# Case Study: Portfolio Optimization, CVaR vs. ST_DEV

** 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_min_cvar_dev_2p9**

Minimize Cvar_dev (minimizing portfolio Cvar deviation)

subject to

Linear = 1 (budget constraint)

Linear ≥ Const (constraint on the portfolio rate of return)

Box constraints (lower bounds on weights)

——————————————————————–

Cvar_dev = CVaR Deviation 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 | 10 | 1,000 | 0.03631774 | <0.01 |

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 | 10 | 1,000 | 0.0363177 | <0.01 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | R code | Data | 10 | 1,000 | 0.0363177 | <0.01 |

**PROBLEM 2: problem_st_dev_covariances_2p9**

Minimize Sqrt_quadratic (minimizing risk measured by standard deviation calculated with the covariance matrix))

subject to

Linear = 1 (budget constraint)

Linear ≥ Const (constraint on the portfolio rate of return)

Box constraints (lower bounds on weights)

——————————————————————–

Sqrt_quadratic = PSG function which implements Standard Deviation calculated with covariance matrix

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 | 10 | 10 * 10 | 0.00874964 | <0.01 |

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 | 10 | 10 * 10 | 0.00874964 | <0.01 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | R code | Data | 10 | 10 * 10 | 0.00874964 | <0.01 |

**PROBLEM 3: problem_st_dev_scenarios_2p9**

Minimize St_dev (minimizing risk measured by standard deviation calculated with the matrix of scenarios)

subject to

Linear = 1 (budget constraint)

Linear ≥ Const (constraint on the portfolio rate of return)

Box constraints (lower bounds on weights)

——————————————————————–

St_dev = Standard Deviation

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 | 10 | 1,000 | 0.008749650 | <0.01 |

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 | 10 | 1,000 | 0.008749650 | 0.02 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | R code | Data | 10 | 1,000 | 0.008749650 | 0.02 |

**CASE STUDY SUMMARY**

This case study compares three setups of a single-period portfolio optimization problem when risk is measured by CVaR Deviation, Standard Deviation calculated with the matrix of scenarios, and Standard Deviation calculated with the covariance matrix. In the third setup we use sqrt_quadratic PSG function. The second and the third setups are equivalent representations of the Markowitz (1952) problem trading-off mean and variance of portfolio return. The original Markowitz problem finds a minimum-variance portfolio under restriction on mean return. Here we keep a similar setup but with the CVaR deviation as a replacement to the Standard deviation.