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

** 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

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset1 | 26 | 3046 | 0.000657006 | 0.08 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab Subroutines | Matlab Code | Data | |||||

R | R Code | Data |

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

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset | 76 | 30,000 | 0.018652555968 | 1.41 | |||
---|---|---|---|---|---|---|---|

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

——————————————————————–

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset | 76 | 30,000 | 0.018652555968 | 1.40 | |||
---|---|---|---|---|---|---|---|

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).