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Original Article

The procyclicality of hedge fund alpha and betaReceived (in revised form): 29th April 2013

François-Éric Racicot

is Associate Professor of Finance at the Telfer School of Management, University of Ottawa. His research interestsfocus on the problems of measurement errors, specification errors and endogeneity in financial models of returns.He is also interested in developing new methods used for forecasting financial time series − especially hedge fundrisk. He has published several books and many articles in quantitative finance and financial econometrics.

Raymond Théoret

is Full Professor of Finance at École des sciences de la Gestion, University of Quebec − Montreal (UQAM). Hisresearch focuses on banking systemic risk and hedge fund risk. He has published many books in asset pricingand many articles in the banking and hedge fund areas in well-known journals.

Correspondence: François-Éric Racicot, Telfer School of Management, CGA-Canada Accounting andGovernance Research Centre, University of Ottawa, 55 Laurier Avenue East, Ottawa, ON, K1N 6N5 CanadaE-mail: [email protected]

ABSTRACT Traditional financial institutions as banks follow procyclical risk strategies,that is, they increase their leverage in economic expansions and reduce it in contractions,which leads to a procyclical behaviour for their betas and other risk and financial perfor-mance measures. We study the cyclical aspects of hedge fund strategies, an issue quiteoverlooked in the literature. We rely on two procedures: conditional modelling and Kalmanfiltering of hedge funds’ alpha and beta. We find that hedge fund betas are usually procycli-cal. Our results also show that the alpha is often high at the beginning of a market upsidecycle but, as the demand pressure increases, it progressively shrinks, which suggests that thealpha puzzle ought to be analysed in a dynamic setting.

Journal of Derivatives & Hedge Funds (2013) 19, 109–128. doi:10.1057/jdhf.2013.3;published online 6 June 2013

Keywords: procyclical risk measures; systemic risk; financial stability; conditional models; KalmanFilter; spectral analysis

INTRODUCTION

Procyclical risk analysis is now one of the mainconcerns for researchers working in the field offinancial institutions, especially in bankingresearch and macroprudential analysis (Amatoand Furfine, 2004; Heid, 2007; Rajan, 2005,2009; Shin, 2009; Adrian and Shin, 2010;

Jacques, 2010; Gennaioli et al, 2011). However,these analyses are often cast in a static setting.Moreover, the hedge fund industry, despite itsgrowing share in the world financial system,1 isquite neglected in these analyses of riskprocyclicality or merged with shadow banking.For instance, the principal factor at the root of the

© 2013 Macmillan Publishers Ltd. 1753-9641 Journal of Derivatives & Hedge Funds Vol. 19, 2, 109–128www.palgrave-journals.com/jdhf/

recent financial crisis in the United Statesemanated from the subprime mortgage sector inwhich hedge funds are much involved. Indeed,according to Adrian and Shin (2010), the share ofhedge funds in the origination of US subprimemortgages by the leveraged financial sector wasthen as high as 32 per cent, which suggests thathedge funds may generate important financialshocks having repercussions throughout thewhole economy.Hedge funds have thus become one of the

pillars of the financial system, especially in theUnited States. The linkages between the bankingsector and the hedge fund industry are so tightthat a financial crisis originating from hedge fundsmay endanger financial stability, especially viathe repos market (for example, the LTCM2

episode). Moreover, pension funds’ savings areincreasingly invested in hedge funds, the returnon conventional savings vehicles being on adownward trend.3 Hence, the social impact ofthe bankruptcy of a major hedge fund should notbe overlooked because it may impair seriouslythe pensioners’ well-being, as was the case duringthe last subprime crisis. Moreover, a failureof an important financial institution may entailexternalities for the whole financial system(Vives, 2010). In this respect, the risk taking of amajor hedge fund may create aggregate shocksthat could give raise to contagion effects (Vives,2010; Gennaioli et al, 2011). Macroprudentialanalysis must give rise to policies that endogenizethese systemic risks.4

To shed more light on the procyclical risk inthe hedge fund industry, we test two hypothesesusing data on the Greenwich5 US hedge funds.First, we document an unknown stylized fact inthe hedge fund industry. Indeed, we observe ahigh volatility at business cycle frequencies in thereturns spectrum of many hedge fund strategies.

We thus check this apparent procyclicality ofrisk management in the hedge fund industry byusing two kinds of approaches: the conditionalcoefficient model and a state-space model basedon the Kalman filter. We use these setups tostudy the cyclical behaviour of the hedge funds’alphas and betas categorized by strategies. Ourmethodology allows us studying the cyclicalcomponents of hedge fund performance for thewhole industry and then by strategy. Financialprocyclicality is related to the amplification ofaggregate risk by financial institutions at thebusiness cycle frequencies (Altman and Saunders,2001; Berger and Udell, 2004; Jacques, 2010).Since hedge fund are known to be veryleveraged, there is a suspicion that this procyclicaleffect exists as the beta is positively related tofinancial leverage, a ratio which usually increasesin economic expansions and decreases incontractions.The second hypothesis we test concerns the

prevalence of the alpha puzzle in a dynamicsetting, especially at business cycle frequencies,a problem also neglected in the literature(Racicot and Théoret, 2007b, 2009). Indeed, thealpha puzzle is mostly studied in a static setting,which may bias the estimated alpha. We rathercast the alpha analysis in a dynamic setting tostudy more closely the alpha puzzle. We test ifthe alpha has also a tendency to be procyclical.In this respect, we conjecture that the alphaincreases at the beginning of an economicrecovery but, as demand pressures increase, itdecreases progressively thereafter, the increase inthe financial institutions’ leverage contributing tothe rise of demand pressures, the rise in asset pricesand the fall in returns (Danthine and Donaldson,2002; Geanakoplos, 2010). Unfavourablemacroeconomic shocks related to a developingrecession or crisis should also depress the alpha.

Racicot and Théoret

110 © 2013 Macmillan Publishers Ltd. 1753-9641 Journal of Derivatives & Hedge Funds Vol. 19, 2, 109–128

We thus suspect that the resulting overestimatedalphas trend downward with the maturation ofan economic recovery. The puzzle related to thebloated alpha must thus be analysed in a dynamicsetting.6 Moreover, this decrease in the alpha atthe end of an economic recovery may be usedas a leading or forward-looking performanceindicator in the hedge fund industry.The organization of this article is as follows.

In the next section, we present the dynamicmodels of hedge fund returns. The subsequentsection provides the stylized facts on hedge fundreturns and our empirical results. The last sectionconcludes.

CYCLICAL FRAMEWORK FOR

ANALYSING HEDGE FUNDRETURNS

The Kalman filtering approach

A Kalman filter model is usually made up of asignal or observation equation and of state ortransition equations for the unobserved variablesor coefficients.7 To explain the expected hedgefund excess returns, we choose, as signalequation, the three-factor Fama and French(1992, 1993 and 1997) model:8

Rp -Rf� �

t = αt + β1t Rm -Rf� �

t + β2SMBt

+ β3HMLt + β4 gprodt + εt ð1Þwhere Rp−Rf is the excess return of a portfolio,Rf being the risk-free return; Rm−Rf, the marketrisk premium; SMB, a portfolio that mimics the‘small firm anomaly’, which is long in the returnsof selected small firms and short in the returns ofselected big firms; HML, a portfolio that mimicsthe ‘value stock anomaly’, which is long inreturns of stocks of selected firms having a highbook value/market value ratio (value stocks)and short in selected stocks having a low book

value/market value ratio (growth stocks); gprod,the annual industrial production growth rateexponentially smoothed with the Holt-Wintermethod. αt is the time-varying alpha, β1t, thetime-varying beta, and εt, the innovation.To explain the excess return of a portfolio,

Fama and French added to the CAPM marketrisk premium two other risk factors: the SMBand HML ones. We omit in this equation themomentum factor proposed by Jegadeesh andTitman (1993) and Carhart (1997) because theinfluence of this factor is weak for the majorityof hedge fund strategies, a state space modelestimated with the Kalman filter having to beparsimonious.9

A positive sign for SMB in equation (1)suggests that a portfolio manager, here a hedgefund, prefers stocks of small firms over stocks ofbigger ones, which is usually the case for hedgefunds. Moreover, a positive sign for the variableHML would be symptomatic of a preference forstocks with a high book-to-market value ratioover stocks with a low book-to-market valueratio, which is also a frequent preference in thehedge fund industry.10

The transition equations give therepresentation of the time-varying alpha (αt) andbeta ( β1t).

11 We first assume that the statevariables follow a pure random walk and then weextend this recursive process with financialvariables which convey valuable information tothe hedge fund manager.

A pure recursive modelWe start our modelization with a purely recursiverepresentation of the time-varying coefficientsof equation (1), which is the simplest way formaking coefficients variable. This is the modelgenerally used in the Kalman filter literature.

The procyclicality of hedge fund alpha and beta

111© 2013 Macmillan Publishers Ltd. 1753-9641 Journal of Derivatives & Hedge Funds Vol. 19, 2, 109–128

The respective equations of the alpha and betaare therefore:

αt = αt - 1 + ξt (2)

β1t = β1;t - 1 + νt (3)

with ξt and νt being respectively the innovationsterms of equations (2) and (3). If ξt∼iid(0,σξ2) andvt∼iid(0,συ2), which amount to pure white noisevariables, these equations would represent purerandom walk processes.The filtering of the time-varying coefficients

in the framework of a pure recursive process iseasy to understand. For instance, in equation (3),the coefficient estimated at time (t−1)serves as a seed value (a guess) for the estimationof the coefficient at time t. However, thefiltered coefficient at time t is computedoptimally with the Kalman filter following theflow of new information which piles up fromone period to the next (Ljungqvist andSargent, 2004). Therefore the estimatedcoefficient β̂1t may be quite different from thecorresponding coefficient estimated one periodearlier, that is β̂1;t - 1, even in a pure randomwalk setting.

Recursive model combined with con-ditioning financial market variablesEconomic and financial information may beadded in equations (2) and (3) by relying onmore elaborate recursive processes. Thesevariables allow monitoring the reaction of thetime-varying coefficients to the conditioningmarket information. In this model, we expressthe time-varying alpha and beta as follows:

αt = αt - 1 +ϕ1rt - 1 +ϕ2 Rm -Rf

� �t - 1

+ ξt (4)

β1t = β1;t - 1 +ϕ3rt - 1 +ϕ4 Rm -Rf� �

t - 1

+ϕ5VIXt - 1 + νt ð5Þ

with r, the level of short-term interest rate;(Rm−Rf), the market risk premium, and VIX, theimplicit volatility of the S&P500 index. Theconditioning variables are lagged on period, ouraim being to track the reaction of the time-varying coefficients to the conditioning marketinformation. The retained financial variables,which are lagged one period, are thus known attime t.Note that equation (4), like equation (5), may

be transformed in first-differences. For instance,equation (4) can be written as:

αt - αt - 1ð Þ=ϕ1rt - 1 +ϕ2 Rm -Rf� �

t - 1 + ξt: (6)

The revision of the alpha from one period to thenext is thus the function of three elements: theinterest rate observed one period earlier, themarket risk premium also observed with a oneperiod lag and an innovation. The coefficientsφ1, φ2 and the innovation result from thesearching process of the filter.Let us specify the expected sign of the

variables incorporated in equations (4) and (5).An increase in the interest rate might beperceived as good news or bad news by hedgefunds. If this increase is seen as a forthcomingdeterioration of the stock market upward trendor as an indicator of inflation, that is bad news.But for hedge funds that follow call-like optionstrategies, which bet on the stock marketvolatility, that might be good news. The signs ofφ1 and φ3 are thus indeterminate in equations (4)and (5). As shown in the empirical section, thesign is quite related to the specific hedge fundstrategies.In equation (4), an increase in the market

risk premium at time (t−1) tends to signalmarket strengthening. Indeed, if the marketrisk premium follows a martingale process,



we can write:

E Rm -Rf� �

t Ωtjh i

= ðRm -Rf Þt - 1 (7)

with Ωt the information set available at time t.According to equation (7), an increase in themarket risk premium a time (t−1) is viewed as astrengthening of the stock market. This mayinduce hedge funds to position themselves for anincrease in their alpha, this positioning beingdependent on the portfolio manager skills. In thiscase, the sign of φ2 is positive. But if the alpha isnot manageable, this coefficient should be closeto 0. However this should not be the case forthe time-varying beta which is viewed as acontrol or decision variable. As a signal of marketstrengthening, an increase in the market riskpremium should encourage hedge funds totake more risk and therefore to increase theirbeta. The sign of φ4 would thus be positive inequation (5).In addition to the standard variables usually

incorporated in models with time-varyingcoefficients, we also include the implicit volatilityof the S&P500, that is, the VIX, to explain thetime-varying beta. Indeed, an increase in marketvolatility12 should usually induce hedge funds tobear less risk, and therefore to decrease their beta.However, an increase in market volatility mightbe welcomed by some hedge fund strategieswhich are very involved in option-basedstrategies. Another specification not used in thisarticle to introduce volatility in the equation ofthe time-varying beta but often found in theliterature on market timing relies on the square ofthe market risk premium as an indicator of stockmarket volatility, the second moment of themarket risk premium:13

β1t = β1;t - 1 +ϕ3rt - 1 +ϕ4 Rm -Rf� �

t - 1

+ϕ6 Rm -Rf� �2

t - 1 + νt ð8Þ

Nevertheless, note that, according to Treynorand Mazuy (1966), the squared market riskpremium might serve to detect good or badmarket timing,14 a good market timing beingassociated to a positive sign for this variable. Thesign of the squared market risk premium is thustheoretically indeterminate in equation (8).

The conditional approach

There is another way to compute time-varyingcoefficients. Following the conditional approach(Ferson and Schadt, 1996; Christopherson et al,1998; Ferson and Qian, 2004), time-varyingalpha and beta are obtained by substitutingequations (4) and (5) in equation (1), replacingαt−1 and β1,t−1 by α0 and β0. Equation (9) obtains:

Rpt -Rft = α0 +φ1rt - 1 +φ2 Rm -Rf� �

t - 1

+ β0ðRm -Rf Þt +φ3rt - 1ðRm -Rf Þt+ � � � +φ4 Rm -Rf

� �t - 1ðRm -Rf Þt

+φ5VIXt - 1ðRm -Rf Þt + β2SMBt

+ β3HMLt + β4gprodt + εt ð9ÞTo estimate equation (9), we resort to aparsimonious approach, the OLS estimation. Theconditional alpha and beta are easily identifiedusing equations (4) and (5). Equation (9) may beviewed as a very good approximation of theKalman filter when using the same explanatoryvariables.Note that there is another current method to

make the regression coefficients time-variable:the rolling or recursive regression. This methodwas used by McGuire et al (2005) to study thedynamics of the hedge funds risk exposures.While this parsimonious method is comparablewith the Kalman filter, it is not an optimalprocedure like the filter since there is no optimalrevision of the covariance matrix period afterperiod.



EMPIRICAL RESULTS

The data

This study is based on a sample of the indices ofUS Greenwich Alternative Investment hedgefund strategies, a leader in hedge fund databasesand the first institute to collect data on thebroad universe of hedge funds. Note that wecompared hedge fund databases in previousstudies (Racicot and Théoret, 2007a, b) and theempirical results are very close, especially withrespect to the Hedge Fund Research Institute(HFR) database.15 Statistical information on thissample is reported in Table 1. Our observationperiod for the monthly returns of these hedgefund indices runs from January 1995 to March

2010, for a total of 183 observations for eachindex (strategy). The risk factors that appear inthe Fama and French equation – which are themarket risk premium and the two mimickingportfolios: SMB and HML – are drawn fromFrench’s website.16 The interest rate used to testthe models is the US three months Treasury billrate and the selected market portfolio index,the S&P500. The period we analyse was plaguedby four major financial crises: (i) the Asianfinancial crisis (1997); (ii) the Russian/LTCMcrisis (1998); (iii) the bursting of the high-techmarket bubble (2000); (iv) the 2007–2009subprime market crisis, related to high riskmortgages. Our period of analysis is thereforerich in major stock market corrections. Despite

Table 1: Descriptive statistics of the Greenwich indices returns, 1995–2010a

Mean Median Max Min SD Skew Kurtosis CAPM-beta

Distressed securities 0.68 1.04 4.79 −7.44 2.06 −1.47 6.86 0.22Equity market neutral 0.87 0.80 8.10 −2.53 1.41 1.33 8.95 0.08Futures 0.67 0.21 7.71 −6.80 3.10 0.18 2.72 −0.08Macro index 0.55 0.66 4.00 −2.95 1.45 0.29 3.06 0.27Market neutral group 0.93 0.92 7.20 −6.06 1.48 −0.61 8.99 0.20Short-sellers 0.18 −0.10 11.41 −6.88 3.61 0.56 3.46 −1.01Value index 0.61 1.11 5.68 −9.65 2.54 −1.21 5.94 0.56Arbitrage index 0.87 0.90 4.10 −8.58 1.38 −2.40 17.99 0.16Convertible arbitrage index 0.31 0.60 6.55 −19.31 2.98 −3.78 26.43 0.40Growth index 1.04 1.19 20.10 −12.99 4.53 0.43 5.50 0.76Long-short 1.09 1.31 13.20 −9.24 3.02 0.04 5.20 0.52

Mean of indices 0.71 0.78 8.44 −8.40 2.51 −0.60 8.65 0.19

Weighted composite 0.56 0.90 4.75 −5.96 1.86 −1.01 5.18 0.37S&P500 0.46 1.29 11.06 −18.47 4.62 −1.13 5.99 1.00

aThe statistics reported in this table are computed on the monthly returns of the Greenwich indices over the

period running from January 1995 to March 2010. The weighted composite index is computed over the whole

set of the Greenwich indices (strategies). The CAPM beta is estimated using the simple market model, that is:

Rit−Rft= α+β(Rmt−Rft)+εit, where Ri is the return of the index i, Rm is the S&P500 return, Rf is the riskless rate

and εi is the innovation.



these market collapses, Table 1 reveals that theGreenwich hedge funds performed quite wellduring this period. The mean monthly return ofthese indices was 0.71 per cent over this period,for an annual rate of 8.5 per cent. This rate ishigher than the annual mean return of theS&P500 over the same period, which amountedto 5.5 per cent. The low performers over thisperiod were the short-sellers, the convertiblearbitrage and macro strategies, whereas the highperformers were the long-short, growth andmarket-neutral group strategies. Moreover, thestandard deviation of returns differs greatly fromone index to the next. The indices returnsstandard deviations are generally below theS&P500 one.Several researchers argue that the strategies

followed by hedge funds are similar to option-based strategies (Fung and Hsieh, 1997, 2004;Weisman, 2002; Agarwal and Naik, 2000, 2004).And effectively, Table 1 reveals that some hedgefund strategies are actually similar to hedgedoption ones, like the covered call and protectiveput ones. These option-based strategies have abeta which is quite low, of the order of 0.6 forat-the money options, and may yet offer all thesame quite high returns which approximate thoseshown in Table 1.17 The equity market-neutraland the arbitrage strategies have a very lowbeta compared with other funds but theirreturn exceeds the overall mean return.Furthermore, plain vanilla puts, to which the

short-seller strategy is linked, have a negativeexpected return. That might explain the lowmean return of the short-seller index over theperiod of analysis. At 0.18 per cent, it is wellbelow the mean return of the whole set ofstrategies. Incidentally, the CAPM beta of theshort-seller index, equal to −1.01, is negative andquite high in absolute value over the sample

period. According to the CAPM, the excessreturn of a portfolio having a negative betashould be low and even negative: this is the caseof the short-seller index.Furthermore, according to Table 1, the

composite index of hedge funds has less kurtosisthan the market index given by the S&P500.However, this characteristic is not shared byall hedge fund strategies, the convertiblearbitrage index having a kurtosis as high as26.43. A high kurtosis means that rare orextreme events are more frequent than for thenormal distribution, which suggests that payoffsare very nonlinear. Once more, we may relatethese statistics to those associated to the cash-flows of option-based strategies. Their payoffshave a relatively low standard deviation but ahigh degree of kurtosis compared with thereturns of the stock market index, which ispriced in their returns.

Stylized facts provided by spectra of

macroeconomic variables

Looking at time series spectra is a good wayto establish their periodicity. In this respect,Figure 1 provides the spectra of selected USmacroeconomic variables, returns factors andreturns of hedge fund strategies. The spectra ofthe two macroeconomic variables used in thisstudy, the industrial production growth and theunemployment rate, have their usual plot.Industrial production growth displays highvolatility at high frequencies while theunemployment rate, being measured in levelsince it is a stationary time series, has the most ofits volatility concentrated at very low frequencies.These plots are consistent with the seminalbusiness cycle analysis performed by Stock andWatson (1990).18



Figure 1 also presents the spectra of the factorsentering in the Fama and French equation(equation (1)). Similarly to the analysis of Stockand Watson (1990), the spectrum of the mostimportant variable explaining hedge fund returns,that is the market risk premium, suggests that thisvariable is procyclical. A simple moving averageof the market risk premium also indicates that itis procyclical (Figure 2). It thus has a predictablecomponent.19 However, according to itsspectrum, the SMB factor is much more cyclicalthan the market index. This may be explained bythe composition of this index which is weightedby small and big firms. The HML variable has ashorter cycle than the SMB one. The differences

between the spectra of SMB and HML factorsare thus interesting from the standpoint ofeconometric analysis. Moreover, the spectrumof the VIX shows dampened fluctuations,which asymptotically vanish.20

The results

Kalman filter resultsIn this section, we consider in detail theestimation of equation (1) using the Kalmanfilter. The specification of the time-varying alphaand beta is given in equations (4) and (5).21

The Kalman filter estimation of this system ofequations, which is reported in Table 2, reveals

(a)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 1/2 Pi Pi

Spectrum US industrial production growth

0.0000.0050.0100.0150.0200.0250.0300.0350.040

0 1/2 Pi Pi

Spectrum unemployment rate

(b)

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0 1/2 Pi Pi

Market risk premium Spectrum

0.0

0.4

0.8

1.2

1.6

2.0

0 1/2 Pi Pi

SMB Spectrum

0.00.20.40.60.81.01.21.41.6

0 1/2 Pi Pi

HML Spectrum

0

10

20

30

40

50

60

70

0 1/2 Pi Pi

VIX Spectrum

Figure 1: (a) Macroeconomic variables spectra; (b) Return factors spectra.



that this system performs quite well. The overallperformance of the model is the highest for theweighted composite index with the highestlikelihood ratio (L). The L statistic is also quitehigh22 for three of the indices: market-neutral,long-short and value index. The model performsmoderately well for the distressed securities andgrowth strategies. The model is less satisfying forthe futures index.In the estimation of the measurement equation

given by equation (1), the market risk premiumis the most important risk factor impacting onhedge fund returns, followed by the SMB one. InTable 2, the final state beta provided by the sv2variable coefficient has a high of 1.10 for thegrowth strategy and a low of 0.16 for the equitymarket-neutral strategy. Most of the strategiesbetas are below the market portfolio one, being 1by definition, hedge funds usually reducingtheir market exposure with hedging operations.Let us notice that the betas reported in Table 2are those corresponding to the final state of ourestimation period. To analyse the dynamics of thealphas and betas corresponding to the Greenwichindices, we must build the state series of thesecoefficients.

Before considering this subject, note inTable 2 that the SMB factor is significant at the99 per cent confidence level for all strategiesanalysed except for the futures one for which it isnot significant. Since hedge funds are in search ofabnormal returns, they prefer to hold stocks ofsmall firms over those of bigger ones.23 TheHML factor is for its part significant at the90 per cent confidence level for only threeindices: the growth index, the long-short indexand the weighted composite one. The impact ofthis factor is generally lower than SMB in mostof the empirical studies using the Fama andFrench model to estimate hedge fund returns.One of our main contributions in this study is

to analyse the procyclicality of risk-taking in thehedge fund industry. In that respect, we haveintroduced the smoothed growth of industrialproduction in the Fama and French model.This variable is significant at the 95 per centconfidence level for the weighted compositeindex, which stands for the performance of therepresentative hedge fund, and for the valueindex. Consistent with economic theory, the signof this variable is negative. Indeed, the growth ofindustrial production may be considered as aninverse proxy for the degree of risk aversion, acountercyclical variable (Cochrane, 2005). Inexpansions, risk aversion decreases following theincrease in the industrial production growth, andthus expected returns decrease ceteris paribus; incontractions, the inverse relationship holds.Let us now consider the estimated states series

of the filtered alpha and beta. We begin ourdiscussion with the beta state series which is apriori more manageable than the alpha statevariable. In Table 2, the final state beta (sv2),associated to the market risk premium is verysignificant for every index as indicated by thep-values of this coefficient. It is equal to 0.48 for

-6

-4

-2

0

2

4

6

95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10

Figure 2: US market risk premium moving

average.

Note: US recessions are shaded.



Table 2: Kalman filter estimationa

Time-varying alpha Time-varying beta SMB HML gprod L

sv1 r(−1) (Rm−Rf)(−1) sv2 r(−1) (Rm−Rf)(−1) VIX

Distressed securities 0.3097 −0.0097 −0.0068 0.3593 −0.0014 0.0000 0.0043 0.1689 0.0403 −0.0121 −344.640.0034 0.5989 0.3735 0.0000 0.9172 0.7714 0.0016 0.0000 0.2641 0.7643

Market neutral −0.1904 −0.0493 −0.0165 0.1596 −0.0145 0.0001 0.0027 0.1299 −0.0022 −0.0183 −295.160.0183 0.0003 0.0056 0.0000 0.2183 0.3054 0.0114 0.0000 0.9384 0.5847

Long-short −0.0571 −0.0525 0.0225 0.6152 −0.0150 0.0001 0.0041 0.2618 −0.0669 −0.0563 −296.720.4823 0.0001 0.0001 0.0000 0.0300 0.0484 0.0001 0.0000 0.0380 0.0588

Value index −0.0367 −0.0560 0.0193 0.3985 0.0072 0.0025 −0.0002 0.2815 0.0364 −0.0861 −300.670.6592 0.0005 0.0021 0.0000 0.5283 0.0321 0.2502 0.0000 0.2049 0.0171

Growth index −0.4483 −0.0700 0.0314 1.1032 −0.0591 0.0055 0.0007 0.3416 −0.2984 −0.0472 −372.190.0003 0.0006 0.0004 0.0000 0.0007 0.0014 0.0014 0.0000 0.0000 0.3196

Futures index 0.1807 0.0145 −0.0134 1.0342 −0.1540 0.0108 0.0019 0.0041 0.0596 0.1132 −503.460.4628 0.6376 0.3856 0.0000 0.0000 0.0001 0.0000 0.9602 0.4725 0.1447

Weighted composite 0.0292 −0.0229 0.0125 0.4782 −0.0216 0.0040 0.0002 0.1995 −0.0424 −0.0449 −260.060.6623 0.0002 0.0084 0.0000 0.0000 0.0000 0.0000 0.0000 0.0937 0.0408

aThe model we use is explained in the second section. The equations are estimated resorting to the Kalman filter. sv1 and sv2 are the final state values of

the alpha and beta. For each strategy, the first line of numbers provides the estimated coefficients of the variables and the second line gives the

corresponding p-values (in italics). The L statistic is the log likelihood associated to an estimation. The sv1 coefficient is the final state of the alpha and the

sv2 coefficient is the final state of the beta.

RacicotandThéoret

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the weighted composite index, the beta of hedgefunds being usually quite moderate relatively totraditional managed stock portfolios like themutual funds ones. The indices having thehighest beta are the growth index and the futuresone. These betas are greater than one. Theindices having the lowest beta are the marketneutral, as expected for such a strategy, and thedistressed securities index.Regarding the conditioning variables, we note

in Table 2 that the interest rate (rf ) has a negativeimpact on hedge funds betas, which support ourhypothesis since an increase in interest rate signalsa market deterioration, which lead hedge fundsto take less risk. In other respects, according tothe market variable (Rm−Rf ), hedge funds takemore risk when the market returns, as measuredby the S&P500 index, increases, but this effect isquite low and not significant for the distressedsecurities and market-neutral strategies. Finally,financial market volatility, as measured byVIX, impacts positively and significantly on themarket returns of all strategies except the valueindex one, for which the exposure to volatilityis negative and insignificant. In light of ourprevious comments about the sign of thisvariable, hedge funds seem conditioned by thepayoffs related to forward market volatility, thevalue of an option being dominated by itsvolatility.Turning to the alpha, we note in Table 2

that the estimated final state alpha given by sv1is generally low and even negative for most ofthe strategies. Indeed, our sample ends at thebeginning of 2010, which corresponds to the startof an economic recovery in the United Statesfollowing the subprime crisis, which explain thelow level of sv1 for the majority of the strategies.This result does not support the alpha puzzle.Note however that the final state alpha of the

distressed securities strategy is quite high, at 0.31,which is related, as shown later, to the specificityof this strategy. Moreover, consistent with theresults obtained for the beta, the alpha respondsgenerally negatively (and significantly) to anincrease in the interest rate and positively (andsignificantly) to a strengthening of the stockmarket conditions. Like the beta, the hedge fundalpha thus seems procyclical.The plots of the betas indicate that they are far

from being constant, as suggested by theconventional CAPM, and exhibit a procyclicalbehaviour. In Figure 3, we compare the profile ofthe state beta of the weighted composite indexwith the US unemployment rate, a well-knowncountercyclical time series. As there is a trainingperiod for the Kalman filter, there are usuallyspikes at the start of the plots of the state betas oralphas. We note that the state beta is negativelycorrelated to the unemployment rate. Moreprecisely, the beta has an inverse cycle comparedwith unemployment: it is thus procyclical.However, remind that the unemployment rate isa lagged indicator of economic activity while thebeta, being the result of an investment decisionprocess, is forward-looking. In that respect, the

2

4

6

8

10

12

0.0

0.1

0.2

0.3

0.4

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1994 1996 1998 2000 2002 2004 2006 2008 2010

unemployment rate

state beta

Figure 3: US unemployment rate and

Greenwich hedge fund composite index.




behaviour of the beta during the 2007–2009subprime crisis is very instructive. During the firsthalf of the crisis, the unemployment rate jumpssignificantly while the beta drops substantially,which suggest that the representative hedge fundreduced greatly its exposure to market risk.But being a lagged indicator, the unemploymentrate continued to increase in the last phase ofthe crisis, while the beta resumed its increase,in expectation of the upcoming economicexpansion. In this sense, the beta is forward-looking.As shown in Figure 4, the state beta of the

weighted composite index decreased duringthe 1997 Asian crisis before resuming its increasein 1998. Thereafter, following the first USrecession of the millennium, the beta decreasedfrom the beginning of 2000 till the end of 2002,which paved the way to a market recovery.It almost doubled from 2003 to the middle of2005. It decreased progressively thereafter inexpectation of an economic slowdown and inreaction to the corporate accounting scandals.This beta dynamics is comparable with the oneobtained by McGuire et al (2005) during theperiod 1997–2004 with respect to hedge fund

risk exposure, which is a leveraging during theupward trend of the stock markets or theeconomic expansions followed by a deleveragingduring the crises.The state alpha related to the weighted

composite index has a profile similar to the betabut is more volatile (Figure 4). The alphadecreased after the Asian crisis, the decreasegaining momentum during the technologicalbubble. During this episode, the estimated alphadropped from a high of 1 per cent monthly toa low close to 0 per cent, which suggests thatthe alpha puzzle must be studied in a dynamicsetting. Our procyclical approach seems thusmore relevant to study the alpha process. As inthe case of beta, the profile of the alpha isparticularly interesting during the 2007–2009subprime crisis. According to Figure 4, itdecreases to a low of −0.5 per cent in the middleof the crisis before recovering thereafter, a profilesimilar to the beta. In summary, alpha and betaco-move positively, a result in line with thecommon factors that drive these twoperformance measures.

Conditional modelsIn this section, we compare in this section thecycle of state and conditional beta and alpha.Figure 5 provides this comparison for theGreenwich weighted composite index. We notethat the conditional beta (equation (5)) has aprocyclical profile that is very similar to the oneof the state beta, which supports the Kalman filterestimation. Both betas are very synchronizedduring an economic slowdown, as during thetechnological bubble burst and the subprimecrisis. But when the economy recovers, we notethat the state beta is forward-looking with respectto the conditional one. For example, during the

-1.0-0.50.00.51.01.52.0

0.0

0.1

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0.5

1994 1996 1998 2000 2002 2004 2006 2008 2010

state beta

state alpha

Figure 4: State series for the alpha and beta

of the Greenwich weighted composite index.




2003–2007 economic recovery, the increasein the conditional beta lagged the increasein the state beta by many months. The samephenomenon is observed during the last periodof the subprime crisis, whereas the state betajumps while the conditional beta increases onlymodestly. Note also that the state beta began itsdecrease well before the subprime crisis, which isless the case for the conditional beta. The Kalmanfilter thus seems better appropriate to model theforward-looking properties of time-varyingbetas.Figure 6, which plots the Kalman filtered and

conditional alphas for the Greenwich weightedcomposite index, sheds more light on the alphapuzzle in a dynamic setting. It seems that thealpha puzzle is much less problematic in ourprocyclical framework for analysing hedge fundreturns. Contrary to the previous empiricalanalyses of hedge fund returns, the alpha maybecome nil or even negative in periods ofeconomic contraction. This pattern is morepronounced for the Kalman filtered series thanthe one obtained for the conditional model. It isthus interesting to note that the alpha is positivein economic expansions, even moving on an

upward trend in the first phase of economicexpansions. Note that in the 1990s, there actuallyseems to be an alpha puzzle but it would berelated to the long period of economic expansionwhich prevailed then. Note also in Figure 7that the cyclical behaviour of the alpha isasymmetric, having a tendency to decrease morein contractions than it increases in expansions.Even if our sample is not long enough to beconclusive, the alpha seems mean-revertingtowards a level which declines slowly.The cyclical properties of the weighted index

return are shared by the returns of the specificstrategies analysed. We note in Figures 7 and 8

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conditional beta

state beta

Figure 5: State series and conditional model

for the beta of the Greenwich weighted

composite index.


-0.8

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conditional alpha

state alpha

Figure 6: State and conditional series for the

alpha of the Greenwich weighted composite

index.


-0.020.000.020.040.060.080.10

-0.3-0.2-0.10.00.10.20.3

95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10

state beta

conditional beta

Figure 7: State and conditional beta of the

equity market-neutral strategy.




that the betas of the equity market-neutral andlong-short strategies display procyclicality, acounterfactual result given the nature of thesestrategies.24 In line with the weighted compositeindex, the betas of both strategies returned onan upward trend well before the end of the2007–2009 subprime crisis, the swing being morepronounced with the Kalman filter. Moreover,the conditional beta seems more sensitive tomarket conditions than the Kalman filteredone, especially for the equity market-neutralstrategy.It is also interesting to study the procyclical

behaviour of two strategies having higher betasthan the representative hedge fund, which, likethe market-neutral and the long-short strategies,have a relatively low market exposure. Figures 9and 10 provide the cyclical behaviour of thevalue index and the growth index whose CAPMbetas are respectively 0.56 and 0.76. We notethat the procyclical behaviour of these twostrategies is quite different. The value indexhas a procyclical profile which is similar to theweighted composite index and is quite sensitiveto market conditions. Moreover, as the weightedcomposite, the value index is forward-looking,especially before and during the 2007–2009

subprime crisis. However, contrary to theweighted composite index, note that the valueindex conditional beta is more forward-lookingthan the Kalman filtered beta, which suggeststhat the value index strategy is very sensitive tofinancial market conditions. Another interestingaspect related to the cyclical profile of the growthindex is that during the expansion 2002–2007,its beta, both computed by the Kalman filter andthe conditional model, began to decrease wellbefore the end of the expansion after havingpeaked at a level near 1, which is not the casefor the value index which follows economicconditions more closely.

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95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10

state beta

conditional beta


long-short strategy.


0.32

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0.68

1994 1996 1998 2000 2002 2004 2006 2008 2010

conditional beta

state beta


value index strategy.


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state beta

conditional beta


growth index strategy.




Finally, we consider two very specializedstrategies: the futures and the distressed securitiesones (Figures 11 and 12). Owing to the instabilityof their returns, it was difficult to apply theKalman filter to these strategies so we resortedonly to the conditional model. The procyclicalprofile of the futures index beta is close to theone given by the weighted composite index.However, the fluctuations of the futures indexbeta are much wider, being in a range of[−0,5, 0.4] in Figure 11. This profile is also verydifferent from the ones of the long-short andmarket-neutral strategies, which also explains

why the Kalman filter has difficulties to capturethe time profile of the futures index. In otherrespects, even if it is quite less volatile than thefutures index, one interesting dimension of thedistressed index is that, contrary to the otherindices, its beta increases during contractions.Actually, the distressed securities strategy focuseson contraction periods so distressed funds takemore risk during economic slowdowns. As itsbeta tends also to decrease in an economicexpansion, the distressed securities strategyseems to be countercyclical.

CONCLUSION

In this article, we revisit hedge fund returnmodels relying on a procyclical setting basedon two dynamic procedures: the Kalman filterand the conditional model approach. Theseprocedures appear more rigorous than thestandard OLS one usually used to compute theconditional alpha and beta. In the Kalman filterapproach, the model coefficients are updatedeach period in an optimization framework takinginto account all information accumulated untilthis period. We also consider the conditionalmodel as an approximation to the Kalman filter.To the best of our knowledge, our study is

the first to consider in detail the cyclical profileof hedge fund returns. Indeed we resort to aprocyclical approach to study hedge fund returns,introducing cyclical variables in our models,such as the growth of industrial production,unemployment rate, the rate of vacancy and theVIX, among others. We find that these variablescapture quite well the cyclical behaviour of thehedge fund risk-return trade-off.Our study shows that the state and conditional

alphas are quite responsive to the businesscycle, usually increasing during expansions and

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Figure 11: Conditional beta of the futures

index strategy.


0.08

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0.24

0.28

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Figure 12: Conditional beta of the distressed

securities strategy.




decreasing during contractions. The Kalmanfilter and the conditional model both support thisprocyclical profile. We may thus conclude thatthe absolute hedge fund return, as measured bythe Jensen alpha, displays a cyclical behaviour,a new aspect for this performance measure notfound in the literature. Our experiments alsoshow that the alpha tends to mean-revert fromone business cycle to the next. In order to checkfor its robustness, this result needs furtherinvestigation.The time-varying beta of hedge fund strategies

also displays a pronounced procyclical profilefor both the Kalman filter and the conditionalmodel. The beta responds positively to themarket risk premium and negatively to theinterest rate level, a quite rational behaviour.Moreover, the beta is conditioned by a cyclewhich may be explained quite easily by thebehaviour of the macroeconomic variables weuse. Incidentally, the performance of modelsrelying on conditioning information is usuallymuch better to explain the expected excessreturns of the hedge funds because thetime-varying beta is significantly related tomacroeconomic information. A promisingresearch avenue is to consider the impact ofprocyclicality of hedge fund risk on the businesscycle itself. Actually, do hedge funds exacerbatethe amplitude of the business cycles? This is avery important question from the point of viewof macroprudential policy that deserves furtherinvestigation.

ACKNOWLEDGEMENTS

We would like to thank Ying L. Becker andseminar participants at the Southern FinanceAssociation Annual Meeting held in Key West,Florida on November 2011 for their helpful

comments. We finally thank Bhaswar Gupta andtwo anonymous referees for their suggestions.

NOTES

1 Hull (2006) reports that one trillion $ wasinvested in this industry in 2004. In 2008, thisamount peaked to 1.93 trillion (Sadka, 2010).

2 LTCM is the acronym of Long Term CapitalManagement, a highly levered US hedgefund that sustained massive losses in 1998.

3 According to Devasabai (2010), the share ofhedge fund investments in US pension funds,which is presently about 2.5 per cent, willincrease to 15 per cent by the end of thedecade.

4 For instance, regulators could constrain thefinancial institutions to pay for the externalitiesthey generate. In that respect, when a bigfinancial institution increases its leverage, it cangive raise to negative externalities for the otherinstitutions, which are not endogenized in itsbehaviour. Actually, these externalities couldentail an important reduction in liquidities forthe whole financial system, liquidity being apublic good.

5 More specifically, the full name of thiscompany is: Greenwich AlternativeInvestment.

6 Note also that the alpha puzzle was detectedover the 1990s, while the Great Moderationwas at its peak. The high observed alphamight be explained by these favourablebusiness conditions.

7 The appendix provides a general descriptionof the Kalman filter procedure. For anintroduction to the Kalman filter and itsuses in finance, see Rachev et al (2007),chap.11 for a hedge fund return modellingapplication. According to L’Habitant (2004),the Kalman filter is similar to a least squares



estimation except that the coefficients ofthe model are updated at every periodfollowing the arrival of new information.The applications of the Kalman filter infinance go back to the beginning of the1980s. At this time, the filter was used toforecast the real rate of interest and the riskpremia in forward and futures markets(Fama and Gibbons, 1982; Hsieh andKulatilaka, 1982). Bassett et al (1991) used theKalman filter to forecast forward prices ofnontraded securities. There are also extensiveapplications in the fields of exchange ratesand term structure of interest rates where theKalman filter is used to forecast volatility andother key variables (Pennacchi, 1991). Thereare nevertheless few studies in the hedge fundarea that analyse their style dynamics,especially the time-varying dimensions ofalpha and beta (Swinkels and van der Sluis,2001; Giamouridis and Vrontos, 2007).

8 Our experiments show that the Fung andHsieh’s (2004) factors have only a marginalexplanatory power on hedge fund returnsand do not affect the cyclical behaviour of thealphas and betas, the main object of ourstudy. Since we resort to a Kalman filterframework in this article, we must use aparsimonious approach so we do not retainthe Fung and Hsieh factors.

9 While the Fama and French device is still apopular choice to model hedge fund returns,other more parsimonious models haveappeared recently in the literature, forinstance the one proposed by Berkelaaret al (2009). These authors suggest thathedge fund strategies could be modelledusing the following parsimonious model:rh;t = αh;t + β

EQh;t r

EQh;t + βFIh;t r

FIh;t + εt. This simple

two-factor model uses the returns on equity

(r EQ) and those on fixed income (r FI)securities, as many studies have shown that asignificant portion of hedge fund returnstrategies is well explained by these twofactors. The parameterαh;t represents thetime-varying alpha of strategy h and the othercomponents represent the systematic betacomponents. Thus, the performance at everytime period of a particular strategy can beexplained by the manager skills (alpha), thereturns on equity (stocks) and the returns onfixed income securities (for example, bonds).Because of the time-varying nature of theirmodel, they rely on the Kalman filter toestimate the model parameters. For a similarapproach, see also Eling and Faust (2010).

10 The signs of the factor loadings are seldomdiscussed in the studies of the Fama andFrench model. This gap must be filledbecause the signification of these signs ispartly a matter of interpretation.

11 For another approach to dynamize alphas andbetas, see Giamouridis and Vrontos (2007),Mamaysky et al (2007, 2008).

12 Note that an increase in market volatilityusually takes place during an economiccontraction, when the volume of markettransactions skyrockets. For more details, seeLe Grand and Ragot (2010).

13 Market volatility is then another factorcontributing to the revision of the time-varying alpha and beta from one period to thenext.

14 In this situation, the squared market riskpremium represents the co-asymmetry of agiven portfolio return with the market return.

15 These results are available upon request.16 The address of the French’s website is: http://

mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.



17 For a discussion of the beta and the meanreturn of option-based strategies, see Whaley(2006), chapter 10.

18 See also Racicot (2011) for a review ofspectral analysis techniques applied tofinancial data.

19 Note that the spectrum of a random variable,being by nature unpredictable, is flat.

20 We also built the spectra on hedge fundreturns. See Racicot and Théoret (2012).

21 Note that, for robustness check, we estimatedrandom walks for the state variables given byEquations (2) and (3). For the sake ofsimplicity, we do not report the results herebut these results were poor compared withthe estimations of Equations (4) and (5) inwhich we add control variables to account forthe cycles and the funds’ forward lookingstrategies (Racicot and Théoret, 2007a, b).

22 To see the correspondence between the R2and the log-likelihood statistic in Table 2,note, for instance, that the equation of theexcess return of the equity hedge strategy hasa R2 of 0.83 when estimated by the OLSmethod and a log-likelihood statistic of −173when estimated by the Kalman filter.

23 Indeed, there is a lack of accountinginformation for small firms compared withthe big ones. This ‘opacity’ might explain thesmall firm anomaly, which is an abnormalreturn on small firms stocks after accountingfor systemic risk.

24 Indeed we might believe that these hedgefunds should ‘hedge procyclicality’.

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APPENDIX

The Kalman filter procedure

Our main procedure relies on the Kalman filter.A brief description of this method follows.Assume an observable time series yt. This variablemay be for instance a financial asset return. Itdepends on the variable ht which is unobservable



or latent. This variable could be the stochasticvolatility of yt. Since we cannot observe ht, wehave to simulate it. The variance of ht, denotedby ωt, is also unobservable. The model can berepresented as follows: yt= θ1+θ2ht+εt; ht+1=θ3+θ4ht+ηt, where θi are the parameters toestimate, εt stands for a Gaussian noise whosevariance is v1t, and ηt is a Gaussian noise withvariance v2t. The first equation is themeasurement or observation equation whereasthe second is the state or transition equation.Let us now consider the case of time-variablecoefficients. At time (t−1), estimations of ht−1 andits variance ωt−1 as well as coefficients θit−1 arepredetermined. At time 0, we must have apreliminary estimation of h0 and ω0. But becausethese values are unknown, the software EViews,used in this study, put a zero value to h0 and ahigh value to ω0 in order to account for theuncertainty related to the estimation of h0. Letus step back to time (t−1) of the simulation orof the filtering. The three steps of the procedurefollowed by the Kalman filter are then:forecasting, updating and parameter estimation.In the first step, we make the following two

forecasts: ht|t−1, that is the forecast of htconditional to the information set at time (t−1),and ωt|t−1, that is the forecast ωt conditional tothe information set at time (t−1). These forecasts,which are unbiased conditional estimations,obtain: ht|t−1= θ3,t−1+θ4,t−1ht−1; ωt|t−1=θ4,t−12 ωt−1+v2,t−1. The second step is the

updating one. At time t, we have a newobservation of y, that is, yt. We can thus computethe prediction error υt= yt−θ1,t−1−θ2,t−1ht|t−1.The variance of υt, denoted by ψt, is given by:ψt= θ2,t−1

2 ωt|t−1+v1,t−1. We use the parameters υtand ψt to update ht and its variance ωt asfollows: ht= ht|t−1+θ2,t−1×ωt|t−1×υt/ψt; ωt=ωt|t−1+θ2,t−1

2 ×ωt|t−12 /ψt. The last two equations

are conditionally unbiased and efficientestimators. The Kalman filter is thus optimalbecause it is the best estimator in the class of linearestimators. The third step deals with parameterestimation. To estimate the parameter θi, we usethe maximum likelihood method. Thelikelihood function can be written as follows:l=−1/2∑tlog(ψt)−1/2∑tυt

2/ψt. To completethe procedure, we go to time (t+1) and repeat thethree-step procedure up to period n.



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