# Case Study: Portfolio Optimization with Mixed CVaR and Mixed VaR Profiles

** 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_Mixed_CVaR_Profile**

maximize Avg_g (maximizing average expected return of the portfolio)

subject to

mixed CVaR ≤A (constraint on the mixed CVaR for the overall portfolio)

CVaR _k ≤ B_k , k = 1,…,K (constraints on individual risks formulated with CVaR)

max_risk_n ≤ B_n, n=1,…,N (set of constraints on individual risks formulated with Maximum Risk)

Box constraints (lower and upper bounds on variables)

——————————————————————–——————

mixed CVaR = weighted sum of CVaRs with different confidence levels

CVaR_k = CVaR of k-th individual risk (including contributions from various contracts)

max_risk_n = Maximum Risk of n-th individual risk (including contributions from various contracts)

Box constraints = constraints on individual decision variables

——————————————————————–——————

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

Dataset1 | 804 | 5,000 | 1,294,974,119.20 | 44.78 | |||
---|---|---|---|---|---|---|---|

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 Rows | Objective Value | Solving Time, PC 2.83GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset2 | Problem statement | Data | Solution | 804 | 10,000 | 1,295,057,851.40 | 24.39 |

Dataset3 | Problem statement | Data | Solution | 804 | 100,000 | 1,243,596,513.73 | 624.17 |

**NOTE:** Problem statements can be simplified using InnerProduct and MultiConstraints.

**PROBLEM 2: problem_Mixed_VaR_Profile**

maximize Avg_g (maximizing average expected return of the portfolio)

subject to

mixed VaR ≤A (constraint on the mixed VaR for the overall portfolio)

VaR _k ≤ B_k , k = 1,…,K (constraints on individual risks formulated with VaR)

max_risk_n ≤ B_n, n=1,…,N (set of constraints on individual risks formulated with Maximum Risk)

Box constraints (lower and upper bounds on variables)

——————————————————————–——————

mixed VaR = weighted sum of VaRs with different confidence levels

VaR_k = VaR of k-th individual risk (including contributions from various contracts)

max_risk_n = Maximum Risk of n-th individual risk (including contributions from various contracts)

Box constraints = constraints on individual decision variables

——————————————————————–——————

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

Dataset1 | 804 | 5,000 | 1,566,047,896.30 | 246.36 | |||
---|---|---|---|---|---|---|---|

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 Rows | Objective Value | Solving Time, PC 2.83GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset2 | Problem statement | Data | Solution | 804 | 10,000 | 1,398,113,477.37 | 290.71 |

Dataset3 | Problem statement | Data | Solution | 804 | 100,000 | 1,487,959,976.69 | 2,448.99 |

**NOTE:** Problem statements can be simplified using InnerProduct and MultiConstraints.

**CASE STUDY SUMMARY**

This case study considers a porfolio optimization problem solved by an insurance company. Two setups of the problem are considered. In the both setups the objective function is the average expected return of the portfolio. There are two groups of constraints: a) constraint on the mixed CVaR (in the first setup) or VaR (in the second setup) for the overall portfolio; b) constraints on individual risks (including contributions from various contracts). Constraints on individual risks are formulated with CVaR (in the first setup), VaR (in the second setup), and Maximum Risk (in both setups) over scenarios.