Cluster Analysis as a Funds of Hedge Funds Portfolio Tool
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Constructing a diversified portfolio of managers in a fund of funds requires a method for determining how the different exposures complement each other. We show how cluster analysis can be an important tool in this process, especially during a period of market turmoil. Cluster analysis complements the qualitative assessment of a manager’s style and can be used to develop a view of changing markets and manager responses to these changes.
Figure 1: Cluster Analysis Results
Using cluster analysis to group investments has been an overlooked tool among money managers, long accustomed to applications of stochastic modeling. As shown in many nonfinance studies (see for instance Hartigan 1975), cluster analysis has the power to reveal patterns that researchers, analysts, and decision makers might not easily uncover in the data—patterns that provide insights that aid problem solving.
Miceli and Susinno (2003, 2004) use cluster analysis to classify various hedge funds based on the fund returns. They argue that clustering can be used to monitor potential style drifts, conduct peer group analysis, and identify a proper benchmark for groups of funds. They argue that clusters are easier to interpret than large correlation matrices and can be helpful during portfolio construction.
Martin (2000) analyzes monthly returns for hedge funds and concludes that there is significant heterogeneity in individual fund returns within clusters, so aggregate data are likely to be only weakly applicable to individual funds. Kazemi, Gupta, and Daglioglu (2003) argue that a principal components analysis on returns of global strategy funds revealed a structure that was different from that of standard equity and hedge fund indices. Das (2003) clustered managers based on asset class, style of hedge fund, incentive fee, risk level, and liquidity. She found that the cluster analysis was more successful than the ZCM/Hedge classification in categorizing manager return histories.
Gibson and Gyger (2007) conclude that clustering analysis shows that certain managers do not follow their investment style consistently over time. They also use fuzzy clustering to illustrate the degree of misclassification that exists in the industry-accepted investment-style classifications.
In this paper, we show how cluster analysis can reveal important insights into a current issue: portfolio management behavior during the financial crisis of 2008–2009. Cluster analysis can supplement the classic tools of qualitative management interviews and this tool can, over time, help us to better understand a manager’s exposures and how a group of managers adapts to changing market opportunities. To permit greater focus on the discussion of our method, we concentrate on strategies utilized by managed futures hedge funds (also referred to as Commodity Trading Advisors, or CTAs). We construct our initial analysis by looking at monthly returns from January 2004 to December 2007 and at our out-of-sample comparison between January 2008 and March 2009.
To conduct our analysis, we employ several statistical tools. We use cluster analysis as our main tool to separate various CTA strategies into a number of categories and examine how those categories change during different time periods.
Data and Methodology
During the last decade, funds of hedge funds have become increasingly popular with the investors who look to allocate capital to hedge funds, but do not have the resources to research, monitor, and manage a number of stand-alone hedge fund investments. Despite the fact that the fund of funds sector has been substantially affected by the current financial crisis, fund of funds’ investment in hedge funds still accounted for an estimated $606 billion of the $2.1 billion hedge fund industry at the end of the fourth quarter of 2009 (according to BarclayHedge Alternative Investment Databases).
Funds of hedge funds vary in their mix of qualitative and quantitative methodologies they use to construct portfolios. The qualitative methods include in-depth interviews of the manager’s investment style and operations. Quantitative methods include the calculation of the Sharpe ratio and other performance measures, portfolio optimization, and analysis of the correlation matrix of returns. Rules governing maximum exposures to specific investment strategies are also often used to encourage portfolio diversification.
Dividing the world into specific investment strategies requires a well-established set of categories or factors that are easily identified and are known to be good predictors of the pattern of returns. In parts of the hedge fund universe these categories have been created and successfully implemented. Jaeger and Wagner (2005) argue that each fund style has three or four factors that explain a statistically significant portion of returns. These factors can be used to build replicating factor strategies at lower fee levels. Additionally, the returns of the replicating strategies can be useful to the fund of funds managers to measure the skill of hedge fund managers.
However, some strategies are heterogeneous, and organizing hedge funds into static groups may be problematic during periods of financial stress. For these reasons, we have found it useful to supplement our qualitative-style due diligence by grouping managers who have similar performance histories. We have found cluster analysis to be especially useful for building peer groups and examining the change in strategies over time.
The hedge fund returns used in this study were extracted from the Barclay Managed Futures (CTA) Database, which is one of the most comprehensive hedge fund return databases providing information on hedge funds in the managed futures space.
We selected 59 CTA managers for our analysis. We wanted to include funds that represented the full spectrum of managed futures strategies, as well as various time frames and markets. To that end, we included managers from each of the three investment strategies:
- Systematic CTA managers conduct their trading based on signals generated by a computer program or system. Most systematic CTAs tend to use technical indicators in their models. Once the models are in place, a pure systematic CTA does not require any additional human input regarding trading decisions.
- Discretionary CTA managers use their own decision-making skills to determine when to enter and exit trades. Some discretionary managers tend to rely more on fundamental analysis than technical analysis.
- Hybrid CTA managers use a combination of systematic and discretionary approaches. This usually consists of screening the investment universe by a computer algorithm, and then using a manager’s discretion to make the final trading decision.
All funds in our study had return histories going back to at least January 2004. Each fund had assets under management of at least $10 million. Additionally, all candidate funds met two criteria: (1) we had interviewed the fund’s manager, and (2) the fund had a high level of familiarity and transparency in regards to the investment process and the stated exposures and expertise of the manager. Further details of the characteristics of the managers selected for the study are presented in Table 1.
Table 1: Descriptive Statistics on the 59 Programs Selected as of January 2008
Clustering methods can be divided into various groups based on their procedures for arriving at clusters and the criterion used to evaluate whether funds cluster together. The clustering methods can be divided into two main categories of hierarchal and nonhierarchical methods. Agglomerative methods and divisive methods are the two hierarchical clustering techniques. Agglomerative methods start with clusters consisting of individual managers and combine similar clusters until all funds are grouped in a single cluster. Divisive methods proceed in the opposite direction, starting with all funds in a single cluster and cleaving until each cluster contains a single fund. Nonhierarchical methods start with a fixed number of target clusters and attempt to group all funds into these target clusters.
To group the funds, we wanted a cluster procedure that did not require us to specify in advance the number of the clusters or their size. Also we wanted a method that would be robust to the specifics of alternative measures of fund similarity and would generate tight clusters in which all members of the cluster were more similar to each other than to any nonmembers. This is in contrast to methods in which members might be very different but be grouped due to common intermediate links.
For these reasons we used a popular method called complete-linkage agglomerative cluster analysis. The analysis proceeds using the following steps:
- Start the process with each manager as its own cluster.
- Join the two closest clusters into a new cluster with the difference between the managers defined as one minus the correlation. The higher the correlation, the closer the clusters: di(j,k) = max(dij, dik)
- Recalculate the distance between the new cluster and all other clusters (maximum distance).
- Repeat steps 2 and 3 until all managers are grouped into a single cluster, or you are satisfied that you have a set of clusters that are most reflective of the data characteristics.
Each clustering method also requires a criterion for determining the similarity of funds so that funds can be clustered. Similarity measures depend on the types of input data and the goals of the clustering. For some characteristics, Euclidean distance measures can be used. For discrete characteristics, a count of the number of overlapping characteristics, also known as the Hamming distance, can be used. We used the one minus the correlation matrix of the pattern of historical returns, because correlation is a commonly used measure of association for comparing performance histories. Correlations also emphasize the timing of the relative best and worst performance controlling for volatility.
The results of the cluster analysis appear in Figure 1 (see above).
We next labeled funds that were grouped together to determine how the groupings lined up with our qualitative assessments of each manager and if there were any indications of the number of groups. We identify five major clusters within our data set. From our due diligence interviews we extracted the main characteristics of the managers comprising each cluster in Figure 1. The groupings allowed us to focus on the bigger picture of different sensitivities to market events.
Table 2: Main Characteristics of Clusters
- Systematic trend followers
- Multiple time frames
- Single sector (e.g., energy)
- Long-term systematic trend followers
- Emphasize financial commodities
- Globally diversified
- Use daily data (open-high-low-close)
- Multiple time frames
- Fundamentally driven systematic approach
- Short-term time horizon
- Mean reversion/pattern recognition/counter trend
- Technical analysis
We next determined which months in the managers’ performance history stood out as being consistently positive or negative for managers in each cluster. We reviewed those key months within each cluster, as well as across clusters. We also determined if the key dates were in line with the strategy of the managers comprising the cluster.
We also reviewed correlations within clusters, as well as changes in historical correlations that took place in 2008. To identify any potential trends present during the time frame of our analysis, we examined the key dates when the returns were particularly high or low.
We constructed our initial analysis looking at monthly returns from January 2004 to December 2007. We selected five months in which the average standardized returns of all managers in the cluster were the lowest. Then we selected the five months in which the returns were the highest.
We then tested to ensure that most managers had positive returns during the top five months in each cluster and had negative returns during the bottom five months in each cluster. This was done to ensure that the results are not skewed by a limited number of over- or underperforming managers. The results, summarized in Figure 2, show that for all months in question the majority of the managers within each cluster moved in the same direction as the average return.
Figure 2: Key Months Based on Standardized Returns
Next, we organized the top five months and the bottom five months for each cluster on a heat map (Figure 3). The top five months are highlighted in blue and the bottom five months are highlighted in red.
Figure 3: Key Months Heat Map: In-Sample Analysis
Next we abstracted comments from managers’ newsletters to compile a description of their monthly positioning and performance. February 2004 was the second best month for Cluster B and Cluster D, and the fifth best month for Cluster A and Cluster C. During that month, the equity market experienced a turnaround after an almost yearlong rally. The S&P 500 fell 2.5% between the end of January and the first week of February. That was a reversal in the long-term trend that many of the multiple-time-frame and medium-term trend followers were able to capitalize on. The only cluster that did not benefit during the month was Cluster E, which is comprised of short-term discretionary managers.
A key positive performance month of February 2004 is shortly followed by a key negative performance month, April 2004. It was the second-worst performance month for Clusters A, B, and C. During that month precious metals experienced a sharp downturn; gold declined by more than 9%, and silver by more than 20%. This had a strong negative effect on the managers who were long on precious metals, including single-sector managers in Cluster B, and longer-term trend followers in Clusters A and C, whose trend-following systems might not have recognized the trend reversal. Another market mover in April 2004 was the US dollar, which rallied more than 3.5%. The currency movement was also detrimental to the trend followers in Clusters A and C.
The next clustering of best and worst months evident from Figure 3 starts in the beginning of 2007 and lasts through September 2007.
This distribution of the top and bottom performance months in this clustering is not surprising given the state of financial markets in 2007. The market turmoil in 2007, 2008, and 2009 has drastically changed the economic landscape in the US and globally. The Dow Jones Industrial Average declined more than 54% from its all-time-high close (14,164 in October 2007) to its lowest point in almost ten years (6,470 in March 2009). The price of oil dropped from $147 in July 2008 to $40 in the beginning of December 2008. Since the beginning of the crisis, the financial services industry has shed more than 220,000 jobs in the US alone, according to Bloomberg.
According to Brunnermeier (2008), the current crisis was sparked when subprime mortgage default rates began to increase in the first quarter of 2007. During February 2007 the financial markets experienced their first monthly decline since May 2006. The S&P 500 fell 2%, while the NASDAQ fell 1.8%. Out of the five clusters on which we conducted our analysis, Cluster D was the one most affected during the first quarter of 2007. The cluster includes managers who incorporated a fundamental approach to their trading and applied their analyses to multiple time frames. January 2007 was the fourth-best performance month for the cluster, followed by the worst performance month during the time period analyzed. The February sell-off was mainly attributed to profit taking in what was believed to be an overheated economy. This phenomenon was reflected in a number of technical indicators, on which the fundamental managers who comprise Cluster D did not necessarily rely.
The beginning of the second quarter of 2007 saw a short-lived market rally. The S&P 500 rose, and many of the largest financial companies retained levels close to their all-time highs. April and May 2007 were the second- and fourth-best months for the managers in Cluster C, which is comprised of managers who emphasized the use of financial futures in their portfolios and tend to use long-term following systems. At the same time, May 2007 was the fourth-worst month for Cluster E. Contrary to the managers in Cluster C, managers in Cluster E followed short-term discretionary strategies and used mean reversion and pattern recognition techniques.
In July and August 2007 many financial institutions were in trouble again. Some of the first hedge funds that ran into subprime-related problems were the two infamous Bear Stearns funds that almost collapsed in July 2007, forcing their parent to invest $3.2 billion to keep them afloat. Next, the biggest US home loan lender, Countrywide Financial, announced an earnings drop of 33% and significantly decreased their future earnings forecasts. As financial companies were affected by the unraveling crisis in July and August 2007, so were the CTA managers who traded financial futures.
Based on our analysis, July and August 2007 were the worst and third-worst months for Cluster C. Managers in that cluster tended to use financial futures and could have been adversely impacted by a drop in market liquidity. Systematic trend followers in Cluster A were also negatively affected in August. The dramatic reversal in the financial markets led many trend followers to significant losses, as their systems did not properly recognize the changes in the market conditions. However, August 2007 was the best performance month for the managers in Cluster D. It appears that the managers who relied on fundamental rather than technical indicators were able to take advantage of the reversals in the financial markets that happened in August 2007, unlike the trend-following managers in Cluster A.
September 2007 was a positive month for the managers in Clusters A, B, and E. Managers in Cluster A used multiple trading time frames, and managers in Cluster E tended to use short-term trading strategies, which was beneficial in September 2007 because the markets did not exhibit any pronounced trends. Cluster B includes single-sector managers who used a certain level of discretion in their trading. One of the most popular sectors in Cluster B was energy, so an increase of approximately 7% in oil prices in September 2007 was one of the main drivers of their returns.
We next examine how the clusters identified using the performance data for the period from January 2004 to December 2007 would have performed from January 2008 to the first quarter of 2009.
For each cluster we used the performance in the fifth-best and fifth-worst months as reference points to determine if any months in 2008 can be identified as key months. Any month in which the performance of the out-of-sample data was higher than the fifth-best month in the in-sample analysis is considered a key positive month. Likewise, any month when the performance was lower than the fifth-worst month in the in-sample analysis is considered a key negative month. The results are presented in the same format as our original key months analysis.
Figure 4: Key Months Heat Map: Out-of-Sample Analysis
In January 2008 the financial markets experienced a worldwide sell-off, triggered by the rating agency Fitch downgrading Ambac, one of the major bond guarantors. The panic that followed forced the Fed to cut the interest rates twice during the month of January, and gold reached a new high of $929 per ounce. Single-sector managers in Cluster B, many of who were long on precious metals, benefited from the increase in gold prices.
February 2008 was characterized by overall strengthening of the commodities that were used as a hedging tool for the weakening stock market. Goldman Sachs Commodity Index rose more than 11% in February 2008, while the S&P 500 declined 3.5%. Managers in Clusters A and B outperformed managers in other clusters. Many of the trend-following managers in Cluster A were long on commodities, and those positions had a positive effect on their returns. Managers in Cluster B also enjoyed positive performances, which can be attributed to a significant increase in the price levels of various commodities.
In March 2008 Bear Stearns, which was smaller and more leveraged than other Wall Street banks, ran into liquidity issues as the global credit crisis continued to unfold. Since Bear Stearns was deemed “too interconnected” to simply go bankrupt, the Federal Reserve had to step in and granted a $30 billion loan to JPMorgan Chase to acquire Bear Stearns. The original purchase price announced was $2 per share. However, due to significant political opposition and pressure from the equity holders, the purchase price was increased to $10 per share. By comparison, in January 2007 the stock peaked at $170. Cluster D was heavily affected by the market turmoil in March 2008. The fundamental indicators, on which the managers in this cluster rely, may not have predicted the rapid market decline.
July 2008 was a month of very poor performance for Cluster A. During that month the energy prices peaked and started their steep decline. In July 2008 the price of crude oil went down by more than 10%. This reversal of commodity prices had a significant negative impact on the performance of the systematic trend-following managers in Cluster A.
September 2008 was characterized by rapidly developing situations and significant government intervention. The US Treasury takeover of Fannie Mae and Freddie Mac, the collapse of Lehman Brothers, the purchase of Merrill Lynch by Bank of America, and the events associated with the market bailout deal all resulted in immense daily trading ranges in the equity and fixed income markets. In October 2008 tremendous market volatility continued and reached unprecedented levels. At the end of October, the CBOE Volatility Index (VIX) closed at over 80, which was a previously unheard of high. As recessionary fears worsened and forced sellers flooded the markets, the vast majority of financial markets experienced massive sell-offs, with the S&P 500 declining almost 17%.
In both September and October 2008, Clusters B (discretionary managers) and D (fundamentally driven managers) underperformed. As the markets were driven by unprecedented volatility and government intervention, it became difficult for those managers to trade according to fundamentals. An uncertain market environment was also challenging for discretionary managers. Additionally, due to significant daily market swings, many managers were stopped out of their positions and faced margin calls.
Unlike September 2008, when extreme market volatility did not have any particular direction, October 2008 saw pronounced trends in both equity and bond markets. Systematic trend followers in Cluster A benefited, and managers who used short- and medium-term systems outperformed.
In December 2008 market volatility indices retreated from the record highs set in October and November. Most of equity markets were almost flat, and deflationary pressures drove up the price of 30-year US bond futures to an incredible 10% intramonth. Many systematic managers had a flat to slightly positive month, because the markets were not characterized by any pronounced trends. However, the discretionary managers in Cluster E outperformed.
In February 2009 the VIX rose above 50 again, while most other asset classes experienced losses. Investor confidence softened, with both the MSCI EAFE and the S&P 500 declining approximately 10% during the month. Sideways trading was persistent in many financial markets, which made it a difficult environment for discretionary managers in Cluster B.
In March 2009 equities rallied sharply reacting to the government’s plan to rescue the banking system. This intervention was unexpected by many fundamental managers in Cluster D, who had a difficult month as their short equity positions experienced significant losses. Other managers in the cluster suffered losses in energy markets because oil trading was especially choppy during the month.
The above analysis shows how a clustering of managers gives focus to both the historical return patterns as well as the exposures of the different groups of managers. The method can be used to help to determine whether the manager’s performance is in line with the “official” strategy when reviewed against its peers and the existing market environment.
Next, we look at how cluster analysis provides a framework for examining the global change over time in the structure of managers.
Changes in Correlations
In this section, we review how the correlation matrices that we derived as part of our analysis for the time period beginning January 2004 to December 2007 changed during the period from January 2008 to March 2009.
To examine the effects of these changing market conditions on the managers and assess their responses, we compared the manager-to-manager correlations in 2008 and the first quarter of 2009 versus their cross correlations in the previous four years.
We used the cluster analysis to order the rows and columns of the correlation matrix and placed the changes in correlation in the cells of the matrix.
We organized the output in the heat map below. White spaces indicate no significant changes in correlations. Blue spots indicate increasing correlations, with darker shades of blue representing more significant increases. Red spots indicate managers becoming less correlated, with darker shades of red representing the lowest correlations.
Figure 5: Correlation Changes
Overall, the pattern of correlations across the two nonoverlapping time periods is quite stable, as most of the matrix is in white. This is especially true for the systematic managers in Clusters C (technically driven systematic managers) and D (fundamentally driven systematic managers), both in terms of the correlations among funds with each cluster and the relationship of these funds to funds outside their respective clusters.
Within Cluster A (managers using multiple time frames), there is a tendency toward lower cross correlations in 2008 versus their historical levels. This may indicate a regression toward the mean or a splitting up of the cluster as managers take different strategies to adapt to shifting market opportunities.
A line of blue in the upper left corner indicates an increase in correlation between many of the managers in Cluster A to the managers in Cluster E. The specific funds within Cluster E employ very short-term strategies. This may indicate a shifting of strategies to short-term frames by funds in Cluster A.
We believe that the analysis of changes in correlations proves to be a useful tool in determining whether the managers stay true to their strategy over time. A fund of funds structure often dictates a strict investment mandate in which a portfolio manager needs to adhere to predetermined allocation limits for each investment strategy. The tool that we described can help managers independently confirm that the desired strategy allocations are being followed. It can also be used as a method of catching early warning signs of style drifts.
As noted before there are many possible clustering methods. To understand the robustness of clustering methods, we compared the complete-linkage clustering against those obtained by applying several popular clustering techniques to the data set.
Robustness of the Clustering Method
We selected three popular methods, covering some of the major paradigms and methods for analyzing similarities. The methods are:
- Average distance hierarchical clustering
- Ward’s method
- Divisive clustering
These methods are described in more detail in (Baird and Noma, 1978, chapter 11) and Hartigan (1975) and include representatives of agglomerative, divisive, and non-hierarchical methods. The methods also include different biases when considering the ability of intermediate funds to attract or separate two clusters.
All methods were applied to the same set of historical returns of the 59 managers from January 2004 to December 2007. All methods used one minus the correlations of the monthly returns as a measure of similarity between pairs of funds. A subjective criterion was used to determine the number of clusters and the constituents of each cluster, but judgments were made without reference to the other clusters or to the underlying manager descriptions. In all cases, we felt that between four and five clusters was the best description of the data.
Table 2 shows the number of funds that are in the row cluster for the complete-linkage hierarchical cluster analysis and the column cluster for the comparison clustering. For instance, for the six funds within Cluster B of the complete-linkage clustering, five are placed in Cluster C of the average-linkage clustering and one is in Cluster B.
Table 3: Comparison of Clustering Methods
The similarity of each clustering pair is assessed using both a statistical model and a communication model. If two clusterings are totally unrelated then the distribution of funds in each cell, nij, in Figure 1 could be estimated by the following formula:
ni. = the number of funds in row cluster i
n.j = the number of funds in column cluster j
n.. = the total number of funds that were clustered
A chi square test (Hays, 1994) determines the probability that the observed pattern of mappings could have occurred by chance. If this random hypothesis is rejected then there is some interrelationship between the clusters. The results in Table 4 reject the random model and show that there is a statistically significant pattern of mappings from one clustering to the other.
Table 4: Statistical Significance of Methods
The Communications Model
With the random hypothesis rejected, the next question is the degree to which the clusterings are similar. One way to assess the similarity is by measuring the degree to which the grouping of the funds gives us information on how the funds are grouped using a second clustering method. A method was developed by Shannon (1984; see also Weaver & Shannon, 1963) to evaluate the quality of a communications channel. They looked at the amount of information (entropy) in each cluster divided by the number of funds in each cluster.
N = the number of clusters
pi = number of funds in cluster i divided by the total number of funds in all clusters
To compare the complete-linkage clustering versus the average-linkage clustering, we compute the entropy for the complete-linkage clustering and for the average-linkage clustering separately. We then apply the same formula to the complete- vs average-linkage matrix in Table 2. This is the total amount of information (entropy) in the entire pair of clusterings. A larger overlap indicates a commonality of the two methods. The higher this overlap (also known as information transmission), the more similar the clusterings are.
Table 4 shows the entropies for each pair of clusters with the total entropy from the transition matrices from Table 1. The degree of similarity in the clustering is compared to the total entropy. For each of the three clusterings, compared to the complete-linkage method, the overlap is between 40–55% of the total entropy. This indicates a healthy degree of communality of the complete-linkage clustering with the alternative clustering methods.
Table 5: Degree of Similarity of Clustering Methods
We have shown how clustering managers based on historical returns can supplement the information obtained from a qualitative review of the manager. We have shown how the clusters highlight which fund characteristics are most salient in understanding a strategy’s ability to diversify a portfolio. By examining the up and down months for funds in each cluster, we can assess the strategy sensitivities. In addition, by using the clustered funds as a baseline, we can develop ideas about how the various strategies are reacting to markets during a period of violent change. Within the set of CTAs we analyzed, we found support for the following hypotheses:
- Systematic strategies that clustered prior to 2008 continued to be correlated in 2008.
- Managers using multistrategy/multiple time frames shift their strategies, as indicated by lower cross correlations, as they adapt to changing market opportunities.
- In 2008 long-term systematic managers appear to be shifting toward models and time frames employed by short-term managers.
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–Elliot Noma, PhD, and Maria Shtrapenina, CFA