# Risk Management under Omega Measure

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

**Lemma**

**1.**

**Proof.**

**Definition**

**1.**

**Definition**

**2.**

## 3. Sharpe Ratio and Omega Measure Equivalence

**Theorem**

**1.**

**Proof.**

## 4. Portfolio Optimization

## 5. Numerical Analysis

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Theorem**

**2**

**.**Let $f\left(x\right)$ be a differentiable quasi-concave function subject to non-negativity constraints. If $\nabla f\left({x}^{*}\right)\ne 0$ and ${x}^{*}$ satisfies the KKT conditions with constants ${\mu}^{*}$, then it is a global optimal solution.

Algorithm 1: Sharpe Ratio active-set (SRAS) algorithm. | |

1: | $i=0$ |

2: | ${w}^{i}=\mathbf{0}$ |

3: | $j=argmax\frac{{e}_{j}}{\sqrt{{\Sigma}_{jj}}}$ |

4: | ${w}_{j}^{i}=\frac{{e}_{j}}{\sqrt{{\Sigma}_{jj}}}$ |

5: | ${W}^{i}=\left\{j\right|{w}_{j}^{i}=0\}$ |

6: | ${P}^{i}=\left\{j\right|{w}_{j}^{i}>0\}$ |

7: | loop |

8: | ${x}_{{P}^{i}}^{i}={\Sigma}_{{P}^{i}}^{-1}{e}_{{P}^{i}}$ |

9: | ${x}_{{W}^{i}}^{i}=\mathbf{0}$ |

10: | ${p}^{i}={x}^{i}-{w}^{i}$ |

11: | if ${p}^{i}=0$ then |

12: | ${\mu}_{{W}^{i}}^{i}=\frac{{w}^{iT}e{(\Sigma {w}^{i})}_{{W}^{i}}}{{({w}^{iT}\Sigma {w}^{i})}^{\frac{3}{2}}}-\frac{{e}_{{W}^{i}}}{\sqrt{{w}^{iT}\Sigma {w}^{i}}}$ |

13: | if ${\mu}_{j}^{i}\ge 0\phantom{\rule{4.pt}{0ex}}\forall j\in {W}^{i}$ then |

14: | ${w}^{*}=\frac{{w}^{i}}{{\sum}_{j=1}^{n}{w}_{j}^{i}}$ |

15: | quit |

16: | else |

17: | ${k}^{i}=\underset{j\in {W}^{i}}{argmin}{\mu}_{j}^{i}$ |

18: | ${W}^{i+1}={W}^{i}\setminus \left\{{k}^{i}\right\}$ |

19: | ${P}^{i+1}={P}^{i}\cup \left\{{k}^{i}\right\}$ |

20: | ${w}^{i+1}={w}^{i}$ |

21: | end if |

22: | else |

23: | ${\alpha}^{i}=min\{1,\underset{j\in {P}^{i},{p}_{j}^{i}<0}{min}\frac{-{w}_{j}^{i}}{{p}_{j}^{i}}\}$ |

24: | if ${\alpha}^{i}<1$ then |

25: | ${h}^{i}=\underset{j\in {P}^{i},{p}_{j}^{i}<0}{argmin}\frac{-{w}_{j}^{i}}{{p}_{j}^{i}}$ |

26: | ${W}^{i+1}={W}^{i}\cup \left\{{h}^{i}\right\}$ |

27: | ${P}^{i+1}={P}^{i}\setminus \left\{{h}^{i}\right\}$ |

28: | end if |

29: | ${w}^{i+1}={w}^{i}+{\alpha}^{i}{p}^{i}$ |

30: | end if |

31: | $i=i+1$ |

32: | end loop |

**Theorem**

**3.**

**Proof.**

## 6. Numerical Results

## 7. Model with Skewness

**Proposition**

**3.**

**Proof.**

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Proof**

**of Lemma 1.**

**Proof**

**of Theorem 1.**

**Proof**

**of Proposition 1.**

**Proof**

**of Proposition 2.**

**Proof**

**of Theorem 3.**

**Lemma**

**A1.**

**Proof**

**of Lemma A1.**

**Lemma**

**A2.**

**Proof**

**of Lemma A2.**

**Proof**

**of Proposition 3.**

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**Figure 1.**Omega measure versus skewness for a skew-normal random variable with $\mu =0.1$, $\sigma =0.3$ and $L=0.01$.

SRAS | Gurobi | |||
---|---|---|---|---|

Time (s) | Solution | Time (s) | Solution | |

Dow 1 Yr | 0.0386 | 2.6769 | 0.6881 | 2.6769 |

Dow 2 Yr | 0.0057 | 3.3883 | 0.5551 | 3.3883 |

Dow 5 Yr | 0.0030 | 15.7604 | 0.5829 | 15.7604 |

S&P 1 Yr | 0.0479 | 7.2073 | 0.6569 | 7.2073 |

S&P 2 Yr | 0.0095 | 4.4550 | 0.6233 | 4.4550 |

S&P 5 Yr | 0.0053 | 5.0557 | 0.5562 | 5.0557 |

Mean | 0.0184 | 0.6104 |

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**MDPI and ACS Style**

Metel, M.R.; A. Pirvu, T.; Wong, J. Risk Management under Omega Measure. *Risks* **2017**, *5*, 27.
https://doi.org/10.3390/risks5020027

**AMA Style**

Metel MR, A. Pirvu T, Wong J. Risk Management under Omega Measure. *Risks*. 2017; 5(2):27.
https://doi.org/10.3390/risks5020027

**Chicago/Turabian Style**

Metel, Michael R., Traian A. Pirvu, and Julian Wong. 2017. "Risk Management under Omega Measure" *Risks* 5, no. 2: 27.
https://doi.org/10.3390/risks5020027