** 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_max_drawdown_0p170**

Maximize Linear (maximizing average annualized portfolio return)

subject to

Drawdown_dev_max ≤ Const (constraint on the maximum drawdown)

Box constraints (lower and upper bounds on weights)

——————————————————————–

Drawdown_dev_max = Drawdown Deviation Maximum

Box constraints = constraints on individual decision variables

——————————————————————–

Data and solution in Run-File Environment

Maximize Linear (maximizing average annualized portfolio return)

subject to

Drawdown_dev_avg ≤ Const (constraint on the average drawdown)

Box constraints (lower and upper bounds on weights)

——————————————————————–

Drawdown_dev_avg = Drawdown Deviation Average

Box constraints = constraints on individual decision variables

Maximize Linear (maximizing average annualized portfolio return)

subject to

Cdar_dev ≤ Const (constraint on the CDaR)

Box constraints (lower and upper bounds on weights)

——————————————————————–

Cdar_dev = CDaR Deviation

Box constraints = constraints on individual decision variables

This case study demonstrates an optimization setup with Conditional Drawdown-at-Risk (CDaR) deviation on a single sample path. For some value of the confidence parameter

• Chekhlov, A., Uryasev S., and M. Zabarankin (2003): Portfolio Optimization with Drawdown Constraints, in

• Chekhlov, A., Uryasev S., and M. Zabarankin (2005): Drawdown Measure in Portfolio Optimization,

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Problem Statement | Data | Solution | 32 | 1,166 | 0.809763 | <0.01 |

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 | 32 | 1,166 | 0.809763 | <0.01 |

Data and solution in R Environment

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

Dataset1 | R code | Data | 32 | 1,166 | 0.809763 | <0.01 |

PROBLEM 2: problem_average_drawdown_0p0307

PROBLEM 2: problem_average_drawdown_0p0307

Maximize Linear (maximizing average annualized portfolio return)

subject to

Drawdown_dev_avg ≤ Const (constraint on the average drawdown)

Box constraints (lower and upper bounds on weights)

——————————————————————–

Drawdown_dev_avg = Drawdown Deviation Average

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 | 32 | 1,166 | 0.763602 | <0.01 |

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 | 32 | 1,166 | 0.763602 | <0.01 |

Data and solution in R Environment

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

Dataset1 | R code | Data | 32 | 1,166 | 0.763602 | <0.01 |

PROBLEM 3: problem_CDAR_0p110

PROBLEM 3: problem_CDAR_0p110

Maximize Linear (maximizing average annualized portfolio return)

subject to

Cdar_dev ≤ Const (constraint on the CDaR)

Box constraints (lower and upper bounds on weights)

——————————————————————–

Cdar_dev = CDaR Deviation

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 | 32 | 1,166 | 0.754324 | <0.01 |

Data and solution in MATLAB Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Matlab code | Data | Solution | 32 | 1,166 | 0.754324 | <0.01 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | R code | Data | 32 | 1,166 | 0.754324 | <0.01 |

**CASE STUDY SUMMARY**

This case study demonstrates an optimization setup with Conditional Drawdown-at-Risk (CDaR) deviation on a single sample path. For some value of the confidence parameter

*α*Conditional Drawdown-at-Risk (CDaR) deviation on a sample path is defined as the mean of worst (1-

*α*) * 100% drawdowns (see Chekhlov

*et al*. (2003, 2005). This deviation measure is considered in active portfolio management. Negative drawdown curve is called the “underwater curve”. Maximal and average drawdowns are limiting cases of CDaR deviation (where

*α*= 0 corresponds to the average drawdown and

*α*= 1 corresponds to maximum drawdown). The optimization problem maximizes annualized portfolio return on a sample path subject to constraints on CDaR deviation with various values of the confidence parameter (including limiting cases: average and maximum drawdown).

References

References

• Chekhlov, A., Uryasev S., and M. Zabarankin (2003): Portfolio Optimization with Drawdown Constraints, in

*Asset and Liability Management Tools*, ed. B. Scherer (Risk Books, London) pp. 263–278.

• Chekhlov, A., Uryasev S., and M. Zabarankin (2005): Drawdown Measure in Portfolio Optimization,

*International Journal of Theoretical and Applied Finance,*Vol. 8, No. 1, pp. 13–58.