The Omega ratio is a risk-return performance measure of an investment asset, portfolio, or strategy. It was devised by Con Keating and William F. Shadwick in 2002 and is defined as the probability weighted ratio of gains versus losses for some threshold return target. The ratio is an alternative for the widely used
Sharpe ratio and is based on information the Sharpe ratio discards.
Omega is calculated by creating a partition in the cumulative return distribution in order to create an area of losses and an area for gains relative to this threshold.
The ratio is calculated as:
:
where
is the
cumulative probability distribution function of the returns and
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
and so would be preferred by an investor. When
is set to zero the gain-loss-ratio by Bernardo and Ledoit arises as a special case.
Comparisons can be made with the commonly used
Sharpe ratio which considers the ratio of return versus volatility. The Sharpe ratio considers only the first two
moments of the return distribution whereas the Omega ratio, by construction, considers all moments.
Optimization of the Omega ratio
The standard form of the Omega ratio is a non-convex function, but it is possible to optimize a transformed version using
linear programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomia ...
.
To begin with, Kapsos et al. show that the Omega ratio of a portfolio is:
The optimization problem that maximizes the Omega ratio is given by:
The objective function is non-convex, so several modifications are made. First, note that the discrete analogue of the objective function is:
For
sampled asset class returns, let
and
. Then the discrete objective function becomes:
Following these substitutions, the non-convex optimization problem is transformed 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 yields the final form of the Omega ratio optimization problem:
where
are the respective lower and upper bounds for the portfolio weights. To recover the portfolio weights, normalize the values of
so that their sum is equal to 1.
See also
*
Modern portfolio theory
Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of Diversificatio ...
*
Post-modern portfolio theory
*
Sharpe ratio
*
Sortino ratio
The Sortino ratio measures the risk-adjusted return of an investment asset, portfolio, or strategy. It is a modification of the Sharpe ratio but penalizes only those returns falling below a user-specified target or required rate of return, while ...
*
Upside potential ratio The upside-potential ratio is a measure of a return of an investment asset relative to the minimal acceptable return. The measurement allows a firm or individual to choose investments which have had relatively good upside performance, per unit of d ...
References
External links
How good an investment is property? *
{{Financial risk
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Financial risk modeling
Investment indicators
Linear programming