** Case study background and problem formulations**

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, and PSG R.

**PROBLEM1: Problem_CVaR_only**

Minimize -Linear+Quadratic ( expected portfolio terminal value + regularization)

subject to

Portfolio value = sum(asset values)

Cash outflow = sum(adjustments)

Asset position value dynamics

Lower and upper bounds on asset positions

CVaR(-cash outflow – annuity yield) < -required cash outflow

Box constraints

——————————————————————–

Quadratic = External Quadratic Penalty Function

Box constraints = constraints on individual decision variables

——————————————————————–

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

Dataset 1 | 113752 | 100 | -3.74178374384 | 292.48 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Data | Solution | |||||

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

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

Dataset 2 | 113752 | 100 | -0.02226697343924 | 353.01 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Data | Solution | |||||

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

**PROBLEM2: Problem_CVaR_plus_monotonicity**

Minimize -Linear+Quadratic ( expected portfolio terminal value + regularization)

subject to

Portfolio value = sum(asset values)

Cash outflow = sum(adjustments)

Asset position value dynamics

Lower and upper bounds on asset positions

CVaR(-cash outflow – annuity yield) = 0 (monotonicity constraint)

Box constraints (non-negativity constraints on positions)

——————————————————————–

Quadratic = External Quadratic Penalty Function

Box constraints = constraints on individual decision variables

——————————————————————–

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

Dataset 1 | 113752 | 100 | -3.74140546345 | 450.37 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Data | Solution | |||||

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

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

Dataset 2 | 113752 | 100 | -0.0222384434066 | 554.95 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Data | Solution | |||||

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

**CASE STUDY SUMMARY**

This case study solves the retirement portfolio selection problem. The objective is to maximize discounted terminal wealth of the investor, while maintaining constant cash outflows from the portfolio by selling some portion of assets, over an entire investment horizon. The cash outflow requirements are imposed using CVaR constraints and additionally, by monotonicity constraint on the cash outflows.