** Case study background and problem formulations**

**PROBLEM 1: problem_1_32**

Minimize Cvar_risk (minimizing portfolio CVaR)

subject to

Linear = 1 (budget constraint)

Linear ≥ Const (constraint on the portfolio mean return)

Box constraints (no short constraints)

——————————————————————–

Cvar_risk = CVaR Risk 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 | 4 | 10,000 | 0.054853 | 0.01 |

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

Dataset1 | Matlab code | Data | Solution | 4 | 10,000 | 0.054853 | 0.02 |

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

Dataset1 | R code | Data | 4 | 10,000 | 0.054853 | 0.02 |

Problem Datasets | |||
---|---|---|---|

Dataset1 | Matlab code | Data | Solution |

**PROBLEM 2: problem_2_32**

Minimize [lambda*xa + Cvar_risk] (minimizing weighted average of risk and penalty)

subject to

Linear = 1 (budget constraint)

Linear + xa ≥ Const (constraint on portfolio mean return combined with penalty variable xa)

Box constraints (no short constraints and non-negativity constraint on penalty variable xa)

——————————————————————–

Cvar_risk = CVaR Risk for Loss

lambda = penalty coefficient

xa = penalty variable

Box constraints = constraints on individual decision variables

——————————————————————–

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

Dataset | 5 | 10,000 | 0.054853 | 0.02 | |||
---|---|---|---|---|---|---|---|

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

Matlab Toolbox | Data | ||||||

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

R | Data |

**PROBLEM 3: problem_3_32**

Maximize Avg_g (maximizing portfolio mean return)

subject to

Cvar_risk ≤ Const (CVaR constraint)

Linear = 1 (budget constraint)

Box constraints (no short constraints)

——————————————————————–

Avg_g = Average Gain

Cvar_risk = CVaR Risk for Loss

Box constraints = constraints on individual decision variables

——————————————————————–

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

Dataset1 | Problem Statement | Data | Solution | 4 | 10,000 | 0.001050 | 0.01 |

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

Dataset1 | Matlab code | Data | Solution | 4 | 10,000 | 0.00105 | 0.01 |

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

Dataset1 | R code | Data | 4 | 10,000 | 0.00105 | 0.01 |

Problem Datasets | |||
---|---|---|---|

Dataset1 | Matlab code | Data | Solution |

**CASE STUDY SUMMARY**

This case study demonstrates a simple (basic) setup of single-period portfolio optimization problem when risk is measured by CVaR. Portfolio optimization has come a long way from when Markowitz’s (1952) seminal work suggested the mean/variance framework. The Markowitz problem addresses the trade-off between mean and variance of a portfolio return. In the original formulation, the problem is to find a minimum-variance portfolio under the constraint of mean return. Variance is not a good measure of risk for asymmetric distributions. Conditional Value-at-Risk (CVaR) (see Rockafellar and Uryasev (2000, 2002)) takes into account only the downside part of the distribution and it has nice mathematical properties (a coherent risk measure). One of the advantages of the CVaR approach is that it solves the portfolio optimization problem without normal assumption on distribution of market risk factors. This assumption violates in most markets, where distribution of market risk factors is characterized by fat tails, skewness and high dependence among extreme events. Moreover, the CVaR approach can be easily implemented within more complex approach to the portfolio optimization problem, where the portfolio manager blends his subjective views on the market with a prior market distribution (see, Meucci (2005a, 2005b)). This case study implements the portfolio optimization problem described by Meucci (2005b). This project also demonstrates how to handle possible infeasibility of the constraints by adding the compensation variables.

**References**

• Rockafellar, R. T. and S. Uryasev (2000): Optimization of Conditional Value-At-Risk, The Journal of Risk, Vol. 2, No. 4, pp. 21-51.

• Rockafellar, R.T. and S. Uryasev (2002): Conditional Value-at-Risk for General Loss Distributions, Journal of Banking and Finance, 27/7.

• Meucci, A. (2005a): Beyond Black-Litterman: Views on Non-Normal Markets, (November 2005), download: http://ssrn.com/abstract=848407.

• Meucci, A. (2005b): Beyond Black-Litterman in Practice: A Five-Step Recipe to Input Views on Non-Normal Markets, (December 28, 2005), download: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=872577.