The Brave New World of Hedge Fund Indexes

© Edhec 2002

The Brave New World of Hedge Fund Indexes

Desperately Seeking Pure Style Indexes

Lionel Martellini

Marshall School of Business

University of Southern California

Chercheur Associé EDHEC martelli@usc.edu

Joint work with Noël Amenc

Mesure de la Performance et des Risques de la Gestion Alternative 18/06/02

Outline

• Motivation

– Performance Mesurement and Asset Allocation

– Problems with Hedge Fund Indexes

• The World of Hedge Fund Indexes

– Overview of Popular Hedge Fund Strategies

– Survey of Existing Hedge Fund Indexes

• Hedge Fund Indexes are not Created Equal

– Heterogeity in Competing Hedge Fund Index Returns

– Implications for Asset Allocation

• Desperately Seeking Pure Style Indexes

– Statistical Approach

– Portfolio Approach

• How Pure is Pure?

– The Two Basic Theorems of Pure Indexing

– Testing Representativeness of Pure Indexes

© Edhec 2002 Mesure de la Performance et des Risques de la Gestion Alternative 18/06/02

Motivation

From an Absolute to a Relative Return Perspective

• HF managers often use risk-free rate as a benchmark

• This absolute return approach is theoretically valid if and only if

– CAPM is the true model

– Hedge fund beta is zero

• Hedge fund indexes and sub-indexes are a natural choice for benchmarking hedge fund returns

• Right benchmarking is a fundamental problem in the presence of incentive fees

• Reliable HF indexes are also needed for

– Strategic Asset Allocation

– Tactical Asset Allocation


Motivation

Equity Universe, Sub-Universes, and Specialized Indexes

Universe of Equity

(20000+)

MSCI Sub-Universe

Russell Sub Universe

Wilshire Sub-Universe (5000 approx)

S&P Indices (value, growth)

Russell Indices (growth,value)

• Composite of World Unknown

• Sub-universe (e.g., MSCI) may or may not represent World Index

• Competing indexes for the same universe


Motivation

HF Universe, Sub-Universes, and Specialized Indexes

Zurich Hedge Fund Indices

Zurich Sub-Universe (1300 approx)

Universe of Hedge

Funds (6000+)

CSFB/Tremont Index

TASS Sub-Universe

HFR Sub Universe

EACM 100

• Composite of World Unknown

• Sub-universe (e.g., HFR, TASS) may or may not represent World Index

• Competing indexes for the same universe


Motivation

Problems are Amplified for Hedge Fund Indexes

• Existing indexes are not fully representative

– Because of the lack of regulation on hedge fund performance disclosure, existing data bases only cover a relatively small fraction of the hedge fund population

– Probably only a little more than half of existing hedge funds choose to selfreport their performance to one of the major hedge fund databases

– One of the most popular hedge fund indexes, the EACM 100, does not account for more than a tiny percentage of all existing hedge funds

– Most HF indexes are equally-weighted (all but CSFB/Tremont)

• Existing indexes are biased

– Most hedge fund indexes are based upon managers' self-proclaimed styles

– Given that hedge fund managers jealously protect the secret of their investment strategies (the so-called black-box problem), relying on managers' self-proclaimed style is actually almost a necessity

– This procedure only makes sense under the following two conditions: (1) a manager follows a unique investment and (2) a manager's self-proclaimed style matches the manager's actual trading strategies

– Style drift problem (see for example Lhabitant (2001))


Motivation

Pure Style Indexes are not Observable

Population of hedge funds following a given strategy

Lack of representativeness

Sample of hedge funds in the database used by a given commercial index

© Edhec 2002

Presence of a style bias


World of HF Indexes – Popular Strategies

Hedge Funds Classification

Hedge Fund Classifications

© Edhec 2002

Relative Value

Bond Hedge Equity Market Neutral

Convertible Arbitrage

Event Driven

Deal Arbitrage

Distressed Debt

Bankruptcy

Multi-Strategy

Equity Hedge

Long/Short Equity Long Bias

Global Macro

Discretionary Systematic


World of HF Indexes – Popular Strategies

Hedge Funds Styles

• Directional strategies: aimed at benefiting from market movements and trends

– Global macro

– Hedge (long bias)

– Long (e.g. “growth” or “value” stocks)

– Short (e.g. “overvalued”, “glamour” stocks)

• Non directional strategies:

– Event driven (corporate events such as takeovers, spin-offs, mergers, etc.)

– Restructuring (buying or shorting securities of companies under Chapter 11 and/or undergoing some form of reorganization)

– Fixed-income arbitrage (long and short via treasuries, corporate and/or asset-backed securities)

– Capital structure arbitrage (buying and selling different securities of the same issuer, e.g., convertibles/common stock)

– Equity market neutral (long-short zero beta strategies)


World of HF Indexes – Competing Providers

There exist at least a Dozen HF Index Providers

Providers

EACM

HFR

CSFB/Tremont

Zurich Capital

Van Hedge

Hennessee Group

Hedgefund.net

LJH Global Investments

MAR

Altvest

Magnum

# of Indexes Launch Date # of Funds

13

15

9

1996

1994

1999

100

1,100

340

Website eacmalternative.com

hfr.com

hedgeindex.com

5

12

22

33

16

15

13

8

2001

1995

1992

1979

1992

1990

2000

1994

60

750

450

1,800

800

1,300

1,400

NA zcmgroup.com

vanhedge.com

hedgefnd.com

hedgefund.net

ljh.com

marhedge.com

altvest.com

magnum.com



Strategies Covered in this Paper

Sub-Universe


Emerging Markets

Equity Market Neutral

Event Driven

Fixed Income Arbitrage

Global Macro

Long/Short

Merger Arbitrage

Relative Value

Short Selling

Distressed Securities

Fund of Funds

List of Competing Indexes

CSFB, HFR, EACM, Zurich, Hennessee, HF Net

CSFB, Altvest, HFR, MAR, Van Hedge, Hennessee, HF Net

CSFB, Van Hedge, HFR, MAR, Hennessee, HF Net

CSFB, Altvest, MAR, EACM, HFR, Hennessee, HF Net, Zurich

CSFB, HFR, Van Hedge, Hennessee, HF Net

CSFB, Altvest, Van Hedge, MAR, HFR, Hennessee, HF Net, Magnum 02/97

CSFB, Altvest, Zurich, EACM, HF Net 01/98

Altvest, HFR, Zurich, Hennessee, EACM, HF Net 01/98

Altvest, HFR, Van Hedge, EACM, HF Net

Altvest, HFR, Van Hedge, MAR, EACM

Van Hedge, Altvest, HFR, EACM, Zurich, HF Net, Hennessee

Van Hedge, Altvest, HFR, Zurich

01/96

01/96

01/98

01/98

Date

01/98

01/96

01/96

01/98

01/96



Strategies Not Covered in this Paper

© Edhec 2002

Sub-Universe

Market Timing

Aggressive Growth

International

Special Situations

Opportunity

List of Competing indexes

HFR, Van Hedge

Magnum, Van Hedge, HF Net

Magnum, MAR, EACM, Hennessee

Magnum, Van Hedge, HF Net

Magnum, Van Hedge, EACM, HF Net

Emerging Markets - Latin America

Fixed Income - High Yield

Regulation D

Sectors - Energy

Sectors - Financial

HFR, Hennessee

HFR, Hennessee

HFR, HF Net

HFR, HF Net

Van Hedge, Altvest, HFR, EACM, Zurich, HF Net

Sectors - Technology

Sectors - Health Care

Statistical Arbitrage

Capital growth

Altvest, HFR, Magnum, HF Net

Altvest, HFR, HF Net, Hennessee

HFR, HF Net

Magnum, Hennessee


Hedge Fund Indexes are not Created Equal

Heterogenity in Hedge Fund Indexes – Max Difference

Sub-Universe


Emerging Markets

Max Difference (with dates and indexes)

4.75% (Oct 98; CSFB (-4.67) / Hennessee (0.08))

19.45% (Aug 98; (MARH -26.65) / Altvest (-7.2))

Equity Market Neutral 5.00% (Dec 99; Hennessee (0.2) / Van Hedge (5.2))

Event Driven 5.06% (Aug 98; CSFB (-11.77%) / Altvest (-6.71))

Fixed Income Arbitrage 10.98% (Oct 98; HF Net (-10.78) / Van Hedge (0.2))

Global Macro

Long/Short

Merger Arbitrage

17.80% (May 00: Van Hedge (-5.80) / HF Net (12))

22.04% (Feb 00: EACM (-1.56) / Zurich (20.48))

1.85% (Sep 98: Altvest (-0.11) / HFR (1.74))

Relative Value

Short Selling

10.47% (Sep 98: EACM (-6.07) / Van Hedge (4.40))

21.20% (Feb 00: Van Hedge (-24.3) / EACM (-3.09))

Distressed Securities 7.38% (Aug 98: HF Net (-12.08) / Van Hedge (-4.70))

Fund of Funds 8.01% (Dec 99: MAR-Zurich (2.41) / Altvest (10.42))



Heterogeity in Hedge Fund Indexes – Correlation

Sub-Universe


Emerging Markets


Event Driven


Global Macro

Long/Short

Merger Arbitrage

Relative Value

Short Selling


Fund of Funds

Average Correlation Lowest Correlation

0.8183

0.6350

0.9284

0.4276

0.9232

0.8301

0.1258

0.8458

0.5407

0.5598

0.4575

0.9193

0.6752

0.8811

0.8645

0.8757

0.2254

0.2698

-0.1901

0.8797

0.3042

0.7796

0.7218

0.7985



Implications for Asset Allocation

14.00%

12.00%

10.00%

8.00%

100% S&P 500

6.00%

4.00%

2.00%

0.00%

0.00% 2.00%

100% Lehm an US aggregate

4.00% 6.00%

CSFB HFR

8.00%

Van Hedge

10.00%

Henessee HF net

12.00% 14.00%

S&P + Lehman

16.00% 18.00%

Efficient frontiers based on monthly data for the period extending from January 1996 to October 2001



Investment in Hedge Funds

• Problems

– Competing indexes disagree

– All are potentially flawed

• Can not tell which is best

– All existing indexes have both advantages and drawbacks

– For example, Zurich indexes may be less biased than some of their competitors, but they are less representative

• Statistical approach

• Portfolio approach

– Maximization of Representativeness

– Minimization of Bias



Statistical Approach – Kalman Filter

• Simple model

R kt



I t

 e kt

– R

–

I t

– e kt kt is the return on competing index k is the (unobservable) return on pure index is the noise, measurement error resulting from presence of biases and absence of representativeness

• Assume normally distributed pure indexes and measurement errors

• Kalman filter is used both to evaluate the likelihood function and to forecast and smooth the unobserved state variables

• Here, our motivation is to estimate (smooth) the unobserved pure index, based on the observed returns on competing indexes:

I

ˆ t , T

:



E



I t

R

1

, R

2

,..., R

T





Statistical Approach – The Results

Sub-Universe Mean Volatility Est. Error Average Correlation

Conv. Arbitrage 0.85% (0.94%) 1.17% (1.34%) 0.28% 0.9068

Emerging Markets 0.08% (0.43%) 5.16% (5.34%) 0.04%

Eq. Market Neutr.

0.91% (0.83%) 0.40% (1.03%) 0.17%

0.9472

0.6632

Event Driven 0.93% (0.86%) 1.94% (2.09%) 0.22%

Fixed Income Arb. 0.56% (0.36%) 1.36% (1.70%) 0.04%

Global Macro 0.92% (0.69%) 2.34% (2.55%) 0.34%

Long/Short 1.13% (1.11%) 1.11% (3.13%) 0.00%

0.9641

0.7260

0.8157

0.6407

Merger Arb.

1.11% (0.89%) 1.19% (1.31%) 0.15%

Relative Value 1.01% (0.90%) 0.91% (1.97%) 0.14%

Short Selling

Distressed

0.16% (0.49%) 7.45% (7.70%)

0.52% (0.73%) 2.21% (2.26%)

1.07%

0.27%

Fund of Funds 0.73% (0.91%) 2.51% (2.52%) 0.26%

0.9649

0.8201

0.9277

0.9353

0.9398

Performance of Kalman filter pure index (in parenthesis, the average of competing index)



Statistical Approach – Convertible Arbitrage

0.1

0.08

0.06

0.04

0.02

0

Fe b-

98

Ap r-9

8

Ju n-9

8

Au g-

98

Oc t-9

8

De c-9

8

Fe b-

99

Ap r-9

9

Ju n-9

9

Au g-

99

Oc t-9

9

De c-9

9

Fe b-

00

Ap r-0

0

Ju n-0

0

Au g-

00

Oc t-0

0

De c-0

0

-0.02

-0.04

-0.06

-0.08

-0.1

© Edhec 2002

Zurich

EACM

CSFB

Kalman

Henessee

Kalman - 2 Sigmas

Mesure de la Performance et des Risques de la Gestion Alternative

HFR

Kalman + 2 Sigmas

18/06/02


Portfolio Approach – The Method

• Black-Box problem

– A pure index generated Kalman filtering techniques has an appealing built-in element of optimality (minimized mean squared error)

– It can not, however, be regarded as index in its own right, since it can not be expressed as a portfolio of individual hedge funds

• Portfolio approach

– Taking a portfolio of existing indexes should intuitively be better than selecting any of them

– Is equally-weighting a good scheme?

• Maximization of representativeness

– Principal component analysis

• Minimization of bias

– Minimum-variance analysis



Portfolio Approach – PCA

• Use factor analysis techniques to generate a set of pure indexes

– They can be thought of as the best possible one-dimensional summaries of information conveyed by competing indexes for a given style, in the sense of the larger fraction of the variance explained.

– Here, we are looking for the portfolio weights that make the combination of competing indexes capture the largest possible fraction of the information contained in the data from the various competing indexes

• The method

– From a mathematical standpoint, it involves transforming a set of K correlated competing indexes into a set of orthogonal variables, or implicit factors, which reproduces the original information present in the correlation structure

– Each implicit factor is defined as a linear combination of original variables

– Use the first component of a PCA of competing indexes as a candidate for a pure style index (typically captures a large proportion of cross-sectional variations because competing styles tend to be at least somewhat positively correlated)



PCA – The Results

© Edhec 2002

Sub-Universe


Emerging Markets


Event Driven


Global Macro

Long/Short

Merger Arbitrage

# of Indexes % of Variance Explained

6 84.91

7

6

91.97

58.91

8

6

8

5

4

85.41

65

74.13

86.8

83.81

Relative Value

Short Selling


Fund of Funds

7

5

5

5

71.26

78.42

77.6

91.19



Portfolio Approach – Min Var Analysis

• Pure hedge fund indexes generated as the first component in a factor analysis should be as representative as possible since there is no other linear combination of competing indexes that implies a lower information loss

• Another approach consists in focusing on minimization of the bias

• Use same model as before

R kt



I t

 e kt

• Add assumptions

– Noise term is zero on average

– Noise term is uncorrelated with return on pure index

– Homoscedastic model



Portfolio Approach – Min Var Analysis

• Define the return on a portfolio of indexes

R pt

 w '



R t



I t

 w '

 e t

• Under previous assumptions, we have that

E

 

Var pt



  pt

E



 

 t

I

2  w '

  e

 w

• The problem of minimizing the bias is

Min w

Var

  pt

 w '

  e

 s .

t .

w '



1



1 w

• Solution is w

* 



1 '

 



1 ee



1 ee





1

1



Portfolio Approach – Additional Assumption

• If one is willing to make the additional assumption of no correlation between noise terms for various competing indexes, then the variance-covariance matrix of residuals is diagonal, and one obtains the following simple solution w i

*  j



 i

2

 j

2

1

 

I

2

1

 

I

2

• May also impose positivity constraints to ensure that resulting index is a long-only portfolio of individual hedge funds



Portfolio Approach – Additional Assumption

© Edhec 2002

Sub-Universe Mean Volatility

Convertible Arbitrage 1.08% (0.94%) 1.13% (1.34%)

Average Correlation

0.90

Emerging Markets 0.80% (0.43%) 3.33% (5.34%)

Equity Market Neutral 0.85% (0.83%) 0.44% (1.03%)

0.95

0.58

Event Driven 0.95% (0.86%) 3.78% (2.09%)

Fixed Income Arbitrage 0.37% (0.36%) 1.33% (1.70%)

Global Macro

Long/Short

0.09% (0.69%) 3.73% (2.55%)

0.78% (1.11%) 0.76% (3.13%)

0.87

0.78

0.70

0.56

Merger Arb.

Relative Value

Short Selling

Distressed

Fund of Funds

0.99% (0.89%) 1.15% (1.31%)

0.82% (0.90%) 1.00% (1.97%)

0.82% (0.49%) 6.95% (7.70%)

0.46% (0.73%) 2.55% (2.26%)

0.94% (0.91%) 2.45% (2.52%)

0.92

0.68

0.47

0.80

0.95


How Pure is Pure?

The Two (Obvious) Theorems of Pure Indexing

• PCA and min-var analysis have appealing element of optimality

• However, any portfolio (e.g., equally-weighted portfolio) should perform better than competing indexes

• Theorem 1: An index of the indexes is always less biased than the average of the set of indexes it is extracted from

 e

2 p

 j n 



1 w j

 e

2 j

• Theorem 2: An index of the indexes is always more representative than any competing index I

1



I

2



I

3

I

3





I

1



I

2



Index 1

I

1





I

2



I

3



I

2





I

1



I

3



Index 2

18/06/02 © Edhec 2002

Index 3

Mesure de la Performance et des Risques de la Gestion Alternative


Testing Representativeness – The Method

• Reduction of bias is very hard to test (chicken-and-egg problem)

• We have tested enhancement of representativeness using the following test experiment

– We have merged the 3 major databases providing information on individual fund return (MAR, TASS and HFR)

– We have also added data on funds which do not report to any data base, that had been directly obtained from administrators

– We have gathered monthly returns on a total of 7,422 hedge funds, including 2,317 funds that do not report their returns to the major data bases

– We have formed equally-weighted portfolios for each style based on managers’ self-proclaimed styles and compute correlation with pure indexes

• These portfolios are

– Arguably biased

– Undoubtly representative



Testing Representativeness – The Results

Sub-Universe Kalman (%) PCA (%) Min. Bias (%) Min. Bias Positive Weights

Convertible Arbitrage 0.84 (67%) 0.84 (67%) 0.76 (33%)

Emerging Markets 0.95 (29%) 0.98 (86%) 0.95 (29%)

0.78 (50%)

0.86 (14%)

Equity Market Neutral 0.38 (83%) 0.41 (83%) 0.38 (67%)

Event Driven 0.95 (75%) 0.96 (100%) 0.92 (37%)

Fixed Income Arbitrage 0.93 (80%) 0.81 (60%) 0.88 (60%)

Global Macro 0.71 (86%) 0.77 (86%) 0.55 (43%)

0.38 (67%)

0.92 (37%)

0.76 (60%)

0.78 (86%)

Long/Short

Merger Arbitrage

Relative Value

Short Selling

Distressed

Fund of Funds

0.63 (40%)

0.85 (67%)

0.88 (83%)

0.69 (50%)

0.95 (100%)

0.93 (75%)

0.98 (100%)

0.86 (67%)

0.89 (83%)

0.73 (50%)

0.94 (86%)

0.94 (100%)

0.41 (20%)

0.86 (67%)

0.61 (17%)

0.44 (0%)

0.82 (14%)

0.93 (75%)

0.24 (20%)

0.84 (67%)

0.88 (83%)

0.75 (100%)

0.91 (43%)

0.83 (25%)


Conclusion

• Contribution

– Document heterogeneity in competing hedge fund index providers

– Attempt to provide remedies to the problem

• Extensions

– We have suggested that a database of pure style indexes be maintained at the EDHEC-MISYS Risk and Asset Management

Research Center, and posted on a dedicated web site

– The index construction methodology

• Step 1: Use the first 3 years of monthly returns to calibrate the model.

• Step 2: Perform a PCA analysis of competing indexes for each strategy on the data used for calibration purposes

• Step 3: These portfolios are held for 3 months, their monthly returns are recorded, and the same process is repeated

– Our results can easily be extended to traditional investment styles such as growth/value, small-cap/large-cap


The Brave New World of Hedge Fund Indexes

The Brave New World of Hedge Fund Indexes

Related documents

Products

Support

The Brave New World of Hedge Fund Indexes

The Brave New World of Hedge Fund Indexes

Related documents

Add this document to collection(s)

Add this document to saved

Suggest us how to improve StudyLib