This case study demonstrates an optimization setup for Conditional Drawdown-at-Risk (CDaR) deviation with multiple sample paths. Here we consider a case study with 180 sample paths of the underlying instruments.

 

Background

Problem 1. Constraint on the maximum drawdown

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

Problem 2. Constraint on the average drawdown

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

Problem 3. Constraint on the CDaR

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

References

 

Background

 

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).

 

Problem 1

 

Simplified Problem Statement

 

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)

 

where

 

Drawdownmulti_dev_max = Drawdown Deviation Maximum Multiple

Box constraints = constraints on individual decision variables

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with PSG MATLAB function tbpsg_run (PSG Subroutine Interface):

 

 

Input Files to run CS:

 

 

Problem 2

 

Simplified Problem Statement

 

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)

 

where

 

Drawdownmulti_dev_avg = Drawdown Deviation Average Multiple

Box constraints = constraints on individual decision variables

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with PSG MATLAB function tbpsg_run (PSG Subroutine Interface):

 

 

Input Files to run CS:

 

 

Problem 3

 

Simplified Problem Statement

 

Maximize Linear (maximizing average annualized portfolio return)

subject to

Cdarmulti_dev ≤ Const (constraint on the CDaR)

Box constraints (lower and upper bounds on weights)

 

where

 

Cdarmulti_dev = CDaR Deviation Multiple

Box constraints = constraints on individual decision variables

 

Mathematical Problem Statement

 

 

Problem dimension and solving time

 

 

Solution in Run-File Environment

 

 

Input Files to run CS:

 

Output Files:

 

Solution in MATLAB Environment

 

Solved with PSG MATLAB function tbpsg_run (PSG Subroutine Interface):

 

 

Input Files to run CS:

 

 

 

References

 

[1]  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.

[2]  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