**PROBLEM 1: maximizing the expected return subject to bounded negative expectile risk**

Maximize Avg_g (maximizing the expected return of financial instruments)

subject to

expectile <= Const1 (constraint on the negative expectile risk of the portfolio)

Linear = Const2 (budget constraint)

Box constraints (box constraints for individual positions)

——————————————————————–

Avg_g = Average Gain

Box constraints = constraints on individual decision variables

———————————————————————

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset 1 | 4 | 10000 | 0.00094986 | 0.06 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab | Matlab Code | Data | |||||

R | R Code | Data |

**PROBLEM 2: minimizing expectile risk subject to bounded from below expected returns**

Minimize expectile (minimize negative expectile risk of the portfolio)

subject to

Avg_g >= Const1 (constraint on the expected return of financial instruments)

Linear = Const2 (budget constraint)

Box constraints (box constraints for individual positions)

——————————————————————–

Avg_g = Average Gain

Box constraints = constraints on individual decision variables

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset 1 | 4 | 10000 | 0.02447960 | 0.01 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab | Matlab Code | Data | |||||

R | R Code | Data |

**PROBLEM 3: maximizing the expected return subject to bounded negative expectile risk**

Maximize Avg_g (maximizing the expected return of financial instruments)

subject to

expectile <= Const1 (constraint on the negative expectile risk of the portfolio)

Linear = Const2 (budget constraint)

Box constraints (box constraints for individual positions)

——————————————————————–

Avg_g = Average Gain

Box constraints = constraints on individual decision variables

——————————————————————–

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

Dataset | Matlab code | Data | Solution | 4 | 10000 | 9.49104E-04 | 0.25 |

**PROBLEM 4: minimizing expectile risk subject to bounded from below expected returns**

Minimize expectile (minimize negative expectile risk of the portfolio)

subject to

Avg_g >= Const1 (constraint on the expected return of financial instruments)

Linear = Const2 (budget constraint)

Box constraints (box constraints for individual positions)

——————————————————————–

Avg_g = Average Gain

Box constraints = constraints on individual decision variables

——————————————————————–

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

Dataset | Matlab code | Data | Solution | 4 | 10000 | 2.52517E-02 | 0.72 |

**CASE STUDY SUMMARY**

This case study demonstrates portfolio optimization problem when risk is measured by negative expectile risk. Two cases of formulation when risk is minimized and bounded subject to expected return of financial instruments are considered. Results are obtained using PSG external functions interface in MATLAB.