# Case Study: Portfolio Optimization with Second-Order Stochastic Dominance Constraints

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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_ssd (with MultiConstraint)
Maximize Avg_g (maximizing portfolio mean return)
subject to
Linear = 1 (budget constraint)
Pm_pen (Vector_of_thresholds) ≤ Vector_of_Constants (Second Order Stochastic Dominance (SSD))
Box constraints (bounds on decision variables)
——————————————————————–
Avg_g = Average Gain
Pm_pen = Partial Moment Penalty for Loss
Box constraints = constraints on individual decision variables
——————————————————————–
MATLAB code preparing Dataset 1

Dataset1 26 3046 0.000657006 0.08 # 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 3.4GHz (sec)
Dataset2 Problem Statement Data Solution 29 3020 0.000334689 0.05
Dataset3 Problem Statement Data Solution 90 3020 0.000865278 0.21
PROBLEM 2: problem_ssd (with MultiConstraint)
Maximize Avg_g (maximizing portfolio mean return)
subject to
Linear = 1 (budget constraint)
Pm_pen (Vector_of_thresholds) ≤ Vector_of_Constants (Second Order Stochastic Dominance (SSD))
Box constraints (bounds on decision variables)
——————————————————————–
Avg_g = Average Gain
Pm_pen = Partial Moment Penalty for Loss
Box constraints = constraints on individual decision variables
——————————————————————–

Dataset 76 30,000 0.018652555968 1.41 # 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

NOTE:
Problem 2 is a simplified variant (with MultiConstraint ) of the original Problem 3 (with many individual constraints) presented below.

PROBLEM 3: problem_ssd (with a list of constraints)

Maximize Avg_g (maximizing portfolio mean return)
subject to
Linear = 1 (budget constraint)
Pm_pen ≤ Constj, j =1,…,J
Box constraints (bounds on decision variables)
——————————————————————–
Avg_g = Average Gain
Pm_pen = Partial Moment Penalty for Loss
Box constraints = constraints on individual decision variables
——————————————————————–

Dataset 76 30,000 0.018652555968 1.40 # 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

CASE STUDY SUMMARY

This case study finds a maximum expected return portfolio dominating the benchmark in the Second Order. Mean-Risk models (standardly used in portfolio optimization) are convenient from the computational point of view and have an intuitive appeal. However, these models use only two statistics to characterize the distribution (and ignore other important information about the distribution). Stochastic dominance, in contrast, takes into account the entire distribution of a random variable. The second-order stochastic dominance is an important criterion in portfolio selection. This case study does optimization with several datasets from Fidan Keçeci et al. and Fabian et al.

SSD constraints may contain many redundant nonlinear constraints. PSG MultiConstraint option does automatic preprocessing and removes redundant constraints.

References
• Fabian, C.I., Mitra, G, Roman, D., and V. Zverovich (2011): An enhanced model for portfolio choice with SSD criteria: a constructive approach. Quantitative Finance, 11(10), 1525-1534.
• Rudolf, G., and A. Ruszczynski (2008): Optimization problems with second order stochastic dominance constraints: duality, compact formulations, and cut generation methods, SIAM J. OPTIM, Vol. 19, No. 3, pp. 1326–1343.
• Roman, D., Darby-Dowman, K., and G. Mitra (2006): Portfolio construction based on stochastic dominance and target return distributions, Mathematical Programming, Series B, Vol. 108, pp. 541-569.
• Ogryczak,W., and A. Ruszczynski (1999): From stochastic dominance to mean–risk models: Semideviations as risk measures. European Journal of Operational Research, Vol. 116, pp. 33–50.
• Fidan Keçeci, N., Kuzmenko, V., and S. Uryasev (2015): Portfolios Dominating Indices: Optimization with Second-Order Stochastic Dominance Constraints. Journal of Risk and Financial Management (to appear).