The case for the Omega function

The Omega function and associated metrics offer a solution to the limitations of parametric approaches to investment performance appraisal.

In the welter of recent criticisms of the Sharpe ratio, it is easy to forget that the insights into performance evaluation due to Bill Sharpe were profound. But it is also obvious that the underlying assumptions of the ratio are far too simplistic for modern investment management, and that a nonparametric generalisation is needed.

To begin, it is worth dwelling briefly on therole of the risk-free rate in the Sharpe ratio. Without this element1, the ratio is scale invariant, that is to say it cannot distinguish between sets of returns where the only difference is a question of leverage or gearing. The problem is that these distributions may have identical efficiency of production of return per unit of risk but the terminal values of the levered set are higher on average, a question of scale. With the interpretation of the risk-free rate as the cost of borrowing, however, this problem resolves.

There are related scale or exposure complexities for portfolios containing short positions.

While it is common to refer to the asymmetries of returns arising from the use of options or complex strategies as justification for more advanced methods, there is a much simpler and more direct rationale for the development of an accurate and discerning technique - our preferences are intrinsically asymmetric; we want the upside and not the downside to any investment.

Searching for a way to compare financial data

The fundamental problem is simple: we need to be able to compare distributions regardless of their empirical properties in a manner that is financially meaningful. The techniques, widely used in academia, known as Stochastic Dominance are not fit for this purpose2.

There have been many attempts3 to advance from the mean-variance world of Sharpe, most notably with the development of the Sortino ratio. The denominator of this ratio is the downside deviation, a second order statistic4, and proves its Achille's heel. Choices are made correctly over downside potential but incorrectly over the upside.

There is also a substantial body of work in academic finance which questions the existence of higher order statistics5 for financial data; we can always calculate a sample statistic from a dataset but the question here is whether this converges to a finite value as we increase the sample size. This concern, while apparently an academic nicety, would have profound consequences for the risk management industry if irrefutably proven. Pragmatically, it means that we should avoid the use of higher order statistics if we can capture their information content in some other manner.

It is worth noting that there is a mathematical equivalence between the standard deviation of a distribution and the first order lower partial moment (at the mean) in the case of Normal distributions - another positive attribute of the Sharpe ratio.

Any definition of risk must capture the idea that it is fundamentally related to the rate of change of the object under consideration. In finance, where we are concerned with wealth and asset and liability values, this rate of change is simply returned. If we wish to think about the risk ness of returns, then we are concerned with the rate of change of returns. In theory at least, the expected return of an asset must be positive for that asset to have value and the meaning of negative values is, to say the least, intellectually challenging6 - these are not liabilities7.

These were the background concerns that led to the development of the Omega function and Omega metrics.

The Omega function

The Omega function for returns is mathematically equivalent to the returns distribution and consequently contains all information contained in the original data. It relies upon only the existence of the mean return and consequently is indifferent as the existence of higher order statistics. It is intuitive, being defined as the ratio of the upside pay-off to the downside pay-off relative to a threshold return, for the entire range of returns. Relative to single threshold this is: what do I win multiplied by how likely is it that I win divided by what do I lose multiplied by how likely is it that I lose - the comparison to the ratio of a put option to a call option is immediate.

Risk is defined as the rate of change of the Omega function. For the purposes of comparison of risk it is convenient to use the relative (or logarithmic) rate of change, in the same manner that we use modified duration rather than Macaulay duration in bond analysis. Here risk is much richer than the global statistic, standard deviation, with which we are all familiar - now we can consider and quantify risk locally to a level of return8.

The question remains of comparing Omega functions. The decision rule is simple. We prefer the higher valued function at any given threshold. We may simply consider the values at a single threshold, as many have done, but this can mislead critically. Only the entire function contains all of the information in the data. To compare Omega functions correctly we need a functional on the Omega function, which reduces it to a single number. One that rewards higher mean return together with higher concentration around the mean that penalises asymmetric returns below the mean and rewards above. It should also reduce to the Sharpe ratio if returns are in reality normal. This is precisely what the Omega F2 metric does - and it demonstrates empirically a strong relation to realized values of wealth or asset values arising from both traditional asset management and alternative investment strategies.

Conclusion: the superiority of non-parametric approaches

A number of researchers9 have derived Omega functions from parametric models fitted to empirical data. If a distribution can be written in closed form, there is always a closed form expression for the Omega function. The problem at root is that the parameter estimation process is costly in terms of information discarded. The justification that these parametric models can easily be used predicatively carries little substance; the choice of (parametric) model is critical and non-trivial. Non-parametric methods are certainly superior and their application holds the prospect of dismissing for ever the regulator's mantra that past performance is no guide to the future.

Footnotes:

1. Without the presence of the risk-free rate, the measure is known as the Information ratio.

2. There are many possible criticisms of the techniques utilised as Stochastic Dominance; perhaps the most elementary is that with increasing complexity, they apply to ever smaller classes of distribution and consequently rapidly lose relevance for the practitioner.

3. These have included many follies, such as the recently proposed ratio of upside to downside information ratios which would encourage downside while discouraging upside risk.

4. Technically this is the second order lower partial moment with respect to the Minimum Acceptable Return.

5. This dates to the work of Mandelbrot, Fama and Samuelson in the 1960s and is also evident in the recent work of Los and McCulloch.

6. Some practitioners for instance urge the non-reporting of negative Sharpe ratios - a poor practice.

7. A liability value distribution is the image about zero of an asset value distribution, which means that the symmetries differ.

8. Similarly we can think about returns locally to asset values - and gain insight into the options 'smile'.

9. Including some who have failed to notice that they have generated nonsensical negative probabilities in the process.

Author : Con Keating is a financial analyst at the Finance Development Centre, who has had extensive experience in commercial and merchant banking, insurance, and investment management. He is a former Chairman of the Committee on Methods and Measures of the European Federation of Financial Analyst Societies and has taught as a visiting scholar at Universities in the USA and Europe. He is a member of the Societe Universitaire Europeene pour Recherche en Finance and a member of the Steering Committee of the Financial Research Centre at City University.

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