# The Omega Function for Portfolio Optimization

The selection of the best allocation of an investor's wealth in various investment alternatives, such that the investor obtains the best possible outcome at the end of one investment period has been the purpose of the single period portfolio optimization theory (Markovitz, 1959; Elton and Gruber, 1987). Performance measure function is one way to establish a preference relation between assets with random outcome. One of the most commonly used investment performance measures is the Sharpe ratio developed by Nobel Laureate William. F. Sharpe (Sharpe, 1994). Another one is the Omega function recently advocated by Shadwick and Keating (2002).Sharpe ratio is calculated by subtracting the risk-free rate from the rate of return for a portfolio and dividing the result by the standard deviation of the portfolio returns. The Omega function is defined as the probability weighted ratio of gains to losses relative to a threshold return level as determined by the investor. In other words, it is the ratio of expectations of gains above the threshold to expected losses below the threshold. This calculation segregates returns into losses and gains above and below a return threshold and then takes the probability weighted ratio of returns above the return threshold divided by the returns below the threshold. The assumption needed for Sharpe ratio is that return distributions are normal distributions. Hence for hedge funds etc where the portfolio returns are not normally distributed, Sharpe ratio is not the best investment performance measure (Favre Bulle and Pache, 2003). For non normal and asymmetrically distributed returns, Omega function is the more suitable performance measure. Even in the case of normally distributed returns omega measure gives extra information .This is because it takes into consideration the preferences of investors about losses and gains unlike Sharpe ratio. Sharpe ratio defines risk in terms of the standard deviation of an assumed normal distribution of individual returns without regard to upside or downside volatility of returns. Omega measure defines risk with respect to the upside and downside volatility of the returns and comprises the entire distribution of a portfolio. Thus it allows for asymmetrically distributed returns also.

Let x_{i,} i = 1,2.....,n be the amount
invested in asset i expressed in percentage term out of an initial
capital W and r = ( r_{1},r_{2},....r_{n})′
the returns vector of each asset over the holding period,
x'E(r) be the expected return on the portfolio defined by the
vector x = (_x_{1},x_{2},....x_{n})′and the
variance of the portfolio return is x'Q(x) where Q is
the matrix of variances-covariance of the vector of returns r. Then
the Omega for the portfolio associated with the decision vector x
and any specified threshold loss s denoted by Ω_{s}(x) can
be articulated as

In the study by Favre-Bulle and Pache(2003),it is tested whether the attractiveness of normal distribution than any other distribution with excess skewness /kurtosis is verified in an Omega framework. For this, normal distributions are missed to simulate four distributions of 200 data points each, with various statistical properties. The omega functions clearly showed the influence of larger moments than skewness and kurtosis on the attractiveness of an asset. The portfolio optimization done in the omega framework is compared with other frameworks like mean variance framework, VaR, adjusted VaR and downside deviation. The efficiency frontier analysis for portfolio optimization techniques under various frameworks has shown that omega frontier dominated the other opportunity sets. Further it is shown that omega optimization technique never gives a smaller expected return than other settings. This is because omega measure makes use of all the moments in a distribution. It does not need any assumption regarding the utility function and considers investor's preferences regarding losses or gains. Other studies like Bachmann and Pache (2003) have also shown that omega measure gives the most consistent results for portfolio optimization.

Thus it can be shown that omega measure provides the most efficient measure for portfolio optimization especially when the returns are non-normally and asymmetrically distributed. Since it makes use of the higher moments of a portfolio, it can provide more consistent results than the traditional Sharpe ratio which focuses only on mean and variance. The Omega function is based on the principle that more money is preferred to less. Unlike the Sharpe ratio which does not define risk in terms of downside and upside volatility, omega measure defines risk in terms of downside and upside volatility. It does not make any assumption regarding the utility function. It considers investor's preferences regarding expected losses and gains. Since Omega measure makes use of all moments, traditional optimization methods are not sufficient .The global optimization techniques like threshold accepting approach is commonly used for optimizing a non convex and non smooth function like Omega function. Moves to the neighbouring solutions that improve the objective function value are accepted in this procedure. The numbers of iterations are fixed and then the neighbourhood is explored with a fixed number of steps during each iteration Another algorithm used for optimizing Omega function is the MCS algorithm which makes use of the global and local search. Many studies have shown that portfolio optimization with Omega measure provides more efficient results than under other frameworks. This is particularly efficient for hedge funds and bonds. For portfolios with normal distributions also, omega measure makes use of the additional information and gives consistent results.