Background

Problem 1. Maximizing  annualized portfolio return on multiple sample paths subject to constraint on CDaR Deviation Multiple

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

Problem 2. Maximizing  annualized portfolio return on the single united sample path subject to constraint on CDaR Deviation

Simplified Problem Statement

Mathematical Problem Statement

Problem dimension and solving time

Solution in Run-File Environment

Solution in MATLAB Environment

References

 

Background

 

This case study considers two optimization setups for Conditional Drawdown-at-Risk (CDaR) deviation. The first setup operates with multiple sample paths as separate units similarly to the case study Portfolio Optimization with Drawdown Constraints on Multiple Paths, (cdarmulti_dev, drawdownmulti_dev_max, drawdownmulti_dev_avg). The second setup combines all multiple sample paths into one united single sample path and operates with it similarly to the case study "Portfolio Optimization with Drawdown Constraints on a Single Path (cdar_dev drawdown_dev_max drawdown_dev_avg)”. This case study compares solutions of two optimization problems:  (1) maximizing  annualized portfolio return on multiple sample paths subject to constraint on CDaR Deviation Multiple, and  (2) maximizing  annualized portfolio return on the single united sample path subject to constraint on CDaR Deviation. In the first problem, for some value of the confidence parameter Conditional Drawdown-at-Risk (CDaR) deviation on a sample path is defined as the mean of worst (1-)*100% drawdowns  taken simultaneously over time and sample paths (see Chekhlov et al. (2003, 2005)). In the second problem Conditional Drawdown-at-Risk (CDaR) deviation on the united sample path is defined as the mean of worst (1-)*100% drawdowns (see Chekhlov et al. (2003, 2005)).

 

Problem 1

Maximizing  annualized portfolio return on multiple sample paths subject to constraint on CDaR Deviation Multiple.

 

Simplified Problem Statement

 

Maximize Linear (maximizing average annualized portfolio return)

subject to

Cdarmulti_dev ≤ Const (constraint on CDaR Deviation Multiple (for multiple paths))

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:

 

 

Problem 2

Maximizing  annualized portfolio return on the single united sample path subject to constraint on CDaR Deviation.

 

Simplified Problem Statement

 

Maximize Linear (maximizing average annualized portfolio return)

 subject to

Cdar_dev ≤ Const (constraint on the CDaR)

Box constraints (lower and upper bounds on weights)

 

where

 

Cdar_dev = CDaR Deviation

 

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.