# Case Study: Portfolio Optimization with Drawdown Constraints on Multiple Paths

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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_max_drawdown_0p08**

Maximize Linear (maximizing average annualized portfolio return)

subject to

Drawdownmulti_dev_max ≤ Const (constraint on the maximum drawdown)

Box constraints (lower and upper bounds on weights)

——————————————————————–

Drawdownmulti_dev_max = Drawdown Deviation Maximum Multiple

Box constraints = constraints on individual decision variables

——————————————————————–

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

Dataset1 | 31 | 12,925 | 0.572829 | 0.02 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

Instructions for importing problems from Run-File to PSG MATLAB.

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

Dataset2 | Problem Statement | Data | Solution | 18 | 211,680 | 0.248021 | 0.16 |

**PROBLEM 2: problem_average_drawdown_0p009**

Maximize Linear (maximizing average annualized portfolio return)

subject to

Drawdownmulti_dev_avg ≤ Const (constraint on the average drawdown)

Box constraints (lower and upper bounds on weights)

——————————————————————–

Drawdownmulti_dev_avg = Drawdown Deviation Average Multiple

Box constraints = constraints on individual decision variables

——————————————————————–

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

Dataset1 | 31 | 12,925 | 0.276401 | 0.19 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

Instructions for importing problems from Run-File to PSG MATLAB.

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

Dataset2 | Problem Statement | Data | Solution | 18 | 211,680 | 0.190913 | 14.34 |

**PROBLEM 3: problem_CDAR_0p03**

Maximize Linear (maximizing average annualized portfolio return)

subject to

Cdarmulti_dev ≤ Const (constraint on the CDaR)

Box constraints (lower and upper bounds on weights)

——————————————————————–

Cdarmulti_dev = CDaR Deviation Multiple

Box constraints = constraints on individual decision variables

——————————————————————–

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

Dataset1 | 31 | 12,925 | 0.384147 | 0.19 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

Instructions for importing problems from Run-File to PSG MATLAB.

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

Dataset2 | Problem Statement | Data | Solution | 18 | 211,680 | 0.247420 | 7.80 |

**CASE STUDY SUMMARY**

This case study demonstrates an optimization setup for Conditional Drawdown-at-Risk (CDaR) deviation with multiple sample paths. For some value of the confidence parameter Conditional Drawdown-at-Risk (CDaR) deviation on multiple paths is defined as the mean of worst (1-) * 100% drawdowns taken simultaneously over time and sample paths (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 = 0 corresponds to the average drawdown and = 1 corresponds to maximum drawdown). The optimization problem maximizes annualized portfolio return subject to constraints on CDaR multiple deviation with various values of the confidence parameter (including limiting cases: average and maximum drawdown).

Each problem in the case study is solved using 2 datasets:

• Dataset1 for “short case study” including 31 variables and 11 and sample paths (12,925 scenarios);

• Dataset2 for “long case study” including 18 variables and 180 sample paths (211,680 scenarios).

• Dataset1 for “short case study” including 31 variables and 11 and sample paths (12,925 scenarios);

• Dataset2 for “long case study” including 18 variables and 180 sample paths (211,680 scenarios).

**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.