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Omega ratio

The ratio is calculated as:

${\displaystyle \Omega (\theta )={\frac {\int _{\theta }^{\infty }[1-F($r)]\,dr}{\int _{-\infty }^{\theta }F(r)\,dr}},}

where ${\displaystyle F}$ is the cumulative probability distribution function of the returns and ${\displaystyle \theta }$ is the target return threshold defining what is considered a gain versus a loss. A larger ratio indicates that the asset provides more gains relative to losses for some threshold ${\displaystyle \theta }$ and so would be preferred by an investor. When ${\displaystyle \theta }$ is set to zero the gain-loss-ratio by Bernardo and Ledoit arises as a special case.^[2]

Comparisons can be made with the commonly used Sharpe ratio which considers the ratio of return versus volatility.^[3] The Sharpe ratio considers only the first two moments of the return distribution whereas the Omega ratio, by construction, considers all moments.

The standard form of the Omega ratio is a non-convex function, but it is possible to optimize a transformed version using linear programming.^[4] To begin with, Kapsos et al. show that the Omega ratio of a portfolio is:$${\displaystyle \Omega (\theta )={w^{T}\operatorname {E} ($$r)-\theta \over {\operatorname {E} [(\theta -w^{T}r)_{+}]}}+1} If we are interested in maximizing the Omega ratio, then the relevant optimization problem to solve is:$${\displaystyle \max _{w}{w^{T}\operatorname {E} ($$r)-\theta \over {\operatorname {E} [(\theta -w^{T}r)_{+}]}},\quad {\text{s.t. }}w^{T}\operatorname {E} (r)\geq \theta ,\;w^{T}{\bf {1}}=1,\;w\geq 0} The objective function is still non-convex, so we have to make several more modifications. First, note that the discrete analogue of the objective function is:$${\displaystyle {w^{T}\operatorname {E} ($$r)-\theta \over {\sum _{j}p_{j}(\theta -w^{T}r)_{+}}}} For ${\displaystyle m}$ sampled asset class returns, let ${\displaystyle u_{j}=(\theta -w^{T}r_{j})_{+}}$ and ${\displaystyle p_{j}=m^{-1}}$. Then the discrete objective function becomes:$${\displaystyle {w^{T}\operatorname {E} ($$r)-\theta \over {m^{-1}{\bf {1}}^{T}u}}\propto {w^{T}\operatorname {E} (r)-\theta \over {{\bf {1}}^{T}u}}} With these substitutions, we have been able to transform the non-convex optimization problem into an instance of linear-fractional programming. Assuming that the feasible region is non-empty and bounded, it is possible to transform a linear-fractional program into a linear program. Conversion from a linear-fractional program to a linear program gives us the final form of the Omega ratio optimization problem:$${\displaystyle {\begin{aligned}\max _{y,q,z}{}&y^{T}\operatorname {E} ($$r)-\theta z\\{\text{s.t. }}&y^{T}\operatorname {E} (r)\geq \theta z,\;q^{T}{\bf {1}}=1,\;y^{T}{\bf {1}}=z\\&q_{j}\geq \theta z-y^{T}r_{j},\;q,z\geq 0,\;z{\mathcal {L}}\leq y\leq z{\mathcal {U}}\end{aligned}}} where ${\displaystyle {\mathcal {L}},\;{\mathcal {U}}}$ are the respective lower and upper bounds for the portfolio weights. To recover the portfolio weights, normalize the values of ${\displaystyle y}$ so that their sum is equal to 1.

• Modern portfolio theory
• Post-modern portfolio theory
• Sharpe ratio
• Sortino ratio
• Upside potential ratio

• How good an investment is property?
• "The Omega Measure: A better approach to measure investment efficacy" (PDF) (Press release). California: Propertini.