Omega ratio

A criticism to the classical Sharpe ratio is that it is only appropriate when the portfolio return is elliptically distributed, as it only takes into account only first and second moments of the returns distribution, neglecting other higher order moments. Especially, when the portfolio return is skewed or exhibits fat tails, then the Sharpe Ratio might result in counterintuitive performance evaluations and rankings.

To address this issue, Omega ratio has been introduced as a measure of the ratio between risk and rewards that takes into consideration all moments of the portfolio returns distribution. It captures both the downside and upside potential of the constructed portfolio, while remaining consistent with utility maximization.

The Omega Ratio makes use of a threshold value \(\tau\), or minimum acceptable return to distinguish the upside from the downside. This means that portfolio returns above \(\tau\) are considered profits, whereas returns below \(\tau\) are considered as losses.

Mathematically, given the cumulative portfolio returns distribution \(F(r)\), Omega ratio is defined as:

\[\begin{equation} \Omega(r) = \dfrac{\int_{\tau}^\infty (1-F(r)) dr}{\int_{-\infty}^{\tau} F(r) dr}. \end{equation}\]

This concept is well illustrated in the following picture, where the returns cumulative distribution is shown. Here the Omega Ratio is defined as the ratio of the green area over the red area. The green area on the right of the threshold \(\tau\) (in this example 0.0015) and above the cumulative distribution represents the upside potential. The red area on the left of the threshold and below the cumulative distribution represents the downside potential (risk):

omega_ratio

After some easy manipulation, the Omega ratio can be operationally defined as:

\[\begin{equation} \Omega(r_P) = \dfrac{\mathbf{w}^T \boldsymbol \mu - \tau}{\left< (\tau - \mathbf{w}^T \mathbf{R})^+ \right >} + 1 \end{equation}\]

In the expression above, at the numerator we have the expected portfolio return, with \(\boldsymbol \mu\) being the vector of expected returns, while at the denominator we consider the average \(\left< \cdot \right>\) of the positive excess returns, over all temporal samples \(t=1,\ldots, T\) with \(\mathbf{R}_t\) the return of all assets at time \(t\), hence a vector of \(N\) assets. The product \(\mathbf{w}^T\mathbf{R}\) is then to be intended as a matrix product between the \(N\) dimensional weights product and the \(N \times T\) dimensional returns matrix.

Similarly to the classical mean-variance frontier, it is possible to draw an efficient frontier also for Omega ratio:

omega_frontier

The frontier is a direct analogue to the mean-variance efficient frontier, whereas in this case the numerators (return) and denominator (volatility) of the classical Markowitz frontiers are replaced by specific formulas for reward (Omega numerator) and risk (Omega denominator), operationally defined as:

\[\begin{split} \begin{equation*} \begin{aligned} \textrm{Omega numerator (reward)} =& \mathbf{w}^T \boldsymbol \mu - \tau \\ \textrm{Omega denominator (risk)} =& \sum_{t=1}^T \max \left( \tau - \mathbf{w}^T \mathbf{R}_t,\, 0 \right ) \end{aligned} \end{equation*} \end{split}\]

The slope of the tangent to the frontier passing through the origin gives the maximum Omega Ratio and indicates the associated portfolio. The affiliated Omega Ratio for each point on the frontier is given by the slope of the line passing through it and the origin. The goal is to find the line with the maximum slope that passes through the origin and a point on the frontier. Since the frontier is non-decreasing and concave, the tangent from the origin to the frontier yields the portfolio with the maximum Omega.

Although Omega ratio is a non-convex function, it is possible to express its maximization in terms of a convex optimization trasforming the problem into a linear one, because Omega is quasi-concave in portfolio weights \(\mathbf{w}\).

For the set of prices defined in skportfolio.data.load_tech_stock_prices the Omega efficient frontier is delineated in the following picture:

omega_frontier_stocks

As in the mean-variance case, the maximum return stock is the one with the maximum risk, as quantified by the Omega risk on the x-axis. Similarly to maximum sharpe ratio and minimum volatility, we identify two points on the frontier and the tangent line from the origin touches the frontier exactly at maximum omega ratio point. Small red points are from random uniform portfolios, while the golden dot represents the equally weighted portfolio.

Maximum Omega ratio

Maximization of the omega ratio is done through linearization of the quasi-convex Omega-ratio problem. Following a recent paper by Kapsos et al 1, the mathematical optimization problem to find the maximum Omega ratio portfolio in skportfolio is the following:

\[\begin{split} \begin{equation*} \begin{aligned} \underset{\mathbf{s}\in \mathbb{R}^n, \mathbf{q}\in \mathbb{R}^m, z \in \mathbb{R}}{\text{max}} & \mathbf{s}^T \boldsymbol \mu - \tau z & \\ & q_t \geq \tau z - \mathbf{s}^T \mathbf{R_t} & \forall t = 1,\ldots,T \\ & q_t \geq 0 & \forall t = 1,\ldots,T \\ & \mathbf{1}^T \mathbf{q} = 1 \\ & \mathbf{1}^T \mathbf{s} = z \\ & z w_{\min} \leq \mathbf{s} \leq z w_{\max} \\ & \mathbf{s}^T \boldsymbol \mu \geq \tau z \\ & z \geq 0 \end{aligned} \end{equation*} \end{split}\]

Once solved, the optimal weights are then recovered from the \(\mathbf{s}\) variable after normalization. The above linear program not only ensures global optimality, but is also simple and fast to solve, even if the number of constraints grows linearly with the number of data points (indicated by \(T\)).

Note

Importantly, here the number of constraints grows linearly with number of temporal samples. The EfficientOmegaRatio class offers a way to subsample the returns at a predetermined fraction of the original, thus making the problem computationally easier to solve. Please refer to the EfficientOmegaRatio class documentation, for the fraction parameter.

Minimum Omega Risk 📖

This portfolio minimizes the denominator in the Omega ratio. It is indeed very similar to the objective function of the Minimum Absolute Deviation portfolio. The minimum Omega risk optimization problem is solved through the EfficientOmegaFrontier interface:

\[\begin{split} \begin{equation*} \begin{aligned} & \underset{\mathbf{w}\in \mathbb{R}^+}{\text{minimize}} & & \left\langle (\tau - \mathbf{w}^T \mathbf{R})^+ \right\rangle \\ & \text{subject to} & & \mathbf{w}^T \mathbf{1} = 1\\ \end{aligned} \end{equation*} \end{split}\]

Efficient risk on omega-ratio frontier

This portfolio minimizes the denominator in the Omega ratio, conditioned on having a return greater than target \(\varrho_{\textrm{target}}\). The following optimization problem is solved through the EfficientOmegaFrontier interface via the .efficient_risk method:

\[\begin{split} \begin{equation*} \begin{aligned} & \underset{\mathbf{w}\in \mathbb{R}^+}{\text{minimize}} & & \left\langle (\tau - \mathbf{w}^T \mathbf{R})^+ \right\rangle \\ & \text{subject to} & & \mathbf{w}^T \mathbf{1} = 1\\ &&& \mathbf{w}^T \boldsymbol\mu - \tau \geq \varrho_{\textrm{target}} \end{aligned} \end{equation*} \end{split}\]

Efficient return on omega-ratio frontier

This portfolio maximizes the numerator of the Omega ratio, conditioned on having a denominator in the Omega ratio greater than target \(\rho_{\textrm{target}}\). The following optimization problem is solved through the EfficientOmegaFrontier interface via the efficient_return method:

\[\begin{split} \begin{equation*} \begin{aligned} & \underset{\mathbf{w}\in \mathbb{R}^+}{\text{minimize}} & & \mathbf{w}^T \boldsymbol\mu - \tau \\ & \text{subject to} & & \mathbf{w}^T \mathbf{1} = 1\\ &&& \left\langle (\tau - \mathbf{w}^T \mathbf{R})^+ \right\rangle \leq \rho_{\textrm{target}} \end{aligned} \end{equation*} \end{split}\]

References

  1. "Optimizing the Omega Ratio using Linear Programming", Kapsos M., Zymler S., Christofides N., Rustem B. (2011) url: ↩