Predictability in Hedge Fund Index Returns and Implications for FoF Style Allocation

7/31/2019 Predictability in Hedge Fund Index Returns and Implications for FoF Style Allocation

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Predictability in Hedge Fund Index Returns and

its application in Fund of Hedge Funds style allocation

By

Philippe Pillonel

Laurent Solanet

Masters ThesisNovember 2004

Keywords: Hedge funds. Return predictability. Portfolio optimization.

JEL Classification: G23, G14, G11.

Both Philippe Pillonel and Laurent Solanet are students in Masters in Banking and Finance(MBF) at the Ecole des HEC of the University of Lausanne. We are responsible for any error.

We are grateful to Mr. Nils Tuchschmid for his continuous help and close supervision. Hisinsightful comments and suggestions have laid the foundation for this work.

We would also like to address a special thank to Mr. Franois-Serge Lhabitant, our thesisDirector.


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Predictability in Hedge Fund Index Returns and

its application in Fund of Hedge Funds style allocation

AABBSSTTRRAACCTT

In this paper, we search for evidence in return predictability of hedge fund indexes. We

assume that the expected future returns can be characterized by a factor model, at first linear

single-factor and subsequently multi-factor and non-linear. Based on these forecasts, we

perform different portfolio optimization problems. The performance of these optimum

portfolios is then compared with that of two benchmarks (equally- and buy-and-hold) made

of the same hedge fund indexes. In a first part, we find that evidence of predictability in hedge

fund index return is mainly due to the persistence in hedge fund style performance. Then, in a

second part, we observe that the benefits for a fund of hedge funds manager in performing

tactical style allocation strategies via our predictive models is jeopardized by numerous

operational/investment constraints.


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Introduction ................................................................................................................................4

1 Evidence of Predictability in Hedge Fund Index Returns...................................................51.1 Review of Literature and Background Theory ............................................................. 6

1.2 Data ............................................................................................................................. 10

1.3 Predictive Variables .................................................................................................... 13

1.4 Methodology...............................................................................................................16

1.5 The Predictive Models ................................................................................................ 21

1.5.1 Linear Single-factor Predictive Models ............................................................... 22

1.5.2 Linear Multi-factor Predictive Models.................................................................31

1.5.3 Non-linear Multi-factor Predictive Models..........................................................42

2 A Practical Application: TSA in Fund of Hedge Funds....................................................52

2.1 Operational Constraint: Redemption Notification ...................................................... 53

2.2 Investment Constraint: Turnover and Diversification ................................................56

3 Conclusion.........................................................................................................................59

Appendix A. Hedge Fund Index Data......................................................................................60

A.1 Hedge Fund Classification..........................................................................................60

A.2 Summary Statistics of Hedge Fund Index Returns..................................................... 65

Appendix B. Predictive Variable Data....................................................................................69

B.1 Summary Statistics of Predictive Variables................................................................ 69

Appendix C. Others.................................................................................................................71

C.1 Methodology Scheme ................................................................................................. 71

C.2 Optimal portfolio weights (without any upper weight constraint).............................. 72

C.3 Optimal portfolio weights (with upper weight constraints of 20%) ........................... 73

References ................................................................................................................................74


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Introduction

In this paper, we focus on the predictability of hedge fund index returns and its eventual

application in a fund of hedge funds. There is now a consensus in empirical finance that

expected asset returns are, to some extent, predictable, at least for traditional asset classes.

However, literature on evidence on return predictability of hedge fund is still in its infancy.

Amenc, El Bied and Martellini (2002) were the first to report both statistical and economic

significance of predictability in hedge funds returns.

Like Amenc et al. (2002), we use factor models to find evidence of predictability in various

hedge fund index returns. Given that the true set of predictive variables is virtually unknown,

we extend Amenc et al. (2002) empirical analysis using various forecasting models to analyze

hedge fund index returns predictability and its impact to tactical style allocation (TSA) 1

strategies. We take into account a larger number of predictive variables reflecting the stage of

the economic cycle, the interest rate environment, and the dynamic trading strategies applied

by hedge funds. These variables are able to predict changes in hedge fund index returns. We

finally expand the sample period until December 2003.

In order to provide some evidence of the economic significance of these predictive models,

we analyze their out-of-sample performance in terms of tactical style allocation. Three

portfolio construction models are performed, all based on traditional optimization (i.e. mean-

variance framework).

Traditional portfolio optimization models require forecasts of the portfolio expected returns

and an estimate of their covariance matrix. In this paper, we estimate the expected returns

using different factor models. The difficulty arise when there is no consensus on the most

appropriate factor model, that is why we attempt in this paper to compare an extensive

number of different factor models beginning with the simplest form of the fund returns (linear

single-factor models) and ending with more complex representation (non-linear multi-factor

models).

1 Amenc et al. (2002) introduce the term "tactical style allocation" (TSA) rather than "tactical asset allocation" (TAA) because hedge fundsmay be better regarded as new investment styles than investment classes. Moreover, in this paper, we precisely look at evidence ofpredictability in hedge fund index returns and, accordingly, at its implications to tactical allocation across hedge fundstyles.


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The paper is organized as follows: the first section provides evidence of return predictability

in hedge fund indexes. The second section explores the practical application of our predictive

models, i.e. their uses in tactical style allocation by fund of hedge funds managers. The third

section concludes.

1 Evidence of Predictability in Hedge Fund Index Returns

There are many studies that show that stock returns at time t can be forecasted with

information based at time t-1. For example, Harvey (1989) shows that up to 18% of the

variation in U.S. stock portfolios can be forecasted on a monthly basis. Harvey (1991) finds

similar results with international data [see also Ferson and Harvey (1991a) and (1991b)].More recently, Amenc, El Bied and Martellini (2002) provide strong evidence of

predictability in hedge fund index returns.

To remove any ambiguity, Amenc et al. (2002) and this paper as well focus on evidence of

predictability in hedge fund returns at the index level and not in individualhedge fund returns.

As each index relates more or less to a particular hedge fund investment style, the return

predictability should be applied to hedge fund styles and not to specific hedge funds (seeAppendix A.1 for a description of the different hedge fund styles).


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1.1 Review of Literature and Background Theory

Over the last few decades, there has been a growing interest in the modeling and forecasting

of economic and financial variables, such as GDP, exchange rates, stock prices or returns.

Most of these earlier works used structural models, trying to explain the fluctuations in the

variable under study with some exogenous macroeconomic variables as the explanatory

variables. Lately, with the advancement of time series econometric techniques, many

researchers resort to time series models in their forecasting endeavor. This approach gain

further popularity when data of higher frequency are becoming available from the equity,

foreign exchange and derivatives markets, which is particularly useful to those with short-

term horizons.

Equivalently, Brooks (2002) makes the distinction between two types of forecasting:

Time series forecasting involves trying to forecast the future values of a series

given its previous values and/or previous values of an error term.

Econometric (structural) forecasting relates a dependent variable to one or more

independent variables. Such models often work well in the long run, since a long-run

relationship between variables often arises from no-arbitrage or market efficiency.

Return prediction derived from arbitrage pricing models is an example of the second type. 2

Time series models have been widely applied in forecasting financial time series for several

reasons. The most important reason is that time series models enjoy greater simplicity as

compared to the econometric structural models without loosing their forecastability. In other

words, the forecasting performance of time series models are at least comparable to structural

models disregarding the fact that the former requires minimum information set. Unlike a

structural model, a time series model demands nothing more than the historical records of the

variable under investigation. It is assumed that the movements of a time series are solely

explained in terms of its own past and therefore forecasts can be made by extrapolation of the

past (Harvey, 1993).

2 In this paper, our forecasting model will be derived from arbitrage pricing models, yet we will use autoregressive terms as well. Tthereforewe are dealing with a mix of these both types of forecasting.


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Econometric (structural) forecasting

Forecasting economic variables is a difficult art. There are actually two ways of considering

this task.

1. One approach consists in forecasting returns by first forecasting the values ofeconomic variables (scenarios on the contemporaneous variables).

E(yt| t-1) = Et-1(yt) = 1 + 2 Et-1(x2t) +3 Et-1(x3t) + + kEt-1(xkt)

2. The other approach to forecasting returns is based on anticipating market reactions to

known economic variables (econometric model with lagged variables).

E(yt| t-1) = Et-1(yt) = 1 +2 Et-1(x2,t-1) +3 Et-1(x3,t-1) + + kEt-1(xk,t-1)

= 1 +2x2,t-1 + 3x3,t-1 + +kxk,t-1

Amenc et al. (2003) write that a number of academic studies (e.g., de Bondt and Thaler

[1985], Thomas and Bernard [1989]) suggest that the reaction of market participants to known

variables is easier to predict than financial and economic factors. The performance of timing

decisions based on an econometric model with lagged variables results from a better ability to

process available information, as opposed to privileged access to private information.

1.1.1 Asset Return Predictive Models

For the review of the asset return predictability and its background theory, we extensively

refer to Ferson (2003), who says :

"Virtually all asset pricing models are special cases of the fundamental equation:

Pt= Et{mt+1(Pt+1+Dt+1)} (1)

wherePt is the price of the asset at time tandDt+1is the amount of any dividends,

interest or other payments received at time t + 1. The market-wide random

variable mt+1is the stochastic discount factor (SDF) 1. The prices are obtained by


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discounting the payoffs using the SDF, or multiplying by mt+1, so that the

expected present value of the payoff is equal to the price.

Assuming nonzero prices, Equation (1) is equivalent to:

E(mt+1 Rt+1 1 | t) = 0 (2)

whereRt+1 is theN-vector of primitive asset gross returns and 1 is an N-vector of

ones. The gross returnRi,t+1 is defined as (Pi,t+t +Di,t+1)/ Pi,t. We say that a SDF

prices the assets if Equations (1) and (2) are satisfied. Empirical tests of asset-

pricing models often work directly with Equation (2) and the relevant definition of

mt+1.

Return predictability

Rational expectation implies that the difference between return realizations and

the expectations in the model should be unrelated to the information that the

expectations in the model are conditioned on. For example, Equation (2) says that

the conditional expectation of the product of mt+1 and Ri,t+1 is the constant 1.

Therefore, 1 mt+1Ri,t+1 should not be predictably different from zero using anyinformation available at time t. If we run a regression of 1 mt+1Ri,t+1 on any

lagged variable,Zt, the regression coefficients should be zero.

Conditional asset pricing presumes the existence of some return predictability.

There should be instruments Zt for which E(Rt+1 | Zt) or E(mt+1 | Zt ) vary over

time, in order for the equation E(mt+1Rt+1 1 | Zt ) = 0 to have empirical bite.

Interest in predicting security-market returns is about as old as the securitymarkets themselves. Fama (1970) reviews the early evidence."


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1.1.2 Hedge Fund Return Predictive Models

What about evidence of predictability in hedge fund returns? Is actively managed portfolios

may be as predictable as buy-and-hold portfolios?

Due to the myriad of strategies employed by hedge funds, their highly dynamic and the

extensive use of derivatives and leverage, models for hedge fund returns are inherently

complex.

In a recent paper, Amenc, El Bied, and Martellini (2002) provide evidence of predictability in

hedge fund index returns, and discuss its implications in terms of tactical style allocation

decisions. See the section 1.5.2.1 for a presentation of their methodology.

As for individual hedge fund return predictability, Martin (1999) writes that there are

difficulties in systematically determining and representing the sources of individual fund

returns3. He adds two important remarks.

1. All the evidence can be taken as a justification for the creation of index-based product

designed to efficiently deliver the returns to particular hedge fund styles.

2. The evidence also provides a rationale for the development of models for the dynamic

allocation of capital across hedge fund styles (tactical style allocation - TSA)

The first remark can be related to the recent emergence of investable hedge fund indices by

several hedge fund index providers (CSFB/Tremont, HFR, ). The second remark is also

backed by a multitude of academic papers [see Amenc and Martellini (2001), Agarwal and

Naik (2003), Alexander and Dimitriu (2004) to cite a few of them]. This last remark is also a

rationale for the use of TSA portfolios as an evaluation tool for our numerous predictive

models.

3 If we really want to predict the returns of an individualhedge fund and disregard the previous remark made by Martin, we can proceed asfollows:

1.perform a style analysis of the hedge fund (see Lhabitant, 2004) and2.forecast the returns of this particular hedge fund based on :

the fund's exposures to the investment styles (point 1) the forecasted returns of the investment styles

In brief, the forecasted returns of an individual hedge fund are the weighted averages of the investment styles' forecasted returns with weightsbeing the fund's exposures to the investment styles. We assume there will be no style drift.


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1.2 Data

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

First Boston/Tremont (CSFB/Tremont). The CSFB/Tremont Hedge Fund indexes have been

used in a variety of studies on hedge fund performance and order several advantages with

respect to their competitors:

They are transparent both in their calculation and composition, and constructed in a

disciplined and objective manner. Starting from the TASS+ database, which tracks

over 2,600 US and offshore hedge funds, the indexes only retain hedge funds that have

at 5 least US $10 million under management and provide audited financial statements.

Only about 300 funds pass the screening process.

They are computed on a monthly basis and are currently the industrys only asset

weighted hedge fund indexes.3 Funds are re-selected quarterly, as necessary, and in

order to minimize the survivorship bias, they are not excluded until they liquidate or

fail to meet the financial reporting requirements. This makes these indexes

representative of the various hedge fund investment styles (see in the annex for more

style information) and useful for tracking and comparing hedge fund performance

against other major asset classes.

The CSFB/Tremont sub-indexes were launched in 1999 with data going back to 1994. They

cover nine strategies: convertible arbitrage (HF1), dedicated short bias (HF2), emerging

markets (HF3), equity market neutral (HF4), event driven (HF5), fixed-income arbitrage

(HF6), global macro (HF7), long/short equity (HF8) and managed futures (HF9). See

Appendix A.1 for a description of the hedge fund strategies.

In this paper, we select the nine CSFB/Tremont Hedge Fund Indexes based on monthly data

over the period January 1994 to December 2003. This period includes the Asian and Russian

crisis along with the LTCM debacle and the Internet bubble burst.

For the nine hedge fund indexes and the various risk factors, monthly data are used

throughout, spanning 120months from January 1994 to December 2003. We use arithmeticreturn for all our result.


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As far as the hedge fund indexes are concerned, we denote by NAV tthe net asset value of a

hedge fund at time t. From the funds net asset values, (arithmetic) returns are derived as

follows:

Rt=1-t

1-ttNAV

NAV-NAV

Table 1 reports the nine Tremont Hedge Fund Indexes performances and statistics over this

period. It is followed by a graph showing the evolution of $100 invested in each of these

indexes.

Table 1. Performances and summary statistics of Tremont hedge fund indexesThis table shows the performance and some summary statistics for the nine CSFB/Tremont hedge funds indexesduring the period 1994-2003. The figures come from the monthly return time series and have been annualized.

There is more descriptive statistics about CSFB/Tremont Hedge Fund Indexes in the

Appendix A.2

ANNUAL RETURNCONV.

ARBITRAGE

SHORT

BIAS

EMERG.

MARKETS

EQUITY

MKT.NTRL

EVENT

DRIVEN

FIXED

INC. ARB.

GLOBAL

MACRO

LONG

SHORT

MANAGED

FUTURES

1994 -8,07% 14,91% 12,51% -2,00% 0,75% 0,31% -5,72% -8,10% 11,95%

1995 16,57% -7,35% -16,91% 11,04% 18,34% 12,50% 30,67% 23,03% -7,10%

1996 17,87% -5,48% 34,50% 16,60% 23,06% 15,93% 25,58% 17,12% 11,97%

1997 14,48% 0,42% 26,59% 14,83% 19,96% 9,34% 37,11% 21,46% 3,12%

1998 -4,41% -6,00% -37,66% 13,31% -4,87% -8,16% -3,64% 17,18% 20,64%

1999 16,04% -14,22% 44,82% 15,33% 22,26% 12,11% 5,81% 47,23% -4,69%

2000 25,64% 15,76% -5,52% 14,99% 7,26% 6,29% 11,67% 2,08% 4,24%

2001 14,58% -3,58% 5,84% 9,31% 11,50% 8,04% 18,38% -3,65% 1,90%

2002 4,05% 18,14% 7,36% 7,42% 0,16% 5,75% 14,66% -1,60% 18,33%

2003 12,90% -32,59% 28,75% 7,07% 20,02% 7,97% 17,99% 17,27% 14,13%

Mean 10.49% -3.17% 7.10% 10,65% 11,40% 6,80% 14,49% 12,16% 7,08%

St. Dev. 4,78% 18,02% 17,76% 3,07% 6,04% 3,95% 12,12% 11,00% 12,14%

Skewness -1,57 0,92 -0,58 0,21 -3,46 -3,25 -0,04 0,22 0,03

Kurtosis 4,05 2,15 3,71 0,24 22,98 16,61 1,98 3,35 0,58

% Up Month 81,67% 45,00% 59,17% 84,17% 80,00% 81,67% 70,83% 66,67% 55,83%

Avg Gain 13,23% 23,31% 27,40% 11,09% 14,24% 9,34% 23,64% 20,39% 19,79%

Avg Loss -3,23% -23,33% -20,97% -0,97% -3,56% -2,85% -10,26% -9,16% -12,16%


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The above table and plot confirm that the hedge fund universe is very heterogeneous: some

hedge fund strategies have relatively high volatility (e.g., dedicated short bias, emergingmarkets, global macro, long/short equity and managed futures); they act as return enhancers

and can be used as a substitute for some fraction of the equity holdings in an investors

portfolio. On the other hand, some other hedge fund strategies have lower volatility (e.g.,

convertible arbitrage, equity market neutral, fixed-income arbitrage and event driven); they

can be regarded as a substitute for some fraction of the fixed-income holdings in an investors

portfolio.

The above statistics show also that most hedge funds returns exhibit negative skewness and

positive excess kurtosis; these two characteristics are not welcome for a risk averse investor.


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1.3 Predictive Variables

We consider hedge funds as investments that provide exposure to several types of risk. Some

of these risk factors come from the traditional part (equity, interest rates, credit), but the

main of them come from the non-traditional part (spreads, volatility, markets trends) of

these alternative investments.

We have selected the following risk factors on the basis of previous evidence of their ability

to predict hedge fund returns and/or their natural influence on them.

US equity market (proxied by the return on the S&P500 index)

World equity market (proxied by the return on the MSCI World index ex US)

Emerging equity market (proxied by the return on the MSCI Emerging market index)

Small capitalization equity market (proxied by the return on the Russel 2000 index)

Dividend yield (proxied by the dividend yield on the S&P500 index).

It has been shown to be associated with slow mean reversion in stock returns across several economic

cycles [Keim and Stambaugh (1986), Campbell and Shiller (1998), Fama and French (1998)]. It serves as

a proxy for time variation in the unobservable risk premium since a high dividend yield indicates that

dividends have been discounted at a higher rate.

Equity market volume (proxied by the change in the volume on the NYSE)

Implicit volatility (proxied by changes in the average of intra-month values of the

VIX).

Short term interest rate (proxied by the yield on 3-month T-Bill )

Fama (1981) and Fama and Schwert (1977) show that this variable is negatively correlated with futurestock market returns. It serves as a proxy for expectations of future economic activity.


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Term spread (proxied by the difference between the yield on 3-month T-Bill and 10-

year Treasuries)

Default spread (proxied by the difference between the yield on Moody's long term

BAA bonds and the yield on Moody's long term AAA bonds)

This captures the effect of default premiums, which track long-term business cycle conditions; higher

during recessions, lower during expansions [Fama and French (1998)].

AAA yield (proxied by the yield on Moody's long term AAA bonds)

US bond market (proxied by the return on the Lehman Aggregate Bond index)

US high-yield bond market (proxied by the return on the Merrill Lynch High Yield)

Currency (proxied by the return on the FED trade-weighted US Dollar index)

Gold (proxied by the return on the gold price)

It is a proxy of inflation

Commodities (proxied by the return on the Goldman Sachs Commodity Index GSCI)

Oil (proxied by the return on the refiner acquisition cost of imported crude oil)

It is closely related to short-term business cycles

By using all these variable, our single- and multi-factor models covering equities, bonds,

currencies and commodities provide a good depiction of the different hedge fund styles.

We will also try to take into account the specific characteristics of the hedge fund return

distributions (due to the use of derivatives and dynamic trading strategies) by introducing

non-linear functions of the above predictive variables.

All the data concerning the risk factors (proxies) have been extracted from DataStream,

except for the VIX (CBOE website) and the oil prices data (EIA website).


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Table 2. Summary statistics for the predictive variablesThis table shows some summary statistics for all the predictive variables during the period 1994-2003. Thefigures come from the monthly return time series and have been annualized.

Mean Std. Dev. Skewness Kurtosis

US equity mkt

12.47% 15.84% -0.60 0.29

World equity mkt 5.92% 15.49% -0.50 0.49

Emerging equity mkt 3.15% 23.76% -0.77 1.98

Small cap equity mkt 11.62% 19.74% -0.49 1.01

Dividend yield 1.79% 0.55% 0.70 -0.69

Implied volatility 21.38% 6.37% 0.49 -0.12

Short term interest rate 4.20% 1.68% -0.91 -0.65

Term spread 1.48% 1.14% 0.23 -0.99

Default spread 0.81% 0.23% 1.06 0.01

US bond mkt -0.07% 4.07% -0.45 0.73

High-yield bond mkt 7.33% 7.29% -0.76 3.45

Currency -0.46% 5.44% -0.54 0.17

Gold 1.39% 12.24% 1.28 4.57

Commodities 5.40% 19.37% 0.18 0.31

Oil 12.22% 25.87% -0.01 0.40

See Annex B.1 for more detailed descriptive statistics of the predictive variables and the

correlation matrix.

In comparison to the returns of the hedge fund indexes seen previously, we can say that our

equity-related indexes show in average lower returns and higher volatility. Yet, they are also

characterized with lower negative skewness and lower excess kurtosis, which make their

return distributions more normal.


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1.4 Methodology

Hereafter are the different steps of our methodology:

Step 1: For each hedge fund index, we select a sub list of explanatory variables or risk

factors. This variable selection is done on the entire sample period (1994-2003) among the

predictive variables described in previous section and through one of these criteria:

statistic model: we consider all combination of these variables (k variables give 2k

combinations) and select the one that allows for a good quality of fit and robustness.

The explanation power is measured in terms of (in-sample) R-squared4 for single

factor models or (in-sample) adjusted R-squared5 for multi-factor models. The

robustness is measured by the (in-sample) Chow test6.

economic model: we consider macroeconomic, financial variables and dynamic

trading strategies that are known to have a influence or a predictive power on the 9

hedge fund indices.

Step 2: For each hedge fund index, we analyse several models (starting with linear and

continuing to more and more non-linear models) based on risk factors (starting with one risk

factor and continuing with several risk factors). We construct the linear specifications for

predicting hedge fund returns that is implied by the different popular asset pricing models.

The forecasts generated allow us to study the ability of different asset pricing models to

explain the predictable variation in returns.

Step 3: Out-of-sample forecasting: the previous selected models are used to forecast the

future returns of the respective hedge fund indexes. This is done by dynamically calibrating

them using a rolling window of 60 data to predict the 61 st and the following 60 returns of the

respective hedge fund indexes. We thus generate 60 out-of-sample forecasts.

A window of 60 data

t1 t60 t61t59t2

Yt

Xt-1

We predict

return in t61(Y61)

4 R-squared: is defined in terms of variance about the mean of Ytso that if a model is reparameterised and the dependent variable changes, Rwill change. R take value between 0 and 1: 0 means the model have no explanatory power, the closest to 1 the higher explanatory power. Rwill always be at least as high for regression with more regressors [Brooks (2002)].5 Adjusted R-squared: in order to get around the R problem with more regressors, a modification is often made which takes into account theloss of degrees of freedom associated with adding extra variables [Brooks (2002)].6 A Chow test consists of dividing the sample into two part of same duration and computing the error sum of squared residuals for each part.Then a Chow statistic is obtained based on the restricted error sum of squares to test the null hypothesis that there is no structural changeusing the F-distribution tables [Chow (1960)].


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Step 4: We perform a Tactical Style Allocation based on the forecasts of our hedge fund

index returns (step 3). We use as variances and covariances matrix input the unconditional

one (i.e. estimated on the 60-month rolling window range). Practically, we solve 3 different

portfolio optimisation problems in order to obtain the allocation (weights) that should be done

across the hedge fund styles. Here are our three optimization programs (optimizers):

P1: Maximization of the expected return of the portfolio subject to a variance

constraint (mean-variance approach)

)(1

...,,1pt

wwREMax

n subject to BenchBenchpp RVarRVar === )()(

and =

=n

ii

w1

1 and 0 wi1

P2: Maximization of the Information Ratio, i.e. the excess return of the portfolio

with respect to a benchmark per unit of tracking error

)(

)(1...,,1

Benchp

Benchpt

ww RRVar

RREMax

n

=IRp subject to

=

=n

ii

w1

1 and 0 wi1

P3: Maximization of the excess return of the portfolio with respect to a

benchmark, subject to a tracking error constraint of 2%

)(1...,,1

Benchptww

RREMaxn

subject to TERRVar Benchp )(

and =

=n

i

iw1

1 and 0 wi1

where Ri is the return of the ith hedge fund index return

wi is the amount invested in the ith hedge fund index, no short

selling allowed

Rp = =

n

i

ii Rw1

is the return of a portfolio invested in each of the

nine hedge fund indexesp is the standard deviation of a portfolio invested in the nine

hedge fund indexes

RBench is the return of a benchmark portfolio defined as an equally-weighted portfolio invested in the nine hedge fund index return

Bench is the standard deviation of a benchmark portfolio

Note that we use only mean-variance optimizers in this paper. Another measure of risk would

have been more appropriate, especially as we deal with hedge fund returns. However, these


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other measures are either not coherent (Value-at-Risk) or our sample size is too small to give

robust results (conditional VaR).

Once these optimizations performed, we end up with several optimal TSA portfolios, each of

them resulting from the joint use of a particular predictive model (i.e. a particular set of

forecasted returns) and a particular portfolio optimizer.

For each of optimizer, we use only the

historical covariance as forecasting

covariance model. Indeed, we expect

covariance to be more stable in the case of

hedge funds compared to unmanaged assets

(see Figure 1),

This stability comes from funds manager who

typically have a target level of volatility:

dynamic hedge fund strategies adapt to

changes in the market environment.

For example, Fixed Income Arbitrage

managers will reduce leverage when equity

index volatility goes up, since equity index

volatility is usually associated with spread

widening in credit markets.

Figure 1. Convertible Arbitrage correlation and

covariance with respect to other CSFB Hedge

Fund indices (rolling windows of 36 months)

Step 5: In order to provide some evidence of the economic significance of these predictive

models, we compare the optimum TSA portfolio obtained in Step 4 (4 optimal portfolios, one

for each optimization problem) with four benchmarks:

Benchmark 1: an equally-weighted benchmark made of the 9 hedge fund indices i.e.

each month you re-allocate it in order to have the same value (1/9) invested in each

Hedge Fund index.

Benchmark 2: a buy-and-hold benchmark made of the 9 hedge fund indices i.e. you

invest 1/9 in each of the Hedge Fund index and do not re-allocate it any more the

following periods.

Benchmark 3: a perfect timer, i.e. each month t, you allocate 100% in the Hedge

Fund index that will best perform at t+1. Benchmark 3 returns are the maximum

returns you can attend from Hedge Fund indices allocation.


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Benchmark 4: a global minimum variance portfolio. We use the covariances matrix

based on the historical observations of the entire rolling windows (60 months).

)(...,,1

pww

RVarMinn

= p subject to =

=n

i

iw1

1 and 0 wi 1

None of these benchmarks use a predictive model.

Step 6: We evaluate the performance of our different TSA portfolios with our benchmarks. In

doing so, we test the economic significance of our predictive models. In practice, the

performance of a tactical asset (or style) allocation strategy is always measured against a

(strategic) benchmark portfolio. Similar to the Sharpe ratio, which takes both total return and

total risk into account, one typical measure in comparing the performance of different tactical

style allocation strategies is the information ratio (see below). Tactical asset/style allocation

strategies are strategies that attempt to deliver a positive information ratio by systematic

asset/style allocation shifts. So, the higher the information ratio, the better.

The following is a description of the statistics we used:

1. Annualized Mean is the arithmetic mean return of the portfolio invested in the

hedge fund indices.

121

1

9

1,,

=

= =

obsnb

t ititi

obs

pRw

nbR

2. Annualized Standard Deviation is the average dispersion of return of the

portfolio around the mean. Mathematically speaking it is calculated as the squared

root of the average squared deviation from the mean.

p = 12)(1

11

2,

=

obsnb

tptp

obs

RRnb

3. Tracking Error evaluates the performance of our portfolio against a benchmark

portfolio. Mathematically speaking it is the standard deviation of the returns

difference between our portfolio and the benchmark.

TEp = =

obsnb

t

benchmark

t

portfolio

t

obs

RR

nb1

2)(

1

1


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4. Information Ratio compares the excess return with its tracking error.

IRp =p

Benchp

Benchp

Benchp

TE

RR

RRVar

RR =

)(

5. Percentage Up is number of time the strategy had a positive return. This gives an

intuitive idea if the strategy is often right or not (right side of the distribution).

6. Average Gain is arithmetic mean of the periods with a gain. This gives an

intuitive idea of intensity of gain when the strategy is right (size of the right tail of

distribution).

7. Average Loss is arithmetic mean of the periods with a loss. This gives an intuitive

idea of intensity of loss when the strategy is wrong (size of the left tail of

distribution).

8. Hit Ratio which is the percentage of time that the return on the tactical style

allocation portfolio is greater than the return on the benchmark

9. Drawdown is the maximum uninterrupted decline in net asset value (in percentage

terms). The drawdown gives an intuitive idea of severity of loss7.

Nevertheless, it should be used with caution. For example, Drawdown will give a

poor interpretation if you have big and frequent interrupted losses. That the reason

why this performance measure must be used in complement of previous ones.

10.Annualized Turnover is the number of time the portfolio change (in percentage

terms)

Turnover = 12)0,(1

1

1

=

obsnb

t

tt

obs

wwMaxnb

7 For a comprehensive description of all these statistics and many more, see Lhabitant (2004)


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1.5 The Predictive Models

The development of a parsimonious factor model that adequately explains, or in our case

predicts, hedge fund returns is a great challenge. Since there is no consensus on the best

model we estimate different types of predictive models in order to forecast the expected future

returns of the hedge fund indexes. They are the predictive versions of popular asset pricing

models (conditional asset pricing models).

Each of our models are estimated by least square regressions over the period Jan-94 to Dec-

2003 on each of the 9 CSFB/Tremont hedge fund indexs returns.

In the following subsections, we present the models from the simplest form (linear single-

factor model) to a more complex representation (non-linear multi-factor model). Each

description of the predictive model is directly followed by an evaluation of their out-of-

sample performances in terms of tactical style allocation.


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1.5.1 Linear Single-factor Predictive Models

First of all we analyze linear single factor models. Usually used as the simplest asset-pricing

model. A linear single factor model is nothing but a linear regression; with the dependent

variable given by the monthly asset return (in our case, a hedge fund index return) and the

independent variable is chosen to represent the market, in the classical asset pricing context;

in our case, the independent variable is chosen to represent whatever could be the most

influent risk factor with respect to the returns in the hedge fund indexes.

This approach is justified by the fact that some hedge funds strategies have returns that look

linearly related to returns of the market.

For example, we found that these returns are positively linearly related to S&P 500 returns in

the case of Equity Long/Short and Event-Driven strategies and negatively related to S&P 500

returns for Dedicated Short Bias strategy.

Figure 2. Relation between returns of hedge funds strategies and returns of the market.

The performance of the Short bias strategy (in % permonth) vs. US equities (S&P 500, in % per month)

The performance of the Event driven strategy (in %per month) vs. US equities (S&P 500, in % per month)

The performance of the Long/short strategy (in % permonth) vs. US equities (S&P 500, in % per month)

We also noticed that some hedge fund indexes display positive serial correlation or

persistence (see Figure 3 next page and Appendix A.2 Statistical Summary Table). This is the

case for Convertible Arbitrage (1st and 2nd lag), Emerging Market (1st lag), Equity Market

Neutral (1st lag), Event-Driven (1st lag) and Fixed Income Arbitrage (1st lag) styles.


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Figure 3. CSFB/Hedge Fund indexes serial correlation


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In the next three sections, we study the predictive power of linear single factor models with:

each of the risk factors we have originally selected (see section 1.3)

the lagged return of the hedge fund index itself as a predictive variable.

1.5.1.1 Single-factor model

We remember the risk factors we are interested in: S&P 500, MSCI ex US, MSCI Emerging

markets, NYSE Volume, term spread, T-bill 3 months, AAA yield, credit spread, Oil, GSCI,

Gold, Currency, Frank Russel 2000 and VIX.

For each hedge fund index and each of 14 factors, we search for a relationship of the type:

Yt= + Xt-1 + t

where

Yt is the explained variable, given by the hedge fund index return at time t,

Xt-1 is the explanatory variables given by the risk factor return at time t-1,

t is the error term at time t.

In order to estimate the coefficients and , we use the Ordinary Lest Squares (OLS) method,

which attempts to minimize the sum of the squared errors.

We chose a confidence interval of 5%: we select models with a R of the regression bigger

than 5%. In Table 3, we present all the results for the period January 1994 to December 2003:

in the first column the risk factor, each column corresponds to a hedge fund index. For each

selected models we measure the quality of the regression:

its explanatory power with the R,

the significance of the coefficients with the p-value (smaller than 0.05 means thecorresponding coefficient is statistically different from 0). It gives the probability that

the t-static exceeds the observed sample under the null hypothesis of a zero population

value.

For each strategy, we highlight the highest R.


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The figures in Table 3 do confirm that for the strategies such as:

Emerging Markets, the prevailing factor is the MSCI Emerging Market index which

alone explains about 10% of the one-month ahead future returns

Event-Driven, the S&P500 index explains about 7.5% of the future returns.

On the other hand, these figures are surprising when it comes to predict strategies such as

Convertible Arbitrage, Dedicated Short Bias or Long/short. Indeed,

Convertible Arbitrage expected returns are explained by both oil and commodity

prices for about 8% each.

Dedicated Short Bias and Long/short strategies do not have any significant single

factor despite their known equity market exposures8

In all these cases, the coefficients are statistically different from 0.

The results of Table 3 presage that the search of lagged relationships (predictive models)

between the returns of hedge fund indexes and the returns of predictive variables will be a

more difficult task than the search of contemporaneous relationships (explanatory models). A

predictive/forecasting model has to capture the dynamics (cycle, persistence, ) of hedgefund index returns, whereas an explanatory model has only to describe the current level of

hedge fund index returns.

The next step of our work is to use these results in forecasting. We proceed as follows:

1. We first select the best risk factor for each hedge fund index.

2. Then, with a rolling window of 60 months, we do an OLS for each hedge fund index

and its corresponding chosen risk factor to get the best predictive models3. We finally use the latter to construct our optimal TSA portfolios and evaluate them by

comparing their performances in term of information ratio with those of our two

benchmark portfolios.

The results are shown and commented in section 1.5.1.4.

8 We noted, however, significant exposures to the equity market indexes when contemporaneous returns were used.


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Table 3.

YtXt-1

CONV.

ARBITRAGESHORT

BIASEMERG.

MARKETS

EQUITY

MKT.

NTRL

EVENT

DRIVEN

FIXED

INC.

ARB

GLOBAL

MACROLONG

SHORTMANAGED

FUTURES

S&P 500 0.0082(0.0000)

0.1085(0.0016)

R 0.0750

Oil -0.0362(0.0063)

0.0151(0.0007)

R 0.0868

VIX 0.0064(0.0000)

-0.0367(0.0039)

R 0.0622

Volume

MSCI

ex. US

-0.0155(0.0172)

0.0001(0.0004)

R 0.0792

Franck

Russel

0.0071(0.0286)

2000 -0.001(0.0076)

R 0.0521

T-Bill3 months

Term

spread

0.0124(0.0000)

-0.0025(0.0007)

R 0.0880

AAA

Credit

Spread

MSCI

Emerging

0.0070

(0.1228)Market 0.005

(0.0005)

R 0.0918

GSCI 0.0083(0.0000)

-0.0386(0.0410)

0.1255(0.0014)

0.0001(0.0060)

R 0.0775 0.0536

Gold

Currency


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1.5.1.2 Single-factor model with own lagged-return

For each hedge fund index we search for a relationship of the type:

Yt= + Yt-1 + t

where

Yt is the explained variable given by the hedge fund index return at time t,

Yt-1 is the explanatory variables given by the hedge fund index return at time t-1,


The reason is that some hedge fund indexes exhibit significant serial correlation. (see

Figure 3). Indeed, managers can smooth portfolio returns when marking illiquid securities

using historical prices or whatever they think it is reasonable. This is particularly true forDistressed Securities strategies where the instruments bought are by nature very illiquid.

Another cause is that some strategies generate constant stream of cash flows, like coupon

payment on bonds or interest earned on the short sale rebate. This is the case for Convertible

Arbitrage and Fixed Income Arbitrage strategies. See Appendix A.2 for more on the smooth

characteristic of hedge fund index return time series.

In order to approximate the coefficients and , we use the Ordinary Lest Squares (OLS)

method, which attempts to minimize the sum of the squared errors.

Table 4.This table shows all the results for the period January 1994 to December 2003: the estimated coefficients and in

brackets their p-value, in all cases with a R bigger than 5%.

Yt

Yt-1

CONV.

ARBITRAGE

DEAD

SHORT

BIAS

EMERG.

MARKETS

EQUITY

MKT.

NTRL

EVENT

DRIVEN

FIXED

INC.

ARB

GLOBAL

MACRO

LONG

SHORT

MANAGED

FUTURES

0.0038(0.0027)

0.0042(0.3520)

0.0061(0.0000)

0.0058(0.0008)

0.0033(0.0029)

0.5525(0.0000)

0.3004(0.0007)

0.2937(0.0010)

0.3460(0.0001)

0.4044(0.0000)

R 0.2996 0.0852 0.0803 0.1147 0.1570

Our results confirm our previous remark that some strategies are positively serially correlated.

In all these cases, the coefficients are statistically different from 0.


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Like before, the next step of our work is to use these results in forecasting:

With a rolling window of 60 months, we do an OLS for each hedge fund index and its own

lagged return in order to get the best predictive models for the expected hedge fund index

returns. We then use them to construct our optimal TSA portfolios. We finally evaluate themby comparing their performances with those of our two benchmark portfolios. In the case of

the 4 indexes for which no predictive model could have been calibrated, we simply use the

unconditional expected return as a forecast of the expected return. This allows regressing the

return on these indexes on a constant variable (equivalent to taking the mean).


1.5.1.3 Single-factor model with own lagged-return (moving average)

We now look if there is more predictive power in a moving average of their own lagged

returns than just in a one-period lag return. Therefore, for each hedge fund index we search

for a relationship of the type:

Yt= + )YY(Yn

1n-t2-t1-t +++ + = +

=

n

i

itYn 1

1+ t

whereYt is the explained variable given by the hedge fund index return at time t,

Yt-n is the explanatory variables given by a moving average of lagged hedge fund index

returns,


The next step of our work is to use these results in forecasting:

With a rolling window of 60 months, we regress each hedge fund index on a moving average

of its lagged returns (3-month moving average, 12-month and historical mean of the rolling

window) in order to get the best predictive models for the expected hedge fund index returns.

We then use them to construct our optimal TSA portfolios. We finally evaluate them by

comparing their performances with those of our two benchmark portfolios. In the case of the 3

indexes for which no forecasting model could be calibrated, we simply use the unconditional

expected return as a forecast of the expected return.



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1.5.1.4 Performance of the linear single-factor predictive models

This table gives the TSA portfolios performances resulting from the use of the previous linear single-fac

expected future returns. The performances are given for a 60-month holding period starting in January 2000 a

Table 5. Linear single-factor model performance

AverageModel \ Annualized (in%) P

1P2

P3

ReturnStandard

Deviation

Track

Error1

Info.

Ratio1

%Up Gain LossHit

Ratio1

D

D

Benchmark 1 (Equally) 9.72 3.29 0.00 n/a 85 0.89 -0.08 0 2

Benchmark 2 (Buy-and-hold) 9.91 3.49 0.25 0.75 81 0.91 -0.08 59 2

Benchmark 3 (Perfect timer) 57.63 10.17 4.65 10.31 100 4.80 0.00 100 0

Benchmark 4 (Minimum variance) 8.97 1.79 0.73 -1.03 95 0.76 -0.01 49 0

Linear model X 11.54 3.87 1.14 1.60 85 1.05 -0.09 61 2

with one regressor X 11.60 3.99 1.13 1.66 85 1.07 -0.10 56 3

X10.83 6.67 1.64 0.68 66 1.20 -0.30 47 5

Linear model X 13.32 6.53 1.51 2.38 81 1.35 -0.24 53 3

with lagged return X 19.51 11.55 2.97 3.29 81 2.02 -0.39 63 6

X 14.56 9.58 2.28 2.12 73 1.70 -0.49 59 5

Linear model X 14.16 6.38 1.44 3.09 86 1.40 -0.22 69 4

with lagged return X 16.73 10.43 2.66 2.64 81 1.79 -0.39 66 6(3-month moving average) X 16.00 8.82 2.06 3.05 83 1.70 -0.37 63 5

Linear model X 11.69 6.57 1.27 1.55 78 1.30 -0.33 63 5

with lagged return X 9.64 11.20 2.66 -0.03 76 1.51 -0.70 63 1(12-month moving average) X 13.97 9.64 2.10 2.03 73 1.73 -0.56 61 9

Linear model X 9.33 4.39 0.97 -0.40 73 0.90 -0.12 39 2

with lagged return X 11.05 3.68 1.15 1.16 85 1.02 -0.10 56 3(60-month moving average) X 9.57 8.64 1.90 -0.08 59 1.19 -0.39 39 8

1. with respect to the Benchmark 1 (equally-weighted)

Note: This table is computed with OLS_TREMONT_RF2_V12.M program.


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The above table shows that all the linear single-factor models, except for the one with the

moving average on the entire window, outperform clearly our benchmarks with respect to the

information ratio.

For the linear single-factor models, we note that with respect to the information ratio, the

model with lagged hedge fund return has more predictive power than regression on lagged

risk factor returns. This again confirms that some strategies exhibit persistence.

We see that the 3-month moving average model is the best of the moving average models

with respect to the information ratio. Furthermore, this 3-month moving average model

clearly outperforms all other linear single-factor models (IR between 2,64 and 3,09). This

nave model has a very good predictive power.

As we stated previously, some hedge fund managers have to some extent the ability to

smooth their portfolio returns. Moreover, some strategies generate constant stream of cash

flows (carry). These good results are therefore not so surprising. However, it is very difficult

to estimate which part of these results are due to those practices?

As far as the turnover is concerned, it is well known that the allocation of an optimal portfolio

is very sensitive to inputs and more particularly to estimates of expected future returns.

Although efficient frontiers that are based only on recent data will reflect current market

conditions more accurately, optimal portfolios will not stay optimal for very long and will

require constant rebalancing. This fact is confirmed by our results. Indeed, the longer the

moving average, the lower the turnover. Still, the turnover for these 3 programs is very high.

The P1 program is in average better than P2 and P3.

The diversification of our portfolios is very poor as well. See Appendix C.2

In brief, we note that the 3-month moving average model is the best predictor in term of

information ratio. That does not appear to be coherent with respect to the Figure 3 on

CSFB/Hedge Fund indexes serial correlation. Yet, in term of predictive power, it turns out

that the 3-month moving average model is the best model to capture the smooth

characteristics of hedge fund index returns. Unfortunately, this model generate quite highturnover, ranging from 400 to 700% annually.


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1.5.2 Linear Multi-factor Predictive Models

The previous section concentrated on linear single-factor models. We go now one step further

to linear multi-factor models. This new approach is justified by the fact that some hedge funds

strategies have returns that are linearly related to more than one risk factor (Figure 4, 5 and 6).

We see that these returns are linearly related to S&P 500, MSCI ex. US and Russel 2000

index returns.

Then we try to capture the different preferred habitats of hedge funds with different

location factors (stocks, bonds, commodities or currencies, domestic or foreign) suggest by

Fung and Hsieh (1998a, 1998b). These location factors are typically linearly related to

conventional asset classes.

Figure 4. Hedge fund index returns versus S&P 500 index returnsThe bars correspond to the market factor monthly returns and the lines to the hedge fund index monthly returns.

The performance of the Short Biasedstrategy versus S&P500.

The performance of the Event Drivenstrategy versus S&P500.

The performance of the Long/Shortstrategy versus S&P500.

Figure 5. Hedge fund index returns versus MSCI ex. US index returns

The bars correspond to the market factor monthly returns and the lines to the hedge fund indexmonthly returns.

The performance of the Short Biasedstrategy versus MSCI ex. US.

The performance of the Event Drivenstrategy versus MSCI ex. US.

The performance of the long/Shortstrategy versus MSCI ex. US.


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Figure 6. Hedge fund index returns versus Russel 2000 index returnsThe bars correspond to the market factor monthly returns and the lines to the hedge fund index monthly returns.

The performance of the Short Biasedstrategy versus RUSSEL 2000.

The performance of the Event Drivenstrategy versus RUSSEL 2000.

The performance of the Long/Shortstrategy vs. RUSSEL 2000.

Based on these figures, in next sections we study:

Statistic multi-factor models suggest by Amenc, Bied and Martellini (2002)

Economic multi-factor models.

As we did in the previous section about single-factor linear models, we use our results in

order to compare performance with respect to our two benchmarks.

1.5.2.1 A statistic multi-factor model: Amenc, Bied and Martellini Model

In this section we use a method developed by Amenc, Bied and Martellini (2002). They built

a model for predicting CSFB/Tremont index return over the period January 1994 to December

2000. In order to be able to compare their results with ours, we replicate these models for the

period January 1994 to December 2003.

They chose 10 risk factors: T-Bill 3-month yield, Dividend yield, Default spread, Term

spread, Implicit volatility (VIX), Market volume (NYSE), Oil price, US equity factor (S&P

500), World equity factor (MSCI World Index ex US), Currency factor.

From these risk factors, they selected a subset of variables that allows a good trade-off

between quality of fit (allowed for at least 5 percent in-sample explanatory power) and

robustness (Chow Statistic test). This based on the explanatory power of the: raw variables : Xi,t,

change in variables : Xi,t - Xi,t-1,

return of raw variables: (Xi,t/Xi,t-1)-1,

one month lag Xi,t-1, two months lag Xi,t-2, three months lag Xi,t-3,

moving average (Xi,t-1+ Xi,t-2+Xi,t-3) 1/3.


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They have found the following set of 6 factors that most closely predict the return on hedge

fund indexes:

The moving average of the return on the S&P 500 over the previous 3 months,

denoted as MA(S&P) t-1

Crude oil price, denoted as Oil t-1

Changes in the 3-month Treasury bill rate, denoted as 3m t-1

Changes in the VIX index, denoted as VIX t-1

Market volume, denoted as Vol t-1

The moving average of the return on the MSCI World Index ex US over the previous

3 months, denoted as MA(MSCI)t-1

Then, for each hedge fund index and these 6 factors, they ran the following Generalized

Least-Squares regressions (GLS):

Yi,t = i + i,0i,0Yt-1 + i,1i,1MA(S&P)t-1 + i,1i,2Oilt-1 + i,1i,23mt-1 +

i,1i,2VIXt-1 + i,1i,2Volt-1 + i,1i,2MA(MSCI)t-1+

Where the coefficient i,k (with i = 1,,9 for the nine indexes and k=1,,7 for the seven

variables) take the value 0 when the variable kis not use in the model for index i or otherwisetake the value 1. They look for all possible combinations of these variables and keep the

model with the highest explanatory power in terms of in the sample R adjusted of regressions

of the nine CSFB/Tremont hedge fund indexes.


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Table 6 gives results of models ( and ) for each hedge fund index with respect to all

possible factors and in parentheses their corresponding p-value for the period January 1994 to

December 2003.

Table 6. Predictive Models (Amenc, Bied and Martellini method)

Const Rt-1 MA(S&P) t-1 Oil t-1 3m t-1 VIX t-1 Vol t-1 MA(MSCI) t-1

CONV. ARBITRAGE -0.0086

(0.0617)

0.4008

(0.0000)

0.1055

(0.0313)

0.0008

(0.0055)

-0.0065

(0.1835)

-0.1672

(0.3054)

DEAD SHORT BIAS

EMERG. MARKETS -0.0180(0.2950)

0.3314(0.0008)

0.0011(0.1908)

0.0016(0.2533)

EQUITY MKT.NTRL. 0.0060

(0.0000)

0.3114

(0.0006)

0.0003

(0.2173)EVENT DRIVEN -0.0039

(0.5348)0.2524

(0.0200)0.2551

(0.0145)0.0004

(0.1501)-0.0072(0.2882)

-0.1726(0.0000)

FIXED INC. ARB. -0.0045(0.2485)

0.3036(0.0006)

0.1120(0.0098)

0.0004(0.0518)

-0.0003(0.3031)

GLOBAL MACRO -0.0206(0.1152)

0.2079(0.1146)

0.0022(0.0041)

-0.7588(0.1105)

LONG/ SHORT

MANAGED FTRS

They performed out-of-sample testing of their models using a rolling window of 60 months.

The Table 7 provides information on the performance of the predictive models for the nine

CSFB/Tremont Hedge Fund indexes for the period January 1994 to December 2003.

Table 7. In the sample and Out of sample performance of the predictive models

IN the sample adjusted R OUT of sample Hit Ratio

CONV. ARBITRAGE 0.34 0.85

DEAD SHORT BIAS 0.47

EMERG. MARKETS 0.10 0.58

EQUITY MKT.NTRL. 0.08 0.92

EVENT DRIVEN 0.14 0.82

FIXED INC. ARB. 0.22 0.78

GLOBAL MACRO 0.06 0.56

LONG/ SHORT 0.61

MANAGED FTRS 0.54

Then, they tested the economic significance of return predictability in terms of over-

performance of style allocation models in a static mean-variance framework. They based their


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tactical style allocation strategies on the conditional expected returns obtained from the

predictive models. In the case of the 3 indexes for which no forecasting model could be

calibrated (adjusted R < 5%), they simply used the unconditional expected return as a

forecast of the expected return. Table 9 gives the results. We replicate the models for the

period January 1994 to December 2003 and use an additional portfolio optimisation programs

P1. The programs P2 and P3 are identical to those used by Amenc et al.

These results are shown and commented in the section 1.5.2.3.

As we can see in these statistical models, oil price is a predictor of convertible and fixed

income arbitrage strategy. Does it make sense economically? For the convertible arbitrage

strategy for instance, Figure 7 shows that the expected return is mainly explain by: the

constant (the non explicative part of the model) and the Oil price. We know that convertible

arbitrage strategy exploit pricing anomalies between convertible securities and their

underlying equity. If the Oil price has an explanatory power statistically speaking, there is

none economically speaking.

That is the reason why in the next section we study models based on economic significance

and we compare it with these models. So in the next section we do not look for the best

combination of all risk factors respect to the explanatory power (R adjusted), but we select

some risk factors with respect to their economic significance.

Figure 7. Expected return decomposition for convertible arbitrage strategy.


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1.5.2.2 An economic multi-factor model

In this section, we search for an economic significance and not statistic one. So we do not

look to the best combination of all risk factors with respect to the explanatory power (adjusted

R), but we select some risk factors with respect to their known economic significance for

each hedge fund style.

This selection is presented in the following table.

Table 8. Matrix of economically significant predictable variables for each hedge fund index

S&P

Oil

Vol

VIX

MSCI

World

Russel

2000

T-Bill

3-month

Term

Spread

AAA

Default

Spread

MSCI

Emerging

GSCI

Comm

Gold

Cur

CONV. ARBITRAGE

DEAD SHORT BIAS

EMERG. MARKETS

EQUITY MKT.NTRL

EVENT DRIVEN

FIXED INC. ARB

GLOBAL MACRO

LONG/ SHORT

MANAGED FTRS

For each hedge fund index and each of 14 factors, we search for a relationship of the type:

Yt= + 1X1, t-1 + 2X2, t-1 + +nXn, t-1 + t

where

Yt is the explained variable at time t, given by the hedge fund index return,

Xt-1 is the explanatory variables at time t-1, given by the risk factor return,


Furthermore, in order to be able to compare these economic models with statistic ones seen in

the previous section (predictive model of Amenc, Bied and Martellini), we create others

models with the same set of risk factors as before to which we add the lagged return of the

hedge fund index.


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This second relationship is then of the type:

Yt= + 0Yt-1 + 1X1, t-1 + 2X2, t-1 + +nXn, t-1 + t

where


Yt-1 is the explanatory variables given by the hedge fund index return at time t-1,



Next step of our work is to use these results in forecasting:

Then with a rolling window of 60 months from January 1994 to December 2004, we do an

OLS for each hedge fund index and his respective risk factors in order to get the best

predictive models for the expected hedge fund index returns. We then use them to construct

our optimal TSA portfolios. We finally evaluate them by comparing their performances with

those of our two benchmark portfolios.

These results are shown and commented in the section 1.5.2.3.


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1.5.2.3 Performance of the linear multi-factor predictive models

This table gives the TSA portfolios performances resulting from the use of the previous linear multi-factor m

future returns. The performances are given for a 60-month holding period starting in January 2000 and ending

Table 9. Linear multi-factor models performance

AverageModel \ Annualized (in%) P1

P2

P3

Return

Standard

Deviation

Track

Error1

Info.

Ratio1

%UpGain Loss

Hit

Ratio1

Dr

Do

Benchmark 1 (Equally) 9.72 3.29 0.00 n/a 85 0.89 -0.08 0 2.

Benchmark 2 (Buy-and-hold) 9.91 3.49 0.25 0.75 81 0.91 -0.08 59 2.

Benchmark 3 (Perfect timer) 57.63 10.17 4.65 10.31 100 4.80 0.00 100 0.

Benchmark 4 (Minimum variance) 8.97 1.79 0.73 -1.03 95 0.76 -0.01 49 0.

Linear model X 14.16 6.38 1.44 3.09 86 1.40 -0.22 69 4.

with lagged return X 16.73 10.43 2.66 2.64 81 1.79 -0.39 66 6.(3-month moving average) X 16.00 8.82 2.06 3.05 83 1.70 -0.37 63 5.

STATISTIC multi-factor model X 12.11 3.45 1.02 2.34 92 1.05 -0.04 53 1.

(Amenc, Bied and Martellini model) X 11.45 4.10 1.27 1.36 88 1.07 -0.12 49 3.

without lagged return X 15.39 6.62 1.49 3.81 80 1.37 -0.09 63 1.

STATISTIC multi-factor model X 12.61 4.01 1.09 2.65 85 1.16 -0.11 59 3.

(Amenc, Bied and Martellini model) X 13.03 3.94 1.30 2.55 85 1.18 -0.09 59 3.

with lagged return X 11.98 6.56 1.42 1.59 73 1.28 -0.29 53 3.

ECONOMIC multi-factor model X 10.25 5.23 1.22 0.44 78 1.10 -0.24 47 4.

without lagged return X 9.76 8.52 2.15 0.02 76 1.25 -0.44 51 6.

X 10.68 8.36 1.92 0.50 71 1.46 -0.57 53 6.

ECONOMIC multi-factor model X 10.90 5.46 1.32 0.90 81 1.19 -0.28 53 4.

with lagged return X 12.19 9.01 2.34 1.06 76 1.50 -0.48 51 6.

X 10.55 8.32 1.92 0.43 69 1.44 -0.56 51 6.

1. with respect to the Benchmark 1 (equally-weighted)

2. This table is computed with OLS_TREMONT_V12.M Matlab program for STATISTIC multi-factor models. ECONOMIC multifactor modeMatlab program.


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We find similar results as Amenc, Bied and Martellini when we replicate their models:

Information ratios are around 2.5 and 1.5 for P2 and P3 respectively.

The above table also shows that all our linear multi-factor models have a higher ex-post

information ratio compared to benchmarks. This confirms Amenc, Bied and Martellini results

that the superior performance of the Tactical Style Allocation programs relative to the

benchmark (equally- and value-weighted) is clear.

We see that Statistic models have better predictive power than Economic ones. It is not

surprising, since statistic models are customized to be better. But the significance predictive

variables are not very clear, as we saw previously.

Finally, we note that the predictive power of the multi-factor models is increased when we

add the lagged return as a predictive variable. These results are not very surprising when you

compare Figures 8 to 12 as the TSA portfolios using these models invest mainly in non-

directional strategies and so, mimic the behaviour of single factor models with 3-month

moving average in lagged returns. Non-directional strategies are preferred in the case of our

allocation optimization programs, since we chose the variance as risk measure9.

The following figures give the average allocation in hedge funds indices for each model (in

the case of P1 optimization).

We see that the 3-month movingaverage model invest mainly(67%) in non-directionalstrategies:

30% in Convertible arbitrage, 15% in Equity market neutral, 13% in Event driven, 9% in Fixed income arbitrage.

Figure 8. Allocation of 3-month

moving average model

9 If we used an alternative measure of risk like the modified VaR which take into account the third and four moments of the returndistribution, the preference for this non-directional strategies would be reduced in favour of high skewness and low kurtosis strategies.


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Figure 9. Allocation of Statistic multi-

factor models with lagged returnFigure 10. Allocation of Statistic multi-

factor models without lagged return

On another hand, we see that

sensitive at adding or remov

Without lagged return, 4

market neutral and 18%

directional strategies).

With lagged return, 38%

arbitrage, 34% is investe

strategy (two non- direct

invested in Long/Short (

On the contrary, the Econom

of adding or removing the la

Figure 11. Allocation of Economic

multifactor models with lagged return

Figure 12. Allocation of Economic

multifactor models without lagged return


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This predictive power of hedge fund index lagged returns confirms the results of the

previous section.

But which part of these multi-factor models comes from the mimic and which part comes

from the predictive power of risk factors? Economical model is not very sensitive at adding or

removing the lagged return. But the superior performance of this model without lagged return

compared to benchmark is not very significant: Information Ratio is between 0.02 and 0.5.

As for the turnover, we note again that all the optimization programs give very high value, P2

and P3 programs being the worst as their turnovers stand between 600 and 900 percent.

In addition to high turnover, all the TSA portfolios we have seen display very poor

diversification (see Appendix C.2).

Still, we note that all the multi-factor models do not beat the 3-month moving average single-

factor model as the latter has higher information ratio and final return. On the other hand, the

3-month moving average model has higher standard deviation and tracking error. Knowing

that investors are interested in return, this naive model appears to be the best predictive

model so far. Also, these same investors will care more about the risk (standard deviation)

during periods of bear market. In this case, we can imagine them to switch to multi-factor

models in order to mitigate their downside risks.


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1.5.3 Non-linear Multi-factor Predictive Models

So far, we have only concentrated ourselves on linear models. We go now one step further by

discussing non-linear models in our quest to search for the best predictive model. This new

approach is justified by the fact that some hedge funds strategies have returns that are not

linearly related to returns of risk (Figure 13). We see that these returns are non linearly related

to S&P 500 returns in the case of global macro and managed futures strategies and also in the

case of global macro non linearly related to our currency index. We also note that the fixed

income strategy is non-linearly related to MSCI ex. US (straddle shape).

Figure 13. (in % per month)

The performance of the global macro strategyvs. US equities (S&P 500) The performance of the global macro strategyvs. currency (major currencies index)

The performance of the fixed income strategyvs. World equities index ex. US (MSCI ex US)

The performance of the managed futuresvs. US equities (S&P 500)

The conclusion is that linear factor models of investment styles using standard asset

benchmarks, as Sharpe (1992), are not designed to capture the non-linear return features

commonly found among hedge funds.


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Fung and Hsieh (1998a, 1998b) suggest that hedge funds returns are the result of three

factors:

Location factors determine where a hedge fund invests on a long-term basis (stocks,

bonds, commodities or currencies, domestic or foreign). They are typically linearly

related to conventional asset classes.

Trading strategy factors are the result of the hedge fund managers active decision and

short-term trades (buy-and-hold, long-short, trend-following). They are typically

nonlinearly related to location factors and harder to identify.

Leverage decisions may differ between individual location and trading strategy

factors, so that identifying leverage decisions precisely may be quite difficult.

Fung and Hsieh (2001a) search across five asset classes (stocks, government bonds,

currencies, three month interest rates and commodities) spanning twenty six different markets

and found that, during extreme equity market movements, trend followers can be explain by a

combination of currencies (deutschemark and Japanese Yen), commodities, three month

interest rates and US bonds): preferred habitats.

These results are from contemporary data. Fung and Hsiehs approach does not result in a

very reliable model.

Since we saw the persistence of hedge fund indices returns, we go now one step further and

we study in next sections:

Quadratic single factor model,

Single factor model as Call or Put payoff,

As we did with the linear models, we use our results in order to compare performance with

respect to our two benchmarks.


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1.5.3.1 Quadratic multi-factor model

The aim of this section is to look for convexity, by searching for each hedge fund index and

each of 14 factors a relationship of the type:

Yt= + 1 (Xt-1) + 2(Xt-1) + t

where




In doing so, we expect to capture the convexity effect or the ability of some Hedge Funds

indices (in average, since we work on hedge fund indices), as long-short equity strategies, to

time the market.

In order to approximate the coefficients , 1 and 2, we use the Ordinary Lest Squares (OLS)

method, which attempts to minimize the sum of the squared errors.

Next, we measure the quality of the regression:

its explanatory power with the R adjusted

significance of coefficients with the p-value.

The performances of these models are shown and commented in the section 1.5.3.3.


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Table 10.Yt

Xt-1CONV.

ARBITRAGEDEAD SHORT

BIASEMERG.

MARKETSEQUITY

MKT. NTRLEVENT

DRIVENFIXED

INC. ARBGLOBA

MACR

S&P 500 0.0082 0.00791 0.1085 0.02772 -1.0154

R adj.

Oil NaN NaN NaN NaN NaN NaN NaN VIX -0.0038 0.0084 0.0100 0.0064

1 0.1774 -0.1622 -0.0737 -0.0367 2

R adj.

Volume NaN NaN NaN NaN NaN NaN NaN

MSCI . 0.0018 0.0038

ex. US 1 -0.8191 0.6465 2 -4.6338

R adj.

FR2K . 0.0022 0.01981 0.4847 0.03522 -1.1127

R adj.

T-Bill

(3 months)

.NaN NaN NaN NaN NaN NaN NaN

Term .

spread 1

2

R adj.

AAA . 0.0125

1 -0.280021

R adj.

Credit

spreadNaN NaN NaN NaN NaN NaN NaN

MSCI . -0.0041 0.0064 0.0086 0.0087 0.0090 0.0185

Emerging 1 -0.4845 0.2203 0.0381 0.1605 0.0227 -0.0232Market 2 0.7938 -1.2283 -0.6533 -1.1435

R adj.

GSCI 0.0083 1 0.1255 1

R adj.

Gold NaN NaN NaN NaN NaN NaN NaN

Currency

12

R adj.


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1.5.3.2 Non-linear multi-factor model with option-like regressors

A standard mean to control for option-like return features is to add a nonlinear function of

factor returns as independent regressors. Using this kind of model, Agarwal and Naik (2001)

obtain R values that are dramatically higher than the ones obtained by Fung and Hsieh using

Sharpes (1992) asset class factor model. These results tend to prove the importance of

including trading in performance evaluation models for hedge funds.

The aim of this section is to look for a relationship that incorporates a Call or Put payoff i.e.

of the type:

Yt= + 3Xt-1 + 4 Max(Xt-1,0) + t

Yt= + 3Xt-1 + 4 Max(-Xt-1,0) + t

where




In doing so, we attend to captureleverage effect (Call payoff) for

directional strategy and insurance

effect (Put payoff) for non-

directional strategy.

In order to approximate the coefficients , 3 and 4, we perform the Ordinary Lest Squares

(OLS) method, which attempts to minimize the sum of the squared errors, for each hedge fund

index and each of 14 factors.

NAVt

NAVt+1

4

Resu

lting

posit

ion


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As usual, we measure the quality of the regression:

its explanatory power with the R adjusted

significance of coefficients with the p-value.

We present all the results in Table 11 for the period January 1994 to December 2003: We

report the estimated coefficients along with their p-value, where the R of the regression is

bigger than 5%. For each strategy, we highlight the highest R.

Next step of our work is to use these results in forecasting:

Then with a rolling window of 60 months from January 1994 to December 2004, we do an

OLS for each hedge fund index and his respective risk factors in order to compare the

obtained performances with those of our two benchmarks.

The results are shown and commented in the section 1.5.3.3.


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Table 11.

Yt

Xt-1CONV.

ARBITRAGEDEAD SHORT

BIASEMERG.

MARKETSEQUITY

MKT. NTRLEVENT

DRIVENFIXED

INC. ARBGLOBA

MACR

S&P 500 -0.0063(0.4414)

0.0082(0.0000)

0.0079(0.0000)

1 -5.0000(1.0000)

0.1085(0.0016)

0.0277(0.2035)

2 -1.0154

(0.0037)3 3.5000

(1.0000)4 -3.5000

(1.0000)R adj. 0.0885 0.0750 0.0756

Oil NaN NaN NaN NaN NaN NaN NaN

VIX -0.0038(0.3766)

0.0084(0.0176)

0.0100(0.0000)

0.0094(0.0000)

1 0.1774(0.0000)

-0.1622(0.0000)

-0.0737(0.0000)

0.0313(0.0039)

2

3 -0.1058(0.0094)

4'

R adj. 0.2119 0.1856 0.3375 0.1084

Volume NaN NaN NaN NaN NaN NaN NaN

MSCI

ex. US

. 0.0026(0.5454)

0.0038(0.3329)

0.0100(0.0000)

1 -0.8038(0.0001)

0.6465(0.0000)

0.0313(1.0000)

2

3 -0.1250(1.0000)

4 -0.2656(1.0000)

R adj. 0.4829 0.3132 0.1608

FR2K . 0.0022(0.5913)

0.0198(0.0000

1 0.4847(0.0000) 0.0352(0.39162 -1.1127

(0.00033 0.1029

(0.3887)4

R adj. 0.2929 0.0913

T-Bill

(3 months)

.NaN NaN NaN NaN NaN NaN NaN


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Term

spread

.

1

2

34

R adj.

AAA . 0.0125

(0.00011 -0.2800

(0.00432

3

R adj. 0.0607

Credit

spreadNaN NaN NaN NaN NaN NaN NaN

MSCI

Emerging

. -0.0041(0.3023)

0.0064(0.1589)

0.0086(0.0011)

-0.0087(0.4144)

0.0090(0.0000)

-0.018(0.0000

Market 1 -0.4845(0.0000)

-0.2203(0.0010)

0.0381(0.0060)

0.1605(0.0033)

0.0227(0.0787)

-0.0232(0.6285

2 0.7938(0.0535)

-0.2624(0.0361)

-0.6533(0.0000)

-1.1435(0.0019

3

4

R adj. 0.4827 0.0817 0.0808 0.4028 0.3550 0.0686GSCI 0.0083

(0.0000)1 0.1255

(0.0014)2

3

R adj.

Gold . 0.0088(0.0000)

1 0.1936(0.0028)

2

3 -0.2355(0.0094)

4

R adj. 0.0596 Currency

1

2

3

4

R adj.


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1.5.3.3 Performance of the non-linear multi-factor predictive models

This table gives the TSA portfolios performances resulting from the use of the previous non-linear multi-fa

expected future returns. The performances are given for a 60-month holding period starting in January 2000 a

Table 12. Non-linear multi-factor models performance

AverageModel / Annualized (in%) P

1P2

P3

Return

Standard

Deviation

Track

Error1

Info.

Ratio1

%UpGain Loss

Hit

Ratio1

Draw

Down

Benchmark 1 (Equally) 9.72 3.29 0.00 n/a 85 0.89 -0.08 0 2.19

Benchmark 2 (Buy-and-hold) 9.91 3.49 0.25 0.75 81 0.91 -0.08 59 2.47

Benchmark 3 (Perfect timer) 57.63 10.17 4.65 10.31 100 4.80 0.00 100 0.00

Benchmark 4 (Minimum variance) 8.97 1.79 0.73 -1.03 95 0.76 -0.01 49 0.37

Linear model X 14.16 6.38 1.44 3.09 87 1.40 -0.22 69 4.24

with lagged return X 16.73 10.43 2.66 2.64 81 1.79 -0.39 66 6.10

(3-month moving average) X 16.00 8.82 2.06 3.05 83 1.70 -0.37 63 5.66

Quadratic multi-factor model X 11.47 4.79 0.97 1.80 78 1.11 -0.16 53 2.16

X 13.15 5.19 1.57 2.18 88 1.22 -0.12 59 3.15

X 12.83 9.03 1.97 1.58 69 1.51 -0.44 47 7.96

Multi-factor model X 12.27 4.61 0.94 2.72 80 1.15 -0.13 56 2.31

with option-like regressors X 14.60 5.77 1.70 2.87 86 1.35 -0.13 61 3.15

X 13.77 8.75 1.90 2.14 68 1.55 -0.40 47 7.

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Predictability in Hedge Fund Index Returns and Implications for FoF Style Allocation

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