Case Study: Portfolio Optimization with Nonlinear Transaction Costs

TEST PROBLEMS
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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_1_Trans_Costs
Maximize Avg_g (maximizing expected portfolio gain)
subject to
Cvar ≤ Const1 (constraint on CVaR)
Polynom_abs + Linear ≤ Const2 (budget constraint (linear transaction costs,
(linear transaction costs, power coefficient in Polynom_abs function for each instrument = 1))
Box constraints (bounds on positions)
——————————————————————–
Avg_g = Average Gain
Polynom_abs = Polynomial Absolute
Box constraints = constraints on individual decision variables
——————————————————————–

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset 100 224 1309.67359 0.02
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data
R R Code Data

PROBLEM 2: problem_2_Trans_Costs
Maximize Avg_g (maximizing expected portfolio gain)
subject to
Cvar_risk ≤ Const3 (constraint on CVaR)
Linear + Linear ≤ Const4
Box constraints (bounds on positions)
——————————————————————–
Avg_g = Average Gain
Cvar_risk = CVaR Risk for Loss
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 200 224 1309.67065 0.02
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 200 224 1309.67 0.04
Data and solution in R Environment
Problem Datasets # of Variables # of Scenarios Objective Value Solving Time, PC 3.50GHz (sec)
Dataset1 R code Data 200 224 1309.67 0.04

PROBLEM 3: problem_3_Trans_Costs
Maximize Avg_g (maximizing expected portfolio gain)
subject to
Cvar_risk ≤ Const51 (constraint on CVaR)
Polynom_abs + Linear ≤ Const6 (budget constraint,
(linear transaction costs, power coefficient in Polynom_abs function for each instrument > 1))
Box constraints (bounds on positions)
——————————————————————–
Avg_g = Average Gain
Cvar_risk = CVaR Risk for Loss
Polynom_abs = Polynomial Absolute
Box constraints = constraints on individual decision variables
——————————————————————–

# of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec)
Dataset 100 224 193.59187 1.04
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Subroutines Matlab Code Data
R R Code Data

CASE STUDY SUMMARY
This case study considers portfolio optimization problem with the average gain objective function, CVaR constraint, and nonlinear transaction cost depending upon the total dollar value of the bought/sold assets, see, for instance Krokhmal et al., (2002, 2006). The basic portfolio optimization methodology with CVaR functions is described in Rockafellar and Uryasev (2000). For a treatment of non-convex transaction costs see Konno and Wijayanayake (1999). This case study presents two equivalent problem formulations. The first problem formulation uses nonlinear function polynomial_abs to account for nonlinear transactions costs involving both short and long positions. The second problem formulation, in case of linear transaction costs, doubles number of variables and uses linear functions for accounting for transaction costs (similar to Krokhmal at al., (2002)). The first formulation may be preferable for portfolios with large number of instruments or scenarios and when transaction costs nonlinearly depend upon investments.

References
• Krokhmal. P., Palmquist, J., and S. Uryasev (2002): Portfolio Optimization with Conditional Value-At-Risk Objective and Constraints. The Journal of Risk, V. 4, No. 2, 11-27.
• Krokhmal P. and S. Uryasev (2006): A Sample-Path Approach to Optimal Position Liquidation. Annals of Operations Research, Published Online, November, 1-33.
• Rockafellar, R. T. and S. Uryasev (2000): Optimization of Conditional ValueAt-Risk, The Journal of Risk, Vol. 2, No. 4, pp. 21-51.
• Konno, H. and A. Wijayanayake (2002): Portfolio optimization under D.C. transaction costs and minimal transaction unit constraints, Journal of Global Optimization, V. 22, Issue 1-4 (January), 137-154.