Procyclicality and diversification in the hedge fund industry in the aftermath of the subprime crisis

Abstract

Using an updated database that extends after the subprime crisis, we revisit the asymmetries of hedge fund behavior in recession compared with economic expansion. In this respect, we study the time-varying α’s and β’s associated with strategy returns using an innovative framework based on the Kalman filter and the multivariate GARCH. We find that hedge fund managers reduce drastically their risk exposure during financial crises while their behavior is much smoother in normal times. We also find that hedge funds continue to provide good prospects for investors in terms of risk-adjusted returns. Actually, the procyclicality of hedge fund strategies’ returns seems to decrease through time. Moreover, the strategies’ behavior in terms of α and β tends to become more heterogeneous in times of crisis. The strategy exposure to adverse shocks seems to recede even after accounting for the subprime crisis. Finally, many hedge fund strategies benefit from an increase in the volatility of stock market returns. Hedge fund strategies may thus constitute a way to offset the lower expected returns observed in the conventional financial markets and may contribute to portfolio diversification.

INTRODUCTION

Many researchers have analyzed the asymmetric behavior of hedge fund during bear markets compared with bull markets, the subprime crisis having fostered this kind of studies (Mitchell and Pulvino, 2001; Agarwal and Naik, 2004; Capocci et al, 2005; Billio et al, 2009; Bollen and Whaley, 2009; Bollen, 2011; Sandvik et al, 2011). According to Dewachter and Wouters (2014), asymmetry is related to the fact that ‘the reaction of agents – here hedge funds – is much more pronounced during periods of recession (crises) while they behave much smoothly during booming business cycle periods’. In this respect, researchers found that hedge funds reduce drastically their risk exposures in periods of crisis. The signs of the factor loadings may even change, these factors accounting for other sources of risk in crises.¹

Most authors study the asymmetric behavior of hedge funds across two states of nature: bull markets and bear markets (for example, Capocci et al, 2005; Sandvik et al, 2011). To perform these studies, the methods used go from simple regressions on these two states to regime-switching models. The analysis is quite static because the behavior of hedge funds is not monitored inside these states. A notable exception on that matter is the study of Bollen and Whaley (2009), who rely on the Kalman filter to track the autoregressive behavior of the hedge fund β across the states of nature.

In this study, we extend the framework of Bollen and Whaley (2009) by monitoring the procyclical behavior of the α’s and β’s of hedge fund strategies. To shed light on this dimension, we feature a parsimonious return model based on the Kalman filter in order to pin down the dynamics of hedge fund strategies. One of our contributions is to relate the time-varying α and β of each strategy to conditioning market information. In this respect, the β of a strategy is related to the payoffs of a lookback straddle defined as the first principal component of Hsieh’s lookback risk factors (Fung and Hsieh, 2001, 2004). This feature of our model allows us to analyze how hedge funds behave when the volatility of financial markets increases (Treynor and Mazuy, 1966; Henriksson and Merton, 1981; Fung and Hsieh, 2001; Billio et al, 2009). The co-movements of hedge fund strategies’ returns, β’s and α’s have also not been studied yet using a comprehensive approach accounting for the impact of business cycles.² These co-movements are related to the level of risk in the hedge fund sector.

In this article, we also focus on the dynamics of diversification benefits provided by the hedge fund strategies. The literature has opposite views on this topic. In this respect, some researchers (for example, Billio et al, 2009) argue that risk is greatly underestimated in the hedge fund industry, particularly in crisis periods. In contrast, other researchers (for example, Sandvik et al, 2011; Brown et al, 2012; Boyson et al, 2013) assess that there is any significant underperformance by the hedge fund industry during financial crises.³ Moreover, some strategies – as the managed futures, global macro and short sellers ones – even offer good diversification benefits when these benefits are needed the most, that is, in periods of financial turmoil. Other strategies – for example, the trend followers – benefit from the stock market volatility that is often associated with a downward trend in stock market returns (Black, 1976; Fung and Hsieh, 1997, 2001, 2004)⁴. It is important to revisit the topic of diversification in a dynamic setting with more recent data and with improved methods since investors – especially pension funds that suffer from chronic undercapitalization – are in search of yield in a world plagued by lower expected returns on traditional assets.

In line with many recent studies (Zhong, 2008; Sandvik et al, 2011), our empirical work shows that the α’s related to hedge fund strategies tend to decrease through time. However, they remain positive, suggesting that hedge funds continue to deliver positive absolute returns. Surprisingly, for many strategies, the α increased during the subprime crisis, suggesting that hedge fund strategies may display a good performance even in times of turmoil (Sandvik et al, 2011). In other respects, the β of hedge funds is quite procyclical but the strategies’ β’s behave more heterogeneously during crises, suggesting again diversification opportunities. Strategies’ returns also tended to move less homogenously during the subprime crisis than during the three preceding ones – that is, the Asian, Russian-LTCM and bubble-tech crises – another indication of increasing diversification benefits in the hedge fund sector. However, our analysis shows that the indicators used to monitor the cross-sectional co-movements of time series may deliver ambiguous signals and thus ought to be interpreted with caution, an issue overlooked in many previous studies.

This article is organized as follows. The next section presents our empirical return model. The subsequent section reports our database and the stylized facts associated with the hedge fund strategies’ returns. The penultimate section analyses the empirical results while the final section concludes.

THE EMPIRICAL RETURN MODEL

Our model aims at studying the procylicality of the hedge fund strategies over the period spanning January 1995–September 2012. To do so, we rely on a hedge fund return model estimated with the Kalman filter.⁵ In such a model, the structure of the signal and state equations ought to be parsimonious so we only introduce key risk-based factors in the signal equation. We therefore do not resort to more elaborated hedge fund return models such as the Fung and Hsieh’s (2004) seven risk-based factor model.

The signal or observation equation, which relates the return of strategy i (R _it ) to its risk factors, is formulated as follows:

where r _ft is the risk-free return; α _it is the time-varying α; β _it is the time-varying β; R _mt is the market portfolio return; SMB _t is the return of a mimicking portfolio that is long in small firm stocks and short in big firm stocks – size being measured by stock market capitalization; Spread _t is the term structure spread, that is the spread between the Federal Reserve 10-year constant maturity yield and the 3-month Treasury bills yield, which may be assimilated to a portfolio that is long in the long-term (10-year) interest rate and short in the short-term interest rate.

In equation (1), (R _mt −r _ft ) and SMB _t are two important risk factors found in most hedge fund return models. Fung and Hsieh (2004) call them the equity ABS (asset-based-style) factors, which stand for the main drivers of the long/short hedge fund strategy – that is, the leading hedge fund strategy. To these two factors, we add the term spread (Spread _t ), a variable that has gained strength in explaining returns in line with the development of shadow banking (Billio et al, 2009). An increase in the spread usually signals an increase in the risk premia on bonds and possibly on stocks, which tends to give rise to an increase in expected returns on these securities since returns usually follow a mean-reverting or Ornstein-Uhlenbeck process. Moreover, an increase in the spread also forecasts an economic recovery, which is associated with higher expected returns (Ang et al, 2004). Note also that a positive relationship seems to link the long-term interest rate and stock risk premia at the statistical level.⁶ Indeed, one of the main drivers of the structural decrease in stock risk premia would be the structural drop in long-term interest rates. According to this argumentation, which is quite unexplored in the hedge fund industry, the sign of the coefficient (γ ₂) of the term spread should be positive in equation (1). These arguments that favor a positive sign for γ ₂ are akin to a ‘price of risk’ approach to the term spread.

This argument is based on the following equation – borrowed from Veronesi (2010) – of the current long term r(0, T) rate observed at time 0 and having a maturity equal to T:

where E(r) is the expected future yield; λ _t is the price of risk, here market risk – that is, risk related to the bond duration – since there is no default risk on government bonds; and σ _t ² is the variance of the interest rate. The last term of equation (2) represents an adjustment term that accounts for the convexity linking the price of a bond to its yield. According to equation (2), an increase in λ _t leads to an increase in the long-term yield but is not associated with an increase in future spot rates as in the expectation theory. According to Veronesi (2010), it is rather associated with an increase in future bond prices or capital gains on bond holdings. Another argument that favors a positive sign for γ ₂ is that hedge funds are big investors in mortgage-backed securities (MBS). Yet, an increase in the term spread is associated with an increase in the yield of MBS, which entails an increase in expected returns for hedge funds holding MBS.

However, according to the ‘expectations approach’ to the term spread – which is associated with the first term of equation (2) – the sign of γ ₂ would be negative. Indeed, the term spread has become an important indicator of monetary policy but is also a proxy for the phases of the business cycle. According to Adrian and Shin (2010), the fact that short-term interest rates are close to zero has induced central banks to change the way they manage monetary policy. The credit channel⁷ is now partly implemented through this spread. An increase in the spread is associated with a tightening of monetary policy. Moreover, the term structure spread is also an important indicator of monetary policy in the literature focusing on a new channel of the transmission of monetary policy, namely the risk-taking channel⁸ (for example, Disyatat, 2010; Gambacorta and Marques-Ibanez, 2011). Finally, the term structure spread is a proxy for the phases of the business cycle, an increase in the spread being associated with an economic contraction. It is thus a countercyclical indicator of business conditions. The expectations approach to the term spread thus states that γ ₂ < 0. The sign of the term spread in equation (1) is thus an empirical issue.⁹

The state space equation for the α may be written as follows:

We relate the state space equations of α and β to macroeconomic and financial variables, given the importance of the timing of the α and β to these variables in the hedge fund literature (Chen and Liang, 2007; Avramov et al, 2011; Cai and Liang, 2012; Cao et al, 2013). We thus postulate that the α follows an autoregressive process augmented with conditioning market information. Equation (3) may be written in first differences, such as:

The updating of the α from one period to the next is thus a function of three elements: the interest rate, the market risk premium and an innovation. The coefficients θ _1i , θ _2i and the variance of the innovation result from the search procedure inherent to the Kalman filter.

Similarly, the state space equation for the β is:¹⁰

In addition to the two conditioning variables included in the state space equation of the α, the state space equation of the β includes the pc_lookback variable. This variable is the first principal component of Fung and Hsieh’s option risk factors, which are lookback straddles¹¹ on stocks, bonds, short interest, commodities and foreign currencies. Fung and Hsieh (1997, 2001, 2004) rely on lookback straddles to study the behavior of trend followers¹² in the hedge fund industry. However, according to these authors, there are substantial differences in trading strategies among trend follower funds, so it may not be possible to pin down a single benchmark that can be used to monitor the performance of trend followers (Fung and Hsieh, 2001). We thus combine the five ABS trend-following factors into one principal component.

We can conjecture the expected signs of the variables included in equations (3) and (5). First, an increase in the interest rate might signal a deterioration of business conditions. It thus leads to a decrease in the α (θ _1i <0) and a decrease in the β (δ _1i <0), hedge funds reducing their exposure to market risk in times of economic slowdown. Second, an increase in the market risk premium (R _mt −r _ft ) is viewed as a strengthening of the stock market. This may induce hedge funds to position themselves for an increase in their α, this behavior being related to the portfolio manager’s skills. In this case, the sign of θ _2i is positive. However, if the α is not manageable, this coefficient should be close to zero. This should not be the case for the time-varying β, which is considered as a control or decision variable. As a signal of market strengthening, an increase in the market risk premium should induce hedge funds to take more risk, and therefore to increase their β. We thus expect δ _2i >0. The sign of the coefficient of the pc_lookback factor in equation (5) will be discussed later.

DATA SOURCES AND STYLIZED FACTS

The data are taken from the database managed by Greenwich Alternative Investment (GAI). GAI has one of the oldest hedge fund databases, containing more than 13 500 records of hedge funds as of March 2010. Returns provided by the database are net of fees. The survivorship bias is accounted for in this database, as index returns for periods since 1994 include the defunct funds.

The data set runs from January 1995 to September 2012, for a total of 213 observations. In addition to the weighted composite index, the database includes 12 indices of well-known hedge fund strategies reported in Table 1.¹³ We also report the indices of GAI strategy groups whose sample starts in January 1995. The market risk premium and the risk factor SMB are drawn from French’s website.¹⁴ The lookback-straddle option factors come from the Hsieh’s database.¹⁵

Table 1 Descriptive Statistics, Greenwich Alternative Investment hedge fund indices, 1995–2012

Table 1 reports the descriptive statistics of our hedge fund database. There is some heterogeneity in the historical returns and risk characteristics of hedge fund strategies. For instance, the monthly mean returns range from −0.07 per cent for the short sellers¹⁶ to 1.07 per cent for the value index, and the standard deviation ranges from 1.29 per cent for the market neutral group to 5.83 per cent for the short sellers.

According to Table 1, the strategies having the lowest mean return before the crisis were the short sellers, macro and futures, their returns being −0.16, 0.57 and 1.01 per cent, respectively. However, these strategies performed the best during the subprime crisis, with returns equal to 0.94, 0.57 and 0.90 per cent, respectively. This asymmetry between the two periods for these strategies was reported by Sandvik et al (2011). Owing to the good performance of these strategies in periods of crisis, these authors consider them as good diversification outlets. However, the picture changes after the crisis. Indeed, the mean returns of the strategies remained below their pre-crisis level. Note that the return of the macro strategy is quite stable through time, a characteristic that is not shared with the other strategies.

In other respects, the hedge funds’ β’s are generally low, the average β computed over all strategies being equal to 0.22. Two strategies display a negative β: the short sellers (−0.91) and the futures strategy (−0.08).¹⁷ The strategy with the highest positive β is the growth one (0.69) while the strategy with the lowest positive β is the equity market neutral one (0.08).

We can classify the hedge fund strategies in three main categories according to the value of their β.¹⁸ Some strategies are directional in the sense that they have a greater exposure to the fluctuations of the overall stock market. They thus tend to have a higher β than the strategies’ average one. In this group, we may include the growth (0.69), long–short (0.49), macro (0.21), futures (−0.08) and short-sellers’ (−0.91) strategies. Note that the futures strategy displays a low β but is usually considered as directional.¹⁹ The value strategy might also be a candidate for this category since its β is quite high (0.53), but actually it is usually classified in the arbitrage category (Connor and Lasarte, 2005). The strategies with the highest β are usually the ones that display the highest adjusted R ² in standard multifactor return models such as the Fama and French model. Conversely, the strategies with the lowest β – equity market neutral (0.08), and market neutral group (0.17) – are often involved in arbitrage activities. Another usual category is the event-driven one. Strategies like the event-driven, distressed securities, diversified event driven and opportunistic enter in this category. Their β is usually moderate. Note that these categories are not exclusive as a strategy may belong to two categories, such as the distressed one that may also be considered as an arbitrage strategy.

The standard deviation of the GAI weighted composite index is less than the S&P500 one over our sample period, the respective levels being 2.18 and 4.59 per cent (Table 1). In fact, the standard deviation of the return of the weighted composite index seems to decline through time, which is not the case for the S&P500 return (Figure 1). More importantly, the standard deviation of the weighted composite index increased less during the subprime crisis than during the bubble tech one, while the standard deviation of the S&P500 return increased much more during the subprime crisis. This is a first evidence of a decline of procyclicality in the hedge fund sector, which is supported by our analysis of the cross-sectional co-movements of the strategies in the section ‘Empirical results’.

Figure 1

Rolling standard deviations: GAI weighted composite return and S&P 500 return.Note: The standard deviation is computed on a rolling window of 12 months.

Not surprisingly, the strategies’ standard deviations are correlated positively to their β’s (Figure 2). Note that short sellers are outside the regression line relating standard deviation to β but actually, their β – when measured in absolute value – is relatively high, consistent with the standard deviation of the returns for this strategy. The hedge fund mean return also co-moves positively with the β (Figure 3). According to the CAPM, the slope of this regression multiplied by the β is equal to the risk premium of the strategy. However, there are two outliers: the macro and short sellers’ strategies. Other risk factors must be relied on to explain their returns.

Figure 2

Strategies’ β and return volatility.

Figure 3

Strategies’ mean return and β.

The strategies displaying the highest mean return are not necessarily those embedded with the highest Sharpe ratio, a risk-adjusted measure of returns. For instance, the value and opportunistic strategies have the highest mean return but their respective Sharpe ratio is close to the strategies’ average. Conversely, the market neutral group has the highest Sharpe ratio (0.60) while its mean return is close to the strategies’ corresponding average (0.81 per cent).²⁰

Many strategy returns display negative skewness: event driven, distress securities, diversified event driven, value index, speciality and the multi-strategy index. Returns of directional strategies tend to display a positive skewness. This contrasts with the market portfolio that displays a negative skewness. Note that our results are more or less in line with Chan et al (2007) and Heuson and Hutchinson (2011) who find that most hedge fund strategies display negative skewness, what they consider as an indication of tail risk. However, a more straightforward measure of tail risk is kurtosis. Most hedge funds present excess kurtosis. For our hedge fund strategies, kurtosis ranges from 3.38 (futures) to 11.81 (opportunistic index). Note also that there is a negative correlation between strategy kurtosis and standard deviation (Figure 4). Since kurtosis is a direct measure of fat -tail risk – that is, risk associated with rare events – a strategy return volatility does not necessarily measure its whole market risk. In this sense, a more reliable risk measure would be the fourth cumulant, which combines standard deviation and kurtosis.

Figure 4

Strategies’ kurtosis and return standard deviation.

Table 2 provides the correlation matrix between hedge fund strategy returns, the hedge fund global index and the three Fama and French risk factors. As a whole, most hedge funds have a high positive correlation with the market proxy and the SMB portfolio. The strategies having the highest correlation with the market are the value index, long–short and growth strategies. Short sellers and futures strategies are negatively correlated with the market, the correlation coefficients being −0.72 and −0.11, respectively. This observation is consistent with other studies (for example, Sandvik et al, 2011). Strategies that have a low correlation with the market tend also to exhibit a low correlation with SMB. It is especially the case for the equity market neutral and the futures strategies. In other respects, the correlation between returns and HML is very low for the following strategies: equity market neutral, event driven, distressed securities and futures. Except for the short sellers, strategies tend to be negatively correlated with HML. As shown in the empirical section, this sign is related to crisis periods.

Table 2 Correlation matrix of strategies’ returns and Fama and French risk factors

Some strategies have a high correlation with the hedge fund global index, the coefficient exceeding 0.8. These strategies are: long–short – that is, the strategy that weights the more in the global index – opportunistic index, event driven, value index and multi-strategy index. These strategies thus offer less diversification benefits when combined together. In contrast, the futures and macro strategies display the lowest correlation with the global index, and offer potentially good diversification opportunities. Sandvik et al (2011) have identified the same strategies as good potential providers of diversification benefits.

EMPIRICAL RESULTS

An overlook at the asymmetries

Table 3 provides the estimation of the three-factor Fama and French model for the GAI global index over subperiods in order to highlight the asymmetric impact of the subprime crisis. Before the crisis – that is, from January 1995 to May 2007 – the α is relatively high at 0.8 per cent monthly and significant at the 1 per cent level. Two risk factors impact positively and significantly on the global index: the market factor and the SMB factor. The market β of the global index is quite low at 0.35, suggesting the moderate exposure of the representative hedge fund to the market. In other respects, the positive sign for the SMB coefficient suggests that hedge funds prefer to buy the stocks of small firms. The HML factor is not significant before the crisis.

Table 3 Fama and French model applied to the GAI weighted composite index over key periods

During the crisis, the α is not significantly different from 0, indicating that the representative hedge fund did not ‘create’ α during this crisis. The market β is higher than the one estimated during the pre-crisis period. However, according to previous studies (Capocci et al, 2005; Sandvik et al, 2011), it is well-known that hedge funds reduce their β (or deleverage) during a crisis. The higher β we observe in the subprime crisis is because of the fact that the β has culminated just before the subprime crisis. The apparent counter-result we obtain is thus because of the averaging implicit in a regression. In line with other studies (for example, Sandvik et al, 2011), the SMB factor loses its explanatory power during the subprime crisis. The exposure of hedge funds to small firms is thus quite reduced during a crisis. Moreover, SMB may be a proxy to liquidity risk in crisis periods (Billio et al, 2009) and hedge funds reduce their exposure to liquidity risk during these periods. Turning to HML, we note that the behavior of hedge funds is also strongly asymmetric in down-market versus up-market conditions. Indeed, its coefficient is significantly negative, at −0.2881, during the subprime crisis. Note also that the R ² is much higher during the crisis than before. This result was also obtained in other studies like Billio et al (2009), Bollen (2011) and Sandvik et al (2011).

To get a better grasp of the time-varying nature of the model factor loadings, we re-estimate our model on a 15-month rolling window. The results appear in Figure 5. Regarding the α, we observe that it decreases during the Asian (June 1997–January 1998) and Russian-LTCM (August 1998–October 1998) crises but that it recovered thereafter. It hits its maximum at the beginning of the second millennium but it collapses toward 0 with the bursting of the bubble tech. Thereafter, the α fluctuated in a lower range than the one observed before the Asian crisis and its decrease was less pronounced during the subprime crisis than during the two preceding ones – especially the crisis associated with the bubble tech. Turning to the β, we note an important deleveraging process preceding the bubble-tech crisis but the β recovered quickly during the subsequent economic expansion. Hedge fund managers also reduced their exposure to the equity market during the subprime crisis, but, in line with the behavior of the α, the decrease of the β was much lower than during the bubble tech whereby it decreased to the 0 level. The reduction of the hedge fund β during crisis periods is mentioned in many studies (Cappoci et al, 2005; Billio et al, 2009; Bollen, 2011 and Sandvik et al, 2011).

Figure 5

Recursive coefficients of the Fama and French model computed on a rolling window.

Notes: We run the simple Fama and French model on the GAI global return using a 15-month rolling window. The dotted lines enclose the confidence interval defined at the 5 per cent level. In this context, when the lower interval falls below zero, the corresponding coefficient is not significant at the 5 per cent level.

In other respects, the behavior of the SMB loading over our sample is quite interesting. Similarly to the β, it collapsed during the bubble tech crisis, but recovered very quickly thereafter. However, its decrease was higher than the one of the β during the subprime crisis and it remained close to 0 thereafter. As noted earlier, SMB may be considered as a proxy for funding liquidity risk (Billio et al, 2009), especially during financial crises. During these periods, hedge funds are induced to reduce liquidity risk, and thus their exposure to SMB. Finally, HML loading also behaves asymmetrically in crisis periods compared with economic expansion ones. While its coefficient tends to be positive during expansion, it clearly turns negative during crises. In line with SMB, the coefficient of HML remained depressed at the end of our sample, that is, September 2012.

Estimation of the benchmark model

Table 4 provides the results of the estimation of our benchmark model given by equation (1). As indicated by the likelihood ratio (L), the fit of the model is quite good for most of the strategies. However, four strategies display a low likelihood ratio: macro, equity market neutral, futures and short sellers. These results, which are shared with many other studies (for example, Sandvik et al, 2011) suggest that other specific risk factors are at play to explain the returns of these strategies whose payoffs seem to be highly non-linear.

Table 4 State space regressions of strategy index returns using the Kalman filter, January 1995–September 2012

In our model, the coefficient of the market risk premium is time varying. Its state space value, which may be associated with its mean value or long-term value, is given by sv2 in Table 4. As expected, the market risk premium is the factor that impacts the most hedge fund returns. The β’s of the strategies are very close to the ones estimated with the standard market model (Table 1). The other factor that stands as an important driver of hedge fund returns is SMB. Actually, hedge funds have a preference for the stocks of small firms over the stocks of big ones. In other words, hedge funds have a greater exposure to stocks with a smaller capitalization. Researchers find the same kind of preferences in the mutual fund industry (Haiss, 2005). According to McGuire et al (2005), this result is consistent with hedge fund investment in technology stocks and startup companies during the dotcom boom (2000). In other respects, Figure 6 shows that the sensitivity of hedge fund strategies to SMB is quite correlated to their market β.

Figure 6

Strategies’ SMB and β.

The term spread – which may be viewed as a portfolio long in the 10-year bond yield and short in the 3-month Treasury bills yield – impacts positively and significantly many hedge fund strategies’ returns. The ‘price of risk’ approach to the term spread thus dominates in this case. For instance, the estimated coefficient of the term spread is equal to 0.7194 in the weighted composite index equation, significant at the 5 per cent level. The strategies that are the most exposed to the term spread are the growth (2.7218), multi-strategy (1.6586), opportunistic (1.0128), long–short (0.7583) and equity market neutral (0.6144). As stated previously, the impact of the spread variable as a driver of performance is quite unexplored in the hedge fund literature but our experiments show that it might be important to explain hedge fund returns – especially in times of low interest rates.

Turning to the factors that explain the time variability of the β, Table 4 shows that pc_lookback contributes the most to this time variability. With the exception of short sellers, its impact is negative and significant for most of the funds. For instance, in the model of the weighted composite index, its estimated coefficient is equal to −0.8560, significant at the 5 per cent level. The strategies that are the most exposed to this factor are: futures (−2.4833), opportunistic (−1.2879), value index (−1.2803) and diversified event driven (−0.9415). Hedge funds thus reduce their market β (systematic risk) when the yield on the pc_lookback increases. In other words, this factor may be associated with the hedging operations of hedge funds when the stock market declines or shows unusual volatility. In this respect, there is a negative conditional covariance between the pc_lookback and the stock market return as measured by the S&P500 (Figure 7). Note that this covariance – which is computed with a multivariate GARCH²¹ (MGARCH) using a BEKK procedure (Bollerslev et al, 1988; Engle and Kroner, 1995) – is particularly high in periods of crisis – especially during the subprime crisis. The behavior of the pc_lookback may therefore be assimilated to a long put one. More precisely, this factor may be viewed as an insurance factor in our return model (Agarwal and Naik, 2004). In line with this interpretation, Figure 8 shows that the MGARCH conditional covariance between the pc_lookback and the GAI weighted composite index is generally positive. This suggests that the pc_lookback may act as a backstop for hedge funds against the fluctuations of the stock market. Note that the covariance between the pc_lookback and the weighted composite index may become negative in times of market turmoil – suggesting that the pc_lookback does not provide a perfect hedge – but this covariance is much less in absolute value than the one linking the pc_lookback and the S&P500 return.

Figure 7

Conditional covariance between the pc_lookback and S&P 500 returnNote: The conditional covariance is computed using a multivariate GARCH based on a BEKK procedure (Bollerslev et al, 1988; Engle and Kroner, 1995).

Figure 8

Conditional covariance between the pc_lookback and GAI weighted composite index.Note: The conditional covariance is computed using a multivariate GARCH based on a BEKK procedure (Bollerslev et al, 1988; Engle and Kroner, 1995).

Consistent with our interpretation, Fung and Hsieh (2001) argue that a portfolio of lookback straddles on currencies, bonds and commodities can reduce the volatility of a typical stock and bond portfolio during extreme market downturns. However, in our study, the lookback factor is the first principal component of the lookback returns on five assets and it is an explanatory variable in the β’s state equation. In Fung and Hsieh (2001), the lookback factors are not combined and constitute individual risk factors in the return equations.

Another interpretation of the link between the pc_lookback factor and a strategy’s β hinges on the following argument. Recall that the pc_lookback factor is built with lookback straddles that provide greater positive payoffs when the financial markets are volatile. Figure 9 plots the conditional covariance between the VIX – a well-known indicator of the implicit volatility of stock returns – and the S&P500 returns. This covariance – which is also computed with a MGARCH – is usually negative, which supports the Black (1976) leverage effect, and it peaks when the market is dropping, its largest drop being observed during the subprime crisis. Figure 10 shows that the MGARCH conditional covariance between the VIX and the GAI weighted composite index shares a similar profile. However, this covariance is less in absolute value than the one linking the VIX to the S&P500. This may be explained by the influence of the pc_lookback. In this respect, Figure 11 shows that the MGARCH conditional covariance between the pc_lookback and the VIX is positive. As expected, it peaks when the market trends downward. Moreover, Figure 12 plots the behavior of the pc_lookback and the VIX. Note that the pc_lookback seems to be a leading indicator with respect to the VIX – especially during the subprime crisis. It does signal a market downturn before the VIX. Consistent with our results, hedge funds are induced to take less systematic risk during these episodes.

Figure 9

Conditional covariance between the VIX and S&P 500 return.Note: The conditional covariance is computed using a multivariate GARCH based on a BEKK procedure (Bollerslev et al, 1988; Engle and Kroner, 1995).

Figure 10

Conditional covariance between the VIX and the GAI weighted composite index.Note: The conditional covariance is computed using a multivariate GARCH based on a BEKK procedure (Bollerslev et al, 1988; Engle and Kroner, 1995).

Figure 11

Conditional covariance between the VIX and the pc_lookback. Note: The conditional covariance is computed using a multivariate GARCH based on a BEKK procedure (Bollerslev et al, 1988; Engle and Kroner, 1995).

Figure 12

Moving average, pc_lookback and VIX. Note: The moving average is computed on a rolling window of 12 months.

To gain a better understanding of the link between the pc_lookback factor and the strategies’ returns, we have computed the time-varying market β of this factor, relying on the simple market model estimated with the Kalman filter:

Figure 13, which plots the estimated β of the pc_lookback, shows that it is usually negative but that it increases in absolute value during a crisis, which suggests that the pc_lookback behaves as a backstop against the decrease in portfolio returns. Substituting equation (6) into equation (5) and then equation (5) into equation (1) leads to the appearance of the following term in a strategy return equation: β _i,t δ _3i β _{t,pc_look} (R _mt −r _ft )². Given our previous results, the coefficient of (R _mt −r _ft )² is positive. The strategies that have a significant δ _3i in equation (5) – especially the futures, opportunistic, value index and diversified event driven – thus benefit when the volatility of the stock market (as measured by (R _mt −r _ft )²) increases. These strategies thus share the nature of the Fung and Hsieh’s (2001, 2004) trend followers. Note that this result is in line with the papers of Treynor and Mazuy (1966) and Henriksson and Merton (1981) on market timing where non-linear functions of the market risk premium are relied on to deal with option-like return features (Fung and Hsieh, 2001).

Figure 13

β of the pc_lookback. Note: The time-varying β is computed with the Kalman filter applied to the simple market model.

The level of the interest rate (r _f ) also impacts negatively and significantly the β of some strategies. These strategies (or group of strategies) are: macro (−6.4763), directional trading group (−3.5814) and speciality strategies group (−3.4073). When the interest rate increases, these strategies thus reduce their β since an increase in interest rate may signal a coming decline of the stock market. Since central banks control short-term interest rates, they can thus rely on the interest rate channel to impact the risk-taking behavior of hedge funds. It is interesting to observe that the β of the macro strategy is the most responsive to the interest rate, this strategy relying on models based on macroeconomic factors. It is thus quite sensitive to monetary policy. Note that short sellers seem to adopt a contrarian position when the interest rate increases, its impact on their β being estimated at 6.2219, significant at the 5 per cent level. Short sellers thus decrease their risk when the interest rate increases. Actually, they follow the same behavior as the other strategies since the short sellers’ β is usually negative.

As indicated in Table 4, only few strategies’ β’s respond significantly to the market risk premium, which stands for the market trend. For one of them – the macro strategy – the estimated coefficient of R _m −R _f is positive and significant at the 5 per cent level. This strategy seems to track closely the market trend. The long–short and opportunistic strategies also display a positive coefficient for R _m −R _f , these coefficients being significant at the 10 per cent level.

Even if our sample includes the subprime crisis, Table 4 shows that the α puzzle seems unsolved over our estimation period (Racicot and Théoret, 2009, 2014). Most of the strategies display significant α’s as measured by their estimated coefficients. Indeed, the average α (sv1) computed over the 12 strategies is equal to 0.27 per cent on a monthly basis. The futures strategy displays the highest α (0.74 per cent) while the growth strategy displays the lowest one (−0.68 per cent). This is the only strategy endowed with a negative α. Note that the rank of a strategy in terms of the level of its mean return (Table 1) does not usually correspond to its rank in terms of the level of its α (Table 4). For instance, the short-sellers’ strategy displays the highest estimated α after the futures strategy but has the lowest mean return over our sample period.

Strategies’ α’s seem quite sensitive to the level of the interest rate. For instance, the estimated coefficient of r _f in the weighted index state equation of sv1 is equal to −0.1070, significant at the 1 per cent level. In the same vein, the α’s of many strategies respond negatively and significantly to interest rates: value index (−0.1969), long–short (−0.1556), equity market neutral (−0.1531), opportunistic (−0.1456), diversified event driven (−0.0878), market neutral group (−0.0833) and multi-strategy index (−0.0736). Therefore, an increase in the interest rate tends to depress a strategy’s α. This may be related to business conditions, an increase in the interest rate signaling a recession or an acceleration of inflation, with a corresponding tightening of monetary policy. These events tend to depress the α.

Like in the case of the β analysis, few strategies display a link between their α and the market risk premium. For the distressed strategy, the coefficient of the market risk premium is equal to −0.0095, significant at the 5 per cent level. This link can be easily explained since a deterioration in business conditions leads to an increase in business failures, a situation that benefits to the distressed securities’ strategy. By contrast, an increase in the market trend benefits to the multi-strategy index (0.0090) and the value index (0.0068).

Kalman-filtered time-varying α and β

Figure 14 plots the Kalman-filtered time-varying α’s and β’s²² for the weighted index and the strategies over the period 1997–2012.²³ For most strategies, the β follows a mean-reverting or Ornstein-Uhlenbeck process. Also for most of them, the β trended upward during the economic expansion, which preceded the subprime crisis. Hedge funds thus take more risk when business conditions are improving. However, the β of these strategies decreased substantially during the subprime crisis, which suggests that hedge funds greatly reduced their market exposure during this period. Thereafter, there was a recovery of their β that moved back near its pre-crisis level at the end of 2012.

Figure 14

Strategies’ time-varying α and β.Note: The time-varying α and β are computed by applying the Kalman filter to the model given by equations (1), 2, 3, 4 and 5.

However, it is interesting to note that some strategies’ β’s do not follow a mean-reverting process. In this respect, the short sellers’ β, usually negative, tends to move on an upward trend during the sample period. We note that the subprime crisis impacted less the short sellers’ β than the ones of the majority of the other strategies. In other respects, the β of the directional trading group displays a low volatility and remains close to zero most of the time. In line with its group, the β of the futures strategy tends also to remains close to zero. However, in contrast with the other strategies, its β increased in absolute value during the subprime crisis, suggesting that the futures strategy was more involved in ‘shorting’ activities.

Turning to the time variability of the α, we first note that some strategies succeed in maintaining a high α through time. This is the case of the following strategies (or group of strategies): growth, specialty strategy group, directional trading group, futures and short sellers. Second, for most of the strategies, the α has trended downward since 1999. The α puzzle thus tends to recede through time, at least over our sample period. This result is shared with many recent studies (for example, Zhong, 2008; Sandvik et al, 2011; Cay and Liang, 2012). It is attributed to decreasing returns to scale, increased competition in the hedge fund sector and the sheer growth of assets under management in this sector. However, the α remains positive for most strategies and it has recovered since the subprime crisis. In this respect, we find that the subprime crisis had little impact on the strategies’ α’s. On the contrary, the α of some strategies increased during the crisis. In this respect, the following strategies benefited from the crisis in terms of their α: distressed securities, event driven, diversified event driven, market neutral group and futures. These strategies are very specialized and based on arbitrage, which may lead to positive payoffs during crises. Interestingly, the α of the futures strategy jumps at each crisis that occurred during our sample period – that is, the Asian, bubble-tech and subprime crises. It is thus quite immune to crises (Sandvik et al, 2011). Finally, since the strategies’ α’s are not mean-reverting like most of their β’s, we can induce than the α is less manageable than the β. It is more related to the particular situation of the hedge fund industry, like the low regulation in this sector.

The return co-movement of hedge fund strategies

The co-movement between security returns in a portfolio is an important indicator of its risk. Indeed, when the co-movement is high, this suggests that the potential for portfolio diversification is quite limited. It is thus interesting to examine the opportunities for diversification in a portfolio built with hedge fund strategies.

We rely on three indicators to track the co-movement of strategies’ returns. The first – which corresponds to the cross-sectional standard deviation – is used by Beaudry et al (2001) to study the co-movement of firm returns on investment.²⁴ Solnik and Roulet (2000) also rely on the cross-sectional dispersion to estimate the co-movement of stock market returns. Sabbaghi (2012) transposed this indicator to the study of the co-movements of the returns on hedge fund indexes. The cross-sectional standard deviation – also named cross-sectional dispersion – is defined as:

where N is the number of strategies, and R _it is the cross-sectional vector of the strategy returns observed at time t. The cross-sectional standard deviation of returns is thus the square root of their cross-sectional realized variance. When the cross-sectional standard deviation of returns increases, the dispersion of returns increases. There is thus a rise in the heterogeneity of the hedge fund strategies in this case. This is good news in regard to portfolio diversification. And when the cross-sectional standard deviation decreases, there is an increase in the homogeneity of the strategies. This is bad news with respect to portfolio diversification because strategies’ returns move closer in this case.

A more straightforward indicator of return co-movement is their cross-sectional covariance defined as:

where N is the number of strategies, R _it is the cross-sectional vector of the strategies’ returns observed at time t, i is the unitary vector, and I is the identity matrix. The cross-sectional covariance is thus defined as the average of the cross-sectional second co-moments (Adrian, 2007).²⁵ An increase in the cross-sectional covariance of the strategies’ returns signals a higher co-movement between these returns, so the degree of homogeneity of the strategies increases. Conversely, a decrease in the cross-sectional covariance signals a decrease in return co-movement, so the degree of heterogeneity of the strategies increases.

It would be desirable that these two indicators of co-movement move in an opposite direction. That is, when cs_sd decreases, cs_cov should increase. This is then an unambiguous signal of an increase in the co-movement of the strategies’ returns, hence an increase in homogeneity. But, as shown later, this is not necessarily the case at the empirical level.

The third indicator of return co-movement is the cross-sectional correlation of returns. It is defined as the ratio of equations (8) and (7) squared:

where cs_var is the cross-sectional variance. When cs_cov increases and cs_var decreases simultaneously, cs_corr increases: the co-movement between strategies’ returns increases unambiguously. Conversely, when cs_cov decreases and cs_var increases simultaneously, cs_corr decreases: the co-movement between returns decreases unambiguously. In the other cases, the signal given by cs_corr is somewhat ambiguous because cs_var and cs_cov do not indicate the same direction regarding the co-movement between strategies’ returns.

Sabagghi (2012) relies on the three indicators of co-movement given by equations (7)–(9), , in order to investigate the return co-movement of the strategy indices provided by Credit Suisse.²⁶ We reproduce this exercise for the GAI strategies. Figure 15 plots our three indicators of strategies’ return co-movement from 1997 to 2012. The cross-sectional covariance registered a big jump during the bubble-tech crisis and a smaller one during the subprime crisis. According to this indicator, the strategies’ return co-movement over the crises shows a tendency to decrease through time, a good news in regard to portfolio diversification. Outside the crises, the co-movement between the strategies’ returns – as measured by the cross-sectional covariance – is low, which suggests that the risk associated with the hedge fund strategies is quite diversifiable.

Figure 15

Cross-sectional correlation (cs_corr) of the strategies’ returns and its components.Note: The cross-sectional time series are computed using a moving average of 12 months.

The signal sent by the cross-sectional deviation in regard to the co-movement of the strategies’ returns is different. First, the time profile of the two co-movement series – that is, cs_var and cs_cov – seems to diverge. The cross-sectional covariance jumps in time of crises and is low and stable otherwise. For its part, the cross-sectional deviation jumped during the bubble-tech crisis and declined progressively thereafter. However, similarly to the cross-sectional covariance, it jumped during the subprime crisis with a lower amplitude than the one observed during the bubble-tech crisis. Contrary to the cross-sectional covariance, the cross-sectional deviation indicates that the behavior of the strategies is more heterogeneous in times of crises and more homogeneous in times of economic expansion. Since the cross-sectional deviation trends downward, it signals that the behavior of the strategies tends to become more homogeneous through time.

A closer look at the two series shows that they are strongly correlated since 2003 (Figure 16). They thus send a different signal in terms of the pattern of diversification in the hedge fund industry. The cross-sectional correlation is the ratio of these two diverging signals sent by its components (Figure 15). First, it tends to increase through time, signaling that the behavior of the strategies becomes more homogeneous, a profile borrowed from the cross-sectional deviation. Second, the cross-sectional correlation increases during crises, suggesting a more homogenous return pattern during these periods (Figure 16). Thus, the impact of the cross-sectional covariance dominates the cross-sectional correlation one during these periods. Contrary to the profile of the cross-sectional covariance, we also note that the cross-sectional correlation was higher during the subprime crisis than during the bubble-tech one, which reflects the lower level of the cross-sectional dispersion during the subprime crisis. Fortunately, the cross-sectional correlation decreased significantly after the crisis, indicating less co-movement between the strategies’ returns.

Figure 16

Financial crises and co-movements between strategies’ returns.Note: The cross-sectional time series are computed using a moving average of 12 months.

A regression of cs_sd on 12 Almon lags of cs_cov shows that the sum of the lags is equal to 0.36, significant at the 1 per cent level. An increase in covariance was an early indicator of the high volatility that took place during the bubble-tech crisis but to a less extent during the subprime crisis (Adrian, 2007). In summary, according to cs_cov, the co-movement between the strategies’ returns has decreased since 1997. Moreover, cs_cov increased less during the subprime crisis than during the bubble-tech one. Hence, the potential for diversification seems to have increased in the hedge fund industry. However, cs_sd shows a tendency to decrease over the sample period, which pushes cs_corr upward. In order to gauge the co-movement of returns, it seems therefore more advisable to rely on cs_cov, a quite straightforward indicator of co-movement.

The α and β co-movements of the of hedge fund strategies

We also computed the same statistics for the strategies’ β’s and α’s (Figure 17).²⁷ In times of economic expansion, the cross-sectional covariance of the strategies’ β’s shows a tendency to increase. The strategies’ behavior thus becomes more homogenous in terms of β. This is the pattern we observed in the previous section in economic expansion. However, in times of crisis, the cross-sectional covariance of the β decreases. This indicates that the risk-taking behavior of the strategies is more heterogeneous in periods of turmoil, which suggests a potential for portfolio diversification. Turning to the cross-sectional standard deviation of the β’s, we note that its reaction to the bubble-tech crisis was very low but that it decreased substantially during the subprime crisis, which contradicts the signal sent by the cross-sectional covariance. Linking together the movements of the cross-sectional covariance and cross-sectional standard deviation, the cross-sectional correlation of the strategies’ β’s increases during economic expansions, which suggests that the risk-taking behavior of the strategies is more homogenous during good times. However, the cross-sectional correlation of the β’s decreased sharply during the subprime crisis, some strategies taking higher risk while others doing the opposite. A closer look at the link between the β’s cs_cov and cs_corr shows that they are strongly and positively correlated (Figure 18). In other words, the cs_sd does not disturb the positive link between cs_cov and cs_corr for the β’s as it was the case for returns.

Figure 17

Cross-sectional correlation of the strategies’ α’s and β’s.

Figure 18

Financial crises and β’s cross-sectional covariance and correlation.

Regarding the α, the behavior of the cross-sectional indicators was quite different during the bubble-tech and subprime crises. The cross-sectional covariance jumped during the bubble-tech crisis, which suggests more homogeneity about the profiles of the strategies’ α’s. The increase observed during the subprime crisis was not significant. After the bubble-tech crisis, the cross-sectional covariance of the α’s was stable and low, which indicates an increase in the α heterogeneity among strategies. The signal sent by the cross-sectional deviation of the α’s differs again. This indicator jumps during the two crises, which suggests less homogeneity in the behavior of the strategies’ α’s. It tends to decrease during the sample period, suggesting more homogeneity.

The cross-sectional correlation of the α’s increased substantially during the bubble-tech crisis but it receded thereafter, which suggests that the behavior of the strategies’ α’s is less homogenous. However, it resumed its increase after the subprime crisis. In summary, in view of the apparent maturation process for the strategies’ α’s observed in the previous section – that is, a downward trend for most α’s – there seems to be more heterogeneity at the α level than in the past. This pattern is shared by the strategies’ β’s, which indicates that the potential for diversification in the hedge fund industry tends to increase, especially in times of crisis.

CONCLUSION

While the returns’ behavior of standard financial instruments over the business cycle is well-known, this is less the case for alternative investments like hedge funds. Yet, contrary to many other financial institutions for which short selling is restricted by the law, hedge funds may adopt investment strategies that allow them to deliver positive payoffs during crises. Some strategies – as the distressed and short sellers’ ones – even benefit from a decline in stock markets. It is thus important to model the behavior of hedge fund strategies over the business cycle in order to pin down the dynamics of their risk-return trade-off.

In this respect, our contribution is two-fold. First, we study the cycles of the strategies’ time-varying α’s and β’s using a Kalman filter approach that embeds conditioning information in the α’s and βs’ state equations. This information allows us to see how hedge funds adjust their risk level to market information. In this respect, our pc-lookback factor is particularly relevant to monitor the hedging operations of hedge funds. Moreover, we track the volatility of hedge fund during the business cycle using an innovative approach in this kind of study, that is, a MGARCH approach. Second, we study the behavior of the cross-sectional dispersions of returns and especially the ones of the α’s and β’s, another innovative approach relying on the Kalman filter to simulate the strategies’ α’s and β’s.

The results of our study indicate that hedge fund strategies continue to provide good diversification benefits over the business cycle. First, the volatility of their returns seems to decrease through time, which suggests a better management of structured products. Second, in spite of the subprime crisis, the α of most strategies remains positive. Some strategies benefited from this crisis, which suggests good opportunities for hedge fund investors even in bad times (Sandvik et al, 2011). Third, our results are consistent with the fact that hedge funds’ portfolio managers modify their β’s in line with the volatility of financial markets – as measured in our return model by a principal component of returns computed with lookback straddles. They can thus benefit from an increase in the volatility of financial markets, which is often associated with a downward trend of the stock indices (Black, 1976). Fourth, the strategies’ α’s and β’s have co-moved less strongly during the subprime crisis – a major financial crisis – than during the preceding ones, which is in line with a learning-by-doing or a maturation process in the hedge fund sector. More precisely, as noted by other researchers, there is no evidence of underperformance in the hedge fund industry during the subprime crisis and no indication that hedge funds induce fire sales in the subprime crisis – fires sales being an important amplification channel during financial crises (Sandvik et al, 2011; Shleifer and Vishny, 2011; Brown et al, 2012; Boyson et al, 2013; Gennaioli et al, 2013). This development may suggest a decrease in systemic risk in the hedge fund industry, which is often because of contagion or herding – that is, a greater homogeneity in the behavior of market participants (Wagner, 2010).

Procyclicality thus seems to decrease in the hedge fund industry, a good news for investors in search for higher yields like pension funds (Racicot and Théoret, 2013). One promising avenue for further research is to model the co-movements of the returns, α’s and β’s of the hedge fund strategies. Indeed, how macroeconomic shocks or uncertainty do impact these co-movements?²⁸ This is an important question that must be addressed to gain a better understanding of the hedge fund time-varying risk-return trade-off.

Appendix

The multivariate GARCH²⁹

We rely on a multivariate GARCH process (MGARCH) to compute the conditional covariances and correlations of key variables. The original autoregressive conditional heteroskedasticity model (ARCH(q)) due to Engle (1982) may be written as:

where h _t is the conditional variance and ɛ _t , the innovation of the regression. Bollerslev (1986) generalizes Engle’s model by allowing the conditional variance to follow an ARMA (p, q) process. The GARCH (p, q) model obtains:

One problem with these formulations is that they neglect the conditional covariances between the innovations. The MGARCH model palliates this limitation. In this framework, assuming a GARCH (1, 1) process, each element of the conditional variance–covariance matrix may be written as:

Generalizing to a GARCH (p, q) process, we obtain the Bollerslev et al (1988) vectorized (vec) model:

where C is an N × N matrix and A and B are N ² × N ² matrices.

The VEC MGARCH model thus requires the estimation of a number of coefficients that may be quite large. Hence we adopt the BEKK (Engle and Kroner, 1995) procedure, a more parsimonious approach in terms of the number of parameters to estimate. It reads:

where C, A, and B are N × N matrices.