MFA Hedge Fund Paper

8/6/2019 MFA Hedge Fund Paper

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Hedge Fund Performance Attribution Analysis when Returns are Non-Normal: An

Evaluation of the Trading and Risk-Management Style of Fixed-Income Relative

Value Arbitrage Hedge Fund Managers

Bernard Murphy, Mark Cummins

Kemmy Business School, University of Limerick

John Frain

Department of Economics, Trinity College Dublin

May 2007

Abstract

In this paper we evaluate hedge fund performance when returns arenon-normal and display non-linear dependencies with underlying assetclasses. We focus in particular on the non-directional and hybrid classes of …xed-income strategies; convertible arbitrage, …xed income arbitrage anddistressed debt where relative-value trading is the dominant style. We…rst employ a statistically rigorous factor analysis procedure to uniquelyidentify a sibling class of strategies whose return-generating processes weposit are driven by a common factor; namely, a buy-and-hold exposure to

global credit risk. We augment this risk factor with a germane set of trad-ing strategy risk-factors which parsimoniously capture the variation in keymarket variables and which can capture the proprietary trading and risk-management style of hedge fund arbitrageurs. Using a standard ordinaryleast squares estimator we provide evidence that the …xed-income sectorhas a neutral exposure to shifts in the market volatility term structure.We interpet this as evidence of a proprietary hedging expertise whichcan account for some of the excess risk-adjusted return generated by thestrategy. We check our results for robustness using a maximum likelihood’alpha-stable’ estimator which is capable of overcoming the statistical in-ference and hypothesis testing limitations of standard linear statisticalmodels and their estimators when returns are non-normal. We providestrong evidence to show that by not properly accounting for non-normalitye¤ects in the regression framework, a signi…cant fraction of a strategy’s

excess risk-adjusted return can be incorrectly attributed to manager skill.Nonetheless, our results suggest that the major proportion of the absolutereturns generated by these hedge funds is justi…ably attributable to therelative value acumen and the proprietary trading and risk-managementexpertise of fund managers. These strategies it would appear generateboth "alpha" and "beta" returns for their investors.

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Keywords : Hedge funds; relative value arbitrage; alpha-stable residu-als; implicit factor models; maximum-likelihood estimation; performance

attribution analysis; trading strategy risk factors.

Corresponding Authors: [email protected] [email protected]

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1 Introduction

Hedge funds are privately-organised investment partnerships which were origi-nally geared towards an elite mix of high net worth private clients, family trustsand institutional investors. Not originally intended for retail investors, hedgefunds have bene…ted therefore from being lightly regulated and not constrainedin the style of investing they follow. As a result, hedge funds can typicallybe dynamically long and short both traditional and alternative asset classes,can use derivatives for both position taking and implementing tactical hedgesand are free to aggressively leverage their investment capital in the search forabsolute returns.

It is estimated that the global hedge fund industry has grown from $950Billion by year-end 2004 [2] to circa $1.3 Trillion (AUM) by year-end 2006 [1].Over the next four years global institutional investments in hedge funds areexpected to triple to $1 Trillion, of which 65% is attributable to pension plans in

search of long-term portfolio diversi…cation bene…ts. In Ireland however, thereis currently a considerable amount of apathy towards increasing allocations toalternatives and to hedge funds in particular. This is due in part to perceived’crowding-out’ in the hedge fund industry [10] and the associated di¢culty inidentifying consistent top 1

4 -tile performing managers or funds. We suspecthowever that other factors are important, such as a lack of understanding of hedge fund strategies and associated tactical hedges, high perceived value-at-risk exposures associated with excessive use of leverage and derivatives andimplicit in heavily skewed and kurtotic returns distributions, and a prevailinglack of trust due to the absence of reporting and compliance requirements -all in all a formidable set of due diligence obstacles as perceived by traditionalinvestment managers.

Nonetheless, as the hedge fund industry continues to grow it will seek autho-risation to solicit investors in the traditional investment world - in particular,pension funds - and will undoubtedly adopt a more accomodative stance to thedisclosure and compliance requirements of …nancial regulators. Against thismarket backdrop the need for institutional asset managers to augment theirin-house due-diligence competencies for the particular case of hedge funds willbecome more obvious. In anticipation of a growing industry need therefore,we present a performance evaluation framework (based on a ’weak-form’ im-plementation of a style-based factor model) in this paper which can overcomethe limitations of standard linear statistical models when faced with a combi-nation of a non-linear1 return dependency on the underlying asset classes and

1 Although hedge fund managers trade the same asset classes as traditional fund managers,it is the dynamical and leveraged nature of their investment strategies, along with the extensive

use of derivatives for both position-taking and risk-management purposes that gives rise tothis non-linear dependency. See Fung and Hsieh [6] who characterise the payo¤ pro…le of trend-following commodity trading advisor (CTA) funds as analogous to that of an equitylookback straddle (long combination of a lookback Put and Call) where the exposure can becharacterised in properietary trading speak as long gamma. The authors further characterisethe return pro…le on Global/Macro funds as analogous to that of an equity collar and theexposure of Fixed Income Arbitrage funds as being essentially short vega, or equity market

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non-normality in the regression residuals. Our focus is on the ’non-directional’class of hedge fund strategies in which market-neutrality, relative-value and ar-

bitrage are the de…ning trading and risk-management objectives of the fund. Toaccomodate heterogeneity in the return generating processes across this familyof strategies, we therefore adopt the tactic of evaluating aggregate hedge fundperformance on a sector-by-sector basis.

1.1 Empirical Contribution of Paper

This paper extends the hedge fund performance evaluation literature in threekey dimensions. First, we employ a statistically rigorous factor analysis proce-dure to uniquely identify a "sibling class" of strategies whose return-generatingprocesses we posit are driven by a common factor; namely, a buy-and-hold ex-posure to global credit risk. Second, adopting the perspective of a …nancialengineer or proprietary derivatives trader, we identify and construct a germaneset of "trading strategy" risk-factors2 which parsimoniously capture the varia-tion in key market variables and which can capture the proprietary trading andrisk-management "style" of …xed-income hedge funds. Third, we then assessthe empirical potential of these factors to explain a strategy’s return-generatingprocess, in particular its market-neutrality, by using a maximum-likelihoodalpha-stable estimator. The latter can implicitly account for non-normalityrisk without the need to construct explicit factor proxies for systematic higher-moment risk exposures.

We present empirical evidence that the …xed-income arbitrage and distresseddebt sectors display signi…cant stand-alone exposures to ’primary’ risk factorssuch as global credit, but not to ’secondary’ risk-factors such as the slope orshape of the yield curve. We show that rather than viewing these members of

the non-directional and hybrid classes of hedge fund strategies as relatively low-risk and bene…ting from the self-proclaimed quantitative expertise and relative-value acumen of the fund manager, they can at times be seen as unhedgedleveraged “bets” - which can go very wrong during extreme market movementswhen the normal ‘Laws of Finance’ break down. Indeed the demise of LongTerm Capital Management in late 1998 provides indisputable ex-post evidencein support of this view.

volatility.2 For the Convertible and Fixed Income Arbitrage investable indices for example, we regress

excess fund returns against germane global credit and mortgage-backed securities index data.For the latter index we also regress against two implicit orthogonal interest rate factors -parallel shift and slope factors derived from a principal components analysis of forward (par)swap rate curves - to determine the exposure to these risk factors. We also regress againstinterest rate swaptions data to capture non-linearity dependencies between strategy returns

and the underlying …xed-income ’asset class’, in e¤ect capturing a strategy’s net exposure tointerest rate volatility.

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1.2 Structure of Paper

The paper is set out as follows. In Section 2 we provide a taxonomic clas-si…cation for the dependent variable data used in the study and comment onboth the relevance of, and the steps taken to mitigate, well-known biases suchas survivorship, selection and back-…ll bias for our hedge fund dataset. De-scriptive statistics for our hedge fund data are presented and various statisticsfor the goodness of …t of the returns series to both a Normal and a non-Normal(-stable distribution) alternative distribution are presented which provide sig-ni…cant insights into the investment styles followed by hedge funds. In order tocapture the well documented consequences of trading strategy[5] and leveragefactors on hedge funds return generating processes, we uniquely propose andconstruct a time-series of implicit statistical factors derived from a principalcomponents analysis of movements in forward swap rate curves. We augmentthese factors with liquid forward-at-the-money Libor swaption data to capture

non-linear dependencies to the underlying interest rate asset class. Finally,we propose extending the set of regressor variables to include both the VIXvolatility index to capture non-linear dependencies with the equity asset class(in e¤ect, the US stock market) and a global high yield index to capture sys-tematic exposures to the credit asset class. We emphasise that the selectionof regressor variable data is linked to the speci…c empirical focus of this paper;namely, to examine the performance of the non-directional or market-neutralclass of ’sibling’ strategies, Convertible Arbitrage, Fixed Income Arbitrage andthe ’hybrid’ strategy Distressed Debt.

In Section 3 we implement a factor analysis screening methodology to iden-tify ’likely-candidate’ regressor variables on which we regress the excess returnsgenerated by various style-speci…c hedge fund strategies. Based on our interpre-tation of the rotated factor loadings we make the general case for including both

trading strategy (market-timing, dynamic hedging) and location choice (buy-and-hold in a given asset class) risk factors when evaluating the performance of the directional / market-timing and the non-directional / market-neutral hedgefund styles, respectively. In the case of the latter class of strategies we pro-vide an ex-post heuristic statistical justi…cation for selecting the explanatoryvariable datasets discussed in Section 2.

In Section 4 we specify a weak-form version of a style-based linear factormodel and present estimates of Jensen’s alpha and of factor coe¢cients basedon ordinary least squares (OLS) estimation. Even using a naive OLS estimatorwe are able to make signi…cant observations concerning the attribution of per-formance to manager skill, in particular to a proprietary hedging competencyin the …xed-income markets.

In Section 5 we provide a robustness check to the normally distributed residu-als assumption underlying the OLS estimator used in Section 4. We propose andimplement an alternative maximum-likelihood / alpha-stable estimator whichcan account for non-normality in the regression residuals and can allow morerobust statistical inference and hypothesis testing be undertaken, assuming agood …t to the residuals series. Notwithstanding optimisation di¢culties en-

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countered in the numerical maximisation of the log-likelihood function, we …ndthat in all cases a signi…cant fraction of excess risk-adjusted return is incorrectly

attributed to manager skill. Based on our interpretation of the …t of an alpha-stable distribution to hedge fund returns, we conclude that this component of excess return is in fact compensation for systematic kurtosis and in particularskewness risk. Nonetheless, we …nd that a signi…cant fraction of the excessreturns generated are due to manager skills. These strategies it would appeargenerate both alpha and beta returns for their investors.

Section 6 concludes with a discussion of the implications of our …ndingsfor traditional investment funds such as pension funds. In light of the moreaccomodative stance being adopted by the hedge fund industry towards disclo-sure and compliance requirements, we propose how the statistical framework of Section 4 can be deployed to robustly estimate the value-at-risk of hedge fundportfolios.

2 Preliminary Data Screening

In the following sections we …rst provide descriptive statistics for the hedgefund / dependent variable data used in our study and introduce the -stabledistribution which underlies the maximum-likelihood based regression analysisin Section 4. We then describe a factor analysis procedure which we use inconjunction with a certain amount of intuition to identify and propose germaneregressor variable data for the regression studies subsequently implemented inSection 4.

2.1 Hedge Fund Dataset

Our hedge fund data is based on the net-of-fee monthly returns on a market-representative subset of the approximately 400 largest funds which have reportedto the 4000-plus fund Credit Suissse / Tremont (CS/T) database over the sampleperiod January 1994 through November 2005. Launched in 1999 but backdatedto January 1994 to mitigate some of the well-known database biases discussedbelow, the CS/T broad-based hedge fund index illustrated in Figure 1 is the…rst asset-weighted index to have been introduced for the hedge fund industry.

Like traditional asset class indices the CS/T investable hedge fund indexis asset-weighted and is transparently replicable through direct holdings in the10 style-di¤erentiated funds that make up the index. The 10 strategy-basedsectors or styles include the 6 largest assets under management (AUM) fundsoperating in each “sector” and can therefore be regarded as being representativeof aggregate performance both across sectors and for the industry generally.

Figure 1 also illustrates the screening methodology employed in the con-struction of both the broad-based and the 10 style-speci…c hedge fund indices.In light of the performance attribution context for our study, we focus only onthe investable subset of funds that are included in the database.

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Figure 1: CSFB / Tremont Hedge Fund Index Database Construction

Although the problem of high attrition rates in the hedge fund industry iswell documented3 , since 1994 (the start of our sample period) the CS/T data-base has included the performance histories of funds that have ceased operationsor have ’blown up’. Consequently, the commonly-reported problems attribut-able to "survivorship bias" have been largely mitigated as a consequence of therigorous rules-based approach to constructing the CS/T index. In regard topossible "selection bias", we emphasise again that the non-inclusion of closedfunds in the database is largely irrelevant in light of the performance attributionfocus of our study. We focus only on investable hedge funds which by de…ni-tion are open to institutional (funds-of-hedge funds, pension funds) and private(high net worth, family trusts) investors, for whom our …ndings are likely tobe of greatest relevance. Finally, the "back…ll / instant history bias" result-ing from the immediate back…lling of a fund’s performance history when …rstincluded in the index is not present in the CS/T database.

Descriptive statistics for the hedge fund dataset are presented in Table 1following4 . Various statistics for the goodness of …t of the return series to botha Normal and a ‡exible non-Normal (-stable, explained in greater detail in

3 See Edwards (1999) [3] who cites attrition rates of about 25% for US-domiciled hedgefunds and greater than 50% for non-US funds over the period 1989 to 1996.

4 The -stable parameters f;;;g have been estimated by numerically maximising therelevant log-likelihood function. The monthly log-densities are recursively numerically gen-erated from an inverse numerical Fourier transform of the -stable characteristic function.Section 5 and Appendix A detail the procedure involved.

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Section 5) alternative distribution are presented. The goodness of …t normal-ity tests indicate considerable problems with the …t of a Normal distribution,

whereas the -stable distribution provides a better …t5 . The symmetry parame-ter (1 1) in the latter drives the skewness of the returns distributionwith a zero value indicating a symmetric distribution. The stability para-meter determines the weight in the tails with a value of 2.0 indicating (inconjunction with = 0) a Normal distribution. The parameter is positiveand measures dispersion, while the parameter is a real number and can bethought of as a location measure.

5 We defer addressing the empirical implications of these …ndings to Section 5 but alert thereader at this juncture to previous research (see [?]) which reveals that non-normality in the

dependent variable data typically also manifests as non-normality in the regression residualsgenerated by a standard ordinary least squares (OLS) estimator. Consequently, key distrib-utional assumptions underlying the OLS estimator are violated which can result in spuriousconclusions being drawn from standard statistical inference and hypothesis tests. Section 5will demonstrate how a maximum likelihood -stable estimator can be implemented to pro-vide a robustness check on standard OLS based inferences about the performance attributioncharacteristics of the hedge fund investment strategies examined in this paper.

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S t a

r t D a t e : J a n - 9 4

E n d D a t e : N o v - 0 5

H F S t r a t e g y

H F I

C B

S B

E

M

E M N

E D

D D

F

I

G M

E L S

M F

M e a n ( % ) 1

1 0 . 4

8 . 4

- 0 . 4

9

. 3

1 0 . 4

1 1 . 0

1 3 . 0

6 . 1

1 3 . 2

1 1 . 7

7 . 2

S t d D e v ( % ) 1

7 . 9

4 . 8

1 7 . 3

1 6 . 5

7 . 9

5 . 7

6 . 5

3 . 8

1 1 . 2

1 0 . 3

1 2 . 1

S k e w

n e s s

0 . 1 2

- 1 . 2 8

0 . 8 1

- 0

. 6 4

0 . 1 2

- 3 . 3 6

- 2 . 8 4

- 3 . 0 2

0 . 0 3

0 . 2 3

0 . 0 2

( E x c e s s ) K u r t o s i s

2 . 0 7

2 . 7 4

1 . 8 7

4 . 2 9

2 . 0 7

2 2 . 9

1 7 . 7

1 5

. 4

2 . 5 4

3 . 6 2

0 . 3 1

G o o d n e s s o f F i t T e s t s f o r N o r m a l d i s t r i b u t i o n

J B T e s t a

K S T e s t b

G o o d n e s s o f F i t T e s t s f o r - s t a b l e d i s t r i b u t i o n

K S

0 . 0 3 8

0 . 0 5 5

0 . 0 8

0 . 0 5

0 . 1 7

0 . 0 5

0 . 0 5

0 . 0 4

0 . 0 5

0 . 0 4

0 . 0 7

p

v a l u e

0 . 9 8 3

0 . 7 7 0

0 . 2 5 0

0 . 7 9 9

0 . 0 0

0 . 8 9

0 . 8 2

0 . 9 4

0 . 8 8

0 . 9 4

0 . 4 5

L R d

2 0 4

3 3

1 6 7

1 9 5

3 1 8

9 0

7 5

3

0

2 0 9

2 9 9

1 4 0

M a x i m u m L i k e l i h o o d e s t i m a t e s o f - s t a b l e d i s t r i b u t i o n

1 . 5

1 . 4 5

1 . 8

1 . 6 7

1 . 9

1 . 8 4

1 . 8

1 . 5

1 . 3 9

1 . 6 8

2 . 0

( 0 . 2 6 )

( 0 . 2 4 )

( 0 . 2 4 )

( 0 . 2 5 )

( 0 . 2 5 )

( 0 . 2 5 )

( 0 . 2 5 )

( 0 . 2 5 )

( 0 . 2 6 )

( 0 . 2 5 )

( 0 . 2 5 )

0 . 1 7

- 0 . 6 9

1 . 0

- 0

. 3 5

0 . 9 9

- 0 . 9 9

- 1 . 0

- 1

. 0

- 0 . 0 4

0 . 3 4

0 . 3 6

( 0 . 4 8 )

( 0 . 3 3 )

( 0 . 3 3 )

( 0 . 6 3 )

( 0 . 6 3 )

( 0 . 6 3 )

( 0 . 6 3 )

( 0 . 6 3 )

( 0 . 4 3 )

( 0 . 6 5 )

( 0 . 0 0 )

0 . 0 1

0 . 0 1

0 . 0 3

0 . 0 3

0 . 0 1

0 . 0 1

0 . 0 1

0 . 0 0

0 . 0 1

0 . 0 1

0 . 0 2

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 0 )

0 . 0 1

0 . 0 1

0 . 0 0

0 . 0 1

0 . 0 1

0 . 0 1

0 . 0 1

0 . 0 0

0 . 0 1

0 . 0 1

0 . 0 1

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 1 )

( 0 . 0 0 )

( 0 . 0 1 )

( 0 . 0 1 )

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 0 )

( 0 . 0 0 )

1

M e a n s a n d S t a n d a r d D e v i a t i o n s a r e a n n u a l i s e d . T h

e m e a n 1 - m o n t h U S $ L i b o r r a t e o v e r

t h e s a m p l e p e r i o d w a s 4 . 6

%

a n n u a l i s e

d .

a

T h e J a r q u e - B e r a t e s t s t a t i s t i c i s a s y m p t o t i c a l l y

2

( 2 ) d i s t r i b u t e d u n d e r t h e a s s u m p t i o n

t h a t t h e N o r m a l i t y h y p o t h e s i s i s t r u e

, w i t h

c r i t i c a l v a l u e s o f 5 . 9 9 a n d 9 . 2 1 a t t h e 5 %

a n d 1 %

l e v e l s r e s p e c t i v e l y .

b

T h e K o l m o g o r

o v - S m i r n o v ( K S ) t e s t s t a t i s t i c i s l e s s t h a n . 0 2 f o r t h e s a m p l e s i z e e x a m i n e d .

c

F i g u r e s i n b r a c k e t s a f t e r e a c h s t a b l e p a r a m e t e r a r e 1 / 2 w i d t h 9 5 %

c o n … d e n c e i n t e r v a l s ( i . e . 1 . 9 6 t i m e s t h e s t d . e r r o r o f t h e e s t i m a t e ) .

d

L i k e l i h o o d R a t i o ( L R ) t e s t o f t h e j o i n t r e s t r i c t i o n

( n o r m a l i t y h y p o t h e s i s ) :

=

2 a n d

=

0 :

T h e t e s t s t a t i s t i c i s a s y m p t o

t i c a l l y

2 ( 2 ) d i s t r i b u t e

d w i t h c r i t i c a l v a l u e s o f 5 . 9 9 a n d 9 . 2 1

a t t h e 5 %

a n d 1 %

c o n … d e n c e l e v e l s r e s p e c t i v e l y .

L e g e n d :

H F I =

b r o a d h e d g e f u n d i n d e x ; C B =

c o n v e r t i b l e a r b i t r a g e ; S B =

d e d i c a t e d s h

o r t b i a s ; E M

=

e m e r g i n g m a r k e t s ; E M N =

e q u i t y m a r k e t n

e u t r a l ; E D

=

e v e n t d r i v e n ; D D

= d i s t r e s s e d d e b t ; F I =

… x e d i n c o m e a r b i t r a g e ; G M

=

g l o b a l / m a c r o ; E L S =

e q u i t y

l o n g / s h o r t a n d M F = m a n a g e d f u t u r e s .

T a b l e 1 : S u m m a r y D i s t r i b u t i o n a l S t a t i s t i c s f o r M o n t h l y H e d g e F u n

d R e t u r n s 1 9 9 4 - 2 0 0 5

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Figure 2: Taxonomy of Hedge Fund Strategies

Closer inspection of Table 1, in conjunction with the classi…cation taxon-omy illustrated in Figure 2, reveals some interesting insights into the dynamicaland leveraged nature of hedge fund investment strategies, and indeed the het-erogeneity in the investment styles followed. For example, the skewness orsymmetry parameter of the -stable distribution (i.e. ) is estimated at or veryclose to the negative limit of the parameter for the Event Driven, Fixed IncomeArbitrage, Convertible Arbitrage and Distressed Debt strategies. Bearing inmind the relatively stable nature of the return-generating processes underlyingthe …xed-income arbitrage funds, Figure 3 below reveals what must be seen asthe very considerable losses incurred by the Fixed Income and Convertible Ar-bitrage strategies over the post ’Russian Debt Crisis’ August-October period inlate 1998 when sharply widening credit spreads (on non-US sovereign and bothUS lower tier investment and non-investment grade corporate debt) took theirtoll. The Distressed Debt index reported monthly returns of -12.45%6 , -1.43%

and +0.89% respectively for the same three-month period.6 Between the start of the Russian debt crisis on August 17th 1998 and September 10th

1998, yields on emerging market debt as re‡ected in the J.P. Morgan emerging market bondindex had risen to a spread of 17.05 full percentage points above comparable-maturity USTreasury yields. Similarly, yields on US B-rated bonds (the very market where convertibleissues were commonplace) rose to almost 11% above Treasury and 5.7% above high investmentgrade corporates. See Edwards ??.

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Being relatively stable strategies the non-directional funds would thereforehave taken longer to recover from these losses than did the more opportunistic

Global / Macro and Emerging Markets directional funds, whose sample re-turn skewness over the 11-year period was therefore not as extreme as their…xed income counterparts. The evidence is indisputable that the credit mar-ket exposures of the …xed-income oriented funds in particular resulted in theirincurring especially high losses during this eventful period in the history of thehedge fund industry. It is not surprising therefore that these strategies shouldpresent with negatively-skewed return distributions for the sample period - adistributional characteristic which is well captured by the stable distributionsymmetry parameter .

Figure 3: Negative Skewness in HF Returns Autumn 1998

In contrast to the …xed-income strategies, the Dedicated Short Bias strategypresents with a signi…cantly positively-skewed returns distribution ( +1:0)for the sample period, which is consistent with the opportunistic and market-timing / directional nature of the fund’s investment style in the equity assetclass. As a …nal observation, the Broad Hedge Fund Index has a relativelysymmetrical (

0) if somewhat fat-tailed (

+1:5) returns distribution,

which is re‡ective of the disparate hedge fund styles (which is implicit in thenegatively and positively skewed return distributions) comprising the index,and indeed is implicitly re‡ective of the highly-leveraged nature of many of these strategies.

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2.2 Preliminary Screening for Explanatory Variable Data

The purpose of traditional style regression (or performance attribution) analysisis to identify the set of ’best’ exposures to traditional asset class benchmarksso as to minimise (mutual) fund tracking variance. A high R2 value from theregression analysis implies that a high proportion of total return variance canbe explained by the return variation in the benchmark asset classes. A highR2 value is therefore con…rmation of high dependent variable correlation to thefactor asset classes. Although this form of analysis works well for traditional’buy-and-hold’ investment strategies, the dynamically-directional and leveragednature of hedge fund investing makes this form of analysis ill-suited7 for evalu-ating the alternative investment strategies deployed by hedge funds.

Mindful of these limitations and the need for a parsimonious factor structure,as a preliminary screening device we factor-analyse a multivariate system of returns observed for a panel of "investment strategies"; comprising buy-and-

hold strategies in the ten style-speci…c hedge fund strategies which report to theCredit Suisse / Tremont database, and a long-only strategy in the equity assetclass (Diversi…ed US Large Capitalisation Equity and Diversi…ed Bond. Thebuy-and-hold equity strategy is proxied by relative movements in the S&P500large-cap index.

Before describing the screening procedure we …rst specify the linear factormodel which forms the basis of the regression analyses carried out in Section 4.The model can be represented as

Rt;HF = +Xk

bkF kt + et (1)

Based on an ’excess return’ speci…cation F k denotes the risk-premium required

(in equilibrium) per unit exposure of the investable hedge fund index to the kthfactor, in other words the "price of risk" for the kth factor. The intercept term (Jensen’s alpha) in Eqn. 1 denotes the excess mean fund return over andabove the ‘fair-value’ or ‘risk-adjusted’ return of the strategy.

Maintaining notational consistency with the factor model notation in Eqn.1, the inputs - outputs to the factor analysis algorithm can then be concisely

7 See Fung and Hsieh [5] who report that almost half (48%) of the hedge funds evaluated intheir database had R2’s below 25%, in contrast to the same proportion of mutual funds withestimated R2’s above 75%.

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summarised as follows

X

= multivariate T -by-d observed excess returns systemX

i = T -by-1 = i +WiF+ ei i = 1;:::;d

F = k-by-1 independent standardised common factors

W = d-by-k matrix of Factor Loadings

d = No. of "Investment Strategies"

= cov(e)

= d-by-d diagonal matrix of speci…c variancesXX

= d-by-d returns covariance matrix

XX= WW

T +

where the elements of the factor loadings matrix W are estimated by maximumlikelihood. Thus, WW

T denotes the variation in the system explained bythe common factors. Note that the factors are not unique in the sense that’principal components’ are and indeed this is re‡ected in the variety of methodsavailable to generate the rotated factor loadings that are presented in Table 1below.

Mindful of the fact that factor analysis seeks to explain as much as possiblethe ’o¤-diagonal’ nature of the variation in a multivariate system, we are con-scious that the multivariate returns system under investigation ideally ought tobe highly correlated. But since hedge funds generate returns which are typicallyuncorrelated8 with traditional asset class returns, and indeed are quite hetero-geneous across the strategies they follow, our factor analysis at best can only

be used as a preliminary screening tool to identify ’likely-candidate’ regressorvariables. Nonetheless, as a form of heuristic diagnostic, factor analysis canprovide useful insights into the nature of the pervasive risk-factors which aredriving the variation in both hedge fund and traditional investment strategies.

The results of the factor analysis for a 3-factor model are presented in Table2. Factor loadings are presented in columns 2 through 4 with speci…c variancespresented in column 5.

To facilitate a meaningful interpretation of the common factors which aredriving hedge fund returns, the latent factors have been optimally rotated sothat each ‘measured variable’ (investment strategy) has only a small number of large factor loadings (i.e. is in‡uenced by a small number of common factors),

8 The elements of the diagonal speci…c variance matrix can be interpreted as indicatorsof the cross-sectional variation in a strategy’s returns not explained by the common factors.

A value of 1.0 for example indicates that the return variation is due entirely to speci…c factors(note that the factor analysis algoritm standardises the input return vectors to have unitvariance) and that common factor in‡uences on the returns to this style are negligible. Theconverse applies in the case where the speci…c variance for an investment style is equal to 0.Given the relatively uncorrelated nature of hedge fund returns - both with traditional assetclasses and across sector-based strategies - the e¤ect on the style-based speci…c variances ismanifest in the majority of the strategies listed in Table 2.

13



No. " Investment Strategy " Factor 1 Factor 2 Factor 3 diag []1 Broad HF Index 0.476 0.095 0.676 0.005

2 CB Arbitrage -0.127 0.632 0.110 0.6313 Short Bias -0.765 -0.139 0.209 0.338

4 Emerging Mkts 0.377 0.346 0.167 0.467

5 Equity Market Neutral 0.234 0.168 0.053 0.850

6 Event Driven 0.200 0.877 -0.047 0.005

7 Distressed Debt 0.131 0.878 -0.098 0.128

8 FI Arbitrage -0.195 0.413 0.377 0.683

9 Global / Macro 0.014 0.054 0.947 0.055

10 Equity Long/Short 0.978 -0.107 0.146 0.064

11 Managed Futures -0.001 -0.286 0.367 0.859

12 US LC Equity 0.588 0.149 -0.056 0.546

Table 2: Factor Loadings 3-Factor Model

preferably only one or two. In a sense the objective is to identify possible’sibling’ hedge fund strategies which are commonly a¤ected by one or two dom-inant factors. Then based on one’s understanding of the dynamic trading and risk-management strategies followed by the siblings, it might be possible to se-lectively link the latent factors to either "location choice" / investable assetclass or "trading strategy" factors which can then be used in a more informed,’case-by-case’ regression analysis of sector-based hedge fund returns.

We emphasise again that we seek to identify not only potential return-generating risk-factors but also germane hedge factors which might explain thesource of a strategy’s marker-neutrality. Figure 4 below illustrates.

Figure 4 allows a more intuitive interpretation of the common factors whichare driving the multivariate return system represented in Table 1. Each "in-vestment strategy" is located by a vector in three-dimensional factor space, withthe orientation and length of each vector indicating the extent of the variationin returns explained by the relevant dominant factor. For example, the factorloadings in Table 1 might appear to suggest that the …rst factor (Factor 1) couldbe a traditional "location choice" style factor (i.e. buy-and-hold a diversi…ed,large-cap exposure to the US equity asset class) due to its sizeable commonin‡uence on the returns of the equity-based investment strategies. However, acloser inspection of Figure 4 would incline one to identify Factor 1 as a dynami-cal "trading strategy" factor (i.e. market-timing in the equity asset class). Thelarge absolute loadings of the dynamic Equity Long/Short and Dedicated Short

Bias strategies (dashed vectors) to this factor contrast sharply with those of both the traditional long-only US equity strategy9 and the Equity Market Neu-

9 With the exception of the bear market of the early 2000’s the US equity market waslargely in an upward-trending bull episode over the 11-year sample period examined. Hence,the signs of the factor loadings for the three directional equity strategies are consistent withthe intuition in this regard.

14



Figure 4: Hedge Fund Sibling Strategies & Common Factors

tral hedge fund strategy in particular. In fact, the "non-directional" style of thelatter strategy is the antithesis of the "directional" or ’market-timing’ style of the short bias and equity long/short strategies. Although market-timing hedgefunds are not the subject of this paper, the relationship of these two siblingstrategies to a common (market-timing) factor is well illustrated by Figure 1.

Proceeding in a similar fashion, inspection of Figure 4 reveals a commonsensitivity shared by the Convertible Arbitrage, Fixed Income Arbitrage andDistressed Debt hedge fund styles (bolded vectors) to what we posit (i.e. sub-

ject to subsequent empirical veri…cation) is an alternative location choice fac-tor; namely, a buy-and-hold strategy in the credit asset class. Distressed Debthedge funds invest in the debt of companies who might typically be restructur-

ing in the aftermath of bankruptcy proceedings, but signi…cantly may also betnon-directionally in the sense of "taking arbitrage positions within a company’scapital structure, typically by purchasing a senior debt tier and short-sellingcommon stock, in the hopes of realizing returns from favourable shifts in thespread between the two tiers" (see [2]). Prevailing interest rate exposure is typi-cally hedged out leaving the strategy exposed therefore to either an improvement

15



or otherwise in the creditworthiness of the target company.Convertible Arbitrage funds typically hedge out prevailing equity and inter-

est rate exposure, often leaving the strategy exposed to the credit risk of theissuing company. In fact, anecdotal reports suggest that ’CB Arb’ funds havegenerated most of their returns in recent years from a long-only buy-and-holdexposure to the credit asset class10 . A location choice factor in the credit assetclass will allow us to empirically test that conjecture. Fixed Income Arbitragefunds typically bet on anticipated widening / narrowing of term spreads alongthe default-free yield curve (hedging out duration and convexity risk for exam-ple) but may also bet on anticipated favourable movements in the quality spreadbetween corporate investment grade and government yields of comparable ma-turity. In all three strategies one might therefore reasonably expect to see acommon sensitivity to the credit asset class. Indeed Figure 4 clearly showsthe strategies to have signi…cant positive loadings to the second factor, infor-mation which might therefore be reasonably interpreted as implying potentialbuy-and-hold exposures for these strategies to a credit ’location choice’ factor.

Before proceeding with the analysis a few comments on the relatively un-correlated nature of the system are in order. The speci…c variances for themajority of strategies indicate a very limited collective ability of the commonfactors to explain individual variation in the system. In contrast, the BroadHedge Fund Index strategy (which is an asset-weighted ’portfolio’ of the sector-based hedge fund strategies listed in Table 1) displays a speci…c variance valuewhich indicates that the likely disparate dominant factors which are driving thereturns of individual strategies are collectively manifest in the broad index.

3 Explanatory Variable Data

Based on the data screening insights derived in the previous section, in this sec-tion we provide a brief description of the various explanatory variable datasetsused in the regression analyses of Sections 5 and 6. For the cases where ourregressor variable data has not been direcftly observable, we also provide a de-scription of the methodologies used to construct the relevant regressor variabletime-series.

3.1 Fixed Income Datasets

In order to properly attribute the performance of …xed-income oriented hedgefunds we di¤erentiate between our selection of traditional "location choice" andalternative "trading strategy" factors. For the Fixed Income Arbitrage sectorfor example, in the former case we chose the Lehman US MBS mortgage-backed

10 We show later in a simple linear regression that the convertible arbitrage strategy appearsto have been largely vega-neutral over the sample period - that is, ’in the aggregate’ and basedon short-term implied volatility (VIX) as the relevant proxy for vega risk. We also show thestrategy to have been largely equity market-neutral but to have had some residual exposureto gamma risk, most probably re‡ecting a simple delta-hedging risk-management strategy inthe underlying equity market.

16



securities index in order to re‡ect the trading activities of relative value hedgefunds in this market. In addition, we included the Lehman Global Treasury

Index to re‡ect the global remit of …xed-income hedge funds. In each of thesecases, the empirical intent was to examine the extent (or otherwise) of traditionalbuy-and-hold investment styles in the respective markets. For the tradingstrategy factors on the other hand, we used a set of implicit statistical factorswhich are intended to capture in a parsimonious fashion the return-generatingcharacteristics of the proprietary trading and risk-management practices whichare manifest in hedge fund returns; namely, low-volatility, absolute (in thiscontext, meaning market-neutral) returns. The rationale for including thesestatistical factors, and their construction methodologies, are explained in moredetail in the following sections.

3.1.1 Libor Swap Data

In deciding on an appropriate set of risk-factors to regress the Fixed IncomeArbitrage returns against, we interpreted the output of a factor analysis of hedge fund returns in Section 2 (Figure 4) to justify why we should investigateboth the credit asset class as a potential "location choice" return-generatingfactor, along with an implicit "trading strategy" factor; namely, (changes in)the slope of an appropriate reference yield curve. The latter factor can bereasonably proxied by the second-highest eigenvalue orthogonal factor extractedfrom a principal components analysis (PCA) of the historical variation in thebenchmark zero curve. This factor is known to drive the ’twists’ that canchange the slope of the zero yield curve and represents a very real exposure forrelative-value oriented …xed income arbitrage funds with ceterus paribus bullishor bearish expectational views on the steepening of the yield curve betweenvarious maturity reference points11 .

Figure 5 below shows both the individual and the cumulative variability inthe US Libor zero curve explained by the …rst three largest-eigenvalue prin-cipal components. The principal components analysis was performed on thestandardised daily changes in the zero yields over the sample period April 1997through April 2007. While the …rst factor accounted for just under 84% of total variation, the second factor accounted for a signi…cant 11% of the globalvariation in the system. Both factors have the standard interpretation of shift and twist factors, respectively.

11 At the start of 1998 Long Term Capital Management was reported to have had a notionalvalue position of c. $697 Billion in swaps (see Edwards [3]), much of it in the form of US yieldcurve swaps which exchanged the di¤erence between a speci…ed Libor yield and a speci…edconstant-maturity Treasury (CMT) yield. The short spread trade was in e¤ect a relative value

bet on the slope of the yield curve, which in LTCM’s case (receive Libor / pay CMT) wouldpay o¤ if spreads narrowed over the contract period. In contrast to LTCM’s expectationsof narrowing yield spreads however, over the course of August and September yield spreadsdramatically widened in the face of a market stampede to more liquid and higher-quality …xedincome securities.

17



Figure 5: Parallel Shift & Slope Factors in US Libor Zero Curve 1997-2007

Given the dynamically hedged nature of …xed income relative value strate-gies, we therefore also investigate the explanatory power of a parallel shift "trad-ing strategy" factor in order to assess the extent of the strategy’s "P&L" sensi-tivity to this factor. However, in contrast with the extant yield curve literature,

we uniquely use the correlation matrix of changes in the forward (par) swaprate curve rather than in the benchmark zero yield curve as the input to ourPCA routine. We show below that forward swap rate curves are the funda-mental market benchmarks used in the marking calculations for the …xed legsof forward-starting swaps (and for European-exercise swaptions). Moreover,proprietary …xed income traders will use swaps to either acquire a favourableexposure12 to anticipated interest rate changes or to hedge a pre-existing (e.g.bond portfolio) interest rate exposure. As they are cheaper but have the samelevel of interest rate exposure13 , …xed income arbitrage funds are more likely to

12 As shown in the following footnote, the …xed leg of a spot-starting swap has initially (andsubsequently) a longer duration than the ‡oating leg. If the Libor-Swap zero curve rises(falls) as anticipated this will lower (increase) the …xed leg more than that of the ‡oating leg,thus raising (lowering) the mark-to-market value of the swap to the buyer (i.e. the holder of

a payer swap who is short the …xed-leg and long the ‡oating-leg) and lowering (raising) thevalue of the swap to the seller (i.e. the holder of a receiver swap who is long the …xed-leg andshort the ‡oating-leg ).

13 For a notional principal N , a plain vanilla interest swap with maturity date T m can bepriced at generic time t as the di¤erence between the …xed and ‡oating legs of the swap. Itcan be shown that the swap can be equivalently priced as the di¤erence between a date T mmaturing coupon bond (default-free) paying the swap …xed rate F (annually) and a discount

18



trade interest rate swaps rather than coupon-paying Treasuries for interest-raterisk management purposes. Finally, mindful of the dynamical nature of …xed

income market-neutral hedge fund strategies, we deliberately eschewed the useof a total return index such as the Lehman Swap Total Return Index beause of the passive buy-and-hold investment style implied by a strategy which trackssuch an index.

Figure 6: Shift & Twist Factors in Forward Swap Rate Curve Apr-97 toNov-05

Figure 6 shows the familiar term-dependency or shape of the factor loadingsfor the …rst two principal components extracted from a principal components

bond maturing on the start date of the swap B(t; T 0):

SWAP t = N:

nXi=1

F i

360B(t; T i) + B(t; T m)

!N:B(t; T 0)

where T i denotes the payment dates on the …xed leg and i denotes the actual/360 day countfraction between …xed leg payment dates.

When the swap is spot -starting (t = T 0) we get

SWAP t = N:

nXi=1

F i

360B(t; T i) + B(t; T m)

!N

noting that the right-most term in the pricing formula is duration-neutral, being simply aconstant. The advantage of using the swap for hedging purposes therefore is that it is pricedmuch cheaper than the embedded F coupon-paying bond while having the same duration.

19



analsyis of the forward par swap rate curve over the sample period April 1997through November 2004. Our analysis shows that the …rst two components

dominate in explaining the global variation in the reference forward swap ratecurve over the sample period examined, with the …rst factor accounting for justunder 90% and the second factor accounting for 10% of the variation in thesystem.

We provide in the following a brief description of the methodology and thedata used to derive the time series of forward swap rate curves used in theprincipal components analsyis. First, we de…ne the date-t value of a forward-starting payer swap contract, which commences at the forward-date T l andterminates at date T L with t < T l < T L; and which makes payments at thedates T j ; j = l + 1;:::;L: Assuming a $-unit principal or notational amountand …xed payments made at the rate F , if D (t; T ) denotes the date-t value of azero-coupon bond with maturity T; then the value of a forward-starting payer swap (i.e. long the ‡oating leg and short the …xed leg) V pay

l;L

(t) may be imputedfrom the following relationship for a newly-issued swap

V payl;L (t) = V flol;L (t) V fixl;L (t) = fD (t; T l) D (t; T L)g F LXj=l+1

j1D (t; T j) ;

where V flol;L (t) is the value of the ‡oating side of the swap; V fixl;L (t) is the value of the …xed side of the swap; and j1 is the market convention for the daycountfraction for the swap payment at date T j : We then de…ne the forward (par)swap rate yl;L (t) to be the …xed rate at which the date t value of the newlyissued forward-starting swap contract is zero, i.e.

yl;L (t) =D (t; T l) D (t; T L)

PL

j=l+1 j1D (t; T j)=

D (t; T l)D (t; T L)

P l+1;L (t)(2)

where the notation P l+1;L (t) denotes the present value of a basis point - i.e. theincrease in the value of the …xed leg in the swap if the swap rate increases.

The forward swap rate curves we have used in this study are constructedfrom a daily time-series (January 1997 through November 2005) of imputed forward swap rates extracted from the US $ Libor / Swap market (Datastream).Because of their greater usage for risk-management purposes we choose the setof 6 month, 1 year, 1.5 year, ..., 10 year (i.e. in 1/2 year forward increments)forward-starting contracts with a …xed underlying swap length of 1 year14 . Aforward swap rate curve for trade date t is then formed by augmenting thefront end of the set of imputed forward swap rates with the swap rate on a spot starting 1-year swap.

14 To impute the forward swap rate time-series corresponding to this set of forward starting

swap contracts, we …rst generated for each trade date in the series the ’Libor-Swap’ zero curveusing quoted Libor rates and relevant maturity …xed rates on spot starting swaps. Sinceit was necessary to have available the imputed 1.5-year, 2.5-year,..., 9.5-year zero rates, thecorresponding spot swap rates were determined by implementing a spline interpolation of thespot-starting 1-year, 2-year,..., 10-year swap (…xed) rates. For each trade date we then usedthe bootstrapped zero curve to compute the required-maturity discount bond prices used inEqn. 2 to impute the daily time-series of forward swap rates.

20



Figure 7: Forward Par Swap Rate Curve Dynamics Autumn 1998

We show in Figure 7 that the US $ forward par swap curve was relatively ‡atand downward trending during August and into September 1998. However, justas the market repercussions of the Russian debt crisis were most dramatically

initially re‡ected in the August returns for the Distressed Debt hedge fundsector (-12.45%), the changing fortunes of …xed-income oriented relative valuehedge funds were subsequently implicitly manifested in the movements in theUS Libor swap market15 during September and October. Having displayed adownward trend over August, during September and into October the forwardswap curve can be seen to shift sharply upwards and steepen out along the termstructure, re‡ecting the general widening of both term and quality spreads inthe post Russian debt crisis period. Figure 7 shows clearly the well-knownparallel shift and twisting (i.e. steepening) factors that are known to drivemost of the observed variation in interest rate term structures. As the …xedrates quoted on newly-issued forward-staring swaps rose sharply16 , the swapbooks of proprietary …xed-income hedge funds would have been directly and

15 It is well known that swap spreads (over Treasury) in‡uence the spreads on commercial

mortgage-backed securities (CMBS) because of the prevailing use of interest rate swaps tohedge CMBS positions by market participants (see [4] p. 212). This was very evident duringAutumn 1998 when credit and liquidity shocks drove both swap spreads and CMBS spreadsto historically high levels. We explain in Section 3.1.3 how hedge funds have exploited theirrelative value acumen in conjunction with their trading and risk-management expertise tobecome big participants in the mortage-backed securities markets.

16 As the reference Libor-Swap zero curve began to shift up sharply in early October 1998,

21



signi…cantly a¤ected. Given the prevailing use of interest rate swaps by hedgefunds in hedging general interest rate exposure, we feel that it is more relevant

to extract statistical factors from a principal components analysis of variationsin the forward swap curve rather than the zero yield curve.

Our empirical objective is to look for evidence of a statistically-signi…cantreturn-generating factor in the case of relative value strategies with an unhedgedexposure to the twist factor, and for evidence of a potential hedging factor todetect evidence of duration-neutrality (i.e. a hedged exposure to the parallelshift factor). However, in light of the swap-based hedging propensity of hedgefunds who trade the mortgage-backed securities markets (see Section 3.1.3 fol-lowing), it must be said that the net exposure to the "twist" factor must awaitempirical veri…cation until our discussion of regression results in Section 4.2.

3.1.2 Interest Rate Swaption Volatility Data

In order to account for possible non-linear dependencies between the returns onpredominantly US-based non-directional …xed-income strategies and the under-lying …xed income asset class, or conversely in order to detect the neutralityof the strategy to interest rate volatility, we extracted a proxy for the (interestrate) market volatility term structure from a US Libor payer-swaptions datasetdownloaded from Datastream. We show in Section 3.1.3 below that mortgage-backed securities arbitrageurs do seek to hedge out interest rate volatility inorder to lock-in excess risk-adjusted spreads on these securities. For these fundsto be truly regarded as relative value vehicles, the cost of being long the impliedvolatility in the interest rate options used for hedging interest rate volatility riskshould be less than the value received from being structurally short the "impliedvolatility received from homeowners"17 . As with the exposure to the "twist" factor in the previous section, however, the net exposure of the Fixed Income

Arbitrage sector to a "vega" factor, and the incremental explanatory power of such a factor, is a subject for empirical veri…cation in Section 4.2.

The data set comprised swaptions quotes on a selection of daily observationsover the period June 1997 through November 2005 (equating to 102 monthly-equivalent trade dates). Maintaining a degree of consistency with the forwardswap rate dataset described in the previous section, the swaptions data representat-the-money-forward quotes for 6-month, 1-year, 2-year, 3-year, 4-year and 5-year forward contracts written on underlying swap contracts of 1 year length inall cases. Quotes are given with the typical market convention of Black impliedvolatilities as embedded in Eqn. 3 following

P S t = P l+1;L (t) [yl;L (t) N (d1) KN (d2)] (3)

because of their longer duration the …xed legs on outstanding spot and forward starting swaps

would have decreased by proportionately more than the ‡oating legs. The …xed rate quotesfor newly issued swaps would therefore have needed to have been increased sharply at thistime so as to maintain equality between the …xed and ‡oating legs on newly issued swaps.

17 If market interest rate volatility rises, then the risk of an increase in prepayment speedswill increase. Other things being equal, this will lower the market value of the MBS securityheld long by the hedge fund (i.e. assuming the MBS was initially underpriced), indicatingthat the hedge fund is therefore structurally short "interest rate vega".

22



where

d1 = ln [yl;L (t) =K ] +1

2 (T l t)p

T l t;

d2 = d1 p

T l t;

yl;L (t) is the currently observable forward swap rate (de…ned in Eqn. 2) fora payer swap commencing on date T l and ending on date T L, BlackT l;T L isthe Black implied volatility of the (lognormally-distributed) forward swap rate,and the present value of a basis point P l+1;L (t) is de…ned as in Eq. (2). Fora …xed-length underlying swap, our swaptions dataset therefore constitutes areasonable proxy for the forward swap rate market volatility term structure.

The informational content of this extensive swaptions dataset can howeverbe parsimoniously reduced into a small number of common explanatory factorsusing the same principal components analysis (PCA) procedure used in Section

3.1.1 above.

Figure 8: PCA Analysis of Log-Di¤erences in ATMF Swaption ImpliedVolatilities Jun97 - Nov05

The maturity characteristics of the …rst two principal components (whichcollectively account for 93% global fraction of the total variance of changes inthe swap volatilities) are illustrated in Figure 8. The PCA was performedon the monthly log-di¤erences of the at-the-money-forward (ATMF) swaptionimplied volatility dataset (see [9]). Figure 8 shows that both factors have

23



standard ‘parallel shift’ and ‘tilt’ interpretations with the …rst factor accountingfor roughly 77% of the global variation in this term structure system.

3.1.3 Mortgage Backed Security Dataset

In order to examine the predominantly domestic (i.e. US) exposure of relativevalue strategies to the mortgage backed securities (MBS) markets18 over thesample period examined, we used the Lehman US Aggregate Index. This in-dex includes the Lehman US Treasury Index, the US Investment Grade CreditIndex (covering the investment-grade corporate bond market) and indices forMBS, ABS (asset-backed securities) and CMBS (commercial mortgage backedsecurities). The aggregate index has become the dominant …xed income bench-mark used by investment practitioners to measure and evaluate the investmentperformance of …xed income portfolios.

Relative value strategies are the dominant hedge fund style in the MBS

and CMBS markets and as a result of aggressive leveraging, hedge funds havebecome big participants. The fundamental investment style is to arbitrageperceived mispricing of securities by isolating the "excess risk-adjusted spread"by in turn hedging out all other risk exposures such as falling interest rate /prepayment risk, changes in the slope of the yield curve, interest rate volatilityas well as negative convexity. We believe that the aforementioned index and itssubcomponents therefore contain potentially signi…cant information about thenature of the return-generating process underlying the Fixed Income Arbitrageand Convertible Arbitrage hedge fund sectors in particular.

3.1.4 Lehman Global Treasury Index

In order to examine the extent (or otherwise) of a passive buy-and-hold invest-

ment style which might be present in the Fixed Income Arbitrage sector, we in-cluded a location choice factor which also re‡ects the global remit of …xed-incomehedge funds; namely, the Lehman Global Treasury Index. The index includeslocal currency-denominated sovereign debt of non-emerging market countries(including the G-7 countries and other ’major’ and ’global’ issuers as de…ned inthe index), in which all issues are …xed-rate and non-convertible.

3.2 Credit Dataset

In choosing datasets for the credit asset class we emphasise that the relevant riskfactor in each case corresponds to a location choice style factor which is germane to the proclaimed investment style of the hedge fund sector under investigation.

18

Long Term Capital Management were one of the …rst big hedge funds to trade this marketin 1994 when they placed short bets on the spread between interest only (IO) passthroughsecurities and principal only (PO) securities converging. At the time IO’s had fallen sharplyin value due to the surge of US mortgage re…nancings that followed in the wake of fallingmortgage rates. Believing the fall was in excess of the fair value predicted by internal models,Long Term accumulated a leveraged IO position estimated at $2 Billion (see [7]) while ithedged out the underlying interest rate exposure by purchasing Treasuries.

24



Besides implying a generic buy-and-hold investment strategy in the credit assetclass, this implies that for all three hedge fund strategies examined, we needed

to include both the ’domestic’ (i.e. US) credit market as well as the possibleglobal high-yield exposures of the sectors.

3.2.1 Investment Grade Credit Index

In order to capture the in‡uence on returns of both a domestic investment gradecorporate credit factor, and of a global sovereign-debt risk factor, we selected theLehman Global Credit Index. The index includes investment grade and high-yield credit securities extracted from the Lehman Global Aggregate and GlobalHigh Yield indices. We feel that this index constitutes an appropriate location choice factor to use for the Convertible Arbitrage hedge fund sector, which (onthe basis of the factor analysis in Figure 4 earlier) we a priori posit involvesa passive buy-and-hold strategy in the global credit asset class. NEED TO

REWRITE TO INCORPORATE INSIGHTS FROM P. 285 IN LHABITANT.

3.2.2 Global High Yield Bond Index

To isolate the possible in‡uence of a ’distressed debt’ risk factor on DistressedDebt hedge fund returns we used the Lehman Global High Yield Index (LGHY).The LGHY index integrates the non-investment grade portions of the LehmanEmerging Markets Index with the US High Yield Index, the Pan-EuropeanHigh Yield Index and also (since July 1999) the CMBS High Yield Index. Ac-cordingly, we feel that this index is an appropriate return generating factor toinclude in our evaluation of the international credit market exposures of the…xed income hedge fund sectors examined. As in the case of all of the Lehmanindices cited above, the LGHY data are represented as excess (over 1-month US

Libor) monthly log-returns and are therefore interpretable as ’risk-premia’ forthe purposes of our factor-based performance evaluation analysis.

25



Monthly Returns on Lehman Global High Yield Index

The August-September 1998 "credit market crash" is clearly evident in thereturns data in Figure 3.2.2, as it was earlier in the returns of various hedgefund sectors (including Distressed Debt and Global / Macro) presented in Table

1.

3.3 Equity Dataset

Although all of the strategies which are the empirical focus of this paper are…xed income oriented, the Convertible Arbitrage sector does have an indirectexposure to underlying equity markets through the embedded warrant of the is-suing company. In light of the predominantly US-based market for convertibleswe therefore focused on the US stock market for evidence of a return-generating(or hedge) factor present in the returns on this strategy over the sample period.

3.3.1 S&P500 Index

As a proxy for the equity market in which US convertible issuers trade, wechose the S&P500 large capitalisation total returns index. The convertiblearbitrage strategy typically delta-hedges the underlying stock market exposureof the purchased (i.e. buy-and-hold) convertibles. However, the strategy doesnot necessarily dynamically hedge against rapidly changing equity prices and somay be a priori expected to display residual ’gamma’ exposure. Consequently,

26



the reader should be aware that a priori this may disguise the empirical evidencein support of the strategy’s ’delta’ neutrality.

3.3.2 VIX Volatility Index

Mindful of the non-linear dependency of many hedge fund strategies to tradi-tional asset classes, the use of the VIX index19 can be viewed as a proxy forthe informational content embedded in a panel of short-dated S&P500 indexoptions. Consequently, the index o¤ers the promise of detecting the embeddedwarrant-attributable non-linear dependency between the returns on the Con-vertible Arbitrage hedge fund sector and on the US equity market.

4 Maximum Likelihood -Stable Estimation

As shown in Table 1 earlier, the stable distribution can model the negativeskewness and excess kurtosis that characterise hedge fund return distributions.The use of the distribution in Finance became popular in the 1960’s (see ) butinterest waned thereafter. This decline in interest was due not only to therelative mathematical complexity, and the considerable computational powerneeded to implement the distribution in practical complications, but was alsodue to the success of the Black-Scholes-Merton Gaussian approach to Financetheory which was developed at the same time. We refer the reader to Ap-pendix C for details of the maximum likelihood estimation procedure appliedto hedge fund return distributions. In Appendix C we show the asymptoticdistributional properties of the estimator as applied to the estimation of the’factor loadings’ or ’slope coe¢cients’. We …rst report regression results ob-tained using a standard ordinary least squares (OLS) estimator. We then check

our results for robustness using the maximum likelihood alpha-stable estimator(MLE/-stable).

4.1 Extended Linear Factor Model

In the style of Sharpe [8] and Fung & Hsieh [5] we use a factor model (Eqn.1 earlier) in an attempt to explain the return-generating processes underlyingeach sibling strategy in the …xed-income family of hedge fund styles discussedin Section 2.2.

Rt;HF = +Xk

bkF kt + et

We use the term "extended" for our linear model in order to di¤erentiate our ap-proach from traditional performance evaluation studies. As mentioned in both

the Introduction and in Section 2, the dynamic trading and risk-managementstrategies adopted by both the "non-directional" and "hybrid" class of strategies

19 The monthly VIX values, which measure short-term volatility expectations for theS&P500 equity market, are de-annualised to monthly-equivalent values for consistency withall other data.

27



requires that we use germane regressor variables in order that we can meaning-fully report on the statistical signi…cance of the ’trading strategy’ factor loadings

in each case. Consequently, both the number (k) and the choice of factors (F k)vary for the particular strategies in accordance with the ’case-by-case’ rationaleprovided in Section 3 earlier.

4.2 OLS Results and Analysis

For the three hedge fund sectors examined we provide estimates of Jensen’salpha and the various factor loadings based on our estimation of the extendedlinear factor model using an ordinary least squares estimator. For each model,we also present the R2 value along with the F test goodness-of-…t statistic(notwithstanding the normality constraint on the residuals) and associated p-value for the full model to test the null that all the factor coe¢cients, notincluding the constant term, are simultaneously zero.

4.2.1 Fixed Income Arbitrage

The estimated factor loadings and standard errors, along with simple ’goodness-of-…t’ statistics are presented in Table 3 following.

Fixed Income Arbitrage [OLS Estimator]Risk Factors F k

1. 2. 3. 4. 5. 6.

Int. Rate Volatility

US Agg. Global Global FPSR Curve Term Structure

Index High Yield Treasury Shift Twist Shift bbk -.013 .20 -.01 0.00 0.00 0.00tstat -.13 5.73 -.73 -1.19 1.13 -1.40 b = .00374 (4.49% annualised) tstat = 3.37

R2 = 33.2%

F test statistic ~26;143 = 7.8

p-value= 0.00

R2 = 31.8% (i.e. without Col. 6 risk factor) b= 4.08% annualised (i.e. without Col. 6 risk factor) tstat = 3.13

Table 3: OLS Results Fixed Income Arbitrage Hedge Fund Sector

Columns 4. and 5. represent factor proxies for parallel shift and twist factorsin the US forward swap rate curve described in Section 3.1.1 earlier. The dataused in the regression analysis are the …rst-di¤erenced factor score time-series forthe 1st and 2nd principal components extracted from a principal componentsanalysis of standardised monthly changes in the aforementioned swap curve.

28



They may therefore be regarded as "conditioning information" factors which canin‡uence the dynamic trading and hedging responses of hedge fund managers in

a dynamically changing marketplace. The column 6. factor contains the …rst-di¤erenced factor score time-series for just the 1st principal component extractedfrom a PCA analysis of standardised changes in the Libor swap volatility ‘termstructure’ (i.e. the set of option-implied volatilies for options on underlying1-year swap contracts maturing 6 months out, 1Y out, 2Y, . . . , up to 5Y out)explained in Section 3.1.2 earlier. As with the interpretation of the factorsrepresented in Columns 4. and 5., this risk factor may similarly be viewed as aconditioning information factor which can in‡uence changes in the proprietarytrading and hedging tactics of hedge fund managers in response to anticipatedchanges in market volatility.

The most signi…cant inference to take from Table 3 is that by adding thefactor proxy for shifts in the benchmark market volatility term structure toour set of regressor variables, the increase in the alpha of approximately 41basis points is attributable to the skill of the (average) hedge fund manager20 .This is a not insigni…cant amount given the low-volatility nature of the return-generating process underlying this strategy. In reality, relative value hedgefunds will dynamically shift their portfolio allocations and/or re-balance theirhedges as markets move. We conclude therefore that this incremental gain inalpha is attributable speci…cally to the proprietary trading and hedging expertiseof the hedge fund managers who trade in this sector.

That is not to say, however, that the strategy is a "true alpha generator".Table 3 clearly shows that the strategy has a signi…cant positive loading to theGlobal High Yield credit risk factor. When not included as a factor in theregression analysis the R2 value drops to just 2.75% and the associated Jensen’salpha drops to 3.84% annualised. Accordingly, we …nd that the sector’s average

compensation for being exposed to a buy-and-hold strategy in the credit assetclass amounts to a signi…cant 65 basis points. Signi…cantly, this …nding con…rmsthe intuition discussed in relation to Figure 4 earlier concerning the propositionthat a credit market "location choice" factor is present in the return-generatingprocess for this strategy.

20 In order to rationalise this di¤erence, consider that on the left hand side of the regressionthe excess returns re‡ect both long (through long option hedges) and short (exposure to risingMBS prepayment speeds) exposures to interest rate volatility as a result of the strategy’sproprietary arbitraging of mortgage backed securities. On the right hand side the strategy’spassive short exposure to interest rate volatility is picked up by the US Aggregate Index(Column 1) which includes a MBS sub-index. By including the interest rate volatility factor(i.e. adding back Column 6) we are in e¤ect allowing the regression to pick up a balancedexposure to interest rate vega, commensurate with left hand side exposure. The 41 basis pointdi¤erence in Jensen’s alpha constitutes the component of "relative value" which is thereforeprotected from adverse interest rate volatility movements and is correctly attributable tomanager skill - i.e. to a proprietary hedging competency in this instance. More generally,this is a signi…cant …nding as it justi…es the use of "trading strategy" factors in correctlyattributing fund performance to manager skill.

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4.2.2 Convertible Arbitrage

The risk factors, the estimated factor loadings and standard errors, and a num-ber of simple ’goodness-of-…t’ statistics are presented in Table 4 following.

Convertible Arbitrage [OLS Estimator]Risk Factors F k

1. 2. 3. 4. 5.

US Agg Global Equity Factors Global

Index Treasury Market Volatility High Yield bbk .124 -.02 0.07 0.05 0.11

tstat .75 -.10 2.34 2.01 2.74

= .0039 (4.69% annualised) tstat = 3.3

R2 = 11.88%F test statistic ~2

5;143 = 3.64

p-value= 0.004

Table 4: OLS Results Convertible Arbitrage Hedge Fund Sector

Columns 1. and 2. contain the excess monthly log-returns data for theLehman indices described earlier. Column 3 contains the excess monthly log-returns data for the US stock market (S&P500 index) while Column 4 is agermane proxy for capturing non-linear dependencies between returns on strat-

egy and on the underlying US stock market. The time-series for this index is inthe form of “log relative index levels” and is included to capture the strategy’sexposure to short-term ‘vega’.

Our analysis of Table 4 shows that Jensen’s alpha is signi…cant at 4.68%annualised. As might be expected given the dynamic delta-hedging natureof the strategy the estimated stock market loading is low at +0.07. Whatthis probably re‡ects is a deliberate exposure to “gamma risk” whereby thehigh convexity of typical convertible securities results in the position not be-ing delta-hedged frequently enough to be neutral to larger than normal, rapidchanges in the underlying stock market. Interestingly, the strategy’s loadingto the volatility factor is +0.06 which suggests that ‘vega’ risk was also largelyhedged or balanced out over the sample period examined. Although this mayappear initially to be counter-intuitive given the long-volatility exposure of the

embedded warrant, many large hedge funds would have been sellers of ’vega’(in e¤ect, hedging out or balancing out the volatility exposure of the strategy)in the aftermath of Autumn 199821 . Moreover, vega has a second-order ef-

21 In the aftermath of the rescue of Long Term Capital Management in 1998, many hedgefunds took advantage by using variance swaps to sell realised volatility at high implied levels.

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fect on option prices and so would not be expected to be a major factor in thereturn-generating process for this strategy anyway.

Table 4 shows that the strategy has a signi…cant but moderate positive load-ing to the Global High Yield ’credit risk’ factor. Given the propensity of convertible arbitrageurs to largely (but not completely22 ) hedge out credit riskthrough the use of asset swaps, this is perhaps not too surprising. On the faceof it, the OLS results seem to support the ’anecdotal’ view that the Convert-ible Arbitrage sector has generated positive returns over the latter half of thesample period largely on the back of favourable movements in the credit assetclass. Mindful of the institutional under-pricing practices in the US convertiblesmarket at the time and the proprietary nature of the trading and hedging mod-els used by the early generations of convertible arbitrageurs, it appears likelythat the strategy over the early years of the sample period was predominantlygenerating alpha returns. In more recent years however as these competitive ad-vantages were eroded and more funds adopted the strategy, it would appear thatthe strategy began to systematically take on directional risk, particularly in theinternational and domestic credit markets. In e¤ect, convertible arbitrageursbecame proprietary traders in the search for acceptable returns. However, weprovide further insights on this …nding and inference in Section 4.3 when we usethe alpha-stable maximum likelihood estimator to quantify the extent to whichthe loading on the credit risk factor shown in Table 4 masks an underlyingsensitivity to a systematic (negative) skewness risk factor.

4.2.3 Distressed Debt

The risk factors, the estimated factor loadings and a number of simple ’goodness-of-…t’ statistics are presented in Table 5 following.

The estimates reported in Table 5 show the strategy to have a very signi…cantloading to the US stock market factor in particular. The sample correlationwith returns on the S&P500 index was +0.546, and along with the respectivesample standard deviations back up this calculation of sample “beta”. In con-trast, and perhaps surprisingly, the estimated Global High Yield factor loading

There was no shortage of “distressed buyers” from structured …nance houses willing to buy athistorically high levels, who were structurally short ’vega’ through sales of guaranteed equity-linked products to retail investors and who were showing signi…cant mark-to-market losses onthe embedded Call options in these products.

22 Convertible arbitrageurs largely hedge out credit risk by selling the convertible security toa credit investor at a signi…cant discount to market - in e¤ect the ’‡oor value’ of the straightbond security taking into account the credit spread on the bond. The discount is paid for bythe credit investor giving the convertible arbirageur a call on the convertible at a strike ‡oorlevel which would require a very signi…cant improvement in the creditworthiness of the issuerfor the call on the convertible to be exercised. In such an event, the arbitrageur would callthe convertible, making good the credit investor’s leg of the asset swap with a tighter creditspread. In e¤ect, both parties stand to gain from the improvement in the creditworthiness of the issuer - with most but not all of the gain accruing to the credit investor.

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Distressed Debt [OLS Estimator]Risk Factors F k

1. 2. 3. 4.Global Global Equity US Agg.

High Yield Treasury Market Index bbk .04 .11 2.68 0.18

tstat 0.77 0.47 7.61 0.97 b = .0157 (18.81% annualised) tstat = 9.32

R2 = 33%

F test statistic ~24;143 = 17.24

p-value= 0.00

Table 5: OLS Results Fixed Income Arbitrage Hedge Fund Sector

was statistically insigni…cant at +0.04 and had a negligible e¤ect on the estima-tion of alpha in the regressions. Given the R2 of c. 33% for the regression modelthis would lead one to agree with anecdotal reports that the strategy largelyhedges out exposure to the credit asset class per se and instead derives its re-turns from astute non-directional betting on anticipated favourable movementsin spread di¤erentials between the returns of senior debt and common equitysecurities in target companies. In fact, the goodness-of-…t F-test for the fullmodel indicates that the hypothesis that all coe¢cients are zero (other than theconstant) can be soundly rejected. Again, given the largely neutral exposure tothe global credit asset class, this is strong evidence in support of the hypothesisthat the strategy does generate excess, risk-adjusted returns which are due to

manager skill; namely, superior security selection (i.e. relative value acumen)and proprietary trading and risk-management expertise.

4.3 MLE / Alpha-stable Results and Analysis

For the three hedge fund sectors examined we provide in the following Tablesthe results of the maximum likelihood regression analyses using a log-densityfunction which can properly account for non-normality e¤ects in the residuals.

4.3.1 Fixed Income Arbitrage

Table 6 shows that by not properly accounting for non-normality in the

return distribution (and hence in the residuals), the reduction in the estimatedJensen’s alpha (denoted bJ in Table 6) to 2.89% relative to the OLS estimatesuggests that the di¤erential of 160 basis points may have been incorrectlyattributed to manager skill. However, the robustness of this claim must for themoment await further empirical investigation as explained below.

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Fixed Income Arbitrage [MLE -Stable Estimator]Risk Factors F k

1. 2. 3. 4. 5. 6. Alpha-StableInt. Rate Vol Parameters

US Agg. Global Global FPSR Curve Term Structure b b b Index High Yield Treasury S hift T wist Shift (SE) (SE) (SE) bbk -.004 .039 -.007 .00 .00 .00 1.47 -1.0 0.00

SE .05 .026 i i i i (.128) (*) (.00)

Z stat -.08 1.5 i i i i 11.4 * 13.2 bJ = .0024 (2.89%) annualised Z stat = 1.23

R2 - 32%Max. Log Likelihood = 327

Table 6: MLE Performance Evaluation Results : Fixed Income Arbitrage

In light of the fact that the estimated value for the skewness parameter( b ) falls on its lower permitted boundary, the traditional maximum likelihoodapproach to calculating standard errors of the estimates is not valid. Whenthe maximum of the likelihood function is on the boundary of the log-likelihoodspace it is not possible to invert the Hessian to estimate con…dence intervals asthe Hessian may not be positive de…nite. Hence standard errors are not easilycalculated using the traditional information matrix approach. As an alternativeto manual manipulation of the optimisation algorithm to overcome this problem,in the interim we propose to estimate the parameter space conditional on theassumption that the true value of beta = -1.0 and estimate con…dence intervalsusing a simulation methodology.

Before proceeding we also draw the reader’s attention to the fact that theskewness parameter measures the relative weight of the negative and positiveheavy tails in the stable distribution. A beta of -1 implies that the negativetail is heavy and that the positive tail decays exponentially (like a normal dis-

tribution). An estimated value of b = 1:0 would imply a large systematicdownside risk with no corresponding upside and so would imply a substantialcompensatory risk premium for systematic negative skewness exposure, if thiswas indeed a true feature of the returns distribution for the strategy. The 160basis point di¤erential cited above does appear to support the latter hypoth-esis. Had the strategy been signi…cantly longer-lived than the sample period1994-2005, a legitimate concern would be that the small sample was pickingup a few extreme values which happened to be negative, for example Autumn1998 when convergence trades diverged rather than converged due to the post

credit crash ’‡ight to quality’ in global …xed income markets, whereas a largersample might have picked up some positive extremes. Nonetheless, the strategy’as we have known it’ since the early 1990’s does appear to have an underlyingreturn generating process which includes a systematic non-normality risk factorwhich in turn has empirically manifested as a heavily negatively skewed returns

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distribution over the sample period.

4.3.2 Convertible Arbitrage

Convertible Arbitrage [MLE -Stable Estimator]Risk Factors F k

Alpha-Stable

1. 2. 3. 4. 5. Parameters

US Agg. Global Equity Factors Global b b b Index Treasury Market Volatility High Yield (SE) (SE) (SE) bbk .07 .12 .02 .017 .03 1.41 -.59 .006

SE .098 .123 .021 .021 .035 (.125) (.175) (.00)

Z stat .71 .99 1.08 .78 .88 11.3 -3.38 10.5

bJ = .00264 (3.16% annualised) Z stat = 1.28

R2 = 11.9%Max. Log Likelihood = 415

Table 7: MLE Performance Evaluation Results : Convertible Arbitrage

As with the Fixed Income Arbitrage results, Table 7 shows that by notproperly accounting for non-normality in the return distribution, the increasein Jensen’s alpha to 4.69% in the OLS case suggests that 153 basis points hasbeen incorrectly attributed to manager skill. However, the robustness of thisclaim must again for the moment await further empirical veri…cation.

4.3.3 Distressed Debt

Distressed Debt [MLE -Stable Estimator]Risk Factors F k

1: 2. 3. Alpha-Stable

Lehman Lehman US Parameters

Global Global Equity b b b High Yield Treasury Market (SE) (SE) (SE) bbk .009 .34 1.85 1.83 -1.0 0.008

SE .04 .10 .30 (.082) * (.00)

Z stat .21 3.26 6.10 22.3 * 15.4

bJ = .0137 (16.40% annualised) . Z stat = 9.06

R2

= 33% +Max. Log Likelihood = 8,417

Table 8: MLE Performance Evaluation Results : Distressed Debt

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As with the other hedge fund indices examined, Table 8 shows the residualsto have a good …t to an alpha-stable distribution, with the skewness ( ) pa-

rameter indicating a particular heavy negative skew in the distribution. Theestimate for the kurtosis () parameter also suggests a somewhat fat-tailedresiduals distribution. The factor loading estimates are qualitatively similar tothe OLS estimates in Table 5 earlier, with two notable exceptions. The load-ing on the Global Treasury factor shows a notable increase in both level andsigni…cance from the standard OLS regressions.

Most signi…cantly, however, the annualised Jensen’s alpha estimate is some2.41% lower than in the case where non-normality in the returns and residualswas not accounted for in the regression framework. By properly accounting fornon-normality e¤ects in the regression framework, rather than through an addi-tional set of regressor variables, we provide evidence that the risk-premium forsystematic (negative) skewness in particular is signi…cant. Conversely, by notaccounting properly for a negative skewness factor when the return distributionis heavily negatively skewed, Jensen’s alpha is overstated by a substantial 241basis points. Although this component of excess return is attributable to risk-taking rather than to manager skill, the major portion of excess risk-adjustedreturns nonetheless appear to be attributable to manager skill, thus con…rmingthe conclusions drawn from the OLS regression results earlier.

5 Conclusions and Future Research

In this paper we evaluated hedge fund performance when returns are non-normaland display non-linear dependencies with underlying asset classes. We focusedin particular on the non-directional and hybrid classes of …xed-income strate-gies; convertible arbitrage, …xed income arbitrage and distressed debt where

relative-value trading is the dominant style. We constructed a germane arrayof implicit statistical risk factors which were designed to parsimoniously capturethe dynamic trading and risk-management strategies followed by ’relative value’hedge funds, as well as the non-linear dependencies of returns on traditional as-set classes. We found that hedge fund returns in this non-directional siblingclass of strategies can be attributed to a mix of "alpha" and "beta" factors.

Signi…cantly, we provided strong evidence to con…rm the intuition that bynot properly accounting for non-normality e¤ects in the regression framework,a signi…cant fraction of hedge fund returns can be incorrectly attributed tomanager skill in certain cases. Nonetheless, our results suggest that the majorproportion of the absolute returns generated by these hedge funds is justi…ablyattributable to the relative value acumen and the proprietary trading and risk-management expertise of fund managers. These relative-value strategies itwould appear generate both "alpha" and "beta" returns for their investors.

The maximum-likelihood alpha-stable estimator is well suited for capturingthe value-at-risk implications of return distributions with fat tails and negativelyskewed characteristics. We propose to advance this research in the near termwith a value-at-risk study of hedge fund returns when the return distribution is

35



well-…tted by an alpha-stable distribution such as that used in this study.

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6

References

[1] Bank of New York (2006).

[2] Credit Suisse First Boston / Tremont Hedge Fund Database.www.hedgeindex.com.

[3] Edwards, F. R., 1999, "Hedge Funds and the Collapse of Long-Term CapitalManagement," Journal of Economic Perspectives, 13, 189-210.

[4] Fabozzi, F., 2006, "The CMBS Market, Swap Spreads and Relative Value".

[5] Fung, W. and D.A. Hsieh, 1997, "Empirical Characteristics of DynamicTrading Strategies," Review of Financial Studies, 10, 275-302.

[6] Fung, W. and D.A. Hsieh, 1999, "A Primer on Hedge Funds," Journal of Empirical Finance, 6 , 309-331.

[7] Lowenstein, R., 1998, "When Genius Failed," Fourth Estate, London.

[8] Sharpe, W.F., 1997, "Asset Allocation: Management Style and Perfor-mance Measurement," Journal of Portfolio Management, 18, 7-19.

[9] Skiadopoulos, G., Hodges, S. and Clewlow, L., 1998, "The dynamics of implied volatility surfaces," Financial Options Research Centre Preprint1998/86, Warwick Business School, University of Warwick.

[10] National Pension Reserve Board

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A A Maximum Likelihood Estimator for -Stable

Return DistributionsB Regression with non-normal -Stable Errors

Consider the standard regression model

yi =kXj=1

xij j + "i; i = 1; : : : ; N (4)

where yi is an observed dependent variable, the xij are observed independentvariables, j are unknown coe¢cients to be estimated and "i are identically andindependently distributed. Equation 4 may be written in matrix form as

y = X + " (5)

where

y =

0BBB@y1y2...

yN

1CCCA ; X =

0BBBBB@x11 x12 : : : x1kx21 x22 : : : x2k

......

. . ....

xN 1 xN 2 : : : xNk

1CCCCCA ; =

0BBB@ 1 2...

k

1CCCA ; " =

0BBB@"1"2...

"N

1CCCA(6)

The standard OLS estimator of is

OLS = (X 0X )1X 0y (7)

Thus OLS = (X 0X )1X 0" (8)

Thus in the simplest case where X is predetermined OLS is a linear sumof the elements of ". If the elements of " are independent identically distributednon-normal -stable variables then OLS has an -stable distribution. Thevariance of "i does not even exist. Thus standard OLS inferences are not valid.([?]) prove the following properties of the asymptotic t-statistic

1. The tails of the distribution function are normal-like at 12. The density has in…nite singularities j1 xj at 1 for 0 < < 1 and

6=

1. When 1 < < 2 the distribution has peaks at

1.

3. As ! 2 the density tends to normal and the peaks vanish

When 1 < < 2 the OLS estimates are consistent but converge ar a rate of n

1

1 rather than n

1

2 in the normal case.

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[?, ?, ?] shows that, subject to certain conditions, the maximum likelihoodestimates of the parameters of an -stable distribution have the usual asymp-

totic properties of a Maximum Likelihood estimator. They are asymptoticallynormal, asymptotically unbiased and have an asymptotic covariance matrixn1I (; ; ; )1 where I (; ; ; ) is Fisher’s Information. [?] examines linearregression in the context of -stable distributions paying particular attention tothe symmetric case. Here the symmetry constraint is not imposed. Assumethat "i = yi

Pkj=1 xij j is -stable with parameters f;;; 0g. If we denote

the -stable density function by s(x; ; ; ; ) then we may write the densityfunction of "i as

s("i; ; ; ; ) =1

s

yi

Pkj=1 xij j

; ; 1; 0

!; (9)

the Likelihood as

L("; ; ; ; 1; 2; : : : ) =

1

n nYi=1

s

yi

Pkj=1 xij j

; ; 1; 0

!; (10)

and the Loglikelihood as

l("; ; ; ; 1; 2; : : : ) =nXi=1

n log( ) + log

s

yi

Pkj=1 xij j

; ; 1; 0

!!!

=nXi=1

("i):

(11)

The maximum likelihood estimators are the solutions of the equations

@l

@ m=nXi=1

0("i)xim = 0; m = 1; 2; : : : ; k

nXi=1

0("i)

"i"ixim = 0; m = 1; 2; : : : ; k

nXi=1

0("i)

"i(y1

kXj=1

xij j)xim = 0; m = 1; 2; : : : ; k

n

Xi=10("i)

"i(yi

k

Xj=1 xij j)xim = 0; m = 1; 2; : : : ; k

nXi=1

0("i)

"iyixim =

nXi=1

0("i)

"i

kXj=1

xij j

(12)

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If W is the diagonal matrix

W =

0BBBBBB@0("1)"1 0 : : : 0

0 0("2)"2

: : : 0...

.... . .

...

0 0 : : : 0("n)"n

1CCCCCCA ; (13)

Using the notation in equation (6)we may write equation (12) in matrix format.

X 0W y = (X 0W X ) (14)

or if X 0W X is not singular

= (X 0W X )1X 0W y (15)

Thus the maximum likelihood regression estimator has the format of a Gen-eralized Least Squares estimator in the presence of heteroscedasticity where

the variance23 of the error term "i is proportional to 0("i)"i

. The e¤ect of the“Generalized Least Squares” adjustment is to give less weight to larger obser-vations. Figure ?? compares the weighting pattern derived from equation (13)for -stable processes with = 1:2 and 1:6 with those of a standard normaldistribution. For compatibility purposes the -stable curves are drawn with = 1=

p 2. As expected the normal distribution gives equal weights to all obser-

vations. The estimator for -stable processes gives higher weights to the centerof the distribution and extremely small weights to extreme values. This e¤ectincreases as is reduced.

This result explains the results obtained by [?] who completed a Monte

Carlo study of the use of truncated means as measures of location in -stabledistributions. They found

When = 1:1 the :25 truncated 24 means are still dominant for all n. For = 1:3 and = 1:5 the :50 truncated means are generally best, and when = 1:9 the distributions of the :75 truncated means are uniformly less disperse than those of other estimators. Finally,when the generating process is Gaussian ( = 2) the mean is the “best” estimator. Of course it is also minimum-variance, unbiased in this case.

The shape of the weight curves in the skewed case is shown in …gure ( ??).The weights are based on the same -stable distributions as those in …gure ??except that is now

0:1. The most surprising aspect of the weighting systems

is the negative weights given to small positive observations. Again the e¤ectsare more pronounces as is reduced.

23 This is only an analogy. The vatiance of the error term does not exist24 A g truncated mean retains 100g% of the data. Thus a :25 truncated mean is an average

of the central 25% of the data

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C Asymptotic Distributional Properties of the

Maximum Likelihood Estimator with

-StableErrors

This will be completed to show the asymptotic distributional properties of theestimator. In particular, we will show the information matrix to be the inverseof the Hessian of the LLF at the solution and show how various goodness of …t tests are possible with this estimator, for example, Lagrangian Multiplier,Likelihood Ratio.

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