Center for Computational Finance Hedge Fund Risk Profiling: A non-linear approach to assess the risk and optimise Funds of Hedge Funds allocation. Carnegie

Center for Computational Finance

Hedge Fund Risk Profiling:A non-linear approach to assess the risk and optimise

Funds of Hedge Funds allocation.

Carnegie Mellon University, March 21, 2005

Raphaël DouadyResearch Director, Riskdata®

[email protected] www.riskdata.com +33 1 44 54 35 00 +1 212 931 5794

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CMU, March 21, 2005 Computational Finance Seminar

INVESTOR’S PROBLEM

The Investor Problem

What is the most likely Hedge Fund behaviour under the various market conditions?

What factor or event can put the Hedge Fund at risk?

Is the risk of a portfolio well diversified across the funds

Goal

Build and Rebalance portfolio of Hedge Funds

Select new Hedge Funds to invest in

3


INVESTOR’S PROBLEM

Risk Transparency

Beyond past performance, can we anticipate market situations which can kill us?

What information can we derive from past returns?

4


Hedge Fund Modelling

Hedge Funds form asset class different from others

Apparent Statistical Instability

Structural Non-linearity stemming from Dynamic Trading

Usual market factors inefficient to explain returns

Seldom and imprecise information:

Net Asset Value (weekly or monthly, delayed in all cases)

Exposure and sensitivity report

Position transparency only in some cases

5


Correlation of Long-Short Equity Funds to TUNA LS Index24M slipping period (end indicated)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

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Seminole C

Crestwood

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Galleon Om

Knoll Capi

Pangaea Ov

Sandler Co

Gordon Hou

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Modelling vs. Index

Long-short Equity Europe

-2%

0%

2%

4%

6%

8%

10%

-25% -20% -15% -10% -5% 0% 5% 10% 15% 20%

STOXX Returns

Fund

Ret

urns

= 0 > 0

Beta = 0 does not imply no exposure to Risk Factor

2002

2003

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Hedge Fund Modelling

General Modelling Methodology

Determine a set of Factors that define the “Market”

Identify, for each Hedge Fund, the Factors that do impact the returns

Build a Proxy of the fund, as a function of each Selected Factor, or of the subset of them

HF return = Proxy + Prediction error

Proxyt = E(HF returnt | Factort U t-1)

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What Statistical Model for H.F.

Single factor vs. Multi-factor

Question: Which factor set?

Linear vs. Non-linear

Question: What type of non-linear modelling?

Instantaneous info vs. Lagged

Question: Number of periods for the Fund? For the Factors?

Return series vs. Integrated series

Extreme moves modelling

Question: Which criterion for "extreme"

9


What Statistical Model for H.F.

Single Factor Multi-Factor

Linear

CAPM: Poor explanation + Misleading

Max correlation: Stable and Robust, but can miss explanation + no aggregation

Factor set for each Class of strategy no aggregation

General Factor Set Spurius analysis

Stepwise Regression Still spurious!

Non-linear

Collection of Pairwise Non-linear Models: Optimal trade-off Explanation vs. Complexity

General Non-linear Multi-factor Representation Too many parameters Spurious Analysis

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State of the Art

Maximum Correlation

Select, in a set of market factors, the factor that is the most correlated to the fund

Proxy the fund by linear regression with respect to this factor

Factor Model / Style Analysis

Determine a fixed factor set

Size limited to the number of data points

Multi-dimensional regression of the Fund returns on this set

Constrain by positive weights for stability (only with directional funds)

Stepwise Regression

Factor set Not Limited

Exposed to Spurious Selections

Still Linear

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Stepwise Regression

Start from large Factor base

Equity indices (country, sector, style…), Fixed income, etc.

Select a small number of factors F1 … Fn such that R2 is maximum

Start with most correlated factor

Include factor that increases R2 the most, etc. Stop when increase is too small.

Remove factors that decrease R2 the less. Stop when decrease is too large

Continue until we can neither include nor remove factors.

Set R2 threshold so that n be in chosen range (3 - 6 factors)

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Evaluation Criteria

Explanatory Power

In-sample modelling error

Fund(t) = f(Factor1(t), …, Factorn(t)) + (t)

calibrated on the whole analysis period

Predictive Power

Out-of-sample modelling error

Fund(t) = f(t-1)(Factor1(t), …, Factorn(t)) + (t)

calibrated on [t0, t - 1]

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Explanation Power

Evaluation Criteria

R2

R2 = Var(Explained) / Var(Return)

Other formula for R2

Var(Return) = Var(Explained) + Var(Error)

R2 = 1 – Var(Error) / Var(Return)

Spurious Selections act Positively

Var(Explained) = Var(Really Explained) + Var(Spurious)

R2 = Real R2 + Var(Spurious) / Var(Return)

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Explanation Power

R-square obtained with a Set of 25 Factors – Linear Reg.

TUNA Hedge Fund Indices

Selection of best combination of 5 factors

Factor set:

S&P500, size/style indicesCorp. Bond and HY indicesUS Libor, bond curve, swap curveMSCI World, Emerging marketsFama-FrenchFX BasketCommodity index, Gold, OilS&P optionsS&P historical and implied VolUS T-bond historical vol

STRATEGY Avg R² 5 fact.#periods

Tuna Long Only Average 96.6% 0.2% 0.1%

Tuna Value Average 91.9% 1.2% 0.5%

Tuna Long/Short Hedged Average 91.2% 13.1% 11.2%

Tuna Equity Hedge Aggregate Average 91.1% 13.1% 11.2%

Tuna Hedge Fund Aggregate Average 90.9% 0.0% 0.0%

Tuna Opportunistic Average 90.8% 1.2% 0.8%

Tuna Aggressive Aggregate Average 90.3% 0.1% 0.0%

Tuna Technology Sector Average 89.2% 0.8% 0.3%

Tuna Short Bias Average 88.2% 0.5% 0.5%

Tuna Event Driven Average 87.7% 2.4% 2.8%

Tuna Aggressive Growth Average 87.5% 0.1% 0.0%

Tuna Fund of Funds Average 86.0% 28.1% 24.4%

Tuna Small/Micro Cap Average 83.5% 0.3% 0.0%

Tuna Healthcare Sector Average 83.2% 0.5% 0.2%

Tuna Other Average 83.2% 1.8% 1.0%

Tuna Emerging Markets Average 82.3% 3.4% 2.4%

Tuna Country Specific Average 81.5% 0.4% 0.7%

Tuna MarketTimer Average 78.9% 1.4% 0.6%

Tuna Fixed Income Average 76.8% 1.7% 2.7%

Tuna Market Neutral Average 74.3% 7.5% 13.0%

Tuna Distressed Average 74.2% 2.2% 3.1%

Tuna Finance Sector Average 73.8% 0.4% 0.1%

Tuna Options Strategies Average 67.4% 0.5% 0.3%

Tuna Short-termTrading Average 66.5% 0.4% 0.2%

Tuna Risk Arbitrage Average 66.1% 1.1% 1.0%

Tuna Regulation D Average 60.9% 0.4% 0.1%

Tuna Relative Value Aggregate Average 60.4% 0.8% 1.6%

Tuna Convertible Arbitrage Average 57.1% 4.3% 6.5%

Tuna Macro Average 54.0% 3.1% 4.3%

Tuna Energy Sector Average 52.1% 0.1% 0.0%

Tuna VC / Private Equity Average 49.8% 0.0% 0.0%

Tuna Fixed Income Arbitrage Average 46.3% 2.4% 4.7%

Tuna Statistical Arbitrage Average 46.0% 0.8% 0.3%

Tuna Other Relative Value Average 44.4% 0.8% 1.6%

Tuna Special Situations Average 43.1% 0.2% 0.0%

Tuna Options Arbitrage Average 38.1% 0.5% 0.3%

Tuna CTA Average 36.2% 4.2% 3.6%

10%

80%

11%

9%

% Funds % AUM

76%

14%

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Prediction Power

Correlation between Predicted Series and Actual Returns

Not influenced by Spurious Selections

Prediction Power P2

P2 = 1 – Var(Error) / Var(Return)

Spurious Selections act Negatively

Var(Error) = Var(Specific) + Var(Spurious)

P2 = Real R2 – Var(Spurious) / Var(Return)

Direction Match Probability

Probability that Actual Return has the same sign as the Prediction Biased if the the Fund average return is ≠ 0

Unbiased measure: Correlation of Sign Series

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Testing Procedure

Test Pannel (250 funds)

Directional: 75

Non directional: 64

Arbitrage: 32

Special/Event: 24

Aggregates: 23

Other: 22

Random: 10

Hedge Fund Analysis

3Y slipping window

Monthly returns

[Jan 99 – Dec 01]

to [Jan 01 – Dec 03]

Factor set

~200 factors

Equity, IR, Commodity, FX…

Volatility, Correlation, Trend…

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Stepwise Regression

Explanation vs. Prediction PowerAverage over 240 Funds

-60%

-40%

-20%

0%

20%

40%

60%

80%

R2

P2

Factor base: ~200 factorsAverage selection: 1.3 factors

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Maximum Correlation

Select, in each time period, the factor that is the most correlated to the fund

Eliminate periods with a correlation below some threshold (positive or negative)

Regress returns on the selected factor

Compute Return Prediction

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Max Correlation

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Selection Rate

Prediction Correlation

P2

Threshold

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Maximum Correlation

Selection by CorrelationAverage over 240 Hedge Funds

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

60%

70%

30% 40% 50% 60% 70% 80% 90% 100%

Correlation Threshold

% Selections

R2

Predict. Correl.

P2

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Maximum Correlation

Selection by CorrelationLong/Short Equity

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%

60%

70%

80%

30% 40% 50% 60% 70% 80% 90% 100%

Correlation Threshold

% Selections

R2

Predict. Correl.

P2

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Stepwise Regression Max Correlation


-60%

-40%

-20%

0%

20%

40%

60%

80%

R2

P2


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Other Selection Methods

Non linear regression: F-test, Log-likelihood

Causality (non linear VARMA): F-test

Cointegration. Non linear factor: ∫ Factt² dt

P2

Direction Match

Joint occurrence of Extreme Moves

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Multiple Pairwise Analysis

Select a factor is at least one of the statistical tests is positive

Compute a different prediction for each factor

Measure the prediction uncertainty

Compute the MLE estimate of the fund return, knowing

Each single-factor prediction + uncertainty

Factor correlation structure

Compare to actual Fund return

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OUT OF SAMPLE TEST


-60%

-40%

-20%

0%

20%

40%

60%

80%

100%

R2 StepwiseP2 StepwiseR2 FOFiXP2 FOFiX


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YES: RISK PROFILING

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Findings

Classical Linear methods are either often spurious (stepwise regression) or miss essential factors (correlation)

Non linear modelling is necessary

Statistical factors, such as Hist. Vol., Correl Index, etc. explain a lot of hedge fund returns

Causality is efficient because of Lagged series

Co-integration is useful to find the “right” factor, but not for prediction capabilities. Dickey-Fuller mean reversion test worsen statistics

Direction match probability test good for “event” type strategies

Large factor shifts should be analysed differently: use the frequency of joint large move occurrence between the fund and the factor.

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Conclusion

Performance Analysis + Correlations are insufficient for the construction of Portfolios of of Hedge Fund

A Complete Set of Risk Factors contains Factors that replicate Dynamic Strategies

Sensitive to Volatility and Correlation of Assets

Include Non-linear Features

Hedge Funds must be Proxied by Non-linear functions of Factors

Building a Risk Profile is the only way to identify Market Conditions under which Funds over/under-perform

This is also the only way to extract Stable information from Return series

Documents

Center for Computational Finance Hedge Fund Risk Profiling: A non-linear approach to assess the risk and optimise Funds of Hedge Funds allocation. Carnegie