# Case Study: VaR vs Probability 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_MaxRet_Prob**

Maximize Linear (maximizing estimated return)

subject to

Pr_pen ≤ Const1 (probability constraint)

Linear = 1 (budget constraint)

Box constraints (upper bounds on positions)

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Pr_pen = Probability Exceeding Penalty for Loss

Box constraints = constraints on individual decision variables

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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 | 0.00120185276974 | 0.01 |

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Data and solution in MATLAB Environment

Data and solution in MATLAB Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Matlab code | Data | Solution | 10 | 1,000 | 0.00120185 | 0.02 |

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Data and solution in R Environment

Data and solution in R Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value | Solving Time, PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | R code | Data | 10 | 1,000 | 0.00120185 | 0.02 |

**PROBLEM 2: problem_MaxRet_VaR**

Maximize Linear (maximizing estimated return)

subject to

Var_risk ≤ Const2 (VaR constraint)

Linear = 1 (budget constraint)

Box constraints (upper bounds on positions)

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Var_risk = VaR Risk for Los

Box constraints = constraints on individual decision variables

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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 | 0.00120185276974 | 0.01 |

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Data and solution in MATLAB Environment

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 | 0.00120185 | 0.01 |

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Data and solution in R Environment

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 | 0.00120185 | 0.01 |

**CASE STUDY SUMMARY**

This case study demostrates the equivalence between chance constraints and VaR constraints, as explained in Sarykalin et al. (2008). Several engineering applications deal with probabilistic constraints such as the reliability of a system or a delivery system likelihood to meet a demand. In portfolio management, often it is required that portfolio loss with high reliability should not exceed some value. In these cases an optimization model can be set up so that constraints are required to be satisfied with some probability level rather than almost surely. Chance constraints and VaR (percentile) constraints are closely related. We will illustrate numerically the equivalence of the constraints:

, i.e., the constraint assuring that the probability that loss exceeding

*w*is less or equal than 1- is equivalent to the constraint that VaR (percentile) with confidence level is less or equal than

*w*.

We solve two portfolio optimization problems. In both cases we maximize the estimated return of the portfolio. In the first problem, we impose a constraint on probability; in the second problem, we impose an equivalent constraint on VaR. We expect to obtain at optimality the same objective function value and similar optimal portfolios for the two problems.

**References**

• Sarykalin, S., Serraino, G., and Uryasev, S. (2008):

*VaR vs CVaR in Risk Management and Optimization*, INFORMS Tutorials in Operations Research, Institute for Operations Research and the Management Sciences, Hanover, MD, 270–94.

• Rockafellar, R.T. (2007):

*Coherent Approaches to Risk in Optimization Under Uncertainty*, INFORMS Tutorials in Operations Research. Institute for Operations Research and the Management Sciences, Hanover, MD, 38–61.