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 = ( r₁,r₂,….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 = (_x₁,x₂,….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
is the cumulative distribution function of the outcome associated to the portfolio configuration x. It is a monotonic decreasing function. The larger the ratio, the higher is the quality of the portfolio returns distribution. By plotting the Omega values of various skewed and asymmetrical distributions it can be seen that portfolios that are more skewed to the right at any loss thresholds with higher Omega values are preferable to other portfolios with lower omega values. Thus Omega helps to compare returns for different asset classes and rank them according to the values of Omega. Omega value makes use of higher moment effects like skewness, kurtosis etc in a portfolio distribution unlike Sharpes ratio, which uses only mean and variance. Further, it requires no estimates of higher moments. Another advantage of Omega measure is that the return threshold is set according to investor preference and situation thus avoiding a typical benchmark (Darsinos and Satchell, 2003).According to Shadwick and Keating (2002), omega function can be considered as a universal performance measure. Though normality can be considered as standard for all distributions, the distributions of the returns like Gilts and Austrian government bonds, hedge funds etc are found to be historically asymmetrical.
For an investor, the optimal Omega portfolio at threshold s can be expressed as follows.
max Ωs(x)
A portfolio can be optimized using the Omega function at a specified threshold which can be selected in a number of ways. Since here, the returns are maximised instead of minimizing variance, the probability weighted losses are minimized while the probability weighted gains are maximised. Omega function is based on the rule that more money is preferred to less. Hence higher omega value is preferable to lower value.
Due to the high moment information which Omega captures when compared to Sharp’s ratio, the programming above is a non convex programming and there can be many local optima for these. One method usually applied in this case is to simplify the specification until traditional methods can be successfully applied to the problem. However, simplifying the problem will lead to the sacrifice of the gains from using the Omega measure itself. Global optimization methods like heuristic or non deterministic approaches are generally used to solve these types of problems (Prigent, 2007).The heuristic methods allows to provide good quality approximations to the global optimum when the repeated runs are increased. They do not rely on strong assumptions about the optimization problem. Using these methods, the objective function needs to be evaluated for a given element of the search space. A different approach consists in applying optimization heuristics such as evolutionary algorithms (Simulated Annealing, Threshold Accepting), Neural Networks, Genetic Algorithms, Tabu Search, hybrid methods and many others, which have been developed over the last two decades. They are divided into constructive methods and local search methods (Prigent,2007).The choice of a neighbour solution classically maximises a criterion. The local search methods make use of the information in the neighbourhood of a current solution. Local search methods are divided into trajectory methods which works on a single solution and population methods which works on a whole set of solutions simultaneously. Trajectory methods involve threshold methods and tabu search, where the second class includes genetic algorithms, differential evolution methods and ant colonies. These methods include non deterministic elements and occasionally accept impairments. Due to this non deterministic nature, local optima can be overcome more easily. The algorithm should have moved into a favourable region of the search space as more iterations have passed and it becomes less tolerant in accepting impairments. However, these methods do not produce high quality solutions with certainty but only stochastic approximations. The main advantage is that when the traditional methods fail as in the case of omega measure, these methods can still provide better approximations particularly in the case of hedge funds etc.
Threshold methods are of two types, first method uses a probabilistic acceptance criterion while the second uses the maximum threshold as deterministic. Threshold accepting approach is one of the most widely used optimization techniques used in this regard (Duek and Sheuer, 1990). This is a deterministic analog of simulated annealing. It is a very flexible technique and it requires no simplification of the problem. It allows for a straightforward implementation of all kinds of problems. 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. A worse solution is accepted if its difference to the current solution is smaller or equal to a threshold. Ultimately, in the last round threshold reaches zero. The quality of results in this case is determined by the definition of the sequence of thresholds and the method used to create neighbouring solution (Duek and Winker, 1992).Applying this algorithm to actual data in many studies have shown that this is applicable to optimization problems of this type (Gilli and Schumann, 2008 etc).The results depend on a number of assumptions regarding the problem. The objective function has to satisfy certain conditions, the search space has to be connected. The results are existence results. There exists suitable
parameters for the threshold accepting implementation such that the global optimum of the objective function can be approximated at arbitrary accuracy with any fixed probability close to one by increasing the number of iterations(Althofer and Koschnik,1991). Although finding the global optimum is certainly preferred, this will not always be feasible for problems of high computational complexity given limited computational resources. Hence, various other measures of convergence might be also of interest for optimization.
Tabu search is another optimization method designed for the exploration of discrete search spaces where the set of neighbour solutions is finite. It involves selecting the neighbourhood solution in a particular way to avoid visiting the same solution more than once. Among the population methods, genetic algorithm is popular one. New candidates for the mechanism are produced with crossover and then applies mutation. The new individual which inherits good characteristics from the parent has a better chance to survive (Prigent, 2007).In this algorithm, first a set of solutions are accepted and then as set of neighbouring solutions are accepted. The ant colonies resemble the way an ant searches for food and find their way back to nest. First a random neighbourhood is explored. Then they guide other ants also to the source of food. As they go on travelling, their behaviour is able to optimize their work. The search area of the ant belongs to the discrete set from which the elements forming the solutions are selected, the amount of food is associated with an objective function and the phremone trail is modelled with an adaptive memory. (Gilli and Schumann, 2008 etc).
Differential evolution has been developed by Storn and Price (1997) for continuous objective functions. It updates a solution of population vectors by addition, subtraction and crossover. Then the fittest solutions are selected among the original and updated population. Particle swarm is a population based optimization technique developed by Eberhart and Kennedy (1995).This algorithm updates a population of solution vectors called particles, by an increment called velocity.
In a general framework, optimization heuristics are also called Meta heuristics (Prigent, 2007).They can be considered as a general skeleton of an algorithm applicable to a wide range of problems. They are made up of different components and help to imagine a wide variety of techniques .A typical example of a big level co evolutionary hybrid is memetic algorithm proposed by Moscato (1989).This combines the advantages of population methods and trajectory approaches. Each agent of a population individually runs a threshold method. The members interact by competition and cooperation. The competition is between neighbours in a circle. Hence a better solution replaces the worst neighbour. Cooperation resembles crossover in genetic algorithms. Here solutions of distant agents are combined to generate new children replacing their parents. Based on some acceptance criterion, the decision on replacing the parents or not is taken. For a properly tuned heuristic method, the amount of computational resources spent on a single run will have a positive influence on the result.
There are other global optimization methods like MCS algorithm of Huyer and Neumaier (1999). It seeks the global minimum of an n variable function in a hyper box. Here, the root box is divided into many sub boxes. There are global and local parts for the algorithm. The sub boxes that explore the big unexplored territory are covered by the global part. Local part divides sub boxes that have good function values. A multilevel approach is used to achieve a balance between local and global parts. Initially, a preliminary set of sub boxes are created using a default procedure. The initial points can either be user defined or default options are available for this. Through the initial points, quadratic interpolants are generated .Using it, the variability of the objective function is estimated. Then each coordinates are ranked based on it. Sweeping through the levels are started by the algorithms in the next stage. The global and local parts of the algorithm are covered then. Each box is divided either by rank or an expected gain. It can be successfully terminated either based on the search reaching a pre specified target function value or by reaching a static limit.MCS splits along a single coordinate at a time, at adaptively chosen points. For splitting a sub box into several children, a new function evaluation is required in most cases. A base point is give to each child different from the parent. Local searches will go on till the base point is not likely in the basin of attraction of a previously funded local minimum.
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.