# 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

——————————————————————–

Dataset1 31 12,925 0.572829 0.02 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data R R Code Data
Download other datasets in Run-File Environment.
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

——————————————————————–

Dataset1 31 12,925 0.276401 0.19 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data R R Code Data
Download other datasets in Run-File Environment.
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

——————————————————————–

Dataset1 31 12,925 0.384147 0.19 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data R R Code Data
Download other datasets in Run-File Environment.
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 α Conditional Drawdown-at-Risk (CDaR) deviation on multiple paths is defined as the mean of worst (1-α)* 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 α = 0 corresponds to the average drawdown and α = 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).

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.