the tactical and strategic value of hedge fund

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Fin Mkts Portfolio Mgmt (2007) 21: 425–444DOI 10.1007/s11408-007-0060-8

The tactical and strategic value of hedge fund

strategies: a cointegration approach

Roland Füss · Dieter G. Kaiser

Published online: 24 August 2007© Swiss Society for Financial Market Research 2007

Abstract This paper analyzes long-term comovements between hedge fund strate-

gies and traditional asset classes using multivariate cointegration methodology. Since

cointegrated assets are tied together over the long run, a portfolio consisting of these

assets will have lower long-term volatility. Thus, if the presence of cointegration

lowers uncertainty, risk-averse investors should prefer assets that are cointegrated.

Long-term (passive) investors can benefit from the knowledge of cointegrating re-

lationships, while the built-in error correction mechanism allows active asset man-agers to anticipate short-run price movements. The empirical results indicate there is

a long-run relationship between specific hedge fund strategies and traditional finan-

cial assets. Thus, the benefits of different hedge fund strategies are much less than

suggested by correlation analysis and portfolio optimization. However, certain strate-

gies combined with specific stock market segments offer portfolio managers adequate

diversification potential, especially in the framework of tactical asset allocation.

Keywords Hedge fund strategies · Stock markets · Tactical and strategic assetallocation · Portfolio optimization · Multivariate cointegration analysis ·

Johansen test

JEL Classification C32 · G11 · G15

R. Füss ()

Department of Empirical Research and Econometrics, University of Freiburg,

Platz der Alten Synagoge, 79085 Freiburg im Breisgau, Germany

e-mail: [email protected]

D.G. Kaiser

Feri Institutional Advisors GmbH, Haus am Park, Rathausplatz 8-10, 61348 Bad Homburg, Germany

e-mail: [email protected]



426 R. Füss, D.G. Kaiser

1 Introduction

Over the past decade, the hedge fund universe has grown from a handful of firmsmanaging a few hundred million dollars to more than 8,000 hedge funds worldwide

managing more than $1 trillion (HFR 2006). As a result, however, hedge funds faceconstant scrutiny from investors and researchers.Behind the torrid growth of the hedge fund universe has been the perceived

diversification benefit hedge funds can provide traditional portfolios because of their low correlation structure (Brooks and Kat 2002; Morley 2001; Zask 2000;

Planta and Banz 2002; Eling 2006). This article, however, posits that the assump-tion of correlation analysis as it applies to hedge funds deserves to be reexaminedand reinterpreted. Kat (2003) has argued that correlation coefficients are only reliableunder a normal distribution of variables. Hedge funds tend to exhibit nonzero skew-ness and/or excess kurtosis (see Kat and Lu 2002; Füss and Kaiser 2007; Galeano and

Favre 2001). Therefore, the lower and upper bounds of a correlation coefficient mightbe narrower than ±1 (Kat 2003). We contend it is unreliable to claim that hedge fundshave low correlations with stock market indices.

Mean–variance methods are also used to analyze return series (Markowitz 1952).

However, with these types of methods: (1) stationarity is assumed, (2) returns must benormally distributed, and (3) return correlations between assets must be stable. Sincefinancial log asset prices are mostly nonstationary (see Nelson and Plosser 1982),they must be transformed into stationary variables before correlation analysis can beapplied. And, this procedure can result in information loss on long-run components

because it removes the possibility of finding common price trends.Cointegration methods, on the other hand, work directly on portfolio values and

make no assumption about stationarity of the asset values (Alexander 2001). There-fore, this article draws on the theory of cointegration processes to discern whether

hedge fund strategies exhibit a long-run equilibrium relationship with the time seriesof conventional financial asset classes.

Cointegration refers to the fact that financial assets share common stochastictrends that cause time series to move toward long-term equilibrium after every ter-minal shock. The built-in error correction mechanism determines how long it will

take to reach equilibrium. The presence of cointegration does not make prices fullypredictable, but it does make it possible for investors to better time their portfolioholdings (Gregoriou and Rouah 2001).

Cointegration affects both tactical and strategic financial decision making (Lucas

1997). Because of the long-term relationship between cointegrated assets, a port-folio of these assets with weights taken from cointegrating vectors will have lowerlong-term volatility. In addition, the error correction mechanism allows active assetmanagers to anticipate price movements over the short term. The speed of adjustmentestimated by the error correction model shows the economic relevance of long-term

comovements.When it takes decades for the asset returns to move toward the common stochastic

trend, the existence of such a trend is of little relevance for an investor with finitehorizon (Kasa 1992). Without cointegration, there is no mean reversion in the price

spread, and uncertainty for active and passive asset managers will be higher. Thus,risk-averse investors will still prefer cointegrated assets.



The tactical and strategic value of hedge fund strategies 427

Most studies on cointegrated assets have applied the approach to traditional stock

market portfolios and indices. For example, Bossaerts (1988) developed a test of

cointegration and applied it to five size-based and five industry-based stock portfolios.

His results rejected the null hypothesis of no cointegration at a very high significance

level, thereby providing substantial evidence of cointegration.Kasa (1992) studied the drivers of equity markets in the US, Japan, England, Ger-

many, and Canada, and found evidence of a single stochastic trend. He used the Jo-

hansen test for common trends on monthly and quarterly data from the Morgan Stan-

ley Capital International (MSCI) equity indices from January 1974 through August

1990.

Other studies have examined the interrelationships among regional stock mar-

kets to find potential gains from international diversification. For example, Corhay

et al. (1993) found cointegration among stock prices in several European countries

(except Italy, which did not influence the long-term relationship). Arshanapalli andDoukas (1993) found links between US and European markets (the UK, Germany,

and France) using the bivariate Engle and Granger (1987) approach.

On the other hand, Taylor and Tonks (1989) found no pairwise cointegration be-

tween US and UK equity markets, a result reinforced by Kanas (1998) using the mul-

tivariate trace statistic, the Johansen approach, and Bieren’s test. This suggests there

is no cointegration between the US market and any major European equity market.

Another strand of studies has focused on stock market links between emerging

markets and their regional areas. Pan et al. (1999) used the multivariate cointegration

approach and found no evidence of common stochastic trends in the equity marketsof Australia, Hong Kong, Japan, Malaysia, or Singapore. Garret and Spyrou (1999)

investigated the existence of common trends in Latin America and Asia-Pacific equity

markets. They noted some common trends,1 but they did not rule out the possibility

of long-term diversification benefits, because some of the countries do not enter the

region’s common trend.

Gregoriou and Rouah (2001) focused on hedge fund investments, examining com-

mon stochastic trends between the ten largest hedge funds of different styles and the

equity market indices of the S&P 500, the MSCI World, the Russell 2000, and the

NASDAQ index from January 1991 through December 2000. The authors found evi-dence of cointegration with the stock market indices for just two of the hedge funds.

They argue that large hedge funds tend to allocate assets over a wide range of invest-

ment instruments, such as futures, options, currencies, swaps, and other derivatives.

Therefore, the performance of these hedge funds will not be strongly correlated to

standard benchmarks.

Füss and Herrmann (2005) studied long-term interdependence between hedge

fund strategies and the stock market indices of France, Germany, Japan, North Amer-

ica, and the UK from January 1994 to December 2003. They found no evidence of

common stochastic trends, except for a weak long-term interrelation between hedgefund strategies and the US stock market.

1For further studies on the interdependence of emerging markets, especially in Asia and Latin America,

see Hung and Cheung (1995), Chaudhuri (1997), and Chen et al. (2002).




Both Gregoriou and Rouah (2001) and Füss and Herrmann (2005) use the Engle

and Granger (1987) two-step cointegration approach, which is considered weaker

than the Johansen method. The results of the Johansen test do not depend on the

normalization selected (Hamilton 1994), while, in comparison, the Dickey–Fuller

(DF) test is numerically dependent upon the precise formulation of the cointegratingregression.2

Füss et al. (2006) test for the presence of cointegration between hedge funds and

traditional and alternative financial assets. Their empirical results suggest that, for a

traditional portfolio, the hedge fund composite index not only enters the cointegrating

vector, but the returns also react to the common trend. Thus, risk-averse investors with

long-term investment horizons do not increase risk by including hedge funds.

On the other hand, for a portfolio consisting only of alternative assets, hedge funds

share a common trend with NASDAQ-listed companies, small-cap stocks, and real

estate equities. However, only hedge fund and emerging equity returns react signifi-cantly to the common trend. The authors conclude that investors with both traditional

and alternative portfolios can benefit from risk diversification.

We interpret the overall absence (existence) of a common stochastic trend in most

of the studies as the existence (failure) of long-run gains from diversification. But to

expand this narrow perspective, we differentiate between tactical and strategic asset

allocation, as per Lucas (1997). Thus, the relevance and the implications of cointe-

gration between asset prices and hedge fund strategies for asset allocation decisions

will depend on the holding period of the investment, the rebalancing frequency of the

portfolio, and investor risk attitude. In contrast to Füss et al. (2006), who represent

the hedge fund universe by an aggregate composite index, we analyze the dynamic

linkages between various hedge fund strategies and traditional asset markets.

The remainder of this article is organized as follows. Section 2 reviews the dif-

ferent asset allocation levels in the context of investment decisions. In Sect. 3, we

discuss stationarity and cointegration more fully, as well as the augmented Dickey–

Fuller and Johansen tests and how they affect asset allocation. Section 4 presents our

empirical findings of cointegration relationships between the hedge fund styles and

conventional financial assets. Section 5 concludes.

2 Asset allocation methods

Most finance literature defines three levels of asset allocation: benchmark, strategic,

and tactical. The aim of asset allocation in all cases is to obtain the best possible

risk/return profile for a portfolio. But the three methods use dramatically different

ways to achieve their objectives, which we will examine further in this section.

In benchmark asset allocation (or what is referred to as indexing), the portfolio

manager makes investment decisions according to the asset weights of the bench-

mark index. Strategic asset allocation, in contrast, is based on a long-term view of

2Dickey et al. (1991) argue that the results of the Engle–Granger cointegration approach may not be

consistent because it is sensitive to the choice of dependent variables. Johansen’s multivariate cointegrationtest is more robust.




Table 1 Asset allocation methods

Source: Dahlquist and Harvey (2001)

performance and usually has a 5 year time horizon. The weights are determined by

long-term forecasts, so there is no need to rebalance the portfolio (see Table 1).

Tactical asset allocation is the process of short-term deviations from the strategic

weights, usually in one month to one quarter increments. Because indexing results

in only slight alterations in portfolio weights, variation in investment weights will

increase with the strategic program. Strategic asset allocation weight changes are

slow and evolving so as to maintain the objective of rebalancing the portfolio within

a year; tactical asset allocation weight changes, in contrast, are highly dynamic.

The use of conditioning information to determine the weights also naturally varies

according to the allocation method. Benchmark allocation requires no conditioning

information at all; in strategic and tactical asset allocation, conditioning information

is generally used. In addition, strategic allocation decisions are sometimes based on

unconditional information by assuming that historical returns are representative of

future returns. In the context of portfolio optimization, using ex post data can lead to

a fixed weight portfolio. On the other hand, Dahlquist and Harvey (2001) note that,

for both tactical and strategic asset allocation, short- and long-term expected returns

induce weight changes unless the conditioning information has no predictive ability.

To establish a consistent and effective investment policy, investors must gauge

the level of future uncertainty by using quantitative models. Time series models are

attractive for this purpose because the future behavior of a times series is explained

by its own past and by the past of related time series. Thus, investors need no (or only

some) prior knowledge about related (exogenous) economic variables.

To be more precise, we focus here on long-term comovements between hedge

funds and financial asset markets, and the effects of short- and long-term planning

horizons. Cointegration will reveal the existence of any long-run equilibrium rela-

tionships between hedge funds and other financial series.

The built-in error correction mechanism illustrates how series react to temporary

deviations from long-term equilibrium. Assuming investors are risk-averse, under

strategic asset allocation, they will benefit from being aware of and understanding

cointegrating relationships (e.g., that asset prices will stay together over the long




term). As Lucas (1997) has noted, cointegrating relationships of financial time seriesshow less long-term variability, and thus, less long-term risk.

For tactical asset allocation, we can incorporate the reaction to temporary statesof disequilibrium into the calculus. If time series are cointegrated, the error correc-

tion mechanism allows the portfolio manager to anticipate some of the near-termdevelopments. This means that the conditioning information is provided by the ad-

justment coefficient of the temporary deviations. However, if this process unfoldsover decades, passive long-term investors would be better served by focusing on er-

ror correction instead of cointegrating vectors, and active investors could probablyignore the presence of cointegration.

3 Stationarity and cointegration in a system with hedge funds

In the context of long-run properties of financial time series, researchers have tendedto focus on asset price characteristics, i.e., random walk (unit root) or mean reversion

(trend stationary) processes. For example, if asset prices are mean-reverting, overtime the price level will revert to its trend path (or mean return). This suggests future

returns are predictable from information on past returns.3

In contrast, a random walk supposes any shock to an asset price is permanent, andthere is no tendency for it to revert back. Thus, in a random walk framework, future

returns are unpredictable based on historical observations, and the most recent returnwould be the best predictor of future returns.4

Another trait of random walks is that the longer the time horizon, the more likelyit is that prices will wander far from their trend path. In the long run, this boundless

growth in volatility of asset prices characterizes the nonstationarity of random walks,which in turn has important implications for asset pricing and portfolio allocation

decisions. Of course, for financial decisions on optimal asset allocation, the aboveresults imply that the correct choice concerning the (non)stationarity property of a

time series is of major relevance.To evaluate whether hedge fund strategies provide long-term diversification bene-

fits for traditional portfolios, we need to test for the presence of common stochastic

trends. Hence, we need to use cointegration, as defined and developed by Granger(1981) and Engle and Granger (1987). Cointegration is a property of some non-stationary time series. If two nonstationary time series are cointegrated, a linear com-

bination relationship that is also stationary is said to exist. In the context of portfoliotheory, if the value series of a fixed weight portfolio of assets with nonstationary

prices is stationary, the assets will form a cointegrated set. The set of asset weightswithin such a portfolio is called the cointegrating vector.

Cointegration means there is some long-term equilibrium relationship tying the

times series together, despite the fact that short-term departures from equilibrium

3Consequently, stationarity is based on the assumption that the effect of present shocks in the time seriesdiminishes or disappears in the distant future. Technically speaking, stationarity means that the time seriesexhibits a constant mean, standard deviation, and autocovariance that depend only on the time lag.

4Hence, in financial literature, the property of nonstationarity often emerges quite naturally as a result of the assumption of efficient markets and the absence of arbitrage (Lucas 1997).




may also be present. Johansen (1988, 1991), Johansen and Juselius (1990, 1991)

developed the multivariate test for cointegration and the error correction model. TheJohansen procedure is a maximum likelihood estimation of a fully specified error

correction model in transitory form:

Xt = µ + Γ 1Xt −1 + · · · +Γ k−1Xt −k+1 + Π Xt −1 + εt (1)

where Xt exhibits the vector of price changes in period t, µ is a constant vector,

Γ represents the short-run dynamics, and Π is the long-run impact matrix, which

will have reduced rank under cointegration.

The number of stationary linear combinations of Xt , the cointegrating vectors isdetermined by the rank of this matrix. If Π is of intermediate rank, 0 < r(Π) =

r < n, so that r linear combinations of nonstationary variables are stationary, and r

cointegrating vectors, or n − r stochastic trends, exist. Because the matrix does not

have full rank, two n × r matrices α and β can be factored so that Π = αβ, where denotes transposition.

Consequently, we rewrite (1) as

Xt = µ +

k−1i=1

Γ i Xt −i + αβXt −1 + εt . (2)

In this factorization, the r columns of β can be defined as cointegrating vectors,i.e., the linearly independent combinations of Xt that are stationary. α is the matrix

of the error correction terms that shows the impact of r cointegrated vectors on Xt .The i th row of α represents the direction and strength of the adjustment process.

Note that the evidence of the error correction mechanism for tactical asset alloca-tion decisions is quite obvious. If Xt is in a transitory state of disequilibrium, e.g.,

β Xt = 0, we can predict some of the future developments of Xt due to the function

of the error correction mechanism (Lucas 1997).

To determine the rank r of estimated matrix Π , we first calculate the eigenval-ues λi . The number of significantly nonzero eigenvalues shows the rank of the ma-

trix Π , and can be evaluated by the trace test and the maximal eigenvalue test. As

Kasa (1992) points out, the two tests differ in their assumptions about the alternativehypothesis. The trace statistic is the result of testing the restriction r ≤ q (q < n)

against the completely unrestricted model r ≤ n:

λtrace =−T

ni=q+1

ln(1 − λi ) (3)

where T is the sample size and λr+1, . . . , λn are the n − r smallest squared canonical

correlations.

We refer to the second restricted maximum likelihood ratio test as the maximaleigenvalue test statistic. The λmax is found by again testing the null hypothesis of at

most q cointegrating vectors against the alternative of one additional cointegrating

vector (i.e., r ≤ q + 1):

λmax = −T ln(1 − λq+1) (4)




where λ∗1, . . . , λ∗q are the largest squared correlations. The maximal eigenvalue test

clearly produces more straightforward results.5

These technical explanations allow us to establish a connection to the optimal asset

allocation framework. To illustrate, consider an investor with a quadratic risk aversion

utility function, where the conditional mean and variance of portfolio returns overthe investment horizon matters. Further assume that the vector of log asset prices Xt

follows a VAR process of order one:

Xt = µ + Π Xt −1 + εt , with εt ∼ iid (0,Ω). (5)

For Π = 0, we obtain the standard model with iid returns, while Π = αβ in (5)

represents a cointegrated system of asset prices.6 Given a constant holding vector x in

the assets over investment horizon H , the conditional mean and conditional varianceare given, respectively, in (6) and (7):

xE0(XH − X0) =

H −1s=0

x(I + Π )s µ + x

(I + Π )H − I

X0, (6)

xV 0(XH − X0)x =

H −1s=0

x(I + Π )s Ω(I + Π )s x. (7)

When there is no cointegration (Π = 0), we obtain the standard formulations H xµ

and H xΩx for the mean and variance, respectively, from which the usual Markowitz

(1952) mean–variance analysis follows. When the asset prices are cointegrated (Π =

αβ), mean–variance analysis still applies, but with different formulas and different

values for conditional moments, which generate a shift in the mean–variance frontier.

Thus, for asset allocation, the relevance of cointegration depends on the holding pe-

riod of the investment, the portfolio’s rebalancing frequency, and the investor’s risk

appetite.

For strategic asset allocation with a static (no-rebalancing) long-term investment

style, as shown above, we construct a portfolio with weights corresponding to coin-

tegrating vectors in βXt −1. The lower long-term volatility of such portfolios makes

them particularly attractive for highly risk-averse long-term investors. On the other

hand, portfolio managers using tactical asset allocation can anticipate the temporary

deviations of asset prices from long-run equilibrium, and can effectively rebalance

the portfolio using the matrix α .

5To be more precise, if λi are evenly distributed, the trace statistic tends to have greater power than λmax

because it considers the range of all n − q of the smallest eigenvalues. Comparatively, λmax tends to

produce better results when λi are either large or small (Kasa 1992).

6For Π = 0, the eigenvalues of (I +Π ) lie on the unit circle, and the model in (9) reduces to a pure random

walk with drift.




4 Data and empirical results

Our data include monthly total return indexes in US dollars from December 1993to December 2005 (there are 145 end-of-month observations in each series).7 Our

hedge fund strategy indices come from Credit Suisse/Tremont. The other financialasset class indices are represented by the S&P 500 Composite Index, the NASDAQComposite (DS calculated), the Wilshire All Growth, the Wilshire All Value, and theJ.P. Morgan Government Bond Index.8

Using hedge fund indices to proxy for hedge fund strategy performance can causevarious biases in the time series. These biases arise from the lack of regulation anddifficulty in estimating prices inherent in the hedge fund structure. Because hedgefunds are not legally required to report return information publicly, managers decidehow and when to present performance information. This can mean that reported val-ues do not reflect actual performance, however. And price estimation can be difficult

because hedge funds invest in instruments that are often not stock exchange oriented.The most common biases in the literature are survivorship bias, backfilling bias,

selection bias, autocorrelation bias, and the multiperiod bias (Fung and Hsieh 2000).These biases generally result in overestimating expected returns and underestimatingexpected variance, which affects the values of the cointegrating vectors and the errorcorrection model coefficients. The overall finding of cointegration or no cointegra-tion, however, will still hold. Note that the Credit Suisse/Tremont indices, which arenoninvestable, are affected particularly by the survivorship and instant history biases(see Lhabitant 2004). Note also that broad-based hedge fund style indexes generally

understate trading style diversity and overstate any risk of style convergence.

4.1 Descriptive statistics

Over our sample period, most of the indices increase continuously, implying a highrate of return. Table 2 gives the summary statistics for the hedge fund strategies, aswell as for the stock, bond, and index returns.

Note first that the annualized average of continuously compounded returns is thehighest for the global macro and distressed securities strategies (12.70% and 12.61%,respectively). For the risk-adjusted return (e.g., the simple Sharpe ratio defined asthe coefficient of mean and the standard deviation), the J.P. Morgan bond index andequity market neutral strategy significantly outperform the others. Comparatively,only the short-seller strategy has a negative annualized mean return (−2.05%), andhigh volatility (17.06%).

The S&P 500, NASDAQ, and value stocks outperform most of the hedge fundstrategies for both absolute and risk-adjusted returns. The NASDAQ composite is themost volatile asset, with an annualized standard deviation of 26.87%.

7Hakkio and Rush (1991) argue that the length of the time series is more important than the frequency of the information for discerning the presence of cointegration. Shiller and Perron (1985) and Perron (1989)support this observation. They find no empirical evidence that changing the frequency of observationswhile keeping the sample length fixed influences the results of cointegration testing, as it is mainly along-term property.

8We use the S&P 500 as a proxy for the overall market, especially for the large-cap sector (correlationof 0.9953). The NASDAQ proxies for “new economy” firms.




Table 2 Times series statistical properties, January 1994–December 2005

Indices Mean Std. dev. Skew- Excess Sharpe J.B.

(in % p.a.) (in % p.a.) ness kurtosis ratio test

Relative value strategies ln (convertible arbitrage) 8.259 4.771 −1.372 3.090 0.500 102.44a

ln (fixed-income arbitrage) 6.089 3.802 −3.207 17.122 0.462 2,005.8a

ln (equity market neutral) 9.458 2.919 0.309 0.300 0.935 2.828

Event-driven strategies

ln (risk arbitrage) 7.436 4.220 −1.369 6.904 0.509 330.96a

ln (distressed securities) 12.609 6.579 −3.213 21.357 0.553 2,984.6a

Opportunistic strategies

ln (long/short equity) 11.244 2.940 −0.027 3.865 0.319 89.64a

ln (short-sellers) −2.049 17.058 0.610 1.136 −0.035 16.67a

ln (emerging markets) 8.033 16.687 −1.125 6.288 0.139 267.61a

ln (global macro) 12.695 11.033 −0.197 2.882 0.332 50.78a

Stock and bond market

ln (S&P 500 composite) 10.007 14.853 −0.730 0.919 0.674 17.85a

ln (NASDAQ composite) 11.412 26.868 −0.628 1.181 0.425 17.83a

ln (Wilshire growth) 7.800 18.218 −0.885 1.470 0.428 31.78a

ln (Wilshire value) 10.826 14.582 −0.892 2.423 0.742 54.31a

ln (J.P. Morgan bond) 6.018 4.755 −0.588 1.081 1.266 15.32a

Based on monthly continuously compounded total returns for 145 observations. a denotes significance atthe 1% level (rejection of the normal distribution). The Sharpe ratio is defined as the coefficient of mean

and standard deviation without adjustment for the risk-free rate

Note that the emerging market strategy has extremely high volatility compared to

its mean return. However, almost all asset classes exhibit asymmetric return patterns

with negative skewness and positive excess kurtosis, except for equity market neutral.

The Jarque–Bera test shows to what degree returns deviate from a normal distri-

bution. A high value suggests returns do not follow a normal distribution at the 1%significance level. The results again show substantial variation between the different

asset classes, so using standard deviation as a single measure of risk may alter the

actual performance. Moreover, in a portfolio optimization context, standard devia-

tion is an incomplete measure of risk and it may lead to suboptimal asset allocation

decisions.

Finding the optimal portfolio weights in a mean–variance analysis also requires

that return correlations between assets be stable.9 However, correlation analysis is

only valid for stationary variables. We can make most of the financial data stationary

by taking first differences of the prices or by de-trending the variables. Thus, assetprices are integrated of order one, but with the disadvantage that valuable information

9For instance, Lhabitant (2002) found that correlation between most hedge fund indices and between US

and European equity markets is much higher in down markets than in up markets.




Table 3 Contemporaneous correlations between monthly returns January 1994–December 2005

Indices ln (S&P ln (NASDAQ ln (Wilshire ln (Wilshire ln (J.P. Mor-

500 comp.) comp.) growth) value) gan bond)

ln (convertible 0.141 0.140 0.112 0.185 0.035

arbitrage)

ln (fixed-income 0.037 0.037 0.035 0.051 0.079

arbitrage)

ln (equity market 0.371 0.275 0.312 0.362 0.106

neutral)

ln (risk arbitrage) 0.460 0.383 0.394 0.547 −0.108

ln (distressed 0.558 0.487 0.492 0.592 −0.067

securities)

ln (long/short 0.600 0.729 0.684 0.515 0.029

equity)

ln (short-sellers) −0.757 −0.814 −0.806 −0.684 0.127

ln (emerging 0.488 0.496 0.479 0.478 −0.110

markets)

ln (global macro) 0.230 0.160 0.194 0.228 0.217

can be lost because de-trending eliminates any possibility of detecting common pricetrends.

Due to their trading strategies,10 hedge funds are typically nonlinear functions of

traditional markets, so using linear correlation coefficients as a measure of depen-

dence is not reliable (Lhabitant 2004). Using the correlation coefficient as an indi-

cator for dependence among random variables is also problematic, however, because

(1) only linear dependence is measured, and (2) the results are only meaningful if the

multivariate distribution is elliptic (Embrechts et al. 1999, 2002; Kat 2003).

As Table 2 shows, most hedge fund strategies and financial assets have a negative

skewness and/or an excess kurtosis, so the joint distribution is far from being elliptic.Also, if the distribution is not elliptic, the correlation coefficient does not exhaust

the full interval [−1,+1], so it can be much smaller for certain distributions. This

can lead to incorrect findings of very low dependence, even though the variables are

perfectly correlated.11

The cointegration approach is again more suitable because it works directly on

asset prices rather than returns and does not require the assumption of stationarity of

the asset value series.

10These trading strategies lead to complex hedge fund portfolios including nonlinear assets such as op-tions, interest rate derivatives, and so on. Such portfolios exhibit both nonnormality fluctuations of the

underlying assets and nonlinear functions in traditional assets.

11Besides the limitation to two variables, other problems may arise when using the correlation coefficients,

such as spurious correlations and nonresistance against outliers. See Lhabitant (2004) for more about these

problems.




Table 3 shows the correlation matrix between the hedge fund strategies and thevarious US stock and bond segments. Note that only the monthly returns for theshort-sellers strategy are negatively correlated with the US stock market. Further-more, the bond index is negatively correlated with risk arbitrage, distressed secu-

rities, and emerging markets. We expected that emerging markets would be morehighly correlated, particularly with the stock market indices. The long/short equityand distressed securities indices are highly positively correlated with the NASDAQindex.

Despite its problems, correlation and cointegration are related but distinct con-cepts. High correlation does not imply high cointegration, and higher correlation isneither necessary nor sufficient for higher cointegration between assets. In fact, coin-tegrated series can actually have low correlations (Füss and Herrmann 2005).

4.2 Testing for unit roots

Before applying the Johansen cointegration methodology, we test whether the timeseries is integrated to the same order, or whether each series requires the same degreeof differencing to achieve stationarity. As Engle and Granger (1987) discuss, a seriesis said to be integrated of order d , I(d), if the d times differenced series has a sta-tionary invertible ARMA representation. Tests of stationarity are often characterizedas unit roots. If the hedge fund strategy and the financial market indices data exhibita unit root, they are considered integrated, I (1). If asset returns exhibit a randomwalk, temporary shocks in the returns persist over time and do not disappear by re-

verting to the mean. However, such behavior inevitably affects the timing of portfoliorebalancing (Gregoriou et al. 2001).12

The results of the augmented Dickey–Fuller (ADF) tests (Dickey and Fuller 1981,1979; Said and Dickey 1984) reported in Table 4, provide strong evidence that allseries in the levels are nonstationary as suggested by the small values of the ADFstatistics.

However, when we use first differences, the null hypothesis is rejected at the 1%level for all asset classes except for equity market neutral. Accordingly, the cointegra-tion results for this strategy should be interpreted with caution, because cointegration

analysis requires that the variables be stationary, of the same order, and significant atthe 1% level.A constant term or drift parameter is present in the return series of hedge funds, as

well as in the S&P 500, except for emerging markets and short-sellers. The constantterm for these series reflects fluctuations around a mean, which may be the resultof overestimating hedge fund returns. Thus, we conclude that all financial series arenonstationary in levels and stationary in returns. This means that all indices are inte-grated of order one, I (1), which is a necessary condition in testing for cointegration.

Even though the hedge fund strategies and financial asset prices follow a randomwalk, we next investigate how independent the random walk components are. In other

words, cointegration exists when there is a mean reversion in the price spread betweenthe strategy indices and the traditional asset categories.

12Gregoriou et al. (2001) note a finding of nonstationarity by the ADF test does not necessarily implyrandom walk behavior, since random walks are only one example of nonstationarity.




Table 4 Unit root tests of prices and returns

Indices Unit root test in level Xt Unit root test in first difference Xt

ADF ADFc ADFct ADF ADFc ADFct

Relative value strategiesConvertible arbitrage 2.856 (3) −5.416a (2)

Fixed-income arbitrage −2.068 (1) −7.843a (0)

Equity market neutral −2.078 (8) −3.445b (7)


Risk arbitrage −1.753 (5) −4.048a (5)

Distressed securities −2.201 (1) −8.939a (0)


Long/short equity −1.270 (6) −4.756a (5)

Short-sellers −3.45b (1) −10.70a (0)

Emerging markets −2.403 (7) −3.874a (6)

Global macro −2.410 (8) −3.546a (7)

Stock and bond market

S&P 500 comp. −1.915 (0) −12.03a (0)

NASDAQ comp. −2.11 (10) −3.710a (5)

Wilshire growth −2.30 (10) −11.29a (0)

Wilshire value −1.469 (0) −10.90a (0)

J.P. Morgan bond −2.554 (3) −9.399a (1)

All test statistics are augmented Dickey–Fuller t -tests, where ADFct denotes the ADF statistic with trendand constant term, ADFc is the ADF statistic with constant term and no trend, and ADF is the ADF statistic

without constant term or trend. For each time series, the appropriate model is chosen by minimizing theAkaike information criterion (AIC) or the Schwartz criterion (SIC) values. a and b indicate significanceat the 1% and 5% levels (unit root is the null hypothesis) on the basis of the critical values given byMacKinnon (1996). Lag length is the order of the augmentation needed to eliminate any autocorrelationin the residuals of the ADF regression. The lag orders in the ADF equations for each time series aredetermined by the significance of the coefficient for the lagged terms and are in parentheses

4.3 Testing for cointegration

Because we are interested in the diversification effects of hedge fund strategies, we

incorporate the corresponding indices of the three hedge fund styles into a traditionalstock and bond portfolio. We can thus make inferences about tactical and strategic

asset allocation decisions for a mixed-asset portfolio, and determine whether hedge

funds become substitutes for equity or bond allocation.

We test for cointegration between the financial series and the three style categories

by using the Johansen maximum likelihood (ML) procedure (Johansen 1988, 1991;

Johansen and Juselius 1990). As we noted earlier, standard tests for cointegration

such as the trace and maximal eigenvalue tests are biased toward nonrejection of the no-cointegration hypothesis when the data are subjected to structural breaks.13

13According to the CUSUM test, no structural breaks are found for any of the series, but the Chow break-point test indicates significant structural breaks for the S&P 500, NASDAQ, and growth stocks from No-




Table 5 VAR lag order selection

Lag 0 1 2 3 4

Relative value strategies

FPE 5.12E−20 5.34E−30 4.09E−30a

5.52E−30 6.25E−30AIC −21.71 −44.70 −44.98a −44.70 −44.62


FPE 8.27E−17 1.29E−25a 1.32E−25 1.70E−25 2.12E−25

AIC −17.02 −36.26a −35.19 −33.90 −32.67


FPE 4.33E−16 9.19E−26a 1.37E−25 1.57E−25 2.06E−25

AIC −9.84 −32.11a −31.73 −31.63 −31.42

Lag 5 6 7 8


FPE 9.28E−30 1.12E−29 1.55E−29 1.88E−29

AIC −44.30 −44.23 −44.07 −44.10


FPE 2.77E−25 2.67E−25 2.94E−25 3.77E−25

AIC −31.40 −30.46 −29.41 −28.24


FPE 2.87E−25 3.81E−25 4.31E−25 4.38E−25

AIC −31.21 −31.12 −31.27 −31.65

aIndicates lag order selected by the Akaike information criterion (AIC) and the final prediction error (FPE)

However, we decided not to split the time series into subsamples, as the length of the

data series is important for discerning cointegration.

We use the Akaike information and final prediction error criteria to specify the

order of the unrestricted VAR model (see Lütkepohl 1991).

Table 5 shows that both information criteria refer to a VAR model of order k = 2

for a portfolio including relative value strategies (where the lag order is 1 for the

event-driven and opportunistic strategies). Hence, the vector error correction model

(VECM) involves terms in differences k − 1 = 1 and k − 1 = 0, respectively.

According to the ADF test, the distribution of test statistics from the trace and

maximal eigenvalue tests depends on the deterministic components drift and trend in

the system. Due to the time series used, and in accordance with the results of the unit

root tests, we can ignore linear and quadratic data trends. But we do need to decide

whether to include a constant in the cointegration equation. A constant implies that

the mean of the cointegration relationships between the time series differs from zero.

However, we assume that the log prices of the different asset classes are driven by the

vember 1998 to February 2001, and for emerging markets hedge fund strategies from July 1998 to April2001. For this time series, the cointegration tests results may not be reliable because parameter stability

becomes questionable over the whole sample period.




Table 6 Johansen’s maximum likelihood test

H0 Eigenvalues Trace test – λtrace Maximal eigenvalue test – λmax

Estimated 5% critical 1% critical Estimated 5% critical 1% critical

statistics value value statistics value value

Relative value strategies (variables: S&P 500, NASDAQ, growth, value, bonds, convertible arbitrage,

fixed-income arbitrage, and equity market neutral)

r = 0 0.3223 162.35a 141.20 152.32 55.63a 47.99 53.90

r ≤ 1 0.2008 106.72 109.99 119.80 32.05 41.51 47.15

r ≤ 2 0.1693 74.68 82.49 90.45 26.52 36.36 41.00

r ≤ 3 0.1187 48.15 59.46 66.52 18.06 30.04 35.17

r ≤ 4 0.0795 30.09 39.89 45.58 11.84 23.80 28.82

r ≤ 5 0.0696 18.25 24.31 29.75 10.32 17.89 22.99

r ≤ 6 0.0453 7.93 12.53 16.31 6.63 11.44 15.69r ≤ 7 0.0090 1.30 3.84 6.51 1.30 3.84 6.51

Event-driven strategies (variables: S&P 500, NASDAQ, growth, value, bonds, risk arbitrage, distressed

securities)

r = 0 0.4421 177.30a 109.99 119.80 84.03a 41.51 47.15

r ≤ 1 0.2245 93.28a 82.49 90.45 36.61b 36.36 41.00

r ≤ 2 0.1405 56.67 59.46 66.52 21.80 30.04 35.17

r ≤ 3 0.1209 34.87 39.89 45.58 18.56 23.80 28.82

r ≤ 4 0.0669 16.31 24.31 29.75 9.97 17.89 22.99

r ≤ 5 0.0396 6.34 12.53 16.31 5.82 11.44 15.69r ≤ 6 0.0036 0.51 3.84 6.51 0.51 3.84 6.51

Opportunistic strategies (variables: S&P 500, NASDAQ, growth, value, bonds, long/short equity, short-

sellers, emerging markets, global macro)

r = 0 0.3739 228.28a 175.77 187.31 67.43a 53.69 59.78

r ≤ 1 0.2665 160.85a 141.20 152.32 44.64 47.99 53.90

r ≤ 2 0.2277 116.22b 109.99 119.80 37.21 41.51 47.15

r ≤ 3 0.1804 79.00 82.49 90.45 28.65 36.36 41.00

r ≤ 4 0.1332 50.35 59.46 66.52 20.59 30.04 35.17

r ≤ 5 0.0966 29.76 39.89 45.58 14.63 23.80 28.82r ≤ 6 0.0614 15.13 24.31 29.75 9.13 17.89 22.99

r ≤ 7 0.0373 6.00 12.53 16.31 5.47 11.44 15.69

r ≤ 8 0.0037 0.53 3.84 6.51 0.53 3.84 6.51

a and b indicate that the null hypothesis of no cointegration can be rejected at the 1% and 5% significance

levels, respectively. The λtrace and λmax statistics are carried out under the assumption of no deterministictrend (i.e., without a constant in the cointegrating vector). Critical values for the Johansen test come fromOsterwald-Lenum (1992); r refers to the number of cointegrating vectors in the model

same factors and thus exhibit the same long-term evolution. So we consider a model

with no deterministic component.

Table 6 reports the results for the three different strategies. For the relative value

portfolio, the trace and maximal eigenvalue tests indicate the presence of one coin-

tegration vector at the 1% level. For the event-driven portfolio, the λtrace indicates




two long-term equilibriums at the 1% significance level, while the λmax shows two

cointegration equations at the 1% level and two at the 5% level.

For the opportunistic portfolio, the test statistics are significantly higher: The trace

statistic suggests no more than two (three) cointegrating vectors, since we fail to

reject r ≤ 2 (r ≤ 3) at the 1% (5%) level. The maximal eigenvalue test indicates onlyone common stochastic trend at the 1% significance level.

We tested in all cases for the absence of a deterministic trend component by com-

paring the restricted model without a constant with the unrestricted model with a con-

stant. We applied the following likelihood ratio (LR) test statistic (Johansen 1995):

LR = T

ri=1

log

1 − λi

1 − λ∗i

w∼χ 2(r) (8)

where λi and λ∗i are the eigenvalues of the restricted and unrestricted models.The test results signify the correct specifications with respect to the deterministic

component in all three cointegration relationships.

The existence of one or more common stochastic trend(s) does not imply that all

assets are a driving force in the common trend. It is possible that some of the financial

or hedge fund series do not enter the common stochastic trends. Furthermore, as

we discussed earlier, the relevance of diversification benefits depends on the speed

of adjustment toward the common trend (Kasa 1992). Thus, if returns do not react

significantly to common trends, their existence will only slightly affect the benefits

of diversification.However, to analyze the nature of the cointegrating vector and the adjustment

coefficients, we perform a formal LR test. Table 7 gives the results from tests of

restrictions on the composition of the cointegrating vector present in each sample,

and tests of restrictions on the reaction of an asset’s returns to the common trend.

For the event-driven and opportunistic strategies, we set binding restrictions on both

cointegrating vectors simultaneously so that the test statistics are χ 2(2) distributed.

Significant values indicate that the specific asset enters at least one common trend.

Results from the relative value strategies reveal that NASDAQ stocks, bonds, and

the equity market neutral strategy do not share a common trend with the remainingasset prices, and they do not adjust to this cointegrating vector. This implies that risk-

averse investors with a long-term investment horizon can lower their level of risk

even with convertible and/or fixed-income hedge fund strategies in their portfolios.

For tactical asset allocation, however, the fixed-income strategy would provide

diversification benefits in the short term, because it also does not react to the existing

common trend. For the event-driven style, we see that only the distressed securities

strategy shares a highly significant common stochastic trend with growth and high-

tech stocks. The short-term adjustment is merely a result of this strategy. For tactical

and strategic asset allocation, a portfolio consisting of these assets has higher long-

term volatility and should be avoided by risk-averse investors.

In contrast, however, active and passive investors can benefit by adding risk ar-

bitrage hedge funds to a conservative equity/bond portfolio. Interestingly, for both

hedge fund styles, bonds do not share a common trend with stock markets and hedge

fund strategies, and thus offer substantial diversification potentials.




Table 7 Testing entering in and adjustment to the relevant cointegrating vector


βS&P 500 = 0 χ 2(1) = 6.5074b αS&P 500 = 0 χ 2(1) = 5.9494b

βNASDAQ = 0 χ 2(1) = 2.1382 αNASDAQ = 0 −

βgrowth cap. = 0 χ 2(1) = 4.7676b αgrowth cap. = 0 χ 2(1) = 6.5170b

βvalue cap. = 0 χ 2(1) = 6.4080b αvalue cap. = 0 χ 2(1) = 3.6031c

βbonds = 0 χ 2(1) = 0.0371 αbonds = 0 −

βbonvertible arbitrage = 0 χ 2(1) = 6.2720b αconvertible arbitrage = 0 χ 2(1) = 3.8029c

βfixed-income arbitrage = 0 χ 2(1) = 8.3018b αfixed-income arbitrage = 0 χ 2(1) = 2.0557

βequity market-neutral = 0 χ 2(1) = 0.2629 αequity market-neutral = 0 −


βS&P 500 = 0 χ 2(2) = 4.1724 αS&P 500 = 0 −

βNASDAQ = 0 χ 2(2) = 7.8501b αNASDAQ = 0 χ 2(2) = 2.7378

βgrowth cap. = 0 χ 2(2) = 11.6103a αgrowth cap. = 0 χ 2(2) = 3.1818

βvalue cap. = 0 χ 2(2) = 5.4879c αvalue cap. = 0 χ 2(2) = 4.5550c

βbonds = 0 χ 2(2) = 0.7323 αbonds = 0 −

βrisk arbitrage = 0 χ 2(2) = 0.3052 αrisk arbitrage = 0 −

βdistressed securities = 0 χ 2(2) = 12.5490a αdistressed securities = 0 χ 2(2) = 27.0101a


βS&P 500 = 0 χ 2(2) = 8.5930b αS&P 500 = 0 χ 2(2) = 7.9169b

βNASDAQ = 0 χ 2(2) = 1.4908 αNASDAQ = 0 −

βgrowth cap. = 0 χ 2(2) = 4.4413 αgrowth cap. = 0 −

βvalue cap. = 0 χ 2(2) = 9.2109a αvalue cap. = 0 χ 2(2) = 6.7589b

βbonds = 0 χ 2(2) = 6.4098c αbonds = 0 χ 2(2) = 7.1081b

βlong/short equity = 0 χ 2(2) = 6.0034b αlong/short equity = 0 χ 2(2) = 17.2960a

βshort-sellers = 0 χ 2(2) = 3.0202 αshort-sellers = 0 −

βemerging markets = 0 χ 2(2) = 7.2260b αemerging markets = 0 χ 2(2) = 3.1893

βglobal macro = 0 χ 2(2) = 4.9844c αglobal macro = 0 χ 2(2) = 8.9304b

a, b , and c indicate that the null hypothesis (no entering into and no adjustment to the cointegrating vec-

tor(s)) can be rejected at the 1%, 5%, and 10% significance levels, respectively

In an opportunistic portfolio, conservative assets like the S&P 500, value stocks,

and bonds share common trends with most of the strategies, except for short-sellers.

The emerging market hedge funds do not react to these common trends, but long/short

equity does exhibit highly significant adjustment, which again confirms the results

from Table 3’s correlation analysis. This implies that risk-averse investors can lower

long-run volatility by investing according to the cointegrating vectors, while active

managers can benefit from the knowledge of short-term movements in the asset

prices. Investors enhance diversification potential by taking high-tech and growthstocks into account in their tactical and strategic asset allocation.

Overall, it is obvious that nearly all hedge fund strategies share at least one com-

mon trend with certain traditional assets. This information is useful for tactical and

strategic asset allocation, and also for forecasting hedge fund strategy prices.




5 Conclusion

Hedge funds have been considered ideal as a way to diversify more traditional stock

and bond portfolios because of their perceived low correlation with these markets.

This article, however, presents another side of the story. We find that alternative in-vestments may provide diversification benefits, but low correlation is hardly the rea-

son. We note that hedge fund returns are significantly nonnormal, which makes cor-

relation analysis unsuitable. In this context, standard deviation does not fully reveal

the risk structure, and the correlation coefficient might have different limits.

We confirm the existence of cointegration between US stock markets and hedge

fund strategy indices. We used the Johansen cointegration test to examine common

stochastic trends that move groups of hedge fund indices and US stock market indices

to common equilibrium after each terminal shock. Our results can be used within a

framework of tactical decision making and strategic planning. Because cointegration

lowers uncertainty, risk-averse investors should prefer cointegrated assets, and in-

deed, most recent research has concluded that non-cointegrated assets are better for

portfolio diversification.

In contrast, however, we find that risk-averse investors may sometimes prefer coin-

tegrated assets because of lower uncertainty about their movements as a group. And

the built-in error correction mechanism allows active investors to anticipate short-

term price movements more effectively. Our empirical results suggest that the equity

market neutral, risk arbitrage, and short-seller strategies do not enter the cointegrat-

ing vectors. In addition, fixed-income arbitrage and emerging markets do not react

to these common trends in the short term. There also seems to be a higher long-termcodependence between conservative assets (the S&P 500, value stocks, and bonds)

and the relative value and opportunistic strategies. NASDAQ and growth stocks, how-

ever, exhibited long-term relationships with event-driven strategies to a greater extent,

particularly with distressed securities.

Overall, we conclude that the long-term diversification benefits of hedge fund

strategies are much less significant than correlation analysis and portfolio optimiza-

tion techniques have previously suggested. However, certain strategies, combined

with specific stock segments, can offer portfolio managers adequate diversification

potential, especially within a tactical asset allocation framework.

Acknowledgements The authors thank the two anonymous referees for their constructive criticism andtheir helpful suggestions on the manuscript. Any remaining errors are, of course, our own.

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Roland Füss is lecturer at the department of Empirical Research and Economet-rics and assistant professor at the department of Finance and Banking at the Uni-

versity of Freiburg, Germany. He holds a Diploma in Business Administrationfrom the University of Applied Science in Lörrach, a Diploma and a Ph.D. degreein Economics from the University of Freiburg. His research interests are in thefield of applied econometrics, alternative investments as well as international andreal estate finance. Roland Füss has authored numerous articles in finance journalsas well as book chapters.

Dieter G. Kaiser is a Director Alternative Investments at FERI Institutional Ad-

visors GmbH in Bad Homburg, Germany where he is responsible for portfoliomanagement and the selection of event driven and commodity hedge funds. From2003 to 2007 he was responsible for institutional research and business devel-opment at Benchmark Alternative Strategies GmbH in Frankfurt. He has writtennumerous articles on Alternative Investments that have been published in both,

academic and professional journals and is the author and editor of seven books.Dieter G. Kaiser holds a B.A. in Business Administration from the University of

Applied Sciences Offenburg, an M.A. in Banking and Finance from the FrankfurtSchool of Finance and Management, and a Ph.D. in Finance from the Universityof Technology Chemnitz.

the tactical and strategic value of hedge fund

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