Constructing Long/Short Portfolios with the Omega ratio

## Keywords

Optimisation heuristics, Threshold Accepting, Portfolio optimisation

# Abstract

We construct portfolios with an alternative selection criterion, the Omega function, which can be expressed as the ratio of two partial moments of the return distribution. Finding Omega-optimal portfolios, in particular under realistic constraints like cardinality restrictions, requires to solve non-convex optimisation problems. Since standard (gradient-based) optimisation methods fail here, we suggest to use a heuristic technique (Threshold Accepting). The focus is on the empirical performance of the selected portfolios, especially the effects of allowing short positions. Many studies on portfolio optimisation assume that short sales are not allowed. This is despite the fact that theoretically, short positions can improve the risk-return characteristics of a portfolio, and practically, institutional investors can and do sell stocks short. We investigate whether removing the non-negativity constraint really improves out-of-sample portfolio performance under realistic assumptions, that is when optimal weights need to be estimated from the data, different transaction costs apply to long and short positions or short selling is restricted to specific assets.

# 1 Introduction

In the framework of modern portfolio theory (Markowitz [8]), a portfolio of assets is completely characterised by a desired property, its ‘reward’, and something undesirable, its ‘risk’. Markowitz identified these two properties with the expectation and the variance of returns, respectively, hence the expression mean-variance optimisation. In recent years, a large number of alternative specifications has been suggested. This development has been driven by an increased theoretical interest in the properties of risk measures, as well as by the growth of alternative instruments (hedge funds, derivatives) with strongly non-symmetric distributions.

In this Article, we look at portfolio selection under such an alternative selection criterion, the Omega function (Keating and Shadwick [6]). This function treats upside and downside risk differently, thus heeding the theoretical criticism against mean-variance optimisation. We investigate in particular the differences in performance between long-only and long-short portfolios. Short-selling has recently received increased attention because of the growth in so-called 130/30-funds (Lo and Patel [7]). These funds are allowed to build up short positions of up to 30% of their assets under management. This limit follows from a regulatory restriction in the United States; there are also 120/20- or 140/40-funds, and other variations (Jacobs [4]).

Practitioners often stress that short positions are different from long positions, that the process of building portfolios need to follow different rules. In portfolio theory, on the other hand, there are no differences between long and short positions (except for the sign); the portfolio mathematics stay the same. In fact, when it comes to optimisation, dropping the inequality constraints even simplifies the model. Practically, however, there arise different microstructure and regulatory issues (Jacobs [5]). Not every asset can easily be sold short, there needs to exist a borrowing market for the security. This borrowing market itself follows certain market dynamics; a drying up of liquidity in certain stocks can lead to ‘short squeezes’, that is a sudden demand to return borrowed shares. With the growth of derivatives markets, this has become less of a problem, as options and future positions can be constructed that, for all practical purposes, behave just like short positions (even though there are, sometimes subtle, differences in cash flows and counterparty risks). Modelling all such peculiarities in detail seems not a viable approach in our view, as it may lead to very model-sensitive solutions. Below, we use a more robust method to obtain realistic portfolios. For one, we assume higher transaction costs for short positions (since a borrowing fee needs to be paid); furthermore, we restrict short sales to a list of large and liquid stocks. Even if the borrowing market for such securities may not function properly, there exist derivatives markets which allow to replicate short positions.

Optimising a portfolio under the Omega function, in particular under realistic constraints, leads to non-convex optimisation problems. We will apply a heuristic method, Threshold Accepting (TA), to solve these models.

# 2 The Investor's Problem

Given an intial wealth $v_0$, an investor selects a portfolio from a universe $\mathcal{A}$ of assets ($\mathcal{A}$ may include a risk-free bond). Let the indexes of the chosen assets be collected in $\mathcal{J}$; the column vector $x$ stores the (integer) numbers of assets held. The investor holds this portfolio until time $T$; final wealth $v_T$ is then given by

(1)
\begin{align} v_T = x'p_T\, , \end{align}

where $p_T$ are the prices at $T$. As a convention, we will compute portfolio losses $\ell$, defined as

(2)
\begin{align} \ell = v_0 - v_T\,. \end{align}

Since final wealth is a weighted sum of random variables (the prices in $T$) and is thus itself a random variable, so are the losses $\ell$. Their cumulative distribution function (CDF) will be defined by the characteristics (the marginal distributions) of the single assets in $\mathcal{J}$ and their dependence structure. An objective function $\Phi$ for the investor will be a function that maps this CDF into a real number.

The objective function that we investigate here is the Omega function, as described in Keating and Shadwick [6]. It is defined as

(3)
\begin{align} \operatorname{Omega} = \frac{{\displaystyle\int_{r_d}^{b}} \left(1-F(r)\right) \mathrm{d}r}{{\displaystyle\int_{a}^{r_d}} F(r) \mathrm{d}r} \end{align}

where $F$ and $f$ are the CDF and probability density function of $r$, and $a$ and $b$ set a relevant return interval. For a given ‘desired return’ threshold $r_d$ this is equivalent to the ratio of the upper and the lower partial moment of order one, hence we may also refer to the function as the Omega ratio. (When Omega is used as an ex post performance measure, it is usually computed for a range of thresholds.)

Partial moments are defined as

(4)
\begin{align} M_{lo} = \int_{-\infty}^{r_d} (r_d - r)^{m_{lo}} \, f(r)\mathrm{d}r \end{align}

and

(5)
\begin{align} M_{up} = \int_{r_d}^{\infty} (r - r_d)^{m_{up}} \, f(r)\mathrm{d}r\,. \end{align}

Partial moments are thus functions of a threshold (the desired return $r_d$), and an exponent, $m_{(\cdot)}$. The subscripts $lo$ and $up$ indicate which tail of the distribution is considered. Setting $r_d$ equal to the mean of $r$ and $m_{lo}=2$, for instance, gives the (lower) semi-variance. Obtained moments are often rescaled by taking the $m_{(\cdot)}$th root.

Thus our optimisation problem can be stated as

(6)
\begin{array} {l} \min_{x} \; \Phi(\ell) \\ x_j^{\inf} \leq x_j \leq x_j^{\sup} \qquad j \in \mathcal{J} \\ K_{\inf} \leq \#\{\mathcal{J}\} \leq K_{\sup} \,. \end{array}

$x_j^{\inf}$ and $x_j^{\sup}$ are vectors of minimum and maximum holding sizes, respectively, for those assets included in the portfolio (ie, those in $\mathcal{J}$). When short-sales are allowed, this constraint becomes $x_j^{\inf} \leq |x_j| \leq x_j^{\sup}$. $K_{\inf}$ and $K_{\sup}$ are cardinality constraints which set a minimum and maximum number of assets in $\mathcal{J}$.

# 3 Methodology & Data

Our optimisation is scenario-based. The easiest way to create the necessary scenarios is to use every historical return as one scenario, hence explicitly modelling the data is not necessary for the algorithm. There is, however, evidence that the method of scenario creation considerably influences the out-of-sample performance of selected portfolios. In this study, we estimate simple factor models for all assets, then we resample from the regressors and the residuals. The resulting return scenarios can easily be aggregated into portfolio losses. The partial moments, and hence the Omega function, can then be estimated from these losses (see the paper for details).

The optimisation algorithms are written in Matlab R2007a and can be downloaded from http://comisef.eu. Some of the computations were distributed with Matlab's Parallel Computing Toolbox on the Myrinet Cluster of the University of Geneva (see http://spc.unige.ch/).

Our data set comprises more than 500 price series of European companies, all denominated in EUR. The data runs from January 1998 to March 2008, thus they include phases of rising and declining share prices. We set up a list of securities that can be sold short. The list comprises rather liquid assets with a large market capitalisation. This may of course introduce a slight bias ‘long small-cap, short large-cap’ into our analysis, but then the algorithm is only allowed, never forced, to include short positions.

We set $x^{\inf}=1\%$ and $x^{\sup}=5\%$, an upper cardinality is 50. The short book is constrained to maximally 30% of assets so to resemble a 130/30-fund. We do not include a riskless asset. Since our algorithm works with actual position sizes, that is integer numbers, a small fraction (order of magnitude of less than 1% of the portfolio) is usually left uninvested.

We implemented a rolling-window backtest. Thus we optimised the model at point in time $t_1$ on data from $t_1-H$ to $t_1-1$ ($H$ was set to around 250 days, that is one year). The resulting portfolio was held until $t_2=t_1+F$, with $F$ set to around 90 days (3 months). At this point, the portfolio was reoptimised, using data from $t_2-H$ until $t_2-1$, and held until $t_3=t_2+F$, and so on. In other words, we constructed a portfolio using data from the last year, held the portfolio for three months, and then rebalanced. In this manner, we ‘walked forward’ through the data to compute a wealth trajectory. All trajectories presented below show resulting out-of-sample paths of wealth, spanning the period from 6 January 1999 to 19 March 2008. Transaction costs were set to 10 basis points for long positions, and 40 basis points for short positions.

## Sensititivy check

To check the sensitivity of our results the following approach was taken: assume a small number of observations were randomly selected and deleted (perhaps there were missing values in the data file). As the historical returns series have changed, the scenarios will be created
differently, and the composition of the optimal portfolio will change. If the portfolio selection method is robust, we would generally expect the resulting portfolio to be similar to the original one, as the change in the historical data is only small. Hence we would expect the new portfolio to exhibit a similar performance as the original one. Repeating this procedure many times, we obtain a collection of out-of-sample
wealth paths. The distribution of these paths gives an indication of the sensitivity of a particular strategy to a particular data set. The whole procedure is described in more detail in the paper.

# 4 Results

As benchmarks, we use $1/N$-portfolios (DeMiguel et al. [1]) and minimum-variance portfolios (long-only). There is evidence that both of these strategies robustly produce good out-of-sample results The results for portfolios optimised on historical data (that is, without resampled scenarios) are shown in the following figure.

As can be seen, the $1/N$-strategy, returning about 7.4% per year, performs rather poorly when compared with the minimum-variance portfolio whose yearly return was about 14.2%.

Next we apply our robustness check to the MV-portfolio (long-only), obtaining 100 paths from jackknifed historical data. The same procedure is then implemented for MV-portfolios that allow short-selling. The resulting ‘bands’ are shown in the figure below.

The median yearly return for the long-only MV-portfolios is about 13.7%, whereas for the long/short-portfolios (constrained to 130/30) it drops to 9.4% per year. Thus allowing short positions leads to a considerably worse performance for the MV-portfolio.

Next we test the portfolios that are constructed by minimising the Omega function. Firstly, we look at the portfolios optimised on historical data (ie, no resampling). As can be seen from the next figure, the selected portfolios perform badly, in particular the long/short portfolio (note the changed scales). For the long-only portfolio, the annualised return is about 6.3%, but the 130/30-portfolio returns a disappointing -3.9% per year.

A well-known pitfall in scenario-based portfolio optimisation is the existence of arbitrage opportunities in the scenario set. This is less of a problem for long-only strategies; when short position are allowed, however, the algorithm may finance seemingly favourable positions by selling short less attractive assets. A practical solution is to increase the number of observations, which is easily done since we assume a model for the data. In fact, switching to a resampling-based scenario-generation method improves the results considerably. The the upper panel of the next figure, we show the distributions of annualised returns from 100 out-of-sample paths. As can be seen, the Omega-portfolios that are constructed with the resampled scenarios show a much higher return than the portfolios obtained from optimising on historical data: the worst case returns are about 13% per year, whereas the median returns increase to about 15.8% for long-only and 17.0% for 130/30.

The Sharpe ratios, depicted in the lower panel of the figure, show that in terms of volatility, the long-only Omega-portfolio generates its higher returns with a higher variability of returns along the trajectory. The long-short Omega-portfolio does so as well (in fact, its average volatility is slightly higher than that of the long-only portfolio), but compensates with an even higher average return.

It needs to be stressed that the higher return of the 130/30-portfolio comes along with a higher sensitivity to the data. For the long-only Omega strategy, the minimum and maximum returns were 13.2% and 18.0% respectively, thus spanning a range of 4.8%. The long-short portfolio, well-constrained to a maximum short book of 30%, had a range of almost 8% (from 12.9% to 20.7%). We also ran backtests in which we allowed the short book to grow to 200% of initial wealth (implying a long book of 300%). Here the median return dropped to about 14%, and the range of outcomes increased to more than 10% (from 8.3% to 18.5%).

# Conclusion

We investigated the performance of portfolios optimised under an alternative selection criterion, the Omega function. Our overall empirical findings can be summarised as follows:

• The data that is used to optimise matters. Solely relying on historical data led to poor performance in our tests. Modelling asset returns (ie, generating scenarios) had a strong impact on portfolio performance.
• Conditional on the applied data modelling procedure, the Omega function selected well-performing portfolios (in terms of final wealth). These portfolios, however, exhibited a relatively higher volatility. This may not be surprising as Omega does not directly penalise the variability of returns.
• Allowing short positions in the portfolios improved performance, but with more estimation risk (ie, a higher sensitivity to the data). In general, constraining position sizes, in particular the total volume of assets to be sold short, led to a better performance than constructing unconstrained long-short portfolios.