# Case Study: Portfolio Optimization with Drawdown Constraints on a Single Path

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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_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

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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 0.809763 <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

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

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

• 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.