Investors traditionally use performance benchmarks to gauge the risk-reward ratio of stocks and unit trusts. The Sharpe Ratio of a stock, for example, is equal to the excess return divided by the standard deviation of the returns. The Sharpe Ratio, however, assumes that the returns are normally distributed, and penalises upside volatility as well as downside volatility.
Many investment vehicles (such as hedge funds), however, are not normally distributed and exhibit skew and kurtosis in the returns distribution. Additionally, many investors want to encourage upside volatility and only penalise downside volatility.
Shadwick and Keaton (2001) define a new performance benchmark called the Omega Ratio. The Omega Ratio is equal to the area of a returns distribution above an acceptable return divided by the area of a returns distribution below an acceptable return. This definition neatly captures all the information in the empirical returns distribution and better describes the risk profile of most investors.
Given a list of historical returns and the minimum acceptable return, the Omega Ratio is easily calculated in Maple. Figure 1 gives an example calculation, which uses Maple’s select statements to separate the returns above and below the minimum acceptable return.
Figure 1. The Omega Ratio in Maple
The Omega Ratio of a portfolio of several investments is calculated from the weighted historical returns. Figure 2 shows a procedure that calculated the Omega Ratio of a portfolio of three investments. The procedure
- accepts three arguments – the weights of three investments (weight1, weight 2 and weight3)
- calculates the weighted historical returns based on three lists of historical returns (returns1, returns2 and returns 3, defined elsewhere)
- and returns the Omega Ratio.
Figure 2. Procedure to calculate the Omega Ratio of three investments in Maple
Mean-variance optimisation is a traditional asset allocation technique, introduced by Harry Markowitz in 1952. However, financial analysts often levy several criticisms against this technique. Mean-variance optimisation
- assumes the returns are normally distributed (and hence does not model the risk of low-probability but high impact events)
- is sensitive to risk-free rates (a small change in risk-free rate can generate very different investment weights).
Financial analysts are now exploring how the Omega Ratio can be used to optimise asset allocation. The problem is simple to state: what are the investment weights that maximise the Omega Ratio of a portfolio of assets.
In reality, this optimisation problem is non-convex and non-smooth. Traditional gradient-based optimisers only give local maxima and will not necessarily converge on the optimal investment weights. Maple’s Global Optimization Toolbox, however, easily processes this class of problem.
Figure 3 shows simulated returns for three assets (the data is derived from Figures 5, 6 and 7 of this paper).
Figure 3. Simulated Asset Returns
Figure 4 demonstrates the use of the Global Optimization Toolbox to find the investment weights that maximise the Omega Ratio, given the procedure in Figure 2 and the returns in Figure 3.
Figure 4. Maximize the Omega Ratio with the Global Optimziation Toolbox
A Maple 15 worksheet with the calculations described above can be download from the link at the bottom of this post. Although only three investments are used, the Global Optimization Toolbox can process a far larger investment space.
Additionally, a very large investment space can be processed in parallel. The space of investment weights is split into multiple sections, with the Global Optimization Toolbox operating in parallel on each section. Once the optimum investment weights for each section are found, the results (i.e. the investment weights and the Omega ratio for each section) are collected. The investment weights that give the largest Omega Ratio are then chosen.
The parallel optimisation is conducted using
- local grid computing (a part of the base Maple package), which enables the user to launch multiple thread-safe processes on a single computer
- or the Grid Computing Toolbox, which allows a calculation to be deployed to a cluster.
Download Maple 15 Worksheet to Optimise the Omega Ratio with the Global Optimization Toolbox
To make sure 
