Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Hedge Fund Replication and Alternative Beta Thierry Roncalli

1 Évry 2 Department

1

Guillaume Weisang

2

University, France

of Mathematical Sciences, Bentley University, MA

Laboratoire J. A. Dieudonné, Université de Nice Sophia-Antipolis, April 29, 2009

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Outline 1

Investing in Hedge Funds The Success of the Hedge Fund Industry Hedge Fund Replication Factor Models Previous Works

2

HF Replication and Tracking Problems Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

3

Alpha Considerations What is Alpha? Explaining the Alpha The Core/Satellite Approach

4

Conclusion HF Replication in Practice Main Ideas

5

Appendix

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

The Success of the Hedge Fund Industry Hedge Fund Replication Factor Models Previous Works

The Success of the Hedge Fund Industry

Hedge-Funds (HF) deliver higher Sharpe ratios than Buy-and-Hold

1

strategies on traditional asset classes : HFRI

SPX

UST

Annualized Return

9.94%

8.18%

5.60%

1Y Volatility

7.06%

14.3%

6.95%

0.77

0.26

0.18

Sharpe

The performance behaviour of the HF industry was very good during the equity bear market between 2000 and 2003.

1 Jan.

1994 - Sep. 2008 Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

The Success of the Hedge Fund Industry Hedge Fund Replication Factor Models Previous Works

Performance between January 1994 and September 2008

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

The Success of the Hedge Fund Industry Hedge Fund Replication Factor Models Previous Works

Motivation Appealing characteristics of Hedge-Fund returns

Superior risk-return proles Moderate or time-varying correlation with standard assets (approximate call option) However, investment in Hedge-Funds is limited for many investors

Regulatory or minimum size constraints (retail, institutional) Main criticisms: Lack of transparency (risk management  black box) Poor liquidity (HF: monthly or quarterly, FoHF monthly, weekly or daily) → particularly relevant in period of stress (redemption problems) Fees (problem of pricing rather than high fees) Emergence of clones

To alleviate HF limitations for investors seeking exposure to HF type returns: daily trading, no size limitations, etc. Clones cannot compete with the best single hedge funds, but they give a signicant part of the performance of the hedge fund industry. Distinction HF Tracker / HF Risk Management. Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

The Success of the Hedge Fund Industry Hedge Fund Replication Factor Models Previous Works

Replication Methods Replication of Strategies

Replication of Payo

Factor Models

Systematic (quantitative) replication of strategies known to be followed by some funds

Distribution (Kat

Reproduce the beta and risk/return prole of (the average of) hedge funds through investment in liquid instruments on standard asset classes

Examples

FX carry trade, selling of volatility, momentum strategy

Strategy reproducing volatility, skewness, kurtosis and correlation of conventional asset classes to HF Strengths

Strengths

Transparency, exposure to non-linear payos and earnings of associated premiums Weaknesses

Approach)

Our Focus !

Weak dependency on current economic and nancial conditions (according to Harry Kat) Weaknesses

allocation methodologies, amount to creation of a new fund

no replication of average returns, no time horizon, big entry ticket (amount to creation of tailor-made HF), exposed to breakdown in distribution and correlation

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

The Success of the Hedge Fund Industry Hedge Fund Replication Factor Models Previous Works

Replication of Strategies These tools are very useful, in particular since the Mado Fraud [Clauss et al., 2009].

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

The Success of the Hedge Fund Industry Hedge Fund Replication Factor Models Previous Works

Replication of Payo Distribution

This approach has been proposed by Harry Kat. Mathematical framework by [Hocquard et al., 2008]. Main idea :

1 2

Because Payo function ⇒ return distribution, one may nd the payo function that implies the desired HF return distribution. One can then generate that distribution by buying this payo or by replicating the payo.

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

The Success of the Hedge Fund Industry Hedge Fund Replication Factor Models Previous Works

Main idea behind Factor Models Assumption The structure of all HF returns can be summarized by a set of risk factors

{r (i ) , i = 1, ..., m}. Comment

Arbitrage Pricing Theory (APT). Extension of Sharpe's style analysis for performance assessment.

A typical replication procedure Step 1: Factor model for HF returns:

rkHF = ∑mi=1 β (i ) rk(i ) + εk

Step 2: Identication of the replicating portfolio strategy

m

rkClone = ∑ βˆ (i ) rk(i ) i =1

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

The Success of the Hedge Fund Industry Hedge Fund Replication Factor Models Previous Works

Previous Works Overview

Linear models with linear assets [Fung and Hsieh, 1997, Amenc et al., 2007, Hasanhodzic and Lo, 2007]

Dierent types of factors have been included depending on the type of strategies followed by hedge funds. e.g. Convertible and Fixed Income Arbitrage, Event Driven, Long/Short Equity, etc. More factors to improve in-sample (and out-sample) t?

Dierence between replication from an academic and a practitioner point of view (explaining vs. replicating as an investment) Option-based models Some authors have introduced options on an equity index as part of the factors [Diez de los Rios and Garcia, 2008]

m

rkHF = ∑ β (i ) rk(i ) + β m+1 max(rk(1) − sk , 0) + εk i =1

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

The Success of the Hedge Fund Industry Hedge Fund Replication Factor Models Previous Works

Previous Works Estimation procedures

Classically, estimation and calibration procedures (in chronological order):

Full factor model regressions, stepwise regressions (versus economic selection of factors), and rolling-windows OLS (to try to capture dynamic allocation). More recently, state-space modeling has been introduced to model and estimate HF returns:

(i ) (i ) Markov Regime-Switching Model: rkHF = ∑m i =1 β (Sk )rk + εk with
2 εk ∼ N 0, σ (Sk ) and Sk is a discrete variable representing the state of the nature [Amenc et al., 2008]; and Kalman Filter [Roncalli and Teiletche, 2008].

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

The Success of the Hedge Fund Industry Hedge Fund Replication Factor Models Previous Works

Previous Works Results

[Amenc et al., 2007] nd that linear factor models fail the test of robustness, giving poor out-of-sample results. It seems that economic selection of factors provides a signicant improvement over other methodologies on the tracking error in the out-of-sample robustness test. Capturing the

unobservable dynamic allocation is very dicult, and

estimates can vary greatly at balancing dates Non-linear models still represent a methodological challenge from a replication point-of-view

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Tracking Problems Denition The following two equations dene a tracking problem (TP) [Arulampalam et al., 2002]:



f (tk , xk −1 , νk ) h(tk , xk , ηk )

= =

xk zk

(Transition Equation) (Measurement Equation)

where: xk

∈ R nx

at step

νk

et

is the state vector, and zk

k.

ηk

∈ R nz

the measurement vector

are mutually independent i.i.d noise processes.

The functions

f

and

h can be non-linear functions.

The goal in a tracking problem is to estimate xk , the current state at step

k

using all available measurements z1:k .

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Global Tactical Asset Allocation as a Tracking Problem m asset classes acting as factors. Let rkF (i ) th be the return of the GTAA fund and rk be the return of the i asset class (`the factors') at time index k . We assume that: Assume a factor model with

m

rkF = ∑ βk(i ) rk(i ) + ηk i =1

(i ) and β (t )

(i )

t ∈ ]tk −1 , tk [ where β((ti)) is the weight of the i th asset class at time t . = βk

for

We associate the following tracking problem:



βk

rkF

where the vector of weights vector, and

rkF

= =

f (tk , βk −1 , νk ) >β

rk



k + ηk (1)

 (m) >

βk = βk , . . . , βk

is the state

is the measurement.

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Solving with Kalman Filter Description

f

is linear:

βk = βk −1 + νk and

νk

and

ηk

are mutually independent i.i.d Gaussian noises.

The Kalman Filter is a recursive algorithm providing the optimal solution in the linear Gaussian case. At each time index

k , among

other things, it provides us with :

(i ) βk |k −1 the prediction of the exposures; (i ) βk |k the ltered estimate of the exposures; (i ) βk |n the smoothed estimate of the exposures.

If

(i )

νk = 0, βk |k

is the recursive OLS estimate whereas

(i )

βk |n

is the OLS

estimate.

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Solving with Kalman Filter An Example

We constructed a GTAA fund allocating wkUS in MSCI USA and wkEU = 1 − wkUS in MSCI EU. Moving OLS: choice of the lag window? Kalman Filter:

β0 ∼ N (b0 , 0) with
b0 = w0US , w0EU > ηk = 0 νk ∼ N (0, Q )

KF #1: Q is a full matrix. KF #2: Q is a diagonal matrix (correlation between (1) (2) βk and βk is zero).

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Solving with Kalman Filter Results

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Link between GTAA and HF replications The return of a hedge fund may be decomposed into two components:

m

p

i =1

i =m+1

rkHF = ∑ βk(i ) rk(i ) + ∑ |

{z

}

|

GTAA factors

(i ) (i )

βk rk {z }

HF factors

The idea of HF replication is to replicate the rst part. Let's note

(i ) (i )

ηk = ∑pi=m+1 wk rk

. The TP system becomes:

 ⇒

βk = βk −1 + νk rkHF = rk> βk + ηk

HF replication is done at the industry level because of the term

which captures the performance of stock picking strategies, illiquid assets, non-linear assets, high frequency strategies, etc.



Aggregation Ecient market hypothesis

⇒

Long/Short Equity

6=

?

=⇒



E [νk ] = 0   σ [νk ]  σ rkHF

Equity Market Neutral.

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

ηk

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Replicating with Kalman Filter Some remarks about model specication

Initialization step:

β0 ∼ N (b0 , V0 ) ⇒

Diuse prior distribution.

Factors specication and system identication: importance of low correlation between factors Specication of the covariance matrix

Q

of the transition equation

(diagonal or not ? Should it be constant ?)

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Replicating with Kalman Filter An example

We consider the replication of the Hedge Fund Research (HFR) Index using 6 asset classes:

an equity exposure in the S&P 500 index (SPX) a L/S position between Russell 2000 and S&P 500 indexes (RTY/SPX) a L/S position between DJ Eurostoxx 50 and S&P 500 indexes (SX5E/SPX) a L/S position between Topix and S&P 500 indexes (TPX/SPX) a bond position in the 10-years US Treasury (UST) and a FX position in the EUR/USD For realistic results, exposures are assumed to be realised using futures (hedged in USD) with a monthly sampling period. Study period: January 1994  September 2008.

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Replicating with Kalman Filter i

()

Results: Evolution of the weights βˆk |k −1

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Replicating with Kalman Filter Decomposing HF returns into alpha and beta

Following [Hasanhodzic and Lo, 2007, Roncalli and Teiletche, 2008], one can introduce the following concepts:

rkHF − rk(0)

= + +

where

  (i ) (0) β¯ (i ) rk − rk ∑m i =1    (i ) (0) (i ) ∑m βˆk |k −1 − β¯ (i ) rk − rk  i =1    (0) (i ) m ˆ (i ) ˆ (i ) rkHF − 1 − ∑m i =1 βk |k −1 rk + ∑i =1 βk |k −1 rk

(Traditional Beta) (Alternative Beta) (Alternative Alpha)

rk(0) represents the risk-free return and β¯ (i ) = n1 ∑nk=1 βˆk(i|k) −1 .

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Replicating with Kalman Filter Performance attribution between alpha and beta

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Replicating with Kalman Filter Decomposition of the yearly performance

Period 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 1994-2008 1997-2008 2000-2008

Traditional Alpha Beta 0.43 1.13 6.99 13.56 11.77 8.35 7.21 8.94 -3.98 6.87 15.56 13.62 3.03 1.90 4.08 0.53 4.39 -5.59 2.99 16.08 1.23 7.71 2.30 6.84 2.32 10.33 5.30 4.43 -5.96 -5.13 3.80 5.92 3.14 5.46 2.20 4.02

Thierry Roncalli, Guillaume Weisang

Alternative Alpha Beta 0.68 0.88 7.00 13.55 12.18 7.95 -2.61 19.93 -4.44 7.39 7.96 21.61 3.63 1.31 2.11 2.47 0.74 -2.18 3.96 15.00 1.83 7.08 1.44 7.74 1.10 11.67 3.35 6.39 -4.90 -6.19 2.22 7.55 1.14 7.55 1.48 4.75

Total 1.56 21.50 21.10 16.79 2.62 31.29 4.98 4.62 -1.45 19.55 9.03 9.30 12.89 9.96 -10.78 9.94 8.77 6.30

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Replicating with Kalman Filter

Performance attribution of the replicated strategy

The main contributor is the long equity exposure. Three other strategies have a good contribution:

the two L/S equity strategies on small caps and Eurozone; and the FX position EUR/USD. TPX/SPX and US Treasury Bonds have little impact. Nonetheless, in the replication process, they can help track the volatility. Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Replicating with Kalman Filter The equity bear market between 2000 and 2003

Highest exposure to equity was in March 2000 (≈ 64%) After March 2000, it appears that the good performance of the HF industry may be explained by two components (cf. bottom right gure):

a decrease of directional equity leverage; (b) a good position on L/S equity on US small caps (RTY) and a good bet on L/S equity between DJ Eurostoxx 50 and SP500 (SX5E). (a)

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

The Problem of Factors Selection Two questions about adding or deleting a factor:

1

Improvement of the performance of the replication ?

2

Pertinence ? (risk management

+ + + + + + + − − − − − −

6F CREDIT GSCI VIX BUND JPY/USD USD/GBP MXEF/SPX SPX RTY/SPX SX5E/SPX TPX/SPX UST EUR/USD 7F

µˆ 1Y

7.55 7.35 7.46 6.55 7.75 7.37 7.48 7.56 6.42 7.08 6.51 7.34 7.86 6.57 7.82

πAB

75.93 73.91 75.07 65.94 77.94 74.18 75.25 76.06 64.56 71.20 65.47 73.82 79.13 66.08 78.64

Thierry Roncalli, Guillaume Weisang

6=

HF tracker)

σTE

3.52 3.51 3.55 4.05 3.54 3.56 3.58 3.03 6.31 4.66 3.73 3.72 3.50 3.60 3.05

ρ

87.35 87.46 87.42 83.71 87.09 87.02 86.81 90.68 47.51 75.92 85.88 85.78 87.47 86.59 90.55

τ

67.10 67.30 68.74 67.29 66.95 66.42 66.66 72.92 32.19 54.02 68.19 64.43 66.92 66.66 72.92

ρS

84.96 85.11 86.52 85.14 84.84 84.23 84.63 89.94 45.82 73.55 85.94 82.30 84.79 84.70 89.95

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Which Strategy may be replicated? Results with the 6F model Name

HFRI Event - Driven (Total)

HFRI ED: Merger Arbitrage HFRI ED: Private Issue/Registered Daily HFRI ED:Distressed / Restructuring HFRI Equity Hedge (Total)

HFRI HFRI HFRI HFRI HFRI

EH: EH: EH: EH: EH:

Energy / Basic Materials Equity Market Neutral Quant. Directional Short Bias Technology / Healthcare

HFRI HFRI HFRI HFRI

Emerging Emerging Emerging Emerging

HFRI Emerging Markets (Total)

Markets: Asia Excluding-Japan Markets: Global Markets: Russia/E Europe Markets:Latin America

HFRI Macro (Total)

HFRI Macro:Syst. Diversied HFRI Relative Value (Total)

HFRI HFRI HFRI HFRI HFRI HFRI HFRI HFRI HFRI

RV: Yield Alternatives RV:Fixed Income - Asset Backed RV:Fixed Income - Conversion Arbeit RV:Fixed Income - Corporate RV:Multi - Strategy FOF: Conservative FOF: Diversied FOF: Market Defensive FOF: Strategic

HFRI Fund of Funds Composite HFRI Fund Weighted Hedge Fund

Thierry Roncalli, Guillaume Weisang

πAB

73.08 62.72 34.11 79.77 53.97 41.41 70.59 69.59 4.48 56.81 71.99 82.47 60.29 35.21 73.96 69.97 49.26 67.94 81.76 50.81 80.26 105.67 90.95 85.13 91.91 68.56 98.25 92.31 75.93

σTE

4.16 2.93 6.73 4.70 4.50 17.15 2.84 5.23 10.18 10.69 10.37 9.99 10.52 25.73 13.61 5.71 6.06 3.07 5.98 4.29 4.36 4.23 3.06 2.98 4.64 5.09 5.62 4.23 3.52

ρ

78.20 65.23 31.15 58.36 87.35 46.20 45.77 92.48 86.34 82.66 69.99 63.84 65.94 43.47 61.68 61.82 65.87 56.46 52.23 7.53 52.48 51.41 56.41 63.09 70.65 44.05 79.14 73.40 87.35

τ

59.58 43.75 24.90 41.50 68.65 30.39 32.31 75.32 70.69 61.39 47.27 47.23 46.14 30.52 39.54 44.21 57.99 40.48 32.30 4.75 30.30 35.73 41.59 47.63 50.62 29.13 58.81 53.74 67.10

HF Replication and Alternative Beta

ρS 78.55 60.87 36.17 57.14 87.12 42.98 44.52 91.63 87.18 78.99 64.78 64.56 62.47 43.60 55.36 62.28 76.70 55.74 45.27 6.77 43.74 49.96 58.42 65.09 68.71 41.17 77.02 71.94 84.96

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Hedge Fund Replication: The Non-Linear Non-Gaussian Case HF Returns are not Gaussians

negative skewness and positive excess kurtosis. Non-Linearities in HF Returns

Non-linearities in hedge-fund returns have been documented from the very start of hedge-fund replication  see, e.g., [Fung and Hsieh, 1997]. It appears that non-linearities are important for some strategies but not for the entire industry [Diez de los Rios and Garcia, 2008]. Non-linearities may be due to positions in derivative instruments or uncaptured dynamic strategies  see, e.g., [Merton, 1981]. No successful hedge fund replication using non-linear models has ever been done

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

The Gaussian Distribution Assumption Framework

We consider the following tracking problem:  HF  rk = rk> βk + ηk β = βk −1 + νk  k ηk ∼ H with H a non Gaussian distribution. TP may be solved using Particle Filters. Assuming that H is a Skew three estimation methods:

t distribution S


µη , ση , αη , νη , we consider

(PF #1) We estimate by ML the parameters of H using the KF tracking errors.

(PF #2) We estimate the m + 3 parameters by GMM with the moment conditions (classical MM + two moments on the skewness and kurtosis). (PF #3) The estimates are those of (PF #2) except for the parameter αˆ η which is forced to -10. Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

The Gaussian Distribution Assumption Results with SIR algorithm and 50000 particles µˆ 1Y

σˆ 1Y

s 0.77 0.45 0.45 0.43 0.31

σTE

87.35 84.71 82.51 77.62

HF LKF2 PF #1 PF #2 PF #3

9.94 7.55 7.76 7.57 6.90

7.06 6.91 7.44 7.28 7.99

LKF PF #1 PF #2 PF #3

75.93 78.09 76.13 69.43

πAB

3.52 4.03 4.25 5.11

ρ

γ1

γ2

-0.57 -0.02 -0.03 -0.11 -0.57

2.76 2.25 2.02 1.93 2.88

τ

84.96 81.94 80.20 73.55

67.10 63.49 61.60 54.75

ρS

Conclusion With linear assets, higher kurtosis and negative skewness come at the cost of a higher tracking error

⇒

σTE .

It is not the right way to do it.

2 LKF

= 6F model + KF. Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Taking into account Non-Linear Assets Using option factors

Problem: Results are data dependent We must be careful with backtests on options. Generally, what is realized is not exactly what has been predicted because of liquidity, bid/ask spread, size amount (e.g., backtest with VIX).

Example We consider a systematic strategy of selling 1M put (respectively call) options with strike 95% (respectively 100%) at the end of the month. Results in a daily basis are reported in the next Figure. Backtests clearly depend on the rebalancing dates (e.g., the end of the month is certainly a most favorable time for selling put options). Results are dependent on the implied volatility data and on skew's and bid/ask spread's assumptions.

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Taking into account Non-Linear Assets Using option factors

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Taking into account Non-Linear Assets Using option factors with exogenous strikes

The TP system becomes



rkHF = ∑mi=1 βk(i ) rk(i ) + βk(m+1) rk(m+1) (sk ) + ηk βk = βk −1 + νk

rk(m+1) (sk ) is the return of a systematic one-month option selling strategy on S&P 500 and sk is the (exogenous) strike of the option at time index k . where

⇒

The TP remains linear with respect to the state variables and may be

solved using Kalman Filter.

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Taking into account Non-Linear Assets Using option factors with exogenous strikes

Table: Results of replicating the HFRI index HF LKF Call Put LKF Call Put

sk

µˆ 1Y

σˆ 1Y

s 0.77 0.45 0.46 0.50 0.54 0.48 0.53 0.55

σTE

87.35 87.12 86.95 87.07 87.68 87.40 87.62

95% 100% 105% 95% 100% 105%

9.94 7.55 7.61 7.92 8.14 7.77 8.15 8.27

7.06 6.91 6.93 6.94 6.88 6.98 6.97 6.92

πAB

95% 100% 105% 95% 100% 105%

75.93 76.62 79.74 81.90 78.20 81.98 83.21

3.52 3.55 3.58 3.55 3.49 3.53 3.48

Thierry Roncalli, Guillaume Weisang

ρ

γ1

γ2

-0.57 -0.02 -0.21 -0.22 -0.06 -0.22 -0.20 -0.04

2.76 2.25 2.91 2.92 2.40 3.35 3.29 2.60

τ

84.96 83.56 83.61 84.57 84.33 84.22 85.21

67.10 65.48 65.53 66.81 66.61 66.60 67.68

ρS

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Taking into account Non-Linear Assets Using option factors with exogenous strikes

Table: Results of replicating the HFRI Relative Value index HF LKF Call Put LKF Call Put

sk

µˆ 1Y

σˆ 1Y

s 1.11 0.52 0.69 0.77 0.92 0.55 0.84 0.85

σTE

56.46 59.68 58.74 54.06 64.42 59.74 54.63

95% 100% 105% 95% 100% 105%

8.50 5.77 6.44 6.71 7.13 6.75 6.97 6.81

3.62 2.75 2.97 3.02 2.97 4.28 3.07 2.85

πAB

95% 100% 105% 95% 100% 105%

67.94 75.76 78.99 83.88 79.49 82.00 80.13

3.07 3.01 3.05 3.20 3.38 3.03 3.15

Thierry Roncalli, Guillaume Weisang

ρ

γ1

γ2

-2.76 0.16 -0.48 -0.46 0.12 -6.49 -0.84 0.24

15.22 1.18 3.29 3.69 1.97 66.57 6.06 1.80

τ

55.74 53.10 51.05 53.30 54.77 52.40 54.44

40.48 38.38 37.10 38.58 38.94 37.60 39.26

ρS

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Taking into account Non-Linear Assets Using option factors with endogenous strikes

The TP system becomes

 (i ) (i ) (m+1) (m+1)  rkHF = (sk ) + ηk ∑m  i= 1 βk r k +βk  rk      βk βk −1 νk sk = sk −1+ εk        νk 0 Q 0   ∼N , εk 0 0 σs2 ⇒

The TP is not linear with respect to the state variables and may be

solved using Particles Filters.

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Taking into account Non-Linear Assets Using option factors with endogenous strikes

Problems with maximum likelihood method: computational time and convergence.

⇒

We prefer to use a grid approach.

Figure: Grid approach applied to the HFRI RV index

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Tactical Asset Allocation and Tracking Problems Hedge Fund Replication: The Linear Gaussian Case Hedge Fund Replication: The Non-Linear Non-Gaussian Case

Taking into account Non-Linear Assets Using option factors with endogenous strikes

Figure: Exposures of the linear assets for the HFRI RV index

Thierry Roncalli, Guillaume Weisang

Figure: Option exposures and strikes for the HFRI RV index

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

What is Alpha? Explaining the Alpha The Core/Satellite Approach

What is Alpha? Notice the alpha is formulated as the unexplained residual of the replication strategy against the benchmarked HF. Thus, the alpha aggregates the performance of all uncaptured eects. Considering the performance of the state-space modeling and the monthly frequency of our replication, we consider that the uncaptured strategies remaining in the alpha could be generated either by dynamic trading in derivatives instruments or trading at ultra high frequencies or in illiquid assets (real estate, private equity, distress securities). A Core-Satellite approach to replication is more appropriated than trying to perfectly replicate HF.

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

What is Alpha? Explaining the Alpha The Core/Satellite Approach

Breakdown on the HF performance 75% corresponds to alternative beta which may be reproduced by the tracker; 25% is the alternative alpha of which:

10% corresponds in fact to alternative beta which may not be implemented and are lost due to the dynamic allocation; 15% makes up a component that we call the pure alternative alpha: optional, high-frequency and/or illiquid strategies

d

HF µˆ 1Y

Clone µˆ 1Y

πAB

σTE

−1

9.94

7.55

75.93

3.52

87.35

67.10

84.96

0

9.94

8.39

84.45

1.94

96.17

80.18

94.55

1

9.94

8.42

84.77

2.05

95.71

80.09

94.42

2

9.94

8.26

83.11

2.22

94.96

78.42

93.59

ρ

τ

ρS

Table: Impact of time lags implementation Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

What is Alpha? Explaining the Alpha The Core/Satellite Approach

The Core/Satellite Approach Problem Improving the quality of replication requires investing in illiquid or optional strategies.

⇒

This is in contradiction with the investment

philosophy of Hedge Fund Replication.

Solution Core: alternative beta Satellite: illiquid or optional strategies Example 70% of alternative beta 10% of opt/quant strategies (SGI Vol premium & JPM Carry Max) 10% of real estate (UK IPD & NCREIF property) 10% of private equity (LPX buyout & LPX venture)

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

What is Alpha? Explaining the Alpha The Core/Satellite Approach

The Core/Satellite Approach

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

HF Replication in Practice Main Ideas

HF Replication in Practice Asset managers and commercial investment banks have launched hedge fund replication products since 2006: BNP Paribas (Innocap Salto), Deutsche Bank (ARB), Goldman Sachs (ART), JPMorgan (HF AltBeta), Merrill Lynch (Torrus Factor Index), Société Générale (SGI Alternative Beta), etc. BlueWhite AI (ABF), Partners Group (ABS), Sgam AI (T-REX), SSgA (Premia), etc.

Products exist in several forms: Fund, Index, Short/Reverse, Leveraged, ETF, etc. The three main applications are: investing, managing liquidity and hedging (especially since the HF liquidity crisis in 2008).

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

HF Replication in Practice Main Ideas

HF Replication in Practice Figure: Comparison3 HFRI / Trackers

3 Hedged

in Euro. Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

HF Replication in Practice Main Ideas

Main Ideas We must distinguish Hedge Fund Replication (HFR) for Risk Management purposes and Investment purposes. Does HFR work?

No for single hedge funds, specic strategies. Yes if you consider the HF industry as a whole and for some specic strategies (long/short equity for example). From an investment point of view, building a HF tracker does not mean building a new hedge fund and it must have the following characteristics:

Liquidity (using only liquid futures). Transparency (for example by giving the portfolio composition). What is the best replication method?

Certainly the Kalman lter. Rolling-window OLS suers of lack of reactivity. Particle lters are not necessary (if you don't need options). Do we have to take into account non-linearities in HFR?

No if you want to build an investment vehicle. Perhaps yes for risk management purposes (but more work is required on the subject). Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Statistics Description µˆ 1Y πAB

is the annualized performance; the proportion of the HFRI index performance explained by the

clone;

σTE is the yearly tracking error; ρ , τ and ρS are respectively the

linear correlation, the Kendall tau

and the Spearman rho between the monthly returns of the clone and the HFRI index;

s

is the sharpe ratio;

γ1 γ2

is the skewness; is the excess kurtosis.

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

For Further Reading I N. Amenc, W. Géhin, L. Martellini, and J-C. Meyfredi. The Myths and Limits of Passive Hedge Fund Replication.

Working Paper, 2007.

N. Amenc, L. Martellini, J-C. Meyfredi and V. Ziemann. Passive Hedge Fund Replication  Beyong the Linear Case.

Working Paper, 2007.

S. Arulampalam, S. Maskell, N.J. Gordon and T. Clapp. A tutorial on particle lters for online nonlinear/non-Gaussian Bayesian tracking.

IEEE Transaction on Signal Processing, 50(2):174-188, Februrary 2002.

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

For Further Reading II P. Clauss, T. Roncalli and G. Weisang. Risk Management Lessons from Mado.

Working Paper, 2009. Available at SSRN:

http://ssrn.com/abstract=1358086.

A. Diez de los Rios and R. Garcia. Assessing and Valuing the Non-Linear Structure of Hedge Fund Returns.

Working Paper, 2008. W. Fung and D. A. Hsieh. Empirical Characteristics of Dynamic Trading Strategies: the Case of Hedge Funds.

Review of Financial Studies, 10:275-302, 1997. Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

For Further Reading III J. Hasanhodzic and A. W. Lo. Can Hedge-Fund Returns Be Replicated?: The Linear Case.

Journal of Investment Management, 5(2):5-45, 2007. A. Hocquard, N. Papageorgiou and B. Rémillard,

Optimal hedging strategies with an application to hedge fund replication.

Wilmott Magazine, Jan-Feb, 62-66, 2008. H.M. Kat Alternative Routes to Hedge Fund Return Replication.

Journal of Wealth Management, 10(3):25-39, 2007. R. C. Merton.

On Market Timing and Investment Performance. I. An Equilibrium Theory of Value for Market Forecasts.

Journal of Business, 54(3):363-406, 1981. Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

For Further Reading IV

T. Roncalli and J. Teiletche. An alternative approach to alternative beta.

Journal of Financial Transformation, 24:43-52, 2008. Available at SSRN:

http://ssrn.com/abstract=1035521.

T. Roncalli and G. Weisang. Tracking Problems, Hedge Fund Replications and Alternative Beta.

Working Paper, 2008. Available at SSRN:

http://ssrn.com/abstract=1325190.

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Kalman Filter If one assumes the tracking problem to be linear and Gaussian, we have 

xk = ck + Fk xk −1 + νk z k = d k + H k x k + ηk

^0 be the initial with νk ∼ N (0, Qk ) and ηk ∼ N (0, Sk ). Let p (x0 ) = φ x0 ,^ x0 , P distribution of the state vector. We have 

with

p (xk | z1:k −1 )

=

p (xk | z1:k )

=



  ^k |k −1 φ xk −1 ,^ xk |k −1 , P   ^k |k φ xk , ^ xk |k , P

 ^ xk |k −1 = ck + Fk^ xk −1|k −1     ^ ^  P = F P k k −1|k −1 Fk> + Qk k |k −1     zk |k −1 = dk + Hk^ xk |k −1  ^ ^ ek = zk − ^zk |k −1   ^k |k −1 Hk> + Sk  Vˆ k = Hk P    ^k |k −1 Hk> Vˆ k−1^  ^ x = ^ x ek  k |k k |k −1 + P   ^ ^k |k −1 − P ^k |k −1 H > Vˆ −1 Hk P ^k |k −1 Pk |k = P

k k

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Particle Filters N

Let xik , wki i =s1 denotes a set of support points xik , i = 1, . . . , Ns and their  associated weights wki , i = 1, . . . , Ns characterizing the posterior density p (xk | z0:k ). The posterior density at time k can then be approximated as 





  Ns p (xk | zk ) ≈ ∑ wki δ xk − xik i =1 s If p (x ) ∝ π (x ) and let x ∼ q (x ) be samples from the importance density q (·), we

have by Bayes rule



p zk | xik × p xik | xik −1   wki ∝ wki −1 q xik | xik −1 , zk

This equation is the core of Particle lters. Considering dierent assumptions leads to dierent numerical algorithms (SIS, GPF, SIR, RPF, etc.).

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta

Investing in Hedge Funds HF Replication and Tracking Problems Alpha Considerations Conclusion Appendix

Particle Filters The GTAA example

Thierry Roncalli, Guillaume Weisang

HF Replication and Alternative Beta