A Systematic Global Macro Fund

Transcription

1 A Systematic Global Macro Fund Correlation and Portfolio Construction January 2013 Working Paper Lawson McWhorter, CMT, CFA Head of Research Abstract Trading strategies are usually evaluated primarily on performance metrics like Sharpe Ratio, with correlation often a secondary consideration at best. But when it comes to portfolio construction, correlation is as important to the final results as the Sharpe Ratio of the underlying strategies. For example, a portfolio of five uncorrelated strategies each with a Sharpe Ratio of 0.5 performs just as well on a portfolio basis as ten highly correlated strategies with a Sharpe of 1.0. Using simple charts, we explore the interaction between correlation, the number of independent strategies, and the Sharpe Ratio realized from building portfolios. This material is intended for information purposes only, it does not constitute investment advice or a recommendation or an offer or solicitation, and is not the basis for any contract to purchase or sell any security, or other instrument, or for Metolius Capital, LLC to enter into or arrange any type of transaction as a consequence of any information contained herein. The information set forth above was compiled from sources Metolius Capital, LLC believes to be reliable; however, Metolius Capital, LLC makes no representation or guarantees hereby with respect to the accuracy or completeness of such data. Past performance is not necessarily indicative of future performance. An investment in the funds will be subject to a variety of risks, and there can be no assurance that its investment objectives will be met or that it will not incur losses.

2 A Thought Experiment In this paper we examine the relationship between correlation, the number of independent trading strategies, and the Sharpe ratio that arises from forming portfolios of these strategies. Our building block in this thought experiment will be a hypothetical random trading system that delivers a Sharpe Ratio of 1.0 over a year of daily returns (252 observations). To facilitate combining trading systems into portfolios, all return streams are standardized to a 10% annualized volatility. Otherwise, if two trading systems with equal Sharpe Ratios but unequal volatilities are combined into a portfolio, the higher volatility system would end up dominating results. By holding volatility constant, we effectively equal weight portfolios. Given our volatility assumption, we can then back into the return required to generate a Sharpe of 1.0 (i.e. 10%, or an average return of % per day). Figure 1 below shows an example of four synthetic trading strategies calibrated to deliver a Sharpe Ratio of 1.0. Other than having a positive drift, these equity curves are entirely random. Nonetheless, they are reasonable facsimiles of what you might encounter in the real world. The first thing that springs to mind is that these would probably look interesting when combined into a portfolio. Figure 1. Four synthetic trading strategies with Sharpe Ratio of Metolius Capital LLC Research Paper Page 2

3 Indeed, when combined into an equally weighted portfolio, they make for an attractive equity curve as shown in Figure 2. The Sharpe Ratio doubles to 2.0, driven entirely by a reduction in volatility which clocks in at 5% for the portfolio versus 10% for each the four underlying trading systems. Unfortunately, there is a catch. While not readily apparent from looking at the individual equity curves, they were actually constructed to have zero correlation. In practice it is difficult to find uncorrelated and profitable trading systems, much less four which are also equally uncorrelated to each other. Figure 2. Combined portfolio Back To Reality To study portfolio construction with slightly more real-world characteristics, a natural starting point is trend-following, the dominant category of CTA s and many global macro funds. Trend following comes in many forms price momentum, moving averages, and technical chart breakout patterns to name but a few. After a significant market move, these diverse, but ultimately very similar, methodologies will all likely hold the same position. Furthermore, operational and business considerations tend to force managers to apply these techniques over similar timeframes and holding periods. Coupled with the fact that most managers typically trade from a relatively limited universe of liquid markets, it is easy to see how correlation arises. Newedge, as prime broker to many of the world s largest CTA and global macro funds, is uniquely positioned to collect data that can help us understand correlation in the real world. Newedge has done considerable research in the areas of index and portfolio construction and publishes the widely followed Newedge CTA Index. The constituents of this index include some of the largest, most sophisticated and established CTA s around. It is used as a benchmark for CTA returns and often as a proxy for trendfollowing. Newedge also calculates a Trend Indicator designed to track the returns of a prototypical trend following strategy. In a research piece entitled Improvements in the Newedge Trend Indicator, they showed a correlation matrix of the individual funds comprising the Newedge CTA Index as well as Metolius Capital LLC Research Paper Page 3

4 their correlation to the Trend Indicator i. Of the funds Newedge categorized as trend followers, the average cross correlation using daily returns in 2011 was Returning to our first example, what if we combine multiple systems into a portfolio, each with a Sharpe of 1.0, but like real-world trend-followers, correlated at 0.78? What improvement in Sharpe can we expect at the portfolio level? Clearly it will be lower than our first example where the Sharpe doubled, but how much lower? Figure 3 shows that the answer is significantly lower. By the fifth system, we have increased the portfolio Sharpe to only 1.10 and each additional strategy thereafter adds less than 0.01 to the Sharpe. Once you have more than a handful of highly correlated return streams there is little benefit to adding another. Figure 3. Portfolio of highly correlated trading strategies. Portfolio Sharpe ( =0.78) Sharpe Ratio Number of Strategies This is a pretty sobering result, but at the same time overly conservative, since most people intuitively grasp that this portfolio provides little true diversification. So what happens if we halve the correlation to 0.39? Now we see a slightly more encouraging picture in Figure 4 where the curve is a little less steep with higher overall Sharpe Ratios. By the time we add the fifth strategy we have achieved an approximately 40% improvement in Sharpe versus a single strategy alone. Metolius Capital LLC Research Paper Page 4

5 Figure 4. Portfolio of moderately correlated trading strategies. Portfolio Sharpe ( =0.39) Sharpe Ratio Number of Strategies Finally to circle back to the beginning and satifsy our curiosity, how does the Sharpe Ratio respond when creating portfolios of perfectly uncorrelated return streams? Figure 5 shows that we can still add considerable value even after the tenth strategy is added to the portfolio. In fact, for truly uncorrelated strategies, the Sharpe Ratio increases at a rate equal to the square root of the number of strategies. Another way to think about this is to recall that our synthetic strategy has an average daily return of %. If we add enough of these return streams, the law of large numbers insures we realize this expectation. In other words, we should never have a losing day even though we would still experience some volatility. Metolius Capital LLC Research Paper Page 5

6 Figure 5. Portfolio of uncorrelated trading strategies. Portfolio Sharpe ( =0) Sharpe Ratio Number of Strategies Trees Don t Grow to the Sky With uncorrelated strategies the maximum Sharpe Ratio is infinite. But as soon as positive correlation is induced, the plot of the Sharpe Ratio rapidly turns concave, suggesting there is an asymptotic maximum. In their paper The Strategy Approval Decision: A Sharpe Ratio Indifference Curve Approach, Bailey, Lopez de Prado, and del Pozo show this maximum to be the average Sharpe Ratio divided by the square root of the correlation ii. Figure 6 below plots the maximum Sharpe Ratio across a range of positive correlations using our base case random trading system with a Sharpe Ratio of 1.0. At a 0.1 correlation, the maximum obtainable portfolio Sharpe is More realistic correlation and Sharpe Ratio assumptions quickly constrain the maximum, probably at a lower level than most people appreciate. Metolius Capital LLC Research Paper Page 6

7 Figure 6. Effective maximum Sharpe Ratio. Maximum Portfolio Sharpe Sharpe Ratio Correlation Interaction All the analysis above examined only one variable at a time, holding either correlation or the number of strategies constant. In Figure 7 we show the interaction between correlation and the number of strategies. At the left-hand side of the chart we see correlation at its upper bound of 1. As the number of strategies increases, the Sharpe Ratio is constant given the perfect correlation. At the opposite extreme, the right hand side of the chart shows how responsive the Sharpe Ratio becomes as strategies with zero correlation are added to the portfolio. Metolius Capital LLC Research Paper Page 7

8 Figure 7. Sharpe Ratio across correlations and strategies Sharpe Strategies Correlation 1.0 The Holy Grail Although the chart above is bounded at zero, correlation can of course be negative as well. The surface chart hints that the holy grail of portfolio construction lies within the realm of negative correlation. At the extreme, two perfectly negatively correlated strategies (assuming a positive expection and appropriately weighted volatilties) would never have a losing period. A similar, but not necessarily more obtainable, result can be found by combining a small number of negatively correlated return streams. For example, it would only take 5 strategies correlated at to have an infinite Sharpe Ratio. While theoretically appealing, negative correlation is difficult to find in practice. Most people tend to focus on building the best stand-alone trading system possible. In other words, they attempt to maximize Sharpe Ratio or a similar metric. Correlation is an afterthought or viewed as a derivative outcome over which they have little or no control. Futhermore, people naturally tend to find a strategy that works (e.g. trend-following), specialize in that area, and then find that it acts as a center of gravity, pulling their research efforts to find different strategies back into its general orbit. In short, it is challenging enough to develop profitable trading systems, let alone truly unique and uncorrelated ones. Conclusion Finally, there are many important caveats that should be mentioned. These simulations should be viewed as a thought experiment, designed to illuminate a concept rather than define physical laws governing portfolio construction in reality. Probably the most obvious shortcoming is that volatility and Metolius Capital LLC Research Paper Page 8

9 correlations are not constant in the real world. That system or market you thought was uncorrelated can suddenly become highly correlated, typically at the most inopportune time (i.e. the correlations going to 1). Our simulated returns were also based on a multivariate normal distribution; however, fund returns and financial data in general tend to have fat tails (i.e. more outliers than expected or the proverbial black swan). The Sharpe Ratio is a perfectly good metric for a study such as this based on normally distributed data; however, a well known limitation is that it fails to capture higher moments of the return distribution. In our own research, we rely on a host of other metrics to measure drawdown, skew, and other important characteristics of trading strategy peformance. But the primary finding of this study is that correlation is as important to the final portfolio as the Sharpe Ratio of the underlying strategies. If you can find five strategies that have zero correlation to each other but an average Sharpe of only 0.50, you end up in the same place as having ten of the best trend followers each with a Sharpe of 1.0 but correlated at This is not necessarily less work, but it is certainly the road less traveled and worthy of further consideration and study. i Newedge AlternativeEdge Research, Improvements in the Newedge Trend Indicator 23 December ii Bailey, David H. and Lopez de Prado, Marcos, The Strategy Approval Decision: A Sharpe Ratio Indifference Curve Approach (February 12, 2012). Algorithmic Finance, Vol. 2, No. 1 (2013), Forthcoming. Available at SSRN: or E. Liberty Drive, Suite 220 Wheaton, Illinois Telephone: Fax: Metolius Capital LLC Research Paper Page 9