12 | 199,554 | 20.033358148852 | 0.28 |

**PROBLEM 3: problem_3**

Maximize Log_eut (maximizing Logarithmic Utility)

subject to

Linear = Const (budget constraint)

Box constraints (bounds on positions)

——————————————————————–

Log_eut = Logarithmic Utility

Box constraints = constraints on individual decision variables

——————————————————————– Data 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 | 0.224365972673 | 0.18 |

Data and solution in MATLAB Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Matlab code | Data | Solution | 12 | 199,554 | 0.224365972673 | 0.14 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | R code | Data | 12 | 199,554 | 0.224365972673 | 0.14 |

**CASE STUDY SUMMARY**

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.