Hedge Fund Persistence Performance

Published Date: 02 Nov 2017

Strategy

Emeline Ozhan & Kevin Bernad

Abstract

Recent literature has found some evidence of performance persistence in

hedge funds. This study investigated whether this persistence varies with fund

characteristics over the time period from January 2000 to December 2012. We

confront hedge funds by a classi_cation based on their strategy issued from

a merged sample from the HFR Hedge funds Indexes databases. We use

the benchmarked hedge fund indexes returns against the S&P500 to obtain

relative returns. Our sample is composed of monthly data, representing 154

observations. Our aim is to analyse the serial correlation of these corrected

dataset by running di_erent tests. After a graphical and an autocorrelation

analysis, we run a Runs test and compute the Hurst exponent. These methods

are both particularly relevant in the analysis of _nancial series. Finally, by

comparing the results of these di_erent approaches, we identify which strategy

generates most persistence.

1

Contents

Introduction 2

1 Sample Selection and Data Mining 5

1.1 Alternative Investment Universe . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Hedge Fund Index Data . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.2 Strategy Classi_cation . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Series Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Data Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Testing for Serial Correlation 13

2.1 Autocorrelation test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Autocorrelation Coe_cient & Randomness . . . . . . . . . . . . . 13

2.1.2 Wald{Wolfowitz Test for Randomness . . . . . . . . . . . . . . . . 14

2.2 Hurst Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 The Rescaled Range Method . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Interpreting the Hurst Index Coe_cient . . . . . . . . . . . . . . . 17

3 Empirical Analysis 19

3.1 Fund characteristics and performance persistence . . . . . . . . . . . . . . 19

3.1.1 Autocorrelogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.2 Runs Con_rmation . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Fractal Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Hurst Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.2 Results by Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Conclusion 26

1

Introduction

Past performance is no guarantee of future performance.1 This dis-

claimer is reminded to each investor for almost all _nancial products. It

supposed that investments performances are very di_cult to predict. Hence

a fund manager's performance is not persistent over several time periods.

Nevertheless, that assumption is not obvious. In term of relative returns,

our hypothesis directly implies that any fund is able to extract reliable in-

formation from the market in the long run. This question refers to the

performance persistence analysis: is a fund able to beat the market over

successive time periods? If so, are past results a good estimator of future

results?

One type of fund has aroused many economists' interest for its rep-

utation of over-performer, particularly highlighted after the 2007 market's

collapse of the sub-primes crisis, Hedge funds. The explosive growth of

Hedge Fund during the 1990's resulted in a proli_c literature about their

performances. Hedge funds are mostly de_ned by their freedom from the

regulatory controls based on the Investment Company Act of 1940. This

exibility regardless the _nancial regulation enables them to undertake a

wider range of investment activities, which _nally achieve a better market

timing than classical mutual funds. This study contributes to this literature

in several ways. First, while there has been a signi_cant amount of research

on the predictability of traditional asset classes, there are few about returns

exhibiting by alternative vehicles such as hedge funds. All the same, examin-

ing Hedge Fund performance predictability as done in Martellini's paper[16]

1Legal disclaimer according to the _nancial regulation law, in french context, emanating from AMF.

2

has led to signi_cant results, particularly in terms of bene_ts. Second, due

to the di_culty of obtaining data about hedge funds, there is limited aca-

demic research in this area.

We study the performance of hedge fund and the persistence over a

time window that encompasses relatively long bullish and bearish period.

As the de_nition of hedge funds covers a multitude of investment allocation,

we distinguish four categories of hedge funds according to their investment

strategy. As the importance of _nding performance persistence rests on the

fact that it would enable investors to beat the market average, consistency

of the performance persistence raises thus questions about market e_ciency.

We confront this hypothesis to historical returns of our four strategies. It

consists in testing if these strategies can result in a performance signi_cantly

greater than those of the whole market, on a semi-permanent basis. Con-

sequently, we use benchmarked returns against the S&P500 for our four

strategies to examine their dynamics relatively to the hypothesis of market

e_ciency.

E_ciency theory suggests that any new information is instantaneously

incorporated into prices, making over-performance only a transitory step. It

also implies the non-existence of ine_ciencies that could lead to arbitrage

opportunities. This phenomenon then faces E_ciency property as more a

series is persistent, more it will be possible to establish a lucrative strategy.

The number of Runs test denied weak-form market e_ciency in several study,

M. Borges found signi_cant results over the European Stock Market [1]. In

addition, aware of the _nancial markets reality, which is usually de_ned as

weakly-e_cient according to Fama2, and giving the de_nition of hedge funds,

2Fama (1970) de_nes, in the article E_cient Capital Markets: A Review of Theory and Empirical

Work, three levels of market e_ciency: a strong e_ciency form referring to a market where all the

3

we can supposed that their strategic freedom regardless _nancial regulation

gives them the possibility to take advantages of these imperfections.

Ardian Harri and B. Wade Brorsen[6] have highlighted this mecha-

nism. The consistency of returns is an important issue in the context of

hedge funds because unlike the traditional mutual funds, investment in hedge

funds involves usually a lock-up period. Altogether, this constrains means

that investors need to have su_cient information about the performance

over a long period before betting their money. Moreover, as hedge funds ex-

hibit a much higher attrition rate compared to mutual funds, phenomenon

examines by Brown[2] and Liang[12] who estimated hedge fund annualized

attrition rates in a range from 4.3% to 15%, the issue of performance per-

sistence becomes especially important in this case. Studying the persistence

of hedge fund performance is thus legitimate through above mentioned ele-

ments.

We develop our analysis into three parts. Firstly, we de_ne the al-

ternative environment of hedge fund and present our database. Particularly,

we insist on the chosen categorisation and explain the consistency of our

approach. We also describe the economic environment of the studied period

and explain the data mining employed. Secondly, we present the principal

existing tests of serial correlation. In this part, we use autocorrelation test

as a _rst approach to identify our process dynamics. Then, we present more

speci_c procedures recognized for their adequacy with _nancial series: the

implementation of the Runs test and the calculation of the Hurst exponent.

Thirdly, by using the above mentioned procedures, we present their results

and identify hedge fund strategies performances in terms of persistence.

information is knew, a semi-strong e_ciency form which only implies public information, and a weak

e_ciency form corresponding to the past information.

4

1 Sample Selection and Data Mining

To represent the alternative investment universe, we chose to use data from

HFR Hedge fund indexes. The HFR indexes have been used in a variety

of studies on hedge fund performance and seem to be reliable indicators of

the alternative investment universe. We use a decomposition of the universe

into four representative strategies of the major investment process.

1.1 Alternative Investment Universe

Into our data selection process, we confront several constrains. Hedge Fund

universe is dark and we face the issue of misrepresentation. Particularly,

we chose our database in respect with recent studies about bias existing on

Hedge Fund indexes to ensure the reliability of our tests results.

1.1.1 Hedge Fund Index Data

Hedge Fund data are di_cult to obtain and usually not highly reliable. This

can be explained because of the freedom of this fund to provide or not their

results. This miss of regulation enables hedge fund to present their results

only when it is relevant for their own interest. There is consequently a sig-

ni_cant information bias between that we can obtain and reality. Also, we

can observe a survivor bias in some database. Survivor bias refers to the fact

that over a long time period, the index observed represents only surviving

funds. As the poorest performers are not represented, there is an arti_cial in-

crease of the index performance. Consequently, the question of the attrition

e_ect3 is often raised about Hedge Fund. Carpenter and Lynch[8] exam-

ine the problem of survivor bias and attrition e_ect and conclude that they

could lead to false results, and particularly, to wrong persistence hypothesis

3Attrition E_ect refers to the high turnover in new-launch and close-down activity in the Hedge Fund

industry. This selection bias is thus directly correlated to survivor bias.

5

acceptance. The choice of the data to use is consequently not obvious and

lead to debates.

In the light of these informations we chose to use data from HFR Hedge

fund indexes to represent the alternative investment universe. We think

that HFR indexes respond to the constrain we mention. In the facts, Funds

are re-selected quarterly, as necessary, and in order to minimize the survivor-

ship bias, they are not excluded until they liquidate or fail to meet the _nan-

cial reporting requirements. This makes these indexes representative of the

various hedge fund investment styles and useful for tracking and comparing

hedge fund performance against other major asset classes. In addition, the

HFR indexes have been used in a variety of studies on hedge fund perfor-

mance. They o_er several advantages with respect to their competitors: we

can say that they are transparent both in their calculation and composition,

and constructed in a disciplined and objective manner.

Finally, our sample is composed of the four strategies indexes returns net

of management fees. We use monthly data from January 2000 to December

2012 representing 154 observations.

1.1.2 Strategy Classi_cation

As the de_nition of Hedge Funds covers a multitude of di_erent types of

investment allocation, we distinguish four categories of hedge funds accord-

ing to their investment strategy. To serve this purpose, we use the strategy

classi_cation constructed by HFR. Hedge Fund Research has constructed a

Strategy Classi_cation System representative of the alternative investment

universe. The classi_cations reect the contemporaneous evolution in the

Hedge Fund industry. This classi_cation is in constant evolution to reect

the reality of the industry. The objective is to de_ne strategy which can

6

Figure 1: Hedge Fund Industry Classi_cation

be used to characterize Hedge Fund return. This classi_cation leads to the

construction of reliable quantitative index. The Hedge Fund industry is

thus divided into four strategies: Equity Hedge, Event Driven, Macro and

Relative Value. We de_ne these strategies as follows: The Equity Hedge

strategy refers to investment positions both long and short in equity and

equity derivative securities. The investment processes included are as well

quantitative as fundamental techniques and the investment universe broadly

diversi_ed or narrowly focused on speci_c sectors.

The Event Driven Hedge Fund apply a strategy that attempts to take ad-

vantage of events such as merger and restructuring that can result in the

short-term mispricing of a company's stock.

The Macro strategy is a strategy that bases its holdings, such as long and

short positions in various equity, _xed income, currency, and futures mar-

kets, primarily on overall economic and political views. The investment

process is predicated on movements in underlying economic variables.

7

The Relative Value strategy consists in purchasing a security that is ex-

pected to appreciate, while simultaneously selling short a related security

that is expected to depreciate. Managers employ a variety of fundamen-

tal and quantitative techniques to establish investment theses, and security

types range broadly across equity, _xed income, derivative or other security

types.

1.2 Series Description

We study the performance of hedge fund and the persistence over a time

window that encompasses relatively long bullish and bearish period to dis-

tinguish between strategies the most performing. As performance persis-

tence raises questions about market e_ciency, we confront this hypothesis

to historical returns of our four strategies. Thus, the more direct way to

experience this theory is to use benchmarked returns against the S&P500.

1.2.1 Data Mining

Our sample encompasses data from January 2000 to December 2012. The

hedge Fund industry grown exponentially during the 90's. Therefore we

start our sample in 2000 as the industry was more stable and perennial and

some strategies were not well develop yet before. This period is also interest-

ing because of its macro-economic context. Financial markets experienced

alternation of expansion and recession with the internet bubble in 2000. Be-

tween 2002 and 2007 with the sub-primes crisis the emergent markets grew.

These information will appear helpful for the interpretation of our results.

Speci_cally, we use the indexes of four Hedge Fund market subsections

which the merger represents the entire industry. Then we calculate returns

for each. Bene_t of using returns, versus prices, is normalization. The simple

8

returns are de_ned such that: Ri =

pi ô€€€ pj

pj

Thus we can write:

1 + Ri =

pi

pj

= exp

log

pi

pj

With this property, we can de_ne our log-returns: log(1 + Ri) = log(

pi

pj

)

Using log-returns, we bene_t of properties as the continuously compounded

return is the sum of the continuously compounded single period returns.

We then process by the same way to compute log-returns of the S&P500.

The simple benchmarked returns would be, with Rm the market returns:

Ribenchmarked = Ri ô€€€ Rm

Using the same decomposition, we obtain:

log(1 + Ribenchmarked) = log(

pi

pj

) ô€€€ log(

pmi

pmj

)

1.2.2 Descriptive Statistics

First of all, a brief history of the period studied is necessary. The data set

begins with the technology bubble inated from 1998 to late 2000, which

precipitated the market break down until 2003. Then, there is a 5 consecu-

tive years period of euphoric from 2003 to 2007 that starts with the second

Gulf War and ending in the summer of 2007. From 2007, it is a bearish pe-

riod for overall industry, accelerated by the _nancial crisis and the ensuing

by the global economic crisis. The four historical plot of the log-returns (see

Figure 14 p.30) show well the context we mentioned.

First, we study the log-returns evolution against the S&P500 log-returns.

Graphically speaking, hedge fund indexes behavior is less volatile than mar-

ket (see Figure 17 p.31). During market collapses, the four strategies min-

imize their losses. These _ndings suggest that there is potentially a deter-

mining process behind. Second, the volatility of the data set is evolving

9

and changing from bearish to bullish period. In bearish state, market and

overall industry is more volatile than during bullish state. In conclusion,

the variance of the series seems to change over time and there are second-

order dependence. Nevertheless, we can interpret descriptive statistics table

because as our data set is monthly, the beaks-down are lifted.

Figure 2: Historical Equity Hegde

log returns Vs S&P500

Figure 3: Historical Relative Value

log returns Vs S&P500

Figure 4: Historical Macro log re-

turns Vs S&P500

Figure 5: Historical Event Driven

log returns Vs S&P500

The lecture of the two descriptive statistics table (see Figure 6 p.11)

con_rms that the hedge fund universe is very heterogeneous: some hedge

fund strategies have relatively high volatility compare to others. The Equity

Hedge Strategy seems to be the more volatile. On the other hand, some

other hedge fund strategies have lower volatility, the Relative value strategy

particularly is very stable compare to the S&P500.

10

Figure 6: Descriptive Statistics: Log-Returns

Figure 7: Descriptive Statistics: Benchmarked Log-Returns

Similarly, the distribution analysis indicates that the strategies are less

volatile than market and that, in addition their expected returns are higher.

In term of Mean-Variance analysis these strategies are thus better investment

than the market. Nevertheless, the comparison with the normal law indicates

large negative skewness for Equity Hedge and Event Driven. The Macro

strategy seems the closest of the normal distribution which would indicate

randomness. However, most of the strategies exhibit high kurtosis indicating

extreme loses event. Finally, monthly returns are not normally distributed,

because they are negatively skewed and leptokurtic.

Figure 8: Equity Hegde

Distribution Vs S&P500

Figure 9: Relative Value

Distribution Vs S&P500

Figure 10: Macro Distri-

bution Vs S&P500

Figure 11: Event Driven

Distribution Vs S&P500

11

2 Testing for Serial Correlation

To determine if performance persists, one of the usual procedures employed

is to model current returns on past returns. We proceed in this study in

the same way than Fama and Solnik. Fama and Solnik used the analysis

of autocorrelation to detect dependence relations. Then, to con_rm their

results, they used a Runs test. In addition, we decided to run a third test

consisting in the calculation of the Hurst coe_cient to validate our results.

2.1 Autocorrelation test

Usual tests for e_cient market hypothesis try to highlight independence into

successive price variations. As autocorrelation is the cross-correlation of a

signal with itself, one of these tests consists in estimating the correlation co-

e_cient between returns and lagged returns. In _nance, the autocorrelation

of a series of returns describes the correlation between returns of the process

at di_erent times.

2.1.1 Autocorrelation Coe_cient & Randomness

Some past empirical studies on _nancial series have run autocorrelation anal-

ysis as the Cootner's study[17], but the major part found insigni_cant co-

e_cient. The best known and more reliable one is the Fama's study[5] on

stock prices walk in 1965. He examines daily returns of Dow-Jones equity

from 1957 to 1962. His study does not suggest any past dependency into

price dynamics. The same study has been undertook by Solnik[22] but

this time using European market data. Solnik's study also con_rms the

weakly-e_cient market hypothesis. By running the same procedure, we try

to identify relations of dependence in our returns which would con_rm the

persistence of the strategy's performance.

12

The relation of dependency between times s and t of our series is the au-

tocorrelation. If the auto-covariance function (h) = cov(Xt;Xtô€€€h) is well-

de_ned, the value of _ must lie in the range [ô€€€1; 1], with _ = 1 indicating

perfect correlation and _ = ô€€€1 indicating perfect anti-correlation such that

for the time series X1;X2; :::XT and known moments of _rst and second

order _i and _i at time i we obtain:

(h) =

E [(Xt ô€€€ _T )(Xtô€€€h ô€€€ _T )]

_t_tô€€€h

Thus _, the correlation coe_cient can be computed

_(h) = corr(Xt;Xtô€€€h) =

(h)

(0)

The simple autocorrelogram is therefore the normalized auto-covariance

function. It satis_es similar properties in comparison to . We have:

_(0) = 1;

j_(h)j _ 1;

_ is a symetrical function: 8h _(ô€€€h) = _(h);

_ is a positive function

Finally, the autocorrelation coe_cient is a measure of the dependence of

a time series. Articles [10] from the International Research Journal of Fi-

nance and Economics test the weak-form market e_ciency though a Wald-

Wolfowitz Test for randomness. Thus, a way to test for the signi_cativity of

this coe_cient is to run this procedure.

2.1.2 Wald{Wolfowitz Test for Randomness

The Runs test for randomness is used to test the hypothesis that a time

series is random. Runs test have been used in several studies about pre-

dicting the Directional Movement of Index Returns [11]. A run is a set of

13

sequential values that are either all above or below the mean. To simplify

computations, the data are _rst centered to their mean. To carry out the

test, the total number of runs is computed along with the number of positive

and negative values. A positive run is then a sequence of values greater than

zero while a negative run is a sequence of values less than zero. We can then

test if the number of positive and negative runs is distributed equally in time.

The test statistic is asymptotically normally distributed, for a time series of

returns X(t), we de_ne Y (t);

Y (t) = X(t) ô€€€ _

The centred series of X(t) with _ the mean of X1; :::Xt. Then we can com-

pute R the number of Runs of Y (t) such that the adjusted-centred statistic

Z is computed as follows;

Z =

(R ô€€€ E [R]) q

_(R)2

And with n the number of positive values and m the number of negative

values, we de_ne the expected value and the standard deviation of R:

E [R] =

2nm

n + m

_(R) =

2nm[2nm{n{m]

2 [n + m]2 ô€€€ 1

Then under the null hypothesis:

H0 : Randomness

Z N(0; 1)

2.2 Hurst Exponent

The intuitively fractal nature of _nancial data has lead a number of math-

ematicians to apply the fractal mathematics to _nancial time series. These

14

fractal properties have been well described by Kale & Butar[9] who have

applied the Hurst coe_cient on _nancial series. They explain that as the

Hurst Index coe_cient is a dimensionless estimator for the self-similarity

of a time series thus, we can measure for serial correlation in terms of the

Hurst exponent which is a convenient summary of statistics of persistence

in time-series data.

2.2.1 The Rescaled Range Method

Stochastic processes with long-term dependence have been introduced _rst in

the context of hydrology and geophysics by Hurst[7]. Recently the technique

has been popularized in economics by B. B. Mandelbrot [14]. Due to its

appearance in E. E. Peters works [18][19], it is now popular for analysis

of _nancial markets [15]. In Finance, the Hurst exponent was used as the

measurement of the degree of e_ciency in several studies about prediction

based on weak-form e_cient market hypothesis [4].

Using the Hurst exponent we can identify time series dynamics and applied

to _nancial data, it can be interpreted as a measure for the trendiness. Here,

we use the

R

S

method presented by R.Racine[21] for estimating the Hurst

Index of our benchmarked hedge fund indexes returns. The Hurst exponent

H can be calculated by rescaled range analysis such that:

E

_

R(t)

S(t)

_

= CnH; n ô€€€! +1

where R(t) is de_ned as follows:

R(t) = max(Z1; ::Ztjt = 1; 2:::n) ô€€€ min(Z1; ::Ztjt = 1; 2:::n)

15

with Y (i) the de-trended time series Xi,

Y (i) = X(i) ô€€€ _ for i = 1; 2:::n with _ the mean of X1; :::Xn such that,

Z(t) =

Pt

i=0

Xi for i = 1; 2:::n

and where S(t) is the standard deviation of Xi for i = 1; 2:::n

S(t) =

vuut

1

t

Xn

i=0

(Xi ô€€€ _)2

and C an arbitrary constant term.

Then, by partitioning the time series into k subsets of size j; j = 1; :::

N

k

,we

can compute _(j) = E

_

R(j)

S(j)

_

to obtain the equation:

log(E(_(j)) = log(C) + Hlog(j)

that can, for several subsets, be solved by a least square _t whose the slope

of the obtained regression is the estimator of H.

2.2.2 Interpreting the Hurst Index Coe_cient

This method of implementation is the original one proposed by Harold Hurst.

There exists a wide range of calculation techniques of the Hurst exponent.

Some studies have found a signi_cant bias in this estimation method. Bo

Qian and Khaled Rasheed[20] identify that the R

S approach is sensible to the

window size of the subsets used and to the length of the time series tested.

Notwithstanding their _ndings, the convergence properties of our estimator

are not contradictory with our interest due to the relative low size of our

series.

As we said at the beginning of this part, the notion of market e_ciency

is closely linked to the absence of memory: the e_ciency hypothesis is then

16

associated with the random walk model. The Hurst exponent enables to

de_ne the trend of our series. We can di_erentiate three cases of dynam-

ics type. With H by construction in the range [0; 1]: a Hurst exponent

towards 0 represents a high fractal dimension, thus the process displays

anti-persistence. On the contrary, an exponent towards 1 represents a small

fractal dimension and thereby the process exhibits persistence. By deduc-

tion, the hypothesis of e_cient market is validated for a Hurst exponent

closes to 0,5, corresponding to a White Noise process.

17

3 Empirical Analysis

We run tests on all benchmarked and non-benchmarked strategies. We ex-

pect though each procedure _nd signi_cant and reliable results.

3.1 Fund characteristics and performance persistence

First of all, we present an autocorrelation analysis that gives an intuition

about the dynamics of our series. Then, we except con_rm these results by

running a Runs Test.

3.1.1 Autocorrelogram

The observation of the autocorrelogram enables to detect for serial corre-

lation. We run the procedure for our four series benchmarked and non-

benchmarked (see Figures 33 p.32). The results of this graphical analysis

are helpful in the identi_cation of the process. The analysis of autocor-

relograms does not give signals of dependence for the benchmarked series,

Pearson's coe_cient of autocorrelation are all not signi_cant. Reversely, the

non-benchmarked series exhibit dependence of _rst and second order. These

signals are particularly accentuated for Relative Value and Event Driven

which the Pearson coe_cients are both signi_cant. The correlation coe_-

cients for Relative value are at the _rst order 0.55 and 0.23 at the second

order. The Event Driven coe_cient for the _rst lag is 0.41 and not signi_-

cant at the second lag. Equity Hedge seems also weakly autocorrelated at

_rst order with a coe_cient of 0.24 and _nally Macro autocorrelogram shows

independence with a Pearson coe_cient at the _rst and second order both

not signi_cant. The scatter plot of the four series show graphical evidences

of these relations of dependence.

18

In addition, according to the moments distribution analysis, the series

are probably heteroskedastic. We propose to run the ARIMA procedure on

the Macro strategy which _ts the most to the normal law. (see Figure 38

p.35). This analysis concludes that HFR Macro cannot be model by ARIMA

process and the model poorly explains the autocorrelation. In fact the vari-

ance of the series changes too much. We cannot do inference in this series

4. Finally, both heteroskedasticity and autocorrelation results represent a

legitimation for the implementation of Runs and Hurst procedure which are

_rstly, measures of persistence and most of all, robust to heteroskedasticity.

4We do not present the results of the ARIMA procedure for all data set as the tests performed on

other series draw to the same conclusion.

19

3.1.2 Runs Con_rmation

The Runs Test statistics is a measure of the goodness of _t with a random

distribution such that the p-value of the Runs test describes the probability

to _t with a random distribution. The table shows results of the Bilateral

Runs Test for a level of risk _ = 0:05:

Figure 12: Runs Test Results

First, the results on the benchmarked strategies are all consistent with the

theory that relative returns do not exhibit persistence. We already high-

light randomness though statistical inference. We suppose that the use of

benchmark deletes the volatility of the data set. In conclusion, even if the

benchmarked series perform better than market in terms of mean returns

(see Figure 6 p.11), we cannot say that the series persist.

Reversely, we _nd consistent results about persistence with the non-benchmarked

data set. Indeed, H0 is signi_cantly rejected for the Event Driven Strategy

with a p-value of 0:0009. The same conclusion is given for Equity Hedge

and Relative Value, with respectively p-value of 0:0184 and 0:00094. The

H0 Hypothesis of randomness is rejected for most of data set. There is still

20

non-reject of H0 for the Macro Strategy with a p-value of 0.25. In addition,

the table shows that the Runs statistics for the tests rejecting H0 is negative.

The point is that the bilateral Runs test can as well reject for persistence

than for anti-persistence. In this case, by considering a level of risk _ = 0:05

for the unilateral test:

H0 : _ = 0 Randomness V s Ha : _ > 0 Persistence

The rejecting zone is W = [ZjZ < 1:64].

In conclusion, we can test directly for persistence with the table. Particu-

larly, in the case of the three others strategies Equity Hedge, Event Driven

and Relative Value; the lecture of the results table intuitively shows per-

sistence. For example, while Event Driven p-value is the lowest, equals to

0:0009 attesting of non-randomness, the number of runs is 56 over 153 obser-

vations. As the number of runs is low compare to the length of the sample,

it means that there is a persistence of the performance.

Finally, the test of number of Runs gives strong signals of persistence for

Equity Hedge, Event Driven and Relative Value (see Figure p.37), which

is consistent with our statistical inference. However, these results do not

enable to measure the persistence.

3.2 Fractal Measure

The statistical inference and the Runs test have provided results about per-

sistence performance but no measure of it. The Hurst exponent solves this

issue as it is a measure of persistence.

3.2.1 Hurst Implementation

The R/S statistic is the range of partial sums of deviations of a time series

from its mean, rescaled by its standard deviation. A log-log plot of the R/S

21

statistic versus the number of points of the aggregated series should be a

straight line with the slope being an estimation of the Hurst exponent(see

Code p.50).

In order to back-test our procedure, we run the Hurst exponent computation

over _rstly a simulated random variable and secondly over the S&P500. We

conclude on the reliability of the method, the Random variable Hurst expo-

nent is equals to 0.503 and the S&P500 Hurst exponent is equals to 0.504,

which is consistent with Fama's _ndings about randomness of the market.

Hurst method discloses the following properties of the statistical data:

clustering, persistence, mixing, anti-persistence, presence of periodical or

non-periodical cycles. In our case, by studying the distribution of the series,

we conclude to heteroskedasticity. Dealing with heteroskedasticity usually

appears di_cult and leads to meaningless results. Giving the constrains we

face, the Hurst exponent is a way to test the robustness of our last _ndings.

22

Indeed, the biggest interest of using the Hurst exponent is its robustness to

heteroskedasticity.

3.2.2 Results by Strategy

As we expected, the Hurst exponent of benchmarked series is for each of

them inferior or equals to 0:5 (see p.30-p.47). Most of them exhibit mixing

properties which is consistent with our Runs Test results (see Figure p.37).

The Hurst coe_cient for benchmarked Equity Hedge strategy is 0:30, for

Event Driven 0:41, for Macro 0:36 and for Relative Value 0:43. Reversely, in

the case of the non-benchmarked strategies, most of them exhibit clustering

feature. The most impressive result is the Relative Value Hurst exponent

equals to 0:77. Then, The Event Driven coe_cient is 0:59 and _nally the

Equity Hedge strategy with a H = 0:55. In addition, consistently with our

_ndings for Macro strategy with the Runs Test, the Hurst exponent is infe-

rior to 0:5 equals to 0:42. Nevertheless, we prevent about these results that

there is not statistical inference existing in the Hurst exponent. The inter-

vals di_erentiating persistent and anti-persistent feature is thus arbitrary.

In this case, we consider a conservative interval of 0:01 points of precision.

To sum up, _rst, the Hurst exponent computation con_rms that there is

no persistence in the benchmarked log-returns. The benchmarked returns

represent the surplus of the strategy in comparison to the market. Thus, we

can conclude that the excess returns are not persistent and that the predic-

tion of the gain over the market is impossible. These results mean that the

probability of overperfoming is random and _nally, the hedge Fund industry,

whatever the strategy considered, does not beat the market with predictabil-

ity. Second, Hurst exponent reversely con_rms that there is a persistence

feature of the Hedge Fund strategies. This feature is particularly underlined

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for the Relative Value Strategy. The strategies which exhibit persistence are

thus enough elaborated to detect reliable information from the market. This

is not the case for the Macro strategy. We suppose that as this strategy is

highly dependent of the market, this correlation could be the cause of its

anti-persistence. The confrontation of these statements draws to the close

that investing in Hedge Fund is rational and that past performances are

correlated to future performances. To go further, the modeling of the most

persistent strategy with ARFIMA procedure could be a development of this

study.

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Conclusion

Using data from four HFR hedge fund indexes, we _nd that there is

strong evidence of very signi_cant predictability in hedge fund returns. The

procedures run among the benchmarked and non-benchmarked log-returns

lead to contrary conclusions. We accept the weak-form market e_ciency

for the S&P500 dataset. Thus, the benchmark we use is random. This

fact could explain the results over the benchmarked strategies. Also, we

_nd dependence relations for the strategy Relative Value, Event Driven and

Equity Hedge. However, this relations are not found for the same bench-

marked series. We conclude that the excess returns are random because of

the randomness of the benchmark. Even if the series are autocorrelated, the

subtraction by a random walk lead to an unpredictable process. Thus these

results do not condemn the Fama's hypothesis.

Our overall conclusions con_rm that there is a mid-term dependence

of returns. In aggregate we concluded that the monthly prices do not fol-

lows random walks. Moreover, the rejection of the random walk for monthly

returns does not support a mean-reverting model of indexes process. We

note that these results are closed to Lo & Mackinlay's [13] conclusion, using

the variance ratio on stock market. The investors can take the stream of

bene_ts through arbitrage process from pro_table opportunities across these

markets. Finally, the study shows that it is possible to predict future per-

formance with past performance for the most autocorrelated strategies.

The knowledge of the past then provides information to predict future

returns. We _nd mid-term memory, thus an interesting extension would be

to test for longer term persistence. We also suggest to test for shorter-term

persistence. Several studies show that the persistence tend to increase with

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the decrease of the considering time window. Then, the _rst development

we propose is to deal with heteroskedasticity by decomposing our sample

between bull and bear market period as in the Cappocci, Corhay and Hub-

ner's study [3]. E_ectively, the statistical analysis suggest structural breaks

caused by the economic environment. As the decomposition leads to sta-

tionary process, running a such procedure could enable to model our series.

Our analysis can be further improved in a number of ways. Due to the

fractal properties of the persistent strategies, we can model a self-similar

process using the Hurst coe_cient. These properties come from the Fractal

Brownian motion theory. The _rst class of process to consider is the ARMA

process. It is often called "short memory process" in light of the rapid de-

crease of their autocorrelation function. A second class of models related

to the fractional brownian motion is the ARFIMA model 5 which is more

adapted to longer dependence relation.

Overall, we _nd that the persistence is linked with the investment

objectives of the di_erent hedge fund strategies. E_ectively, among the di-

rectional strategy, Macro does not exhibit substantial dependency while,

among the non-directional strategies, Event Driven, Relative value and Eq-

uity Hegde, we _nd signi_cant autocorrelation. Finally, we believe that these

evidences of predictability for the non-directional strategy considerably im-

prove our understanding of hedge funds. Also, our empirical results provide

the _rst exploration of the characteristics of hedge fund strategies through

non-restrictive procedures, a topic that needs more attention in the _eld of

investment management.

5Fractionnaly AutoRegressive Integrated Moving Average. These constitute a generalization of the

ARIMA (p; d; q) process in which the di_erenciation parameter d = H ô€€€ 0:5 is the fractional di_erence

parameter. The d parameter then explains the behavior of long term of the series, the short term being

explained by Autoregressive and moving average components.

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